text
stringlengths 1
2.25M
|
|---|
---
author:
- 'Lihua You [^1] Man Yang [^2] Jinxi Li[^3] Liyong Ren[^4]'
date: |
[School of Mathematical Sciences, South China Normal University,\
Guangzhou, 510631, P.R. China\
]{}
title: 'Spectrum of a class of matrices and its applications[^5]'
---
.2cm
[**Abstract** ]{} In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research.
[*[**AMS Classification:**]{}* ]{} 05C50, 05C35, 05C20, 15A18
[*[**Keywords:**]{}*]{} Matrix; Quotient matrix; Graph; Digraph; Spectrum; Spectral radius
Introduction
============
.5cm We begin by recalling some definitions. Let $M$ be an $n\times n$ matrix, $\lambda_1, \lambda_2, \ldots, \lambda_n$ be the eigenvalues of $M$. It is obvious that the eigenvalues may be complex numbers since $M$ is not symmetric in general. We usually assume that $|\lambda_1|\geq |\lambda_2|\geq\ldots \geq |\lambda_n|$. The spectral radius of $M$ is defined as $\rho(M)=|\lambda_1|$, i.e., it is the largest modulus of the eigenvalues of $M$. If $M$ is a nonnegative matrix, it follows from the Perron-Frobenius theorem that the spectral radius $\rho(M)$ is a eigenvalue of $M$. If $M$ is a nonnegative irreducible matrix, it follows from the Perron-Frobenius theorem that $\rho(M)=\lambda_1$ is simple.
Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $A(G)=(a_{ij})$ denote the adjacency matrix of $G$, where $a_{ij}$ is equal to the number of edges $v_iv_j$. The spectral radius of $A(G)$, denoted by $\rho(G)$, is called the spectral radius of $G$. Let $diag(G)=diag(d_1, d_2, \ldots, d_n)$ be the diagonal matrix with degree of the vertices of $G$ and $Q(G)=diag(G)+A(G)$ be the signless Laplacian matrix of $G$, $L(G)=diag(G)-A(G)$ be the Laplacian matrix of $G$. The spectral radius of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. The spectral radius of $L(G)$, denoted by $\mu(G)$, is called the Laplacian spectral radius of $G$.
For $u,v\in V(G)$, the distance between $u$ and $v$, denoted by $d_G(u,v)$ or $d_{uv}$, is the length of the shortest path connecting them in $G$. For $u\in V(G)$, the transmission of vertex $u$ in $G$ is the sum of distances between $u$ and all other vertices of $G$, denoted by $Tr_G(u)$.
Let $G$ be a connected graph with vertex set $V(G)=\{v_1, v_2, \ldots, v_n\}$. The distance matrix of $G$ is the $n\times n$ matrix $\mathcal{D}(G)=(d_{ij})$ where $d_{ij}=d_{v_iv_j}$. The distance spectral radius of $G$, denoted by $\rho^{\mathcal{D}}(G)$, is the spectral radius of $\mathcal{D}(G)$, which is the largest eigenvalue of $\mathcal{D}(G)$.
In fact, for $1\leq i\leq n$, the transmission of vertex $v_i$, $Tr_G(v_i)$ is just the $i$-th row sum of $\mathcal{D}(G)$. Let $Tr(G)=diag(Tr_G(v_1), Tr_G(v_2), \ldots, Tr_G(v_n))$ be the diagonal matrix of vertex transmission of $G$. M. Aouchiche and P. Hansen [@2013LAA7] introduced the Laplacian and the signless Laplacian for the distance matrix of a connected graph. The matrix $\mathcal{L}(G)=Tr(G)-\mathcal{D}(G)$ is called the distance Laplacian of $G$, while the matrix $\mathcal{Q}(G)=Tr(G)+\mathcal{D}(G)$ is called the distance signless Laplacian matrix of $G$. It is obvious that $\mathcal{Q}(G)$ is irreducible, nonnegative, symmetric and positive semidefinite. The distance signless Laplacian spectral radius of $G$, denoted by $q^{\mathcal{D}}(G)$, is the spectral radius of $\mathcal{Q}(G)$, which is the largest eigenvalue of $\mathcal{Q}(G)$. The spectral radius of $\mathcal{L}(G)$, denoted by $\mu^{\mathcal{D}}(G)$, is called the distance Laplacian spectral radius of $G$.
Since $G$ is a connected graph, then $A(G)$, $Q(G)$, $\mathcal{D}(G)$ and $\mathcal{Q}(G)$ are nonnegative irreducible matrices, it follows from the Perron Frobenius Theorem that $\rho(G)$, $q(G)$, $\rho^{\mathcal{D}}(G)$ and $q^{\mathcal{D}}(G)$ are real numbers and there is a positive unit eigenvector corresponding to $\rho(G)$, $q(G)$, $\rho^{\mathcal{D}}(G)$ and $q^{\mathcal{D}}(G)$, respectively.
Let $\overrightarrow{G}=(V(\overrightarrow{G}), E(\overrightarrow{G}))$ be a digraph, where $V(\overrightarrow{G})=\{v_1, v_2,
\ldots, v_n \}$ and $E(\overrightarrow{G})$ are the vertex set and arc set of $\overrightarrow{G}$, respectively. A digraph $\overrightarrow{G}$ is simple if it has no loops and multiple arcs. A digraph $\overrightarrow{G}$ is strongly connected if for every pair of vertices $v_i, v_j\in V(\overrightarrow{G})$, there are directed paths from $v_i$ to $v_j$ and from $v_j$ to $v_i$. In this paper, we consider finite, simple strongly connected digraphs. Let $\overrightarrow{G}$ be a digraph. If two vertices are connected by an arc, then they are called adjacent. For $e=(v_i, v_j)\in E(\overrightarrow{G})$, $v_i$ is the tail (the initial vertex) of $e$, $v_j$ is the head (the terminal vertex) of $e$. Let $N^-_{\overrightarrow{G}}(v_i)=\{v_j\in V(\overrightarrow{G})|(v_j, v_i)\in E(\overrightarrow{G})\}$ and $N^+_{\overrightarrow{G}}(v_i)=\{v_j\in V(\overrightarrow{G})|$ $(v_i, v_j) \in E(\overrightarrow{G})\}$ denote the in-neighbors and out-neighbors of $v_i$, respectively.
For a digraph $\overrightarrow{G}$, let $A(\overrightarrow{G})=(a_{ij})$ denote the adjacency matrix of $\overrightarrow{G}$, where $a_{ij}$ is equal to the number of arcs $(v_i, v_j)$. The spectral radius of $A(\overrightarrow{G})$, denoted by $\rho(\overrightarrow{G})$, is called the spectral radius of $\overrightarrow{G}$.
Let $diag(\overrightarrow{G})=diag(d^+_1, d^+_2, \ldots, d^+_n)$ be the diagonal matrix with outdegree of the vertices of $\overrightarrow{G}$ and $Q(\overrightarrow{G})= diag(\overrightarrow{G})+A(\overrightarrow{G})$ be the signless Laplacian matrix of $\overrightarrow{G}$, $L(\overrightarrow{G})=diag(\overrightarrow{G})-A(\overrightarrow{G})$ be the Laplacian matrix of $\overrightarrow{G}$. The spectral radius of $Q(\overrightarrow{G})$, $\rho(Q(\overrightarrow{G}))$, denoted by $q(\overrightarrow{G})$, is called the signless Laplacian spectral radius of $\overrightarrow{G}$. For $u,v\in V(G)$, the distance from $u$ to $v$, denoted by $d_{\overrightarrow{G}}(u,v)$ or $d_{uv}$, is the length of the shortest directed path from $u$ to $v$ in ${\overrightarrow{G}}$. For $u\in V({\overrightarrow{G}})$, the transmission of vertex $u$ in ${\overrightarrow{G}}$ is the sum of distances from $u$ to all other vertices of ${\overrightarrow{G}}$, denoted by $Tr_{{\overrightarrow{G}}}(u)$.
Let ${\overrightarrow{G}}$ be a connected digraph with vertex set $V({\overrightarrow{G}})=\{v_1, v_2, \ldots, v_n\}$. The distance matrix of ${\overrightarrow{G}}$ is the $n\times n$ matrix $\mathcal{D}({\overrightarrow{G}})=(d_{ij})$ where $d_{ij}=d_{\overrightarrow{G}}(v_i,v_j)$. The distance spectral radius of $\overrightarrow{G}$, denoted by $\rho^{\mathcal{D}}(\overrightarrow{G})$, is the spectral radius of $\mathcal{D}(\overrightarrow{G})$. In fact, for $1\leq i\leq n$, the transmission of vertex $v_i$, $Tr_{\overrightarrow{G}}(v_i)$ is just the $i$-th row sum of $\mathcal{D}(\overrightarrow{G})$. Let $Tr(\overrightarrow{G})=diag(Tr_{\overrightarrow{G}}(v_1), Tr_{\overrightarrow{G}}(v_2), \ldots, Tr_{\overrightarrow{G}}(v_n))$ be the diagonal matrix of vertex transmission of $\overrightarrow{G}$. The distance signless Laplacian matrix of ${\overrightarrow{G}}$ is the $n\times n$ matrix defined similar to the undirected graph by Aouchiche and Hansen as
$$\mathcal{Q}({\overrightarrow{G}})=Tr({\overrightarrow{G}})+\mathcal{D}({\overrightarrow{G}}).$$
Let $\mathcal{L}(\overrightarrow{G})=Tr({\overrightarrow{G}})-\mathcal{D}({\overrightarrow{G}})$ be the distance Laplacian matrix of $\overrightarrow{G}$. The distance signless Laplacian spectral radius of $\overrightarrow{G}$, $\rho(\mathcal{Q}(\overrightarrow{G}))$, denoted by $q^{\mathcal{D}}(\overrightarrow{G})$, is the spectral radius of $\mathcal{Q}(\overrightarrow{G})$. Since $\overrightarrow{G}$ is a simple strongly connected digraph, then $A(\overrightarrow{G})$, $Q(\overrightarrow{G})$, $\mathcal{D}({\overrightarrow{G}})$ and $\mathcal{Q}({\overrightarrow{G}})$ are nonnegative irreducible matrices. It follows from the Perron Frobenius Theorem that $\rho(\overrightarrow{G})$, $\rho(Q(\overrightarrow{G}))=q(\overrightarrow{G})$, $\rho^{\mathcal{D}}(\overrightarrow{G})$ and $\rho(\mathcal{Q}(\overrightarrow{G}))=q^{\mathcal{D}}(\overrightarrow{G})$ are positive real numbers and there is a positive unit eigenvector corresponding to $\rho(\overrightarrow{G})$, $q(\overrightarrow{G})$, $\rho^{\mathcal{D}}(\overrightarrow{G})$ and $q^{\mathcal{D}}(\overrightarrow{G})$, respectively.
For a connected graph $G=(V(G), E(G))$, the vertex connectivity of a graph denoted by $\kappa(G)$, is the minimum number of vertices whose deletion yields the resulting graph disconnected. Clearly, let $G$ be a connected graph on $n$ vertices, then $1\leq \kappa(G)\leq n-1$. Similarly, for a strongly connected digraph $\overrightarrow{G}=(V(\overrightarrow{G}), E(\overrightarrow{G}))$, the vertex connectivity of a digraph denoted by $\kappa(\overrightarrow{G})$, is the minimum number of vertices whose deletion yields the resulting digraph non-strongly connected. Clearly, let $\overrightarrow{G}$ be a strongly connected digraph with $n$ vertices, then $1\leq \kappa(\overrightarrow{G})\leq n-1$.
There are many literatures about graphs’ and digraphs’ connectivity. For early work, see [@2009LAA], Ye-Fan-Liang characterize the graphs with the minimal least eigevalue among all graphs with given vertex connectivity or edge connectivity. In 2010, Ye-Fan-Wang [@2010LAA] characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. Liu [@2010b] characterized the minimal distance spectral radius of simple connected graphs with given vertex connectivity, or matching number, or chromatic number, respectively. Brualdi [@2010LAA1] wrote a stimulating survey on this topic.
In 2012, Lin-Shu-Wu-Yu [@2012DM] establish some upper or lower bounds for digraphs with some given graph parameters, such as clique number, girth, and vertex connectivity, and characterize the corresponding extremal graphs, give the exact value of the spectral radii of those digraphs. Besides, Lin-Yang-Zhang-Shu [@2012DM2] characterize the extremal digraphs (graphs) with minimum distance spectral radius among among all digraphs (graphs) with given vertex (edge) connectivity.
In 2013, Lin-Drury [@2013DM] characterize the extremal digraphs which attain the maximum Perron root of digraphs with given arc connectivity and number of vertices. Lin-Shu [@2013DAM] determine the extremal digraph with the minimal distance spectral radius with given arc connectivity. Xing-Zhou [@2013] determine the graphs with minimal distance signless Laplacian spectral radius among the connected graphs with fixed number of vertices and connectivity. Oscar Rojo and Eber Lenes [@2013LAA3] obtained a sharp upper bound on the incidence energy of graphs in terms of connectivity. Furthermore, some upper or lower bounds were obtained by the outdegrees and the average 2-outdegrees [@2013ARS; @2013LAA].
In 2014, Hong-You [@2014] determine the digraphs with maximal signless Laplacian spectral radius among the strongly connected digraphs with given vertex connectivity. On the other hand, some extremal digraphs which attain the maximum or minimum spectral radius, the signless Laplacian spectral radius, the distance spectral radius, or the distance signless Laplacian spectral radius of digraphs with given parameters, such as given vertex connectivity, given arc connectivity, given dichromatic number, given clique number, given girth and so on, were characterized, see e.g. [@2013DM; @2013DAM; @2012DM; @2012DM2; @2013].
In this paper, we give the spectrum of a matrix using the quotient matrix, and also apply these results to various matrices associated with graphs and digraphs as mentioned above. Some know results are improved.
Some preliminaries
==================
\[defn21\][([@1979; @1986])]{} Let $A=(a_{ij}), B=(b_{ij})$ be $n\times n$ matrices. If $a_{ij}\leq b_{ij}$ for all $i$ and $j$, then $A\leq B$. If $A\leq B$ and $A\neq B$, then $A< B$. If $a_{ij} < b_{ij}$ for all $i$ and $j$, then $A \ll B$.
\[lem22\][([@1979; @1986])]{} Let $A, B$ be $n\times n$ matrices with the spectral radius $\rho(A)$ and $\rho(B)$. If $0\leq A\leq B$, then $\rho(A) \leq \rho(B)$. Furthermore, if $B$ is irreducible and $0\leq A<B$, then $\rho(A) < \rho(B)$.
By Lemma \[lem22\], we have the following results in terms of digraphs.
\[cor23\] Let $\overrightarrow{G}$ be a digraph and $\overrightarrow{H}$ be a spaning subdigraph of $\overrightarrow{G}$. Then
[(i) ]{}$\rho(\overrightarrow{H})\leq \rho(\overrightarrow{G})$, $q(\overrightarrow{H})\leq q(\overrightarrow{G})$.
[(ii) ]{} If $\overrightarrow{G}$ is strongly connected, and $\overrightarrow{H}$ is a proper subdigraph of $\overrightarrow{G}$, then $\rho(\overrightarrow{H})< \rho(\overrightarrow{G})$, $q(\overrightarrow{H})< q(\overrightarrow{G})$.
[(iii) ]{} If $\overrightarrow{G}$ and $\overrightarrow{H}$ are strongly connected, then $\rho^{\mathcal{D}}(\overrightarrow{H})\geq \rho^{\mathcal{D}}(\overrightarrow{G})$, $q^{\mathcal{D}}(\overrightarrow{H})\geq q^{\mathcal{D}}(\overrightarrow{G})$.
[(iv) ]{} If $\overrightarrow{H}$ is a proper subdigraph of $\overrightarrow{G}$, then $\rho^{\mathcal{D}}(\overrightarrow{H})> \rho^{\mathcal{D}}(\overrightarrow{G})$, $q^{\mathcal{D}}(\overrightarrow{H})> q^{\mathcal{D}}(\overrightarrow{G})$.
The theorem for undirected graph is also established.
\[lem24\][([@1979])]{} If $A$ is an $n\times n$ nonnegative matrix with the spectral radius $\rho(A)$ and row sums $r_1, r_2, \ldots, r_n$, then $\min\limits_{1\leq i\leq n}r_i\leq \rho(A)\leq \max\limits_{1\leq i\leq n}r_i$. Moreover, if $A$ is irreducible, then one of the equalities holds if and only if the row sums of $A$ are all equal.
By Lemma \[lem24\], we have $\rho(\overset{\longleftrightarrow}{K_n})=\rho^{\mathcal{D}}(\overset{\longleftrightarrow}{K_n})=n-1$, $q(\overset{\longleftrightarrow}{K_n})=q^{\mathcal{D}}(\overset{\longleftrightarrow}{K_n})=2(n-1)$; $\rho(\overrightarrow{C_n})=1$, $q(\overrightarrow{C_n})=2$, $\rho^{\mathcal{D}}(\overrightarrow{C_n})=\frac{n(n-1)}{2}$, $q^{\mathcal{D}}(\overrightarrow{C_n})=n(n-1)$. Then by Corollary \[cor23\], we have
\[cor25\] Let $\overrightarrow{G}$ be a strongly connected digraph with $n$ vertices. Then $$\rho(\overrightarrow{G})\leq n-1, \quad q(\overrightarrow{G})\leq 2n-2, \quad \rho^{\mathcal{D}}(\overrightarrow{G})\geq n-1,
\quad q^{\mathcal{D}}(\overrightarrow{G})\geq 2n-2,$$ with equality holds if and only if $\overrightarrow{G}\cong \overset{\longleftrightarrow}{K_n}$.
\[cor26\] Let $\overrightarrow{G}$ be a strongly connected digraph with $n$ vertices. Then $$\rho(\overrightarrow{G})\geq 1, \quad q(\overrightarrow{G})\geq 2, \quad \rho^{\mathcal{D}}(\overrightarrow{G})\leq \frac{n(n-1)}{2},
\quad q^{\mathcal{D}}(\overrightarrow{G})\leq n(n-1),$$ with equality holds if and only if $\overrightarrow{G}\cong \overrightarrow{C_n}.$
In [@2013DAM], Theorem 3.2 show that $\rho^{\mathcal{D}}(\overrightarrow{G})\leq \frac{n(n-1)}{2},$ and the equality holds if and only if $\overrightarrow{G}\cong \overrightarrow{C_n}.$ Now we only show $q^{\mathcal{D}}(\overrightarrow{G})\leq n(n-1)$ and the equality holds if and only if $\overrightarrow{G}\cong \overrightarrow{C_n}.$
If $\overrightarrow{G}$ has a Hamiltonian dicycle, we have $ q^{\mathcal{D}}(\overrightarrow{G})\leq q^{\mathcal{D}}(\overrightarrow{C_n})$ and the equality holds if and only if $\overrightarrow{G}\cong \overrightarrow{C_n}$ by Corollary \[cor23\].
If $\overrightarrow{G}$ does not contain a Hamiltonian dicycle. Noting that $\max\limits_{1\leq i\leq n}r_i\leq n(n-1)$. If $\max\limits_{1\leq i\leq n}r_i< n(n-1)$, then $q^{\mathcal{D}}(\overrightarrow{G})\leq \max\limits_{1\leq i\leq n}r_i<n(n-1)= q^{\mathcal{D}}(\overrightarrow{C_n})$ by Lemma \[lem24\].
If $\max\limits_{1\leq i\leq n}r_i= n(n-1)$, then $\overrightarrow{G}$ contains a vertex $v_1$ such that $2Tr_{\overrightarrow{G}}(v_1)= n(n-1),$ then $\overrightarrow{G}$ contains a Hamiltonian dipath $P$ initiating at $v_1$. Suppose that $P= v_1\rightarrow v_2\rightarrow \ldots\rightarrow v_n$ is the Hamiltonian dipath initiating at $v_1$. Then there is no arc $(v_i, v_j)\in E(\overrightarrow{G})$ if $j-i\geq 2$ since $Tr_{\overrightarrow{G}}(v_1) = \frac{n(n-1)}{2}$. Since $\overrightarrow{G}$ is strongly connected and does not a Hamiltonian dicycle, there exists a dipath $P'$ from $v_n$ to $v_1$ and thus there exists some vertex, namely, $v_k (k\neq n)$, is adjacent to $v_1$, that is $(v_k, v_1)\in E(\overrightarrow{G})$. Since $v_k$ is on the Hamiltonian dipath $P$, we have $(v_k, v_{k+1})\in E(\overrightarrow{G}).$ Hence $$r_k \leq 2(1+1+2+\ldots +n-2)<2(1+2+\ldots +n-1)= 2Tr_{\overrightarrow{G}}(v_1)=n(n-1),$$ it implies that the row sums of $\mathcal{Q}(\overrightarrow{G})$ are not equal. Then by Lemma \[lem24\], we have $$q^{\mathcal{D}}(\overrightarrow{G})< q^{\mathcal{D}}(\overrightarrow{C_n}).$$
Combining the above arguments, we complete the proof.
The spectrum of a matrix
========================
.6cm Let $I_p$ be the $p\times p$ identity matrix and $J_{p,q}$ be the $p\times q$ matrix in which every entry is $1$, or simply $J_p$ if $p=q$. Let $M$ be a matrix of order $n$, $\sigma(M)$ be the spectrum of the matrix $M$, $P_M(\lambda)=det(xI_n-M)$ be the characteristic polynomial of matrix $M$.
\[defn31\][([@2014a])]{} Let $M$ be a real matrix of order $n$ described in the following block form
$$\label{eq31}
M = \left(\begin{array}{ccc}
M_{11} & \cdots & M_{1t}\\
\vdots & \ddots &\vdots \\
M_{t1}& \cdots & M_{tt}\\
\end{array}\right),$$
where the diagonal blocks $M_{ii}$ are $n_i\times n_i$ matrices for any $i\in\{1,2,\ldots, t\}$ and $n=n_1+\ldots+n_t$. For any $i,j\in\{1,2,\ldots, t\}$, let $b_{ij}$ denote the average row sum of $M_{ij}$, i.e. $b_{ij}$ is the sum of all entries in $M_{ij}$ divided by the number of rows. Then $B(M) = (b_{ij})$ (simply by $B$) is called the quotient matrix of $M$. If in addition for each pair $i, j$, $M_{ij}$ has constant row sum, then $B(M)$ is called the equitable quotient matrix of $M$.
\[lem32\]Let $M=(m_{ij})_{n\times n}$ be defined as (\[eq31\]), and for any $i,j \in\{ 1,2\ldots,t\}$, the row sum of each block $M_{ij}$ be constant. Let $B=B(M)=(b_{ij})$ be the equitable quotient matrix of $M$, and $\lambda$ be an eigenvalue of $B$. Then $\lambda$ is also an eigenvalue of $M$.
Let $By = \lambda y$ where $y = (y_1,y_2,\ldots,y_t)^T$. Define $Y = (y_{11},\ldots,y_{1,n_1},\ldots,y_{t1},\ldots,y_{t,n_t})^T$ by the relation $y_{i1} = y_{i2} = \ldots = y_{i,n_i}=y_i$ for each $i\in\{1,2,\ldots,t\}$. For any $i\in\{1,2,\ldots,t\}$ and $k\in\{1,2,\ldots, n_i\}$, let $M_i(k)$ be the $k$-th row of the $i$-th row blocks $(M_{i1}, \ldots, M_{it})$, that is, $M_i(k)$ is the $l$-th row of $M$ where $l=n_1+\ldots+n_{i-1}+k$, then by $M_i(k)Y=(MY)_l=\sum\limits_{j=1}^{n_1}m_{lj}y_1+\sum\limits_{j=n_1+1}^{n_1+n_2}m_{lj}y_2+\ldots
+\sum\limits_{j=n_1+\ldots+n_{t-1}+1}^{n_1+\ldots+n_t}m_{lj}y_t$ and the definition of $b_{ij}$ for each $i,j\in\{1,2,\ldots t\}$, we have $$\lambda Y_l=\lambda y_{ik}=\lambda y_i=(By)_i= \sum \limits_{j=1 }^{t}b_{ij}y_j=M_i(k)Y=(MY)_{l},$$ thus we have $MY = \lambda Y,$ and we complete the proof.
\[exam33\] Let $G=(V,E)$ be the Petersen graph as Figure 1. Let $\{V_1,V_2\}$ be a partition of $V=\{1,2,\ldots, 10\}$, where $V_1=\{1,2,3,4,5\}$ and $V_2=\{6,7,8,9,10\}$. Then the equitable quotient matrices $B(A), B(L), B(Q), B(\mathcal{D}),
B(\mathcal{L}), B(\mathcal{Q})$ corresponding to the adjacency matrix $A(G)$, the Laplacian matrix $L(G)$, the signless Laplacian matrix $Q(G)$, the distance matrix $\mathcal{D}(G)$, the distance Laplacian matrix $\mathcal{L}(G)$, the distance signless Laplacian matrix $\mathcal{Q}(G)$, respectively, are as follows:
(15,10)
(4,4) (10,4) (2,8) (12,8) (7,12) (5,6) (9,6) (4,8) (10,8) (7,10)
(4,4)[(1,0)[6]{}]{} (4,4)[(-1,2)[2]{}]{} (10,4)[(1,2)[2]{}]{} (2,8)[(5,4)[5]{}]{} (12,8)[(-5,4)[5]{}]{} (2,8)[(1,0)[2]{}]{} (10,8)[(1,0)[2]{}]{} (7,10)[(0,1)[2]{}]{} (4,4)[(1,2)[1]{}]{} (10,4)[(-1,2)[1]{}]{} (5,6)[(1,2)[2]{}]{} (5,6)[(5,2)[5]{}]{} (4,8)[(1,0)[6]{}]{} (9,6)[(-5,2)[5]{}]{} (9,6)[(-1,2)[2]{}]{}
(7,12.5)[1]{} (1,8)[2]{} (4,3)[3]{} (10,3)[4]{} (12.5,8)[5]{} (7.5,10)[6]{} (3.5,8.2)[7]{} (4,6)[8]{} (9.5,6)[9]{} (9.5,8.2)[10]{} (1,1.5)[Figure $1$. The Petersen graph ]{}
-0.5cm
$$B(A) = \left(\begin{array}{lcr}
2 & 1\\
1 & 2\\
\end{array}\right), \qquad
B(L) = \left(\begin{array}{lcr}
1 & -1\\
-1 & 1\\
\end{array}\right), \qquad
B(Q) = \left(\begin{array}{lcr}
5 & 1\\
1 & 5\\
\end{array}\right),$$
$$B(\mathcal{D}) = \left(\begin{array}{lcr}
6 & 9\\
9 & 6\\
\end{array}\right), \qquad
B(\mathcal{L})= \left(\begin{array}{lcr}
9 & -9\\
-9 & 9\\
\end{array}\right), \qquad
B(\mathcal{Q}) = \left(\begin{array}{lcr}
21 & 9\\
9 & 21\\
\end{array}\right).$$
Then $$\rho(B(A))=3, \rho(B(L))=2, \rho(B(Q))=6, \rho(B(\mathcal{D}))=15, \rho(B(\mathcal{L}))=18, \rho(B(\mathcal{Q}))=30,$$ but by directly calculating, we have $$\rho(G) = 3, \mu(G)=5, q(G) = 6, \rho^{\mathcal{D}}(G) = 15, \mu^{\mathcal{D}}(G))=18,
q^{\mathcal{D}}(G) = 30.$$ We see that the largest eigenvalue of the equitable quotient matrix $B(M)$ is the largest eigenvalue of $M$ when $M$ is the adjacency matrix $A(G)$, the signless Laplacian matrix $Q(G)$, the distance matrix $\mathcal{D}(G)$, the distance Laplacian matrix $\mathcal{L}(G)$ or the distance signless Laplacian matrix $\mathcal{Q}(G)$ of a graph $G$, and the result is totally different when $M$ is the Laplacian matrix $L(G)$ of a graph $G$.
\[lem34\] Let $M$ be defined as (\[eq31\]), and for any $i,j \in\{ 1,2\ldots,t\}$, $M_{ii} = l_iJ_{n_i} + p_iI_{n_i},$ $M_{ij} = s_{ij}J_{n_i,n_j}$ for $i\not= j$, where $l_i, p_i, s_{ij}$ are real numbers, $B=B(M)$ be the quotient matrix of $M$. Then $$\label{eq32}
\sigma(M)=\sigma(B)\cup \{p_i^{[n_i-1]} \mid i = 1,2\ldots,t\},$$ where $\lambda^{[t]}$ means that $\lambda$ is an eigenvalue with multiplicity $t$.
It is obvious that for any $i,j \in\{ 1,2\ldots,t\}$, $M_{ij}$ has constant row sum, so $B$ is the equitable quotient matrix of $M$. Then $\sigma(B)\subseteq \sigma(M)$ by Lemma \[lem32\].
On the other hand, we note that $\sigma(l_iJ_{n_i} + p_iI_{n_i}) = \{l_in_i + p_i, p_i^{[n_i-1]}\}$, where $l_iJ_{n_i} + p_iI_{n_i}$ has the all-one vector $J_{n_i,1}$ such that $(l_iJ_{n_i} + p_iI_{n_i})J_{n_i,1} = (l_in_i + p_i)J_{n_i,1}$, and its all other eigenvectors corresponding to eigenvalue $p_i$ are orthogonal to $J_{n_i,1}$.
Let $x$ be an any eigenvector such that $(l_iJ_{n_i} + p_iI_{n_i})x = p_ix$, then $x^TJ_{n_i,1} = 0$, and $(\mathbf{0}_{1,n_1},\ldots,x^T,\ldots,\mathbf{0}_{1,n_t})^T$ is an eigenvector of $M$ corresponding to eigenvalue $p_i$. Therefore the $p_i$ is an eigenvalue of $M$ with multiplicities at least $n_i-1$. And thus we obtain at least $\sum_{i=1}^{t}(n_i - 1) = n-t$ eigenvalues of $M$, that is, $\{p_1^{[n_1-1]}, \ldots, p_t^{[n_t-1]}\}\subseteq \sigma (M)$.
Therefore $\sigma(B)\cup \{p_1^{[n_1-1]}, \ldots, p_t^{[n_t-1]}\}\subseteq \sigma (M)$ by Lemma \[lem32\], and $|\sigma (M)|\leq |\sigma(B)|+|\{p_1^{[n_1-1]}, \ldots, p_t^{[n_t-1]}\}|=n$ by $|\sigma(B)|=t$ and $|\{p_1^{[n_1-1]}, \ldots, p_t^{[n_t-1]}\}|=n-t$.
If there exists some $p_i$ such that $p_i\in \sigma(B)$ where $i\in\{1,2,\ldots, t\}$, by the proof of Lemma \[lem32\], we have $My=p_iy$ with $y=(y_{11},\ldots,y_{1,n_1},\ldots,y_{t1},\ldots,y_{t,n_t})^T$, where $y_{i1} = y_{i2} = \ldots = y_{i,n_i}=y_i$ for each $i\in\{1,2,\ldots,t\}$. Then we have $(\mathbf{0}_{1,n_1},\ldots,x^T,\ldots,\mathbf{0}_{1,n_t})y$$=y_i(x^TJ_{n_i,1})=0$, it implies that the eigenvectors corresponding to the eigenvalue $p_i$ of $B$ and the eigenvalue $p_i$ in $\{p_1^{[n_1-1]}, \ldots, p_t^{[n_t-1]}\}$ are all orthogonal, then $|\sigma (M)|= |\sigma(B)|+|\{p_1^{[n_1-1]}, \ldots, p_t^{[n_t-1]}\}|=n$ and thus (\[eq32\]) holds.
\[exam35\] Let $G = K_{n_1,n_2,\ldots,n_t}$ be a complete t-partite graph with $n$ vertices for $ t \geq 2$, the adjacency matrix $A=A(G)$, the Laplacian matrix $L=L(G)$, the signless Laplacian matrix $Q=Q(G)$, the distance matrix $\mathcal{D}=\mathcal{D}(G),$ the distance Laplacian matrix $\mathcal{L}=\mathcal{L}(G)$ and the distance signless Laplacian matrix $\mathcal{Q}(G)$ of $G=K_{n_1,n_2,\ldots,n_t}$ are as follows:
(1). $A = M$, where $l_i = p_i = 0, s_{ij} = 1$ for $i\not= j$ where $i,j\in\{ 1,2,\ldots,t\}.$
(2). $L =M$, where $l_i = 0, p_i = n-n_i, s_{ij} = -1$ for $i\not= j$ where $i,j\in\{ 1,2,\ldots,t\}.$
(3). $Q = M$, where $l_i = 0, p_i = n-n_i, s_{ij} = 1$ for $i\not= j$ where $i,j\in\{ 1,2,\ldots,t\}.$
(4). $\mathcal{D} = M$, where $l_i = 2, p_i = -2, s_{ij} = 1$ for $i\not= j$ where $i,j\in\{ 1,2,\ldots,t\}.$
(5). $\mathcal{L} = M$, where $l_i = -2, p_i = n + n_i, s_{ij} = -1$ for $i\not= j$ where $i,j\in\{ 1,2,\ldots,t\}.$
(6). $\mathcal{Q} = M$, where $l_i = 2, p_i = n + n_i - 4, s_{ij} = 1$ for $i\not= j$ where $i,j\in\{ 1,2,\ldots,t\}.$
It is obvious that for any $i,j \in\{ 1,2\ldots,t\}$, $M_{ij}$ has constant row sum. Then the corresponding equitable quotient matrices are as follows:
$$B(A) = \left(\begin{array}{cccc}
0 & n_2 &\cdots & n_t\\
n_1 & 0 & \cdots & n_t\\
\vdots & \vdots &\ddots & \vdots \\
n_1 & n_2 & \cdots &0 \\
\end{array}\right), \qquad
B(L) = \left(\begin{array}{cccc}
n-n_1 & -n_2 &\cdots &-n_t\\
-n_1 & n-n_2 & \cdots &-n_t\\
\vdots & \vdots &\ddots & \vdots \\
-n_1 & -n_2 & \cdots & n-n_t \\
\end{array}\right),$$
$$B(Q) = \left(\begin{array}{cccc}
n-n_1 & n_2 &\cdots &n_t\\
n_1 & n-n_2 & \cdots &n_t\\
\vdots & \vdots & \ddots &\vdots \\
n_1 & n_2 & \cdots &n-n_t \\
\end{array}\right),
\quad
B(\mathcal{D})= \left(\begin{array}{cccc}
2n_1-2 & n_2 &\cdots &n_t\\
n_1 & 2n_2-2 & \cdots &n_t\\
\vdots & \vdots & \ddots &\vdots \\
n_1 & n_2 & \cdots &2n_t-2\\
\end{array}\right),$$
$$B(\mathcal{L}) = \left(\begin{array}{cccc}
n-n_1 & -n_2 &\cdots &-n_t\\
-n_1 & n-n_2 & \cdots & -n_t\\
\vdots & \vdots & \ddots &\vdots \\
-n_1 & -n_2 & \cdots &n-n_t \\
\end{array}\right),$$
$$B(\mathcal{Q})= \left(\begin{array}{cccc}
n+3n_1-4 & n_2 &\cdots & n_t\\
n_1 & n+3n_2-4 & \cdots & n_t\\
\vdots & \vdots &\ddots & \vdots \\
n_1 & n_2 & \cdots &n+3n_t-4 \\
\end{array}\right).$$
By Lemma \[lem34\], we have
(1). $ P_A(\lambda) =\lambda^{n-t} P_{B(A)}(\lambda) = \lambda^{n-t}[\prod\limits_{i=1}^t(\lambda + n_i) - \sum\limits_{i=1}^tn_i \prod \limits_{j=1,j \neq i}^t(\lambda + n_j)].$
(2). $ P_L(\lambda) =\prod\limits_{i=1}^t(\lambda - n + n_i)^{n_i-1} P_{B(L)}(\lambda) =\lambda(\lambda-n)^{t-1} \prod\limits_{i=1}^t(\lambda - n + n_i)^{n_i-1}.$
(3). $ P_Q(\lambda) = \prod\limits_{i=1}^t(\lambda - n + n_i)^{n_i-1}P_{B(Q)}(\lambda)$
2.0cm $= \prod\limits_{i=1}^t(\lambda - n + n_i)^{n_i-1}[\prod\limits_{i=1}^t(\lambda - n + 2n_i) - \sum\limits_{i=1}^tn_i \prod \limits_{j=1,j \neq i}^t(\lambda - n + 2n_j)].$
(4). $ P_{\mathcal{D}}(\lambda) = (\lambda + 2)^{n-t}P_{B(\mathcal{D})}(\lambda)$
2.0cm $= (\lambda + 2)^{n-t}[\prod\limits_{i=1}^t(\lambda - n_i + 2) - \sum\limits_{i=1}^tn_i \prod \limits_{j=1,j \neq i}^t(\lambda - n_j + 2)]. \hskip.4cm {\rm(\cite{2013LAA2})}$
(5). $P_{\mathcal{L}}(\lambda) =\prod\limits_{i=1}^t(\lambda - n - n_i)^{n_i-1} P_{B(\mathcal{L})}(\lambda) =\lambda(\lambda-n)^{t-1} \prod\limits_{i=1}^t(\lambda - n - n_i)^{n_i-1}.$
(6). $P_{\mathcal{Q}}(\lambda) = \prod\limits_{i=1}^t(\lambda - n - n_i + 4)^{n_i-1}P_{B(\mathcal{Q})}(\lambda)$
2.0cm $= \prod\limits_{i=1}^t(\lambda - n - n_i + 4)^{n_i-1}[\prod\limits_{i=1}^t(\lambda - n - 2n_i + 4) - \sum\limits_{i=1}^tn_i \prod \limits_{j=1,j \neq i}^t(\lambda - n - 2n_j + 4)].$
It is obvious that we obtain the spectrums of $L$ and $\mathcal{L}$ immediately. In fact, $\sigma(L) =\{0, n^{[t-1]}, (n-n_i)^{[n_i-1]}, i\in\{1, 2, \ldots, t\}\}$, and $\sigma(\mathcal{L}) = \{0, n^{[t-1]}, (n+n_i)^{[n_i-1]}, i\in\{1, 2, \ldots, t\}\}.$
.3cm
A block of $G$ is a maximal connected subgraph of $G$ that has no cut-vertex. A graph $G$ is a clique tree if each block of $G$ is a clique. We call $\mathbb{K}_{u,n_2,\ldots,n_{k+1}}$ is a clique star if we replace each edge of the star $K_{1,k}$ by a clique $K_{n_i}$ such that $V(K_{n_i})\cap V(K_{n_j}) = u$ for $i \neq j$ and $i,j\in\{ 2,\ldots,k+1\}.$
\[exam36\] Let $G=\mathbb{K}_{u,n_2,\ldots,n_{k+1}}$, where $n_1=|\{u\}| = 1$, $n_i \geq 2$ for any $i\in\{2,\ldots,k+1\}$ and $n = n_1 + n_2 + n_3 + \ldots + n_{k+1} - k$. Then the adjacency matrix $A=A(G)$, the Laplacian matrix $L=L(G)$, the signless Laplacian matrix $Q=Q(G)$, the distance matrix $\mathcal{D}=\mathcal{D}(G),$ the distance Laplacian matrix $\mathcal{L}=\mathcal{L}(G)$ and the distance signless Laplacian matrix $\mathcal{Q}(G)$ of $G=\mathbb{K}_{u,n_2,\ldots,n_{k+1}}$ are as follows.
(1). $A=M$, where $l_1 = p_1 = 0$ and $l_i = 1, p_i = -1$ for $i\not=1$, $s_{ij} = 1$ for $i = 1 \mbox{or } j = 1$, and $s_{ij} = 0$ for any $i,j\in\{2,\ldots,k+1\}$ and $i \neq j$.
(2). $L = M$, where $l_1 = n-1, p_1 = 0$ and $l_i = -1, p_i = n_i$ for $i\not=1$, $s_{ij} = -1$ for $i = 1 \mbox{or } j = 1$, and $s_{ij} = 0$ for any $i,j\in\{2,\ldots,k+1\}$ and $i \neq j$.
(3). $Q = M$, where $l_1 = n-1, p_1 = 0$ and $l_i = 1, p_i = n_i-2$ for $i\not=1$, $s_{ij} = 1$ for $i = 1 \mbox{or } j = 1$, and $s_{ij} = 0$ for any $i,j\in\{2,\ldots,k+1\}$ and $i \neq j$.
(4). $\mathcal{D} = M$, where $l_1 = 0, p_1 = 0$ and $l_i = 1, p_i = -1$ for $i\not=1$, $s_{ij} = 1$ for $i = 1 \mbox{or } j = 1$, and $s_{ij} = 2$ for any $i,j\in\{2,\ldots,k+1\}$ and $i \neq j$.
\(5) $\mathcal{L} = M$, where $l_1 = n-1, p_1 = 0$ and $l_i = -1, p_i = 2n-n_i$ for $i\not=1$, $s_{ij} = -1$ for $i = 1 \mbox{or } j = 1$, and $s_{ij} = -2$ for any $i,j\in\{2,\ldots,k+1\}$ and $i \neq j$.
(6). $\mathcal{Q} = M$, where $l_1 = n-1, p_1 = 0$ and $l_i = 1, p_i = 2n-n_i-2$ for $i\not=1$, $s_{ij} = 1$ for $i = 1 \mbox{or } j = 1$, and $s_{ij} = 2$ for any $i,j\in\{2,\ldots,k+1\}$ and $i \neq j$.
It is obvious that for any $i,j\in\{2,\ldots,k+1\}$, $M_{ij}$ has constant row sum. Then the corresponding equitable quotient matrices are as follows:
$$B(A) = \left(\begin{array}{cccc}
0 & n_1-1 &\cdots &n_k-1\\
1 & n_1-2 & \cdots & 0\\
\vdots & \vdots & \ddots &\vdots \\
1 & 0 & \cdots &n_k-2 \\
\end{array}\right), \qquad
B(L) = \left(\begin{array}{cccc}
n-1 & 1-n_1 &\cdots & 1-n_k\\
-1 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots\ddots &\vdots \\
-1 & 0 & \cdots &1 \\
\end{array}\right),$$
$$B(Q) = \left(\begin{array}{cccc}
n-1 & n_1-1 &\cdots & n_k-1\\
1 & 2n_1-3 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots \\
1 & 0 & \cdots &2n_k-3 \\
\end{array}\right), \qquad
B(\mathcal{D}) = \left(\begin{array}{cccc}
0 & n_1-1 &\cdots &n_k-1\\
1 & n_1-2 & \cdots & 2£¨n_k-1£©\\
\vdots & \vdots & \ddots & \vdots \\
1 & 2£¨n_1-1£© & \cdots &n_k-2\\
\end{array}\right),$$
$$B(\mathcal{L}) = \left(\begin{array}{cccc}
n-1 & 1-n_1 &\cdots &1-n_k\\
-1 & 2n-2n_1+1 & \cdots &-2(n_k-1)\\
\vdots & \vdots & \ddots &\vdots \\
-1 & -2(n_1-1) & \cdots &2n-2n_k+1 \\
\end{array}\right),$$
$$B(\mathcal{Q}) = \left(\begin{array}{cccc}
n-1 & n_1-1 &\cdots & n_k-1\\
1 & 2n-3 & \cdots &2£¨n_k-1£©\\
\vdots & \vdots & \ddots &\vdots \\
1 & 2(n_1-1) & \cdots & 2n-3 \\
\end{array}\right).$$
By Lemma \[lem34\], we have
(1). $P_A(\lambda) = (\lambda+1)^{n-k-1}P_{B(A)}(\lambda)$
$= (\lambda+1)^{n-k-1}[\lambda\prod\limits_{i=2}^{k+1}(\lambda - n_i + 2) - \sum\limits_{i=2}^{k+1}(n_i-1) \prod \limits_{j=2,j \neq i}^{k+1}(\lambda - n_j +2)].$
(2). $P_L(\lambda) =(\lambda-n_i)^{n_i-2}P_{B(L)}(\lambda)= \lambda(\lambda-n)(\lambda-1)^{k-1}(\lambda-n_i)^{n_i-2}.$
(3). $P_Q(\lambda)=\prod\limits_{i=2}^{k+1}(\lambda - n_i + 2)^{n_i-2}P_{B(Q)}(\lambda)$
$= \prod\limits_{i=2}^{k+1}(\lambda - n_i + 2)^{n_i-2}[\lambda\prod\limits_{i=2}^{k+1}(\lambda - 2n_i + 3) - \sum\limits_{i=2}^{k+1}(n_i-1) \prod \limits_{j=2,j \neq i}^{k+1}(\lambda - 2n_j +3)].$
(4). $ P_{\mathcal{D}}(\lambda) =(\lambda + 1)^{n-k-1}P_{B(\mathcal{D})}(\lambda)$
$= (\lambda + 1)^{n-k-1}[\lambda\prod\limits_{i=2}^{k+1}(\lambda + n_i) - (2\lambda+1)\sum\limits_{i=2}^{k+1}(n_i-1)\prod \limits_{j=2,j \neq i}^{k+1}(\lambda + n_j)].$
(5). $P_{\mathcal{L}}(\lambda) =(\lambda -2n+n_i)^{n_i-2}P_{B(\mathcal{L})}(\lambda)= \lambda(\lambda-n)(\lambda-2n+1)^{k-1}(\lambda-2n+n_i)^{n_i-2}.$
(6). $P_{\mathcal{Q}}(\lambda) =\prod\limits_{i=2}^{k+1}(\lambda - 2n + n_i +2)^{n_i-2}P_{B(\mathcal{Q})}(\lambda)$
$= \prod\limits_{i=2}^{k+1}(\lambda - 2n + n_i +2)^{n_i-2}[(\lambda-n+1)\prod\limits_{i=2}^{k+1}(\lambda - 2n + 2n_i + 1)$
$-(2\lambda-2n+3)\sum\limits_{i=2}^{k+1}(n_i-1) \prod \limits_{j=2,j \neq i}^{k+1}(\lambda - 2n + 2n_j + 1)].$
It is obvious that we can obtain the spectrum of $L$ and $\mathcal{L}$ immediately. In fact, $\sigma(L) =\{0, n, 1^{[k-1]}, n_i^{[n_i-2]}, i\in\{2, 3, \ldots, k+1\}\}$ and $\sigma(\mathcal{L}) = \{0, n, (2n-1)^{[k-1]},(2n-n_i)^{[n_i-2]}, i\in \{2, 3, \ldots, k+1\}\}.$
By Lemma \[lem32\], Example \[exam33\] and Examples \[exam35\]–\[exam36\], we proposed the following conjecture for further research.
\[con31\] Let $M$ be a nonnegative matrix, $B(M)$ be the equitable quotient matrix of $M$. Then the largest eigenvalue of $B(M)$ is the largest eigenvalue of $M$.
Consider two sequences of real numbers: $\lambda_{1}\geq \lambda_{2} \geq ... \geq\lambda_{n}$, and $\mu_{1}\geq \mu_{2}\geq ...\geq \mu_{m}$ with $m<n$. The second sequence is said to interlace the first one whenever $\lambda_{i}\geq \mu_{i}\geq \lambda_{n-m+i}$ for $i=1,2,...,m$. The interlacing is called tight if there exists an integer $k \in [1,m]$ such that $\lambda_{i}=\mu_{i}$ hold for $1\leq i \leq k$ and $\lambda_{n-m+i}= \mu_{i}$ hold for $k + 1 \leq i \leq m$.
[([@1995H])]{}\[lem33\] Let $M$ be a symmetric matrix and have the block form as (\[eq31\]), $B$ be the quotient matrix of $M$. Then
[(1) ]{} The eigenvalues of $B$ interlace the eigenvalues of $M$.
[(2) ]{} If the interlacing is tight, then $B$ is the equitable matrix of $M$.
By Lemmas \[lem32\]-\[lem33\], we have the following result immediately.
\[thm34\]Let $M=(m_{ij})_{n\times n}$ be a symmetric matrix and defined as (\[eq31\]), $B=B(M)$ be the quotient matrix of $M$, and $\mu_{1}\geq \mu_{2}\geq ...\geq \mu_{m}$ be all eigenvalues of $B$. Then $\mu_{1}, \mu_{2}, ..., \mu_{m}$ are eigenvalues of $M$ if and only if $B$ is the equitable matrix of $M$.
.3cm
Spectral radius of strongly connected digraphs with given connectivity
======================================================================
.6cm Let $\Omega(n,k)$ be the set of all simple strong connected digraphs on $n$ vertices with vertex connectivity $k$. Let $\overrightarrow{G}_1 \bigtriangledown \overrightarrow{G}_2$ denote the digraph $G=(V,E)$ obtained from two disjoint digraphs $\overrightarrow{G}_1$, $\overrightarrow{G}_2$ with vertex set $V=V(\overrightarrow{G}_1)\cup V(\overrightarrow{G}_2)$ and arc set $E = E(\overrightarrow{G}_1) \cup E(\overrightarrow{G}_2) \cup \{(u,v),(v,u)|u \in V(\overrightarrow{G}_1), v \in V(\overrightarrow{G}_2)\}.$
Let $p,k$ be integers with $1\leq k\leq n-2, 1\leq p\leq n-k-1,$ $\overrightarrow{K}(n,k,p)$ denote the digraph $\overrightarrow{K_k} \bigtriangledown (\overrightarrow{K_p} \cup \overrightarrow{K}_{n - p - k}) \cup E,$ where $E = \{(u,v)|u \in \overrightarrow{K_p}, v \in \overrightarrow{K}_{n - p - k}\}$. Clearly, $\overrightarrow{K}(n,k,p)\in \Omega(n,k).$ Then the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix of $\overrightarrow{K}(n,k,p)$ are as follows, where $q=n-p-k$.
$$A(\overrightarrow{K}(n,k,p)) = \left(\begin{array}{lcr}
J_p-I_p & J_{p,k} & J_{p,q}\\
J_{k,p} & J_k-I_k & J_{k,q}\\
\mathbf{0}_{q,p} & J_{q,k} & J_q-I_q\\
\end{array}\right),$$ $$Q(\overrightarrow{K}(n,k,p)) = \left(\begin{array}{lcr}
J_p+(n-2)I_p & J_{p,k} & J_{p,q}\\
J_{k,p} & J_k+(n-2)I_k & J_{k,q}\\
\mathbf{0}_{q,p} & J_{q,k} & J_q+(n-p-2)I_q\\
\end{array}\right),$$ $$\mathcal{D}(\overrightarrow{K}(n,k,p)) = \left(\begin{array}{lcr}
J_p-I_p & J_{p,k} & J_{p,q}\\
J_{k,p} & J_k-I_k & J_{k,q}\\
2J_{q,p} & J_{q,k} & J_q-I_q\\
\end{array}\right),$$ $$\mathcal{Q}(\overrightarrow{K}(n,k,p)) =\left(\begin{array}{lcr}
J_p+(n-2)I_p & J_{p,k} & J_{p,q}\\
J_{k,p} & J_k+(n-2)I_k & J_{k,q}\\
2J_{q,p} & J_{q,k} & J_q+(n+p-2)I_q\\
\end{array}\right).$$
.3cm
\[prop41\][([@1976])]{} Let $\overrightarrow{G}$ be a strongly connected digraphs with vertex connectivity $k$. Suppose that $S$ is a $k$-vertex cut of $\overrightarrow{G}$ and $\overrightarrow{G}_1, \overrightarrow{G}_2,\ldots, \overrightarrow{G}_t$ are the strongly connected components of $\overrightarrow{G}-S$. Then there exists an ordering of $\overrightarrow{G}_1, \overrightarrow{G}_2,\ldots, \overrightarrow{G}_t$ such that for $1\leq i\leq t$ and $v\in V(\overrightarrow{G}_i)$, every tail of $v$ is in $\bigcup \limits_{j=1}^{i-1}\overrightarrow{G}_j$.
\[rem42\] By Proposition \[prop41\], we know that $\overrightarrow{G}_1$ is the strongly connected component of $\overrightarrow{G}-S$ where the inneighbors of vertices of $V(\overrightarrow{G}_1)$ in $\overrightarrow{G}-S-\overrightarrow{G}_1$ are zero. Let $\overrightarrow{G}_2=\overrightarrow{G}-S-\overrightarrow{G}_1$. We add arcs to $\overrightarrow{G}$ until both induced subdigraph of $V(\overrightarrow{G}_1)\cup S$ and induced subdigraph of $V(\overrightarrow{G}_2)\cup S$ attain to complete digraphs, add arc $(u,v)$ for any $u\in V(\overrightarrow{G}_1)$ and any $v\in V(\overrightarrow{G_2})$, the new digraph denoted by $\overrightarrow{H}$. Clearly, $\overrightarrow{H}=\overrightarrow{K}(n,k,p)\in \Omega(n,k)$ for some $p$ such that $1\leq p\leq n-k-1$. Since $\overrightarrow{G}$ is the spanning subdigraph of $\overrightarrow{H}$, then by Corollary \[cor23\], we have $\rho(\overrightarrow{G})\leq \rho(\overrightarrow{K}(n,k,p))$, $q(\overrightarrow{G})\leq q(\overrightarrow{K}(n,k,p))$, $\rho^{\mathcal{D}}(\overrightarrow{G})\geq \rho^{\mathcal{D}}(\overrightarrow{K}(n,k,p))$ and $q^{\mathcal{D}}(\overrightarrow{G})\geq q^{\mathcal{D}}(\overrightarrow{K}(n,k,p))$. Thus the extremal digraphs which achieve the maximal (signless Laplacian) spectral radius and the minimal distance (signless Laplacian) spectral radius in $\Omega(n,k)$ must be some $\overrightarrow{K}(n,k,p)$ for $1\leq p\leq n-k-1$.
Let $n,k$ be given positive integers with $1\leq k\leq n-2$, $\overrightarrow{G}\in \Omega(n,k).$ Then
(i). [([@2012DM])]{} $$\label{eq41}
\rho(\overrightarrow{G})\leq \frac{n-2+\sqrt{(n-2)^2+4k}}{2},$$ with equality if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,1)$ or $\overrightarrow{G}\cong \overrightarrow{K}(n,k,n-k-1).$
(ii). [([@2014])]{} $$\label{eq42}
q(\overrightarrow{G})\leq \frac{2n+k-3+\sqrt{(2n-k-3)^2+4k}}{2},$$ with equality if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,n-k-1).$
(iii). [([@2012DM2])]{} $$\label{eq43}
\rho^\mathcal{D}(\overrightarrow{G})\geq \frac{n-2+\sqrt{(n+2)^2-4k-8}}{2},$$ with equality if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,1)$ or $\overrightarrow{G}\cong \overrightarrow{K}(n,k,n-k-1)$.
(iv). $$\label{eq44}
q^\mathcal{D}(\overrightarrow{G})\geq \frac{3n-3+\sqrt{(n+3)^2-8k-16}}{2},$$ with equality if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,1).$
Now we show (i) holds. We apply Lemma \[lem34\] to $A=A(\overrightarrow{K}(n,k,p))$. Since $t=3$, $l_i=1,p_i=-1$ for $1\leq i\leq 3$, $s_{31}=0$ and $s_{ij}=1$ for others $i, j\in\{1,2,3\}$ and $i\not=j$, we have $\sigma(A)=\sigma(B(A))\cup\{(-1)^{[n-3]}\}$, where the corresponding equitable quotient matrix of $A$ is
$$B(A)= \left(\begin{array}{ccc}
p-1 & k & q\\
p & k-1 & q\\
0 & k & q-1\\
\end{array}\right),$$ the eigenvalues of $B(A)$ are $-1,\frac{n-2\pm\sqrt{4p^2-4(n-k)p+n^2}}{2}$. Thus $\rho(A)=\frac{n-2+\sqrt{4p^2-4(n-k)p+n^2}}{2}$.
It is obvious that $\frac{n-2+\sqrt{4p^2-4(n-k)p+n^2}}{2}\leq \frac{n-2+\sqrt{(n-2)^2+4k}}{2}$, and equality holds if and only if $p=1$ or $p=n-k-1$. Thus (\[eq41\]) holds and equality holds if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,1)$ or $\overrightarrow{G}\cong \overrightarrow{K}(n,k,n-k-1).$
Now we show (ii) holds. Similarly, we apply Lemma \[lem34\] to $Q=Q(\overrightarrow{K}(n,k,p))$. Since $t=3$, $l_i=1$ for $1\leq i\leq 3$, $p_1=p_2=n-2$, $p_3=n-p-2,$ $s_{31}=0$ and $s_{ij}=1$ for others $i, j\in\{1,2,3\}$ and $i\not=j$, we have $\sigma(Q)=\sigma(B(Q))\cup\{(n-2)^{[p+k-2]},(n-p-2)^{[q-1]}\},$ where the corresponding equitable quotient matrix of $Q$ is
$$B(Q)= \left(\begin{array}{ccc}
p+n-2 & k & q\\
p & k+n-2 & q\\
0 & k & q+n-p-2\\
\end{array}\right),$$ the eigenvalues of $B(Q)$ are $n-2,\frac{(3n-p-4)\pm\sqrt{(n-3p)^2+8pk}}{2}$. Thus $\rho(Q)=\frac{3n-p-4+\sqrt{(n-3p)^2+8pk}}{2}$.
By the same proof of Theorem 7.6 in [@2014], we can show (\[eq42\]) holds by proving $$\frac{3n-p-4+\sqrt{(n-3p)^2+8pk}}{2}\leq \frac{2n+k-3+\sqrt{(2n-k-3)^2+4k}}{2}$$ for $1\leq p\leq n-k-1$, and equality holds if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,n-k-1).$
Now we show (iii) holds. We apply Lemma \[lem34\] to $\mathcal{D}=\mathcal{D}(\overrightarrow{K}(n,k,p))$. Since $l_i=1, p_i=-1$ for $1\leq i\leq 3$, $s_{31}=2$ and $s_{ij}=1$ for others $i, j\in\{1,2,3\}$ and $i\not=j$, we have $\sigma(\mathcal{D})=\sigma(B(\mathcal{D}))\cup\{(-1)^{[n-3]}\},$ and the corresponding equitable quotient matrix of $\mathcal{D} $ is $$B(\mathcal{D})= \left(\begin{array}{ccc}
p-1 & k & q\\
p & k-1 & q\\
2p & k & q-1\\
\end{array}\right),$$ the eigenvalues of $B(\mathcal{D})$ are $-1,\frac{(n-2)\pm\sqrt{-4p^2+4(n-k)p+n^2}}{2}$. Thus $\rho(\mathcal{D})=\frac{n-2+\sqrt{-4p^2+4(n-k)p+n^2}}{2}$.
It is obvious that $\frac{n-2+\sqrt{-4p^2+4(n-k)p+n^2}}{2}\geq \frac{n-2+\sqrt{(n+2)^2-4k-8}}{2}$, and equality holds if and only if $p=1$ or $p=n-k-1$. Thus (\[eq43\]) holds and equality holds if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,1)$ or $\overrightarrow{G}\cong \overrightarrow{K}(n,k,n-k-1).$
Now we show (iv) holds. We apply Lemma \[lem34\] to $\mathcal{Q}=\mathcal{Q}(\overrightarrow{K}(n,k,p))$. Since $l_1=l_2=l_3=1,$ $p_1=p_2=n-2, $ $p_3=n+p-2,$ $s_{31}=2$ and $s_{ij}=1$ for others $i, j\in\{1,2,3\}$ and $i\not=j$, we have $\sigma(\mathcal{Q})=\sigma(B(\mathcal{Q}))\cup\{(n-2)^{[p+k-2]},(n+p-2)^{[q-1]}\},$ and the corresponding equitable quotient matrix of $\mathcal{Q} $ is $$B(\mathcal{Q})= \left(\begin{array}{ccc}
p+n-2 & k & q\\
p & k+n-2 & q\\
2p & k & q+n+p-2\\
\end{array}\right),$$ the eigenvalues of $B(\mathcal{Q})$ are $n-2,\frac{(3n+p-4)\pm\sqrt{(n+3p)^2-16p^2-8kp}}{2}$. Thus $$\rho(\mathcal{Q})= \frac{3n+p-4 + \sqrt{(n+3p)^2-16p^2-8kp}}{2}.$$
Now we show $\frac{3n+p-4 + \sqrt{(n+3p)^2-16p^2-8kp}}{2}\geq \frac{3n-3+\sqrt{(n+3)^2-8k-16}}{2}$ for $1\leq p\leq n-k-1$, and the equality holds if and only if $p=1$. Let $f(p)=\frac{3n+p-4 + \sqrt{(n+3p)^2-16p^2-8kp}}{2}$. Then $$\frac{\partial ^{2}f(p)}{\partial p^2} = \frac{-4((2n-k)(n-k)+k^2)}{((n+3p)^2-16p^2-8kp)^{\frac{3}{2}}}<0.$$
Thus, for fixed $n$ and $k$, the minimal value of $f(p)$ must be taken at either $p=1$ or $p=n-k-1$. Let $\alpha=k^2-6k-7+8n$ and $\beta=n^2+6n-7-8k$. Then $$2[f(n-k-1) - f(1)]=n-k-2+\sqrt{\alpha}-\sqrt{\beta}=(n-k-2)(1-\frac{n+k}{\sqrt{\alpha}+\sqrt{\beta}}).$$
We can assume that $n > k+2$ since in case $k=n-2$ there is only one value of $p$ under consideration. Now suppose that $f(n-k-1) - f(1)\leq 0$. We will produce a contradiction. We have $$\sqrt{\alpha}+\sqrt{\beta}\leq n+k , \sqrt{\alpha}-\sqrt{\beta}\leq -n+k+2.$$ Whence $\sqrt{\alpha}\leq k+1$ and $\alpha\leq (k+1)^2$ which reduces to $k\geq n-1$ which is out of range. Thus $f(n-k-1) > f(1)$ and $q^\mathcal{D}(\overrightarrow{G})\geq f(1)\texttt{}=q^\mathcal{D}(\overrightarrow{K}(n,k,1))=\frac{3n-3+\sqrt{(n+3)^2-8k-16}}{2}$, with equality if and only if $\overrightarrow{G}\cong \overrightarrow{K}(n,k,1).$
It is natural that whether there exists similar result for the Laplacian spectral radius or the distance Laplacian spectral radius in $\Omega(n,k)$ or not? In fact, we can obtain the spectrum of the Laplacian matrix or the distance Laplacian matrix of $\overrightarrow{K}(n,k,p)$ immediately.
\[prop44\] Let $\overrightarrow{K}(n,k,p)$ defined as before. Then
(i). $\sigma(L(\overrightarrow{K}(n,k,p)))=\{0, n^{[p+k-1]}, (n-p)^{[q]}\}.$
(ii). $\sigma(\mathcal{L}(\overrightarrow{K}(n,k,p)))=\{0, n^{[p+k-1]}, (n+p)^{[q]}\}.$
Firstly, the Laplacian matrix $L(\overrightarrow{K}(n,k,p))$ and the distance Laplacian matrix $\mathcal{L}(\overrightarrow{K}(n,k,p))$ of $\overrightarrow{K}(n,k,p)$ are the following matrices, where $q=n-p-k$.
$$L=L(\overrightarrow{K}(n,k,p)) = \left(\begin{array}{ccc}
-J_p+nI_p & -J_{p,k} & -J_{p,q}\\
-J_{k,p} & -J_k+nI_k & -J_{k,q}\\
\mathbf{0}_{q,p} & -J_{q,k} & -J_q+(n-p)I_q\\
\end{array}\right),$$
$$\mathcal{L}=\mathcal{L}(\overrightarrow{K}(n,k,p)) = \left(\begin{array}{ccc}
-J_p+nI_p & -J_{p,k} & -J_{p,q}\\
-J_{k,p} & -J_k+nI_k & -J_{k,q}\\
-2J_{q,p} & -J_{q,k} & -J_q+(n+p)I_q\\
\end{array}\right).$$
Then the corresponding equitable quotient matrices are as follows: $$B(L)= \left(\begin{array}{ccc}
n-p & -k & -q\\
-p & n-k & -q\\
0 & -k & k\\
\end{array}\right), \qquad
B(\mathcal{L})= \left(\begin{array}{lcr}
n-p & -k & -q\\
-p & n-k & -q\\
-2p & -k & n+p-q\\
\end{array}\right).$$
Then by Lemma \[lem34\] and directly calculating, we obtain (i) and (ii).
.3cm
Spectral radius of connected graphs with given connectivity
===========================================================
.6cm Let $\mathcal{C}(n,k)$ be the set of all simple connected graphs on $n$ vertices with vertex connectivity $k$. Let ${G_1}\bigtriangledown {G_2}$ denote the graph $G=(V,E)$ obtained from two disjoint graphs ${G_1}$, ${G_2}$ by joining each vertex of $G_1$ to each vertex of $G_2$ with vertex set $V=V({G}_1)\cup V({G}_2)$ and edge set $E = E({G}_1) \cup E({G}_2) \cup \{uv | u \in V( {G}_1), v \in V( {G}_2)\}$.
Let $p,k$ be integers with $1\leq k\leq n-2, 1\leq p\leq n-k-1,$ and ${K}(n,k,p)$ be the graph ${K_k} \bigtriangledown ({K_p} \cup {K}_{n-p-k})$. Clearly, ${K}(n,k,p)\in \mathcal{C}(n,k).$ Then the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix of ${K}(n,k,p)$ are as follows, where $q=n-p-k$.
$$A({K}(n,k,p)) = \left(\begin{array}{lcr}
J_p-I_p & J_{p,k} & \mathbf{0}_{p,q}\\
J_{k,p} & J_k-I_k & J_{k,q}\\
\mathbf{0}_{q,p} & J_{q,k} & J_q-I_q\\
\end{array}\right),$$ $$Q({K}(n,k,p)) = \left(\begin{array}{lcr}
J_p+(p+k-2)I_p & J_{p,k} & \mathbf{0}_{p,q}\\
J_{k,p} & J_k+(n-2)I_k & J_{k,q}\\
\mathbf{0}_{q,p} & J_{q,k} & J_q+(n-p-2)I_q\\
\end{array}\right),$$ $$\mathcal{D}({K}(n,k,p)) = \left(\begin{array}{lcr}
J_p-I_p & J_{p,k} & 2J_{p,q}\\
J_{k,p} & J_k-I_k & J_{k,q}\\
2J_{q,p} & J_{q,k} & J_q-I_q\\
\end{array}\right),$$ $$\mathcal{Q}({K}(n,k,p)) =\left(\begin{array}{lcr}
J_p+(n+q-2)I_p & J_{p,k} & 2J_{p,q}\\
J_{k,p} & J_k+(n-2)I_k & J_{k,q}\\
2J_{q,p} & J_{q,k} & J_q+(n+p-2)I_q\\
\end{array}\right).$$
\[rem51\] Let ${G}$ be a connected graphs with vertex connectivity $k$. Suppose that $S$ is a $k$-vertex cut of ${G}$, and $G_1$ is a connected component of $G - S$. Let ${G_2}={G}-S-{G_1}$, we add edges to ${G}$ until both induced subgraph of $V({G_1})\cup S$ and induced subgraph of $V({G_2})\cup S$ attain to complete graphs, the new graph denoted by ${H}$. Clearly, ${H}={K}(n,k,p)\in \mathcal{C}(n,k)$ for some $p$ such that $1\leq p\leq n-k-1$. Since ${G}$ is the spanning subgraph of ${H}$, then by Corollary \[cor23\], we have $\rho({G})\leq \rho({K}(n,k,p))$, $q({G})\leq q({K}(n,k,p))$, $\rho^{\mathcal{D}}({G})\geq \rho^{\mathcal{D}}({K}(n,k,p))$ and $q^{\mathcal{D}}({G})\geq q^{\mathcal{D}}({K}(n,k,p))$. Thus the extremal graphs which achieve the maximal (signless Laplacian) spectral radius and the minimal distance (signless Laplacian) spectral radius in $\mathcal{C}(n,k)$ must be some $K(n,k,p)$ for $1\leq p\leq n-k-1$.
Let $n,k$ be given positive integers with $1\leq k\leq n-2$, ${G}\in \mathcal{C}(n,k).$ Then
\(i) [([@2010LAA])]{} $\rho({G})\leq \rho(K(n,k,1)),$ and $\rho({{K}(n,k,1)}$ is the largest root of equation (\[eq51\]): $$\label{eq51}
\lambda^3 - (n - 3)\lambda^2 - (n + k - 2)\lambda + k(n - k - 2) = 0,$$ with equality holds if and only if $G = K(n,k,1).$
\(ii) [([@2010LAA])]{} $$\label{eq52}
q({G})\leq q(K(n,k,1)) = \frac{2n+k-4+\sqrt{(2n-k-4)^2+8k}}{2},$$ with equality holds if and only if $G = K(n,k,1).$
\(iii) [([@2012DM2])]{} $\rho^\mathcal{D}({G})\geq \rho^\mathcal{D}(K(n,k,1)),$ and $\rho^\mathcal{D}({{K}(n,k,1)}$ is the largest root of equation (\[eq53\]): $$\label{eq53}
\lambda^3 - (n - 3)\lambda^2 - (5n - 3k - 6)\lambda + kn - k^2 + 2k - 4n + 4= 0,$$ with equality holds if and only if $G = K(n,k,1).$
\(iv) [([@2013])]{} $q^\mathcal{Q}({G})\geq q^\mathcal{Q}(K(n,k,1)),$ and $q^\mathcal{Q}({{K}(n,k,1)}$ is the largest root of equation (\[eq54\]): $$\label{eq54}
\lambda^3 - (5n - k -6)\lambda^2 + (8n^2 - 19kn - 24n + 8k + 16)\lambda - 4n^3 + 2(k + 10)n^2 - 2(5k + 16)n + 12k + 16 = 0,$$ with equality holds if and only if $G = K(n,k,1).$
Firstly, we show (i) holds. We apply Lemma \[lem34\] to $A=A({K}(n,k,p))$. Since $t=3$, $l_1=l_2=l_3=1,$ $p_1=p_2=p_3=-1,$ $s_{13}=s_{31}=0$ and $s_{12}=s_{21}=s_{23}=s_{32}=1$, we have $\sigma(A)=\sigma(B(A))\cup\{(-1)^{[n-3]}\},$ where the corresponding equitable quotient matrix of $A$ is $B(A)=\left(\begin{array}{ccc}
p-1 & k & 0\\
p & k-1 & q\\
0 & k & q-1\\
\end{array}\right),$ the eigenvalues of $B(A)$ are the roots of the equation $$\label{eq55}
\lambda^3 - (n - 3)\lambda^2 + (pq - 2n + 3)\lambda + pq - n + pqk + 1 = 0.$$ It is obvious that $\rho(A({K}(n,k,p)))$ is the largest root of the equation (\[eq55\]).
Now we show $\rho(A({K}(n,k,1)))=\max\{\rho(A({K}(n,k,p)))| 1\leq p\leq n-k-1\}$. We note that $p+q=n-k$ and the adjacency matrix is symmetric, without loss of generality, we assume that $q \geq p \geq 1$. Let $f_{p,q}(\lambda)=\lambda^3 - (n - 3)\lambda^2 + (pq - 2n + 3)\lambda + pq - n + pqk + 1$. Let $H= {K_k} \bigtriangledown ({K_{p-1}} \cup {K}_{q+1})={K}(n,k,p-1)$. Obviously, $H \in \mathcal C(n,k)$ and $\rho(H)$ is the largest root of $f_{p-1,q+1}(\lambda) = 0$, then
$ f_{p,q}(\lambda) - f_{p-1,q+1}(\lambda)$ $=pq\lambda+pq+pqk-(p-1)(q+1)\lambda-(p-1)(q+1)-(p-1)(q+1)k$
3.65cm $=(q+1-p)(\lambda+k+1) > 0$,
and
$f_{p,q}(\rho(H)) = f_{p,q}(\rho(H)) - f_{p-1,q+1}(\rho(H))>0=f_{p,q}(\rho({K}(n,k,p))).$
It implies $\rho(H)=\rho({K}(n,k,p-1)) > \rho({K}(n,k,p))$. Thus $\rho({G})\leq \rho(K(n,k,1))$, $\rho({{K}(n,k,1)})$ is the largest root of the equation (\[eq51\]), $\rho({G})=\rho(K(n,k,1))$ if and only if $G = K(n,k,1).$
Second, we show (ii) holds. We apply Lemma \[lem34\] to $Q=Q({K}(n,k,p))$. Since $t=3$, $l_1=l_2=l_3=1,$ $p_1=p+k-2, $ $p_2=n-2,$ $p_3=n-p-2,$ $s_{13}=s_{31}=0$ and $s_{12}=s_{21}=s_{23}=s_{32}=1$, we have $\sigma(Q)=\sigma(B(Q))\cup\{(p+k-2)^{[p-1]},(n-2)^{[k-1]},(n-p-2)^{[q-1]}\}$ where the corresponding equitable quotient matrix of $Q$ is $$B(Q)=\left(\begin{array}{ccc}
2p+k-2 & k & 0\\
p & k+n-2 & q\\
0 & k & q+n-p-2\\
\end{array}\right),$$ the eigenvalues of $B(Q)$ are $n-2, n-2+\frac{k}{2}\pm\frac{1}{2}\sqrt{(k-2n)^2+16p(k-n+p)}$. Thus $\rho(Q)=n-2+\frac{k}{2}+\frac{1}{2}\sqrt{(k-2n)^2+16p(k-n+p)}.$
Let $f(p)=n-2+\frac{k}{2}+\frac{1}{2}\sqrt{(k-2n)^2+16p(k-n+p)}$, then $f(1)=f(n-k-1)=\max\{f(p) | 1\leq p\leq n-k-1\}.$ Therefore, $q({G})\leq \frac{2n+k-4+\sqrt{(2n-k-4)^2+8k}}{2},$ and we complete the proof of (ii) by ${K}(n,k,1)\cong {K}(n,k,n-k-1)$.
Third, we show (iii) holds. We apply Lemma \[lem34\] to $\mathcal{D}=\mathcal{D}({K}(n,k,p))$. Since $t=3$, $l_1=l_2=l_3=1,$ $p_1=p_2=p_3=-1,$ $s_{13}=s_{31}=2$ and $s_{12}=s_{21}=s_{23}=s_{32}=1$, we have $\sigma(\mathcal{D})=\sigma(B(\mathcal{D}))\cup\{(-1)^{[n-3]}\}$ where the corresponding equitable quotient matrix of $\mathcal{D}$ is $$B(\mathcal{D})=\left(\begin{array}{ccc}
p-1 & k & 2q\\
p & k-1 & q\\
2p & k & q-1\\
\end{array}\right),$$ the eigenvalues of $B(\mathcal{D})$ are the roots of the equation: $$\label{eq56}
\lambda^3-(n-3)\lambda^2-(3pq+2n-3)\lambda+pqk-3pq-n+1=0.$$ It is obvious that $\rho(\mathcal{D}({K}(n,k,p)))$ is the largest root of the equation (\[eq56\]).
Similar to the proof of (i), we can show (iii) holds, we omit it.
Finally, we show (iv) holds. We apply Lemma \[lem34\] to $\mathcal{Q}=\mathcal{Q}({K}(n,k,p))$. Since $t=3$, $l_1=l_2=l_3=1,$ $ p_1=n+q-2,$ $p_2=n-2,$ $p_3=n+p-2,$ $ s_{13}=s_{31}=2$ and $s_{12}=s_{21}=s_{23}=s_{32}=1$, we have $\sigma(\mathcal{Q})=\sigma(B(\mathcal{Q}))\cup\{(n+q-2)^{[p-1]},(n-2)^{[k-1]},(n+p-2)^{[q-1]}\} $ where the corresponding equitable quotient matrix of $\mathcal{Q}$ is $$B(\mathcal{Q})=\left(\begin{array}{lcr}
n+p+q-2 & k & 2q\\
p & n+k-2 & q\\
2p & k & n+p+q-2\\
\end{array}\right),$$ the eigenvalues of $B(\mathcal{Q})$ are the roots of the equation: $\lambda^3-(5p+5q+4k-6)\lambda^2+(8p^2+8q^2+5k^2+12pq+13pk+13qk
-20p-20q-16k+12)\lambda-4p^3-4q^3-2k^3-8p^2q-8pq^2-10p^2k
-10q^2k-8pk^2-8qk^2-16pqk+16p^2+16q^2+10k^2+24pq+26pk+26qk
-20p-20q-16k+8=0.$
Similar to the proof of (i), we can show (iv) holds, we omit it.
It is natural that whether there exists similar result for the Laplacian spectral radius or the distance Laplacian spectral radius in $\mathcal{C}(n,k)$ or not? In fact, we can obtain the spectrum of the Laplacian matrix or the distance Laplacian matrix of $K(n,k,p)$ immediately.
\[prop52\] Let $K(n,k,p)$ defined as before. Then
(i). $\sigma(L(K(n,k,p)))=\{0, k, n^{[k]},(p+k)^{[p-1]}, (q+k)^{[q-1]}\}.$
(ii). $\sigma(\mathcal{L}(K(n,k,p)))=\{0, n+p+q, n^{[k]}, (n+q)^{p-1},(n+p)^{q-1}\}.$
Firstly, the Laplacian matrix $L(K(n,k,p))$ and the distance Laplacian matrix $\mathcal{L}(K(n,k,p))$ of $K(n,k,p)$ are the following matrices, where $q=n-p-k$.
$$L=L(K(n,k,p)) = \left(\begin{array}{ccc}
-J_p+(p+k)I_p & -J_{p,k} & \mathbf{0}_{p,q}\\
-J_{k,p} & -J_k+nI_k & -J_{k,q}\\
\mathbf{0}_{q,p} & -J_{q,k} & -J_q+(q+k)I_q\\
\end{array}\right),$$ $$\mathcal{L}=\mathcal{L}(K(n,k,p)) = \left(\begin{array}{ccc}
-J_p+(n+q)I_p & -J_{p,k} & -2J_{p,q}\\
-J_{k,p} & -J_k+nI_k & -J_{k,q}\\
-2J_{q,p} & -J_{q,k} & -J_q+(n+p)I_q\\
\end{array}\right).$$
Then the corresponding equitable quotient matrices are as follows: $$B(L)= \left(\begin{array}{ccc}
k & -k & 0\\
-p & n-k & -q\\
0 & -k & k\\
\end{array}\right), \qquad B(\mathcal{L})= \left(\begin{array}{ccc}
n+q-p & -k & -2q\\
-p & n-k & -q\\
-2p & -k & n+p-q\\
\end{array}\right).$$
Then by Lemma \[lem34\] and directly calculating, we obtain (i) and (ii).
M. Aouchiche, P. Hansen, Two Laplacian for the distance matrix of a graph, Linear Algebra Appl. 439 (2013) 21–33.
J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, London, 1976.
Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, New York: Academic Press, 1979.
R. Brualdi, Spectra of digraphs, Linear Algebra Appl. 432 (2010) 2181–2213.
Andries E. Brouwer, Willem H. Haemers, Spectra of graphs - Monograph -, Springer, 2011.
S. Burcu Bozkurt and Durmus Bozkurt, On the signless Laplacian spectral radius of digraphs, Ars Combinatoria, 108 (2013) 193–200.
D. Cvetkovic, P. Rowlinson, S. Simic, An Introduction to the Theory of Graph Spectra, London, 2010.
Y.Y. Chen, H.Q. Lin, J.L. Shu, Sharp upper bounds on the distance spectral radius of a graph, Linear Algebra Appl. 439 (2013) 2659–2666.
S.W. Drury, H.Q. Lin, Extremal digraphs with given clique number, Linear Algebra Appl. 439 (2013) 328–345.
Roman Drnovsêk, The spread of the spectrum of a nonnegative matrix with a zero diagonal element, Linear Algebra Appl. 439 (2013) 2381–2387.
X. Duan, B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl. 439 (2013) 2961–2970.
W.H. Haemers, Interlacing eigenvalues and graph, Linear Algebra Appl. 226–228 (1995) 593–616.
W.X. Hong, L.H. You, Spectral radius and signless Laplacian spectral radius of strongly connected digraphs, Linear Algebra Appl. 457 (2014) 93–113.
R.A. Horn, Charles R. Johnson, Matrix Analysis, Cambridge University Press, 1986.
G. Indulal, I. Gutman, On the distance spectra of some graphs, Mathematical Communications, 13 (2008) 123–131.
J.B. Jensen, G. Gutin, Digraphs Theory, Algorithms and Applications, Springer, 2001.
H.Q. Lin, Y. Hong, J.F. Wang, J.L. Shu, On the distance spectrum of graphs, Linear Algebra Appl. 439 (2013) 1662–1669.
H.Q. Lin, S.W. Drury, The maximum Perron roots of digraphs with some given parameters, Discrete Math. 313 (2013) 2607–2613.
H.Q. Lin, R.F. Liu, X.W. Lu, The inertia and energy of the distance matrix of a connected graph, Linear Algebra Appl. 467 (2015) 29–39.
H.Q. Lin, J.L. Shu, Y.R. Wu, G.L. Yu, Spectral radius of strongly connected digraphs, Discrete Math. 312 (2012) 3663–3669.
H.Q. Lin, J.L. Shu, The distance spectral radius of digraphs, Discrete Applied Math. 161 (2013) 2537–2543.
H.Q. Lin, W.H. Yang, H.L. Zhang, J.L. Shu, Distance spectral radius of digraphs with given connectivity, Discrete Math. 312 (2012) 1849–1856.
Chia-an Liu, Chih-wen Weng, Spectral radius of bipartite graphs, arXiv:1402.5621v1 \[math.CO\] 23 Feb 2014.
Z.Z. Liu, On spectral radius of the distance matrix, Appl. Anal. Discrete Math. 4 (2010) 269–277.
O. Rojo, E. Lenes, A sharp upper bound on the incidence energy of graphs in terms of connectivity, Linear Algebra Appl. 438 (2013) 1485–1493.
R. Xing, B. Zhou and J. Li, On the distance signless Laplacian spectral radius of graphs, Linear and Multilinear Algebra, 62 (2014) 1377–1387.
M.L. Ye, Y.Z. Fan, D. Liang, The least eigevalue of graphs with given connectivity, Linear Algebra Appl. 430 (2009) 1375–1379.
M.L. Ye, Y.Z. Fan, H.F. Wang, Maximizing signless Laplacian or adjacency spctral radius of graphs subject to fixed connectivity, Linear Algebra Appl. 433 (2010) 1180–1186.
X.L. Zhang, The inertia and energy of the distance matrix of complete $k$-partite graphs, Linear Algebra Appl. 450 (2014) 108–120.
[^1]: [*[Corresponding author:]{}*]{}[email protected].
[^2]: [*[Email address:]{}*]{}[email protected].
[^3]: [*[Email address:]{}*]{}[email protected].
[^4]: [*[Email address:]{}*]{}[email protected].
[^5]: L. You’s research is supported by the Zhujiang Technology New Star Foundation of Guangzhou (Grant No. 2011J2200090) and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (Grant No.2012gjhz0007).
|
---
abstract: 'We propose that quantum physics is the continuous approximation of a more fundamental, discrete graph theory (theory X). Accordingly, the Euclidean transition amplitude $Z$ provides a partition function for geometries over the graph, which is characterized topologically by the difference matrix and source vector of the discrete graphical action. The difference matrix and source vector of theory X are related via a graphical self-consistency criterion (SCC) based on the boundary of a boundary principle on a graph ($\partial_1\cdot \partial_2 = 0$). In this approach, the SCC ensures the source vector is divergence-free and resides in the row space of the difference matrix. Accordingly, the difference matrix will necessarily have a nontrivial eigenvector with eigenvalue zero, so the graphical SCC is the origin of gauge invariance. Factors of infinity associated with gauge groups of infinite volume are excluded in our approach, since $Z$ is restricted to the row space of the difference matrix and source vector. Using this formalism, we obtain the two-source Euclidean transition amplitude over a ($1+1$)-dimensional graph with $N$ vertices fundamental to the scalar Gaussian theory.'
address:
- |
$^1$ Department of Physics\
Elizabethtown College\
Elizabethtown, PA 17022
- |
$^2$ Department of Mathematical Sciences\
Elizabethtown College\
Elizabethtown, PA 17022
- |
$^3$ Department of Philosophy\
Elizabethtown College\
Elizabethtown, PA 17022
author:
- 'W.M. Stuckey$^1$, T.J. McDevitt$^2$ and M. Silberstein$^3$'
title: 'Gauge Invariance from a Graphical Self-Consistency Criterion'
---
[*Keywords* ]{}:graph theory, path integral, gauge invariance, transition amplitude
Introduction {#section1}
============
Those who emphasize the incompleteness of quantum field theory (QFT) over its successes often focus on the many ad hoc and, for some, troubling “fixes” involved in the practice of QFT [^1]. For example, since QFT is independent of overall factors in the transition amplitude, such factors are simply “thrown away” even when these factors are infinity as is the case when the volume of the gauge symmetry group in Faddeev-Popov gauge fixing is infinite[@zee]. In a petition to philosophers of science, Glashow stated[@glashow], “in a sense it really is a time for people like you, philosophers, to contemplate not where we’re going, because we don’t really know and you hear all kinds of strange views, but where we are. And maybe the time has come for you to tell us where we are.” Rovelli went further stating[@rovelli], “As a physicist involved in this effort, I wish that the philosophers who are interested in the scientific description of the world would not confine themselves to commenting and polishing the present fragmentary physical theories, but would take the risk of trying to look ahead.”
Of course, ignoring factors of infinity in the transition amplitude $Z$ per Faddeev-Popov gauge fixing is easily understood in terms of (infinitely) over counting gauge degrees of freedom in the classical field being quantized[@kaku], so there is no problem in that respect. We believe the real issue is the fact that QFT involves the quantization of a classical field[@wallace] when one would rather expect QFT to originate independently of classical field theory, the former typically understood as fundamental to the latter. Herein we accept Glashow and Rovelli’s challenges and respond, not philosophically, but mathematically, and propose a new, fundamental origin for QFT. Specifically, we follow the possibility articulated by Wallace[@wallace] that (p 45), “QFTs as a whole are to be regarded only as approximate descriptions of some as-yet-unknown deeper theory,” which he calls “theory X,“ and we propose a new discrete path integral formalism over graphs for “theory X” underlying QFT. Accordingly, sources ${\bi{J}}$ , space and time are self-consistently co-constructed per a graphical self-consistency criterion (SCC) based on the boundary of a boundary principle[@misner] on the graph ($\partial_1\cdot
\partial_2 = 0$). \[In a graphical representation of QFT, part of ${\bi{J}}$ represents field disturbances emanating from a source location (Source) and the other part represents field disturbances incident on a source location (sink).\] We call this amalgam ”spacetimematter." The SCC constrains the difference matrix and source vector in Z, which then provides the probability for finding a particular source-to-source relationship in a quantum experiment, i.e., experiments which probe individual source-to-source relations (modeled by individual graphical links) as evidenced by discrete outcomes, such as detector clicks. Since, in QFT, all elements of an experiment, e.g., beam splitters, mirrors, and detectors, are represented by interacting sources, we confine ourselves to the discussion of such controlled circumstances where the empirical results evidence individual graphical links. \[Hereafter, all reference to “experiments” will be to “quantum experiments.”\] In this approach, the SCC ensures the source vector is divergence-free and resides in the row space of the difference matrix, so the difference matrix will necessarily have a nontrivial eigenvector with eigenvalue zero, a formal characterization of gauge invariance. Thus, our proposed approach to theory X provides an underlying origin for QFT, accounts naturally for gauge invariance, i.e., via a graphical self-consistency criterion, and excludes factors of infinity associated with gauge groups of infinite volume, since the transition amplitude $Z$ is restricted to the row space of the difference matrix and source vector.
While the formalism we propose for theory X is only suggestive, the computations are daunting, as will be evident when we present the rather involved graphical analysis underlying the Gaussian two-source amplitude which, by contrast, is a trivial problem in its QFT continuum approximation. However, this approach is not intended to replace or augment QFT computations. Rather, our proposed theory X is fundamental to QFT and constitutes a new program for physics, much as quantum physics relates to classical physics. Therefore, the motivation for our theory X is, at this point, conceptual and while there are many conceptual arguments to be made for our approach[@stuckey], we restrict ourselves here to the origins of gauge invariance and QFT.
We understand the reader may not be familiar with the path integral formalism, as Healey puts it[@healey], “While many contemporary physics texts present the path-integral quantization of gauge field theories, and the mathematics of this technique have been intensively studied, I know of no sustained critical discussions of its conceptual foundations.” Therefore, we begin in section \[section2\] with an overview and interpretation of the path integral formalism, which is particularly well-suited for the study of gauge invariance.
The Discrete Path Integral Formalism {#section2}
====================================
In this section we provide an overview and interpretation of the path integral approach, showing explicitly how we intend to use “its conceptual foundations.” We employ the discrete path integral formalism because it embodies a 4Dism that allows us to model spacetimematter. For example, the path integral approach is based on the fact that[@feynman] “the \[S\]ource will emit and the detector receive,” i.e., the path integral formalism deals with Sources and sinks as a unity while invoking a description of the experimental process from initiation to termination. By assuming the discrete path integral is fundamental to the (conventional) continuum path integral, we have a graphical basis for the co-construction of time, space and quantum sources via a self-consistency criterion (SCC). We will show in section 3 how the graphical amalgam of spacetimematter underlies QFT.
Path Integral in Quantum Physics
--------------------------------
In the conventional path integral formalism[@zee2] for non-relativistic quantum mechanics (NRQM) one starts with the amplitude for the propagation from the initial point in configuration space $q_I$ to the final point in configuration space $q_F$ in time $T$ via the unitary operator $e^{-iHT}$, i.e., $\displaystyle \left \langle q_F \left | e^{-iHT} \right | q_I \right \rangle$. Breaking the time $T$ into $N$ pieces $\delta t$ and inserting the identity between each pair of operators $e^{-iH\delta t}$ via the complete set $\int dq | q \rangle \langle q | =1$ we have $$\fl
\left \langle q_F \left | e^{-iHT} \right | q_I \right \rangle = \left [ \prod_{j=1}^{N-1} \int dq_j \right ]
\left \langle q_F \left | e^{-iH\delta t} \right | q_{N-1} \right \rangle
\left \langle q_{N-1} \left | e^{-iH\delta t} \right | q_{N-2} \right \rangle
\ldots$$ $$\left \langle q_2 \left | e^{-iH\delta t} \right | q_1 \right \rangle
\left \langle q_1 \left | e^{-iH\delta t} \right | q_I \right \rangle.$$ With $H=\hat{p}^2/2m + V(\hat{q})$ and $\delta t \rightarrow 0$ one can then show that the amplitude is given by $$\left \langle q_F \left | e^{-iHT} \right | q_I \right \rangle =
\int Dq(t) \exp \left [ i \int_0^T dt L(\dot{q},q) \right ],
\label{eqn1}$$ where $L(\dot{q},q) = m \dot{q}^2/2-V(q)$ . If $q$ is the spatial coordinate on a detector transverse to the line joining Source and detector, then $\displaystyle \prod_{j=1}^{N-1}$ can be thought of as $N-1$ “intermediate” detector surfaces interposed between the Source and the final (real) detector, and $\int dq_j$ can be thought of all possible detection sites on the $j^{\mbox{th}}$ intermediate detector surface. In the continuum limit, these become $\int Dq(t)$ which is therefore viewed as a “sum over all possible paths” from the Source to a particular point on the (real) detector, thus the term “path integral formalism” for conventional NRQM is often understood as a sum over “all paths through space.”
To obtain the path integral approach to QFT one associates $q$ with the oscillator displacement at a [*particular point*]{} in space ($V(q) = kq^2/2$). In QFT, one takes the limit $\delta x \rightarrow
0$ so that space is filled with oscillators and the resulting spatial continuity is accounted for mathematically via $q_i(t)
\rightarrow q(t,x)$, which is denoted $\phi(t,x)$ and called a “field.” The QFT transition amplitude $Z$ then looks like $$Z = \int D\phi \exp \left [ i \int d^4 x L( \dot{\phi}, \phi ) \right ]
\label{eqn2}$$ where $L(\dot{\phi},\phi) = (d\phi)^2/2 - V(\phi)$ . Impulses $J$ are located in the field to account for particle creation and annihilation; these $J$ are called “sources” in QFT and we have $L(\dot{\phi},\phi) = (d\phi)^2/2 - V(\phi) + J(t,x) \phi(t,x)$, which can be rewritten as $L(\dot{\phi},\phi) = \phi D \phi/2 +
J(t,x) \phi(t,x)$, where $D$ is a differential operator. In its discrete form (typically, but not necessarily, a hypercubic spacetime lattice), $D \rightarrow {\bi{K}}$ (a difference matrix), $J(t,x)\rightarrow {\bi{J}}$ (each component of which is associated with a point on the spacetime lattice) and $\phi \rightarrow {\bi{Q}}$ (each component of which is associated with a point on the spacetime lattice). Again, part of ${\bi{J}}$ represents field disturbances emanating from a source location (Source) and the other part represents field disturbances incident on a source location (sink) in the conventional view of path integral QFT and, in particle physics, these field disturbances are the particles. We will keep the partition of ${\bi{J}}$ into Sources and sinks in our theory X, but there will be no vacuum lattice structure between the discrete set of sources. The discrete counterpart to (\[eqn2\]) is then[@zee3] $$Z = \int \ldots \int dQ_1 \ldots dQ_N \exp \left[ \frac {i}{2} {\bi{Q}}\cdot {\bi{K}}\cdot{\bi{Q}}+ i {\bi{J}}\cdot{\bi{Q}}\right ]. \label{eqn3}$$ In conventional quantum physics, NRQM is understood as $(0+1)-$dimensional QFT.
Our Interpretation of the Path Integral in Quantum Physics
----------------------------------------------------------
We agree that NRQM is to be understood as $(0+1)-$dimensional QFT, but point out this is at conceptual odds with our derivation of (\[eqn1\]) when $\int Dq(t)$ represented a sum over all paths in space, i.e., when $q$ was understood as a location in space (specifically, a location along a detector surface). If NRQM is $(0+1)-$dimensional QFT, then $q$ is a field displacement at a single location in space. In that case, $\int Dq(t)$ must represent a sum over all field values at a particular point on the detector, not a sum over all paths through space from the Source to a particular point on the detector (sink). So, how [*do*]{} we relate a point on the detector (sink) to the Source?
In answering this question, we now explain a formal difference between conventional path integral NRQM and our proposed approach: our links only connect and construct discrete sources ${\bi{J}}$, there are no source-to-spacetime links (there is no vacuum lattice structure, only spacetimematter). Instead of $\delta x \rightarrow
0$, as in QFT, we assume $\delta x$ is measureable for (such) NRQM phenomenon. More specifically, we propose starting with (\[eqn3\]) whence (roughly) NRQM obtains in the limit $\delta t \rightarrow 0$, as in deriving (\[eqn1\]), and QFT obtains in the additional limit $\delta x \rightarrow 0$, as in deriving (\[eqn2\]). The QFT limit is well understood as it is the basis for lattice gauge theory and regularization techniques, so one might argue that we are simply [*clarifying*]{} the NRQM limit where the path integral formalism is not widely employed. However, again, we are proposing a discrete starting point for theory X, as in (\[eqn3\]). Of course, that discrete spacetime is fundamental while “the usual continuum theory is very likely only an approximation[@Feinberg]” is not new.
Discrete Path Integral is Fundamental
-------------------------------------
The version of theory X we propose is a discrete path integral over graphs, so (\[eqn3\]) [is not a discrete approximation of (\[eqn1\]) & (\[eqn2\])]{}, but rather [*(\[eqn1\]) & (\[eqn2\]) are continuous approximations of (\[eqn3\])*]{}. In the arena of quantum gravity it is not unusual to find discrete theories[@loll] that are in some way underneath spacetime theory and theories of “matter” such as QFT, e.g., causal dynamical triangulations[@ambjorn], quantum graphity[@konopka] and causets[@sorkin1]. While these approaches are interesting and promising, the approach taken here for theory X will look more like Regge calculus quantum gravity (see Bahr & Dittrich [@bahr] and references therein for recent work along these lines) modified to contain no vacuum lattice structure.
Placing a discrete path integral at bottom introduces conceptual and analytical deviations from the conventional, continuum path integral approach. Conceptually, (\[eqn1\]) of NRQM represents a sum over all field values at a particular point on the detector, while (\[eqn3\]) of theory X is a mathematical machine that measures the “symmetry” (strength of stationary points) contained in the core of the discrete action $$\frac 12 {\bi{K}}+ {\bi{J}}\label{eqn4}$$ This core or [*actional*]{} yields the discrete action after operating on a particular vector ${\bi{Q}}$ (field). The actional represents a [*fundamental/topological, 4D description of the experiment*]{} and $Z$ is a measure of its symmetry. \[In its Euclidean form, which is the form we will use, $Z$ is a partition function.\] For this reason we prefer to call $Z$ the symmetry amplitude of the 4D experimental configuration. Analytically, because we are [*starting*]{} with a discrete formalism, we are in position to mathematically explicate trans-temporal identity, whereas this process is unarticulated elsewhere in physics. As we will now see, this leads to our proposed self-consistency criterion (SCC) underlying $Z$.
Self-Consistency Criterion
--------------------------
Our use of a self-consistency criterion is not without precedent, as we already have an ideal example in Einstein’s equations of general relativity (GR). Momentum, force and energy all depend on spatiotemporal measurements (tacit or explicit), so the stress-energy tensor cannot be constructed without tacit or explicit knowledge of the spacetime metric (technically, the stress-energy tensor can be written as the functional derivative of the matter-energy Lagrangian with respect to the metric). But, if one wants a “dynamic spacetime” in the parlance of GR, the spacetime metric must depend on the matter-energy distribution in spacetime. GR solves this dilemma by demanding the stress-energy tensor be “consistent” with the spacetime metric per Einstein’s equations. For example, concerning the stress-energy tensor, Hamber and Williams write[@Hamber], “In general its covariant divergence is not zero, but consistency of the Einstein field equations demands $\nabla^{\alpha} T_{\alpha \beta} = 0$ .” This self-consistency hinges on divergence-free sources, which finds a mathematical underpinning in $\partial \partial = 0$. So, Einstein’s equations of GR are a mathematical articulation of the boundary of a boundary principle at the classical level, i.e., they constitute a self-consistency criterion at the classical level, as are quantum and classical electromagnetism[@misner2]. We will provide an explanation for this fact in section \[section3\], but essentially the graphical SCC of our theory X gives rise to continuum counterparts in QFT and classical field theory.
In order to illustrate the discrete mathematical co-constuction of space, time and sources ${\bi{J}}$, we will use graph theory a la Wise[@wise2] and find that $\partial_1\cdot \partial_1^T$, where $\partial_1$ is a boundary operator in the spacetime chain complex of our graph satisfying $\partial_1\cdot \partial_2 = 0$ , has precisely the same form as the difference matrix in the discrete action for coupled harmonic oscillators. Therefore, we are led to speculate that ${\bi{K}}\propto
\partial_1\cdot \partial_1^T$. Defining the source vector ${\bi{J}}$ relationally via ${\bi{J}}\propto \partial_1\cdot {\bi{e}}$ then gives tautologically per $\partial_1\cdot \partial_2 = 0$ both a divergence-free ${\bi{J}}$ and ${\bi{K}}\cdot {\bi{v}}\propto {\bi{J}}$, where ${\bi{e}}$ is the vector of links and ${\bi{v}}$ is the vector of vertices. ${\bi{K}}\cdot {\bi{v}}\propto {\bi{J}}$ is our SCC following from $\partial_1\cdot\partial_2 = 0$, and it defines what is meant by a self-consistent co-construction of space, time and divergence-free sources ${\bi{J}}$, thereby constraining ${\bi{K}}$ and ${\bi{J}}$ in $Z$. Thus, our SCC provides a basis for the discrete action and supports our view that (\[eqn3\]) is fundamental to (\[eqn1\]) & (\[eqn2\]), rather than the converse. Conceptually, that is the basis of our discrete, graphical path integral approach to theory X. We now provide the details.
The Formalism {#section3}
=============
The General Approach
--------------------
Again, in theory X, the symmetry amplitude $Z$ contains a discrete action constructed per a self-consistency criterion (SCC) for space, time and divergence-free sources ${\bi{J}}$. As introduced in section \[section2\] and argued later in this section, we will codify the SCC using ${\bi{K}}$ and ${\bi{J}}$; these elements are germane to the transition amplitude $Z$ in the Central Identity of Quantum Field Theory[@zee4], $$\fl
Z = \int D {\bphi}\exp \left [ - \frac 12 {\bphi}\cdot {\bi{K}}\cdot {\bphi}- V({\bphi}) + {\bi{J}}\cdot {\bphi}\right ] \\
= \exp \left [ -V \left ( \frac {\delta}{\delta J} \right ) \right ] \exp \left [\frac 12 {\bi{J}}\cdot {\bi{K}}^{-1} \cdot {\bi{J}}\right ].
\label{eqn5}$$ While the field is a mere integration variable used to produce $Z$, it must reappear at the level of classical field theory. To see how the field makes it appearance per theory X, consider (\[eqn5\]) for the simple Gaussian theory ($V(\phi) = 0$). On a graph with $N$ vertices, (\[eqn5\]) is $$Z = \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} dQ_1
\ldots dQ_N \exp \left [-\frac 12 {\bi{Q}}\cdot {\bi{K}}\cdot {\bi{Q}}+ {\bi{J}}\cdot {\bi{Q}}\right ] \label{eqn6}$$ with a solution of $$Z = \left ( \frac {(2\pi)^N}{\det {\bi{K}}} \right )^{1/2} \exp \left [\frac 12 {\bi{J}}\cdot {\bi{K}}^{-1} \cdot {\bi{J}}\right ].
\label{eqn7}$$ It is easiest to work in an eigenbasis of ${\bi{K}}$ and (as will argue later) we restrict the path integral to the row space of ${\bi{K}}$, this gives $$Z = \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} d{\tilde{Q}}_1
\ldots d{\tilde{Q}}_{N-1} \exp \left [\sum_{j=1}^{N-1} \left (-\frac 12
{\tilde{Q}}_j^2 a_j + {\tilde{J}}_j {\tilde{Q}}_j \right ) \right ] \label{eqn8}$$ where ${\tilde{Q}}_j$ are the coordinates associated with the eigenbasis of ${\bi{K}}$ and ${\tilde{Q}}_N$ is associated with eigenvalue zero, $a_j$ is the eigenvalue of ${\bi{K}}$ corresponding to ${\tilde{Q}}_j$, and ${\tilde{J}}_j$ are the components of ${\bi{J}}$ in the eigenbasis of ${\bi{K}}$. The solution of (8) is $$Z = \left ( \frac {(2\pi)^{N-1}}{\prod_{j=1}^{N-1} a_j} \right )^{1/2} \prod_{j=1}^{N-1} \exp \left ( \frac {{\tilde{J}}_j^2}{2a_j} \right ).
\label{eqn9}$$ On our view, the experiment is described fundamentally by ${\bi{K}}$ and ${\bi{J}}$ on our topological graph. Again, per (\[eqn9\]), there is no field ${\tilde{Q}}$ appearing in $Z$ at this level, i.e., ${\tilde{Q}}$ is only an integration variable. ${\tilde{Q}}$ makes its first appearance as something more than an integration variable when we produce probabilities from $Z$. That is, since we are working with a Euclidean path integral, $Z$ is a partition function and the probability of measuring ${\tilde{Q}}_k={\tilde{Q}}_0$ is found by computing the fraction of $Z$ which contains ${\tilde{Q}}_0$ at the $k^{\mbox{th}}$ vertex[@lisi]. We have $$\fl
P \left ( {\tilde{Q}}_k = {\tilde{Q}}_0 \right ) = \frac {Z \left ( {\tilde{Q}}_k = {\tilde{Q}}_0 \right )}{Z} = \sqrt{\frac {a_k}{2\pi}} \exp \left ( - \frac 12 {\tilde{Q}}_0^2 a_k + {\tilde{J}}_k {\tilde{Q}}_0 - \frac {{\tilde{J}}_k^2}{2a_k} \right )
\label{eqn10}$$ as the part of theory X approximated in the continuum by QFT. The most probable value of ${\tilde{Q}}_0$ at the $k^{\mbox{th}}$ vertex is then given by $$\fl
\delta P \left ( {\tilde{Q}}_k = {\tilde{Q}}_0 \right ) = 0 \Longrightarrow \delta \left ( - \frac 12 {\tilde{Q}}_0^2 a_k + {\tilde{J}}_k {\tilde{Q}}_0 - \frac {{\tilde{J}}_k^2}{2a_k} \right ) = 0 \Longrightarrow a_k {\tilde{Q}}_0 = {\tilde{J}}_k.
\label{eqn11}$$ That is, ${\bi{K}}\cdot {\bi{Q}}_0 = {\bi{J}}$ is the part of theory X that obtains statistically and is approximated in the continuum by classical field theory. We note that the manner by which ${\bi{K}}\cdot
{\bi{Q}}_0 = {\bi{J}}$ follows from $P({\tilde{Q}}_k = {\tilde{Q}}_0) = Z({\tilde{Q}}_k = {\tilde{Q}}_0)/Z$ parallels the manner by which classical field theory follows from QFT via the stationary phase method[@zee5]. Thus, one may obtain classical field theory by the continuum limit of ${\bi{K}}\cdot {\bi{Q}}_0
= {\bi{J}}$ in theory X (theory X $\rightarrow$ classical field theory), or by first obtaining QFT via the continuum limit of $P({\tilde{Q}}_k =
{\tilde{Q}}_0) = Z({\tilde{Q}}_k = {\tilde{Q}}_0)/Z$ in theory X and then by using the stationary phase method on QFT (theory X $\rightarrow$ QFT $\rightarrow$ classical field theory). In either case, QFT is not quantized classical field theory in our approach. In summary:
1. $Z$ is a partition function for an experiment described topologically by ${\bi{K}}/2+ {\bi{J}}$ (Figure \[fig1\]a).
2. $P({\tilde{Q}}_k = {\tilde{Q}}_0) = Z({\tilde{Q}}_k = {\tilde{Q}}_0)/Z$ gives us the probability for a particular geometric outcome in that experiment (Figures \[fig1\]b and \[fig2\]b).
3. ${\bi{K}}\cdot {\bi{Q}}_0 = {\bi{J}}$ gives us the most probable values of the experimental outcomes which are then averaged to produce the geometry for the experimental procedure at the classical level (Figure \[fig2\]a).
4. $P({\tilde{Q}}_k = {\tilde{Q}}_0) = Z({\tilde{Q}}_k = {\tilde{Q}}_0)/Z$ and ${\bi{K}}\cdot {\bi{Q}}_0 = {\bi{J}}$ are the parts of theory X approximated in the continuum by QFT and classical field theory, respectively.
The Two-Source Euclidean Symmetry Amplitude/Partition Function
--------------------------------------------------------------
Typically, one identifies fundamentally interesting physics with symmetries of the action in the Central Identity of Quantum Field Theory, but we have theory X fundamental to QFT, so our method of choosing fundamentally interesting physics must reside in the topological graph of theory X. Thus, we seek a constraint of ${\bi{K}}$ and ${\bi{J}}$ in our graphical symmetry amplitude $Z$ and this will be in the form of a self-consistency criterion (SCC). In order to motivate our general method, we will first consider a simple graph with six vertices, seven links and two plaquettes for our $(1+1)-$dimensional spacetime model (Figure \[fig3\]). Our goal with this simple model is to seek relevant structure that might be used to infer an SCC. We begin by constructing the boundary operators over our graph.
The boundary of ${\bi{p}}_1$ is ${\bi{e}}_4 + {\bi{e}}_5 - {\bi{e}}_2 - {\bi{e}}_1$, which also provides an orientation. The boundary of ${\bi{e}}_1$ is ${\bi{v}}_2 - {\bi{v}}_1$, which likewise provides an orientation. Using these conventions for the orientations of links and plaquettes we have the following boundary operator for $C_2 \rightarrow C_1$, i.e., space of plaquettes mapped to space of links in the spacetime chain complex: $$\partial_2 = \left [ \begin{array}{rr}
-1 & 0 \\
-1 & 1 \\
0 & -1 \\
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & -1 \end{array} \right ]
\label{eqn12}$$ Notice the first column is simply the links for the boundary of ${\bi{p}}_1$ and the second column is simply the links for the boundary of ${\bi{p}}_2$. We have the following boundary operator for $C_1
\rightarrow C_0$, i.e., space of links mapped to space of vertices in the spacetime chain complex: $$\partial_1 = \left [ \begin{array}{rrrrrrr}
-1 & 0 & 0 & -1 & 0 & 0 & 0 \\
1 & -1 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 \end{array} \right ]
\label{eqn13}$$ which completes the spacetime chain complex, $C_0 \leftarrow C_1
\leftarrow C_2$. Notice the columns are simply the vertices for the boundaries of the edges. These boundary operators satisfy $\partial_1\cdot\partial_2 = 0$, i.e., the boundary of a boundary principle.
The potential for coupled oscillators can be written $$V(q_1,q_2) = \sum_{a,b} \frac 12 k_{ab} q_a q_b = \frac 12 k q_1^2 + \frac 12 k q_2^2 + k_{12} q_1 q_2
\label{eqn14}$$ where $k_{11} = k_{22} = k>0$ and $k_{12} = k_{21}<0$ per the classical analogue (Figure \[fig4\]) with $k = k_1 + k_3 = k_2 +
k_3$ and $k_{12} = -k_3$ to recover the form in (\[eqn14\]). The Lagrangian is then $$L = \frac 12 m \dot{q}_1^2 + \frac 12 m \dot{q}_2^2 - \frac 12
kq_1^2 - \frac 12 k q_2^2 - k_{12} q_1q_2 \label{eqn15}$$ so our NRQM Euclidean symmetry amplitude is $$\fl
Z = \int Dq(t) \exp \left [ - \int_0^T dt \left ( \frac 12 m \dot{q}_1^2 + \frac 12 m \dot{q}_2^2 + V(q_1, q_2) - J_1 q_1 - J_2 q_2 \right )\right ]
\label{eqn16}$$ after Wick rotation. This gives $$\fl {\bi{K}}= \left [ \begin{array}{rrrrrr}
\left ( \frac m{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} & 0 & k_{12} \Delta t & 0 & 0 \\
-\frac m{\Delta t} & \left ( \frac {2m}{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} & 0 & k_{12} \Delta t & 0 \\
0 & -\frac m{\Delta t} & \left ( \frac m{\Delta t} + k \Delta t \right ) & 0 & 0 & k_{12} \Delta t \\
k_{12} \Delta t & 0 & 0 & \left ( \frac m{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} & 0 \\
0 & k_{12} \Delta t & 0 & -\frac m{\Delta t} & \left ( \frac {2m}{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} \\
0 & 0 & k_{12} \Delta t & 0 & -\frac m{\Delta t} & \left ( \frac m{\Delta t} + k \Delta t \right ) \end{array} \right ]
\label{eqn17}$$ on our graph. Thus, we borrow (loosely) from Wise[@wise3] and suggest ${\bi{K}}\propto \partial_1\cdot\partial_1^T$ since $$\partial_1\cdot\partial_1^T = \left [ \begin{array}{rrrrrr}
2 & -1 & 0 & -1 & 0 & 0 \\
-1 & 3 & -1 & 0 & -1 & 0 \\
0 & - 1& 2 & 0 & 0 & -1 \\
-1 & 0 & 0 & 2 & -1 & 0 \\
0 & -1 & 0 & -1 & 3 & -1 \\
0 & 0 & -1 & 0 & -1 & 2 \end{array} \right ]
\label{eqn18}$$ produces precisely the same form as (\[eqn17\]) and quantum theory is known to be “rooted in this harmonic paradigm[@zee6].” \[In fact, these matrices will continue to have the same form as one increases the number of vertices in Figure \[fig3\].\] Now we construct a suitable candidate for ${\bi{J}}$, relate it to ${\bi{K}}$ and infer our SCC.
Recall that ${\bi{J}}$ has a component associated with each vertex so here it has components, $J_n$, $n = 1, 2, \ldots, 6$; $J_n$ for $n =
1, 2, 3$ represents one source and $J_n$ for $n = 4, 5, 6$ represents the second source. We propose ${\bi{J}}\propto
\partial_1\cdot{\bi{e}}$, where $e_i$ are the links of our graph, since $$\fl
\partial_1\cdot{\bi{e}}=
\left [ \begin{array}{rrrrrrr}
-1 & 0 & 0 & -1 & 0 & 0 & 0 \\
1 & -1 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 \end{array} \right ]
\left [ \begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \\ e_6 \\ e_7 \end{array} \right ]
= \left [\begin{array}{c} -e_1-e_4 \\ e_1 - e_2-e_3 \\ e_3 - e_7 \\ e_4 - e_5 \\ e_2 + e_5 - e_6 \\ e_6 + e_7 \end{array}\right ]
\label{eqn19}$$ automatically makes ${\bi{J}}$ divergence-free, i.e., $\displaystyle
\sum_i J_i = 0$. Such a relationship on discrete spacetime lattices is not new. For example, Sorkin showed that charge conservation follows from gauge invariance for the electromagnetic field on a simplicial net[@sorkin2].
With these definitions of ${\bi{K}}$ and ${\bi{J}}$ we have, ipso facto, ${\bi{K}}\cdot {\bi{v}}\propto {\bi{J}}$ as the basis of our SCC since $$\fl
\partial_1\cdot\partial_1^T \cdot {\bi{v}}= \left [ \begin{array}{rrrrrr}
2 & -1 & 0 & -1 & 0 & 0 \\
-1 & 3 & -1 & 0 & -1 & 0 \\
0 & - 1& 2 & 0 & 0 & -1 \\
-1 & 0 & 0 & 2 & -1 & 0 \\
0 & -1 & 0 & -1 & 3 & -1 \\
0 & 0 & -1 & 0 & -1 & 2 \end{array} \right ] \left [
\begin{array}{c} v_1 \\ v_2 \\ v_3 \\ v_4 \\ v_5 \\ v_6 \end{array}
\right ] = \left [\begin{array}{c} -e_1-e_4 \\ e_1 - e_2-e_3 \\ e_3 - e_7 \\
e_4 - e_5\\ e_2 + e_5 - e_6 \\ e_6 + e_7 \end{array} \right ] =
\partial_1\cdot{\bi{e}}\label{eqn20}$$ where we have used $e_1 = v_2 - v_1$ (etc.) to obtain the last column. You can see that the boundary of a boundary principle underwrites (\[eqn20\]) by the definition of “boundary” and from the fact that the links are directed and connect one vertex to another, i.e., they do not start or end ‘off the graph’. Likewise, this fact and our definition of ${\bi{J}}$ imply $\displaystyle \sum_i
J_i = 0$, which is our graphical equivalent of a divergence-free, relationally defined source (every link leaving one vertex goes into another vertex). Thus, the SCC ${\bi{K}}\cdot {\bi{v}}\propto {\bi{J}}$ and divergence-free sources $\displaystyle \sum_i J_i = 0$ obtain tautologically via the boundary of a boundary principle. The SCC also guarantees that ${\bi{J}}$ resides in the row space of ${\bi{K}}$ so, as will be shown, we can avoid having to “throw away infinities” associated with gauge groups of infinite volume as in Faddeev-Popov gauge fixing. ${\bi{K}}$ has at least one eigenvector with zero eigenvalue which is responsible for gauge invariance, so [*the self-consistent co-construction of space, time and divergence-free sources entails gauge invariance.*]{}
Moving now to $N$ dimensions, the Wick rotated version of (\[eqn3\]) is (\[eqn6\]) and the solution is (\[eqn7\]). Using ${\bi{J}}= \alpha
\partial_1\cdot{\bi{e}}$ and ${\bi{K}}= \beta \partial_1\cdot
\partial_1^T$ ($\alpha, \beta \in \mathbb{R}$) with the SCC gives ${\bi{K}}\cdot {\bi{v}}= (\beta/\alpha) {\bi{J}}$, so that ${\bi{v}}=
(\beta/\alpha) {\bi{K}}^{-1}\cdot {\bi{J}}$. However, ${\bi{K}}^{-1}$ does not exist because ${\bi{K}}$ has a nontrivial null space, therefore the row space of ${\bi{K}}$ is an $(N-1)-$dimensional subspace of the $N-$dimensional vector space[^2]. The eigenvector with eigenvalue of zero, i.e., normal to this hyperplane, is $\left[
\begin{array}{ccccc} 1 & 1 & 1 & \ldots & 1 \end{array} \right ]^T$, which follows from the SCC as shown supra. Since ${\bi{J}}$ resides in the row space of ${\bi{K}}$ and, on our view, $Z$ is a functional of ${\bi{K}}$ and ${\bi{J}}$ which produces a partition function for the various ${\bi{K}}/2+{\bi{J}}$ associated with different 4D experimental configurations, we restrict the path integral of (\[eqn6\]) to the row space of ${\bi{K}}$. Thus, our approach revises (\[eqn7\]) to give (\[eqn9\]).
We find in general that half the eigenvectors of ${\bi{K}}$ are of the form $\displaystyle \left [ \begin{array}{c} {\bi{x}}\\ {\bi{x}}\end{array} \right ]$ and half are of the form $\displaystyle \left [ \begin{array}{c} {\bi{x}}\\
-{\bi{x}}\end{array} \right ]$. The eigenvalues are given by $\lambda \pm 1$ where $\lambda - 1$ is the eigenvalue for $\displaystyle
\left [ \begin{array}{c} {\bi{x}}\\ {\bi{x}}\end{array} \right ]$, $\lambda + 1$ is the eigenvalue for $\displaystyle \left [ \begin{array}{c} {\bi{x}}\\ -{\bi{x}}\end{array} \right ]$, and $\lambda_j = 3-2\cos (2j\pi/N)$, $j=0,\ldots,N/2-1$. The $k$ components of ${\bi{x}}$ for a given $\lambda_j$ are $\displaystyle x_{jk} = \sqrt{\frac 2N} \cos \left (
\frac {j(2k-1)\pi}{N} \right )$, $k=1,\ldots,N/2$ for $j>0$ and $x_{0k} = 1/\sqrt{N}$, $k=1,\ldots,N/2$ for $j = 0$ ($j = 0
\rightarrow$ eigenvalues of ${\bi{K}}$ are $0$ and $2$). As you can see, there are no degeneracies within the $\displaystyle \left [ \begin{array}{c}
{\bi{x}}\\ {\bi{x}}\end{array} \right ]$ subspace or the $\displaystyle \left [ \begin{array}{c} {\bi{x}}\\ -{\bi{x}}\end{array} \right ]$ subspace. Therefore, the only degeneracies occur between subspaces, so we know all degenerate eigenvalues are associated with unique eigenvectors, as alluded to in a previous footnote.
We have $N$ vertices and $(3N/2 - 2)$ links. Define the temporal (vertical) links $e_i$ in terms of vertices $v_i$ in the following fashion: $e_i = v_{i+1}-v_i$, $i=1, \ldots, N/2-1$ and $\displaystyle e_{N/2+i-1} = v_{N/2+i+1}-v_{N/2+i}$, $i=1, \ldots,
N/2-1$. Define the spatial (horizontal) links via: $\displaystyle
e_{N + i - 2} = v_{N/2+i} - v_i$, $i=1,\ldots,N/2$. This gives $${\bi{J}}= \left [ \begin{array}{cc}
-e_1 - e_{N-1} & \\
-e_i + e_{i-1} - e_{N+i-2} & i=2,\ldots \frac N2 -1 \\
e_{N/2-1} - e_{N+N/2-2} & \\
e_{N-1} - e_{N/2} & \\
e_{N/2+i-2}+e_{N+i-2}-e_{N/2+i-1} & i=2,\ldots,\frac N2 -1 \\
e_{N+N/2-2} + e_{N-2}
\end{array} \right ].
\label{eqn21}$$ We then need to find the projection of ${\bi{J}}$ on each of the orthonormal eigenvectors of ${\bi{K}}$ that have non-zero eigenvalues. Call each projection ${\tilde{J}}_i = \langle i | J \rangle$, where $\langle
i |$ is the $i^{\mbox{th}}$ orthonormal eigenvector. Let $a_i$ ($i =
1, \ldots, N-1$) be the non-zero eigenvalues of ${\bi{K}}$ associated with the eigenvectors $\langle i |$ , ($i = 1,\ldots, N-1$), respectively. To complete the two-source Euclidean symmetry amplitude we need to compute the exponent $$\Phi = \sum_{i=1}^{N-1} \frac {\left ( {\tilde{J}}_i \right )^2}{2 a_i \hbar \beta}
\label{eqn22}$$ where $\hbar$ is viewed as a fundamental scaling factor with the dimensions of action. We find $\Phi = (\Phi_S + \Phi_T +
\Phi_{ST})/(2\hbar \beta)$, where $$\Phi_S = \frac {2 \alpha^2}{N} \left ( \sum_{k=1}^{N/2} e_{k+N-2} \right )^2
\label{eqn23}$$ involves only spatial links $$\Phi_T = \frac {2 \alpha^2}{N} \sum_{j=1}^{N/2-1} \left [ \sum_{k=1}^{N/2-1} \left ( e_k + e_{k+N/2-1} \right )\sin \left ( \frac {2jk\pi}N \right ) \right ]^2
\label{eqn24}$$ involves only temporal links and $$\fl
\Phi_{ST} = \sum_{j=1}^{N/2-1} \frac {4\alpha^2}{N \left ( 1 + 2 \sin^2 \left (\frac {j\pi}N \right)\right )} \left [ \sin \left (\frac {j \pi}N\right)\sum_{k=1}^{N/2-1} \left ( e_k - e_{k+N/2-1} \right ) \sin \left ( \frac {2jk\pi}N \right ) \right .$$ $$\left.+\sum_{k=1}^{N/2}e_{k+N-2}\cos\left(\frac{(2k-1)j\pi}{N}\right)\right]^2
\label{eqn25}$$ involves a mix of spatial and temporal links. (\[eqn23\]) comes from the eigenvalue $2$ associated with $\displaystyle
\left [ \begin{array}{c} {\bi{x}}\\ -{\bi{x}}\end{array} \right ]$, which exists for all $N$ under consideration. (\[eqn25\]) comes from the remaining eigenvalues associated with $\displaystyle \left [ \begin{array}{c} {\bi{x}}\\
-{\bi{x}}\end{array} \right ]$. (\[eqn24\]) comes from the eigenvalues associated with $\displaystyle \left [ \begin{array}{c} {\bi{x}}\\ {\bi{x}}\end{array} \right ]$ having omitted zero, which exists for all $N$ under consideration.
Conclusion
==========
We have assumed the existence of a discrete theory (X) fundamental to quantum physics, the characteristics of which we articulated and explored via a path integral formalism over graphs. Mathematically, one can summarize our proposed theory X as follows: $$\fl
{\bi{K}}\cdot {\bi{v}}\propto {\bi{J}}\rightarrow \frac 12 {\bi{K}}+ {\bi{J}}\rightarrow Z \rightarrow P \left ( {\tilde{Q}}_k = {\tilde{Q}}_0 \right ) = \frac
{Z\left ( {\tilde{Q}}_k = {\tilde{Q}}_0 \right )}{Z} \rightarrow {\bi{K}}\cdot {\bi{Q}}_0 =
{\bi{J}}$$ with QFT and classical field theory understood as the continuum approximations to $\displaystyle P \left ( {\tilde{Q}}_k = {\tilde{Q}}_0 \right ) =
\frac {Z\left ( {\tilde{Q}}_k = {\tilde{Q}}_0 \right )}{Z}$ and ${\bi{K}}\cdot {\bi{Q}}_0 =
{\bi{J}}$, respectively. Thus, the graphical SCC ${\bi{K}}\cdot {\bi{v}}\propto
{\bi{J}}$ statistically reproduces its counterpart ${\bi{K}}\cdot {\bi{Q}}_0 =
{\bi{J}}$ whence classical field theory. While the mathematical details of theory X provided herein are too simplistic to unify physics formally, we do believe they provide a respectable conceptual response to Glashow and Rovelli’s challenges presented in section 1. Our proposed new approach to theory X underlying QFT accounts naturally for gauge invariance via a self-consistency criterion and deals effectively with factors of infinity associated with gauge groups of infinite volume, since the transition amplitude Z is restricted to the row space of the difference matrix and source vector.
While positing a discrete theory at bottom is hardly unique in fundamental physics, and our formal development is tentative, our overall approach to theory X is novel in that it is adynamical and acausal, in contrast to other fundamental theories such as M-theory, loop quantum gravity, causets, etc. Such theories may deviate from the norm by employing radical new fundamental entities (branes, loops, ordered sets, etc.), but the game is always dynamical, broadly construed (vibrating branes, geometrodynamics, sequential growth process, etc.). While itself adynamical, the SCC guarantees the graph will produce divergence-free classical dynamics in the appropriate statistical and continuum limits, and provides an acausal global constraint that results in a self-consistent, co-construction of space, time and matter that is *de facto* background independent. Thus in our approach, one has an acausal, adynamical unity of “spacetimematter” at the fundamental level that results statistically in the causal, dynamical “spacetime + matter” of classical physics. Consequently, fundamental explanation is in terms of a global, adynamical organizing principle. And, ultimate explanation in physics is not in terms of some thing or dynamical entity (obeying a new dynamical equation) “at the bottom” conceived at higher energies and smaller spatiotemporal scales, begging for justification from something at some yet “deeper” scale, but self-consistency writ large for the explanatory “process” as a whole.
![(a) Topological Graph - This spacetimematter graph depicts four sources, i.e., the columns of squares. The graph’s actional ${\bi{K}}/2+{\bi{J}}$, such that ${\bi{K}}\cdot {\bi{v}}\propto {\bi{J}}$, characterizes the graphical topology, which underwrites a partition function $Z$ for spatiotemporal geometries over the graph. (b) Geometric Graph - The topological graph of (a) is endowed with a particular distribution of spatiotemporal geometric relations, i.e., link lengths as determined by the field values $Q$ on their respective vertices. Clusters 1 & 2 are the result of this geometric process for a particular distribution of field values $Q$.[]{data-label="fig1"}](Figure1.eps "fig:"){height="30mm"} ![(a) Topological Graph - This spacetimematter graph depicts four sources, i.e., the columns of squares. The graph’s actional ${\bi{K}}/2+{\bi{J}}$, such that ${\bi{K}}\cdot {\bi{v}}\propto {\bi{J}}$, characterizes the graphical topology, which underwrites a partition function $Z$ for spatiotemporal geometries over the graph. (b) Geometric Graph - The topological graph of (a) is endowed with a particular distribution of spatiotemporal geometric relations, i.e., link lengths as determined by the field values $Q$ on their respective vertices. Clusters 1 & 2 are the result of this geometric process for a particular distribution of field values $Q$.[]{data-label="fig1"}](Figure2.eps "fig:"){height="60mm"}
![(a) Classical Physics - Classical Objects result when the most probable field values ${\bi{Q}}_0$ yield spatiotemporally localized Clusters 1 & 2 as in Figure \[fig1\]b. The lone link in this figure represents the average of the link lengths obtained via the most probable field values ${\bi{Q}}_0$. The most probable values ${\bi{Q}}_0$ are found via ${\bi{K}}\cdot {\bi{Q}}_0 = {\bi{J}}$, so this is the origin of classical physics. (b) Quantum Physics - A particular outcome ${\tilde{Q}}_0$ of a quantum physics experiment allows one to compute the $k^{\mbox{th}}$ link length of the geometric graph in the context of the classical Objects comprising the experiment, e.g., Source, beam splitters, mirrors, and detectors. The partition function provides the probability of this particular outcome, i.e., $\displaystyle P({\tilde{Q}}_k = {\tilde{Q}}_0) = \frac {Z({\tilde{Q}}_k = {\tilde{Q}}_0)}{Z}$.[]{data-label="fig2"}](Figure3.eps "fig:"){height="50mm"} ![(a) Classical Physics - Classical Objects result when the most probable field values ${\bi{Q}}_0$ yield spatiotemporally localized Clusters 1 & 2 as in Figure \[fig1\]b. The lone link in this figure represents the average of the link lengths obtained via the most probable field values ${\bi{Q}}_0$. The most probable values ${\bi{Q}}_0$ are found via ${\bi{K}}\cdot {\bi{Q}}_0 = {\bi{J}}$, so this is the origin of classical physics. (b) Quantum Physics - A particular outcome ${\tilde{Q}}_0$ of a quantum physics experiment allows one to compute the $k^{\mbox{th}}$ link length of the geometric graph in the context of the classical Objects comprising the experiment, e.g., Source, beam splitters, mirrors, and detectors. The partition function provides the probability of this particular outcome, i.e., $\displaystyle P({\tilde{Q}}_k = {\tilde{Q}}_0) = \frac {Z({\tilde{Q}}_k = {\tilde{Q}}_0)}{Z}$.[]{data-label="fig2"}](Figure4.eps "fig:"){height="50mm"}
![Graph with six vertices, seven links $e_i$ and two plaquettes $p_i$.[]{data-label="fig3"}](Figure5.eps){height="50mm"}
![Coupled harmonic oscillators.[]{data-label="fig4"}](Figure6.eps){height="30mm"}
References {#references .unnumbered}
==========
[35]{} Zee, A.: [*Quantum Field Theory in a Nutshell*]{}. Princeton University Press, Princeton (2003), p 170. Glashow, S.: Does quantum field theory need a foundation?: In: Cao, T. (ed.) Conceptual Foundations of Quantum Field Theory, pp 74-88, Cambridge University Press, Cambridge (1999), p 83 Rovelli, C.: ’Localization’ in quantum field theory: how much of QFT is compatible with what we know about space-time?: In: Cao, T. (ed.) Conceptual Foundations of Quantum Field Theory, pp 207-232, Cambridge University Press, Cambridge (1999), pp 228-229 Kaku, M.: Quantum Field Theory: A Modern Introduction. Oxford University Press, New York (1993), pp 298-304. Wallace, D.: In defence of naiveté: The conceptual status of Lagrangian quantum field theory. Synthese 151, 33-80 (2006). Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman, San Francisco (1973), p 364 Stuckey, W.M., Silberstein, M., Cifone, M.: Reconciling spacetime and the quantum: Relational Blockworld and the quantum liar paradox. Foundations of Physics 38(4), 348-383 (2008). quant-ph/0510090; Silberstein, M., Stuckey, W.M., Cifone, M.: Why quantum mechanics favors adynamical and acausal interpretations such as Relational Blockworld over backwardly causal and time-symmetric rivals. Studies in History & Philosophy of Modern Physics 39(4), 736-751 (2008) Healey, R.: Gauging What’s Real: The Conceptual Foundations of Gauge Theories. Oxford University Press, Oxford (2007), p 141 Feynman, R.P.: The development of the space-time view of quantum electrodynamics: In: Ekspong, G. (ed.) Physics: Nobel Lectures 1963-1970, pp 155-178 World Scientific, Singapore (1988) We follow the notational conventions of Zee, 2003 Zee, 2003, p 22 Feinberg, G., Friedberg, R., Lee, T.D., and Ren, H.C.: Lattice Gravity Near the Continuum Limit. Nuclear Physics B245, 343-368 (1984) Loll, R.: Discrete Approaches to Quantum Gravity in Four Dimensions. www.livingreviews.org/Articles/Volume1/1998-13loll, Max-Planck-Institute for Gravitational Physics Albert Einstein Institute, Potsdam (15 Dec 1998) Ambjorn, J., Jurkiewicz, J., and Loll, J.: Quantum gravity as sum over spacetimes. arXiv: 0906.3947 (2009) Konopka, T., Markopoulou, F., Smolin, L.: Quantum Graphity. hep-th/0611197 (2006); Konopka, T., Markopoulou, F., Severini, S.: Quantum Graphity: a model of emergent locality. hep-th/0801.0861, 10.1103/PhysRevD.77.104029 (2008) Sorkin, R.D.: Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School). gr-qc/0309009 (2003) Bahr, B. and Bianca Dittrich, B.: Regge calculus from a new angle. arXiv: 0907.4325 (2009) Hamber, H.W. and Williams, R.: Nonlocal Effective Gravitational Field Equations and the Running of Newton’s G. arXiv:hep-th/0507017 (2005) Misner, Thorne and Wheeler, 1973, p 369; Wise, D.K.: p-Form electromagnetism on discrete spacetimes. Classical and Quantum Gravity 23, 5129-5176 (2006) Wise, 2006 Zee, 2003, p 167 Lisi, A.: Quantum mechanics from a universal action reservoir. arXiv: physics/0605068 (2006) Zee, 2003, p 15 Wise, 2006 Zee, 2003, p 5 See section IV.C of Sorkin, R.: The electromagnetic field on a simplicial net. Journal of Mathematical Physics 16, 2432-2440 (1975)
[^1]: We are focusing on the “textbook variant of QFT." Fraser, D.: Quantum Field Theory: Underdetermination, Inconsistency, and Idealization. Philosophy of Science 74, 536-565 (October 2009).
[^2]: This assumes the number of degenerate eigenvalues always equals the dimensionality of the subspace spanned by their eigenvectors, which we will see is true for ${\bi{K}}$ in this example.
|
---
abstract: 'The famous Pósa conjecture states that every graph of minimum degree at least $2n/3$ contains the square of a Hamilton cycle. This has been proved for large $n$ by Komlós, Sarközy and Szemerédi. Here we prove that if $p \ge n^{-1/2+{\varepsilon}}$, then asymptotically almost surely, the binomial random graph $G_{n,p}$ contains the square of a Hamilton cycle. This provides an ‘approximate threshold’ for the property in the sense that the result fails to hold if $p\le n^{-1/2}$.'
author:
- Daniela Kühn and Deryk Osthus
title: 'On Pósa’s conjecture for random graphs'
---
Introduction
============
The $k$th power of a cycle $C$ is obtained by including an edge between all pairs of vertices whose distance on $C$ is at most $k$. The Pósa-Seymour conjecture states that every graph $G$ on $n$ vertices with minimum degree at least $kn/(k+1)$ contains the $k$th power of a Hamilton cycle. (Here the case $k=2$ was conjectured by Pósa and the general case was later conjectured by Seymour.) This beautiful conjecture was proved for large $n$ by Komlós, Sarközy and Szemerédi [@KSSz98]. The case $k=1$ of course corresponds to Dirac’s theorem [@Dirac] on Hamilton cycles. For $k=2$, there have been significant improvements in the bound on $n$ that is required (see e.g. [@CDH]). More generally, many other recent advances have been made on embedding spanning subgraphs in dense graphs (see e.g. [@BCCsurvey] for a survey). For instance, recall that $G$ has an *$F$-factor* if $G$ contains $\lfloor |G|/|F|\rfloor$ vertex-disjoint copies of $F$. The famous Hajnal-Szemerédi theorem [@HSz] states that every graph with minimum degree at least $kn/(k+1)$ contains a $K_{k+1}$-factor. More generally, Kühn and Osthus [@KOmatch] determined the minimum degree that $G$ needs to have to ensure the existence of an $F$-factor in $G$ (up to an additive constant depending on $F$).
It is natural to ask for probabilistic analogues of these results, i.e. given a graph $H$ on $n$ vertices, how large does $p$ have to be to ensure that $G_{n,p}$ a.a.s. contains a copy of $H$? Here $G_{n,p}$ denotes the binomial random graph on $n$ vertices with edge probability $p$ and we say that a property $A$ holds a.a.s. (asymptotically almost surely), if the probability that $A$ holds tends to $1$ as $n$ tends to infinity. (Note that formally one actually needs to ask the above question for a sequence of graphs $H_i$ whose order tends to infinity.)
This turns out to be a surprisingly difficult problem, and the answer is known for very few (families of) graphs $H$. A notable exception is the seminal result of Johansson, Kahn and Vu [@JKV08], who determined the ‘approximate’ threshold for the existence of an $F$-factor. So this is a probabilistic version of the result in [@KOmatch] mentioned above. Also, Riordan [@riordan] obtained a very general result, which gives a bound that can be applied to every graph $H$. As a corollary, he obtained the threshold for the existence of a spanning hypercube in $G_{n,p}$ and several kinds of spanning lattices, e.g. the square grid. His result can be applied to powers of Hamilton cycles to give the following result (see Section \[sec:riordan\] for the straightforward details):
\[thm:riordan\] Let $k\geq 2$ be fixed. Suppose that $p n^{1/k} \to \infty$ and $p n^{1/3} \to \infty$. Then a.a.s. $G_{n,p}$ contains the $k$th power of a Hamilton cycle.
A simple first moment argument shows that this result gives the correct threshold for $k \ge 3$. Indeed, note that the number of edges in the $k$th power of a cycle of length $n > 2k$ is $kn$. So if $n > 2k$ and $p \le n^{-1/k}$, it follows that the expected number of appearances of the $k$th power of a Hamilton cycle in $G_{n,p}$ is at most $n!p^{kn} \leq ( n p^{k}/2)^n = o(1)$.
However, for squares (i.e. when $k=2$) Theorem \[thm:riordan\] does not give the correct answer. Indeed, the above first moment argument suggests that the threshold should be close to $n^{-1/2}$. Our main result is an ‘approximate’ threshold, i.e. our bound on $p$ is tight up to a factor of $n^{{\varepsilon}}$, where ${\varepsilon}>0$ is arbitrary. Our argument works for higher powers in the same way as it does for squares, so we formulate our proof for arbitrary $k \ge 2$.
\[thm:main\] Let ${\varepsilon}>0$ and $k\geq 2$ be fixed. Suppose that $p=p(n)\geq n^{-1/k + {\varepsilon}}$. Then a.a.s. $G_{n,p}$ contains the $k$th power $C^k$ of a Hamilton cycle.
Note that Theorems \[thm:riordan\] and \[thm:main\] as well as the result on $F$-factors in [@JKV08] (see Theorem \[thm:JKV\]) imply that the threshold for a $K_{k+1}$-factor is much smaller than that for the $k$th power of a Hamilton cycle. So this is different from the ‘deterministic’ setting described earlier, where the minimum degree conditions are the same.
We now discuss some further related results on embedding spanning subgraphs in $G_{n,p}$. The case of Hamilton cycles (i.e. when $k=1$) has been studied successfully and in great detail. In particular, a classical result of Komlós and Szemerédi [@KomSze] and Korshunov [@Korsh] implies that the threshold function for the existence of a Hamilton cycle is $\log n/n$. In fact, much more is true: a celebrated result of Bollobás [@BolHam] and Ajtai, Komlós and Szemerédi [@AKSHam] states that the hitting time for the emergence of a Hamilton cycle on $n$ vertices coincides a.a.s. with the hitting time of the property that the minimum degree is at least 2. (An algorithmic version of this result was later proved by Bollobás, Fenner and Frieze [@BFF].) On the other hand, the expected number of Hamilton cycles already tends to infinity when $np \rightarrow \infty$. So the existence of vertices of degree less than two in $G_{n,p}$ can be viewed as a ‘local obstruction’ to the existence of a Hamilton cycle in $G_{n,p}$. For $k \ge 3$, Theorem \[thm:riordan\] shows that there are no ‘local obstructions’. It seems natural to conjecture that the case of squares is similar, i.e. that the threshold for the square of a Hamilton cycle in $G_{n,p}$ is at $p=n^{-1/2}$.
Another class of subgraphs which has received much attention is that of spanning trees. The best general result is due to Krivelevich [@krivtrees], who showed (amongst other results) that if $T$ is any tree on $n$ vertices of bounded maximum degree and $p \ge n^{-1+{\varepsilon}}$, then a.a.s. $G_{n,p}$ contains a copy of $T$. It seems likely that the term $n^{\varepsilon}$ in this result can be replaced by a much smaller function. This is supported by several results on certain classes of trees. For instance, the threshold for a Hamilton path is $p=\log n/n$ by the above results on Hamilton cycles. As another example, Hefetz, Krivelevich, and Szabó [@HKS] showed that $p=\log n/n$ is the (sharp) threshold for a tree $T$ having a linear number of leaves.
In the probabilistic setting, it is also natural to ask for ‘universality’ results. Again, this is a question where much progress has been made recently. Given a graph $G$ and a family of graphs ${\mathcal{H}}$, we say that a graph $G$ is *${\mathcal{H}}$-universal* if $G$ contains every member of ${\mathcal{H}}$ as a subgraph. An important case is when ${\mathcal{H}}={\mathcal{H}}(n,\Delta)$ consists of all graphs on $n$ vertices with maximum degree at most $\Delta$. Here the best bound is due to Dellamonica, Kohayakawa, Rödl and Ruciński [@DKRR], who showed that if $p \gg n^{-1/2\Delta} \log ^{1/\Delta}n$, then a.a.s. $G_{n,p}$ is ${\mathcal{H}}(n,\Delta)$-universal. Note that the $k$th power of a Hamilton cycle on $n>2k$ vertices has maximum degree $2k$. So the bounds one obtains for this case are significantly weaker than the ones given by Theorems \[thm:riordan\] and \[thm:main\].
The proof in [@riordan] is based on the second moment method. Instead, our proof is based on the ‘absorbing method’, which was introduced as a general method by Rödl, Ruciński and Szemerédi [@RRS] (the underlying idea was also used earlier, e.g. by Krivelevich [@Krivtriangles]). The method has proved to be an extremely versatile tool for embedding various types of spanning subgraphs in dense graphs. Though additional difficulties arise in the context of (sparse) random graphs, we believe that the method has significant further potential in this setting.
This paper is organized as follows. After introducing some notation, we define an ‘absorber’, which will be the crucial concept for extending the $k$th power of an almost spanning cycle into the $k$th power of a Hamilton cycle. We then describe the proof of Theorem \[thm:main\] in Section \[sec:mainproof\], under the assumption that Lemmas \[lem:Absorber\_Factor\], \[lem:linking\] and \[lem:pathfactor\] hold. Section \[sec:mainproof\] also contains an informal overview of the proof. These lemmas are proved in the subsequent sections. In the short final section, we show how Theorem \[thm:riordan\] follows from the more general result in [@riordan].
Notation
========
We write $|G|$ and sometimes also $v_G$ for the number of vertices of a graph $G$. We write $e(G)$ and sometimes also $e_G$ for the number of edges of $G$. We say that two graphs $H$ and $G$ are disjoint if they are vertex-disjoint. Given graphs $G$ and $H$, an $H$-factor in $G$ is a collection of $\lfloor |G|/|H|\rfloor$ pairwise disjoint copies of $H$ in $G$.
We denote the path on $n$ vertices by $P_n$. The *distance* between two vertices $x$ and $y$ in a graph $G$ is the length (i.e. the number of edges) of the shortest path between $x$ and $y$. The *$k$th power* of a graph $G$ is the graph $G^k$ whose vertex set is $V(G)$ and in which two vertices $x,y\in V(G)$ are joined by an edge if and only if the distance between $x$ and $y$ in $G$ is at most $k$. So $P^k_n$ denotes the $k$th power of $P_n$. Suppose that $n\ge 2k$ and that $P_n=x_1\dots x_n$. We will view $x_1$ as the first vertex of $P_n$ and $x_n$ as its final vertex. The *initial endsequence* of $P^k_n$ is the sequence $x_1,\dots,x_k$ and the *final endsequence* of $P^k_n$ is the sequence $x_{n-k+1},\dots,x_n$.
Suppose that $A=(a_1,\dots,a_k)$ and $B=(b_1,\dots,b_k)$ are two sequences of vertices such that all these $2k$ vertices are distinct from each other. An *$(A,B)$-linkage* $R$ is defined as follows: let $R'$ be the $k$th power of a path such that the initial endsequence of $R'$ is $A$ and the final endsequence of $R'$ is $B$. Then we obtain $R$ by removing all edges within $A$ and within $B$. We will use the notion of linkages to join up $k$th powers of paths into longer ones. More precisely, suppose that $Q$ and $Q'$ are $k$th powers of paths which are pairwise disjoint, that $A$ is the final endsequence of $Q$, that $B$ is the initial endsequence of $Q'$ and that $R$ is an $(A,B)$-linkage which meets $V(Q)\cup V(Q')$ only in $A\cup B$. Then $Q\cup R\cup Q'$ is again the $k$th power of a path.
We will omit floors and ceilings whenever this does not affect the argument. We write $\log n$ for the natural logarithm and $\log ^a n:=(\log n)^a$.
Absorbers {#sec:absorber}
=========
The aim of this section is to define an *absorber*, which is the main tool in our proof of Theorem \[thm:main\]. Roughly speaking, an absorber $A$ will be the union of the $k$th power $P^k$ of a path $P$ and the $k$th power $(P')^k$ of a path $P'$ such that the following two properties are satisfied:
- The two endsequences of $P^k$ are the same as the two endsequences of $(P')^k$.
- $V(P')$ is obtained from $V(P)$ by adding one extra vertex $v$ (which we call the absorbtion vertex).
Thus if we can find the $k$th power $C^*$ of some cycle which contains $P^k$ as a subgraph but does not contain $v$, then we can ‘absorb’ $v$ into $C^*$ by replacing $P^k$ with $(P')^k$. When defining the absorber, we have to make sure that our random graph $G_{n,p}$ a.a.s. contains many disjoint copies of this absorber. A result of Johansson, Kahn and Vu (Theorem \[thm:JKV\] below) implies that the latter will be the case if the $1$-densities of all subgraphs of the absorber are not too large. (This will turn out to be true if the parameters $j$ and $\ell$ below satisfy $k\ll j\ll \ell$.)
More precisely, for all $k\ge 2$, $j\geq 3$ and $\ell \geq 2k$, we will now define the $(j,\ell,k)$-*absorber* $A_{j,\ell,k}$. Consider first a path $P$ on $s$ vertices, where $s:=j(2\ell+4)+\ell$, and a vertex $v$ that does not belong to $P$. We call $P$ the *spine* of the absorber and $v$ its *absorbtion vertex*. We will view one endvertex of $P$ as its first vertex and the other endvertex of $P$ as its last vertex. This induces an order of the vertices on $P$. Split $P$ into $j+1$ consecutive disjoint segments $S_1,\dots, S_{j+1}$ such that $S_i$ has $2\ell + 4$ vertices for each $i=1,\dots,j$ and $S_{j+1}$ consists of the final $\ell$ vertices of $P$. For $i=1,\dots,j$, in $S_i$ we label the $(\ell +1)$st, the $(\ell+2)$nd, the $(2\ell + 3)$rd and the $(2\ell+4)$th vertices by $a_{i,1}$, $a_{i,2}$, $b_{i,1}$ and $b_{i,2}$, respectively. We call these special vertices *junctions*.
We add the edges $a_{1,1}v$ and $vb_{1,2}$. For every $i=1,\dots,j-2$, we add the edges $a_{i,2}b_{i+1,2}$ and $b_{i,1}a_{i+1,1}$. Finally, we add the edges $a_{j,2}b_{j,2}$, $a_{j-1,2}a_{j,1}$ and $b_{j-1,1}b_{j,1}$. We will be referring to the resulting graph (consisting of the spine $P$, the absorbtion vertex $v$ and the edges incident to the junctions and to $v$ which we added) as the *skeleton* of the absorber.
It is not hard to see that the graph $P'$ obtained from the skeleton by deleting the edges $a_{i,1}a_{i,2}$ and $b_{i,1}b_{i,2}$ for all $i=1,\dots,j$ is a path with $V(P')=V(P) \cup \{v\}$ and with the same endvertices as the spine $P$ (see Figure \[fig:absorber\] for the case when $j=4$).
![The skeleton of a $(4,\ell)$-absorber. The path $P'$ is indicated by the arrows.[]{data-label="fig:absorber"}](Absorber.pdf)
![Junctions and access vertices of a $(4,\ell)$-absorber.[]{data-label="fig:absorberdetail"}](AbsorberDetail.pdf)
We call $P'$ the *augmented path* of the absorber and the edges in $E(P')\setminus (E(P)\cup\{a_{1,1}v,vb_{1,2}\})$ the *junction edges*. We define the $(j,\ell,k)$*-absorber* $A_{j,\ell,k}$ to be $P^k \cup (P')^k$. The *first endsequence of* $A_{j,\ell,k}$ is the first endsequence of $P^k$ (and thus of $(P')^k$) and the *final endsequence of* $A_{j,\ell,k}$ is the final endsequence of $P^k$ (and thus of $(P')^k$).
Given a junction $a$, let ${\rm Acc}(a)$ be the set consisting of $a$ as well as all the $k-1$ vertices that have distance at most $k-1$ from $a$ in *both* $P$ and $P'$ (see also Figure \[fig:absorberdetail\], where these sets are marked for four of the junctions). Call the vertices in ${\rm Acc}(a)$ *access vertices* associated with $a$. Note that the following properties hold:
- Let $ab$ be any junction edge, where $a$ is the predecessor of $b$ on $P'$. Then the subpath $Q_a$ of $P'$ induced by $a$ and the $\ell$ vertices preceding $a$ on $P'$ is also a subpath of $P$ and ${\rm Acc}(a)$ is the set of all those vertices having distance at most $k-1$ from $a$ on $Q_a$. Similarly, the subpath $Q_b$ of $P'$ induced by $b$ and the $\ell$ vertices succeeding $b$ on $P'$ is also a subpath of $P$ and ${\rm Acc}(b)$ is the set of all those vertices having distance at most $k-1$ from $b$ on $Q_b$.
- $a_{1,1}vb_{1,2}$ is a subpath of $P'$. The subpath $Q_{a_{1,1}}$ of $P'$ induced by $a_{1,1}$ and the $\ell$ vertices preceding $a_{1,1}$ on $P'$ is also a subpath of $P$ and ${\rm Acc}(a_{1,1})$ is the set of all those vertices having distance at most $k-1$ from $a_{1,1}$ on $Q_{a_{1,1}}$. Similarly, the subpath $Q_{b_{1,2}}$ of $P'$ induced by $b_{1,2}$ and the $\ell$ vertices succeeding $b_{1,2}$ on $P'$ is also a subpath of $P$ and ${\rm Acc}(b_{1,2})$ is the set of all those vertices having distance at most $k-1$ from $b_{1,2}$ on $Q_{b_{1,2}}$.
- The graph consisting of all junction edges, of the path $a_{1,1}vb_{1,2}$ and of all the edges $a_{i,1}a_{i,2}$, $b_{i,1}b_{i,2}$ (for all $i=1,\dots,j$) is a cycle.
(A1) and (A2) together with the fact that $\ell\ge 2k$ imply that every edge $e\in E(A_{j,\ell,k})\setminus E(P^k)$ satisfies precisely one of the following conditions:
- There is precisely one junction edge $ab$ such that $e$ joins some vertex in ${\rm Acc}(a)$ to some vertex in ${\rm Acc}(b)$.
- $e$ joins some vertex in ${\rm Acc}(a_{1,1})\cup \{v\}$ to some vertex in ${\rm Acc}(b_{1,2})\cup \{v\}$.
In the first case we say that $e$ is *associated with $ab$* (so $ab$ itself is associated with $ab$) and in the second case we say that $e$ is *associated with* $v$. Note that for every junction edge $ab$ there are precisely $\binom{k+1}{2}$ edges associated with $ab$. Indeed, let $a_k:=a$ and for each $i=1,\dots,k-1$ let $a_i$ be the vertex of distance $i$ from $a$ on $Q_a$, where $Q_a$ is as defined in (A1). (So ${\rm Acc}(a)=\{a_1,\dots,a_k\}$.) Then $a_i$ has precisely $i$ neighbours in ${\rm Acc}(b)$. Similarly, precisely $\binom{k+1}{2}+k$ edges are associated with $v$. Since there are $2j-1$ junction edges, altogether this shows that $$\label{eq:edgesabs}
e(A_{j,\ell,k})= e(P^k)+2j\binom{k+1}{2}+k.$$
Proof of Theorem \[thm:main\] {#sec:mainproof}
=============================
Since the property of containing the $k$th power of a Hamilton cycle is monotone it suffices to show that a.a.s. $G_{n,p^*}$ contains the $k$th power of a Hamilton cycle, where $$p^*=p^*(n):= n^{-1/k + {\varepsilon}_*}.$$ Here ${\varepsilon}_*$ is fixed and we assume that $$\label{eq:eps*}
{\varepsilon}_*\le \frac{1}{10^4 k}.$$ So in particular $p^*=o(1)$. We shall consider a multiple round exposure of $G_{n,p^*}$. More precisely, we will expose $G_{n,p^*}$ in four rounds considering four independent random graphs $G_{n,p^*_1},\dots, G_{n,p^*_4}$, where $p^*_1=\dots=p^*_4$. Thus $p^*_i =(1+o(1))p^*/4\ge n^{-1/k + {\varepsilon}_*/2}$ for all $i=1,\dots,4$.
Roughly speaking, the strategy of our proof is as follows. We will first use $G_{n,p^*_1}$ to find a collection $\mathcal{A}$ of pairwise disjoint absorbers which cover about $n/3$ vertices. Let $A$ denote the set consisting of all absorbtion vertices of all these absorbers. We use $G_{n,p^*_2}$ to connect the $k$th powers of the spines of the absorbers in ${\mathcal{A}}$ into the $k$th power $Q_{\mathcal{A}}$ of a path. To do this we will only use vertices which are not covered by the absorbers in ${\mathcal{A}}$. Moreover, $V(Q_{\mathcal{A}})\cup A$ will contain at most $2n/3$ vertices. Let $S':=[n]\setminus (V(Q_{\mathcal{A}})\cup A)$ denote the set of uncovered vertices. We will use $G_{n,p^*_3}$ to cover $S'$ by a collection $\mathcal{P}$ consisting of not too many $k$th powers of pairwise disjoint paths. Finally, we will use $G_{n,p^*_4}$ connect all the paths in $\mathcal{P}$ as well as $Q_{{\mathcal{A}}}$ into the $k$th power $C^*$ of a cycle. To do this we will only use vertices in $A$. Let $A''\subseteq A$ be the vertices not used in this step. Since $A''$ consists of absorbtion vertices, we can ‘absorb’ all the vertices of $A''$ into $C^*$ to obtain the $k$th power of a Hamilton cycle. More precisely, for each $v\in A''$ let $A_v$ denote the unique absorber in ${\mathcal{A}}$ that contains $v$. Then the subgraph obtained from $C^*$ by replacing the $k$th power of the spine of $A_v$ with the $k$th power of the augmenting path of $A_v$ (for each $v\in A''$) is the $k$th power of a Hamilton cycle in $G_{n,p^*}$.
After outlining our strategy, let us now return to the actual proof. We will use the next lemma (which is proved in Section \[sec:Absorber\_Factor\]) in order to find the collection ${\mathcal{A}}$ of absorbers in $G_{n,p^*_1}$.
\[lem:Absorber\_Factor\] For each ${\varepsilon}>0$ and each integer $k\geq 2$, there exist integers $j\ge 3$ and $\ell_0 \geq 2k$ such that whenever $\ell\ge \ell_0$ and $p=p(n)\ge n^{-1/k + {\varepsilon}}$, then a.a.s. $G_{n,p}$ contains an $A_{j,\ell,k}$-factor.
Let $j=j(k,{\varepsilon}_*/2)$ and $\ell_0=\ell_0(k,{\varepsilon}_*/2)$ be as in Lemma \[lem:Absorber\_Factor\]. Set $$\label{eq:ell}
\ell:=\max\{ \ell_0, \lceil 1/{\varepsilon}_*^2\rceil\}.$$ Then Lemma \[lem:Absorber\_Factor\] implies that a.a.s. $G_{n,p^*_1}$ contains an $A_{j,\ell,k}$-factor. So we may assume that such a factor exists. Let $s:=j(2\ell+4)+\ell$ and note that $|A_{j,\ell,k}|=s+1$. Let ${\mathcal{A}}$ be a collection of $n/(3(s+1))$ copies of $A_{j,\ell,k}$ in this $A_{j,\ell,k}$-factor and let $A$ denote the set of absorbtion vertices in all these copies. (So the assertion of Lemma \[lem:Absorber\_Factor\] is far stronger than we need it to be – see the discussion after Theorem \[thm:JKV\].) Note that the absorbers in ${\mathcal{A}}$ cover $n/3$ vertices of $G_{n,p^*}$. Let $S$ be a set of $n/3$ vertices not covered by these absorbers. As indicated above, our next aim is to use $G_{n,p^*_2}$ in order connect the absorbers in ${\mathcal{A}}$, using some of the vertices in $S$. To do this, we will use the following lemma (which we prove in Section \[sec:linking\]).
\[lem:linking\] Suppose that $k\ge 2$, that $0<{\varepsilon}<1/k$, that $p=p(n) \ge n^{-1/k+{\varepsilon}}$ with $p(n)=o(1)$ and that $f \le {\varepsilon}n/(60 k)$. For each $i=1,\dots,f$ let $A_i$ and $B_i$ be sequences, each consisting of $k$ vertices in $[n]$, such that these $2f$ sequences are pairwise disjoint. Then a.a.s. $G_{n,p}$ contains pairwise disjoint $(A_i,B_i)$-linkages $R_i$ with $|R_i| \le \lceil 30k/{\varepsilon}\rceil$ (for all $i=1,\dots,f$).
Choose an order of the absorbers in ${\mathcal{A}}$. For each $i=1,\dots,|{\mathcal{A}}|-1$ let $A_i$ denote the final endsequence of the $i$th absorber in ${\mathcal{A}}$ and let $B_i$ be the initial endsequence of the $(i+1)$st absorber in ${\mathcal{A}}$. Let $S^*$ denote the union of $S$ together with all the vertices contained in one of these endsequences $A_i$ or $B_i$. Note that $$|{\mathcal{A}}|=\frac{n}{3(s+1)}= \frac{|S|}{s+1}\le \frac{|S|}{\ell}\stackrel{(\ref{eq:ell})}{\le} {\varepsilon}_*^2 |S|\stackrel{(\ref{eq:eps*})}{\le} \frac{{\varepsilon}_* |S|}{180k} \le \frac{{\varepsilon}_* |S^*|}{180k}$$ and $p^*_2\ge n^{-1/k+{\varepsilon}_*/2}\ge |S^*|^{-1/k+{\varepsilon}_*/3}$. So we may apply Lemma \[lem:linking\] (with ${\varepsilon}_*/3$ playing the role of ${\varepsilon}$) to see that a.a.s. the random subgraph of $G_{n,p^*_2}$ induced by $S^*$ contains pairwise disjoint $(A_i,B_i)$-linkages $R_i$ with $|R_i| \le \lceil 90k/{\varepsilon}_* \rceil$ for all $i=1,\dots,|{\mathcal{A}}|-1$. So we may assume that such linkages exist. Let $Q_{{\mathcal{A}}}$ be the union of $R_1,\dots,R_{|{\mathcal{A}}|-1}$ and of the $k$th powers of the spines of all absorbers in ${\mathcal{A}}$. Then $Q_{{\mathcal{A}}}$ is the $k$th power of a path whose initial endsequence is the initial endsequence of the first absorber in ${\mathcal{A}}$ and whose final endsequence is the final endsequence of the last absorber in ${\mathcal{A}}$. Moreover, $Q_{{\mathcal{A}}}$ avoids the set $A$ of absorbtion vertices.
Let $S':=[n]\setminus (V(Q_{\mathcal{A}})\cup A)$ be the set of uncovered vertices. Thus $|S'|\ge n/3$. Our next aim is to cover $S'$ with not too many $k$th powers of paths. To simplify this step, first let $t:=|S'| \mod s^2$. Now remove $s^2-t$ vertices from $A$ and call the resulting set $A'$. Add these $s^2-t$ vertices to $S'$ and call the resulting set $S''$. So $|S''|$ is divisible by $s^2$.
The next lemma (which will be proved in Section \[sec:pathfactor\]) implies that a.a.s. the random subgraph of $G_{n,p^*_3}$ induced by $S''$ contains a $P^k_{s^2}$-factor $\mathcal{P}$. So we may assume that such a factor exists.
\[lem:pathfactor\] Suppose that ${\varepsilon}>0$, that $k,r\ge 2$ and that $p=p(n) \ge n^{-1/k+{\varepsilon}}$. Then a.a.s. $G_{n,p}$ has a $P_r^k$-factor.
Since $|S''|$ is divisible by $s^2$, all the vertices in $S''$ are covered by $\mathcal{P}$. We will now use $G_{n,p^*_4}$ to connect all the copies of $P^k_{s^2}$ in $\mathcal{P}$ as well as $Q_{\mathcal{A}}$ into the $k$th power of a cycle, using some of the vertices in $A'$. To do this, we choose an order of the copies of $P^k_{s^2}$ in $\mathcal{P}$. For each $i=1,\dots,|\mathcal{P}|-1$ let $A'_i$ denote the final endsequence of the $i$th copy of $P^k_{s^2}$ in $\mathcal{P}$ and let $B'_i$ be the initial endsequence of the $(i+1)$st copy. Let $A'_{|\mathcal{P}|}$ denote the final endsequence of the last copy of $P^k_{s^2}$ in $\mathcal{P}$ and let $B'_{|\mathcal{P}|}$ denote the initial endsequence of $Q_{{\mathcal{A}}}$. Finally, let $A'_{|\mathcal{P}|+1}$ denote the final endsequence of $Q_{{\mathcal{A}}}$ and let $B'_{|\mathcal{P}|+1}$ denote the initial endsequence of the first copy of $P^k_{s^2}$ in $\mathcal{P}$. Let $A^*$ denote the union of $A'$ together with all the vertices contained in one of the endsequences $A'_i$ or $B'_i$ with $i=1,\dots,|\mathcal{P}|+1$. Recall that $|A|=|{\mathcal{A}}|=n/(3(s+1))$ and so $|A^*|\ge |A'| \ge |A|-s^2 \ge n/(4(s+1))$. Moreover, $s\ge \ell$. Thus $$|\mathcal{P}|+1=\frac{|S''|}{s^2}+1\le \frac{n}{s^2}\stackrel{(\ref{eq:eps*}),(\ref{eq:ell})}{\le}
\frac{{\varepsilon}_* n}{720k(s+1)} \le \frac{{\varepsilon}_* |A^*|}{180k}.$$ Moreover, $p^*_4\ge n^{-1/k+{\varepsilon}_*/2}\ge |A^*|^{-1/k+{\varepsilon}_*/3}$. So we may apply Lemma \[lem:linking\] (with ${\varepsilon}_*/3$ playing the role of ${\varepsilon}$) to see that a.a.s. the random subgraph of $G_{n,p^*_4}$ induced by $A^*$ contains pairwise disjoint $(A'_i,B'_i)$-linkages $R'_i$ with $|R'_i|\le \lceil 90k/{\varepsilon}_* \rceil$ for all $i=1,\dots,|\mathcal{P}|+1$. So we may assume that such linkages exist. Thus the union of $C^*$ of all these linkages $R'_i$, of all the copies of $P^k_{s^2}$ in $\mathcal{P}$ and of $Q_{{\mathcal{A}}}$ forms the $k$th power of a cycle which covers all vertices apart from some vertices in $A'$.
Let $A''\subseteq A' \subseteq A$ denote the set of all uncovered vertices. For each $v\in A''$, let $A_v\in {\mathcal{A}}$ denote the unique absorber containing $v$. Let $P_v$ denote the spine of $A_v$ and let $P'_v$ denote its augmented path. Note that $C^*$ contains the $k$th power $P^k_v$ of $P_v$ as a subgraph. But the $k$th power $(P'_v)^k$ of $P'_v$ has the same endsequences as $P^k_v$. Thus the graph obtained from $C^*$ by replacing $P^k_v$ with $(P'_v)^k$ for each $v\in A''$ is the $k$th power of a Hamilton cycle in $G_{n,p^*}$. (Note that our construction implies that a.a.s. $G_{n,p*}$ contains $C^*$ as well as $(P'_v)^k$ for every $v\in A$.)
Finding a factor of $k$th powers of paths: Proof of Lemma \[lem:pathfactor\] {#sec:pathfactor}
============================================================================
The *$1$-density* of a graph $H$ on at least two vertices is defined to be $$d_1 (H) := {e_H \over v_H -1},$$ where $e_H$ and $v_H$ denote the number of edges and the number of vertices of $H$. Let $$d_1^{\rm max} (H):=\max_{H'\subseteq H, \ v_{H'}\ge 2} d_1(H').$$ Lemma \[lem:pathfactor\] will be an easy consequence of the following deep result of Johansson, Kahn and Vu [@JKV08], which was already mentioned in the introduction.
\[thm:JKV\] Fix $\varepsilon >0$ and a graph $H$. Suppose that $p(n) \ge n^{-1/d_1^{\rm max}(H) + \varepsilon}$. Then a.a.s. $G_{n,p}$ contains an $H$-factor.
Thus in order to prove Lemma \[lem:pathfactor\], it suffices prove the following proposition.
\[lem:Balanced\] Let $k, r\ge 2$ be integers. Then $d_1^{\rm max} (P^k_r)\le k$.
Consider any $H\subseteq P^k_r$ on $v_H\ge 2$ vertices. Thus there is an ordering $x_1,\dots,x_{v_H}$ of the vertices of $H$ such that for all $i=2,\dots,v_H$ every $x_i$ has at most $k$ neighbours amongst $x_1,\dots,x_{i-1}$. Since $d_1(H[\{x_1,x_2\}])\le 2\le k$, it follows that $d_1(H)\le k$.
It seems likely that our use of Theorem \[thm:JKV\] is not essential and our arguments can be extended to avoid its use. Indeed, first note that we only use Theorem \[thm:JKV\] to prove Lemmas \[lem:Absorber\_Factor\] and \[lem:pathfactor\]. As mentioned earlier, instead of Lemma \[lem:Absorber\_Factor\], we only need an assertion which guarantees a linear number of disjoint absorbers. Such an assertion can be deduced from Lemma \[lem:Absorber\_1\_Density\] and a ‘non-partite’ version of Lemma \[DGbound\]. Moreover, instead of the factor covering all vertices of $S''$ guaranteed by Lemma \[lem:pathfactor\], one can use this version repeatedly to cover almost all the vertices of $S''$. The strategy would then be to use Lemmas \[lem:linking\] and \[DGbound\] to cover the remaining vertices of $S''$ by powers of paths which are also allowed to use some vertices in $A$. But relying on Theorem \[thm:JKV\] makes these steps unnecessary.
Finding a factor of absorbers: Proof of Lemma \[lem:Absorber\_Factor\] {#sec:Absorber_Factor}
======================================================================
The aim of this section is to show that there are integers $j\ge 3$ and $\ell\ge 2k$ such that the 1-density of any subgraph of $A_{j,\ell,k}$ is not much larger than $k$ (see Lemma \[lem:Absorber\_1\_Density\] below). Together with Theorem \[thm:JKV\] this immediately implies Lemma \[lem:Absorber\_Factor\].
\[lem:Absorber\_1\_Density\] For every $k\geq 2$ and every $\delta > 0$, there exist integers $j\geq 3$ and $\ell_0 \geq 2k$ such that whenever $\ell\ge \ell_0$ every subgraph $H$ of $A_{j,\ell,k}$ satisfies $d_1(H) \leq k + \delta$.
Choose $j\ge k/\delta+3$ and $\ell_0\ge 2jk^4/\delta$. Pick $\ell\ge \ell_0$ and let $P$ and $P'$ be the spine and the augmented path of $A_{j,\ell,k}$. So $A_{j,\ell,k}=P^k\cup (P')^k$. Consider any subgraph $H$ of $A_{j,\ell,k}$ on $v_H\ge 2$ vertices. Let $H^*:=H\cap P^k$. We will distinguish the following two cases. Roughly speaking, in the first case $H^*$ ‘spans’ a large interval of $P^k$, in which case we can easily deduce that $d_1(H)$ is at most $k+\delta$.
**Case 1.**
We assume that the first property holds. The argument for the second property is similar. Note that the distance between $a_{i,2}$ and $b_{i,1}$ on $P$ is $\ell+1$ and so the distance between ${\rm Acc}(a_{i,1})\cup {\rm Acc}(a_{i,2})$ and ${\rm Acc}(b_{i,1})\cup {\rm Acc}(b_{i,2})$ on $P$ is $\ell+1-2(k-1)$. Thus $|C|\ge (\ell-2k)/k=\ell/k-2$. Moreover, Proposition \[lem:Balanced\] implies that $d_1(H^*)\le k$. Thus $$\begin{aligned}
d_1(H)& = & \frac{e_H}{v_H-1}= \frac{e_{H^*}}{v_H-1}+ \frac{e_{H\setminus E(H^*)}}{v_H-1}
\le \frac{e_{H^*}}{v_H-1}+\frac{e(A_{j,\ell,k})-e(P^k)}{v_H-1}\\
& \stackrel{(\ref{eq:edgesabs})}{\le} & k+\frac{2j\binom{k+1}{2}+k}{\ell/k-3}
\le k+\frac{2jk^4}{\ell}\le k+\delta,\end{aligned}$$ as required.
**Case 2.** *There is no component of $H^*$ as in Case 1.*
Let $H'$ be the spanning subgraph of $H$ whose edge set is $E(H)\setminus E(H^*)$. So every edge of $H'$ lies in $E((P')^k)\setminus E(P^k)$. Our first aim is to choose a suitable orientation of the edges of $H$. If $xy\in E(H^*)$ we orient $xy$ towards $y$ if and only if $y$ succeeds $x$ on $P$. Recall from (A3) in Section \[sec:absorber\] that the subgraph $D$ of $A_{j,\ell,k}$ consisting of all junction edges, of the path $a_{1,1}vb_{1,2}$ and of all the edges $a_{i,1}a_{i,2}$, $b_{i,1}b_{i,2}$ (for all $i=1,\dots,j$) is a cycle. In order to orient the edges in $E(H')=E(H)\setminus E(H^*)$, we will use an orientation of this cycle $D$, which we will now choose. Orient $a_{1,1}v$ towards $v$ and $vb_{1,2}$ towards $b_{1,2}$. Since $D$ contains the path $a_{1,1}vb_{1,2}$ we can orient all edges of $D$ to obtain a directed cycle. We now use this orientation of $D$ in order to orient the edges in $E(H')$ as follows. Recall from Section \[sec:absorber\] that every edge in $E(A_{j,\ell,k})\setminus E(P^k)\supseteq E(H')$ is either associated with a unique junction edge or with the absorbtion vertex $v$ of $A_{j,\ell,k}$. For every edge $xy\in E(H')$ which is associated with some junction edge $ab$, orient $xy$ towards $y$ if and only if $x\in {\rm Acc}(a)$ and $y\in {\rm Acc}(b)$, where $ab$ is oriented towards $b$ (in the orientation of $D$). Finally, for every edge $xy\in E(H')$ which is associated with $v$, orient $xy$ towards $y$ if and only if $x\in {\rm Acc}(a_{1,1})\cup\{v\}$ and $y\in {\rm Acc}(b_{1,2})\cup\{v\}$.
Note that for every $i=2,\dots,j$, one of the junctions $a_{i,1},a_{i,2}$ sends out a junction edge while the other junction receives a junction edge (in the orientation of $D$). Let $a(+,i)$ denote the former junction and let $a(-,i)$ denote the latter one. Similarly, for every $i=2,\dots,j$ one of the junctions $b_{i,1},b_{i,2}$ sends out a junction edge while the other junction receives a junction edge. Let $b(+,i)$ denote the former junction and let $b(-,i)$ denote the latter one. Let $a(+,1):=a_{1,1}$, $a(-,1):=a_{1,2}$, $b(+,1):=b_{1,1}$ and $b(-,1):=b_{1,2}$. Then the following property holds for all $i=1,\dots,j$: No vertex in ${\rm Acc}(a(-,i))$ sends out an edge in $H'$ while no vertex in ${\rm Acc}(a(+,i))$ receives an edge in $H'$. Similarly, no vertex in ${\rm Acc}(b(-,i))$ sends out an edge in $H'$ while no vertex in ${\rm Acc}(b(+,i))$ receives an edge in $H'$. & (\*)
For each $i=1,\dots,j$, let $C(i,a)$ denote the union of all components of $H^*$ which intersect ${\rm Acc}(a_{i,1})\cup {\rm Acc}(a_{i,2})$ and let $C(i,b)$ denote the union of all components of $H^*$ which intersect ${\rm Acc}(b_{i,1})\cup {\rm Acc}(b_{i,2})$. (Some of the $C(i,a)$ and $C(i,b)$ might be empty.) Let $C^*$ denote the union of all components of $H^*$ which do not intersect any of ${\rm Acc}(a_{i,i'})$ or ${\rm Acc}(b_{i,i'})$ for $i'=1,2$ and $i=1,\dots,j$. Our assumption of Case 2 implies that the vertex sets of graphs $C(1,a),\dots,C(j,a), C(1,b),\dots,C(j,b),C^*$ form a partition of $V(H^*)=V(H)\setminus \{v\}$.
Consider the vertices of $C^*$ in their order on $P$. In the graph $H^*$ each of these vertices sends out at most $k$ edges (in our chosen orientation). However, the last vertex of $C^*$ does not send out any edges in $H^*$. Thus if $C^*\neq \emptyset$ then $$\label{eq:edgesC*}
e(C^*)\le k|C^*|-k.$$ Note also that none of the vertices in $C^*$ are incident to any edges of $H'$, so (\[eq:edgesC\*\]) bounds the number of all edges of $H$ incident to vertices of $C^*$.
Let $r(i,a):= \min\{|C(i,a)|,k\}$. Consider the vertices of $C(i,a)$ in their order on $P$. In the graph $H^*$ each of these vertices sends out at most $k$ edges (in our chosen orientation). However, the last vertex of $C(i,a)$ does not send out any edges in $H^*$. More generally, for each $r=0,\dots,r(i,a)-1$ the vertex of $C(i,a)$ at position $|C(i,a)|-r$ sends out at most $r$ edges in $H^*$. Thus if $C(i,a)\neq \emptyset$ then $$\label{eq:edgesCia}
e(C(i,a))\le k|C(i,a)|-(k+(k-1)+\dots + (k-r(i,a)+1)).$$ Let us now count the number of edges in $H'$ sent out by vertices of $C(i,a)$. $(*)$ implies that no vertex in $C(i,a)-{\rm Acc}(a(+,i))$ sends out an edge in $H'$. But $a(+,i)$ sends out at most $k$ edges in $H'$. More generally, for each $r=0,\dots,r(i,a)-1$ the unique vertex $x$ in ${\rm Acc}(a(+,i))$ which has distance $r$ from $a(+,i)$ on $P$ is incident to $k-r$ edges in $E(A_{j,\ell,k})\setminus E(P^k)\supseteq E(H')$. So $x$ sends out at most $k-r$ edges in $H'$ . (Note that some of these $r(i,a)$ vertices $x$ of $P$ might not lie in $C(i,a)$.) Thus we have the following property: Altogether the vertices in $C(i,a)$ send out at most $k+(k-1)+\dots + (k-r(i,a)+1)$ edges lying in the graph $H'$. &(\*\*)
Clearly, the analogues of (\[eq:edgesCia\]) and $(**)$ also hold for the $C(i,b)$. Moreover, the absorbtion vertex $v$ of $A_{j,\ell,k}$ sends out at most $k$ edges in $H$. Let $I^*:=1$ if $C^* \neq \emptyset$ and $I^*:=0$ otherwise. Altogether the above shows that $$\label{eq:edgeH}
e_H\le kv_H - k I^*.$$ We now distinguish three subcases.
**Case 2a.** *$C^* \neq \emptyset$.*
In this case we have $$d_1(H)\stackrel{(\ref{eq:edgeH})}{\le} \frac{kv_H-k}{v_H-1}=k,$$ as required.
**Case 2b.** *$H$ contains at least one edge associated with $v$ as well as at least one edge associated with every junction edge $ab$.*
In this case we have that $v_H\ge 2j$ since there are $2j-1$ junction edges. Thus $$d_1(H)\stackrel{(\ref{eq:edgeH})}{\le} \frac{kv_H}{v_H-1}=k+\frac{k}{v_H-1}\le k+\frac{k}{2j-1}\le k+\delta,$$ as required.
**Case 2c.** *$C^*=\emptyset$. Moreover, $H$ avoids all edges associated with $v$ or there exists a junction edge $ab$ such that $H$ avoids all edges associated with $ab$.*
We will first show that in this case at least one of the following four properties hold:
- There is an $i$ with $2\le i\le j$ such that $C(i,a)\neq \emptyset$ and $H$ avoids all edges associated with the junction edge sent out by $a(+,i)$.
- There is an $i$ with $1\le i\le j$ such that $C(i,b)\neq \emptyset$ and $H$ avoids all edges associated with the junction edge sent out by $b(+,i)$.
- $C(1,a)\neq \emptyset$ and $H$ avoids all edges associated with $v$.
- $C(1,b)= \emptyset$ and $v\in V(H)$.
To prove that one of (a)–(d) holds, let $D'$ be the cycle obtained from $D$ by replacing the path $a_{1,1}vb_{1,2}$ with a single edge $e_v$ from $a_{1,1}$ to $b_{1,2}$ and contracting each edge $a_{i,1}a_{i,2}$ into a new vertex $(i,a)$ as well as contracting each edge $b_{i,1}b_{i,2}$ into a new vertex $(i,b)$ (for all $i=1,\dots,j$). Thus every edge of $D'$ apart from $e_v$ corresponds to a junction edge. Moreover, our orientation of $D$ induces one of $D'$. So we will view $D'$ as a directed cycle. Colour the vertex $(i,a)$ of $D'$ red if $C(i,a)\neq \emptyset$ and colour $(i,b)$ red if $C(i,b)\neq \emptyset$. Colour the edge $e_v$ of $D'$ red if $H$ contains some edge associated with $v$. Colour each (junction) edge $e\neq e_v$ of $D'$ red if $H$ contains some edge associated with $e$. Since $v_H\ge 2$ and since we are assuming that $C^*=\emptyset$, it follows that at least one vertex of $D'$ is red. Moreover, our assumption that Case 2c holds implies that not all edges of $D'$ are red.
Let us first consider the case when $v\notin V(H)$. Then both endvertices of a red edge of $D'$ are red. Thus $D'$ contains a red vertex $w$ such that the edge from $w$ to its successor on $D'$ is not red. If $w=(i,a)$ for some $i>1$ then (a) holds. If $w=(i,b)$ for some $i\ge 1$ then (b) holds. If $w=(1,a)$ then (c) holds. So suppose next that $v\in V(H)$. In this case we can only guarantee that both endvertices of a red edge $e\neq e_v$ of $D'$ are red. We may also assume that $(1,b)$ is red (otherwise (d) holds). If not all edges in $D'-e_v$ are red then $D'$ contains a red vertex $w\neq (1,a)$ such that the edge from $w$ to its successor on $D'$ is not red. Similarly as before this implies that (a) or (b) holds. So suppose that all edges in $D'-e_v$ are red. This implies that $(1,a)$ is red and $e_v$ is not red. Thus (c) holds. This completes the proof that one of (a)–(d) holds.
Suppose first that (a) holds. Then the vertices in $C(i,a)$ send out no edges lying in the graph $H'$. Together with (\[eq:edgesCia\]) this implies that instead of (\[eq:edgeH\]) we have that $$e_H\le kv_H-(k+(k-1)+\dots + (k-r(i,a)+1))\le kv_H-k$$ and so $d_1(H)\le k$ as required. The arguments for (b) and (c) are similar. So let us now assume that (d) holds. Then $v$ does not send out any edges in the graph $H$. So instead of (\[eq:edgeH\]) we have $e_H\le kv_H-k$ and so $d_1(H)\le k$ as required.
Linking up $k$th powers of paths: Proof of Lemma \[lem:linking\] {#sec:linking}
================================================================
A result of Kreuter [@Kreuter] determines the threshold for the existence of a linear number of disjoint copies of a given graph $Q$ in a random graph $G_{n,p}$. (This threshold is roughly the same as the one in Theorem \[thm:JKV\].) We will prove an analogue of this result for partite multigraphs (see Lemma \[DGbound\]). We will then apply Lemma \[DGbound\] to find disjoint copies of a partite multigraph $Q$, where each copy of $Q_i$ of $Q$ will correspond to an $(A_i,B_i)$-linkage $R_i$ (see Lemma \[partial\]). This allows us to link up a positive fraction of the pairs $(A_i,B_i)$ we are required to link up. Roughly speaking, in the proof of Lemma \[lem:linking\] we will apply Lemma \[partial\] repeatedly to eventually obtain disjoint linkages for all the pairs $(A_i,B_i)$ that we are required to link up.
We write $[k]:=\{1,\dots,k\}$ and $[-k]:=\{-1,\dots,-k\}$. Suppose that $p=p(n)$. We define the random graph ${\mathcal{G}}={\mathcal{G}}(n_0,n,t,k,p)$ as follows: Consider the complete $(t+1)$-partite multigraph $K$ with vertex classes $V_1,\dots,V_t$ of size $n$ and one vertex class $V_0$ of size $n_0$, and where each edge from $V_0$ to $V_i$ has multiplicity exactly $2k$ (for all $i=1,\dots,t$) and all other edges have multiplicity one. Moreover, for all pairs of vertices $x\in V_0$ and $y\in V_1\cup\dots\cup V_t$ the $2k$ edges between $x$ and $y$ in $K$ have labels in $[-k]\cup [k]$ and these labels are distinct for different edges between $x$ and $y$. We obtain ${\mathcal{G}}$ by including each edge of $K$ into ${\mathcal{G}}$ with probability $p$, independently of all other edges.
Let $G$ be any $(t+1)$-partite multigraph with vertex classes $Y_0,\dots,Y_t$ such that $|Y_0|\le n_0$ and $|Y_i|\le n$ for all $i=1,\dots,t$, and where each edge between $Y_0$ and $Y_i$ has multiplicity at most $2k$ (for all $i=1,\dots,t$) and all other edges have multiplicity one. Moreover, for all pairs of vertices $a\in Y_0$ and $b\in Y_1\cup\dots\cup Y_t$ the edges between $a$ and $b$ in $G$ have labels in $[-k]\cup [k]$ and these labels are distinct for different edges between $a$ and $b$.
We say that a (not necessarily induced) copy of $G$ in $K$ is a *good copy of $G$* if for all $i=0,\dots,t$ each vertex in $Y_i$ is mapped to a vertex in $V_i$ and if each edge of $G$ with label $j$ between some pair $a\in Y_0$ and $b\in Y_1\cup\dots\cup Y_t$ of vertices is mapped to the edge of $K$ with label $j$ between the images of $a$ and $b$ in $K$. Let $X_{G}$ denote the number of good copies of $G$ in ${\mathcal{G}}$. Let $D_G$ denote the maximum size of a set of disjoint good copies of $G$ in ${\mathcal{G}}$. Set $$\Phi_G:= \min \{ {\mathbb{E}}( X_{G'} ) \colon G' \subseteq G, e_{G'}>0 \}$$ and $$\Phi^v_G:= \min \{ {\mathbb{E}}( X_{G'} ) \colon G' \subseteq G, v_{G'}>0 \}.$$ Note that we allow $G'$ to consist of a single vertex in the second definition. Also note that $\Phi^v_G \le \Phi_G$.
Throughout this section, when using the $O(.)$, $\Theta(.)$ and $\Omega(.)$ notation, we mean that the size $n$ of the vertex classes $V_1,\dots,V_t$ tends to infinity. In most cases $n_0$ will be a function of $n$, but we sometimes also allow $n_0=1$. The number of vertices in the graph $G$ will always be bounded.
\[DGbound\] Suppose that $\Phi^v_G \to \infty$ as $n\to \infty$. Then there is a constant $c>0$ (depending only on $G$) such that with probability $1-O(1/\Phi^v_G)$ we have $D_G \ge c\Phi_G^v$.
The proof of Lemma \[DGbound\] is essentially the same as that of Theorem 3.29 in [@JLR], which in turn is based on an argument of Kreuter [@Kreuter]. The difference is that in Theorem 3.29 $X_G$ counts disjoint copies of $G$ in $G_{n,p}$ (rather than good copies of $G$ in ${\mathcal{G}}$). For completeness, we will give a sketch which only highlights the (very minor) adjustments one has to make. The proof of Lemma \[DGbound\] needs the following proposition, which is proved using a standard application of Chebyshev’s inequality (see Lemma 3.5 and Remark 3.7 in [@JLR] for a similar and more detailed argument). Note that Proposition \[XGbound\] does not assume any bounds on $n_0$. In particular, we will later also apply it in the case when $n_0=1$.
\[XGbound\]
- $Var(X_G)=O \left( {\mathbb{E}}(X_G)^2/\Phi_G \right)$.
- Suppose that $\Phi_G \to \infty$ as $n\to \infty$ and that ${\varepsilon}>0$ is fixed. Then with probability $1-O(1/\Phi_G)$ we have $X_G = (1\pm {\varepsilon}){\mathbb{E}}(X_G)$.
In the next two proofs we will use the following notation: Suppose that $H$ is a $(t+1)$-partite multigraph on a bounded number of vertices with vertex classes $Y_0,\dots Y_t$ such that $|Y_0|\le n_0$. Then we define $$\label{Psi}
\Psi_H:=p^{e_H} n_0^{|Y_0|} n^{v_H-|Y_0|}.$$ Note that ${\mathbb{E}}(X_H)= \Theta (\Psi_H)$.
55[**Proof of Proposition \[XGbound\].** ]{} Given a good copy $G'$ of $G$ in $K$, let $I_{G'}$ denote the indicator function that $G'$ is contained in ${\mathcal{G}}$. Below, the summations are always over good copies of the relevant graphs in $K$. With the above notation, we have $$\begin{aligned}
{\mathbb{E}}(X_G^2) & =\sum_{G',G''} {\mathbb{E}}(I_{G'}I_{G''})\le {\mathbb{E}}(X_G)^2 + \sum_{E(G') \cap E(G'') \neq \emptyset} {\mathbb{E}}(I_{G'}I_{G''}) \\
& ={\mathbb{E}}(X_G)^2 + O \left( \sum_{ H \subseteq G, e_H >0 } \frac{\Psi_G^2}{ \Psi_H} \right) \\
& ={\mathbb{E}}(X_G)^2 + O \left( \sum_{ H \subseteq G, e_H >0 } \frac{{\mathbb{E}}(X_G)^2}{{\mathbb{E}}(X_H)} \right) \\
& ={\mathbb{E}}(X_G)^2 + O \left( {\mathbb{E}}(X_G)^2/\Phi_G \right).
$$ So $Var(X_G)=O( {\mathbb{E}}(X_G)^2/\Phi_G)$. This proves (i). A straightforward application of Chebyshev’s inequality now completes the proof of (ii).
55[**Proof of Lemma \[DGbound\].** ]{} The proof begins by considering an auxiliary graph $\Gamma$, where the vertices of $\Gamma$ correspond to good copies of $G$ in ${\mathcal{G}}$ (rather than to copies in $G_{n,p}$ as in the proof of Theorem 3.29 in [@JLR]), with an edge between two vertices of $\Gamma$ if the corresponding copies of $G$ share at least one vertex. So $\Gamma$ has $X_G$ vertices and $\sum_F X_F$ edges, where the sum is taken over all unions $F= G_1 \cup G_2$ of two copies of $G$ sharing at least one vertex, and where $F$ is viewed as a $(t+1)$-partite multigraph whose $i$th vertex class $Y_i^F$ is the union of the $i$th vertex classes of $G_1$ and $G_2$ (but we include any vertex in $G_1\cap G_2$ only once). Since $X_F=0$ if $|Y_i^F|>n_0$, we only sum over all those $F$ for which $|Y_0^F|\le n_0$.
Note that any independent set of vertices in $\Gamma$ corresponds to a collection of pairwise disjoint good copies of $G$ in ${\mathcal{G}}$. So one can use Turán’s theorem to show that $$\label{eq:turan}
D_G \ge \frac{X^2_G}{X_G+2 \sum_F X_F}.$$ Proposition \[XGbound\](ii) implies that with probability $1-O(1/\Phi_G)$, we have ${\mathbb{E}}(X_G)/2 \le X_G \le 2{\mathbb{E}}(X_G)$. Together with (\[eq:turan\]) this implies that it suffices to show that with probability $1-O(1/\Phi^v_G)$ we have $$\label{eq:XF}
X_F =O \left( \frac{({\mathbb{E}}X_G)^2}{\Phi_G^v} \right)=O\left( \frac{\Psi^2_G}{\Phi_G^v} \right),$$ where $\Psi_G$ is as defined in (\[Psi\]). To prove (\[eq:XF\]), the first step is to observe that if $F= G_1 \cup G_2$ is as above and $H:=G_1\cap G_2$, then $$\label{eq:exF}
{\mathbb{E}}(X_F)=\Theta(\Psi_F)=\Theta\left(\frac{\Psi^2_G}{\Psi_H}\right)=O\left( \frac{\Psi^2_G}{\Phi_G^v} \right).$$ Next, note that Proposition \[XGbound\](i) implies that $$\label{VarXF}
Var(X_F)=O(\Psi^2_F/\Phi_F).$$ To bound this expression, we need the following log-supermodularity property, where $H_1$ and $H_2$ are arbitrary $(t+1)$-partite multigraphs. This property follows easily from the definition of $\Psi_H$ (indeed, the overlap between $H_1$ and $H_2$ contributes twice to both the left and right hand side). $$\Psi_{H_1 \cup H_2} \Psi_{H_1 \cap H_2}= \Psi_{H_1}\Psi_{H_2}.$$ Now one can proceed exactly as in the proof of Theorem 3.29: Using repeated applications of the log-supermodularity, one can show that the right hand side of (\[VarXF\]) is $O(\Psi_G^4/(\Phi_G^v)^3)$. With this bound, Chebyshev’s inequality now implies that $$\mathbb{P} \left( X_F \ge {\mathbb{E}}(X_F) + \frac{\Psi_G^2}{\Phi_G^v} \right) \le Var(X_F)\cdot \frac{(\Phi^v_G)^2}{\Psi^4_G}
=O \left( 1/\Phi_G^v \right).$$ Together with (\[eq:exF\]) this implies that (\[eq:XF\]) holds with the required probability.
We now apply the above results to find powers of paths. Let $k\ge 2$ and $s\ge 4k$. Recall that $P^k_s$ denotes the $k$th power of a path $P_s=x_1 \dots x_s$ on $s$ vertices. Let $Q$ be the multigraph obtained from $P^k_s$ by contracting $x_1,\dots,x_{k}, x_{s-(k-1)}, \dots ,x_s$ into a single vertex $x_0$ and deleting any resulting loops at $x_0$ (but not removing any of the multiple edges). So $Q$ is a multigraph on $t+1$ vertices, where $t:=s-2k$ and where $x_0$ has degree $k(k+1)$ and all other vertices have degree $2k$. We view $Q$ as a $(t+1)$-partite multigraph with vertex class $Y_0:=\{x_0\}$ and each other vertex class $Y_1,\dots,Y_t$ also consisting of a single vertex. Note that every edge of $Q$ corresponds to a unique edge of $P^k_s$. We now assign each edge of $Q$ at $x_0$ a label as follows: For all $i\in [k]$ and every $x\in Y_1\cup \dots \cup Y_t$ we label an edge of $Q$ between $x_0$ and $x$ which corresponds to an edge of $P^k_s$ between $x_i$ and $x$ with $i$. Similarly, for all $i\in [-k]$ and every $x\in Y_1\cup \dots \cup Y_t$ we label an edge of $Q$ between $x_0$ and $x$ which corresponds to an edge of $P^k_s$ between $x_{s+1+i}$ and $x$ with $i$. So for each $i\in [k]$ there are $k-i+1$ edges with labels $i,\dots,k$ between $x_0$ and $x_{k+i}$. Similarly, for each $i\in [-k]$ there are $k+1+i$ edges with labels $-k,\dots, i$ between $x_0$ and $x_{s-(k-1)+i}$.
\[Phibound\] Let $s> 8k^2$ and define $Q$ as above. Suppose that $1 \le n_0 \le n$ and $p=p(n) \ge n^{-1/k+8k/s}$. Then
- $\Phi_Q =\Omega( n^{8k^2/s})$;
- $\Phi_Q^v =\Omega(n_0)$.
Note that both assertions follow if we can show that any submultigraph $Q'$ of $Q$, which contains at least one edge, satisfies ${\mathbb{E}}(X_{Q'})=\Omega( n^{8k^2/s} n_0)$. Let $v:=v_{Q'}$ and $e:=e_{Q'}$.
First suppose that $v \ge s/(2k)$. In this case, it suffices to note that at most one vertex of $Q'$ has degree at most $k(k+1)$ and all others vertices of $Q'$ have degree at most $2k$. Thus $e \le kv +k^2$ with room to spare. So recalling that $n \ge n_0 \ge 1$, we have $${\mathbb{E}}(X_{Q'})=\Omega\left( p^e n_0 n^{v-1}\right) =\Omega\left( n^{(v+k)(-1+8k^2/s)+v-1}\right).$$ But $$(v+k)(-1+8k^2/s)+v-1\ge 8k^2v/s-k-1 \ge 2,$$ and so the required result follows in this case, with room to spare.
So we may assume that $v \le s/(2k)$. Consider the ordering $x_0,x_{k+1},x_{k+2}, \dots, x_{s-k}$ of the vertices of $Q$. The assumption on $v$ implies that there are $k$ consecutive vertices $x_a,\dots,x_{a+k-1}$ with $k < a \le s-2k+1$ which are not contained in $Q'$. Write $x_{s+1}:=x_0$. Now for each edge $x_ix_{i'}$ of $Q'$ with $i <i'$ we either have $0\le i<i' <a$ or $a+k \le i<i' \le s+1$. In the first case, we orient $x_ix_{i'}$ towards $x_i$ and in the second case we orient $x_ix_{i'}$ towards $x_{i'}$. Now it is easy to see that for every vertex $x_i$ of $Q'$, the outdegree of $x_i$ in this orientation of $Q'$ is at most $k$. Moreover, the above process yields an orientation of all edges of $Q'$ and there is at least one vertex in $Q'$ which has outdegree $0$. (If $Q'$ contains $x_0=x_{s+1}$, then this will be one such vertex. If $Q'$ is disconnected, there will be several such vertices.) Thus $e \le k(v-1)$. So using that $n \ge n_0$ and $v \ge 2$, we have $${\mathbb{E}}(X_{Q'})=\Omega\left( p^e n_0 n^{v-1}\right) =\Omega\left( n_0 (p^kn)^{v-1}\right)=\Omega\left( n_0 p^k n\right)
=\Omega\left( n_0 n^{8k^2/s}\right),$$ as required.
We can now combine Lemma \[DGbound\] and Lemma \[Phibound\](ii) to obtain the following result.
\[covercor\] Let $s> 8k^2$ and define $Q$ as in Lemma \[Phibound\]. Suppose that $n_0 \le n$, that $n_0 \to \infty$ and that $p=p(n) \ge n^{-1/k+8k/s}$. Then there is a constant $c>0$ (depending only on $Q$) such that with probability $1-O(1/n_0)$, we have $D_Q \ge cn_0$.
Our aim is now to apply Corollary \[covercor\] to link up given sets of vertices in $G_{n,p}$ by powers of paths. Suppose that $A=(a_1,\dots,a_k)$ and $B=(b_1,\dots,b_k)$ are two (ordered) sequences of vertices which are disjoint from each other. Recall that a graph $R$ is an $(A,B)$-linkage if $R$ is obtained from the $k$th power of a path whose initial endsequence is $A$ and whose final endsequence is $B$ by deleting all edges within $A$ and within $B$. We call ${\mathcal{A}}:=\{ (A_1,B_1),\dots,(A_f,B_f) \}$ a *set of pairwise disjoint $k$-sequence pairs* if each $A_i$ and each $B_i$ is a sequence of $k$ vertices and all these $2f$ sequences are pairwise disjoint. A *partial ${\mathcal{A}}$-linkage of size $f'$ and parameter $s$* consists of ${\mathcal{R}}=\{R_1,\dots,R_{f'}\}$ where
- for each $i=1,\dots,f'$ there is a $j=j(i)\in [f]$ such that $R_i$ is an $(A_j,B_j)$-linkage;
- the $R_i$ are pairwise disjoint;
- if $j' \neq j(i)$, then $R_i$ avoids $A_{j'}\cup B_{j'}$;
- $|R_i|=s$ for all $i=1,\dots,f'$.
If $j'\neq j(i)$ for all $i=1,\dots,f'$, we say that $(A_{j'},B_{j'})$ *is unlinked by ${\mathcal{R}}$*.
\[partial\] For every $0<{\varepsilon}<1/k$ and every $k\ge 2$ there is a constant $c>0$ such that the following holds: Suppose that $p=p(n) \ge n^{-1/k+{\varepsilon}}$ and that $\log^2 n \le f \le n/(4k)$. Let ${\mathcal{A}}=\{ (A_1,B_1),\dots,(A_f,B_f) \}$ be a set of $f$ pairwise disjoint $k$-sequence pairs. Then with probability $1-O(1/\log ^2n)$, we have that $G_{n,p}$ contains a partial ${\mathcal{A}}$-linkage ${\mathcal{R}}=\{R_1,\dots,R_{f'}\}$ of size $f':=c f$ and parameter $\lceil 10k/{\varepsilon}\rceil$.
Set $s:=\lceil 10k/{\varepsilon}\rceil$. So each $R_i$ will consist of $s$ vertices (including those in the endsequences of $R_i$). Note that the number of vertices contained in some $A_i$ or $B_i$ is $2kf \le n/2$. We will view $G_{n,p}$ as a subgraph of $K_n$. Let $N'$ consist of $n/2$ vertices of $K_n$ which are not contained in any of the $A_i$ or $B_i$. Let $t:=s-2k$. Partition $N'$ into $t$ classes $V_1,\dots,V_t$ of equal size $n':=n/(2t)$. For all $j\in [k]$, let $V'_j$ consist of the $j$th vertex in each of the $A_i$. So $|V'_j|=f$. For all $j\in [-k]$, let $V'_j$ consist of the $(k+1+j)$th vertex in each of the $B_i$. Again $|V'_j|=f$. Let $K'$ be the complete $s$-partite subgraph of $K_n$ induced by the vertex classes $V_1,\dots,V_t$, and all the $V'_j$ for $j\in [-k]\cup [k]$. Let $K$ be the $(t+1)$-partite multigraph obtained from $K'$ by contracting all the vertices in $A_i\cup B_i$ into a single vertex $y_i$, where any resulting loops at $y_i$ are removed (but we do not remove any multiple edges). So the vertex classes of $K$ are $V_0:=\{y_1,\dots,y_f\}$ and $V_1,\dots,V_t$. Note that each edge $e$ of $K$ corresponds to a unique edge $e'$ of $K'$. We now label the edges of $K$ as follows: For all $i\in [f]$ and all $j\in [-k]\cup [k]$ we label an edge $e$ of $K$ between $y_i$ and some vertex $x\in V_1\cup\dots\cup V_t$ with $j$ if the corresponding edge $e'$ of $K'$ joins some vertex in $V'_j$ to $x$.
Now define a random graph ${\mathcal{G}}$ as follows: ${\mathcal{G}}$ is a spanning subgraph of $K$, where we include an edge $e$ of $K$ into ${\mathcal{G}}$ if and only if the corresponding edge $e'$ of $K'$ is included in $G_{n,p}$. This means that each edge of $K$ is included in ${\mathcal{G}}$ with probability $p$, independently of all other edges. So this corresponds exactly to the setting described at the beginning of the section, with $n'$ playing the role of $n$ and $f$ playing the role of $n_0$.
Let $Q$ be as defined before Lemma \[Phibound\]. Then a good copy of $Q$ in $K$ containing $y_i$ corresponds to an $(A_i,B_i)$-linkage in $K_n$. (Thus a good copy of $Q$ in ${\mathcal{G}}$ containing $y_i$ corresponds to an $(A_i,B_i)$-linkage in $G_{n,p}$.) Similarly, a set of $\ell$ disjoint good copies of $Q$ in $K$ (or in ${\mathcal{G}}$) corresponds to a partial ${\mathcal{A}}$-linkage of size $\ell$ and parameter $s$ in $K_n$ (or in $G_{n,p}$).
Also note that $p(n) \ge n^{-1/k+{\varepsilon}} \ge (n')^{-1/k+4{\varepsilon}/5}\ge (n')^{-1/k+8k/s}$. So we can apply Corollary \[covercor\] with $n'$ and $f$ playing the roles of $n$ and $n_0$ to see that, with with probability $1-O(1/\log ^2n)$, $G_{n,p}$ contains a partial ${\mathcal{A}}$-linkage of parameter $s$ and size $cf$, where $c$ depends only on $Q$ (and thus only on $k$ and ${\varepsilon}$).
A simple consequence of the previous arguments is that we can link up a given sequence $A$ of $k$ vertices to a given sequence $B$ of $k$ vertices via the $k$th power of a sufficiently long path.
\[lem:singlelink\] Let $0<{\varepsilon}<1/k$ and $k \ge 2$. Suppose that $p \ge n^{-1/k+{\varepsilon}}$ and that $A=(a_1 \dots a_k)$ and $B=(b_1 \dots b_k)$ are pairwise disjoint sequences of vertices. Then with probability $1-O(1/\log^3 n)$, $G_{n,p}$ contains an $(A,B)$-linkage $R$ with $|R|=\lceil 10k/{\varepsilon}\rceil$.
Let ${\mathcal{A}}:=\{(A,B)\}$ and $s:=\lceil 10k/{\varepsilon}\rceil$. We now define $K'$, $K$, ${\mathcal{G}}$ and $Q$ exactly as in the proof of Lemma \[partial\]. In particular, for all $j\in [k]$, let $V'_j$ consist of the $j$th vertex in $A$. For all $j\in [-k]$, let $V'_j$ consist of the $(k+1+j)$th vertex in $B$. So $V_0$ consists of a single vertex $y$ and $n_0=1$. Again, a good copy of $Q$ in $K$ containing $y$ corresponds to an $(A,B)$-linkage in $K_n$ (with a similar correspondence between ${\mathcal{G}}$ and $G_{n,p}$). Moreover, again we have $p(n) \ge (n')^{-1/k+8k/s}$, where $n':=n/(2t)$ and $t:=s-2k$.
Now Lemma \[Phibound\](i) together with Proposition \[XGbound\](ii) imply that with probability $1-O(n^{-8k^2/s})$, we have $X_Q>0$. So the error bound is at most $O(1/\log ^3 n)$ (with room to spare), as required.
We can now combine Lemmas \[partial\] and \[lem:singlelink\] in order to prove Lemma \[lem:linking\].
55[**Proof of Lemma \[lem:linking\].** ]{} Since $p=p(n)=o(1)$, we can view $G_{n,p}$ as a union of $2\log^2 n$ independent random graphs $G_{n,p_i}$, with $p_i =p'$, where $p'\ge (1+o(1))p/(2 \log^2 n)\ge n^{-1/k+{\varepsilon}/2}$. Let $s:=\lceil 30k/{\varepsilon}\rceil$ and ${\mathcal{A}}:=\{(A_1,B_1),\dots,(A_f,B_f)\}$. Our strategy is to first apply Lemma \[partial\] repeatedly to obtain partial linkages until the number of unlinked pairs in ${\mathcal{A}}$ is less than $\log^2n$ (using a different $G_{n,p_i}$ each time). We will then apply Lemma \[lem:singlelink\] repeatedly in order to link the remaining pairs in ${\mathcal{A}}$ one by one (again, using a different $G_{n,p_i}$ each time).
Let $c=c(k,{\varepsilon})$ be as in Lemma \[partial\] and let ${\mathcal{A}}_0:={\mathcal{A}}$. Suppose that we have obtained a set ${\mathcal{A}}_i$ consisting of $(1-c)^i f$ unlinked pairs from ${\mathcal{A}}$ and that we have found a partial ${\mathcal{A}}$-linkage ${\mathcal{R}}_i$ with parameter $s$ which links precisely all the pairs in ${\mathcal{A}}\setminus {\mathcal{A}}_i$. Let $N_i$ be obtained from $[n]$ by deleting all the vertices in linkages from ${\mathcal{R}}_{i}$. Thus $|N_i|=n- (|{\mathcal{A}}|-|{\mathcal{A}}_i|)s\ge n-fs \ge n/2$ and so $p_i=p'\ge n^{-1/k+{\varepsilon}/2}\ge |N_i|^{-1/k+{\varepsilon}/3}$. Hence if $|{\mathcal{A}}_i|=(1-c)^if>\log^2 n$, we can apply Lemma \[partial\] with ${\varepsilon}/3$ playing the role of ${\varepsilon}$ and with the random subgraph of $G_{n,p_i}$ induced by the set $N_i$ playing the role of $G_{n,p}$. With probability $1-O(1/\log^2 n)$ this yields a partial linkage ${\mathcal{R}}'_i$ of size $c|{\mathcal{A}}_i|$ and parameter $s$. Let ${\mathcal{R}}_{i+1}:={\mathcal{R}}_i\cup {\mathcal{R}}'_i$ and let ${\mathcal{A}}_{i+1}$ denote the set of pairs which are still unlinked. So $|{\mathcal{A}}_{i+1}|=(1-c)^{i+1} f$.
Let $i^*\ge 0$ be the smallest integer for which $(1-c)^{i^*}f\le \log^2 n$. Thus $i^*\le \log_{1/(1-c)} n$. Our argument shows that with probability at least $1-O(i^*/\log ^2 n)$ we can find a partial linkage ${\mathcal{R}}_{i^*}$ of parameter $s$ such that the set ${\mathcal{A}}_{i^*}$ of unlinked pairs has size $|{\mathcal{A}}_{i^*}|= (1-c)^{i^*} f$.
We will now link up the remaining pairs one by one. For this, write ${\mathcal{A}}_{i^*}=\{(A^*_1,B_1^*),\dots,(A^*_{f^*},B^*_{f^*}) \}$. Thus $f^* \le \log ^2n$. Let $N_{i^*}$ be obtained from $[n]$ by deleting all the vertices in linkages from ${\mathcal{R}}_{i^*}$. Suppose that $1 \le j \le f^*$ and that we have obtained an $(A^*_i,B^*_i)$-linkage $R_i^*$ for all $i=1,\dots, j-1$ such that all the $R_i^*$ are pairwise disjoint, $|R_i^*|=s$, $V(R^*_i)\subseteq N_{i^*}$ and such that $R_i^*$ avoids $(A^*_{i'},B_{i'}^*)$ for all $i'\neq i$. Let $$N^*_j:=\left(N_{i^*}\setminus \left(V(R^*_1)\cup \dots\cup V(R^*_{j-1})\cup \bigcup_{i=1}^{f^*} (A^*_i\cup B^*_i)\right)\right) \cup A^*_{j} \cup B^*_{j}.$$ Thus $|N^*_j|\ge n - fs \ge n/2$ and so $p'\ge n^{-1/k+{\varepsilon}/2}\ge |N^*_j|^{-1/k+{\varepsilon}/3}$. Hence we can apply Lemma \[lem:singlelink\] with ${\varepsilon}/3$ playing the role of ${\varepsilon}$ and with the random subgraph of $G_{n,p_{i^*+j}}$ induced by the set $N^*_j$ playing the role of $G_{n,p}$. With probability $1-O(1/\log ^3 n)$ this yields a $(A_j^*,B_j^*)$-linkage $R_j^*$ with $|R^*_j|=s$ and $V(R^*_j)\subseteq N^*_j$.
Since $f^* \le \log ^2n$, this means that altogether, a.a.s. we can find pairwise disjoint $(A^*_i,B^*_i)$-linkages for all $i=1,\dots,f^*$ which only use vertices in $N_{i^*}$ and so are disjoint from the linkages in ${\mathcal{R}}_{i^*}$.
Deriving Theorem \[thm:riordan\] {#sec:riordan}
================================
Given a graph $H$ on at least three vertices, we define $$d_2 (H) := {e_H \over v_H -2} \ \ \text{ and } \ \ d_2^{\rm max} (H):=\max_{H'\subseteq H, \ v_{H'}\ge 3} d_2(H').$$ The purpose of this section is to derive Theorem \[thm:riordan\] from the following result of Riordan [@riordan]. Actually the result in [@riordan] is more general than the version below, as its formulation in [@riordan] does not require the maximum degree of the $H_n$ to be bounded. Moreover, it is stated for $G_{n,m}$ with $m=p\binom{n}{2}$ instead of $G_{n,p}$. But Theorem 2.2(ii) of [@Bollobasbook] allows us to apply it to $G_{n,p}$.
\[thm:riordan2\] Let $(H_n)_{n=1}^\infty$ be a fixed sequence of graphs such that $n=v_{H_n}$, $e_{H_n}\ge n$ and such that the maximum degree of the $H_n$ is bounded. Let $p=p(n)$ be such that $$\label{eq:riordan}
np^{d_2^{\rm max} (H)}\to \infty, \ \ \ pn^2\to \infty\ \ \text{ and }\ \ (1-p)\sqrt{n}\to \infty.$$ Then a.a.s. $G_{n,p}$ contains a copy of $H_n$.
Thus in order to derive Theorem \[thm:riordan\] from this, it suffices to prove the following proposition. Note that the third condition in (\[eq:riordan\]) does not hold if $p$ is very close to $1$. But since the property of containing the $k$th power of a Hamilton cycle is monotonically non-decreasing under the addition of edges, this case follows immediately from the fact that in our case there is some $p$ satisfying all three conditions in (\[eq:riordan\]).
\[prop:2dens\] Suppose that $n\ge 4k$ and $k\ge 3$. Then $d_2^{\rm max} (C^k_n)\le k+\frac{(k+1)k^2}{n}$. Moreover, $d_2^{\rm max} (C^2_n)=3$ if $n\ge 18$.
Let us first consider the case when $k\ge 3$. Consider any $H\subseteq C^k_n$ on $v_H\ge 3$ vertices. Suppose first that $H\subseteq P^k_n$. Thus there is an ordering $x_1,\dots,x_{v_H}$ of the vertices of $H$ such that for all $i=2,\dots,v_H$ every $x_i$ has at most $k$ neighbours amongst $x_1,\dots,x_{i-1}$. Since $d_2(H[\{x_1,x_2,x_3\}])\le 3\le k$, it follows that $d_2(H)\le k$. Now suppose that $H\not\subseteq P^k_n$. Then $v_H\ge n/k$ and by deleting at most $\binom{k+1}{2}$ edges from $H$ one can obtain a subgraph $H'$ with $H'\subseteq P^k_n$. Thus $$d_2(H)\le d_2(H')+\frac{\binom{k+1}{2}}{v_H-2}\le d_2(H')+\frac{\binom{k+1}{2}}{n/k-2}\le k+\frac{(k+1)k^2}{n}$$ since $n\ge 4k$. A similar argument shows that $d_2^{\rm max} (C^2_n)=3$ if $n\ge 18$.
Acknowledgements
================
We are extremely grateful to Nikolaos Fountoulakis for helpful discussions throughout the project and comments on the manuscript. We are also indebted to the referees for pointing out an error in an earlier version.
[8]{}
M. Ajtai, J. Komlós and E. Szemerédi, The first occurrence of Hamilton cycles in random graphs, *Annals of Discrete Mathematics* [**27**]{} (1985), 173–178.
B. Bollobás, The evolution of sparse graphs, In *Graph Theory and Combinatorics*. Proc. Cambridge Combinatorial Conf. in honour of Paul Erdős (B. Bollobás, Ed.). Academic Press, 1984, pp. 35–57. B. Bollobás, *Random Graphs*, Academic Press, London, 1985.
B. Bollobás, T.I. Fenner and A.M. Frieze, An algorithm for finding Hamilton paths and cycles in random graphs, *Combinatorica* [**7**]{} (1987), 327–341.
P. Ch${\rm \hat a}$u, L. DeBiasio and H. Kierstead, Pósa’s conjecture for graphs of order at least $2 \times 10^8$, *Random Structures & Algorithms* [**39**]{} (2011), 507–525.
D. Dellamonica, Y. Kohayakawa, V. Rödl, A. Ruciński, Universality of random graphs, *SIAM J. Discrete Math.* [**26**]{} (2012), 353–374.
G. A. Dirac, Some theorems on abstract graphs, *Proc. London Math. Soc.* **2** (1952), 69–81.
A. Hajnal and E. Szemerédi, Proof of a conjecture of Erdős, *Combinatorial Theory and its Applications (Vol. 2)* (P. Erdős, A. Rényi and V. T. Sós eds.), Colloq. Math. Soc. J. Bolyai 4, North-Holland, Amsterdam (1970), 601–623.
D. Hefetz, M. Krivelevich and T. Szabó, Sharp threshold for the appearance of certain spanning trees in random graphs, preprint.
S. Janson, T. Łuczak and A. Ruciński, *Random graphs*, Wiley-Interscience, 2000.
A. Johansson, J. Kahn and V. Vu, Factors in Random Graphs, *Random Structures & Algorithms* [**33**]{} (2008), 1–28.
J. Komlós, G. N. Sárközy and E. Szemerédi, Proof of the Seymour conjecture for large graphs, *Ann. Combin.* **2** (1998), 43–60. J. Komlós and E. Szemerédi, Limit distributions for the existence of Hamilton cycles in a random graph, *Discrete Math.* [**43**]{} (1983), 55–63.
A.D. Korshunov, A solution of a problem of P. Erdős and A. Rényi about Hamilton cycles in non-oriented graphs, *Metody Diskr. Anal. Teoriy Upr. Syst., Sb. Trudov Novosibirsk* [**31**]{} (1977), 17–56 (in Russian).
B. Kreuter, Threshold functions for asymmetric Ramsey properties with respect to vertex colorings, *Random Structures & Algorithms* [**9**]{} (1996), 335–348.
M. Krivelevich, Triangle factors in random graphs, *Combinatorics, Probability & Computing* [**6**]{} (1997), 337–347.
M. Krivelevich, Embedding spanning trees in random graphs, *SIAM J. Discrete Math.* [**24**]{} (2010), 1495–1500.
D. Kühn and D. Osthus, Embedding large subgraphs into dense graphs, in *Surveys in Combinatorics* (S. Huczynska, J.D. Mitchell and C.M. Roney-Dougal eds.), *London Math. Soc. Lecture Notes* **365**, 137–167, Cambridge University Press, 2009.
D. Kühn and D. Osthus, The minimum degree threshold for perfect graph packings, *Combinatorica* **29** (2009), 65–107.
O. Riordan, Spanning subgraphs of random graphs, *Combinatorics, Probability & Computing* [**9**]{} (2000), 125–148.
V. Rödl, A. Ruciński, and E. Szemerédi, Perfect matchings in large uniform hypergraphs with large minimum collective degree. *J. Combinatorial Theory A* [**116**]{} (2009), 613–636.
Daniela Kühn & Deryk Osthus School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT UK
[*[E-mail addresses]{}:* ]{}
|
---
abstract: |
We present updated measurements of -violating asymmetries in the decays $\Bz\to\Dstarpm\Dmp$ and $\Bz\to\Dp\Dm$ using $(383\pm 4)
\times 10^{6} \BB$ pairs collected by the detector at the PEP-II $B$ factory. We determine the time-integrated asymmetry ${\mathcal{A}}_{\Dstarpm\Dmp}=0.12\pm 0.06\pm 0.02$, and the time-dependent asymmetry parameters to be $C_{\Dstarp\Dm} =0.18\pm
0.15\pm 0.04$, $S_{\Dstarp\Dm}=-0.79\pm 0.21\pm 0.06$, $C_{\Dstarm\Dp}
=0.23\pm 0.15\pm 0.04$, $S_{\Dstarm\Dp} =-0.44\pm 0.22\pm 0.06$, $C_{\Dp\Dm} =0.11\pm 0.22\pm 0.07$, and $S_{\Dp\Dm} =-0.54\pm 0.34\pm
0.06$, where the first uncertainty is statistical and the second is systematic.
author:
- 'B. Aubert'
- 'M. Bona'
- 'D. Boutigny'
- 'Y. Karyotakis'
- 'J. P. Lees'
- 'V. Poireau'
- 'X. Prudent'
- 'V. Tisserand'
- 'A. Zghiche'
- 'J. Garra Tico'
- 'E. Grauges'
- 'L. Lopez'
- 'A. Palano'
- 'G. Eigen'
- 'B. Stugu'
- 'L. Sun'
- 'G. S. Abrams'
- 'M. Battaglia'
- 'D. N. Brown'
- 'J. Button-Shafer'
- 'R. N. Cahn'
- 'Y. Groysman'
- 'R. G. Jacobsen'
- 'J. A. Kadyk'
- 'L. T. Kerth'
- 'Yu. G. Kolomensky'
- 'G. Kukartsev'
- 'D. Lopes Pegna'
- 'G. Lynch'
- 'L. M. Mir'
- 'T. J. Orimoto'
- 'M. T. Ronan'
- 'K. Tackmann'
- 'W. A. Wenzel'
- 'P. del Amo Sanchez'
- 'C. M. Hawkes'
- 'A. T. Watson'
- 'T. Held'
- 'H. Koch'
- 'B. Lewandowski'
- 'M. Pelizaeus'
- 'T. Schroeder'
- 'M. Steinke'
- 'D. Walker'
- 'D. J. Asgeirsson'
- 'T. Cuhadar-Donszelmann'
- 'B. G. Fulsom'
- 'C. Hearty'
- 'T. S. Mattison'
- 'J. A. McKenna'
- 'A. Khan'
- 'M. Saleem'
- 'L. Teodorescu'
- 'V. E. Blinov'
- 'A. D. Bukin'
- 'V. P. Druzhinin'
- 'V. B. Golubev'
- 'A. P. Onuchin'
- 'S. I. Serednyakov'
- 'Yu. I. Skovpen'
- 'E. P. Solodov'
- 'K. Yu. Todyshev'
- 'M. Bondioli'
- 'S. Curry'
- 'I. Eschrich'
- 'D. Kirkby'
- 'A. J. Lankford'
- 'P. Lund'
- 'M. Mandelkern'
- 'E. C. Martin'
- 'D. P. Stoker'
- 'S. Abachi'
- 'C. Buchanan'
- 'S. D. Foulkes'
- 'J. W. Gary'
- 'F. Liu'
- 'O. Long'
- 'B. C. Shen'
- 'L. Zhang'
- 'H. P. Paar'
- 'S. Rahatlou'
- 'V. Sharma'
- 'J. W. Berryhill'
- 'C. Campagnari'
- 'A. Cunha'
- 'B. Dahmes'
- 'T. M. Hong'
- 'D. Kovalskyi'
- 'J. D. Richman'
- 'T. W. Beck'
- 'A. M. Eisner'
- 'C. J. Flacco'
- 'C. A. Heusch'
- 'J. Kroseberg'
- 'W. S. Lockman'
- 'T. Schalk'
- 'B. A. Schumm'
- 'A. Seiden'
- 'D. C. Williams'
- 'M. G. Wilson'
- 'L. O. Winstrom'
- 'E. Chen'
- 'C. H. Cheng'
- 'F. Fang'
- 'D. G. Hitlin'
- 'I. Narsky'
- 'T. Piatenko'
- 'F. C. Porter'
- 'R. Andreassen'
- 'G. Mancinelli'
- 'B. T. Meadows'
- 'K. Mishra'
- 'M. D. Sokoloff'
- 'F. Blanc'
- 'P. C. Bloom'
- 'S. Chen'
- 'W. T. Ford'
- 'J. F. Hirschauer'
- 'A. Kreisel'
- 'M. Nagel'
- 'U. Nauenberg'
- 'A. Olivas'
- 'J. G. Smith'
- 'K. A. Ulmer'
- 'S. R. Wagner'
- 'J. Zhang'
- 'A. M. Gabareen'
- 'A. Soffer'
- 'W. H. Toki'
- 'R. J. Wilson'
- 'F. Winklmeier'
- 'Q. Zeng'
- 'D. D. Altenburg'
- 'E. Feltresi'
- 'A. Hauke'
- 'H. Jasper'
- 'J. Merkel'
- 'A. Petzold'
- 'B. Spaan'
- 'K. Wacker'
- 'T. Brandt'
- 'V. Klose'
- 'M. J. Kobel'
- 'H. M. Lacker'
- 'W. F. Mader'
- 'R. Nogowski'
- 'J. Schubert'
- 'K. R. Schubert'
- 'R. Schwierz'
- 'J. E. Sundermann'
- 'A. Volk'
- 'D. Bernard'
- 'G. R. Bonneaud'
- 'E. Latour'
- 'V. Lombardo'
- 'Ch. Thiebaux'
- 'M. Verderi'
- 'P. J. Clark'
- 'W. Gradl'
- 'F. Muheim'
- 'S. Playfer'
- 'A. I. Robertson'
- 'Y. Xie'
- 'M. Andreotti'
- 'D. Bettoni'
- 'C. Bozzi'
- 'R. Calabrese'
- 'A. Cecchi'
- 'G. Cibinetto'
- 'P. Franchini'
- 'E. Luppi'
- 'M. Negrini'
- 'A. Petrella'
- 'L. Piemontese'
- 'E. Prencipe'
- 'V. Santoro'
- 'F. Anulli'
- 'R. Baldini-Ferroli'
- 'A. Calcaterra'
- 'R. de Sangro'
- 'G. Finocchiaro'
- 'S. Pacetti'
- 'P. Patteri'
- 'I. M. Peruzzi'
- 'M. Piccolo'
- 'M. Rama'
- 'A. Zallo'
- 'A. Buzzo'
- 'R. Contri'
- 'M. Lo Vetere'
- 'M. M. Macri'
- 'M. R. Monge'
- 'S. Passaggio'
- 'C. Patrignani'
- 'E. Robutti'
- 'A. Santroni'
- 'S. Tosi'
- 'K. S. Chaisanguanthum'
- 'M. Morii'
- 'J. Wu'
- 'R. S. Dubitzky'
- 'J. Marks'
- 'S. Schenk'
- 'U. Uwer'
- 'D. J. Bard'
- 'P. D. Dauncey'
- 'R. L. Flack'
- 'J. A. Nash'
- 'M. B. Nikolich'
- 'W. Panduro Vazquez'
- 'M. Tibbetts'
- 'P. K. Behera'
- 'X. Chai'
- 'M. J. Charles'
- 'U. Mallik'
- 'N. T. Meyer'
- 'V. Ziegler'
- 'J. Cochran'
- 'H. B. Crawley'
- 'L. Dong'
- 'V. Eyges'
- 'W. T. Meyer'
- 'S. Prell'
- 'E. I. Rosenberg'
- 'A. E. Rubin'
- 'A. V. Gritsan'
- 'Z. J. Guo'
- 'C. K. Lae'
- 'A. G. Denig'
- 'M. Fritsch'
- 'G. Schott'
- 'N. Arnaud'
- 'J. Béquilleux'
- 'M. Davier'
- 'G. Grosdidier'
- 'A. Höcker'
- 'V. Lepeltier'
- 'F. Le Diberder'
- 'A. M. Lutz'
- 'S. Pruvot'
- 'S. Rodier'
- 'P. Roudeau'
- 'M. H. Schune'
- 'J. Serrano'
- 'V. Sordini'
- 'A. Stocchi'
- 'W. F. Wang'
- 'G. Wormser'
- 'D. J. Lange'
- 'D. M. Wright'
- 'I. Bingham'
- 'C. A. Chavez'
- 'I. J. Forster'
- 'J. R. Fry'
- 'E. Gabathuler'
- 'R. Gamet'
- 'D. E. Hutchcroft'
- 'D. J. Payne'
- 'K. C. Schofield'
- 'C. Touramanis'
- 'A. J. Bevan'
- 'K. A. George'
- 'F. Di Lodovico'
- 'W. Menges'
- 'R. Sacco'
- 'G. Cowan'
- 'H. U. Flaecher'
- 'D. A. Hopkins'
- 'S. Paramesvaran'
- 'F. Salvatore'
- 'A. C. Wren'
- 'D. N. Brown'
- 'C. L. Davis'
- 'J. Allison'
- 'N. R. Barlow'
- 'R. J. Barlow'
- 'Y. M. Chia'
- 'C. L. Edgar'
- 'G. D. Lafferty'
- 'T. J. West'
- 'J. I. Yi'
- 'J. Anderson'
- 'C. Chen'
- 'A. Jawahery'
- 'D. A. Roberts'
- 'G. Simi'
- 'J. M. Tuggle'
- 'G. Blaylock'
- 'C. Dallapiccola'
- 'S. S. Hertzbach'
- 'X. Li'
- 'T. B. Moore'
- 'E. Salvati'
- 'S. Saremi'
- 'R. Cowan'
- 'D. Dujmic'
- 'P. H. Fisher'
- 'K. Koeneke'
- 'G. Sciolla'
- 'S. J. Sekula'
- 'M. Spitznagel'
- 'F. Taylor'
- 'R. K. Yamamoto'
- 'M. Zhao'
- 'Y. Zheng'
- 'S. E. Mclachlin'
- 'P. M. Patel'
- 'S. H. Robertson'
- 'A. Lazzaro'
- 'F. Palombo'
- 'J. M. Bauer'
- 'L. Cremaldi'
- 'V. Eschenburg'
- 'R. Godang'
- 'R. Kroeger'
- 'D. A. Sanders'
- 'D. J. Summers'
- 'H. W. Zhao'
- 'S. Brunet'
- 'D. Côté'
- 'M. Simard'
- 'P. Taras'
- 'F. B. Viaud'
- 'H. Nicholson'
- 'G. De Nardo'
- 'F. Fabozzi'
- 'L. Lista'
- 'D. Monorchio'
- 'C. Sciacca'
- 'M. A. Baak'
- 'G. Raven'
- 'H. L. Snoek'
- 'C. P. Jessop'
- 'J. M. LoSecco'
- 'G. Benelli'
- 'L. A. Corwin'
- 'K. Honscheid'
- 'H. Kagan'
- 'R. Kass'
- 'J. P. Morris'
- 'A. M. Rahimi'
- 'J. J. Regensburger'
- 'Q. K. Wong'
- 'N. L. Blount'
- 'J. Brau'
- 'R. Frey'
- 'O. Igonkina'
- 'J. A. Kolb'
- 'M. Lu'
- 'R. Rahmat'
- 'N. B. Sinev'
- 'D. Strom'
- 'J. Strube'
- 'E. Torrence'
- 'N. Gagliardi'
- 'A. Gaz'
- 'M. Margoni'
- 'M. Morandin'
- 'A. Pompili'
- 'M. Posocco'
- 'M. Rotondo'
- 'F. Simonetto'
- 'R. Stroili'
- 'C. Voci'
- 'E. Ben-Haim'
- 'H. Briand'
- 'G. Calderini'
- 'J. Chauveau'
- 'P. David'
- 'L. Del Buono'
- 'Ch. de la Vaissière'
- 'O. Hamon'
- 'Ph. Leruste'
- 'J. Malclès'
- 'J. Ocariz'
- 'A. Perez'
- 'L. Gladney'
- 'M. Biasini'
- 'R. Covarelli'
- 'E. Manoni'
- 'C. Angelini'
- 'G. Batignani'
- 'S. Bettarini'
- 'M. Carpinelli'
- 'R. Cenci'
- 'A. Cervelli'
- 'F. Forti'
- 'M. A. Giorgi'
- 'A. Lusiani'
- 'G. Marchiori'
- 'M. A. Mazur'
- 'M. Morganti'
- 'N. Neri'
- 'E. Paoloni'
- 'G. Rizzo'
- 'J. J. Walsh'
- 'M. Haire'
- 'J. Biesiada'
- 'P. Elmer'
- 'Y. P. Lau'
- 'C. Lu'
- 'J. Olsen'
- 'A. J. S. Smith'
- 'A. V. Telnov'
- 'E. Baracchini'
- 'F. Bellini'
- 'G. Cavoto'
- 'A. D’Orazio'
- 'D. del Re'
- 'E. Di Marco'
- 'R. Faccini'
- 'F. Ferrarotto'
- 'F. Ferroni'
- 'M. Gaspero'
- 'P. D. Jackson'
- 'L. Li Gioi'
- 'M. A. Mazzoni'
- 'S. Morganti'
- 'G. Piredda'
- 'F. Polci'
- 'F. Renga'
- 'C. Voena'
- 'M. Ebert'
- 'T. Hartmann'
- 'H. Schröder'
- 'R. Waldi'
- 'T. Adye'
- 'G. Castelli'
- 'B. Franek'
- 'E. O. Olaiya'
- 'S. Ricciardi'
- 'W. Roethel'
- 'F. F. Wilson'
- 'R. Aleksan'
- 'S. Emery'
- 'M. Escalier'
- 'A. Gaidot'
- 'S. F. Ganzhur'
- 'G. Hamel de Monchenault'
- 'W. Kozanecki'
- 'G. Vasseur'
- 'Ch. Yèche'
- 'M. Zito'
- 'X. R. Chen'
- 'H. Liu'
- 'W. Park'
- 'M. V. Purohit'
- 'J. R. Wilson'
- 'M. T. Allen'
- 'D. Aston'
- 'R. Bartoldus'
- 'P. Bechtle'
- 'N. Berger'
- 'R. Claus'
- 'J. P. Coleman'
- 'M. R. Convery'
- 'J. C. Dingfelder'
- 'J. Dorfan'
- 'G. P. Dubois-Felsmann'
- 'W. Dunwoodie'
- 'R. C. Field'
- 'T. Glanzman'
- 'S. J. Gowdy'
- 'M. T. Graham'
- 'P. Grenier'
- 'C. Hast'
- 'T. Hryn’ova'
- 'W. R. Innes'
- 'J. Kaminski'
- 'M. H. Kelsey'
- 'H. Kim'
- 'P. Kim'
- 'M. L. Kocian'
- 'D. W. G. S. Leith'
- 'S. Li'
- 'S. Luitz'
- 'V. Luth'
- 'H. L. Lynch'
- 'D. B. MacFarlane'
- 'H. Marsiske'
- 'R. Messner'
- 'D. R. Muller'
- 'C. P. O’Grady'
- 'I. Ofte'
- 'A. Perazzo'
- 'M. Perl'
- 'T. Pulliam'
- 'B. N. Ratcliff'
- 'A. Roodman'
- 'A. A. Salnikov'
- 'R. H. Schindler'
- 'J. Schwiening'
- 'A. Snyder'
- 'J. Stelzer'
- 'D. Su'
- 'M. K. Sullivan'
- 'K. Suzuki'
- 'S. K. Swain'
- 'J. M. Thompson'
- 'J. Va’vra'
- 'N. van Bakel'
- 'A. P. Wagner'
- 'M. Weaver'
- 'W. J. Wisniewski'
- 'M. Wittgen'
- 'D. H. Wright'
- 'A. K. Yarritu'
- 'K. Yi'
- 'C. C. Young'
- 'P. R. Burchat'
- 'A. J. Edwards'
- 'S. A. Majewski'
- 'B. A. Petersen'
- 'L. Wilden'
- 'S. Ahmed'
- 'M. S. Alam'
- 'R. Bula'
- 'J. A. Ernst'
- 'V. Jain'
- 'B. Pan'
- 'M. A. Saeed'
- 'F. R. Wappler'
- 'S. B. Zain'
- 'W. Bugg'
- 'M. Krishnamurthy'
- 'S. M. Spanier'
- 'R. Eckmann'
- 'J. L. Ritchie'
- 'A. M. Ruland'
- 'C. J. Schilling'
- 'R. F. Schwitters'
- 'J. M. Izen'
- 'X. C. Lou'
- 'S. Ye'
- 'F. Bianchi'
- 'F. Gallo'
- 'D. Gamba'
- 'M. Pelliccioni'
- 'M. Bomben'
- 'L. Bosisio'
- 'C. Cartaro'
- 'F. Cossutti'
- 'G. Della Ricca'
- 'L. Lanceri'
- 'L. Vitale'
- 'V. Azzolini'
- 'N. Lopez-March'
- 'F. Martinez-Vidal'
- 'D. A. Milanes'
- 'A. Oyanguren'
- 'J. Albert'
- 'Sw. Banerjee'
- 'B. Bhuyan'
- 'K. Hamano'
- 'R. Kowalewski'
- 'I. M. Nugent'
- 'J. M. Roney'
- 'R. J. Sobie'
- 'J. J. Back'
- 'P. F. Harrison'
- 'J. Ilic'
- 'T. E. Latham'
- 'G. B. Mohanty'
- 'M. Pappagallo'
- 'H. R. Band'
- 'X. Chen'
- 'S. Dasu'
- 'K. T. Flood'
- 'J. J. Hollar'
- 'P. E. Kutter'
- 'Y. Pan'
- 'M. Pierini'
- 'R. Prepost'
- 'S. L. Wu'
- 'H. Neal'
title: 'Measurement of -Violating Asymmetries in $\Bz\to D^{(*)\pm}\Dmp$'
---
-PUB-07/024\
SLAC-PUB-12506\
[^1]
In the Standard Model (SM), violation arises from a complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix, $V$ [@CKM]. Measurements of asymmetries in $\Bz\to(\ccbar)K^{(*)0}$ decays [@conjugate] by the [@Aubert:2002ic] and Belle [@Abe:2002px] collaborations have firmly established this effect and precisely determined the parameter , where $\beta$ is ${\rm arg}[-V_{\rm cd}V^*_{\rm
cb}/V_{\rm td}V^*_{\rm tb}]$. Another way to measure is to use decays whose amplitudes are dominated by a tree-level, color-allowed $b\to\ccbar d$ transition, such as $\Bz\to D^{(*)\pm}\Dmp$. Within the framework of the SM, the time-dependent -asymmetries of $\Bz\to
D^{(*)\pm}\Dmp$ are directly related to when corrections due to penguin diagram contributions are neglected. The penguin-induced corrections have been estimated in models based on the factorization approximation and heavy quark symmetry and are predicted to be a few percent [@Xing:1998ca; @Xing:1999yx]. However, contributions from non-SM processes may lead to a large shift [@Grossman:1996ke]. A significant deviation in the measurement from that of the $\Bz\to(\ccbar)K^{(*)0}$ decays would be evidence involving new physics beyond the SM.
Studies of the violation in $b\to\ccbar d$ transitions have been carried out by both the and Belle collaborations. Most recently, the Belle collaboration reported evidence of large direct violation in $\Bz\to\Dp\Dm$ where $C_{\Dp\Dm} = -0.91\pm 0.23\pm
0.06$ [@Fratina:2007zk], in contradiction to the SM expectation. However, such a large direct violation has not been observed in previous measurements with $\Bz\to D^{(*)\pm} D^{(*)\mp}$ decays, involving the same quark-level weak decay [@Aubert:2005rn; @Aubert:2005hs; @Aushev:2004uc; @Miyake:2005qb].
In this Letter, we present an updated measurement of -violating asymmetries in the decays $\Bz\to\Dstarp\Dm$, $\Bz\to\Dstarm\Dp$ and $\Bz\to\Dp\Dm$. The data used in this analysis comprise $(383\pm 4)
\times 10^{6}$ decays collected by the detector at the PEP-II storage rings. The detector is described in detail elsewhere [@Aubert:2001tu]. Monte Carlo (MC) simulation based on GEANT4 [@Agostinelli:2002hh] is used to validate the analysis procedure and to study the relevant backgrounds.
The decay rate $f_+ (f_-)$ for a neutral $B$ meson decay to a common final state accompanied by a $\Bz(\Bzb)$ tag is given by $$\begin{aligned}
f_\pm(\deltat) = &{\rm e}^{ - | \deltat |/\tau_{B^0}}/4\tau_{B^0} \left\{ (1\mp\Delta w)
\pm (1-2w) \times \right. \nonumber \\
&\left. \left[ S\sin(\Delta m_d\deltat)
-C\cos(\Delta m_d\deltat)\right] \right\},
\label{eq:CP}\end{aligned}$$ where $\Delta t \equiv t_{\rm rec} - t_{\rm tag}$ is the difference between the proper decay time of the reconstructed $B$ meson ($B_{\rm
rec}$) and that of the tagging $B$ meson ($B_{\rm tag}$), $\tau_{\Bz}$ is the lifetime, and is the difference between the heavy and light mass eigenstates determined from the -oscillation frequency [@Yao:2006px]. The average mistag probability $w$ describes the effect of incorrect tags, and $\Delta w$ is the difference between the mistag probabilities for $\Bz$ and $\Bzb$. Since $\Dstarp\Dm$ and $\Dstarm\Dp$ are not -eigenstates, we can define a time-integrated asymmetry $\mathcal{A}_{\Dstarpm\Dmp}$ between the rate of $\Bz\to\Dstarp\Dm$ and $\Bz\to\Dstarm\Dp$, calculated as: $$\mathcal{A}_{\Dstarpm\Dmp}=
\frac{N_{\Dstarp\Dm}-N_{\Dstarm\Dp}}
{N_{\Dstarp\Dm}+N_{\Dstarm\Dp}},$$ where $N$ is the signal event yield.
For $\Bz\to\Dstarpm\Dmp$, the general relations are $S_{\Dstarpm\Dmp}=-\sqrt{1-C^2_{\Dstarpm\Dmp}}\sin(2\beta_{\rm
eff}\pm\delta)$, where $\delta$ is the strong phase difference between $\Bz\to\Dstarp\Dm$ and $\Bz\to\Dstarm\Dp$ [@Aleksan:1993qk]. Under the assumption of negligible penguin contribution, $\beta_{\rm
eff}=\beta$, $\mathcal{A}_{\Dstarpm\Dmp}= 0$ and $C_{\Dstarp\Dm}=-C_{\Dstarm\Dp}$. For $\Bz\to\Dp\Dm$ and in the case of negligible penguin contribution, $C_{\Dp\Dm}$ measures direct violation and is zero, while $S_{\Dp\Dm}$ is $-\stwob$.
The selections of $\Bz\to\Dstarpm\Dmp$ and $\Bz\to\Dp\Dm$ candidates are similar to those of our previous analysis [@Aubert:2005hs]. We reconstruct in its decay to $\Dz\pi^+$. We reconstruct candidates for and mesons in the modes $\Dz\to\Km\pip$, $\Km\pip\piz$, $\Km\pip\pip\pim$, $\KS\pip\pim$ and $\Dp\to\Km\pip\pip$, $\KS\pip$. We reconstruct $\Bz\to\Dp\Dm$ candidates only through the decay $\Dpm\to K^{\mp}\pi^{\pm}\pi^{\pm}$. We require the reconstructed masses of the and candidates to be within 20[${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}of their respective nominal masses [@Yao:2006px], except for the $\Dz\to\Km\pip\piz$ candidate, where we use a looser requirement of 40[${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}. We apply a mass-constrained fit to the selected and candidates and combine candidates with a track, with momentum below 450[${\mathrm{\,Me\kern -0.1em V\!/}c}$]{}in the frame, to form candidates. We reconstruct the candidates from two oppositely charged tracks with an invariant mass within 20[${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}of the nominal mass [@Yao:2006px]. The $\chi^2$ probability of the track vertex fit must be greater than $0.1\,\%$. We require charged kaon candidates to be identified as such using a likelihood technique based on the Cherenkov angle measured by the Cherenkov detector and the ionization energy loss measured by the charged-particle tracking systems [@Aubert:2001tu]. We form neutral pion candidates from two photons detected in the electromagnetic calorimeter [@Aubert:2001tu], each with energy above 30[$\mathrm{\,Me\kern -0.1em V}$]{}. The invariant mass of the pair must be within 30[${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}of the nominal mass [@Yao:2006px], and we require their summed energy to be greater than 200[$\mathrm{\,Me\kern -0.1em V}$]{}. In addition, we further apply a mass-constrained fit to the candidates.
To suppress the $\epem\to\qqbar \;(q=u,d,s,\,{\rm and}\; c)$ continuum background, we exploit the contrast between the spherical shape of events and the more jet-like nature of continuum events. We require the ratio of the second to the zeroth order Fox-Wolfram moments [@Fox:1978vu] to be less than 0.6. We also use a Fisher discriminant, constructed as an optimized linear combination of 11 event shape variables [@Asner:1995hc]: the momentum flow in nine concentric cones around the thrust axis of the reconstructed $\Bz$ candidate, the angle between that thrust axis and the beam axis, and the angle between the line-of-flight of the $\Bz$ candidate and the beam axis. In addition, we employ a combined $D$ flight-length significance variable, derived from the sum of flight lengths of the two $D$ candidates [@Aubert:2006ia], to reduce background.
For each $\Bz\to D^{(*)\pm}\Dmp$ candidate, we construct a likelihood function $\mathcal{L}_{\rm{mass}}$ from the masses and mass uncertainties of the $D$ and candidates [@Aubert:2006ia]. The $D$ mass resolution is modeled by a Gaussian whose variance is determined on a candidate-by-candidate basis from its mass uncertainty before the mass-constrained fit. The -$D$ mass difference resolution is modeled by the sum of two Gaussian distributions whose parameters are determined from simulated events. The values of $\mathcal{L}_{\rm{mass}}$ and $\Delta E\equiv E_B^*-E_{\rm{Beam}}$, the difference between the candidate energy $E_B^*$ and the beam energy $E_{\rm{Beam}}$ in the frame, are used to reduce the combinatoric background. From the simulated events, we optimize the maximum allowed values of $-\ln\mathcal{L}_{\rm{mass}}$ and $|\Delta
E|$ for each individual final state to obtain the highest expected signal significance.
We extract the signal yield from the events satisfying the selection criteria using the energy-substituted mass, $m_{\rm{ES}}\equiv
\sqrt{E^2_{\rm{Beam}}-p^{*2}_B}$, where $p^*_B$ is the $B^0$ candidate momentum in the frame. We select the candidates that have $m_{\rm{ES}}\ge5.23\,{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$. On average, we have $1.5$ and $1.1$ candidates per event for $\Bz\to\Dstarpm\Dmp$ and $\Bz\to\Dp\Dm$ respectively. If more than one candidate is reconstructed in an event, we select the candidate with the smallest value of $-\ln\mathcal{L}_{\rm{mass}}$. Studies using MC samples show that this procedure results in the selection of the correct candidate more than 95% of the time.
We perform an unbinned maximum likelihood fit to the $\mes$ and distributions to extract the asymmetries. We fit the events from $\Bz\to\Dstarp\Dm$ and $\Bz\to\Dstarm\Dp$ decays simultaneously. The probability density function (PDF) of the distribution consists of a Gaussian for the signal and a threshold function [@Albrecht:1990cs] for the combinatorial background. We expect some background events to peak in the signal region due to cross feed from other decay modes. We estimate the fraction of events in the signal Gaussian due to this peaking background to be $(8.8\pm4.4)\,\%$ for $\Bz\to\Dstarpm\Dmp$ and $(4.8\pm7.4)\,\%$ for $\Bz\to\Dp\Dm$ using detailed MC simulations of inclusive decays.
The technique used to fit the distribution is analogous to that used in previous measurements described in Ref. [@Aubert:2002rg; @Aubert:2004zt]. We use information from the other meson in the event to tag the flavor of the fully reconstructed $\Bz\to D^{(*)\pm}\Dmp$ candidate [@Aubert:2002rg]. The signal $\Delta t$ PDF in Eq. \[eq:CP\] is convolved with an empirical $\Delta t$ resolution function [@Aubert:2002rg]. The is calculated from the measured separation $\Delta z$ between the decay vertices of $B_{\rm
rec}$ and $B_{\rm tag}$ along the collision ($z$) axis [@Aubert:2002rg]. The $B_{\rm tag}$ decay vertex is determined by fitting charged tracks not belonging to the $B_{\rm
rec}$ candidate to a common vertex, employing constraints from the beam spot location and the $B_{\rm rec}$ momentum [@Aubert:2002rg]. Only events with a $\Delta t$ uncertainty less than $2.5\,\mbox{ps}$ and a measured $|\Delta t|$ less than $20\,\mbox{ps}$ are accepted for the fit to the distribution. Both the signal mistag probability and the $\Delta t$ resolution function are determined from a large sample of neutral $B$ decays to flavor eigenstates, $B_{\rm flav}$. The combinatoric background $\Delta t$ distributions are parameterized with an empirical description that includes zero and non-zero lifetime components [@Aubert:2002rg]. The non-zero lifetime background is allowed to have effective asymmetries, and these float in the likelihood fit. By default, we assume that the peaking backgrounds have the same $\Delta t$ PDF as the signal but zero asymmetries.
The fits to the data yield $280\pm 19$ signal events for $\Bz\to\Dstarp\Dm$, $219\pm 18$ signal events for $\Bz\to\Dstarm\Dp$, and $131\pm 14$ signal events for $\Bz\to\Dp\Dm$, where the quoted uncertainties are statistical only. In the region of $m_{\rm{ES}}> 5.27\,{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$, the signal purity is approximately 41% for $\Bz\to\Dstarp\Dm$, 34% for $\Bz\to\Dstarm\Dp$, and 46% for $\Bz\to\Dp\Dm$. The fitted violating parameters are $$\begin{aligned}
\mathcal{A}_{\Dstarpm\Dmp} &= \;\;\;0.12\pm 0.06 \pm 0.02 \nonumber \\
C_{\Dstarp\Dm} &= \;\;\;0.18\pm 0.15 \pm 0.04 \nonumber \\
S_{\Dstarp\Dm} &= -0.79\pm 0.21 \pm 0.06 \nonumber \\
C_{\Dstarm\Dp} &= \;\;\;0.23\pm 0.15 \pm 0.04 \nonumber \\
S_{\Dstarm\Dp} &= -0.44\pm 0.22 \pm 0.06 \nonumber \\
C_{\Dp\Dm} &= \;\;\;0.11\pm 0.22 \pm 0.07 \nonumber \\
S_{\Dp\Dm} &= -0.54\pm 0.34 \pm 0.06 \,,
$$ where the first uncertainty is statistical and the second is systematic.
Projections of the fits onto for the three different samples are shown in Figure \[fig:mesplots\]. Figure \[fig:dtplots\] shows the $\Delta t$ distributions and asymmetries in yields between events with $B^0$ and $\Bzb$ tags, overlaid with the projection of the likelihood fit result. As a cross check, we repeat the fit by allowing the lifetime to float. The obtained lifetime is in good agreement with its world average [@Yao:2006px].
The systematic uncertainty of the time-integrated -asymmetry $\mathcal{A}_{\Dstarpm\Dmp}$ is dominated by the potential differences in the reconstruction efficiencies of the positively and negatively charged tracks ($0.014$). Other sources that contribute to the systematic error include the estimate of the peaking background fraction ($<0.001$), the uncertainty in the resolution for the $\Bz\to\Dstarpm\Dmp$ signal events ($0.005$), and a possible fit bias ($0.004$).
The systematic uncertainties on $C$ and $S$ are evaluated separately for each of the decay modes. Their sources and estimates are summarized in Table \[tab:systematics\]. The systematic uncertainties arise from the amount of possible background that tends to peak under the signal and its asymmetry, the assumed parameterization of the $\Delta t$ resolution function, the possible differences between the $B_{\rm flav}$ and signal mistag fractions, the knowledge of the event-by-event beam-spot position, the uncertainties from the finite MC sample used, the possible interference between the suppressed $\bar{b}\to\bar{u}c\bar{d}$ and the favored $b\to c\bar{u}d$ amplitudes in some tag-side decays [@Long:2003wq], and the uncertainty in the resolution for the signal events. All of the systematic uncertainties are found to be much smaller than the statistical uncertainties.
Source $C_{\Dstarp\Dm}$ $S_{\Dstarp\Dm}$ $C_{\Dstarm\Dp}$ $S_{\Dstarm\Dp}$ $C_{\Dp\Dm}$ $S_{\Dp\Dm}$
---------------------------------- ------------------ ------------------ ------------------ ------------------ -------------- --------------
Peaking backgrounds 0.026 0.041 0.027 0.031 0.044 0.042
resolution parameterization 0.011 0.021 0.013 0.012 0.015 0.020
Mistag fraction differences 0.014 0.011 0.016 0.012 0.023 0.013
Beam-spot position 0.004 0.006 0.007 0.036 0.005 0.002
$\deltamd$, $\tau_B$ 0.002 0.003 0.003 0.004 0.001 0.004
MC statistics 0.011 0.015 0.011 0.015 0.036 0.023
Tag-side interference and others 0.016 0.025 0.017 0.020 0.020 0.013
Total 0.037 0.056 0.040 0.056 0.066 0.055
Since $\Dstarp\Dm$ and $\Dstarm\Dp$ are not -eigenstates, it is also illustrative to express the measured -violating parameters $C$ and $S$ in a slightly different parametrization [@Aubert:2003wr]: $C_{\Dstar D}=(C_{\Dstarp\Dm}+C_{\Dstarm\Dp})/2$, $\Delta C_{\Dstar
D}=(C_{\Dstarp\Dm}-C_{\Dstarm\Dp})/2$, $S_{\Dstar
D}=(S_{\Dstarp\Dm}+S_{\Dstarm\Dp})/2$ and $\Delta S_{\Dstar
D}=(S_{\Dstarp\Dm}-S_{\Dstarm\Dp})/2$. The quantities $C_{\Dstar D}$ and $S_{\Dstar D}$ parametrize flavor-dependent direct violation, and mixing-induced violation related to the angle $\beta$, respectively. The parameters $\Delta C_{\Dstar D}$ and $\Delta
S_{\Dstar D}$ are insensitive to violation. $\Delta C_{\Dstar D}$ describes the asymmetry between the rates $\Gamma(\Bz\to\Dstarp\Dm)+\Gamma(\Bzb\to\Dstarm\Dp)$ and $\Gamma(\Bz\to\Dstarm\Dp)+\Gamma(\Bzb\to\Dstarp\Dm)$, while $\Delta
S_{\Dstar D}$ is related to the strong phase difference, $\delta$. We find $$\begin{aligned}
C_{\Dstar D} &= \;\;\;0.21\pm 0.11 \pm 0.03 \nonumber \\
S_{\Dstar D} &= -0.62\pm 0.15 \pm 0.04 \nonumber \\
\Delta C_{\Dstar D} &= -0.02\pm 0.11 \pm 0.03 \nonumber \\
\Delta S_{\Dstar D} &= -0.17\pm 0.15 \pm 0.04 \,,
$$ where the first uncertainty is statistical and the second is systematic.
In summary, this letter reports updated measurements of the violating asymmetries for the decays $\Bz\to\Dstarpm\Dmp$ and $\Bz\to\Dp\Dm$. These measurements supersede the previous results [@Aubert:2005hs], with a more than $50\,\%$ reduction in the statistical uncertainties. The time-dependent asymmetries are consistent with the SM predictions within their statistical uncertainties. We do not see evidence of large direct violation in the decay $\Bz\to\Dp\Dm$ as reported by the Belle Collaboration [@Fratina:2007zk].
We are grateful for the excellent luminosity and machine conditions provided by our 2 colleagues, and for the substantial dedicated effort from the computing organizations that support . The collaborating institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MIST (Russia), MEC (Spain), and STFC (United Kingdom). Individuals have received support from the Marie Curie EIF (European Union) and the A. P. Sloan Foundation.
[99]{}
N. Cabibbo, , 531 (1963); M. Kobayashi and T. Maskawa, [[Prog. Theor. Phys. [****49****]{}]{}]{}, 652 (1973).
We imply charge conjugate modes throughout the paper. Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. Lett. [**89**]{}, 201802 (2002).
Belle Collaboration, K. Abe [*et al.*]{}, Phys. Rev. D [**66**]{}, 071102 (2002).
Z. Z. Xing, Phys. Lett. B [**443**]{}, 365 (1998).
Z. Z. Xing, Phys. Rev. D [**61**]{}, 014010 (2000).
Y. Grossman and M. P. Worah, Phys. Lett. B [**395**]{}, 241 (1997).
Belle Collaboration, S. Fratina [*et al.*]{}, Phys. Rev. Lett. [**98**]{}, 221802 (2007).
Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 151804 (2005).
Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 131802 (2005).
Belle Collaboration, T. Aushev [*et al.*]{}, Phys. Rev. Lett. [**93**]{}, 201802 (2004).
Belle Collaboration, H. Miyake [*et al.*]{}, Phys. Lett. B [**618**]{}, 34 (2005).
Collaboration, B. Aubert [*et al.*]{}, Nucl. Instr. Meth. Phys. Res., Sect. A [**479**]{}, 1 (2002).
GEANT4 Collaboration, S. Agostinelli [*et al.*]{}, Nucl. Instr. Meth. Phys. Res., Sect. A [**506**]{}, 250 (2003).
W. M. Yao [*et al.*]{} (Particle Data Group), J. Phys. G [**33**]{} (2006) 1.
R. Aleksan, A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys. Lett. B [**317**]{}, 173 (1993).
G. C. Fox and S. Wolfram, Phys. Rev. Lett. [**41**]{}, 1581 (1978).
CLEO Collaboration, D. M. Asner [*et al.*]{}, Phys. Rev. D [**53**]{}, 1039 (1996).
Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. D [**73**]{}, 112004 (2006).
ARGUS Collaboration, H. Albrecht [*et al.*]{}, Z. Phys. C [**48**]{}, 543 (1990).
Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. D [**66**]{}, 032003 (2002).
Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. Lett. [**94**]{}, 161803 (2005).
O. Long, M. Baak, R. N. Cahn, and D. Kirkby, Phys. Rev. D [**68**]{}, 034010 (2003).
Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. Lett. [**91**]{}, 201802 (2003).
[^1]: Deceased
|
---
abstract: 'The spectrum from the black hole X-ray transient GRO J1655-40. obtained using the $Chandra$ High Energy Transmission Grating (HETG) in 2005 is notable as a laboratory for the study of warm absorbers, and for the presence of many lines from odd-$Z$ elements between Na and Co (and Ti and Cr) not previously observed in X-rays. We present synthetic spectral models which can be used to constrain these element abundances and other parameters describing the outflow from the warm absorber in this object. We present results of fitting to the spectrum using various tools and techniques, including automated line fitting, phenomenological models, and photoionization modeling. We show that the behavior of the curves of growth of lines from H-like and Li-like ions indicate that the lines are either saturated or affected by filling-in from scattered or a partially covered continuum source. We confirm the conclusion of previous work by [@Mill06] and [@Mill08] which shows that the ionization conditions are not consistent with wind driving due to thermal expansion. The spectrum provides the opportunity to measure abundances for several elements not typically observable in the X-ray band. These show a pattern of enhancement for iron peak elements, and solar or sub-solar values for elements lighter than calcium. Models show that this is consistent with enrichment by a core-collapse supernova. We discuss the implications of these values for the evolutionary history of this system.'
author:
- 'T. R. Kallman,M. A. Bautista, Stephane Goriely,Claudio Mendoza, Jon M. Miller,Patrick Palmeri, Pascal Quinet,John Raymond'
title: 'Spectrum Synthesis Modeling of the X-ray Spectrum of GRO J1655-40 Taken During the 2005 Outburst.'
---
Introduction.
=============
X-ray spectra reveal that warm absorbers (absorption by partially ionized gas) are a common feature in compact objects. Although warm absorbers were first detected in the spectra of active galaxies using the $Einstein$ satellite [@Halp84], $Chandra$ and $XMM-Newton$ grating observations have shown that these occur in X-ray binaries as well, and information about the dynamics and other properties of this gas can have important implications for our understanding of accretion in these systems. A notable example is the 900 ks observation using the $Chandra$ High Energy Transmission Grating (HETG) of the Seyfert galaxy NGC 3783 [@Kasp02], but this has been surpassed in signal-to-noise by the 2005 spectrum of the galactic black hole transient GRO J1655-40 [@Mill06]. The statistical quality of this spectrum makes it one of the best available for testing and refinement of models for warm absorber flows.
GRO J1655-40 was discovered using BATSE onboard the Compton Gamma Ray Observatory [@Harm95]. Radio observations show apparently superluminal jets [@Hjel95]. Optical observations obtained during quiescence show the companion star is a F3-5 giant or sub-giant in a 2.62 day orbit around a 5-8 $M_\odot$ compact object [@Oros97]. Deep absorption dips have been observed [@Balu01] suggesting inclination $\sim 70^\circ$ [@Vand98]. GRO J1655-40 shows the highest-frequency quasi-periodic oscillations (QPOs) seen in a black hole [@Stro01]. Narrow X-ray absorption lines from highly ionized Fe were detected by @Yama01 and @Ueda98. Similar features were detected from all the bright dipping low mass X-ray binaries (LMXBs) observed with XMM-Newton [@Diaz06], and from the LMXB GX 13+1 [@Ueda01]. This suggests that ionized absorbers are a common feature of LMXBs, although they may not be detected in objects viewed at lower inclination.
The properties of the GRO J1655-40 warm absorber have been explored by @Mill06 and @Mill08. The spectrum resembles warm absorbers observed from other compact objects in that the lines are blueshifted, and that the inferred Doppler blueshift velocities are in the range 300 – 1600 km/s. The lines are identified primarily as H- and He-like species of elements with nuclear charge $8\leq Z \leq 28$, and there is no clear evidence for absorption by low ionization material such as the iron M-shell UTA [@Beha03]. The unprecedented signal-to-noise may account for the presence of lines from many trace elements previously not detected in X-rays, essentially all elements between Na and Co. The detection of the 11.92 Å line, arising from the $2p_{3/2}$ metastable level of Fe XXII, implies a gas density $\geq 10^{14}$ cm$^{-3}$. The spectrum is richer in features than another spectrum taken earlier in the same outburst, which showed only absorption by Fe XXVI L$\alpha$. The reason for this richness may be due to the higher column density of absorber and to the softer continuum during this particular observation, although this has not been tested quantitatively.
The observed line strengths can be used to infer the ionization balance, i.e. the ratio of abundances of H-like, He-like and lower ion stages for various elements. Models for the ionization balance in the wind then yield the ionization parameter: the ratio of the ionizing flux to the gas density, and the density is constrained by the Fe XXII line detection. This, together with the observed luminosity and the observed outflow speed, led @Mill06 to conclude that the radius where the outflow originates is too small to allow a wind driven by thermal pressure. That is, the likely ion thermal speeds in the gas are less than the escape velocity at the inferred radius. On the other hand, the mass flux can be inferred from the line strengths, and the estimated rate exceeds what is expected from outflows driven by radiation pressure. This implies that the outflow must be driven by a different mechanism, such as magnetic forces, but this result has been controversial [@Netz06].
The mass flux in the wind is important for our understanding of the mass and energy budget in accreting compact objects and it is clear that accurate models for the ionization balance and synthetic spectrum are needed in order to reliably determine the properties of the wind. It is the goal of this paper to study this, and several issues which were not considered by @Mill06 [@Mill08]: (1) What element abundances are required to account for the lines from the many iron peak elements observed in the spectrum, and what might this tell us about the origin of the gas in the wind and accretion flow? This is the only spectrum obtained from a warm absorber which clearly detects lines from odd-$Z$ elements with $Z \geq 10$ and from iron peak elements other than Fe and Ni. The abundances are of interest since they may contain clues to the evolutionary origin of the system; @Isra99 find evidence for enhanced O/H, Mg/H, Si/H, S/H, and relative to solar, but not Fe/H, and suggest that this may be due to enrichment by the supernova which produced the compact object. (2) What is the possible influence of radiative transfer effects (line emission, partial covering, or finite energy resolution) on the inferred wind properties? Analyses of X-ray warm absorber flows so far have not attempted to account in detail for these processes (although hints to their importance in AGN come from the UV; @Gabe05). They could systematically skew the derived column densities and ionization conditions. It is straightforward, although time consuming, to construct models which will test these effects in various scenarios. (3) Are there correlations between line shape or centroid and the ionization conditions where that line is expected to dominate? A flow with a predominantly ordered velocity field and a central radiation source is likely to show a gradient in ionization balance with position, and therefore with velocity. This should be manifest as correlations between line width or offset with the ionization degree of its parent ion. This effect is not found in Seyfert galaxy warm absorbers [@Kasp02], but examination of the GRO J1655-40 spectrum shows differences among the line profile shapes. After careful attention to item (2) above, we can test this quantitatively.
In the remainder of this paper we present our model fits and interpretation of the $Chandra$ HETG spectrum of GRO J1655-40, attempting to address the above questions. This includes various fitting techniques, testing of radiative transfer effects, and discussion of the constraints on element abundances. Finally, we discuss the implications of these results for the dynamics of the outflow, and possible evolutionary history of the source.
Fitting: Notch Models
=====================
The fits in this paper were performed using the same extraction of the HETG data as was used by [@Mill08]. $Chandra$ observed GRO J1655-40 for a total of 44.6 ks starting at 12:41:44 UT on 2005 April 1. Data was taken from the ACIS-S array dispersed by the High Energy Transmission Grating Spectrometer (HETGS). Continuous clocking mode was used in order to prevent photon pile-up. As described by [@Mill08], a gray filter was used in order to reduce the zero order counting rate. Data was processed using the CIAO reduction package, version 3.2.2. The event file was filtered to accept only standard event grades, good-time intervals, and to reject bad pixels. Streaking was removed using the “destreak” tool, and spectra were extracted using “tg\_resolve\_events” and “tg\_extract”. Arfs were produced using the “fullgarf” tool along with canned rmfs. First order HEG spectra and arfs and first-order MEG spectra and arfs were added using the “add\_grating\_spectra” tool. In this paper we do not fit to the RXTE data obtained simultaneously with the $Chandra$ HETG observation, but we do employ the continuum shape derived from the [@Mill08] fits which include the RXTE data.
Our procedure for analyzing the spectrum of GRO J1655-40 consists of three separate parts. First, we fit the spectrum using [xspec]{} together with simple analytic models describing the continuum and the lines. This consists of a continuum shape which is the same as that used by [@Mill08]: a power law plus a disk blackbody and cold absorption. Based on the fits by [@Mill08], we use a power law index of 3.54 and a disk blackbody temperature with temperature 1.34 keV. We allow the normalizations of the components to vary, and find a best fit normalization of 515.7$\pm$1.5 for the disk blackbody and $\leq$0.15 for the power law. The flux is 2.02 $\times 10^{-8}$ ergs/cm$^2$/s 2-10 keV. We ignore photons with wavelength greater than 15 Å, since there are few counts in this range and its inclusion has negligible effect on the results at shorter wavelength. We refer to this as model 1. The fitting parameters and $\chi^2$ for this and the other fits discussed in this paper are presented in Table \[chi2table\]. The various physical parameters for all the models we test are listed in the first column. Model 1, the best fit to the continuum only, gives $\chi^2/\nu=120618/8189$.
[llllllll]{} log($\xi_1$) &erg cm s$^{-1}$&NA &NA &NA &4.$^+_-0.1$ &3.8$^+_-0.1$ &4.0$^+_-0.1$\
log(N$_1$) &cm$^{-2}$ &NA &NA &NA &23.8$^+_-0.02$ &22.64$^+_-0.02$ &24.0$^+_-0.02$\
v$_{turb}$ &km s$^{-1}$ &NA &NA &NA &50 &200 &200\
v$_{off}$ &km s$^{-1}$ &NA &NA &NA &-375 &-375 &-375\
log($\xi_2$) &erg cm s$^{-1}$&NA &NA &NA &NA &4.6$^+_-$0.1 &NA\
log(N$_2$) &cm$^{-2}$ &NA &NA &NA &NA &23.90 &NA\
EW$_{fe26}$ &keV &NA &NA &NA &0.03 &NA &0.03\
v$_{off,fe26}$ &km s$^{-1}$ &NA &NA &NA &1451. &NA &1451.\
v$_{turb,fe26}$&km s$^{-1}$ &NA &NA &NA &0.03$_{-0.02}^{+0.01}$&NA &0.03$_{-0.02}^{+0.01}$\
NH &cm$^{-2}$ &21.49$^+_-0.01$&21.49$^+_-0.01$&NA &21.49$^+_-0.01$ &21.48$^+_-0.01$ &21.48$^+_-0.01$\
$\gamma$ & &3.54 &3.54 &3.54 &3.54&3.54&3.54\
pl norm & &$\leq$0.01 &$\leq$0.01 &$\leq$0.15&$\leq$0.01 &$\leq$0.01 &$\leq$0.01\
diskbb norm & &513$^+_-$1 &538&534$+_-$1 &548$_{-1.5}^{+0.5}$&533$_{-1}^{+1.5}$&533$_{-1}^{+1.5}$\
$kT_{diskbb}$ &keV &1.35 &1.35&1.35&1.35&1.35&1.35\
$f_{scatt}$ & &0 &0 &0 &0 &0 &0.37\
$\chi^2$ & &120618 &18474 &20671 &33591 &34283 &24561\
dof & &8189 &8108 &8118 &8187 &8189 &8184\
We then add the effects of absorption lines by using negative Gaussian models for the lines, as was done by [@Mill08]. The list of lines and their properties is the primary topic of the rest of this paper. Initially we use the same lines as given in Table 1 of [@Mill08]. There are 71 lines with well-determined wavelengths and identifications in this list. We allow the values of the wavelengths, widths, and line normalizations to be determined by the [xspec]{} minimization. This results in detections of essentially all the lines with parameters consistent with those of [@Mill08]. In addition, we propose identifications for some of the lines which were not identified by those authors. We point out that this procedure differs from those authors in that we fit to a single analytic global continuum, while they fit to a piece-wise powerlaw. The procedure used here is chosen for comparison with the fits to photoionization models in the next section. The best-fit to the continuum plus Gaussians gives $\chi^2/\nu=$18474/8108. We refer to this as model 2 (cf. Table \[chi2table\]). This fit is marginally acceptable based on standard arguments derived from $\chi^2$ statistics, and it is the best of the models presented in this paper. This serves to illustrate the level of systematic errors, or errors in our continuum, which provide an effective limit to our ability to fit the spectrum. Model 2, along with other models discussed below, is plotted in figures \[fita\] – \[fitn\]. In this figure the vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other. These figures show qualitatively the agreement between the model and the data which we achieve.
The second approach to line fitting is to construct an automated line fitting program. This takes the same continuum employed in the continuum-only fit, and then experiments with random placements of lines throughout the 1-15 Å range. These experiments begin with an initial wavelength which is chosen randomly within this range (but excluding a region within 2 Doppler widths of previously found lines), and then the line wavelength, width, and optical depth are varied in an attempt to find a best fit (the wavelength is restricted to a region near the initial wavelength in this procedure). The fit is considered valid if the $\chi^2$ improves by 3 with the inclusion of the line. Line widths are limited to be less than 8 Doppler widths when compared with a turbulent velocity of 100 km/s. The lines are assumed to be Doppler broadened only, and the absorption is a true Gaussian absorption profile. This is in contrast to the [xspec]{} Gaussian line model, which treats absorption as a negative emission. As we will show, many of our line fits result in large optical depths, and in this case a Voigt profile fit would be preferable. The pure Doppler profile likely results in an over-estimate of the line optical depth, since it cannot produce as strong absorption line wings as would a Voigt profile. However, the Voigt profile damping parameter value depends on the line identification, and cannot be conveniently used as a fitting parameter. In our synthetic spectral modeling, in the following section, we fit to Voigt profiles using accurate atomic rates for the damping parameters. For convenience we refer to this as the notch model, although it assumes Gaussian absorption lines rather than true notches.
This procedure yields a total of 292 lines after a total of 20000 attempts at placing random lines. We neglect lines with equivalent widths less than 3.4 $\times 10^{-4}$ eV. We do not consider this to be necessarily an exhaustive list of lines in the GRO J1655-40 spectrum, but likely contains the strongest or least ambiguous features, and it is objective. This procedure detects all the features in the [@Mill08] table, plus many others which are weaker or blended. Some of these are undoubtedly artifacts of the shortcomings of our continuum model, and others may coincide with regions where bound-free continuum opacity is important. These are discussed in turn in the following. This fit yields $\chi^2/\nu=20671/8118$ using the continuum from the [xspec]{} Gaussian fit. We refer to this as model 3 (cf. Table \[chi2table\]). The slightly worse $\chi^2$ value for this model when compared with model 2 is likely due to the limitations of our automated fitting procedure, particularly when two strong lines are close together or partially overlapping. In addition to deriving wavelengths, line center optical depths, and widths ($\sigma$), we also derive errors on these quantities based on the $\Delta\chi^2$=3 criterion of [@Cash79], and we calculate the line equivalent widths by integrating numerically over the best-fit model profile.
Line Identifications
--------------------
The list of lines we detect is given in Table \[linelist\]. This includes the wavelength, width ($\sigma$) and equivalent width derived from the automated fitting leading to model 3. We also provide identifications for the lines. This is done by searching the linelist in the [xstar]{} [@Kall01; @Baut01] database and choosing the line which has the greatest ratio of optical depth to Doppler shift within a Doppler shift of $\leq^+_-$1500 km s$^{-1}$. The optical depth used in this determination is calculated using the [warmabs]{} analytic model, which is an implementation of [xstar]{} use as an analytic model within the [xspec]{} X-ray spectral fitting package, for the conditions in model 4 described in the following section. We have updated both [xstar]{} and [warmabs]{} to include all the previously neglected elements with $Z \leq 30$; this is described in the Appendix. Note that the criterion for line identification is used to choose among the [xstar]{} lines which fall within Doppler shifts of $^+_-$1500 km s$^{-1}$, but does not prevent a line from being included if there is no [xstar]{} line within that interval. If there is an ID, then the parent ion, [xstar]{} wavelength, and upper and lower level designations are also given in Table \[linelist\]. The identification, together with the optical depth derived from the model 3 fits allows the equivalent hydrogen column density of the absorbing ion to be derived. These are discussed in more detail in the following subsection, and are given in the table. The elemental abundances used in the calculation of equivalent hydrogen abundances are those of [@Grev96; @Alle73]. Figures \[fita\] – \[fitn\] show the count spectrum observed by the HETG (relative to the continuum only model 1) together with the lines and identifications from Table \[linelist\]. Also shown in these figures is our fit to model 2 and to models 4, 5 and 6 which will be discussed in more detail below.
![\[fita\]Spectrum $\lambda\lambda$ 1 – 2 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1a.pdf)
![\[fitb\]Spectrum $\lambda\lambda$ 2 – 3 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1b.pdf)
![\[fitc\]Spectrum $\lambda\lambda$ 3 – 4 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1c.pdf)
![\[fitd\]Spectrum $\lambda\lambda$ 4 – 5 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1d.pdf)
![\[fite\]Spectrum $\lambda\lambda$ 5 – 6 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1e.pdf)
![\[fitf\]Spectrum $\lambda\lambda$ 6 – 7 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1f.pdf)
![\[fitg\]Spectrum $\lambda\lambda$ 7 – 8 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1g.pdf)
![\[fith\]Spectrum $\lambda\lambda$ 8 – 9 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1h.pdf)
![\[fiti\]Spectrum $\lambda\lambda$ 9 – 10 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1i.pdf)
![\[fitj\]Spectrum $\lambda\lambda$ 10 – 11 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1j.pdf)
![\[fitk\]Spectrum $\lambda\lambda$ 11 – 12 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1k.pdf)
![\[fitl\]Spectrum $\lambda\lambda$ 12 – 13 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1l.pdf)
![\[fitm\]Spectrum $\lambda\lambda$ 13 – 14 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1m.pdf)
![\[fitn\]Spectrum $\lambda\lambda$ 14 – 15 Å. Spectrum is shown as ratio relative to pure power law model (model 1). Various models 2,4,5,6 are labeled. The vertical axis is the ratio of the model and data to the continuum-only model, model 1. Successive models are offset by unity from each other.](f1n.pdf)
[rrrrrrrrrrrr]{} 1.544& 17264& 1.542&Mn XXV & -357.5 & 0.0006&459.3& 0.71$^{+0.56}_{-0.41}$& 0.029 &1s.2S &4p.2P$^o$ & 2.9$^{+2.3}_{-1.7}\times 10^{23}$\
1.567& 128239& 1.573&Fe XXV & 1187.0 & 0.0029&772.1& 2.62$^{+1.44}_{-0.88}$& 0.152 &1s2.1S &1s.3p.1P\* & 4.4$^{+2.4}_{-1.5}\times 10^{21}$\
1.582& 132917& 1.589&Ni XXVII & 1232.6 & 0.0084&506.7&23.06$^{+9.99}_{-8.75}$& 0.684 &1s2.1S &1s.2p.1P & 1.0$^{+0.7}_{-0.4}\times 10^{23}$\
1.601& & & no ID & & 0.0022&707.2& 1.92$^{+0.91}_{-0.62}$& & & &\
1.613& 17063& 1.615&Cr XXIV & 297.6 & 0.0004&234.9& 1.18$^{+0.80}_{-0.60}$& 0.008 &1s.2S &6p.2P$^o$ & 1.4$^{+0.9}_{-0.7}\times 10^{24}$\
1.626& 17263& 1.626&Mn XXV & 77.5 & 0.0010&412.7& 1.22$^{+0.51}_{-0.40}$& 0.080 &1s.2S &3p.2P$^o$ & 1.7$^{+0.7}_{-0.6}\times 10^{23}$\
1.646& 128543& 1.649&Co XXVII & 484.8 & 0.0006&407.0& 0.66$^{+0.34}_{-0.28}$& 0.422 &1s.2S &2p.2P$^o$ & 7.4$^{+3.8}_{-3.1}\times 10^{22}$\
1.675& 17061& 1.674&Cr XXIV & -107.5 & 0.0004&204.0& 1.22$^{+0.79}_{-0.63}$& 0.029 &1s.2S &4p.2P$^o$ & 3.7$^{+2.4}_{-1.9}\times 10^{23}$\
1.693& & & no ID & & 0.0007&458.1& 0.58$^{+0.27}_{-0.23}$& & & &\
1.709& 128440& 1.712&Co XXVI & 544.2 & 0.0015&539.4& 1.24$^{+0.40}_{-0.31}$& 0.693 &1s2.1S &1s.2p.1P & 8.2$^{+2.6}_{-2.1}\times 10^{22}$\
1.726& & & no ID & & 0.0012&701.0& 0.65$^{+0.24}_{-0.22}$& & & &\
1.742& & & no ID & & 0.0003&225.4& 0.71$^{+0.49}_{-0.36}$& & & &\
1.772& 128358& 1.780&Fe XXVI & 1325.6 & 0.0053&453.6&10.56$^{+3.68}_{-2.72}$& 0.408 &1s.2S &2p.2P$^o$ & 5.9$^{+2.1}_{-1.5}\times 10^{21}$\
1.838& & & no ID & & 0.0007&195.2& 2.35$^{+0.75}_{-0.55}$& & & &\
1.851& 128293& 1.850&Fe XXV & -81.0 & 0.0157&503.9&39.38$^{+9.99}_{-8.98}$& 0.775 &1s2.1S &1s.2p.1P\* & 1.1$^{+0.3}_{-0.3}\times 10^{22}$\
1.865& 127741& 1.864&Fe XXIV & -209.1 & 0.0006&179.5& 2.57$^{+0.96}_{-0.70}$& 0.149 &1s2.2s &1s2s2p.2P0.5 & 3.7$^{+1.4}_{-1.0}\times 10^{21}$\
1.875& 127760& 1.873&Fe XXIV & -320.0 & 0.0003&203.7& 0.41$^{+0.28}_{-0.24}$& 0.015 &1s2.2s &1s2s2p.4P1.5 & 5.7$^{+3.9}_{-3.4}\times 10^{21}$\
1.923& 17262& 1.926&Mn XXV & 480.5 & 0.0007&401.6& 0.44$^{+0.13}_{-0.12}$& 0.420 &1s.2S &2p.2P$^o$ &10.0$^{+2.9}_{-2.7}\times 10^{21}$\
1.951& 16659& 1.948&Ti XXII & -412.1 & 0.0002&299.7& 0.14$^{+0.14}_{-0.09}$& 0.014 &1s.2S &5p.2P$^o$ & 4.1$^{+4.0}_{-2.6}\times 10^{23}$\
1.963& & & no ID & & 0.0003&212.7& 0.44$^{+0.23}_{-0.20}$& & & &\
1.993& 16658& 1.995&Ti XXII & 313.1 & 0.0003&207.6& 0.36$^{+0.21}_{-0.19}$& 0.029 &1s.2S &4p.2P$^o$ & 5.0$^{+2.9}_{-2.7}\times 10^{23}$\
2.004& 17159& 2.006&Mn XXIV & 329.4 & 0.0018&566.0& 0.87$^{+0.15}_{-0.14}$& 0.711 &1s2.1S &1s.2p.1P & 1.1$^{+0.2}_{-0.2}\times 10^{22}$\
2.020& & & no ID & & 0.0007&493.5& 0.32$^{+0.10}_{-0.09}$& & & &\
2.036& & & no ID & & 0.0003&454.9& 0.17$^{+0.09}_{-0.09}$& & & &\
2.062& & & no ID & & 0.0003&442.3& 0.14$^{+0.08}_{-0.08}$& & & &\
2.088& 17059& 2.092&Cr XXIV & 502.9 & 0.0014&416.2& 0.84$^{+0.13}_{-0.12}$& 0.421 &1s.2S &2p.2P$^o$ & 1.4$^{+0.2}_{-0.2}\times 10^{22}$\
2.114& & & no ID & & 0.0002&161.4& 0.26$^{+0.20}_{-0.17}$& & & &\
2.153& & & no ID & & 0.0005&156.3& 0.73$^{+0.28}_{-0.26}$& & & &\
2.179& 16956& 2.182&Cr XXIII & 426.8 & 0.0020&526.2& 0.87$^{+0.11}_{-0.11}$& 0.721 &1s2.1S &1s.2p.1P & 8.2$^{+1.0}_{-1.0}\times 10^{21}$\
2.193& 17016& 2.193&Cr XXIII & -68.4 & 0.0004&181.7& 0.52$^{+0.20}_{-0.18}$& 0.152 &1s2.1S &1s.2p.3P & 2.3$^{+0.9}_{-0.8}\times 10^{22}$\
2.205& & & no ID & & 0.0002&158.5& 0.22$^{+0.20}_{-0.14}$& & & &\
2.312& & & no ID & & 0.0003&158.4& 0.27$^{+0.19}_{-0.17}$& & & &\
2.361& & & no ID & & 0.0003&396.3& 0.10$^{+0.06}_{-0.05}$& & & &\
2.417& & & no ID & & 0.0004&403.5& 0.15$^{+0.06}_{-0.05}$& & & &\
2.491& 16656& 2.492&Ti XXII & 150.5 & 0.0007&410.0& 0.22$^{+0.05}_{-0.05}$& 0.419 &1s.2S &2p.2P$^o$ & 1.7$^{+0.4}_{-0.4}\times 10^{22}$\
2.502& 16183& 2.514&Ca XIX & 1414.8 & 0.0003&151.4& 0.30$^{+0.16}_{-0.14}$& 0.027 &1s2.1S &1s.5p.1P\* & 2.5$^{+1.3}_{-1.2}\times 10^{22}$\
2.545& 16282& 2.549&Ca XX & 515.1 & 0.0011&136.9& 1.69$^{+0.29}_{-0.26}$& 0.078 &1s.2S &3p.2P$^o$ & 4.8$^{+0.8}_{-0.7}\times 10^{22}$\
2.555& & & no ID & & 0.0002&140.0& 0.20$^{+0.17}_{-0.13}$& & & &\
2.701& 16143& 2.705&Ca XIX & 488.7 & 0.0008&400.9& 0.21$^{+0.05}_{-0.05}$& 0.152 &1s2.1S &1s.3p.1P\* & 2.9$^{+0.7}_{-0.7}\times 10^{21}$\
2.857& & & no ID & & 0.0003&117.3& 0.18$^{+0.13}_{-0.11}$& & & &\
2.877& & & no ID & & 0.0003&124.1& 0.23$^{+0.16}_{-0.15}$& & & &\
2.982& 13532& 2.987&Ar XVIII & 548.3 & 0.0007&131.4& 0.48$^{+0.15}_{-0.13}$& 0.029 &1s.2S &4p.2P$^o$ & 1.5$^{+0.5}_{-0.4}\times 10^{22}$\
3.016& 16262& 3.020&Ca XX & 427.7 & 0.0050&415.2& 1.24$^{+0.08}_{-0.07}$& 0.411 &1s.2S &2p.2P$^o$ & 5.7$^{+0.4}_{-0.3}\times 10^{21}$\
3.145& 13528& 3.151&Ar XVIII & 534.2 & 0.0016&304.3& 0.40$^{+0.06}_{-0.05}$& 0.078 &1s.2S &3p.2P$^o$ & 4.3$^{+0.6}_{-0.6}\times 10^{21}$\
3.172& 16146& 3.177&Ca XIX & 491.8 & 0.0024&203.9& 1.08$^{+0.12}_{-0.12}$& 0.770 &1s2.1S &1s.2p.1P\* & 2.5$^{+0.3}_{-0.3}\times 10^{21}$\
3.355& 13463& 3.365&Ar XVII & 938.9 & 0.0009&476.4& 0.11$^{+0.03}_{-0.03}$& 0.153 &1s2.1S &1s.3p.1P\* & 5.5$^{+1.8}_{-1.7}\times 10^{20}$\
3.690& 11876& 3.696&S XVI & 479.7 & 0.0012&107.1& 0.49$^{+0.14}_{-0.15}$& 0.014 &1s.2S &5p.2P$^o$ & 8.0$^{+2.4}_{-2.4}\times 10^{21}$\
3.727& 13535& 3.733&Ar XVIII & 474.9 & 0.0072&351.1& 1.08$^{+0.07}_{-0.06}$& 0.412 &1s.2S &2p.2P$^o$ & 1.9$^{+0.1}_{-0.1}\times 10^{21}$\
3.779& 11875& 3.784&S XVI & 429.5 & 0.0019& 86.5& 1.18$^{+0.30}_{-0.24}$& 0.029 &1s.2S &4p.2P$^o$ & 9.0$^{+2.3}_{-1.8}\times 10^{21}$\
3.943& 13446& 3.949&Ar XVII & 441.3 & 0.0025& 85.2& 1.70$^{+0.33}_{-0.28}$& 0.766 &1s2.1S &1s.2p.1P\* & 1.5$^{+0.3}_{-0.2}\times 10^{21}$\
3.985& 11878& 3.991&S XVI & 466.8 & 0.0044&173.4& 1.20$^{+0.14}_{-0.12}$& 0.079 &1s.2S &3p.2P$^o$ & 3.2$^{+0.4}_{-0.3}\times 10^{21}$\
4.183& 12039& 4.188&Cl XVII & 329.9 & 0.0012& 90.5& 0.41$^{+0.16}_{-0.15}$& 0.416 &1s.2S &2p.2P$^o$ & 7.4$^{+3.0}_{-2.7}\times 10^{21}$\
4.722& 11871& 4.729&S XVI & 457.4 & 0.0184&407.6& 1.27$^{+0.08}_{-0.07}$& 0.413 &1s.2S &2p.2P$^o$ & 5.4$^{+0.3}_{-0.3}\times 10^{20}$\
4.764& 9813& 4.770&Si XIV & 403.0 & 0.0008& 72.9& 0.20$^{+0.22}_{-0.12}$& 0.008 &1s.2S &6p.2P$^o$ & 2.2$^{+2.4}_{-1.4}\times 10^{21}$\
4.824& 9810& 4.831&Si XIV & 441.5 & 0.0031&128.4& 0.48$^{+0.14}_{-0.13}$& 0.014 &1s.2S &5p.2P$^o$ & 2.9$^{+0.9}_{-0.8}\times 10^{21}$\
4.941& 9811& 4.947&Si XIV & 358.2 & 0.0053& 86.8& 1.98$^{+0.51}_{-0.37}$& 0.029 &1s.2S &4p.2P$^o$ & 5.6$^{+1.4}_{-1.1}\times 10^{21}$\
5.032& 11734& 5.039&S XV & 401.2 & 0.0044& 66.2& 2.07$^{+0.46}_{-0.37}$& 0.761 &1s2.1S &1s.2p.1P\* & 4.5$^{+1.0}_{-0.8}\times 10^{20}$\
5.209& 9817& 5.217&Si XIV & 472.2 & 0.0115&237.9& 0.92$^{+0.09}_{-0.08}$& 0.079 &1s.2S &3p.2P$^o$ & 9.1$^{+0.9}_{-0.8}\times 10^{20}$\
5.375& & & no ID & & 0.0022&244.9& 0.11$^{+0.05}_{-0.05}$& & & &\
5.600& 132830& 5.606&Ni XXVI & 332.2 & 0.0023& 62.5& 0.46$^{+0.22}_{-0.20}$& 0.007 &1s2.2s &1S2\_7p & 6.2$^{+2.9}_{-2.7}\times 10^{22}$\
6.004& & & no ID & & 0.0010& 78.3& 0.12$^{+0.09}_{-0.07}$& & & &\
6.018& & & no ID & & 0.0025& 64.8& 0.37$^{+0.13}_{-0.13}$& & & &\
6.045& 8043& 6.053&Al XIII & 392.0 & 0.0063&414.3& 0.14$^{+0.02}_{-0.02}$& 0.079 &1s.2S &3p.2P$^o$ & 1.8$^{+0.3}_{-0.3}\times 10^{21}$\
6.065& & & no ID & & 0.0014& 55.7& 0.23$^{+0.14}_{-0.13}$& & & &\
6.077& & & no ID & & 0.0031& 50.3& 0.66$^{+0.23}_{-0.21}$& & & &\
6.103& 132826& 6.108&Ni XXVI & 226.1 & 0.0088&407.7& 0.22$^{+0.02}_{-0.02}$& 0.024 &1s2.2s &1s2.5p & 7.4$^{+0.8}_{-0.7}\times 10^{21}$\
6.135& & & no ID & & 0.0020& 62.9& 0.28$^{+0.12}_{-0.12}$& & & &\
6.154& 9816& 6.182&Si XIV & 1374.7 & 0.0034&164.6& 0.17$^{+0.04}_{-0.04}$& 0.414 &1s.2S &2p.2P$^o$ & 2.7$^{+0.7}_{-0.6}\times 10^{19}$\
6.172& 9816& 6.182&Si XIV & 495.8 & 0.0414&419.4& 1.35$^{+0.04}_{-0.04}$& 0.414 &1s.2S &2p.2P$^o$ & 2.1$^{+0.1}_{-0.1}\times 10^{20}$\
6.295& & & no ID & & 0.0075&332.6& 0.20$^{+0.03}_{-0.02}$& & & &\
6.352& 128163& 6.360&Fe XXIV & 392.0 & 0.0138&425.8& 0.28$^{+0.02}_{-0.02}$& 0.002 &1s2.2s &1S2\_9p & 7.8$^{+0.6}_{-0.6}\times 10^{21}$\
6.375& & & no ID & & 0.0025& 58.5& 0.35$^{+0.13}_{-0.13}$& & & &\
6.413& & & no ID & & 0.0020& 58.8& 0.27$^{+0.13}_{-0.12}$& & & &\
6.440& 128161& 6.446&Fe XXIV & 270.2 & 0.0139&274.1& 0.41$^{+0.03}_{-0.03}$& 0.004 &1s2.2s &1S2\_8p & 6.5$^{+0.5}_{-0.5}\times 10^{21}$\
6.489& 7856& 6.497&Mg XII & 389.7 & 0.0043&162.7& 0.19$^{+0.04}_{-0.04}$& 0.008 &1s.2S &6p.2P$^o$ & 1.7$^{+0.4}_{-0.4}\times 10^{21}$\
6.550& & & no ID & & 0.0019& 67.4& 0.20$^{+0.10}_{-0.10}$& & & &\
6.569& 128159& 6.575&Fe XXIV & 260.3 & 0.0204&242.7& 0.74$^{+0.04}_{-0.04}$& 0.007 &1s2.2s &1S2\_7p & 6.7$^{+0.3}_{-0.4}\times 10^{21}$\
6.590& 7853& 6.580&Mg XII & -455.2 & 0.0013& 59.5& 0.15$^{+0.11}_{-0.09}$& 0.014 &1s.2S &5p.2P$^o$ & 6.9$^{+5.4}_{-4.4}\times 10^{20}$\
6.640& 9691& 6.648&Si XIII & 361.4 & 0.0163&291.7& 0.42$^{+0.03}_{-0.03}$& 0.748 &1s2.1S &1s.2p.1P\* & 3.4$^{+0.3}_{-0.3}\times 10^{19}$\
6.729& 7852& 6.738&Mg XII & 401.2 & 0.0100& 54.7& 3.40$^{+0.92}_{-0.59}$& 0.029 &1s.2S &4p.2P$^o$ & 7.7$^{+2.1}_{-1.3}\times 10^{21}$\
6.778& 128157& 6.784&Fe XXIV & 256.7 & 0.0269&289.2& 0.76$^{+0.03}_{-0.03}$& 0.012 &1s2.2s &1S2\_6p & 3.7$^{+0.2}_{-0.1}\times 10^{21}$\
6.805& 132821& 6.811&Ni XXVI & 264.5 & 0.0198&416.0& 0.34$^{+0.02}_{-0.02}$& 0.032 &1s2.2s &1s2.4p & 7.7$^{+0.5}_{-0.4}\times 10^{21}$\
6.842& & & no ID & & 0.0013& 75.2& 0.11$^{+0.07}_{-0.07}$& & & &\
6.869& & & no ID & & 0.0090&271.0& 0.20$^{+0.03}_{-0.02}$& & & &\
6.956& & & no ID & & 0.0015& 73.4& 0.12$^{+0.08}_{-0.07}$& & & &\
7.013& & & no ID & & 0.0026& 58.3& 0.26$^{+0.10}_{-0.10}$& & & &\
7.058& & & no ID & & 0.0144&445.5& 0.20$^{+0.02}_{-0.02}$& & & &\
7.093& 7855& 7.106&Mg XII & 557.0 & 0.0426&495.7& 0.61$^{+0.02}_{-0.02}$& 0.079 &1s.2S &3p.2P$^o$ & 4.8$^{+0.2}_{-0.2}\times 10^{20}$\
7.159& 128021& 7.169&Fe XXIV & 419.1 & 0.0556&392.7& 1.17$^{+0.02}_{-0.02}$& 0.026 &1s2.2s &1s2.5p & 2.5$^{+0.1}_{-0.1}\times 10^{21}$\
7.223& & & no ID & & 0.0022& 48.2& 0.25$^{+0.12}_{-0.11}$& & & &\
7.242& & & no ID & & 0.0058&312.1& 0.11$^{+0.01}_{-0.04}$& & & &\
7.265& & & no ID & & 0.0024& 68.3& 0.18$^{+0.07}_{-0.06}$& & & &\
7.283& & & no ID & & 0.0030& 60.1& 0.27$^{+0.08}_{-0.08}$& & & &\
7.373& & & no ID & & 0.0051& 47.7& 0.64$^{+0.14}_{-0.13}$& & & &\
7.387& & & no ID & & 0.0029& 65.1& 0.22$^{+0.07}_{-0.07}$& & & &\
7.400& & & no ID & & 0.0020& 61.6& 0.15$^{+0.07}_{-0.07}$& & & &\
7.415& & & no ID & & 0.0029& 61.9& 0.23$^{+0.08}_{-0.07}$& & & &\
7.432& & & no ID & & 0.0026& 54.2& 0.24$^{+0.09}_{-0.09}$& & & &\
7.464& 126341& 7.472&Fe XXIII & 321.5 & 0.0272&348.1& 0.48$^{+0.02}_{-0.02}$& 0.056 &2s2 &2s.5p & 4.7$^{+0.2}_{-0.2}\times 10^{20}$\
7.490& & & no ID & & 0.0036& 58.0& 0.31$^{+0.09}_{-0.09}$& & & &\
7.513& & & no ID & & 0.0023& 52.0& 0.21$^{+0.11}_{-0.11}$& & & &\
7.533& & & no ID & & 0.0034& 59.9& 0.27$^{+0.10}_{-0.09}$& & & &\
7.555& & & no ID & & 0.0020& 59.7& 0.15$^{+0.08}_{-0.08}$& & & &\
7.575& & & no ID & & 0.0042& 53.8& 0.38$^{+0.11}_{-0.10}$& & & &\
7.617& & & no ID & & 0.0022& 43.5& 0.23$^{+0.15}_{-0.14}$& & & &\
7.664& & & no ID & & 0.0067&278.2& 0.10$^{+0.03}_{-0.02}$& & & &\
7.700& 126899& 7.733&Fe XXIII & 1285.7 & 0.0027& 48.3& 0.25$^{+0.12}_{-0.11}$& 0.035 &2s.2p &2s.5d & 3.6$^{+1.8}_{-1.7}\times 10^{20}$\
7.722& 126899& 7.733&Fe XXIII & 427.3 & 0.0018& 46.4& 0.17$^{+0.12}_{-0.11}$& 0.035 &2s.2p &2s.5d & 2.5$^{+1.8}_{-1.6}\times 10^{20}$\
7.743& 126899& 7.733&Fe XXIII & -387.5 & 0.0030& 72.0& 0.18$^{+0.07}_{-0.06}$& 0.035 &2s.2p &2s.5d & 2.6$^{+1.0}_{-0.9}\times 10^{20}$\
7.850& & & no ID & & 0.0056& 48.5& 0.55$^{+0.13}_{-0.13}$& & & &\
7.863& & & no ID & & 0.0037& 53.6& 0.29$^{+0.10}_{-0.10}$& & & &\
7.875& & & no ID & & 0.0037& 57.2& 0.27$^{+0.09}_{-0.09}$& & & &\
7.890& & & no ID & & 0.0028& 51.8& 0.22$^{+0.10}_{-0.10}$& & & &\
7.904& & & no ID & & 0.0035& 59.5& 0.24$^{+0.09}_{-0.08}$& & & &\
7.946& 127946& 7.983&Fe XXIV & 1396.9 & 0.0019& 51.1& 0.14$^{+0.09}_{-0.08}$& 0.062 &1s2.2s &1s2.4p & 1.2$^{+0.7}_{-0.6}\times 10^{20}$\
7.980& 127946& 7.983&Fe XXIV & 112.8 & 0.0829&409.0& 1.24$^{+0.04}_{-0.02}$& 0.062 &1s2.2s &1s2.4p & 1.0$^{+0.0}_{-0.0}\times 10^{21}$\
8.081& 120327& 8.090&Fe XXII & 348.9 & 0.0043& 52.5& 0.33$^{+0.10}_{-0.09}$& 0.050 &2s2.2p &2s2.5d & 3.3$^{+1.0}_{-0.9}\times 10^{20}$\
8.296& 125709& 8.303&Fe XXIII & 249.5 & 0.0431&311.2& 0.67$^{+0.03}_{-0.03}$& 0.144 &2s2 &2s.4p & 2.3$^{+0.1}_{-0.1}\times 10^{20}$\
8.393& 7883& 8.421&Mg XII & 1000.8 & 0.0042& 50.3& 0.29$^{+0.12}_{-0.11}$& 0.414 &1s.2S &2p.2P$^o$ & 3.6$^{+1.5}_{-1.4}\times 10^{19}$\
8.409& 7883& 8.421&Mg XII & 428.1 & 0.0704&285.7& 1.28$^{+0.04}_{-0.04}$& 0.414 &1s.2S &2p.2P$^o$ & 1.6$^{+0.0}_{-0.0}\times 10^{20}$\
8.705& 119538& 8.715&Fe XXII & 344.6 & 0.0061& 54.4& 0.36$^{+0.11}_{-0.11}$& 0.062 &2s2.2p &2s.2p(3P).4p & 2.7$^{+0.8}_{-0.8}\times 10^{20}$\
8.721& 119538& 8.715&Fe XXII & -206.4 & 0.0057& 57.7& 0.31$^{+0.10}_{-0.10}$& 0.062 &2s2.2p &2s.2p(3P).4p & 2.3$^{+0.8}_{-0.7}\times 10^{20}$\
8.963& 119527& 8.977&Fe XXII & 478.6 & 0.0077& 41.5& 0.61$^{+0.19}_{-0.18}$& 0.122 &2s2.2p &2s2.4d & 2.2$^{+0.7}_{-0.6}\times 10^{20}$\
9.048& 132824& 9.061&Ni XXVI & 431.0 & 0.0602&244.6& 0.88$^{+0.05}_{-0.04}$& 0.248 &1s2.2s &1s2.3p & 1.9$^{+0.1}_{-0.1}\times 10^{21}$\
9.092& 132807& 9.105&Ni XXVI & 428.9 & 0.0409&180.2& 0.76$^{+0.06}_{-0.05}$& 0.130 &1s2.2s &1s2.3p & 3.2$^{+0.2}_{-0.2}\times 10^{21}$\
9.158& 7744& 9.169&Mg XI & 353.8 & 0.0038& 36.1& 0.28$^{+0.21}_{-0.18}$& 0.738 &1s2.1S &1s.2p.1P & 1.8$^{+1.3}_{-1.1}\times 10^{19}$\
9.336& 132741& 9.340&Ni XXV & 118.9 & 0.0414&304.6& 0.42$^{+0.03}_{-0.03}$& 0.462 &2s2 &2s.3p & 4.7$^{+0.3}_{-0.4}\times 10^{20}$\
9.373& 132694& 9.390&Ni XXV & 534.5 & 0.0289&164.9& 0.48$^{+0.05}_{-0.05}$& 0.231 &2s2 &2s.3p & 1.1$^{+0.1}_{-0.1}\times 10^{21}$\
9.469& 6241& 9.481&Ne X & 380.2 & 0.0299&181.8& 0.42$^{+0.04}_{-0.04}$& 0.014 &1s.2S &5p.2P$^o$ & 1.9$^{+0.2}_{-0.2}\times 10^{21}$\
9.695& 6218& 9.708&Ne X & 402.3 & 0.0403&189.3& 0.54$^{+0.05}_{-0.05}$& 0.029 &1s.2S &4p.2P$^o$ & 1.1$^{+0.1}_{-0.1}\times 10^{21}$\
9.723& 6218& 9.708&Ne X & -462.8 & 0.0039& 58.1& 0.14$^{+0.12}_{-0.09}$& 0.029 &1s.2S &4p.2P$^o$ & 3.0$^{+2.5}_{-1.9}\times 10^{20}$\
9.783& & & no ID & & 0.0037& 40.3& 0.19$^{+0.18}_{-0.12}$& & & &\
10.010& 6380& 10.021&Na XI & 323.7 & 0.0531&183.2& 0.72$^{+0.07}_{-0.06}$& 0.416 &1s.2S &2p.2P$^o$ & 9.6$^{+0.9}_{-0.8}\times 10^{20}$\
10.100& & & no ID & & 0.0122& 59.5& 0.43$^{+0.16}_{-0.14}$& & & &\
10.150& & & no ID & & 0.0234& 88.0& 0.60$^{+0.12}_{-0.11}$& & & &\
10.210& 6207& 10.240&Ne X & 881.5 & 0.0047& 48.1& 0.18$^{+0.16}_{-0.11}$& 0.079 &1s.2S &3p.2P$^o$ & 1.3$^{+1.2}_{-0.8}\times 10^{20}$\
10.220& 6207& 10.240&Ne X & 587.1 & 0.0879&250.4& 0.87$^{+0.05}_{-0.05}$& 0.079 &1s.2S &3p.2P$^o$ & 6.5$^{+0.4}_{-0.4}\times 10^{20}$\
10.340& 124604& 10.349&Fe XXIII & 255.3 & 0.0086& 57.1& 0.28$^{+0.16}_{-0.14}$& 0.006 &2s2 &2p.3d & 1.8$^{+1.1}_{-0.9}\times 10^{21}$\
10.490& 124935& 10.505&Fe XXIII & 443.3 & 0.0233& 66.5& 0.77$^{+0.21}_{-0.16}$& 0.021 &2s2 &2p.3s & 1.4$^{+0.4}_{-0.3}\times 10^{21}$\
10.570& 127980& 10.619&Fe XXIV & 1390.7 & 0.0309&479.7& 0.12$^{+0.03}_{-0.03}$& 0.247 &1s2.2s &1s2.3p & 1.8$^{+0.5}_{-0.5}\times 10^{19}$\
10.610& 127980& 10.619&Fe XXIV & 254.5 & 0.2323&490.0& 1.30$^{+0.05}_{-0.06}$& 0.247 &1s2.2s &1s2.3p & 2.0$^{+0.1}_{-0.1}\times 10^{20}$\
10.650& 127947& 10.663&Fe XXIV & 366.2 & 0.3370&716.3& 1.13$^{+0.05}_{-0.05}$& 0.129 &1s2.2s &1s2.3p & 3.3$^{+0.1}_{-0.2}\times 10^{20}$\
10.780& 124988& 10.785&Fe XXIII & 128.0 & 0.0346&478.1& 0.12$^{+0.04}_{-0.03}$& 0.002 &2s2 &2s.3d & 2.5$^{+0.7}_{-0.7}\times 10^{21}$\
10.890& 124606& 10.903&Fe XXIII & 352.6 & 0.0402&435.8& 0.15$^{+0.04}_{-0.03}$& 0.082 &2s.2p &2p.3p & 6.8$^{+1.6}_{-1.5}\times 10^{19}$\
10.930& 125174& 10.980&Fe XXIII & 1372.3 & 0.0079& 41.0& 0.31$^{+0.25}_{-0.19}$& 0.414 &2s2 &2s.3p & 2.7$^{+2.2}_{-1.7}\times 10^{19}$\
10.970& 125174& 10.980&Fe XXIII & 273.5 & 0.1490&406.7& 0.74$^{+0.05}_{-0.04}$& 0.414 &2s2 &2s.3p & 6.6$^{+0.4}_{-0.4}\times 10^{19}$\
11.010& 125372& 11.018&Fe XXIII & 218.0 & 0.1051&221.2& 0.99$^{+0.07}_{-0.08}$& 0.254 &2s2 &2s.3p & 1.4$^{+0.1}_{-0.1}\times 10^{20}$\
11.150& & & no ID & & 0.0350&382.1& 0.15$^{+0.04}_{-0.04}$& & & &\
11.280& 125155& 11.299&Fe XXIII & 508.0 & 0.0289&253.5& 0.15$^{+0.06}_{-0.06}$& 0.751 &2s.2p &2s.3d & 7.3$^{+2.8}_{-2.8}\times 10^{18}$\
11.310& 125155& 11.299&Fe XXIII & -289.1 & 0.0224&170.2& 0.18$^{+0.08}_{-0.08}$& 0.751 &2s.2p &2s.3d & 8.5$^{+4.0}_{-3.9}\times 10^{18}$\
11.350& 124975& 11.337&Fe XXIII & -348.9 & 0.0504&481.3& 0.16$^{+0.06}_{-0.05}$& 0.552 &2s.2p &2s.3d & 1.0$^{+0.4}_{-0.3}\times 10^{19}$\
11.420& 124980& 11.442&Fe XXIII & 585.8 & 0.1575&500.7& 0.52$^{+0.08}_{-0.07}$& 0.615 &2s.2p &2s.3d & 3.0$^{+0.4}_{-0.4}\times 10^{19}$\
11.470& 124991& 11.457&Fe XXIII & -337.4 & 0.0621&253.4& 0.34$^{+0.08}_{-0.07}$& 0.110 &2s.2p &2s.3d & 1.1$^{+0.3}_{-0.2}\times 10^{20}$\
11.540& & & no ID & & 0.0357&437.0& 0.11$^{+0.04}_{-0.04}$& & & &\
11.760& 119502& 11.769&Fe XXII & 242.3 & 0.0710&192.6& 0.51$^{+0.09}_{-0.08}$& 0.673 &2s2.2p &2s2.3d & 2.6$^{+0.5}_{-0.4}\times 10^{19}$\
11.910& 120311& 11.921&Fe XXII & 277.1 & 0.0546&145.6& 0.50$^{+0.12}_{-0.11}$& 0.597 &2s2.2p &2s2.3d & 2.9$^{+0.7}_{-0.6}\times 10^{19}$\
12.120& 6204& 12.134&Ne X & 344.1 & 0.2336&358.4& 1.09$^{+0.08}_{-0.08}$& 0.415 &1s.2S &2p.2P$^o$ & 1.3$^{+0.1}_{-0.1}\times 10^{20}$\
12.300& 89984& 12.282&Fe XXI & -431.7 & 0.0226& 66.1& 0.40$^{+0.26}_{-0.21}$& 1.305 &2s2.2p2 &2s2.2p.3d &10.0$^{+6.6}_{-5.2}\times 10^{18}$\
12.600& & & no ID & & 0.0550&163.0& 0.36$^{+0.12}_{-0.11}$& & & &\
12.640& & & no ID & & 0.0319&208.2& 0.15$^{+0.09}_{-0.09}$& & & &\
12.820& & & no ID & & 0.0703&500.8& 0.15$^{+0.12}_{-0.09}$& & & &\
12.870& & & no ID & & 0.0566&213.6& 0.25$^{+0.20}_{-0.16}$& & & &\
13.110& & & no ID & & 0.0700&175.5& 0.38$^{+0.19}_{-0.17}$& & & &\
13.240& & & no ID & & 0.0744&300.9& 0.21$^{+0.14}_{-0.12}$& & & &\
13.310& & & no ID & & 0.0464&118.2& 0.35$^{+0.29}_{-0.22}$& & & &\
13.450& & & no ID & & 0.2285&553.4& 0.41$^{+0.13}_{-0.11}$& & & &\
13.540& & & no ID & & 0.1563&564.8& 0.25$^{+0.12}_{-0.11}$& & & &\
13.580& & & no ID & & 0.0313& 30.1& 1.27$^{+9.99}_{-0.80}$& & & &\
13.660& & & no ID & & 0.0443& 29.2& 3.28$^{+9.99}_{-2.07}$& & & &\
13.680& & & no ID & & 0.0365& 26.1& 2.42$^{+9.99}_{-1.52}$& & & &\
13.770& & & no ID & & 0.1267&268.8& 0.39$^{+0.19}_{-0.17}$& & & &\
13.820& & & no ID & & 0.0393& 29.6& 1.92$^{+9.99}_{-1.21}$& & & &\
13.880& & & no ID & & 0.1279&450.4& 0.24$^{+0.13}_{-0.12}$& & & &\
13.930& & & no ID & & 0.0883&150.1& 0.49$^{+0.29}_{-0.23}$& & & &\
14.010& & & no ID & & 0.1935&482.0& 0.35$^{+0.14}_{-0.13}$& & & &\
14.090& 16073& 14.097&Ca XVIII & 149.0 & 0.1243&302.6& 0.40$^{+0.17}_{-0.15}$& 0.062 &1s2.2s &1s2.4p & 2.6$^{+1.1}_{-1.0}\times 10^{21}$\
14.160& & & no ID & & 0.0626& 72.3& 0.77$^{+0.73}_{-0.49}$& & & &\
14.190& & & no ID & & 0.1305&246.3& 0.40$^{+0.21}_{-0.18}$& & & &\
14.250& & & no ID & & 0.0399& 27.8& 1.85$^{+9.99}_{-1.17}$& & & &\
14.300& & & no ID & & 0.1415&160.0& 0.77$^{+0.34}_{-0.27}$& & & &\
14.360& & & no ID & & 0.0404& 27.8& 1.81$^{+9.99}_{-1.14}$& & & &\
14.390& & & no ID & & 0.1554&266.4& 0.43$^{+0.19}_{-0.17}$& & & &\
14.430& & & no ID & & 0.0576& 78.8& 0.56$^{+0.59}_{-0.36}$& & & &\
14.470& & & no ID & & 0.0527&112.3& 0.32$^{+0.38}_{-0.20}$& & & &\
14.550& & & no ID & & 0.0577& 26.3& 7.71$^{+9.99}_{-4.86}$& & & &\
14.610& 4675& 14.645&O VIII & 724.9 & 0.1763&189.3& 0.75$^{+0.30}_{-0.25}$& 0.008 &1s.2S &6p.2P$^o$ & 1.8$^{+0.7}_{-0.6}\times 10^{20}$\
14.770& 4634& 14.832&O VIII & 1255.2 & 0.0501& 26.7& 3.02$^{+9.99}_{-1.90}$& 0.014 &1s.2S &5p.2P$^o$ & 4.1$^{+9.9}_{-2.6}\times 10^{20}$\
14.820& 4634& 14.832&O VIII & 238.9 & 0.1715&454.9& 0.26$^{+0.15}_{-0.13}$& 0.014 &1s.2S &5p.2P$^o$ & 3.5$^{+2.0}_{-1.8}\times 10^{19}$\
The range between 1-2.5 Å is dominated by the H-like L$\alpha$ and He-like allowed $n=$1 – 2 lines from the elements Fe (1.77, 1.85 Å ), Mn (1.92, 2.01 Å ) and Cr (2.08, 2.18 Å ). Wavelengths shortward of the Fe XXVI L$\alpha$ line at 1.77 Å contain the K lines of elements heavier than Fe, in addition to the higher series lines of Fe. Although there are marginal detections of some of these lines, there are too few counts in this region to strongly constrain the abundances of the heavier elements Co, Cu, Zn. Ni is an exception to this, since it also has lines from Li-like Ni XXVI at longer wavelength. It is important to emphasize that, although Ly$\alpha$ lines are not clearly detected for Co, Cu and Zn, we cannot set meaningful upper limits to their strength because our model atoms for these elements do not include $n=$2 – 3 lines from the Li-like and lower ionization stages.
The 2 – 3 Å region includes the He-like resonance line of Mn XXIV, the K lines from H- and He-like Cr (2.088 and 2.179 Å ), He-like Sc (2.877 Å ), H-like Ti (2.491 Å ). The apparent absence of lines from V, H-like Sc and He-like Ti provides relatively secure upper limits on the abundances of these elements. We point out that the existence of evaluated wavelengths for these lines from NIST provides added validity to this conclusion. In this range are also $n=$1 – 3 lines from Ca, which are stronger than the $n=$1 – 2 lines from Sc, Ti, Cr and Mn. In this range are also $n=$1 – 3 lines from Ca, Ar, S, Si. For Si XIV Rydberg series lines are detected up to $n=6$.
At 4.18 Å we marginally detect the L$\alpha$ line from H-like Cl XVII. The He-like line from this element is not detected, but we note the absence of evaluated wavelength for this line. Similar comments apply to the H-like line from P, near 5.38 Å. The ratio of the He and H-like lines from S near 4.72 and 5.04 Å follows the same behavior as for Ca and Ar. The 6 –7Å region is dominated by lines from Si XIII and XIV and Li-like Fe and Ni; for Fe these are detected up to $1s^22s - 1s^210p$. At 10.67 and 10.88Å are the lines from Fe XXII which provide the density diagnostics. These lines together extend the range of ionization of observed species to include ions which are indicative of lower ionization parameter gas than the hydrogenic and helium-like ions which predominate.
Table \[linelist\] includes 175 lines of which 100 have IDs. The table of [@Mill08] includes 102 lines, of which 15 have no IDs. Our table has 44 lines which are not in the [@Mill08] table, although we note that they used a 5$\sigma$ criterion for line detection, while ours is 3$\sigma$. Of the unidentified lines in [@Mill08] we propose IDs for 5.6 Å Ni XXVI 2s – 7p and 9.372 Å Ni XXV 2s$^2$ – 2s3p. We have no IDs for the unidentified lines from [@Mill08] at $\lambda\lambda$ 6.25, 7.0555, 7.0851, 9.9509 Å. These latter 3 are crudely consistent with lines arising from the 2p excited levels of Ni XXVI (rest wavelengths 7.048, 7.0950 and 9.6383). But the results of [warmabs]{} suggest optical depths for these lines which are smaller than the resonance lines by factors $\geq 10^4$ assuming a density of 10$^{15}$ cm$^{-3}$ and no trapping of line radiation. All the 75 lines from our notch model for which we have no identifications, have small equivalent widths, within a factor of 2 of our cutoff of 3.5 $\times 10^{-4}$. We speculate that some of them could be due to the notch algorithm attempting to correct for discrepancies between the model continuum and the data, perhaps due to bound-free continuum absorption. In addition, visual inspection of the spectrum shows lines which do not appear in our table. Many of these have possible IDs as lines from excited levels, but none appears in the [warmabs]{} models at sufficient strength to be included in our table. Other lines in this category include: 2.60 Å possibly Ti XXI He-like 1 – 2, 2.15 Å possibly Sc XX He-like 1 – 3, features at 3.61, 5.375, 5.62, 5.65 Å with no obvious ID, and 5.72 Å possibly Al XIII 1 – 4. Features at 6.22 and 6.24 Å could be due to Si XIII 1s2s – 2s2p. Discernible features at $\lambda\lambda$ 6.7, 6.87, 7.06, 9.24, 9.78, 9.96, 10.1, 10.15 and 11.15 Å have no obvious IDs. Many of these correspond to lines in our list which are from excited levels, or from ion stages which are too low to coexist with the dominant ions in the [warmabs]{} models. We also note an emission line at 13.38 Å which coincides with the $\lambda$ 13.387 Å2s – 3p transition in Ti XXII listed in @Shir00. If this is correct, it is hard understand why this line should appear in emission when other analogous lines from Fe XXIV and Ni XXVI, for instance, appear in absorption. Other interesting lines not identified by [@Mill08] are the Fe XXIII lines between 11.28 Å and 11.47 Å, since they arise from the metastable 2s2p$^3P$ level. Their presence corroborates the density estimate from the Fe XXII lines.
The Doppler shift of the lines relative to the lab wavelength are shown graphically in figure \[vofffig\]. In this figure, the color corresponds to the element, and the size of the dot corresponds to the line optical depth. This shows that the lines cluster in a range around 400 km s$^{-1}$ from zero offset (here and in what follows we quote blueshifted velocities as positive, and conversely). This corresponds to $\sim$2 – 10 mÅ for most lines. Many of the laboratory wavelengths are uncertain at this level. Given this, the most notable aspect of the line shifts is the large velocity offset of the Fe XXVI L$\alpha$ and the Ni XXVII lines. These are among the strongest lines in the spectrum, and are blueshifted by $\simeq$1300 km/s, which is significantly greater than any other lines in the spectrum. This, together with the fact that these are the two highest-ionization lines in the spectrum, suggests that these lines are partly formed in a separate component of the flow. If so, this component would have higher ionization, and higher velocity, than the component responsible for the rest of the lines. On the other hand, we consider the possibility that this is an artifact of shortcomings in the HETG calibration when applied to fitting absorption of the steeply sloping continuum under these lines. In what follows we will examine alternative explanations for this in our fits to the spectrum.
The line widths in Table \[linelist\] range from 1 – 10 mÅ. We have searched and found no significant correlation between this quantity and simple quantities characterizing the parent ion, such as the ionization potential or isoelectronic sequence. There is a weak tendency for the largest widths to be associated with lower ionization potential ions. An example is the L$\alpha$ line from Si XIV. However, there are also narrower lines from ions with comparable ionization potential, so this does not constitute a statistically significant correlation.
![\[vofffig\]Doppler velocities obtained by comparing measured notch wavelengths with lab wavelengths from [xstar]{} database. Element color coding is given in the legend. The dot size is proportional to the line optical depth.](f2.pdf)
Ion Column Densities
--------------------
Ion column densities can be derived from the line identifications in Table \[linelist\], assuming that the equivalent widths lie on the linear part of the curve of growth. That is, we calculate
$$N_{ion}=\frac{\tau_{line}}{\frac{\pi e^2}{m_e c} \frac{f}{\Delta\nu_D}}$$
where $\Delta\nu_D$ is the total Doppler width in frequency units, including both thermal and turbulent contributions. $\tau_{line}$ is the line depth measured from the notch fit to the spectrum, and since it is based on Gaussian line fits its accuracy diminishes when values become large. In Table \[linelist\] we list the value of the ratio $N_{ion}/Y_{element}$, where $Y_{element}$ is the elemental solar [@Grev96; @Alle73] abundance relative to hydrogen. $N_{ion}/Y_{element}$, is the equivalent hydrogen column density implied by a given line if its ion fraction were unity, if the elemental abundance were solar, and if it lies on the linear part of the curve of growth. In figure \[colunitfig\] we plot this quantity. The horizontal axis is related to the atomic number of the parent element by $Z_{ion}=Z_{element}+1-0.1*(ion stage)$, where $(ion stage)$=1 for hydrogenic, 2 for He-like, etc. Error bars are plotted and, in most cases, are small compared with the the interval between points. If the assumptions of unit ionization fraction, cosmic abundances, and linear curve of growth were correct, then all the points in this diagram would lie along one horizontal line. Since most elements have lines from more than one ion, the assumption of unit ionization fraction cannot be correct; for these the true column should be the sum of the columns for various ions if the other assumptions were correct. However, this neglects the possible contributions from ions which do not produce observed lines, such as fully stripped species, which are likely to be important for the lower-$Z$ elements.
It is clear from figure \[colunitfig\] that there is a greater dispersion of column densities within elements than can be accounted for by ionization effects. This will be discussed in greater detail in the next section, and suggests the limitation of our second assumption, that of the linear curve of growth. Figure \[colunitfig\] also shows departures from solar abundance ratios, in the sense of apparently enhanced abundances for most elements between Sc and Mn, relative to elements with $Z \leq 16$. This conclusion is dependent quantitatively on the excitation and ionization conditions in the absorber, and in the following section we will attempt to quantify the abundances for a variety of assumptions about the state of the gas.
We expect qualitatively that the lower-$Z$ elements will be more highly ionized than the higher-$Z$ elements, for most plausible ionization mechanisms. This would predict that the apparent elemental abundances would be systematically greater for the low-$Z$ elements than for the high-$Z$ elements in figure \[colunitfig\], since the low-$Z$ elements would have a greater fraction of their ions in the unobservable fully stripped stage. This is the opposite behavior to what we observe, and so reinforces the conclusion that the elements with $Z \geq 16$ have enhanced abundances relative to those with lower $Z$.
![\[colunitfig\]Element column densities obtained assuming linear curve of growth, cosmic element abundances, and unit ion fractions for all lines in Table \[linelist\]. Each data point represents one line. The horizontal axis is the quantity $Z_{ion}$ where we distinguish between contributions of various ions by assigning $Z_{ion}=Z_{element}+1-0.1*(ion stage)$, where $(ion stage)$=1 for hydrogenic, 2 for He-like, etc. Element color coding is the same as in figure \[vofffig\].](f3.pdf)
Curve of Growth
---------------
The validity of the procedure used to derive the columns in figure \[colunitfig\] depends on the assumption that the lines lie on the linear part of the curve of growth. If this is correct, then the line equivalent widths should be proportional to the transition oscillator strengths, and comparison of various lines from the same ion should show this proportionality. In figure \[coghlike\] we test this procedure using the Lyman series lines from H-like ions of Ne, Mg, Al, Si, S, Ar, Ca, Cr, and Mn. We also include the $2s-np$ lines from Li-like Fe XXIV and Ni XXVI. Solid lines are linear regression fits to data with errors less than 0.1. Also shown on this figure are diagonal lines (dashed) corresponding to the proportionality expected for linear curve of growth. This shows that, although some ions appear to follow the linear trend, the strongest lines in particular grow more slowly than linearly. This is particularly apparent for the lines of Ne X, Fe XXIV and Ni XXVI. Each ion has at least 5 lines, and the trend is apparent across the line strengths. This shows that the simple analysis provided in figure \[colunitfig\] and Table \[linelist\] are likely not adequate for the purposes of inferring abundances and the lines may be saturated. On the other hand, weaker lines such as those of Si XIV do apparently follow the linear trend.
![\[coghlike\]Curve of growth for H-like and Li-like ions. Element color coding is the same as in figure \[vofffig\]. Dashed diagonal lines show simple proportionality behavior expected for unsaturated lines. Solid lines are linear regression fits to data with errors less than 0.1.](f4.pdf)
![\[hheplot\]Ratio of column densities derived from linear curve of grow analysis for H-line to He-line ions, versus the nuclear charge. Element color coding is the same as in figure \[vofffig\].](f5.pdf)
Another illustration of the effects of saturation is shown in figure \[hheplot\]. This shows the ratios of He-like to H-like ion abundances inferred from the line equivalent widths and identifications in Table \[linelist\]. These ratios are independent of element abundance, owing to the fact that each ratio is taken between ions from the same element. This shows that the ratios are all in the range between 10$^{-1}$ and 10$^{+1}$ for $ 12 \leq Z \leq 26$, but that there is no systematic trend with $Z$. We might expect the ratio to increase with $Z$, as higher $Z$ elements would be generally less highly ionized, for plausible ionization mechanisms. Also shown in figure \[hheplot\] are the contours traced by an [xstar]{} photoionization model described in more detail in the Appendix. These are labeled by the value of log($\xi$), where $\xi=L/(nR^2)$ is the ionization parameter as defined by [@Tart69]; $L$ is the source energy luminosity integrated from 1 – 1000 Ry, $n$ is the gas number density, and $R$ is the distance from the continuum source to the absorber. These show that a given value of ionization parameter predicts that the He/H abundance ratio should increase between adjacent elements by a factor $\sim$5. The figure clearly shows that a single ionization parameter cannot account for the ratios displayed by all the elements. This could indicate the existence of a broad range of ionization parameters in the source, spanning values indicated by this figure. On the other hand, it could be associated with radiative transfer effects, such as saturation, which make ion fractions inferred from a linear curve of growth unreliable.
Possible explanations for the departures from the linear curve of growth include the influence of saturation which causes the curve to flatten when the lines become optically thick in the Doppler core. Other possibilities include filling-in of the lines by an additional continuum emission component which is not seen in transmission through the warm absorber, and also radiation transfer effects in the absorber itself. The latter includes forward scattering, which may also depend on the relative size of the continuum source and the absorber. Also, thermal emission can fill in the lines. In the following subsection we discuss these in turn.
Radiation Transfer and Curve of Growth
--------------------------------------
The standard curve of growth for a resonance line can be written:
$$\label{curgrowth}
EW=\int{d\varepsilon \left(1 - e^{-\tau(\varepsilon)}\right)}$$
$$\tau(\varepsilon)=\frac{\pi e^2}{m_ec} f x_{ij}Y_j N \frac{\lambda}{v_{turb}}
\phi(\frac{\varepsilon}{\Delta\varepsilon_{turb}})$$
where $\varepsilon$ is the photon energy, $f$ is the oscillator strength, $x_{ij}$ is the ion fraction, $Y_j$ is the element abundance, $N$ is the total column density, $\lambda$ is the line wavelength, $v_{Turb}$ is the turbulent velocity (including the thermal ion speed), and $\phi$ is the profile function, which includes both a Doppler core and damping wings. This is shown in figure \[figcog\], for various choices of the velocity characterizing the Doppler broadening $v_{turb}$, and for a natural width corresponding to that for the Si XIV L$\alpha$ line. This shows that, for a given equivalent width, the effects of saturation are more likely to be important when the Doppler broadening is smallest. That is, the equivalent width where the curve flattens is approximately proportional to the Doppler width. As we will show, the widths of many lines from GRO J1655-40 are not constrained from below by the HETG spectrum. Thus, a possible explanation for the slower than linear curve of growth is that the line Doppler widths are small enough such that the strongest lines in figure \[coghlike\] are affected by saturation.
![\[figcog\]Curve of growth for various choices of the velocity characterizing the Doppler broadening $v_{turb}$, and for a damping parameter corresponding to that for the Si XIV L$\alpha$ line, at a Doppler broadening velocity of 20 km s$^{-1}$. ](f6.pdf)
Filling in by continuum from a separate source which is not absorbed by the warm absorber would preferentially affect lines with larger optical depths. This could qualitatively explain the flattening of the curve of growth which is observed. However, the filling in would also have the wavelength dependence of the additional continuum component. Although this is unknown, the simplest assumption would be that it is the same as the primary continuum. We have tested this possibility quantitatively using direct fitting, and will discuss this further in the following section.
The standard curve of growth in equation (\[curgrowth\]) assumes the simplest possible geometry and microphysics affecting the line: (i)The continuum source has negligible size; (ii) The absorber exists only in a narrow region along the line of sight to the continuum source; (iii) The excitation and deexcitation of the transition responsible for the line is affected only by the radiative excitation and spontaneous decay connecting the two atomic levels. In addition, the gas is assumed to have a velocity field which can be characterized by a Maxwellian distribution with a well-defined thermal or turbulent velocity.
If the continuum source has finite size, or if the warm absorber exists outside the line of sight to the source, then the line profiles will be affected by photons which scatter into our line of sight. Although there is no simple general expression for the intensity observed from an extended scattering atmosphere in this case, it is straightforward to show that in the limit of small optical depth, the scattered emission contribution scales proportional to $(D/R)^2$, where $D$ is the characteristic size of the continuum source and $R$ the characteristic distance from the continuum source to the absorber. This contribution is independent of optical depth in this limit, and so will not affect the shape of the curve of growth at small $\tau$.
Departures from assumption (iii), concerning the population kinetics, can be expected if the radiative decay of the upper level is suppressed or if the upper level can be populated by another process such as recombination or collisional excitation. Some of these possibilities were discussed by [@Masa04]. If the upper level can decay via alternate channels, such as branching to other levels radiatively, collisional deexcitation, or autoionization, the basic absorption profile properties will be unchanged (although these processes can affect the damping parameter). Suppression of the upper level decay by collisional deexcitation could occur if the product of line optical depth and electron density, $n_e$, is large, but this requires optical depth $\sim 10^{18}/n_e$ for iron. This would affect our limits on possible filling-in of the line by collisional or recombination radiation which we present in the next section. Populating the upper level by collisions or recombination requires suitable temperature and ionization conditions. Efficient population by collisional excitation requires temperatures greater than can be accounted for by photoionization heating and radiative cooling, and this in turn will affect the ionization balance. It also implies the existence of an additional heating mechanism. We have tested these possibilities quantitatively using direct fitting, and discuss the results below.
Direct Fitting
==============
Many of the issues discussed in the previous section can be tested using direct fitting. These include: detector resolution, counting statistics, and scattered and thermal (either collisional or recombination) emission. In order to do so, we use the [warmabs]{} analytic model which interfaces with the [xspec]{} spectral fitting package. [warmabs]{} makes use of a stored, precomputed table of level populations for a family of photoionization models. It uses these to calculate a synthetic spectrum ‘on the fly’ within [xspec]{}. The advantage of this is that it calculates the synthetic spectrum automatically for the energy grid of the data such that it matches the energy resolution of the instrument. Interpolation can introduce significant numerical errors when applied to absorption spectra, since the absorption coefficient can change rapidly over a narrow range in energy. [Warmabs]{} calculates all absorption lines using a Voigt function including damping due to both radiative and Auger decays where applicable. Line profiles are calculated using sub-gridding, on an energy scale which is a fraction of a Doppler width, and the opacity and transmittivity then mapped onto the detector grid. Bound-free absorption is also included. [Warmabs]{} uses the full database and computational routines from the [xstar]{} package [@Kall01; @Baut01], and differs from [xstar]{} in that the level populations are pre-calculated rather than calculated simultaneously with the spectrum. The level populations are calculated using a full ‘collisional-radiative’ calculation, albeit with relatively simple model ions in most cases. Thus they include the effects of upper level depletion due to thermalization and photoionization automatically. We have extended the [xstar]{} database to include all the trace elements seen in the GRO J1655-40 spectrum. A description is provided in the Appendix.
[Warmabs]{} does not calculate the flux transmitted by a fully self-consistent slab of gas, as [xstar]{} does. Rather, it calculates the opacity from the illuminated face of such a slab, and then calculates optical depth and transmitted flux analytically by assuming the opacity is uniform throughout the slab. A real slab will shield its interior, which will then have lower ionization and greater opacity than a slab whose opacity is assumed to be uniform. Thus, the results of [warmabs]{} fitting will be absorbing columns which may be greater than would be produced by a real slab. The importance of this effect depends sensitively on the assumed slab structure, i.e. its geometrical thickness, and so its importance cannot be estimated in general. Also, this effect is negligible at high ionization parameters. As we will show, our fits require large ionization parameters. In this case, the advantages of the energy resolution provided by [warmabs]{} outweigh its limitations.
Model 4: Narrow Lines
---------------------
Our basic fit assumes that the absorption is provided by a single component of photoionized gas. We adopt the ionizing continuum used by [@Mill08] which consists of a power law plus disk blackbody, as described in the previous section. The power law photon index is 3.54 and the disk inner temperature is kT=1.35 keV. In our fit we account for the curve of growth by adopting a small turbulent velocity, 50 km s$^{-1}$, so that the ratios of these lines are on the saturated part of the curve of growth. We then search for the single ionization parameter which most nearly accounts for all the lines in the spectrum. We do this using [warmabs]{} and the [xspec]{} package and stepping through ionization parameter in intervals $\Delta$log($\xi$)=0.1. [Warmabs]{} accept the turbulent velocity as an input, but the actual line width is calculated including both this turbulent velocity and the thermal ion velocity corresponding to the equilibrium temperature. As shown in the Appendix figures \[ionbala\] – \[ionbald\], in order to simultaneously produce the ions of iron ranging from B-like (Fe XXII) to H-like, an ionization parameter in the range 3$\leq$log($\xi$)$\leq$4 is needed. We find the best fit at log($\xi$)=4.0$^+_-$0.1, log(N)=23.8 and a blueshift for the absorber of 375 km s$^{-1}$. This can be compared with the radial velocity of the system of 141$^+_-$1 km s$^{-1}$ and the orbital semi-amplitude of the secondary star of 215.5 $^+_-$2.4 km s$^{-1}$ [@Shah02]. This suggests that the outflow is not moving fast when compared with the maximum velocities characterizing the orbital motion in the system. We have also performed equivalent fits for this set of assumptions using the [isis]{} fitting package [@Houc00], and have verified that the results are independent of which fitting package is used.
We refer to this single component [warmabs]{} with $v_{turb}=50 $ km s$^{-1}$ as model 4. We assume a gas density of 10$^{15}$ cm$^{-3}$ in calculating the level populations used by [warmabs]{}. We have not extensively tested our results at lower densities, and we point out that it is likely that densities as low as $10^{13.8}$ cm$^{-3}$ can produce the Fe XXII lines. We employ the same ionizing continuum as derived in the model 1 fit when calculating the populations and gas temperature using [xstar]{}. In our fitting for model 4 and subsequent models we allow the normalizations of the two continuum components to vary. The best fit flux is 1.8 $\times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$ in the 2 – 10 keV band. In Figures \[fita\]-\[fitn\] we show the observed count rate spectrum (black) together with all the models discussed in this section as red curves, labeled according to the model number. Strong lines are marked in blue along with parent ion.
As discussed above, and shown in figure \[vofffig\], the majority of lines have centroid wavelengths which are consistent with a single Doppler blueshift, approximately 400 km s$^{-1}$ with respect to their laboratory values. The notable exception is the Fe XXVI L$\alpha$ line near 1.77 Å which has a centroid wavelength corresponding to a Doppler blueshift of $\simeq$1300 km s$^{-1}$. In model 4, we adopt the hypothesis that this line arises in a separate velocity component of the flow, which must have a much greater ionization parameter so that it does not show up in the other lines. Since the [xstar]{} grid of model level populations does not extend to values where Fe XXVI is the only ion of significant abundance, we do this by adding a single Gaussian component for the high velocity part of this line. A consequence of this is that the 375 km s$^{-1}$ component of the flow, which we model with [warmabs]{} and an ionization parameter log($\xi$)=4, accurately fits to the red part of the Fe XXVI L$\alpha$ profile, while the Gaussian accounts for the rest. This is shown in figure \[fitfeblowup\], which shows the region surrounding the line. The data is shown as the black bars, and model 4 is shown as the dark blue points. The green points show the continuum + Fe XXVI L$\alpha$ Gaussian model alone. The equivalent width of the Fe XXVI Gaussian is 29.6 eV. The $\chi^2$ for this model is 34211 for 8187 degrees of freedom. These parameters are summarized in Table \[chi2table\].
An alternative possibility is that the Doppler blueshift of the Fe XXVI line is affected by instrumental effects and shortcomings in the available HETG response function. This could be due to the fact that the continuum count spectrum is steeply decreasing in the line region, as shown in figure \[fitfeblowup\]. Internal scattering could cause photons from the continuum adjacent to the longer wavelength wing of the line to scatter into the line core, and this effect would be stronger on the long wavelength side than on the short wavelength side. The width of the line spread function is comparable to the line blueshift. In addition, the wavelength calibration may be affected by the use of continuous clocking mode. On the other hand, other lines in the 1.5 – 2 Å region, although at slightly longer wavelength, do not show this effect, as exemplified by the Fe XXV $1s^2-1s2p$ line. Furthermore, we have confirmed that the response matrix we use reproduces the line spread function obtained from the $Chandra$ calibration database [^1]. So, although there is no obvious shortcoming in the response matrix which would explain the discrepancy between the Fe XXVI L$\alpha$ wavelength shift and those of other lines, we will examine both hypotheses: that the high velocity part of the Fe XXVI line is associated with a separate kinematic component (as in model 4), and that it is not. The latter case also corresponds to the assumptions of [@Mill08]. The primary conclusions of this paper, with respect to dynamics, abundances, turbulence, and geometry, are not dependent on this assumption.
The model 4 fit accurately reproduces the strength and shape of the Fe XXVI and Fe XXV lines, although we predict a slightly stronger red wing on the Fe XXV than is observed. This is the contribution of Fe K$\alpha$ from lower ion stages of iron, Fe XXIV and XXIII. The strengths of these lines provides a lower bound on the ionization of iron. We also slightly overestimate the strength of the Mn XXV L$\alpha$ line, relative to the Mn He-like K$\alpha$ line. This may be due to an error in the ionization balance for this element; as shown in figure \[hheplot\], there are apparent departures from a simple monotonic trend in the H/He ratio with nuclear charge. The region longward of 3Å includes the lines from Ca (3.02, 3.18 Å) and Ar (3.73, 3.94 Å). Our single component ionization balance is too low for both these elements, adequately accounting for the He-like lines but under-predicting the H-like lines. We also point out that our assumed line width of 50 km s$^{-1}$ adequately fits the profiles of essentially all the lines shortward of 6 Å.
Model 4 is based on a physical model for the absorber in GRO J1655-40, and it represents the best fit that we can obtain to the spectrum using a single [warmabs]{} component. As shown in figure \[fita\]-\[fitn\], this accounts for the depths and positions of essentially all the lines in the spectrum. Discrepancies between the model and the data fall in three categories: errors in the continuum, errors in line widths, and errors in line strengths (including lines which are missing). In what follows we will explore these in turn. As we will show, the best fits we obtain to a physical model have $\chi^2/\nu$ $\sim$ 2.9. This can be compared with the phenomenological notch fit, model 3, which has reduced $\chi^2/\nu$ $\sim$ 2.2. The difference likely represents the ability of the phenomenological notch fit to account for features which may not be physical lines but rather part of the continuum or consequences of calibration uncertainties. This, we think, represents the ultimate limit in our ability to fit the spectrum. In what follows, we proceed and attempt to interpret the $\chi^2$ values from physical models when compared with each other, but without relying on standard interpretations of these values and their relation to probability of random occurrence, etc. That is, we acknowledge that our fits are not acceptable based on these standard arguments about $\chi^2$, but nevertheless interpret confidence intervals on the fitting parameters by evaluating $\Delta\chi^2$ as if they were.
Model 4 is designed to fit to the curves of growth of lines from Fe XXIV and Ni XXVI by having a relatively small turbulent velocity, so that the lines from these ions are at least partially saturated. This works for these ions, as is apparent from figure \[fitf\], and it also is consistent with the observed widths of the lines which are not resolved. However, it does not account for the observed widths of the broader lines, such as Si XIII and Si XIV, and also for the 2s – 3p lines of Fe XXIV near 10.6 Å. [@Mill08] adopt a line width of 300 km s$^{-1}$, which is a better fit to the observed widths for many lines. We conclude that saturation with a single profile component, although it accounts for the curves of growth of the lines of Fe XXIV, is not consistent with the curves of growth of lines from Si XIV, nor is it consistent with observed widths of some of the strong lines. A possible explanation is that the lines consist of multiple narrow components closely spaced in velocity so they appear blended together in the HETG, and where the components furthest from line center are unsaturated and so appear only in the low members of the Li-like 2s – np series. UV warm absorber lines in AGN appear to follow this behavior [@Gabe05].
![\[fitfeblowup\]detail of spectrum in the 1.5 – 2 Å region. Crosses are data, blue is model 4, red is model 5, green is the Fe XXVI L$\alpha$ Gaussian contribution to model 4.](f7.pdf)
Model 5: Two components plus broad lines
----------------------------------------
Another way to fit the Fe XXVI L$\alpha$ line is to add a higher ionization parameter [warmabs]{} component to the fit. We do this in model 5, which has a [warmabs]{} component with the same ionization parameter as model 4, but with the addition of a second component at an ionization parameter which can make the Fe XXVI line sufficiently strong without over-producing the lower ionization lines. This model also has a turbulent velocity of 200 km s$^{-1}$. We vary the column densities and element abundances of both components in order to find the best fit. In doing so, we force both components to have the same bulk outflow velocity, the same turbulent velocity, and the same elemental abundances. Thus, the two components are allowed to differ only in their ionization parameters and column densities. This produces a fit with $\chi^2/\nu=36281/8189$. This fit is also shown in figures \[fita\] – \[fitn\]. In figure \[fitfeblowup\] we show a blowup of the iron K region, comparing this model with model 4. Model 4 is decomposed into the part accounted for by the Gaussian (green) and the total (blue). Model 5 is shown as the red points. This shows that in spite of the nearly equivalent $\chi^2$, model 5 does less well than model 4 in fitting the iron lines; it generally over-predicts their strengths. This is particularly true in the case of the Fe XXVI L$\alpha$ line. Since this model has the same 375 km s$^{-1}$ outflow velocity as model 4, the best fit accounts for the blue edge of the Fe XXVI L$\alpha$ by over-predicting the line near line center and the red edge. Thus, the conclusions from this model differ from model 4 (and subsequent models) in that they do not depend on the existence of two kinematic components.
In other ways, the comparison between models 5 and 4 reflects the fact that model 5 has on average a higher ionization parameter. As a result, model 5 tends to over-predict the ratio of H-like to He-like lines, compared with both model 4 and with the observation. Also, model 5 has a larger turbulent velocity, and therefore predicts a steeper curve of growth for most lines. This fact is apparent from the Fe XXIV lines in the 6 – 7 Å region. On the other hand, model 5 fits better than model 4 to lines which appear to follow the linear curve of growth, such as the L$\beta$/L$\alpha$ lines of Si XXIV, and also to lines which are detectably broadened, such as the 10.6 Å doublet of Fe XXIV.
Model 6: Partial Covering
-------------------------
An additional possible reason for the apparent saturation of the curves of growth could be due to systematic effects or calibration uncertainties associated with the $Chandra$ telescope, grating and detector. This could lead to apparent scattering of photons in the continuum into the cores of absorption lines which is not accounted for by the detector response matrix, thus preventing the residual flux in the lines from going to zero. This effect cannot be evaluated accurately without calibration data which includes narrow absorption line features, but we can get an indication of its plausibility by fitting to a model which includes some ‘leakage’ of photons into lines from adjacent continuum regions. We do this by fitting to a partial covering model in which the [warmabs]{} component is partially diluted, i.e. $model=((1-C)+C \times {\sc warmabs}) \times continuum$ where $1-C$ is the fraction of scattered continuum at each energy. We also adopt a turbulent velocity of 200 km s$^{-1}$ for this model, since we do not need to account for saturated curves of growth; the dilution of the [warmabs]{} lines tends to flatten the curves of growth. We show this as model 6 in Table \[chi2table\] and in figures \[fita\] – \[fitn\]. This model has $\chi^2/\nu=24561/8184$, and so is the best of the fits using the [warmabs]{} model. It also employs a turbulent velocity which is sufficient to account for the widths of resolved lines while at the same time fitting to the curves of growth for Fe XXIV and Ni XXVI.
The best-fit value of $C$ is 0.37. We can interpret this as being due to true partial covering in the object, i.e. if the continuum source is more extended than the absorber, or as being due to instrumental scattering which is not accounted for by the response matrix we used. The latter can be further subdivided into shortcomings in the response in accounting for the line spread function (LSF) for the HETG which occur in the core region of the LSF, and shortcomings which occur in the wings. The core region of the LSF [^2]. can be approxmiately represented by a Gaussian with full-width-half-maximum (FWHM) of $\simeq$1300 km s$^{-1}$ at the wavelength of the iron line, 1.77 Å. So two intrinsically narrow lines of equal intensity will fill the region between them to 30$\%$ of their peak intensities if they are separated by $\simeq$ 900 km s$^{-1}$. This is comparable to the typical line widths we find in our fits. However, as discussed previously, the response matrix we use does accurately reproduce the LSF in the core region, so it would require a large error in the calibration files, the LSF and the resulting response matrix, in order to account for the filling in of the lines in this way.
Another possibility is that there exists significant scattering in the line wings which is not accounted for by the calibration. This is difficult to evaluate quantitatively, except to point out that the uncertainty in the LSF at the extremes of the wings appears to be at most $\sim$2 – 4 $\%$ of the maximum value. In order to make up the covering fractions we require, the integrated area under the wings would have to be $\sim$30 – 40 $\%$ of the area in the core of the line. The wings would have to extend to $\simeq$10 times the $\sigma$ of the core, or $\simeq$7600 km s$^{-1}$. We cannot evaluate the likelihood of this possibility reliably, and cannot conclusively rule it out, although such a large departure from the calibration would be surprising. This observation differs from typical observations in its use of continuous clocking mode. This prevents the use of detector regions adjacent to the readout strip for background subtraction, but the counting rate in this case is high so that background should not be important at the levels considered here. The fact that the response matrix agrees with the LSF in the core of the line is further evidence that the use of continuous clocking is not the source of the apparent partial covering.
We conclude that true partial covering in the source is more likely than instrumental effects, and thus plays a role in determining the line curves of growth and the overall quality of the fit. Owing to its superior $\chi^2$ we adopt this model as the one which most nearly accounts for the observed spectrum and consider the physical assumptions to be most nearly correct for the GRO J1655-40 outflow. We emphasize that the partial covering does not account for the high velocity component in Fe XXVI L$\alpha$, and model 6 includes the same high velocity Gaussian contribution to this line as model 4.
Other Models
------------
We also have examined the possible influence of thermal emission filling in lines on the one-component [warmabs]{} fit, i.e. model 4. That is, we have included an additional emission component, which emits due to ‘thermal’ (i.e. collisional and recombination) processes rather than resonance scattering. This component is calculated using the [photemis]{} model, and is the emission analog of [warmabs]{}. It is assumed to have the same ionization parameter, abundances, redshift, and turbulent velocity as the warm absorber component. We allow the normalization of the emission component, which is proportional to its emission measure, to vary, along with the optical depth of the warm absorber. In doing so, we find the best fit to be that with zero thermal emission, and the statistical upper limit on the thermal emission component corresponds to an emission measure $EM \leq 1.2 \times 10^{56}$ cm$^{-3}$. We will discuss the implications of this in the next section.
We have also examined the possibility that the absorption is produced in a plasma which is in coronal equilibrium instead of photoionized. This is what might be expected if the gas is heated mechanically, and if the outflow is due to thermal expansion of such a mechanically heated wind. This is done using the [xspec]{} analytic model [hotabs]{}, which calculates the absorption spectrum of partially ionized gas if the ionization is due to electron impact. The free parameter describing the ionization balance in this gas is the electron temperature. A key difference between a photoionized gas and a gas in coronal ionization equilibrium is that the ionization abundance distributions from photoionized gas have more overlap in parameter space than does a coronal gas. That is, at a given temperature each element in a coronal gas is most likely to exist in a pure ionization state, while at a given ionization parameter in a photoionized gas, each element is likely to have a mixture of two or more ionization states. In addition, the ions which can coexist in a coronal plasma all tend to have similar ionization potential. A consequence is that it is impossible for coronal equilibrium to allow the coexistence of H-like or He-like ions of elements with very different nuclear charge, eg. Fe XXV and S XVI. In contrast, a photoionized plasma at log($\xi$)=4 does allow this. For this reason, coronal equilibrium models do not fit the GRO J1655-40 spectra as well as photoionized models.
Abundances
----------
We have also varied the elemental abundances and explored the limits for these allowed by the $\chi^2$ statistical criterion. In doing this, we rely on the criterion of @Cash79, where a 99$\%$ confidence interval is defined by the values of the parameter which fall within $\Delta\chi^2 \leq 10$ of the best fit value. We display these results graphically in figure \[abundfig\] for the elements O – Ni. The abundances here are taken relative to the solar values of @Grev96 for abundant elements, and @Alle73 for the new elements added for this calculation. In all of our models we fix the abundance of Fe at 1 relative to solar. In Figure \[abundfig\] the results of model 4 are shown in black, and the results of model 6 are shown in red. This shows that the models agree on the abundances of elements Sc through Co, such that the abundances of Cr, Mn and Co are all enhanced relative to Fe by at least 50 $\%$. For Ca model 4 predicts values greater than solar by 0.5-1 dex while model 6 is consistent with solar. As discussed above, we conclude that model 6 most nearly fits the overall properties of the spectrum, and in what follows we discuss the implications of the abundance pattern from this model. Limits on the abundance of O come from the O VIII 1s – 5p line at 14.8 Å and 1s – 6p at 14.64 Å. This results in $Y_O$ values in the range 0.2 – 1.5.
![\[abundfig\]Element abundances: model 4(black) and model 6 (red). log(abundance)=0 corresponds to solar [@Grev96] values.](f8.pdf)
The abundance patterns shown in figure \[abundfig\] are crudely consistent between models 4 and 6, and show enhanced abundances of Cr and Mn relative to Fe. Results for Ca-V are ambiguous, and suggest no strong enhancement, while the abundances of Na-Cl are smaller than the solar ratio, relative to Fe.
The overabundances of Fe-group elements strongly suggest that the observed matter has been subject to high-temperature burning conditions. For this reason, we have compared the observed abundances against nucleosynthesis models in massive stars, and more particularly resulting from the hydrostatic C-, O- and Si-burning phases. We use the presupernova nucleosynthesis yields calculated in model stars of 15, 20, and 25 solar masses with solar metallicity [@Limo00]. Each of the C-, O- or Si-burning zone has been weighted by the relative masses needed to best reproduce the observations. The relative mass ratio (C:O:Si) obtained is (6:1:1) for the three model stars. As shown in Fig. \[fig\_abth1\], the predicted abundances (relative to solar) are in rather close agreement with the observed pattern for models 4 and 6.
As far as the light elements are concerned, the agreement between theory and observation is rather good. The O overabundance predicted reaches 1.3 (1.8 for the 20 M$_{\odot}$ star) in agreement with the 1.5 upper limit determined in the 2-component model. Note, however, that the O abundance is not well constrained by the observed spectrum owing to strong interstellar absorption. Furthermore, we consider it possible that the low $Z$ elements are affected by a separate ionizing continuum component. This could suppress absorption from O even if the abundance were greater than solar, although this is contrived given the absence of observations of such radiation. We also point out the possibility that the Ne and O lines are at least partly of interstellar origin (e.g. @Juet04). This would require a coincidence in velocity between the GRO J1655 outflow and the intervening gas. It would also place more severe constraints on the nucleosynthetic models, since it would decrease the inferred abundances of these elements intrinsic to the source. Concerning the underproduction of Na, the disagreement may be due to uncertainties affecting the nuclear reaction rates or the thermodynamical conditions in the combustion zones. For the heavier elements above Ca, the agreement is still good though discrepancies can be seen in particular for Ti, V, and Co. It should be mentioned here that, in the nucleosynthesis simulations, all unstable nuclei produced have been assumed to $\beta$-decay (except the long-lived $^{53}$Mn).
![Comparison of observed and predicted overabundances. The symbols with errors bars correspond to the 1-component (black) and 2-component (red) models. The lines correspond to model calculations in a 15 M$_{\odot}$ (solid), 20 M$_{\odot}$ (dashed) and 25 M$_{\odot}$ (dotted) star, as described in the text. []{data-label="fig_abth1"}](f9.pdf)
It is interesting to compare with the results of the optical abundance determination by @Isra99. These authors find evidence for enhanced O/H, Mg/H, Si/H, S/H relative to solar, but Fe/H and Cr/H are approximately solar. This differs from our results for models 4 and 6, insofar as they can be compared, since @Isra99 do not measure abundances for Mn, Co, and Ni. The conclusions of @Isra99 have not been confirmed in an investigation by [@Foel07]. This also underscores the fact that the abundances we measure are relative to each other, and all are for elements heavier than O. We have assumed that iron is solar and quote other abundances relative to that, but we have no constraints on abundances for light elements, or for any abundances relative to hydrogen.
Discussion
==========
Other implications of our models include the fact that the curves of growth, in the absence of partial covering, indicate a velocity structure which has a small turbulent velocity, $\simeq$50 km s$^{-1}$ at the same time as a larger bulk velocity, $\simeq$400 km s$^{-1}$ . Since the radial velocity of the GROJ1655-40 system is 141$^+_-$1 km s$^{-1}$, this implies an outflow consisting of material which is cold or still compared with the bulk flow. This is unusual when compared with flows which are well studied in stars and non-thermal outflows from compact objects.
All of our models fit to the iron K region by assuming the existence of a separate, higher ionization component which is primarily responsible for producing the Fe XXVI L$\alpha$ line. This is because essentially all the other lines in the spectrum are consistent with a single ionization parameter and outflow velocity. Models 4 and 6 fit the Fe XXVI L$\alpha$ line using an ad hoc Gaussian, while model 5 fits it using a separate [warmabs]{} component. Although the latter approach is more physically consistent, it does not fit the feature as well because for this model we force the two components to have the same outflow velocity.
In either case, we find that the majority of lines fit to a single ionization component, log($\xi$)$\simeq$4. In addition to this, we have the observed luminosity $L\simeq 5 \times 10^{37}$ erg s$^{-1}$ [@Mill08] and density constraints from the Fe XXII metastable line implying $n \geq 10^{14} {\rm cm}^{-3}$. Taken together, these determine the location of the absorber: $R=\sqrt{L/n/\xi} \simeq 10^9 L_{37}^{0.5} n_{15}^{-0.5} \xi_4^{-0.5}$ cm, and its size: $\Delta R=N/n \simeq 10^9 N_{24} n_{15}^{-1}$ cm. In these equations $L_{37}$ is the luminosity in units of $10^{37}$ erg s$^{-1}$, $n_{15}$ is the density in units of 10$^{15}$ cm$^{-3}$ and $\xi_4=\xi/10^4$. That is, for plausible values, $R \leq 7 \times 10^9$ cm, $\Delta R \leq 10^{10}$ cm. This location can be compared with the ‘Compton radius’ within which photoionized gas cannot escape the gravity of the black hole [@Bege83], which is $R_{IC}=10^{10} (M/M_\odot) T_{IC8}^{-1}$, where $T_{IC8}$ is the temperature of the photoionized gas in units of 10$^8$K, and for the GRO J1655-40 spectrum this value is $\leq$0.03 and the mass of the black hole is 5 – 8 $M_\odot$. If so, $R_{IC} \simeq 2 \times 10^{12}$ cm. Thus the inferred wind location is well within the Compton radius and is inconsistent with an outflow driven by thermal expansion, even though weak flows are possible at $\sim 0.1 R_{IC}$ as stated by [@Wood96]. This conclusion is consistent with that of [@Mill08].
On the other hand, we note that the virial radius corresponding to the outflow speed we measure is 1.3 $\times 10^{12}$ cm, considerably greater than the position we infer from the X-ray ionization balance. This, together with the fact that the outflow speed is comparable to the orbital speed in the system raise the possibility that the outflow could be associated with the secondary star or a region of the binary which is at comparable distance from the black hole. If so, then the radius we infer from the ionization balance arguments is an underestimate. This is difficult to understand in view of the density constraints from the Fe XXII lines; it would require that the metastable level of Fe XXII were populated by radiative pumping rather than by collisions. This, in turn, would require that the intensity of the radiation in the $\sim 100 \AA$ region to be close to LTE, and there is no known source for such radiation far from the black hole. In addition, the observed outflow speed does not appear to vary during the observation [@Mill08], which spanned $\simeq$20$\%$ of an orbital period, which argues against an origin associated with the companion star or accretion stream. Thus, we consider this possibility to be unlikely.
Another astrophysical system which appears to show a comparable contrast between line turbulent width and outflow velocity are FU Orionis stars. In these systems the terminal velocity of the wind is 300-400 km/s, and the rotational broadening (seen in absorption) of the wind is about 50 km/s [@Calv93; @Hart95]. The intrinsic turbulent velocity needed to produce the lines is likely much smaller. If so, the contrast between outflow and turbulent velocities is comparable to the contrast between the observed width and the virial speed at the inferred position for the GRO J1655-40 outflow.
We note also that the flow timescale is $t_{dyn}=R/v_{out}\simeq 27$s. This can be compared with the recombination timescale $t_{rec}=(n \alpha)^{-1} \simeq 10^{-4}$s, showing that the assumption of ionization equilibrium is valid for these simple assumptions about the density and location of the outflow. This does not change qualitatively if the absorber is located at the virial radius.
We can also explore the implications of the emission measure upper limit derived in the previous section, $EM \leq 1.2 \times 10^{56}$ cm$^{-3}$. The emission measure expected from a constant density shell with size derived from the ionization parameter, density and thickness is $EM=10^{57} \Omega L_{37} N_{24} \xi_4^{-1} {\rm cm}^{-3}$, where $\Omega$ is the solid angle of the warm absorber/emitter. So we infer $\Omega \leq 0.12 L_{37}^{-1} N_{24}^{-1} \xi_4$ or $\Omega\leq 0.024$ for the most likely value of $L_{37}$. This can be compared with the constraint derived by [@Mill06] which is $\Omega\leq 1.4$ based on the observed limit on the Fe XXIV 2p – 3s emission line at 11.43 Å. We can also then estimate the mass loss rate in the outflow: $= \Omega R^2 n m_H v_{out} = 2 \times 10^{15} N_{24}^{-1} v_7$ gm s$^{-1}$ where $v_7$ is the outflow speed in units of 100 km s$^{-1}$; so $ \simeq 8 \times 10^{15}$ gm s$^{-1}$ for our fits. This is small compared with the mass accretion rate required to fuel the X-ray source, $_{acc} \simeq L/(\eta c^2)\simeq 6 \times 10^{17}$ gm $^{-1}$. The [@Mill06] limit on the solid angle allows a considerably larger mass loss rate, $\leq$10$\%$ of the accretion rate.
Another conclusion of our work is the existence of two velocity components, one with ionization parameter log($\xi$)=4 and speed $v$=375 km s$^{-1}$ relative to the Earth, and the other with higher speed and much greater ionization parameter. We find for this second component a speed $\simeq$1400 km s$^{-1}$, and infer that the ionization parameter be high enough that Fe XXVI and Ni XXVIII are the only ions with appreciable abundance, say log($\xi)\geq 6$. Then the mass flux is = $10^{17} \xi_{6}^{-1} v_8 \frac{\Omega}{4 \pi}$ gm s$^{-1}$, where we have assumed a source luminosity 5 $\times 10^{37}$ erg s$^{-1}$ and $v_8$ is the speed in units of 1000 km s$^{-1}$. It is more difficult to constrain the fractional solid angle of this component than it is for the lower ionization component because any emission may be masked by the lower velocity component, and we have not attempted to test this. It is clear that this component can carry more mass than the low ionization component if its solid angle exceeds 0.02. However, the radial location of this component cannot be reliably established, since the features in the spectrum which provide density constraints do not share the outflow velocity of this component.
The results found here agree qualitatively with those of [@Mill08] in the sense that the ionization parameter and density of the flow containing most of the lines imply a wind location which is too close to the black hole to be easily explained by a driving thermal mechanism. We favor a scenario in which there are two components to the flow, and the lower ionization component is somewhat less ionized and the higher ionization component is more ionized than the single component found by [@Mill08]. Our spectral fitting allows us to constrain elemental abundances, and we find enhanced abundances for several elements in the iron peak. We do not find a pattern of systematic enhancements for $\alpha$-capture elements.
This work was funded in part by a grant from the Chandra theory program. We thank the referee, Frits Paerels, for many constructive comments.
Appendix: Photoionization Models
================================
In order to model the spectrum of GRO J1655-40 we have made modifications and enhancements of the [xstar]{} [^3] photoionization code [@Kall01]. This code calculates the transfer of X-rays and other ionizing radiation, and the ionization balance, opacity and reprocessed emission from gas under a variety of physical conditions. It contains a relatively complete and up-to-date collection of atomic cross sections, rate coefficients and atomic energy levels. The code and atomic database, along with the ‘[warmabs/photemis]{}’ analytic models for [xspec]{}, are freely available, distributed as part of the ftools package, and have been widely used in interpreting X-ray spectra.
The standard [xstar]{} distribution includes all abundant even-$Z$ elements (plus N), i.e.: H, He, C, N, O, Ne, Mg, Si, S, Ar, Ca, Fe and Ni. In order to model GRO J1655-40 we have added all the other elements with $Z \leq 30$, with the exception of Li, Be, and B. The atomic data needed to do this, in addition to the resulting ionization balance for an optically thin gas, are discussed in this appendix. These data include: energy levels, line wavelengths, oscillator strengths, photoionization cross sections, recombination rate coefficients, and electron impact excitation and ionization rates. Distorted wave dielectronic and radiative recombination rate coefficients, photoionization cross sections, and photoexcitation-autoionization rates have been calculated [@Badn03] for ions of these elements belonging to the H-like through Na-like isoelectronic sequences and are also available in the ORNL ADAS database [@Schu00]. Concerning the other atomic parameters, a survey of atomic databases [@Kall07] reveals that they are far from complete for any of these elements. Therefore we rely heavily on hydrogenic scaling for many of these quantities.
In order to model the spectrum we have modified the [xspec]{} analytic model [warmabs]{} in order to account for the new elements. [warmabs]{} uses the optically thin ionization balance calculated by [xstar]{}, along with the same atomic database as [xstar]{}, to calculate the synthetic spectrum expected for a gas seen in transmission with a given ionization parameter, column density, turbulent velocity and abundance set. This takes into account all of the absorption processes, i.e. lines and photoionization cross sections, included in [xstar]{}. A complementary model, [photemis]{}, calculates the emission spectrum expected from such a gas, but that is not used in the calculations presented here.
Rates and atomic structure for elements H, He, C, N, O, Ne, Mg, Si, S, Ar, Ca, Fe and Ni are the same as those described in [@Kall01; @Baut01], and available in version 2.1l of the [xstar]{} code (http://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html). For the odd-$Z$ elements, and the iron peak elements not included in the release version of [xstar]{}, we use atomic structure scaled according to the following prescription: for the H- and He-like isoelectronic sequences we scale all energies and rates according to that expected for hydrogenic quantities: energies scale according to $Z^2$, photoionization cross sections scale according to $Z^{-2}$, transition probabilities scale according to $Z^4$. For these isosequences we scale the full atomic structure and rate set from [xstar]{}, including all levels through $n=6$ for hydrogenic and through $n=5$ for helium-like ions. For other isoelectronic sequences we adopt hydrogenic scaling of a highly simplified hydrogen-like ion, consisting of two spectroscopic levels, 1s and 2p, along with a super-level and continuum. The energies, transition probabilities, and other rates are scaled according to a hydrogenic prescription with an effective $Z=\sqrt{E_{th}/13.6{\rm eV}}$, where $E_{th}$ is the first ionization potential taken from [@Alle73]. Following this, we correct the energies of the n=2 levels using the evaluated wavelengths of the 2 – 1 transitions from the NIST database. These are available for F VIII, Al XII, Al XIII, Sc XX, Sc XXI, Ti XXI, Ti XXII, V XXII, V XXIII, Cr XXIII, Cr XXIV, Mn XXIV, Mn XXV, Co XXVI, Co XXVII, Cu XXVIII, Cu XXIX. We do not have evaluated wavelengths for the K lines of Na, P, Cl, and K.
For ions not previously included in [xstar]{}, ground state photoionization cross sections are taken from the calculations of [@Vern95].
Recombination rate coefficients were taken from the calculations reported in the series of the papers by Badnell and coworkers [@Badn03] and available from the website [^4]. These are available for all elements He – Zn and for isoelectronic sequences H – Mg-like for both dielectronic (DR) and radiative (RR) recombination. For isoelectronic sequences Al-like through Ni-like, we make use of rates from [@Aldr73; @Arna92; @Shul82], which cover ions of Si, S, Ar, Ca, Fe, and Ni in these isosequences. For ions of other elements in these isosequences we interpolate along isoelectronic sequence. [xstar]{} does not directly use the total recombination rate, but rather calculates rates onto a set of spectroscopic levels, typically with principle quantum numbers $n\leq$6, using photoionization cross sections and the Milne relation. Then it calculates a photoionization cross section for one or more fictitious superlevels such that the total recombination rate for the superlevel(s) plus the spectroscopic levels adds to the desired total rate taken from one of the above compilations. The super levels are generally assumed to decay directly to ground without the emission of any observable cascade radiation. The exception is the decay of the H- and He- isoelectronic sequences, for which we have explicitly calculated the decay of the superlevels to the spectroscopic levels using a full cascade calculation [@Kall01; @Baut01].
Ionization Balance
------------------
A key input affecting the ionization balance is the spectral energy distribution (SED) assumed for the source. We take the spectrum adopted by @Mill08, consisting of a disk black body with inner temperature kT=1.35 keV, plus a power law with photon index $\Gamma=3.54$. The normalizations are chosen such that the low-energy cross-over between the two components is at 1 keV. Below this energy we assume that the spectrum is flat (in photons). We assume that the spectrum incident on the gas in the GROJ1655 system is not affected by interstellar absorption. This spectrum is much steeper in the X-ray band than the conventional $\Gamma\simeq$2 power law applied to AGN. The ionization balance is correspondingly quite different in the sense that the mean ionization is lower at a given value of the ionization parameter in the steep spectrum case.
Our choice of spectrum is deficient in photons between $\sim$1 and 100 Ry, relative to higher energies. This leads the gas to be thermally unstable, as originally described by [@Buff74]. This manifests itself as a region of ionization parameter where the temperature and ionization balance of the gas can be multi-valued. In the case of the spectrum we have chosen, this occurs at an ionization parameter log($\xi)\simeq$ 2. This is less than the range of ionization parameters which can produce the ions observed from GRO J1655, and so it will not affect our results. In what follows we will not discuss this further.
Figures \[ionbala\]-\[ionbald\] show the ionization balance for our choice of incident spectrum, as a function of ionization parameter $\xi=L/(nR^2)$. In our case, we adopt cgs units, so $\xi$ has units erg cm s$^{-1}$. It is clear from this figure that the dominant observed ions require ionization parameters log($\xi)\geq$4, but that production of Li-like and B-like iron and Ni requires ionization parameter values at the low end of this range.
![\[ionbala\]ionization balance: H-Ne](f10a.pdf)
![\[ionbalb\]ionization balance: Na-Cl](f10b.pdf)
![\[ionbalc\]ionization balance: Ar-Cr](f10c.pdf)
![\[ionbald\]ionization balance: Mn-Zn, and temperature.](f10d.pdf)
Aldrovandi, S. M. V., & Pequignot, D. 1973, Ast. Ap., 25, 137
Allen, C. W. 1973, London: University of London, Athlone Press, |c1973, 3rd ed.,
Arnaud, M., & Raymond, J. 1992, , 398, 394
Badnell, N. R., O’Mullane, M. G., Summers, H. P., Altun, Z., Bautista, M. A., Colgan, J., Gorczyca, T. W., Mitnik, D. M., Pindzola, M. S. and Zatsarinny, O., 2003, Astron. Astrophys. 406 1151-65
Badnell, N., 2005, http://amdpp.phys.strath.ac.uk/tamoc/DATA/
Baluci[ń]{}ska-Church, M. 2001, Advances in Space Research, 28, 349
Bautista, M., and Kallman, T., 2001 Ap. J. Supp. 134, 139
Begelman, M. C., McKee, C. F., & Shields, G. A. 1983, , 271, 70
Behar, E., et al., 2003, ApJ, 598, 232
Buff, J., & McCray, R. 1974, , 189, 147
Calvet, N., Hartmann, L., & Kenyon, S. J. 1993, , 402, 623
Cash, W. 1979, , 228, 939
D[í]{}az Trigo, M., Parmar, A. N., Boirin, L., M[é]{}ndez, M., & Kaastra, J. S. 2006, , 445, 179
Foellmi, C., Dall, T. H., & Depagne, E. 2007, , 464, L61
Gabel, J. R., et al. 2005, , 623, 85
Grevesse, N., Noels, A., & Sauval, A. J. 1996, Cosmic Abundances, 99, 117
Halpern, J., 1984 Ap J 281 90.
Harmon, B. A., et al. 1995, , 374, 703
Hartmann, L., & Calvet, N. 1995, , 109, 1846
Hjellming, R. M., & Rupen, M. P. 1995, , 375, 464
Houck, J. C., & Denicola, L. A. 2000, Astronomical Data Analysis Software and Systems IX, 216, 591
Israelian, G., et al., 1999, , 401, 142
Juett, A. M., Schulz, N. S., & Chakrabarty, D. 2004, , 612, 308
Kallman, T. R. and Bautista, M., 2001 Ap. J. Supp 133,221
Jackman, T.R., and Palmeri, P., 2007 Rev. Mod. Phys. 79, 79.
Kaspi, S., et al., 2002, Ap. J., 574, 643
Limongi, M., Straniero, O., & Chieffi, A. 2000, , 129, 625
Masai, K., & Ishida, M. 2004, , 607, 76
Miller, J. M., Raymond, J., Fabian, A., Steeghs, D., Homan, J., Reynolds, C., van der Klis, M., & Wijnands, R. 2006, , 441, 953
Miller, J. M., Raymond, J., Reynolds, C. S., Fabian, A. C., Kallman, T. R., & Homan, J. 2008, , 680, 1359
Netzer, H. 2006, , 652, L117
Orosz, J. A., & Bailyn, C. D. 1997, , 477, 876
Shahbas, T., van der Hooft, F., Casares, J., Charles, P. A., and van Paradijs, J., MNRAS 306, 89
Schultz, D., et al., 2000, http://www-cfadc.phy.ornl.gov/xbeam/xbmintro.html
Shirai, T., Sugar, J., Musgrove, A.,2000 J. Phys. Chem. Ref. Data, Monograph No. 8
Shull, J. M., & van Steenberg, M. 1982, , 48, 95
Strohmayer, T. E. 2001, , 552, L49
Tarter, C. B., Tucker, W. H., & Salpeter, E. E. 1969, , 156, 943
Ueda, Y., et al., 1998, , 492, 782
Ueda, Y., et al., 2001, , 556, L87
Yamaoka, K., et al., 2001, , 53, 179
van der Hooft, F., et al., 1998, , 329, 538
Verner, D. A., & Yakovlev, D. G. 1995, , 109, 125
Woods, D. T., Klein, R. I., Castor, J. I., McKee, C. F., & Bell, J. B. 1996, , 461, 767
[^1]: http://cxc.harvard.edu/caldb/calibration/gratings.html
[^2]: http://cxc.harvard.edu/proposer/POG/html/chap8.html\#fg:hetg-heglrf
[^3]: http://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html
[^4]: http://amdpp.phys.strath.ac.uk/tamoc/DATA/
|
---
abstract: 'The generalised longitudinal susceptibility $\chi({\bf q}, \omega)$ affords a sensitive measure of the spatial and temporal correlations of magnetic monopoles in spin ice. Starting with the monopole model, a mean field expression for $\chi({\bf q}, \omega)$ is derived as well as expressions for the mean square longitudinal field and induction at a point. Monopole motion is shown to be strongly correlated, and both spatial and temporal correlations are controlled by the dimensionless monopole density $x$ which defines the ratio of the magnetization relaxation rate and the monopole hop rate. Thermal effects and spin lattice relaxation are also considered. The derived equations are applicable in the temperature range where the Wien effect for magnetic monopoles is negligible. They are discussed in the context of existing theories of spin ice and the following experimental techniques: dc and ac-magnetization, neutron scattering, neutron spin echo, and longitudinal and transverse field $\mu$SR. The monopole theory is found to unify diverse experimental results, but several discrepancies between theory and experiment are identified. One of these, concerning the neutron scattering line shape, is explained by means of a phenomenological modification to the theory.'
address: 'London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, 17-19 Gordon Street, London WC1H 0AJ, UK.'
author:
- 'Steven T. Bramwell'
date: today
title: Generalised Longitudinal Susceptibility for Magnetic Monopoles in Spin Ice
---
Introduction
============
Following the paper of Castelnovo, Moessner and Sondhi [@CMS] on emergent magnetic monopoles, there has been renewed interest in the properties of spin ice [@Harris; @BramwellHarris; @Ramirez; @BramwellGingras]. Magnetic monopole currents were first envisaged by Ryzhkin [@Ryzhkin] while Jaubert and Holdsworth [@Jaubert; @Jaubert2] studied the closely related problem of magnetic relaxation in spin ice by means of numerical simulations of the dipolar spin ice model [@Siddharthan; @Hertog] and of a dual monopole electrolyte. Evidence for the characteristic non-Ohmic conductivity signature of a weak monopole electrolyte - the Wien effect - was reported in Refs. [@Nature; @NatPhys], where the term ‘magnetricity’ was coined.
Experimental evidence indicates that magnetic monopoles afford an economical description of spin ice at temperatures below $\sim 10$ K [@Fennell; @Morris; @Kadowaki]. In particular, down to about 0.3 K the equilibrium specific heat is well described by Debye-Hückel theory [@Morris; @Zhou; @CMS2]. However, to account in detail for the monopole currents and magnetic relaxation is generally a tricky problem, especially in the regime of slow dynamics at subkelvin temperatures, and this is an ongoing subject of investigation [@NatPhys; @Klemke; @Matsuhira-new].
Prior to the recent wave of interest, the spin dynamics of spin ice were studied in detail by Matsuhira [*et al.*]{} [@Mats] and Snyder [*et al.*]{} [@Snyder] by ac-susceptibility, by Ehlers [*et al.*]{} [@Ehlers1; @Ehlers2] using neutron spin echo, by Lago [*et al.*]{} [@Lago] using muon spin relaxation ($\mu SR$), by Orendáč [*et al.*]{} [@Kitagawa] using bulk magnetocalorimetric methods, by Kitagawa [*et al.*]{} [@Kitagawa] using nuclear quadrupole resonance, and by Sutter [*et al.*]{} [@Sutter] using nuclear forward scattering. Previous studies of the spatial spin correlations by neutron scattering may be found, for example, in Refs. [@Harris; @Bramwell1; @Kanoda; @Fennell1; @Fennell2; @Yavorskii; @Clancy].
The spin correlations in the spin ice state are characterised by two remarkable features [@Fennell; @Morris; @Chang; @Chang2]. The first is a property common to many ice-type models, that transverse magnetization (or polarisation) fluctuations are essentially unrestricted while longitudinal fluctuations are strongly suppressed at low temperature. This behaviour is captured in the phenomenological theory of Youngblood and Axe [@YA] (formulated to describe ice rule ferroelectrics), in which the deconfined defects do not carry any Coulombic charge. The second remarkable property, of course, is that in spin ice these defects do carry a magnetic Coulomb charge [@CMS; @Ryzhkin]. However, they are also associated with a ‘Dirac string network’ of spin configurations, that while not pairing the monopoles, does restrict their motion to some extent [@Jaubert; @CMS2; @Quench].
Any complete theory of spin ice needs to account for the difference between longitudinal and transverse correlations, the Coulombic interactions of monopoles and the effect of the Dirac string network. However, different experiments may pick out one or another of these three properties, so approximate models are useful. If monopoles are the focus then it is of most interest to discuss the longitudinal response as this is a highly sensitive measure of the spatial and temporal correlation of magnetic monopoles, as shown below. The simplest approach to treating the monopole correlations is a ‘magnetolyte’ model of freely diffusing magnetic charges, in which the effect of Dirac strings is subsumed into the transport coefficients, and electrolyte properties such as Debye-Hückel screening [@CMS2], Bjerrum pairing [@Nature; @Zhou; @CMS2] and the Wien effect [@Nature; @NatPhys] may be naturally formulated. Another (and earlier) approach [@Ryzhkin] accounts for the ignored spin degrees of freedom in the form of an effective reaction field, and this approach has recently been developed to include magnetic charge screening [@Ryzhkin-new].
The aim of the present work is to calculate a generalised longitudinal susceptibility $\chi({\bf q}, \omega)$ for magnetic monopoles in spin ice and to explore its application to experiment. The theory described here is only a modest extension of the earlier approaches of Ryzhkin [@Ryzhkin] and Castelnovo [*et al.*]{} [@CMS2], but one that is necessary for the purpose of comparing theory with experiment. A useful by-product of this work is a clarification of the relationship between these two approaches, and their relation to that of Youngblood and Axe [@YA]. The equations discussed here are valid at temperatures that are sufficiently high to avoid the non equilibrium physics of the Wien effect [^1] for magnetic monopoles ($>0.4 $ K for Dy$_2$Ti$_2$O$_7$) [@Nature], but not so high that high energy relaxation processes become important $(> 10$ K for Dy$_2$Ti$_2$O$_7$) [@CMS2]. The results of the present paper are applicable to zero and weak applied field ($\mu_0 H \ll 1$ T) [^2].
Ryzhkin’s Approximation
=======================
Ryzhkin [@Ryzhkin] explored the monopole dynamics of spin ice by applying the Jaccard theory of water ice defects in the water ice-spin ice analogy. He showed that the magnetic current density ${\bf J} = {\bf J}_++{\bf J}_-$ is related to the rate of change of magnetization ${\bf M}$ by the equation (in our notation): $$\label{flux}
{\bf J} = \frac{\partial {\bf M}}{\partial t}$$ and he derived the rate of entropy production associated with the flow of the magnetic charges: $$\label{entropy}
\left(T \frac{\partial S}{\partial t}\right)_{\rm irreversible} = \mu_0 {\bf J} \cdot ({\bf H} - \chi_T^{-1}{\bf M}),$$ where $S$ is entropy and ${\bf H}$ is magnetic field. He finally used these relations to derive a thermodynamic equation of motion: $$\label{Ryzhkin}
{\bf J} = \kappa ({\bf H} - \chi_T^{-1}{\bf M}).$$ Here $\kappa = u c Q \mu_0$ is the monopole conductivity, $u = u_+ = -u_-$ is the monopole mobility, $c = c_+ + c_-$ is the total concentration of free monopoles and $Q=Q_+ = -Q_-$ is the monopole charge. The isothermal susceptibility is predicted to be $\chi_T = 2\chi_C$ where $\chi_C$ is the nominal Curie susceptibility for the spin ice system $(\chi_C \approx 3.95/T$ for Dy$_2$Ti$_2$O$_7$). Eqn. \[Ryzhkin\] contains much physics and is deserving of subtle appreciation. The term in ${\bf H}$ represents the normal drift current of the charges $Q_\pm$ in the applied field ${\bf H}$, and if there were only this term, then spin ice would be represented as a true conductor, precisely equivalent to an electrolyte. However the term in ${\bf M}$ opposes the direct current and indeed extinguishes it completely at equilibrium, where ${\bf M} = \chi_T {\bf H}$. This reaction field does not originate in the magnetic monopoles themselves but rather in the configurational entropy of the monopole vacuum: magnetization of the system reduces that entropy and hence provides a thermodynamic force that opposes the current [@Ryzhkin]. It should also be noted that what stops the current in Ryzhkin’s formulation is not the sample boundaries: this is correct under the approximation that the system is homogenous and linear. If one further allows the competition of diffusion and drift to set up charge density gradients then the boundaries immediately become relevant and one must additionally consider boundary conditions that do not allow the passage of monopoles. However this is not necessary in the approximation considered. Finally it should be emphasised that the extinction of the current implied by Eqn. \[Ryzhkin\] holds only for very small field and magnetization, for it is only in this limit that Eqn. \[entropy\] is valid.
At first sight the right hand side of Eqn. \[Ryzhkin\] is zero but this is only true at infinite time. By introducing a frequency Fourier transform of the magnetization and field and combining Eqn. \[flux\] with Eqn. \[Ryzhkin\], Ryzhkin found that $$\label{acchi}
\chi(\omega) = \frac{\chi_T}{1 -i \omega\tau},$$ where the inverse relaxation time $\tau^{-1} = \kappa/\chi_T$. Eqn. \[Ryzhkin\] can also be integrated to predict the magnetic response to the sudden application or removal of a uniform field (see Section. 12). Assuming an ellipsoidal sample, when a uniform external field ${\bf H}_{ext}$ is applied, the internal field ${\bf H}_{int}$ is reduced by the demagnetizing field $\mathcal{D}{\bf M}$, such that ${\bf H}_{int} = {\bf H}_{ext}- \mathcal{D} {\bf M}$. In spin ice the demagnetizing field arises from the magnetic pole density associated with uncompensated surface monopoles. As the spin ice sample is magnetized, an imbalance of magnetic charge develops at opposite surfaces as a result of the transient monopole current described by Eqn. \[Ryzhkin\]. However, as a result of the entropic ‘reaction field’ discussed above, the monopoles are not sufficiently free to achieve complete screening of the internal field. The incomplete screening of of the internal field due to magnetic monopoles has been discussed in detail by Ryzhkin and Ryzhkin [@Ryzhkin-new].
Definition of the two characteristic rates $\nu$ and $\nu_0$
============================================================
We may define the relaxation rate $\nu$ by: $$\label{rate0}
\nu = \tau^{-1} = \frac{\kappa}{\chi_T} = \frac{\mu_0 u Q x}{V_0 \chi_T},$$ where the concentration $c$ has been substituted for the total dimensionless monopole density or mole fraction $x = c V_0$, and $V_0$ is the volume per site of the diamond lattice inhabited by the magnetic monopoles: $V_0 = (8/3\sqrt{3}) a^3$, where $a$ is the near neighbour spacing on the diamond lattice.
Using the Nernst-Einstein relation, the mobility $u$ may be replaced by the diffusivity $D$: [^3] $$\label{mobility}
u =\frac{DQ}{kT},$$ and hence $$\nu
%= \frac{\mu_0 D Q^2 x}{V_0 kT \chi_T}
= \left(\frac{D}{\chi_T}\right) l_D^{-2},$$ where $l_D$ is the Debye length [^4]: $$l_D = \left(\frac{kTV_0}{\mu_0Q^2x}\right)^{1/2}.$$
In turn $D$ is determined by the monopole hopping frequency $\nu_0$. In a simple lattice diffusion approximation we may write [@NatPhys]: $$\label{ddd}
D = \frac{a^2 \nu_0}{6},$$ where $a$ is the diamond lattice constant (the numerical factor of 6 may be modified very slightly when the fact that a monopole may only hop in three out of four local directions is accounted for [@CMS2]). Using the definitions $Q = 2 \mu/a$ where $\mu$ is the rare earth magnetic moment, and $\chi_C = \mu_0 \mu^2/3V_0kT$, Eqns. \[rate0\], \[mobility\] and \[ddd\] may be rearranged to give: $$\label{x}
\nu = g \nu_0 x,$$ where $g = \chi_C/\chi_T = 1/2$ in Ryzhkin’s theory but is more generally weakly temperature dependent and varies between $g=1$ at high temperature and $g=2$ at low temperature [@TSF]. Equation \[x\] will be seen to be very important for the interpretation of experiments on spin ice.
Coulombic Correlation of the Monopole Current
=============================================
We define the flux of positive and negative monopoles as ${\bf j}_+$ and ${\bf j}_-$ respectively. Assuming there is no temperature gradient in the system, then the thermodynamic equations of motion are (with $i,j=+,-$): $${\bf j}_i = L_{ii} {\bf X}_i + L_{ij} {\bf X}_j,$$ where ${\bf X}$ denotes a generalised thermodynamic force. If we assume that the monopole density is small, then following the theory of weak electrolytes we would expect the cross terms $L_{ij}$ with $i\ne j$ to be zero. However the monopole motion should be strongly correlated in the sense that it always acts to maintain local charge neutrality: $$\label{cur}
Q_+{\bf j}_+ + Q_-{\bf j}_- = 0,$$ $$Q_+c_+ + Q_-c_- = 0.$$ It is important to emphasise that this thermodynamic force is not the same as Ryzhkin’s reaction field which is a purely spin phenomenon peculiar to spin ice. In fact, Eqn. \[Ryzhkin\] does not account for Coulombic correlation between magnetic monopoles and in the next level of description this needs to be accounted for.
The left hand side of Eqn. \[cur\] is simply the magnetic diffusion current ${\bf J}_{\rm diffusion}$ which contributes to the total magnetic current ${\bf J} = \partial {\bf M}/\partial t$. At equilibrium in zero applied magnetic field, the monopole diffusion is such that it does not change the local magnetization of the system. Thus positive and negative monopoles tend to move in the same direction. After a perturbation, the local magnetic current relaxes to zero, even though the monopoles continue to hop around the system. The magnetic diffusion current is given by: $${\bf J}_{\rm diffusion} = \sum_{i=\pm} Q_i{\bf j}_i({\bf r}) = - DQ \nabla \delta c({\bf r}),$$ where $\delta c({\bf r}) = c_+({\bf r})-c_-({\bf r})$. In the zero field equilibrium state the average local gradient of charge density is everywhere zero.
Spatial Dependence of Longitudinal Magnetization {#Sec5}
================================================
The Coulombic correlations create a diffusion force that tends to smooth the local longitudinal magnetization. This may be seen as follows. By Helmholtz’ theorem the vector field ${\bf M}({\bf r})$ can be decomposed into an irrotational or longitudinal ($i$) part and a solenoidal or transverse ($s$) part: $${\bf M}({\bf r}) = {\bf M}^i({\bf r}) + {\bf M}^s({\bf r}).$$ The spin ice ground state is defined by the condition ${\bf M}^i({\bf r}) = 0$, which gives $${\bf M}^{\rm ground}({\bf r}) = {\bf M}^s({\bf r}).$$ Physically, this is a consequence of the spin ice ground state consisting of closed loops of spins. The irrotational or longitudinal part is finite only as a result of thermal excitation of magnetic monopoles. As $\nabla \times{\bf M}^i({\bf r}) = 0$, the vector Laplacian is simply $$\label{Lap}
\nabla^2 {\bf M}^i({\bf r}) =\nabla(\nabla \cdot {\bf M}^i({\bf r})),$$ a result that will be used below. Henceforth (unless otherwise stated) we shall only deal with the longitudinal magnetization and the superscript $i$ will be dropped.
Defining $\phi({\bf r})$ as the magnetic scalar potential, the local magnetic field is: $${\bf H}({\bf r}) = -\nabla \phi({\bf r}).$$ and by Poisson’s equation and Maxwell’s equation, the local magnetic charge density is: $$Q\delta c({\bf r}) = Q(c_+({\bf r})- c_-({\bf r})) = \nabla\cdot{\bf H}({\bf r}) = -\nabla^2 \phi({\bf r}) = -\nabla \cdot {\bf M}({\bf r}),$$ Thus, using Eqn. \[Lap\], the local charge density gradient is: $$Q\nabla \delta c{\bf (r)} = -\nabla^2 {\bf M}({\bf r}).$$ The magnetic diffusion current associated with a finite charge density gradient is: $${\bf J}_{\rm diffusion} = -D Q\nabla \delta c{\bf (r)} = D\nabla^2 {\bf M}({\bf r}).$$ This term can then be added to Eqn. \[Ryzhkin\] to describe relaxation of the spatial charge arrangements: $$\label{newcurrent}
{\bf J}({\bf r}) = \kappa \left({\bf H}({\bf r}) - \chi_T^{-1}{\bf M}({\bf r})\right) + D \nabla^2 {\bf M({\bf r})}$$ In recent work Ryzhkin and Ryzhkin [@Ryzhkin-new] stated such an equation to facilitate a calculation of magnetic screening effects in spin ice.
Free Energy Functional
======================
The same equation can be derived from a Landau-type free energy functional as follows. If we apply a local field ${\bf H}({\bf r})$ then this induces a longitudinal response ${\bf M}({\bf r})$. The local magnetization is opposed by the entropy cost of ordering the spins of the sample as well as the entropy cost of creating a local charge imbalance. Note that these two factors are distinct: it is possible to increase order in the sample without creating a local charge imbalance.
From general chemical thermodynamics we expect the entropic cost of charge imbalance to make the following contribution to the local Gibbs free energy: $$G'_{\rm local} = \frac{kT}{2} \frac{(\delta c({\bf r}))^2}{c}.$$ Hence using Poisson’s equation and Maxwell’s equation again: $$G'_{\rm local} = \frac{kT}{2} \frac{\left(\nabla\cdot {\bf M}\right)^2}{Q^2c_0} = \frac{\mu_0}{2} l_D^2 \left(\nabla\cdot {\bf M}\right)^2
= \frac{\mu_0D}{2\kappa} \left(\nabla\cdot {\bf M}\right)^2$$ We may then write down a free energy functional for the system: $$\label{Landau}
G({\bf M}({\bf r})) = \int \frac{\mu_0}{2\chi_T}M({\bf r})^2 - \mu_0 {\bf M}({\bf r})\cdot{\bf H}({\bf r}) + \frac{\mu_0D}{2\kappa} \left(\nabla\cdot {\bf M}\right)^2 d {\bf r},$$ where the first term on the right represents the Jaccard entropy [@Ryzhkin] which in this representation is seen to be equal to the entropy of a cooperative paramagnet [@Henley]. The rate of change of longitudinal magnetization ${\bf M}({\bf r})$ may be found in a linear response approximation: $$\label{func}
\frac{\partial {\bf M} ({\bf r})}{\partial t} = -\frac{\kappa}{\mu_0}\left[\frac{\delta G({\bf M}({\bf r}))}{\delta {\bf M}({\bf r})} \right]
= \kappa \left[{\bf H}({\bf r})-\chi_T^{-1} {\bf M}({\bf r})\right] + D \nabla^2 {\bf M({\bf r})}.$$ which gives Eqn. \[newcurrent\]. The derivation of the right hand term of Eqn. \[func\] is given in the footnote [^5], from which it can be seen that the contribution of surface charge to the Gibbs energy is neglected.
Under conditions of fixed temperature and field, the rate of dissipation is: $$\left(T \frac{\partial S}{\partial t}\right)_{\rm irreversible} = -\left( \frac{\partial G}{\partial t}\right) =
- \frac{\partial {\bf M}({\bf r}, t)}{\partial t} \frac{\delta G({\bf r}, t)}{\delta {\bf M}}.$$ Hence, using Eqn. \[flux\], $$\label{func}
\left(T \frac{\partial S}{\partial t}\right)_{\rm irreversible} = \mu_0
{\bf J}\cdot
\left[ {\bf H}({\bf r})-\chi_T^{-1} {\bf M}({\bf r})+ \frac{D}{\kappa} \nabla^2 {\bf M({\bf r})}\right],$$ which is the extension of Eqn. \[entropy\] to include monopole diffusion.
Owing to the neglect of surface charge, these equations are generally applicable only under conditions of small field and small magnetization, or else at short time. When these conditions are violated the build up of surface charge may have a decisive influence on the internal fields and on the magnetization process, and the preceding equations would need to be modified to account for this.
Generalised Longitudinal Susceptibility
=======================================
We introduce the spatial and time dependent Fourier transforms of the longitudinal magnetization and longitudinal magnetic field: $${\bf M}({\bf r},t) = \frac{1}{2\pi V} \sum_{\bf q} \int {\bf M}({\bf q,\omega})e^{i{\bf q}\cdot{\bf r}-i\omega t} d\omega,$$ $${\bf H}({\bf r},t) = \frac{1}{2\pi V} \sum_{\bf q} \int {\bf H}({\bf q,\omega})e^{i{\bf q}\cdot{\bf r}-i\omega t} d\omega,$$ and the generalised susceptibility (assuming translational invariance): $$\chi({\bf q}, \omega) = \frac{{\bf M}({\bf q},\omega) }{{\bf H}({\bf q},\omega) }.$$ Substituting these definitions into Eqn. \[newcurrent\] and Eqn. \[flux\], we find: $$\label{cqo}
\chi({\bf q}, \omega) = \frac{\kappa}{\nu_{\bf q} -i \omega}.$$ where: $$\label{nuq}
\nu_{\bf q} = D q^2 + \nu.$$ Note that a generalised susceptibility of this sort could also be derived by solving a Langevin equation incorporating the Landau free energy, as described in Ref. [@Goldenfeld].
Using Eqn. \[x\], the generalised susceptibility can also be written: $$\label{chiqw}
\chi({\bf q},\omega) = \frac{\chi_T}{1+(a^2q^2/6gx)- i \omega \tau} = \frac{\xi^{-2} \chi_T}{\xi^{-2}+q^2- 6g i \omega \tau_0}
%= \frac{1}{\chi_T^{-1}+ q^2l_D^2/6 -i\omega \kappa^{-1}},$$ where $\tau_0 = 1/\nu_0$ and the correlation length is: $$\label{xi}
\xi = \frac{a}{\sqrt{6gx}}.$$
It should be emphasised that this is an equation for the longitudinal susceptibility only (${\bf M} \parallel {\bf H} \parallel {\bf q}$). Whether at equilibrium or not, the latter is finite only if there is a finite density of monopoles. In contrast (see Section \[Sec5\]) the transverse susceptibility of spin ice could take a paramagnetic value at equilibrium even in the complete absence of monopoles, as it does in the monopole-free spin ice ground state. In principle the transverse susceptibility could relax through the flipping of closed loops of spins [@BramwellHarris], though in reality it is more likely that it relaxes through the transient passage of magnetic monopoles. A more complete description of the wavevector dependence of the susceptibility is given in Refs. [@YA; @Henley; @Henley-C; @Fennell], and the possibility of quantum mechanical effects giving rise to transverse dynamics distinct from magnetic monopoles (so called ‘photons’) has been discussed in Refs. [@Shannon; @Benton].
There is potentially a problem with Eqn. \[nuq\] and the subsequent equations. To see this we rewrite $\nu_{\bf q}$ as follows: $$\nu_{\bf q} = \nu_0\left(\frac{q^2a^2}{6}+ gx\right),$$ and assume that the maximum possible equilibrium value of the density is $x = 1/2$. Since it seems implausible that $\nu_{\bf q}$ would ever exceed $\nu_0$ (and indeed $\nu_0$ should generally be less than it), then it appears that Eqn. \[nuq\] breaks down at $q^2a^2 > 3 g$. To guarantee this never happens we can write: $$\nu'_{\bf q} = \nu_0\left(\frac{q^2a^2}{6(1+ A q^2a^2/3)}+ gx\right),$$ where $A$ is introduced as a phenomenological (dimensionless) parameter. Applying Eqn. \[cqo\] we find: $$\label{chinew}
\chi({\bf q},\omega) = \frac{\xi^{-2} \chi_T}{\xi^{-2}+q^2/(1+ A q^2a^2/3)- 6g i \omega \tau_0}.$$ As discussed further below (Section \[Neutron\]), it would seem more realistic to use this phenomenological equation than Eqn. \[chiqw\] in order to describe experiment.
Equilibrium Field Fluctuations due to Monopoles
===============================================
Consider now the spin ice state in zero applied magnetic field, where the internal field ${\bf H}({\bf r})$, which originates from the magnetic monopoles, is no longer constrained but instead relaxes self consistently with the magnetization. An approximation to the problem is that close to equilibrium, the field takes the value ${\bf H}({\bf r}) = \chi_T^{-1} {\bf M}({\bf r})$ everywhere, and that this relation is maintained for small fluctuations away from equilibrium. The internal field therefore costs spin entropy: $$-TS_{\rm field} = \mu_0\frac{\chi_T}{2} {H}({\bf r})^2,$$ and energy $$U_{\rm field} = \frac{1}{2} {\bf B}\cdot{\bf H} = .\frac{\mu_0}{2}H^2(1+ \chi_T),$$ but this is offset by the energy gain in magnetizing the sample: $$U' = -\mu_0 {\bf M}\cdot{\bf H} = -\mu_0\chi_TH^2.$$ Summing these contributions and allowing the magnetic charge density to fluctuate along with the field we find a free energy functional for field fluctuations: $$\label{Gfield}
F({\bf H}({\bf r})) = \int \frac{\mu_0}{2}H({\bf r})^2 + \frac{\mu_0D}{2\kappa} \left(\nabla\cdot {\bf H}\right)^2 d {\bf r}.$$ This functional is entirely equivalent to that for electric field fluctuations in an electrolyte and in fact is equivalent to Debye-Hückel theory [@Oosawa]. Thus we see that by suppressing spin fluctuations (i.e. setting ${\bf H}({\bf r}) = \chi_T^{-1} {\bf M}({\bf r})$) we recover the Debye-Hückel approximation of Castelnovo [*et al.*]{} [@CMS2].
Introducing the Fourier transformed field ${\bf H}({\bf r}) = V^{-1} \sum_{\bf q} {\bf H}({\bf q})e^{i {\bf q} \cdot {\bf r}}$ and substituting into Eqn. \[Gfield\] we find: $$F = \frac{\mu_0V}{2}\sum_{\bf q} { H} ({\bf q})^2 (1+ q^2 l_D^2),$$ where $V$ is the volume and we have used $D/\kappa= l_D^2$ and the fact that the field, being derived from a scalar potential, is longitudinal to the wavevector. Since the probability of a fluctuation is $\propto e^{-F/kT}$, we immediately see that the mean square amplitude of a mode is: $$\langle ({H}_{\bf q})^2\rangle = \frac{kT}{(1+q^2 l_D^2)\mu_0V}.$$ The mean square field at a point is: $$\label{mean}
\langle H({\bf r})^2\rangle = \sum_{\bf q} \langle ({H}_{\bf q})^2\rangle
\approx \frac{V}{(2\pi)^3}\int_0^{\pi/a} \frac{4\pi q^2kT}{(1+q^2 l_D^2)\mu_0V} dq.$$ The integral is dominated by short wavelength modes and the mean square field approximately takes the value: $$\label{meansq}
\langle H({\bf r})^2\rangle
\approx \frac{kT}{2\pi l_D^2 a\mu_0}.$$
Mean square induction at a point
================================
Using Eqn. \[meansq\], the mean square induction at a point is: $$\langle B^2 \rangle = \frac{\mu_0 kT}{2\pi l_D^2 a}(1+ \chi_T)^2.$$ Obviously this equation depends on the induction being averaged over a sufficiently large volume that contains locally magnetized spins. If the point with which we are concerned experiences no local induction from the magnetized spins, and sees only a far field, then the mean square induction is simply $$\label{far}
\langle B^2 \rangle_{\rm far} = \frac{\mu_0 kT}{2\pi l_D^2 a}.$$ Recalling that $l_D^{-2} = \mu_0 Q^2 x/ kT V_0$ where $V_0 \sim a^3$ it may be shown that $$\label{near}
\langle B^2 \rangle_{\rm far} \approx B_0^2 x$$ where $$B_0 \sim \frac{\mu_0 Q}{a^2},$$ and the symbol $\sim$ is used to indicate that factors of order unity are dropped. In the case that there is local induction arising from magnetized spins, this equation may be modified to: $$B_0' \sim \frac{\mu_0 Q}{a^2}(1+\chi_T).$$
It is instructive to derive Eqn. \[near\] in direct space. At a point in the sample the squared field may be averaged over contributions at distance $r$ weighted by the probability of finding a monopole at that distance: $$\langle B^2\rangle \approx \sum \left(\frac{\mu_0 Q}{4\pi r^2}\right)^2 \left(\frac{4\pi r^2 x}{a^2}\right)
\sim \frac{B_0^2 x}{a} \int_a^{\infty} \frac{a^2}{r^2} dr = B_0^2 x,$$ which neglects correlation between the field contributions. The fields are correlated over a distance of $l_D$, but the average number of monopoles within a distance $l_D$ is typically of order unity, so correlation may be neglected to a first approximation. The field $B_0$ is approximately that due to a monopole or a spin at a distance $a$. Thus if a defect is viewed at a distance $a$ it looks like a spin, but if viewed at a much greater distance it looks like a monopole. For this reason monopoles are best detected by measuring their far fields [@Nature].
Relaxation of the Field fluctuations
====================================
The equivalent electrolyte theory has been formulated and worked out in detail by Oosawa [@Oosawa], who found that the relaxation rate of a mode labelled by ${\bf q}$ is: $$\label{rate1}
\nu_{\bf q} = D(q^2 + l_D^{-2}) = \nu_0\left(\frac{a^2q^2}{6}\right)+ \kappa.$$ Thus, short wavelength modes relax at a rate of $\nu_0$, the monopole hop rate, and long wavelength modes relax at a rate $\kappa$, the monopole conductivity. Fluctuations are important on all scales between the lattice constant and the Debye length, so there is a dispersion of relaxation rates from the monopole hop rate $\nu_0$ to the bulk field relaxation rate, or monopole conductivity $\kappa$.
From Eqn. \[mean\], it may be seen that monopoles at distance $a$ and those at distance $l_D$ make similar contributions to the mean square field, while those at much greater distance may be neglected (in zero applied field). Monopoles at distance $a$ reverse the local field at a rate of approximately $\nu_0$, whereas the cloud of monopoles at distance $l_D$ only reverses the field at the much slower rate $\kappa$, although it gives rise to small field fluctuations at a rate $\nu_0$.
Comparison of Eqn. \[rate1\] with Eqn. \[rate0\] suggests that the field correlations relax at a rate $\kappa$ while the spin correlations relax at a rate $\nu = \kappa/\chi_T$. Although this difference may reflect the approximations made, it also seems plausible that the spin correlations relax more slowly than the field correlations at low temperature. Thus the spin system can only relax by the passage of monopoles, so the time taken to find the most probable spin arrangement will generally be longer than the time taken to find the most probable monopole arrangement.
Spin Lattice Relaxation
=======================
It is obvious that the relaxation rate $\nu$ is equal to the spin-lattice relaxation rate $\tau_1^{-1}$ but it is useful to see how this arises in detail. Initial application of a magnetic field $H$ should result in almost instantaneous magnetization, in which energy is stored within the system of effective spins. The spin temperature $T_s$ is therefore initially higher than the applied bath temperature $T$. Following Casimir and du Pré [@Casimir; @Morrish] the temperature difference $\theta = T_s-T$ determines the rate of exchange of heat $\mathcal{Q}$ with the thermal bath, and the consequent return of the spin system to thermal equilibrium at temperature $T$: $$\label{alph}
\frac{d\mathcal{Q}}{dt} = -\alpha \theta.$$ We may also write: $$d\mathcal{Q} = C_H dT_s$$ where $C_H$ is the heat capacity at constant field. Hence we find: $$\frac{d T_s}{dt} = -\frac{\alpha}{C_H} \left(T_s - T\right),$$ and the spin temperature relaxes at a rate $\tau_1^{-1} = \alpha/C_H$.
To link this to the magnetization relaxation we use the thermodynamic relations $$d\mathcal{Q} = TdS = C_MdT-\mu_0 T\left(\frac{\partial H}{\partial T}\right)_M \chi dH,$$ and $$\chi dH = \chi_T dH + \left(\frac{\partial M}{\partial T}\right)_H dT,$$ where $\chi dH = dM$ and $C_M$ is the heat capacity at constant magnetization. If then we apply a field $H = H_1e^{i\omega t}$ and elicit a response $\theta = \theta_1 e^{i\omega t}$ the above two equations may be solved with the substitutions $dH \rightarrow H, dT \rightarrow \theta$, $d\mathcal{Q} \rightarrow -\alpha \theta dT$ (Eqn. \[alph\]) and use of the thermodynamic relation $C_H-C_M = -\mu_0 T (\partial H/\partial T)_M (\partial M/\partial T)_H$. This gives the well-known result [@Morrish]: $$\chi = \left[\chi_S+ \frac{\chi_T-\chi_S}{1+\omega^2\tau_1^2}\right] + i \left[ \frac{(\chi_T-\chi_S)\omega \tau_1}{1+ \omega^2\tau_1^2}\right],$$ where $\chi_S = \chi_T C_M/C_H$ is the adiabatic susceptibility. By comparison with Eqn. \[acchi\] we see that in Ryzhkin’s approximation $\nu = \tau_1^{-1}$ and the adiabatic susceptibility is assumed to be zero. The former is easily understood as any magnetization involves the passage of a monopole current accompanied by dissipation. However, it is conceivable that the adiabatic response could be finite in the real material, and involve the ‘stretching’ of the excited state magnetic moment along the field direction, in which case the adiabatic susceptibility $\chi_S$ could be a direct measure the density of excited states or monopoles. This idea needs to be checked in detail.
Phonon Bottleneck
-----------------
We may also modify this approach to include a ‘phonon bottleneck’. The spin system is considered to be connected to the phonon system at temperature $T_p$, and the phonon system is connected to the bath at temperature $T$. The thermal relaxation between phonon system and bath is characterised by a thermal conductivity $\alpha'$. If we make a steady state approximation to the phonon temperature, then the rate of heat exchange between phonon system and bath is simply: $$\frac{d\mathcal{Q}}{dt} = \frac{dE}{dt} = -\alpha(T_p-T)$$ where $E = U-\mu_0 M H$ is the magnetic enthalpy. Under the approximation that the monopole internal energy $U$ is constant, we find simply that: $$\label{bottle}
T_p = T + (\alpha')^{-1} \mu_0 H \left(\frac{dM}{dt}\right).$$ Thus a thermometer placed on the sample could be used to measure $T_p$ and hence gain an alternative measure of the magnetic current $dM/dt$ after transients have died away.
The rise in temperature of the sample (Eqn. \[bottle\]) occurs when the rate of flow of heat between the spin and phonon systems exceeds the rate of flow of heat from the phonon system to the bath. In the steady state approximation the criterion for this is $\alpha' < \alpha = \nu C_H$. In the low temperature limit we find $C_H \approx (|\mu|/kT^2V_0)x$ [@Zhou], where $\mu$ is the monopole chemical potential [@CMS; @Jaubert] and hence: $$\frac{x}{T} > \sqrt{\frac{\alpha' k}{|\mu| g\nu_0}},$$ is a criterion for observation of this effect. The ratio $x(T, H = 0)/T$ is always sufficiently small that this analysis suggests that the bottleneck can never be observed in zero applied field, and hence any observation of a bottleneck is likely to reflect a significant field-induced increase in $x(T, H)$ (the Wien effect). This conclusion is consistent with the experimental observations of Slobinsky [*et al.*]{} [@Slobinsky] who observed a phonon bottleneck, albeit in fields much stronger than those appropriate to the theory discussed here.
Thermal Quench
--------------
If bound monopole pairs equilibrate sufficiently quickly with the monopole vacuum, then the magnetic monopoles may be regarded as in direct equilibrium with the vacuum: $(0) = (+) + (-)$. The equiiibrium constant is $$K_{eq} = x_0^2$$ where we temporarily label the equilibrium density as $x_0$. By definition the thermodynamic equilibrium constant is given by $K_{eq} = e^{2 \mu/T}$ where $\mu < 0$ is the monopole chemical potential [@CMS; @Jaubert].
Neglecting Bjerrum pairs [@NatPhys], the kinetic rate equation for the change in monopole density is: $$\label{kin}
\frac{dx}{d t} = - \nu_0 \left(\frac{x^2}{4} - e^{2\mu/T}\right).$$ where the first term on the right accounts for monopole recombination and the second for monopole generation. The recombination rate constant has been assumed to be equal to the monopole hop rate. If the temperature is lowered at a rate $dT/dt = -r$ then it follows that $$\label{cool}
\frac{dx}{dT} = \nu_0 r^{-1} \left(\frac{x^2}{4} - e^{2\mu/T}\right).$$ Numerical solution of this equation shows that monopole density reaches a finite approximate steady state of the order $x=r/\nu_0$ at low temperatures. Putting in reasonable parameters for ${\rm Dy_2Ti_2O_7}$ spin ice (e.g. $\nu_0 =1000~{\rm s}^{-1}, \mu = -4.6$ K), a rate of cooling of $r = 10^{-5}$ K s$^{-1}$ (about 1 K per day) would result in a residual density of about $x = 10^{-7}$ at temperatures lower that 0.25 K (see Fig. \[fig1\]). Arrest of the cooling at a base temperature significantly less than 0.25 K, where $\exp(2\mu/T)$ becomes entirely negligible, then results in a very slow power law decay of the monopole density according to (see Eqn. \[kin\]) : $$\label{kin}
x(t) = \frac{x(0)}{1+ x(0)\nu_0 t/4},$$ and even one day of waiting would barely reduce the density by a further power of 10 (Fig. \[fig2\]). Therefore, with any realisable rate of cooling and time of waiting it is not possible to completely rid the system of monopoles on experimental time scales.
![ \[fig1\] Thermal evolution of the monopole density $x$ according to Eqn. \[cool\] (red line) versus the equilibrium density $2\exp(\mu/T)$ (blue line), with the cooling process starting at 1.0 K at a rate $10^{-5}$ K s$^{-1}$ (here $\nu_0 = 10^3$ s$^{-1}$, $\mu = -4.6$ K). ](Monopole_anneal_plot1.pdf){width="0.9\linewidth"}
![ \[fig2\] Temporal evolution of the monopole density $x$ starting at $x=10^{-7}$, according to Eqn. \[kin\] (parameters as in Fig. \[fig1\]).](Monopole_anneal_plot2.pdf){width="0.9\linewidth"}
The preceding analysis neglects many factors that may become important at low temperatures, including possible thermal evolution of the hop rate, extrinsic factors and kinetic constraints arising from the Dirac strings. However, most of these factors will tend to tend to reduce, rather than increase, the rate of relaxation, so it is safe to conclude that the analysis is correct in its conclusion that a monopole-free state in zero applied magnetic field remains inaccessible to experiment [^6]. A detailed analysis of idealised thermal quenches in spin ice [@Quench] has identified the important role of monopole-antimonopole pairs that cannot immediately annihilate by a single spin flip. These ‘noncontractable’ pairs form long lived metastable states at low temperature.
It is also pointed out in Ref. [@Quench] that at sufficiently low temperatures, owing to the divergence of the mobility as $1/T$ (Eqn, \[mobility\]), monopoles will recombine at the maximum speed allowed by the monopole hop rate. However, in the electrolyte theory, this effect is accounted for by the concept of the Bjerrum pair [@Nature; @NatPhys]. Such nonlinear response occurs only within the pair, that is when the monopole-monopole separation is less than $l_T = \mu_0Q^2/8\pi kT$. This fast nonlinear response then appears like the flipping of giant dipoles of magnetic moment $Ql_T$ [@NatPhys], but for monopoles at greater separation, the ordinary recombination kinetics of Eqn. \[kin\] are obeyed. The average monopole separation grows with decreasing temperature much faster than the Bjerrum pair radius, so in a ‘slow’ quench of the sort described above, the divergence of the mobility should not significantly speed up the rate of recombination. The role of Bjerrum pairs, which is closely connected to the Wien effect [@Nature; @NatPhys] is not considered further here.
Application to Experiment: General
==================================
In the following sections I discuss the application of these ideas to different experimental measurements. The equations quoted and derived here should be applicable at sufficiently small applied field and at temperatures ($> 0.4$ K for Dy$_2$Ti$_2$O$_7$) where the Wien effect is absent, so the dimensionless monopole density $x$ depends only on temperature.
It is important to emphasise that the experimental response in all cases depends on $x(T) = \nu/g\nu_0$. There is a general belief that the monopole hop rate $\nu_0$ is temperature independent [@Jaubert]. Assuming this, $x(T)$ can be calculated by numerical simulation [@Jaubert], by Debye-Hückel theory [@CMS2], or approximately inferred from the specific heat [@Zhou]. For Dy$_2$Ti$_2$O$_7$ the monopole density is roughly constant below $10$ K and decreases rapidly as the temperature is lowered below 2 K (see Fig. \[fig3\]). The corresponding relaxation time therefore shows a plateau between $10$ K and $2$ K, and increases rapidly as the temperature is further lowered [@Jaubert]. The picture of monopoles hopping at a constant rate $\nu_0$ breaks down at temperatures above about 10 K, where an Orbach-like relaxation process involving an excited crystal field level becomes important [@Ehlers1].
![\[fig3\] Approximate evolution of the dimensionless monopole density $x$ with temperature [@Bloxsom] for parameters appropriate to Dy$_2$Ti$_2$O$_7$. The values of $x$ have been inferred from fitting experimental specific heat data to Debye-Hückel theory, according to the method of Ref. [@Zhou].](Chplot.pdf){width="0.9\linewidth"}
Magnetization Measurements
==========================
dc-Magnetization
----------------
To treat a bulk magnetization measurement we can set $q = 0$ in the above equations. In any real sample demagnetizing fields need to be accounted for. If we assume an ellipsoidal sample and write ${\bf H}_{\rm int} = {\bf H}_{\rm ext} - \mathcal{D} {\bf M}$ where $\mathcal D$ is the demagnetizing factor, then Ryzhkin’s equation becomes: $$\frac{\partial{\bf M}}{\partial t} = \kappa ({\bf H_{\rm ext}} - {\bf M}(\chi_T^{-1} + \mathcal{D})).$$ For the case of a steady field this equation may be integrated to find: $${\bf M}(t) = \frac{{\bf H_{\rm ext}}(1-e^{-t/\tau})}{\chi_T^{-1} + \mathcal{D}} + {\bf M}(0)e^{-t/\tau},$$ so the relaxation of the magnetization is purely exponential. It may be seen that the susceptibility $\chi_T$ behaves as an effective demagnetizing field and that the apparent susceptibility is $$\chi_a \equiv \frac{M}{H_{\rm ext}} = \frac{1}{\mathcal{D}+\chi_T^{-1}},$$ which tends towards $1/\mathcal{D}$ as $T \rightarrow 0$.
This equation may be used to describe field cooled (FC) and zero field cooled (ZFC) magnetization measurements. It is assumed that in the FC experiment, the sample is cooled sufficiently slowly that it always remains in equilibrium (although we have shown that this cannot be strictly true), but that in the ZFC experiment it is heated at a sufficient rate to be observed on a timescale $t_{obs} \ll \nu, \kappa$. With these approximations the FC and ZFC magnetizations are: $$\label{eqzfc}
{\bf M}_{ZFC} = \frac{{\bf H_{\rm ext}}(1-e^{-t_{obs}/\tau})}{\chi_T^{-1} + \mathcal{D}},$$ $$\label{fc}
{\bf M}_{FC} = \frac{{\bf H}_{\rm ext}}{\chi_T^{-1} + \mathcal{D}}.$$ Using reasonable parameters, these equations predict a large FC-ZFC splitting in $M(T)/H$, as shown in Fig. \[fig4\].
![ \[fig4\] Field cooled (FC, blue) versus zero field cooled (ZFC, red) splitting according to Eqn. \[eqzfc\], \[fc\], based on Ryzhkin’s theory of monopole current [@Ryzhkin]. The observation time has been set at $t_{obs} = 100$ s, the monopole hop rate at $\nu_0 = 10^3$ s$^{-1}$ and the demagnetizing factor at $\mathcal{D} = 1/3$. The monopole density has been roughly approximated by $x \approx 2 e^{\mu/T}/\left(1+ 2 e^{\mu/T} \right)$ with $\mu = -4.6$ K, appropriate to ${\rm Dy_2Ti_2O_7}$. ](MHplot.pdf){width="0.9\linewidth"}
Here it has been assumed that there is a single observation time of about 100 s, which must be a rather crude approximation. Nevertheless, a dramatic FC-ZFC splitting, qualitatively similar to that shown, was observed in experiment by Snyder [*et al.*]{} [@Snyder]. There appear to be two principal ways in which the experimental result differs from Fig. \[fig4\]. First, the experimental FC magnetization below the splitting temperature, becomes temperature independent at a value smaller than the theoretical $M=H/\mathcal{D}$ [@dimensional]. Second, the experimental splitting temperature ($\sim$0.65 K) for Dy$_2$Ti$_2$O$_7$ is higher than that which can be reasonably justified by Ryzhkin’s model. The higher than expected splitting temperature appears to be related to an anomalous slowing down of relaxation seen in ac-magnetization [@Matsuhira-new; @Quilliam], as well as in numerical simulations [@Jaubert]. Possible causes of the experimentally observed slowing down include the constraints imposed by of the Dirac string network [@Jaubert; @Quench], thermal coupling effects [@Slobinsky] and a transition in the monopole density [@Ryzhkin-verynew]. Also, the Wien effect is important in this regime and will play a role in the transient response [@NatPhys; @dimensional].
As regards Ryzhkin’s prediction [@Ryzhkin] that $\chi_T = 2 \chi_C$, a recent theoretical study [@TSF], using parameters appropriate to ${\rm Ho_2Ti_2O_7}$ spin ice, has shown that there is a very slow crossover between $\chi_T = \chi_C$ at high temperature ($\sim 100$ K) to $\chi_T = 2 \chi_C$ in the low temperature limit. Experimental measurements appear to be consistent with this prediction [@TSF]. This ‘Curie law crossover’ has not yet been experimentally confirmed for ${\rm Dy_2Ti_2O_7}$.
ac-Magnetization
----------------
For ac-magnetization measurements, Ryzhkin’s equation (Eqn. \[Ryzhkin\]) can be applied, using a demagnetization correction. As described above, the rate $\nu= 1/\tau_1$, the spin-lattice relaxation rate that arises in the Bloch equations.
Although Matsuhira [*et al.*]{} have shown that the relaxation is never a simple exponential at the temperatures of interest [@Mats], it appears that the characteristic relaxation time does behave according to Ryzhkin’s theory. Thus at high temperature we would expect a characteristic relaxation time $\tau = 1/g\nu_0 x$ and this is born out in experiment in the temperature range $> 2$ K for Dy$_2$Ti$_2$O$_7$ where $x \approx 1/2$ [@Jaubert]. However at lower temperatures, it is evident that the relaxation rate may not be simply proportional to the monopole density [@Jaubert; @Matsuhira-new; @Quilliam].
Neutron Scattering {#Neutron}
==================
Conventional Neutron Scattering
-------------------------------
Having accounted for the atomic form factor and assuming sufficiently small energy transfer, the partial differential cross section of conventional neutron scattering ($\sigma''$) is proportional to the imaginary part of the generalised susceptibility: $$\sigma'' \propto \frac{kT}{\hbar \omega} {\rm Im}[\chi^{\alpha,\beta}({\bf K}. \omega)].$$ Here $\alpha, \beta = x,y,z$ and ${\bf K}$ is the scattering vector. As shown in Ref. [@Fennell], a polarised neutron scattering experiment may be used to isolate the longitudinal ($zz$) susceptibility discussed here by scanning through a Brillouin zone centre perpendicular to the reciprocal lattice vector ${\bf K}_0$. It is particularly useful to use ${\bf K}_0 = (0,0,2)$ in the face centred cubic basis as there is no nuclear Bragg peak at that wavevector [@Fennell].
For scans along this direction (which corresponds to a scan across the “pinch point” [@Fennell], using Eqn. \[cqo\] and setting ${\bf q = K-K}_0$, we find $$\label{sig}
\sigma''({\rm longitudinal}) \propto \frac{\kappa T}{(Dq^2+ \nu)^2+ \omega^2}.$$ Unfortunately the dynamics of spin ice are generally too slow to test this expression. Instead it is possible to energy integrate and measure in the static approximation whereby the differential quasi-elastic cross section $\sigma'$ is given by: $$\sigma'({\rm longitudinal}) \propto T\chi(\bf q).$$
Using Eqn. \[cqo\] with $\omega = 0$ we find: $$\label{cqo2}
\sigma'({\rm longitudinal}) \propto \frac{\xi^{-2} T \chi_T}{\xi^{-2}+q^2},$$ where as already stated, $\xi = a/\sqrt{6gx}$.
In general $x$ is well approximated by: $$\label{n}
x \approx \frac{\exp(-n(T) J_{\rm eff}/kT)}{1+ \exp(-n(T)J_{\rm eff}/kT)},$$ where $J_{\rm eff}$ is the effective exchange parameter for a given spin ice [@BramwellGingras]. Here the prefactor $n \rightarrow 4$ in the low temperature limit and $n \rightarrow 2$ in the high temperature limit as a result of Debye-Hückel screening [@Jaubert]. For Ho$_2$Ti$_2$O$_7$ spin ice $J_{\rm eff}/k \approx 1.8$ K, so there is a regime at intermediate temperature where $\xi \sim \exp(1.8/T)$. In the experiments of Fennell [*et al.*]{} [@Fennell], the neutron data along wavevectors perpendicular to 002 were fitted to the sum of a Lorentzian function and a flat background. The inverse Lorentzian width was indeed found to depend on temperature as predicted here ($\exp(1.8/T)$) although its absolute value was much larger than predicted. The flat background was also found to depend on temperature according to Eqn. \[n\] at high temperatures (with $n=2$ and a correction for ‘double charge’ monopoles).
### Possible Explanation of the Discrepancy
There are two potential corrections to Eqn. \[cqo2\] that we did not consider in Ref. [@Fennell]. The first stems from the modification of Eqn. \[chiqw\] to give Eqn. \[chinew\], as discussed above. Applying this gives: $$\sigma' ({\rm longitudinal}) \propto \frac{\xi^{-2} T \chi_T}{\xi^{-2}+q^2/(1+ Aq^2a^2/3)}.$$ The second would account for the wavevector dependent misalignment between ${\bf M} ({\bf q})$ and ${\bf q}$. However, this is a relatively minor correction and is not considered further here. Writing $a = (\sqrt{3}/4) a_{\rm fcc}$ where $a_{\rm fcc}$ is the face centred cubic lattice constant and $q = \sqrt{2} (2\pi/a_{fcc}) h $, we find (for a scan along $hh0$) $$\sigma' ({\rm longitudinal}) \propto \frac{(\sqrt{8} \pi \xi/a)^{-2} T \chi_T}{(\sqrt{8} \pi \xi/a)^{-2}+h^2/(1+ (A\pi^2/2)h^2)}.$$ Using Eqn. \[xi\] this may also be written: $$\sigma' ({\rm longitudinal}) \propto \frac{(3gx/4\pi^2)T \chi_T}{(3gx/4\pi^2)+h^2/(1+ (A\pi^2/2)h^2)}.$$ These expressions produce a lineshape and temperature dependence that is very similar to that observed in Ref.[@Fennell], in that they incorporate both the apparent Lorentzian (making it appear anomalously sharp) and the flat background, and they also predict the correct temperature dependence in both cases. It would be interesting to compare them in detail to the experimental data.
Neutron Spin Echo
-----------------
Neutron spin echo measures the intermediate scattering function $S({\bf K},t)$ which is proportional to the frequency Fourier transform of the right hand side of Eqn. \[sig\]. Thus we predict $$S({\bf q},t) \sim \kappa T \exp(-\nu_{\bf q} t),$$ with $\nu_{\bf q}$ given by Eqn. \[rate0\]. A test of this expression would require measuring neutron spin echo for scattering transverse to the pinch point, as above. Experiments so far [@Ehlers1; @Ehlers2] have integrated over larger ranges of ${\bf q}$, including transverse fluctuations, and in a temperature range where $\nu_{\bf q} \approx \nu_0$. A temperature independent relaxation rate has been observed [@Ehlers1; @Ehlers2], but for Ho$_2$Ti$_2$O$_7$ this was several order of magnitude faster than that derived by ac-susceptibility on Dy$_2$Ti$_2$O$_7$. Notwithstanding a possible variation between materials it seems likely that the measured relaxation rate is technique dependent, even though its temperature dependence is not. This suggests a high frequency component to the monopole response that is not contained in the present approximations.
Muon spin relaxation and rotation
=================================
Longitudinal Field $\mu$SR
--------------------------
In a $\mu$SR experiment the muon is self trapped by the lattice distortion it creates. In a dense magnetic oxide like spin ice it is therefore prone to distort the local magnetic environment that it is aiming to probe. Despite this, the published results of $\mu$SR experiments are reasonably explained by the monopole model.
Thus a longitudinal field $\mu$SR experiment on Dy$_2$Ti$_2$O$_7$ was performed by Lago [*et al.*]{} [@Lago], who analysed the long time muon depolarisation rate as a measure of the field fluctuation rate. Hence this should have been a measure of $\nu$ or $1/\tau_1$. The temperature dependence of the corresponding relaxation time is indeed very close to that expected, and it seems very likely that the experiment was observing magnetic monopoles. However the magnitude of the relaxation time was an order of magnitude smaller than that inferred from ac-magnetization measurements. This would again suggest a high frequency component to the monopole response, as noted above.
Transverse Field $\mu$SR
------------------------
If a muon implants into spin ice at a site of large local field, then transverse field $\mu$SR is an uninteresting probe of the spin ice system. Hence we will assume that the muon is at a site of zero local field, either within the sample or exterior to the sample, but near the surface. While the assumption of zero field sites within the spin ice sample gives a highly consistent description of experiment [@Nature; @NatPhys], their existence has been contested on theoretical grounds [@Dunsiger] and the issue has been debated [@Comment; @Claudio; @Blundell].
At sufficiently high temperature (T $>$10 K) we might expect the TF-$\mu$SR dephasing rate $\lambda$ to give a measure of $1/\tau_2$, the spin-spin relaxation rate, which may be specified by a BPP type [@BPP] expression: $\tau_2^{-1}$: $$\frac{1}{\tau_2} = \frac{1}{2\tau_1}+ \frac{\gamma^2 \langle (\delta B^z)^2 \rangle}{\nu'},$$ where $\nu'$ is approximately the spin flip rate and $\delta B^z$ is the scale of the fluctuations of the field component parallel to the applied field. As the latter term tends to dominate, we shall only consider this term from now on.
In the spin ice regime, where $x = \nu/g\nu_0$ is the dominating parameter of the system, the TF-$\mu$SR response is found to have a form that is unfamiliar in the context of $\mu$SR on paramagnets. To explain this it is useful to first consider the dimensional analysis of the problem.
Dimensional Analysis for TF $\mu$SR
-----------------------------------
$\mu$SR theory for a simple paramagnet may be formulated in terms of two parameters: $\Delta$ and $\nu'$. Here $\Delta = \gamma \sqrt{\langle B^2\rangle}$ where the right hand term is the instantaneous root mean square field at the muon site, $\gamma$ is the muon gyromagnetic ratio, and $\nu'$ is the relaxation rate of this local field. In terms of dimensional analysis we would say that $\nu'$ and $\Delta$ constitute two governing parameters, both with the dimensions of $[1/time]$. The quantity of interest in transverse field $\mu$SR is the characteristic rate of muon dephasing, $\lambda$. The formal solution to the problem is: $$\lambda = \Delta f\left(\frac{\nu'}{\Delta}\right),$$ where $f$ is an undetermined function.
In the slow fluctuation limit $\nu'/\Delta \rightarrow 0$ and for $\lambda$ to be finite we have $f \rightarrow constant$. In the fast fluctuation limit $\Delta/\nu' \rightarrow 0$ and we expect $\lambda\rightarrow 0$. The asymptotic form is in fact linear in the small parameter, $f(1/\epsilon) \sim \epsilon$. The two solutions thus become: $$\label{one}
\lambda_{\rm slow} \sim \Delta,$$ $$\label{two}
\lambda_{\rm fast} \sim \frac{\Delta^2}{\nu'},$$ formulae that are often used for the analysis of $\mu$SR data.
These formulae may be rationalised by the following heuristic argument. If the field sensed by the muon is approximately static on the muon lifetime, then the muons precess in phase at a Larmor frequency $\gamma B_a$ where $B_a$ is the applied transverse field, but accumulate a phase difference $\Delta \phi = \Delta t$ in time $t$. If $1/\lambda$ is equated with the time to dephase by order 1 radian then we obtain $\lambda \sim \Delta$. If, on the other hand, the field jumps randomly at rate $\nu$, with jump magnitude $\Delta$, then the phase difference accumulated between flips is $\Delta \phi = \Delta/\nu'$ and the phase undergoes a random walk with end to end distance $\nu t (\Delta/\nu)^2$ in time $t$, yielding Eqn. \[two\].
The case of spin ice is unusual in that there are three, not two, governing parameters. The origin of the third governing parameter is in the thermodynamics of the Coulomb gas in the grand canonical ensemble where the monopole number $N$ is the sole extensive system parameter. We have defined $x=N/N_0$ as a dimensionless monopole density (where $N_0$ is the number of diamond lattice sites) and $\nu_0$ is the temperature independent monopole hop rate. As discussed above, the relaxation rate of the local magnetic field is $\nu = g x \nu_0$ and we may define a scale for the field $\Delta_0$, that depends only on fixed microscopic parameters.
The formal solution of dimensional analysis can be written: $$\lambda_{\rm slow} = \Delta_0 f\left(\frac{\Delta_0}{\nu_0}, \frac{\nu}{\nu_0}\right).$$ The physical picture we wish to explore is that low temperature ($x \rightarrow 0$) corresponds to slow fluctuations, and high temperature ($x\approx 1$) corresponds to fast fluctuations. Taking the slow fluctuation limit $\nu_0/\Delta_0, \nu/\Delta_0 \rightarrow 0$ now does not necessarily eliminate $\nu/\nu_0 = g x$ from the problem. Whether it does so or not depends on the function $f$. If muons detect monopolar fields only (that is, the longitudinal susceptibility), then we would expect $\lambda$ to go to zero as a power law in $x$, for in the absence of monopoles there should be no dephasing. In contrast, in the fast fluctuation limit $\nu/\nu_0$ does drop out of the problem and we again recover Eqn. \[two\]. The two solutions appropriate to the detection of monopolar fields are therefore: $$\label{three}
\lambda_{\rm slow} \sim \Delta_0 \left(\frac{\nu}{\nu_0}\right)^y,$$ $$\label{four}
\lambda_{\rm fast} \sim \frac{\Delta_0^2}{\nu_0}.$$
Thus in the slow fluctuation limit we expect $\lambda \rightarrow \nu^y$, while in the fast fluctuation limit, we expect $\lambda \rightarrow constant$. The former is an unusual result in the context of $\mu$SR and applies to the case where the muons sense only monopolar fields.
TF$-\mu$SR at Low Temperature
-----------------------------
At low temperature the monopole gas is sparse ($ x \ll 1$) and muons that are close to monopoles are rare. The muon experiment acts to some extent as a spectroscopy, associating different field contributions with different times of observation. Hence to use the average field may not be quite correct. The muon signal at long times measures only typical muons, which, are far from magnetic monopoles. The typical distance of a muon to a monopole is approximately $r^{\ast}/a \approx x^{-1/3}$ and the field sensed by the muon is $
|B| \approx B_0 x^{2/3}$. Since this field is random in direction we get the same result for the mean square field as above, but with the exponent $4/3$ on $x$ instead of $1$. In general we might expect the apparent mean square field to be given by the equation $\langle B \rangle^2 = B_0^2x^y$, with $y\approx 1$.
In this limit the Debye length $l_D$ is very large and the conductivity $\kappa$ is very small. Although $\kappa$ scales with $x$, if $y < 2$ then $\gamma \sqrt{\langle B^2\rangle }$ is always larger than it, and the fields are quasistatic (here $\gamma$ is the muon gyromagnetic ratio). If we approximate the fields as completely static on the muon lifetime, then the muons sense a $z-$component of the local field that is of the order of the root mean square field. The field sensed by the muons is approximately $B_0 \pm \sqrt{\langle B^2\rangle }$ and the muons precess coherently at a Larmor frequency $\gamma B_0$ but are dephased by the spread in local fields. Introducing $\Delta = \gamma \sqrt{\langle B^2\rangle }$ the spread of phases accumulated in time $t$ is $$\Delta \phi = t \Delta.$$ The dephasing time $1/\lambda$ is equated with the time taken for $\Delta \phi$ to become of order one radian, with the result [^7] : $$\lambda^{\rm low~T} = \Delta \approx \gamma B_0 x^{y/2}.$$ Hence $\lambda$ is $$\lambda^{\rm low~T} = \gamma B_0 \left(\frac{\nu}{g\nu_0}\right)^{y/2} = \gamma B_0 \left(\frac{\kappa}{g\chi_T \nu_0}\right)^{y/2}.$$
The muon dephasing function depends on the actual field distribution. However it is always of the form: $$P = P(\lambda t).$$ This form (with $\lambda \propto \nu$) was assumed in Ref. [@Nature] and gave a highly consistent description of experiment. Although this applies the current ideas in the Wien effect regime, one would expect this to be reasonable on the grounds of the dimensional arguments given above. Note also that the method of Ref. [@Nature] is insensitive to the precise form of the local field distribution. The typical value of $\lambda$ observed in Ref. [@Nature] was of the order $10^5$ s$^{-1}$. For DTO spin ice $\gamma B_0 \approx 10^8$ s$^{-1}$, so a $\lambda$ of $10^5$ s$^{-1}$ corresponds to x = $10^{-6}$ if $y = 1$ and the monopole field at a typical muon site is about $10^{-3}$ T. The temperature at which the monopole density is expected to fall to this value is 0.3 K, which is consistent with the observations of Ref. [@Nature].
TF$-\mu$SR at High Temperature
------------------------------
In the high temperature limit $x$ becomes of order unity so $\nu \approx \nu_0$. Thus as we pass from low to high temperature, monopoles hopping at a rate $\nu_0$ located near to the muon become increasingly important, but as remarked above, these monopoles cannot be distinguished from spins, and we return to a model of spin flipping at rate $\nu_0$. In this case the ordinary equations of $\mu$SR apply.
Conclusion
==========
The main conclusion of the present work is that magnetic monopoles in spin ice largely determine the longitudinal response of the system. The sole system variable for both static and dynamic response is the dimensionless monopole density $x$, which is determined in a complex way by the four fixed parameters of the problem: $a,Q,\mu$ and $\nu_0$. In contrast, the transverse response does not directly mirror monopole correlations.
The main theoretical results of this paper are contained in Eqns. \[x\], \[newcurrent\], \[Landau\], \[nuq\], \[xi\], \[Gfield\], \[rate1\], \[far\], \[near\].
Temporal and spatial correlations are linked by $x$ and the Eqns. \[x\] and \[xi\] combine to establish a dynamic scaling relation: $$\nu \sim \xi^{-z}$$ with $z = 2$, as would be expected for a problem of Brownian motion. It follows (see Eqn. \[nuq\]) that there is a dispersion of relaxation rates on all scales from the monopole hop rate $\nu_0$ to the magnetization relaxation rate $\nu$. Some evidence has been noted to suggest that field fluctuations relax more quickly than spin fluctuations (see Eqns. \[nuq\], \[rate1\]) but more work is needed to establish this.
The exponents $\nu$ and $z$ defined in this way, and the correlation length $\xi$, are not conventional quantities as they reflect monopole rather than spin correlations. The spin correlations obey static scaling in the following sense. The correlation function $g(r)$, being pseudo dipolar [@YA; @Henley] decays as $g(r)\sim r^{-3}$. Applying the scaling relation $g(r)~r^{-(d-2+\eta)}$ we find $\eta = 2$. As the susceptibility diverges as $1/T$, the susceptibility exponent $\gamma$ takes the value $\gamma=1$. Applying the scaling relation $\nu = \gamma/(2-\eta)$ we find $\nu$ is infinite, meaning that the spin-spin correlation length remains finite at all temperatures. Thus $T=0$ marks an unusual critical point with algebraic decay of spin correlations, a divergent spin susceptibility, but a non-divergent spin correlation length. It is interesting to observe however, that the monopole correlation length does diverge at $T=0$.
Free energy functionals for the magnetization and field fluctuations have been derived (Eqns. \[Landau\], \[Gfield\]) and shown to relate closely to Eqn. \[newcurrent\], previously stated by Ryzhkin and Ryzhkin [@Ryzhkin-new]. In future work it would be interesting to express these as functionals of the density ($x$) and to add further terms to account for energy fluxes in the system, as well as Wien dissociation, which both play a role at low temperature [@Nature; @NatPhys; @Klemke].
The generalised susceptibility (Eqn. \[nuq\]) at the level of Ryzhkin’s description [@Ryzhkin] has been derived, as well as the field fluctuation at the level of Debye-Hückel theory [@CMS2]. The latter was used to calculate the longitudinal field fluctuation at a point in the system (Eqns. \[far\], \[near\]), which may be compared to established results for electrolytes [@Oosawa]. The expressions for the mean square field distribution have been used to show that a point probe such as a muon at a ‘spin free’ site (either inside the sample or just outside) will give a direct measure of the monopole density as assumed in Ref. [@Nature].
It has been shown that according to the electrolyte theory, a non-equilibrium population of monopoles is always frozen into the sample, regardless of the rate of cooling. However in sufficiently weak magnetic field there is never a phonon bottleneck. The former effect should generally be considered when treating low temperature experimental data.
In general the theory discussed here works qualitatively well for real spin ice materials, capturing the temperature, wavevector and time dependence of a diverse range of experimental responses. However, there are three clear discrepancies. First, while the temperature and wavevector dependence of the neutron scattering cross section are well accounted for, the amplitude of the correlation length is not, being an order of magnitude longer in experiment compared to theory [@Fennell]; a possible explanation of this has been proposed here. Second, while different experiments [@Snyder; @Mats; @Ehlers1; @Ehlers2; @Lago] agree on the temperature dependence of the relaxation rate, they exhibit a wide range of relaxation rates: it appears that there is a high frequency response, not accounted for in the hopping model. Third, the ac-susceptibility relaxation in the high temperature limit is more strongly dispersed [@Mats] than predicted by the simple approximations discussed here.
It seems very unlikely that the monopole theory will have to be abandoned to explain these discrepancies. More likely it needs to be refined. In addition to the possible revision of the neutron scattering line shape discussed above, one might also need to consider the effect of quantum fluctuations [@Shannon; @Benton] or minor terms in the spin ice Hamiltonian [@Yavorskii]. Also, microscopic factors affecting the rate of local spin flipping or monopole hopping probably remain to be identified. However, it should also be emphasised that the most distinctive aspect of Coulombic correlation - the tendency to Bjerrum pairing - has not been accounted for here, or in other ‘high temperature’ theories, but will certainly play a role. Thus Bjerrum pairs have been argued to be important in the low temperature non equilibrium regime [@NatPhys] and have been identified in specific heat measurements [@Zhou]. Finally, the Wien effect [@Onsager], though weak at the ‘high’ temperatures considered here, still exists in a screened form [@Liu], and should be accounted for in a more accurate description. Although there is much work to be done, it is clear that the monopole theory of spin ice [@CMS; @Ryzhkin] is a remarkably simple and effective description of a complex condensed matter system.
[99]{}
C. Castelnovo, R. Moessner, and S. L. Sondhi, [*Nature*]{} [**451**]{}, 42 (2008).
M. J. Harris [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**79**]{}, 25547 (1997).
S. T. Bramwell and M. J. Harris. [*J. Phys. Condensed Matter*]{} [**10**]{}, L215 (1998).
A. P. Ramirez [*et al.*]{}, [*Nature*]{} [**399**]{}, 333 (1999).
S. T. Bramwell and M. J. P. Gingras, [*Science*]{} [**294**]{}, 1495 (2001).
I. A. Ryzhkin, [*J. Exp. and Theor. Phys*]{} [**101**]{}, 481 (2005).
L. D. C. Jaubert and P. C. W. Holdsworth, [*Nature Phys.*]{} [**5**]{}, 258 (2009).
L. D. C. Jaubert and P. C. W. Holdsworth, [*J. Phys Condens Matter.*]{} [**23**]{}, 164222 (2011).
R. Siddharthan [*et al.*]{}, [*Phys. Rev. Lett.* ]{} [**83**]{}, 1854 (1999).
B. C. den Hertog and M. J. P. Gingras, [*Phys. Rev. Lett.*]{} [**84**]{}, 3430 (2000).
S. T. Bramwell [*et al.*]{}, [*Nature*]{} [**461**]{}, 956 (2009).
S. R. Giblin [*et al.*]{}, [*Nature Physics*]{} [**7**]{}, 252 (2011).
T. Fennell, [*et al.*]{}, [*Science*]{} [**326**]{}, 415 (2009).
D. J. P. Morris [*et al.*]{}, [*Science*]{} [**326**]{}, 411 (2009).
H. Kadowaki [*et al.*]{}, [*J. Phys. Soc. Jpn.*]{} [**78**]{}, 103706 (2009).
H. D. Zhou [*et al.*]{}, [*Nature Communications*]{} [**2**]{}, 478 (2011).
C. Castelnovo, R. Moessner and S. L. Sondhi, [*Phys. Rev. B*]{} [**84**]{}, 144435 (2011).
B. Klemke [*et al.*]{}, [*Journal of Low Temperature Physics*]{} [**163**]{}, 345 (2011).
K. Matsuhira [*et al.*]{}, [*J. Phys. Soc. Japan*]{}, [**80**]{}, 123711 (2011).
K. Matsuhira [*et al.*]{}, [*J. Phys.: Condens. Matter*]{} [**13**]{}, L737 (2001).
J. Snyder [*et al.*]{}, [*Phys. Rev. B*]{} [**69**]{}, 064414 (2004).
G. Ehlers [*et al.*]{}, [*J. Phys.: Condens. Matter*]{} [**15**]{}, L9 (2003)
G. Ehlers [*et al.*]{}, [*J. Phys.: Condens. Matter*]{} [**16**]{}, S635 (2004).
J. Lago, S. J. Blundell and C. Baines, [*J. Phys. Condens. Matter*]{} [**19**]{}, 326210 (2007).
M. Orendáč [*et al.*]{}, [*Phys. Rev. B*]{} [**75**]{}, 104425 (2007).
K. Kitagawa et al., [*Phys Rev B*]{} [**77**]{}, 214403 (2008).
J. P. Sutter et al., [*Phys Rev B*]{} [**75**]{}, 140402(R) (2007).
S. T. Bramwell [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**87**]{}, 047205 (2001).
M. Kanada, [*et al.*]{}, [*J. Phys. Soc. Jpn.*]{} [**71**]{}, 313 (2002).
T. Fennell [*et al.*]{}, [*Phys. Rev. B*]{} [**70**]{}, 134408 (2004).
T. Fennell [*et al.*]{}, [*Phys. Rev. B*]{} [**72**]{}, 224411 (2005).
T. Yavors’kii [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**101**]{}, 037204 (2008).
J. P. Clancy [*et al.*]{}, [*Phys. Rev. B*]{} [**79**]{}, 014408 (2009).
L. J. Chang [*et al.*]{}, [*Journal of Physics: Conference Series*]{} [**211**]{}, 012013 (2010).
L. J. Chang [*et al.*]{}, [*Phys. Rev. B*]{} [**82**]{}, 172403 (2010).
R. W. Youngblood and J. D. Axe, [*Phys. Rev. B*]{} [**23**]{}, 232 (1981).
C. Castelnovo, R. Moessner and S. L. Sondhi, [*Phys. Rev. Lett.*]{} [**104**]{}, 107201 (2010).
S. T. Bramwell, [*J. Phys.: Condens. Matter*]{} [**23**]{}, 112201 (2011).
I. A. Ryzhkin and M. I. Ryzhkin, [*JETP Letters*]{} [**93**]{}, 384 (2011).
L. D. C. Jaubert [*et al.*]{}, arXiv:1204.6266 (2012).
T. Sakakibara [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**90**]{}, 207205 (2003).
A. V. Shtyk and M. V. Feigel’man, [*Pis’ma v ZhETF*]{} [**92**]{}, 884 (2010).
V. Kaiser and P. C. W. Holdsworth, private communication.
G. H. Wannier, [*Statistical Physics*]{}, New York: Wiley (1966).
N. Goldenfeld, [*Lectures on Phase Transitions and the Renormalization Group*]{}, Addison-Wesley (1992).
C. L. Henley, [*Phys.Rev. B*]{} [**71**]{}, 014424 (2005).
C. L. Henley, [*Annual Review of Condensed Matter Physics*]{} [**1**]{}, 179 (2010).
N. Shannon [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**108**]{}, 067204 (2012).
O. Benton [*et al.*]{}, arXiv:1204.1325 (2012).
F. Oosawa, [*J. Theor. Biol.*]{} [**39**]{}, 373 (1973)
H. B. G. Casimir and F. K. du Pré, [*Physica*]{} [**5**]{}, 507 (1938).
A. H. Morrish, [*The Physical Principles of Magnetism*]{} (Wiley and Sons, 1965).
D. Slobinsky [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**105**]{}, 267205 (2010).
J. Bloxsom and S. T. Bramwell, unpublished.
J. A. Quilliam [*et al.*]{}, [*Phys. Rev. B*]{} [**83**]{}, 094424 (2011)
I. A. Ryzhkin [*et al.*]{}, [*JETP Letters*]{} [**95**]{}, 330 (2012).
S. R. Dunsiger [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**107**]{}, 207207 (2011).
S. T. Bramwell and S. R. Giblin, arXiv:1111.4168v1 (2011).
G. Sala [*et al.*]{} arXiv:1112.3363v1 (2011).
S. J. Blundell, [*Phys. Rev. Lett.*]{} [**108**]{}, 147601 (2012).
N. Bloembergen, E. M. Purcell and R. V. Pound, [*Phys. Rev.* ]{} [**73**]{}, 670 (1948).
C. Henley, private communication (2011).
L. Onsager, [*J. Chem. Phys.*]{} [**2**]{}, 599-615 (1934).
C. T. Liu, [*Ph.D. Thesis*]{}, Yale University (1965).
[^1]: In spin ice the direct current Wien effect takes the form of a transient increase in charge density under applied field, before equilibration to a value lower than the zero field value [@Vojtech]; the alternating current Wien effect should take the form of a steady state increase in current.
[^2]: A detailed discussion of the dynamical susceptibility of spin ice in the vicinity of the critical point induced by a field of $\mu_0 H \approx 1$ T applied along the $[111]$ axis [@Sakakibara] has been given by Shtyk and Feigel’man [@Shtyk].
[^3]: The Nernst-Einstein equation, which may be derived from the Boltzmann transport equation, is one of the basic equations of electrochemistry. According to Wannier [@Wannier], it was used by Nernst in 1888 to make the first direct measurement of the elementary electronic charge - at a time when the electron had not yet been identified and even the existence of atoms or ions was controversial. The Nernst-Einstein equation may be used in this way because diffusivity and mobility are independently measurable for an electrolyte. However, for magnetic monopoles in spin ice no way has yet been identified to measure these quantities independently.
[^4]: In the temperature range considered here (e.g. 0.4 - 10.0 K for ${\rm Dy_2Ti_2O_7}$ ), at zero applied field, the Debye length is is the dominant length scale in the system, and as shown below, magnetic inhomogeneities develop on this length scale. In the lower temperature range, not considered here, inhomogeneities on the scale of the Bjerrum length $l_T = \mu_0Q^2/8\pi kT$ become particularly important, and probably dominate: see Refs. [@Nature; @NatPhys; @dimensional].
[^5]: The functional derivative of $\tilde{G}=\int (\nabla\cdot {\bf M})^2 d{\bf r}$ is found as follows: $$\left\langle \frac{\delta \tilde{G}[{\bf M}({\bf r})]}{\delta {\bf M}}, {\bf f}\right\rangle
=\frac{d}{d \epsilon}
\int \{\nabla \cdot ({\bf M}+ \epsilon {\bf f})\}^2 d{\bf r} ~~~|_{\epsilon=0}=\int 2(\nabla\cdot{\bf f})(\nabla \cdot {\bf M}) d{\bf r},$$ where the angular brackets indicate a distribution with vector test function ${\bf f}({\bf r})$. Now $$\nabla\cdot[({\bf f})(\nabla \cdot {\bf M})]=(\nabla\cdot{\bf f})(\nabla \cdot {\bf M})+({\bf f}) (\nabla (\nabla\cdot{\bf M}) ),$$ so using Eqn. \[Lap\] and the divergence theorem, $$\left\langle \frac{\delta \tilde{G}[{\bf M}({\bf r})]}{\delta {\bf M}}, {\bf f}\right\rangle
=2 \int_S [({\bf f})(\nabla \cdot {\bf M})]\cdot{\bf n} da - \int 2 ({\bf f}) (\nabla^2{\bf M})
~d{\bf r},$$ where $S$ denotes a surface integral, ${\bf n}$ a unit normal to the surface and $a$ a surface element. If the surface charge is everywhere zero then we have: $$\left\langle \frac{\delta \tilde{G}[{\bf M}({\bf r})]}{\delta {\bf M}}, {\bf f}\right\rangle
= - \int 2 ({\bf f}) (\nabla^2{\bf M})
~d{\bf r},$$ and finally $$\frac{\delta \tilde{G}[{\bf M}({\bf r})]}{\delta {\bf M}}
= - 2 (\nabla^2{\bf M}).$$
[^6]: One may contrast the case of pure water at room temperature, where a density of $\sim10^{-9}$ H$^{+}$ ions per water molecule is easily maintained at equiliibrium on experimental time scales: however the diffusion constant of H$^{+}$ in water is some $10^8$ times larger than that of magnetic monopoles in spin ice.
[^7]: Henley [@Henley2] has considered relaxation functions $G(t)$ for nmr for nuclei in sites of zero local field, and in the dilute monopole limit has shown that $G(t) = \exp \left(-n (t/\tau_i)^{\beta}\right)$ where $\beta$ is a positive exponent of order unity and $\tau_i (i=1,2)$ is a characteristic timescale for longitudinal (1) and transverse (2) relaxation. The results obtained here (for zero local field sites) are consistent with the $\mu$SR relaxation function taking this general form.
|
---
abstract: 'In SAR domain many application like classification, detection and segmentation are impaired by speckle. Hence, despeckling of SAR images is the key for scene understanding. Usually despeckling filters face the trade-off of speckle suppression and information preservation. In the last years deep learning solutions for speckle reduction have been proposed. One the biggest issue for these methods is how to train a network given the lack of a reference. In this work we proposed a convolutional neural network based solution trained on simulated data. We propose the use of a cost function taking into account both spatial and statistical properties. The aim is two fold: overcome the trade-off between speckle suppression and details suppression; find a suitable cost function for despeckling in unsupervised learning. The algorithm is validated on both real and simulated data, showing interesting performances.'
address: |
$^{1}$ Dipartimento di Ingengeria, Università di Napoli Parthenope\
$^{2}$Dipartimento di Scienze e Tecnologie, Università di Napoli Parthenope
bibliography:
- 'refs.bib'
title: A New Ratio Image Based CNN algorithm for SAR Despeckling
---
SAR, deep learning, speckle, cnn, denoising
Introduction {#sec:intro}
============
Interpretation and understanding of remote sensing images is always an open issue. There are a lot of applications such as object detection, classification, land use segmentation and denoising aiming to extract information by remote sensing data. A very challenging environment is the Synthetic Aperture Radar (SAR) imaging system. SAR images are affected by multiplicative noise, called speckle, that impairs the performance in all applications. In the last decades several despeckling filters have been proposed. Generally, even if there is not a clear-cut classification, the speckle filters are divided in two groups: local and non local filters. The formers produce filtered images where each output pixel is given by averaging values in its neighbourhood, assuming that closer pixels should bring similar information. Filters like Lee, its enhanced version and Kuan filter belong to this class. These filters suffer of edges blurring. On the edge between two structures the pixel values can be very different and averaging the neighbourhood produces smoothness in the filtered image. In order to overcome this issue, non local filters like PPB, SAR-BM3D, FANS and NL-SAR look for similarity in a larger windows search instead of a close neighbourhood. For a review of former method refer to [@Argenti2013], instead for latters to [@Deledalle2014] and for FANS to [@Cozzolino2014]. In the last years, deep learning is spreading in several image processing fields achieving very good results, not less in the remote sensing community. Deep learning methods have been proposed in applications like classification, detection, data fusion and despeckling. In [@Chen2014] a CNN based approach for SAR target detection is proposed. In [@Scarpa2018] and [@Scarpa2018RS] deep learning methods for pansharpening and SAR-optical data fusion are used.
Regarding despeckling, the lack of a clean reference is still an open issue. In order to overcome this problem, [@Wang2017] train a CNN using simulated data, instead Chierchia et al [@Chierchia2017] trains the network on multilook real SAR images. Following [@Wang2017], in this work we propose a supervised CNN for despeckling, using a cost function given by a linear combination of a per-pixel and a statistical loss. The aim is to show how extracting statistical information helps the network to solve the usual trade-off between speckle suppression and details preservation. Moreover, the use of a statistical loss can help in future to build a neural network for unsupervised despeckling.
Convolutional Neural Networks
=============================
As said, deep learning and convolutional neural networks are becoming fundamental part of several image processing applications. There is not a predefined structure for a CNN, but in general it is a neural network that, in addition or substitution of fully connected layers use convolutional layers. Generally, different layers are combined: convolutional, pooling, batch-normalization, soft-max, non linearities. The number, the kind and the way in which they are combined depend on the final task.
A generic convolutional layer is define by a set of $ M $ kernels of dimension $(K \times K)$. So the $l$-th generic convolutional layer, for $N$-bands input $\bx^{(l)}$, yields an $M$-band output $\bz^{(l)}$ $$\bz^{(l)} = \bw^{(l)} \ast \bx^{(l)} + \bbb^{(l)},$$\
whose $m$-th component is a combination of 2D convolutions:
$$\bz^{(l)}(m,\cdot,\cdot) = \sum_{n=1}^N \bw^{(l)}(m,n,\cdot,\cdot) \ast \by^{(l)}(n,\cdot,\cdot)+ \bbb^{(l)}(m).$$ The tensor $\bw$ is a set of $M$ convolutional $(K\times K)$ kernels, while $\bbb$ is a $M$-vector bias.
Let $\Phi_l\triangleq\left(\bw^{(l)},\bbb^{(l)}\right)$ be the learnable parameters of $l$-th layer. Usually, the output of the layer is followed by an activation function $g_l(\cdot)$ in order to introduce non-linearities. In this work all the convolutinal layers, except the first and the last, are followed by a pointwise ReLU activation function $g_l(\cdot)\triangleq \max(0,\cdot)$ producing the intermediate layer outputs ( the set of $ M $ so-called [*feature maps*]{}) $$\by^{(l)}
\triangleq f_l(\bx^{(l)},\Phi_l) =
\begin{cases}
\max(0,\bw^{(l)} \ast \bx^{(l)} + \bbb^{(l)}), & l<L\\
\bw^{(l)} \ast \bx^{(l)} + \bbb^{(l)}, & l=L
\end{cases}$$ whose concatenation gives the overall CNN function\
$$f(\bx,\Phi) = f_L(f_{L-1}(\ldots f_1(\bx,\Phi_1),\ldots,\Phi_{L-1}),\Phi_L)
\nonumber
\label{eq:chain}$$\
where $\Phi\triangleq(\Phi_1,\ldots,\Phi_L)$ is the whole set of parameters to learn.
Inorder to train the parameters, several training couples of input and reference output samples must be provided, a cost function and a optimization process must be chosen. The cost function $L(\cdot,\Phi)$ compares the similarity between predicted and reference outputs. An optimization process tries to minimize $L$ and depending on it the parameters $\Phi$ are updated.
PROPOSED METHOD {#sec:print}
===============
Speckle Data Simulation
-----------------------
As said before, SAR images are affected by a multiplicative noise called speckle. Let $Y$ be a intensity SAR image, it can be expressed as [@Argenti2013]:\
$$Y = f(X,N) = X\cdot N
\label{eq: sar formation}$$\
where $X$ is the noise-free image and $N$ is the multiplicative speckle. In the hypothesis of fully developed speckle, $N$ has a Gamma distribution [@touzi2002]:\
$$p(N) = \frac{1}{\Gamma(L)} N^L e^{-NL}
\label{eq: speckle distribution}$$\
where $L$ is the number of looks of the SAR image. An ideal despeckling filter will remove the noise without introducing artefacts and preserving the spatial informations.
In this work simulated data are used following the scheme in equations (\[eq: sar formation\])-(\[eq: speckle distribution\]). We consider three datasets of clean images [@Wang2017]: scraped Google Maps that provides urban images, UCID and BSD that provide generic images. We simulated speckle with Gamma distribution and apply it on this three datasets.
Training with Kullback-Leibler divergence
-----------------------------------------
In the proposed method the network (Fig.\[fig: net\]) is composed by 10 convolutional layers. In order to speed up the convergence, each layer except the first and last is followed by a Rectified Linear Unit (ReLu). Moreover for the same reason, batch normalization is performed for each layer except the last. The training process is performed by the Stochastic Gradient Descent with momentum, with learning rate $ \eta = 2 \cdot 10^{-6}$ on $30000 \times (65 \times 65)$ training patches and $12000 \times (65 \times 65)$ for the validation.
In this work we focus our attention on a customized cost function which aim is two fold: firstly have better speckle suppresion and edge preservation; secondly moving towards unsupervised despeckling.
Given a single band noisy image $Y$, and the noise-free reference image $X$, the predicted output is $ \hat X=f(\bx,\Phi)$ (see in Fig.\[fig: net\]) and the predicted noise is $\hat{N} = Y/\hat{X}$ In this work we propose as cost function $L$ a linear combination given by $$L(x,\Phi) = ||f(x,\Phi) - X||_2^2 + \lambda \sum_{i} \log_2 \frac{p_{\hat{N}}(i)}{p_N(i)} \cdot p_{\hat{N}}(i)
\label{eq: cost}$$
where the first term is the Mean Square Error (MSE) between output and its reference, while the second one is the Kullback-Leibler divergence (KL) between the predicted noise probabilities distribution $p_{\hat{N}}$ and that of simulated speckle noise $p_N$ (that follow the Gamma distribution)
With this cost function the network predicts the clean image taking into account the statistical speckle properties. MSE forces the network to predict directly the image by a per pixel comparison with the reference, and KL ensures that the removed noise has probability distributions as close as possible to the Gamma distribution.
As said in previous sections, all the methods show a trade-off between speckle reduction and edge preservation. With the combination in equation (\[eq: cost\]) we try to preserve both spatial and spectral informations. Moreover, using a cost function like $L_2$ means to use a cost function that is independent from the reference and it can be an elegible cost function for despeckling in unsupervised neural networks. Based on the adopted methodologies, the proposed technique will be referred as Kullbacl-Leibler Despeckling Neural Network (KL-DNN) algorithm
-- -- -- -- --
-- -- -- -- --
Experiments
===========
-- -- -- --
-- -- -- --
-- -- -- --
-- -- -- --
The proposed method is tested on both simulated and real SAR images and a comparison two of the most accredited despeckling algorithms, PPB [@Deledalle2009] and FANS [@Cozzolino2014], is conducted. Fig. \[fig: res\_simul\] shows the results on two simulated clips. KL-DNN seems to have better performance compared with PPB and FANS. PPB tends to be oversmoothed on both clips loosing a lot of spatial details. FANS preserves the details better respect PPB but suffers of oversmoothing mainly on small objects. For example, in the first clip FANS misses all the cars in the parking lot, instead in the second clip the edge of the buildings are blurred. Moreover some distortions appear on the roof. Regarding KL-DNN, on clip1 it better preserves small objects like cars. Moreover the image seems more detailed than PPB and FANS. In clip2 KL-DNN has similar performances to FANS, even if the edge of building and spatial details like foliages of trees are better preserved. The disadvantage of KL-DNN is that it suffers of distortion on smooth surfaces: in the parking lot and on the building’s roof some dark spots appear.
In order to have a numerical assessment, for the simulated data in which the clean reference is available, the SSIM, PSNR and SNR indexes are computed. Tab. \[tab: res\] confirms the previous consideration: KL-DNN shows an improvement with respect to PPB and FANS mainly on the first clip; regarding the second clip the gain respect FANS is more slight.
In Fig. \[fig: res\_vele\], results on real SAR image are shown. The image presents both man-made objects and natural areas.
Generally, the high value given by multiple reflections in man-made areas are very challenging to filter and all the three methods shows some difficulties. PPB filters the big scale object like building but it tends to be over smoothed in flat zones, suppressing a lot of details. FANS and KL-DNN have very close performance: FANS seems to better preserve building details, however it introduces some artefacts and oversmooths homogeneous areas; on the other hand KL-DNN finds challenging the man made structures while showing good performances in other areas.\
To have a more clear idea of the filtering quality, ratio images of the a central patch of image are shown in Fig. \[fig: ratio\_vele\]. The ratio image of a perfect filter should contain only speckle. Considering the ratio image of PPB, it is evident that PPB suppresses too many details. As we expect, KL-DNN faces difficulties with strong backscattering (multiple bouncing). However, it is important to note that, except for these points, the ratio image of KL-DNN appears characterized by homogeneously distributed speckle, differently from FANS where some structures appear. It means that KL-DNN has a better ability in suppressing speckle while preserving the details.\
In order to have a numerical assessment the M-index [@Gomez2017] and KL divergence are listed in Tab.\[tab: res\]. KL-DNN achieves the best performance for both indexes, validating the qualitative analysis.
[c]{}
SSIM PSNR SNR
-------- -------- --------- --------
PPB 0.6715 24.9778 5.4899
FANS 0.7731 27.2326 7.7447
KL-DNN
: Numerical Results: a) evaluation on simulated clip1. b) evaluation on simulated clip2. c) evaluation on real SAR image[]{data-label="tab: res"}
\
a)\
SSIM PSNR SNR
-------- -------- --------- --------
PPB 0.6356 23.0970 6.0597
FANS 0.762 25.6149 8.5775
KL-DNN
: Numerical Results: a) evaluation on simulated clip1. b) evaluation on simulated clip2. c) evaluation on real SAR image[]{data-label="tab: res"}
\
b)\
[lcccc]{}
& M-index & &&KL div\
PPB & 12.04&&&0.0147\
FANS & 11.22&&&0.0089\
KL-DNN & &&&\
\
c)\
Conclusions {#sec:foot}
===========
In this work a convolutional neural network for despeckling is proposed. We use a cost function that relies on per-pixel distance between output and reference and, at the same time, on the statistical properties of the noise. The use of this cost function helps the network to suppress the noise while preserving spatial details. Despite some spatial distortions, the proposed method seems to have better detail preservation than other methods, mainly on small object. In fact, more small details are preserved which is a key feature for the scene interpretation of the remote sensing image.
|
---
abstract: 'The $l$^th^ partial barycentric subdivision is defined for a $(d-1)$-dimensional simplicial complex $\Delta$ and studied along with its combinatorial, geometric and algebraic aspects. We analyze the behavior of the $f$- and $h$-vector under the $l$^th^ partial barycentric subdivision extending previous work of Brenti and Welker on the standard barycentric subdivision – the case $l = 1$. We discuss and provide properties of the transformation matrices sending the $f$- and $h$-vector of $\Delta$ to the $f$- and $h$-vector of its $l$^th^ partial barycentric subdivision. We conclude with open problems.'
address:
- 'COMSATS Institute of Information Technology, Lahore, Pakistan'
- 'Fachbereich Mathematik und Information, Philipps-Universität Marburg, 35032 Marburg, Germany'
author:
- Sarfraz Ahmad$^1$
- Volkmar Welker
title: On Partial Barycentric Subdivision
---
Introduction
============
For a $(d-1)$-dimensional simplicial complex $\Delta$ on the ground set $V$ the barycentric subdivision $\sd(\Delta)$ of $\Delta$ is the simplicial complex on the ground set $V\setminus \{\emptyset\}$ with simplices the flags $A_0\subset A_1 \subset\cdots\subset A_i$ of elements $A_j\in \Delta\setminus\{\emptyset\},$ $0\leq j \leq i$. For $1\leq l\leq d$, we define the $l$^th^ partial barycentric subdivision of $\Delta$. This is a geometric subdivision, in the sense of [@St], such that $\sd^{l-1}(\Delta)$ is a refinement of $\sd^l(\Delta)$, $\sd^d(\Delta) = \Delta$ and $\sd^1(\Delta)=\sd(\Delta)$. Roughly speaking, the $l$^th^ partial barycentric subdivision arises when only the simplices of dimension $\geq l$ are barycentrically subdivided. In the paper, we provide a detailed analysis of the effect of the $l$^th^ barycentric subdivision operation on the $f$- and $h$-vector of a simplicial complex. Most enumerative results will be related to refinements of permutation statistics for the symmetric group. Our results extend the results from [@BW] for the case $l =1$. We refer the reader also to [@D] and [@N] for more detailed information in that case.
The paper is organized as follows. We start in Section 2 with geometric and combinatorial descriptions of the $l$^th^ partial barycentric subdivision and its implications on the generators of the Stanley-Reisner ideal of the complex. In Section 3 we study the enumerative combinatorics of the $l$^th^ partial barycentric subdivision. In particular, we relate in Lemma \[le:fvector\] and Theorem \[thm:main\] the effect of the $l$^th^ barycentric subdivision on the $f$- and $h$-vector of the simplicial complex $\Delta$ to a permutation statistics refining the descent statistics. In Section 4 we analyze the transformation matrices sending the $f$- and $h$-vector of the simplicial complex $\Delta$ to the corresponding vector for the $l$^th^ barycentric subdivision. We show that both maps are diagonizable and provide the eigenvalue structure. Note that by general facts the two matrices are similar. The main result of this section, Theorem \[th:eigenvector\], shows that the eigenvector corresponding to the highest eigenvalue of the $h$-vector transformation can be chosen such that it is of the form $(0,b_1,\ldots,b_{d-1},0)$ for strictly positive numbers $b_i$, $1\leq i\leq d-1$. In Section 5 we present some open problems. We ask for explicit descriptions of the eigenvectors and then shift the focus to the local $h$-vector which has been introduced by Stanley [@St]. The local $h$-vector is a measure for the local effect of a subdivision operation. In particular, general results by Stanley, predict that the local $h$-vector for the $l$^th^ partial barycentric subdivision is non-negative. For $l =1$ the local $h$-vector was computed by Stanley in terms of the excedance statistics on derangements. We exhibit some computations and possible approaches to the local $h$-vector for the $l$^th^ barycentric subdivision in general.
The $l$^th^ partial barycentric subdivision
===========================================
Geometric definition
--------------------
We first give a geometric definition of the $l$^th^ partial barycentric subdivision. For that we recall some basic facts about the reflection arrangement of the symmetric group $S_d$ permuting the $d$ letters from $[d] := \{ 1,2,\ldots,d\}$. The reflection arrangement $\mathcal{B}_d$ in $\RR^d$ of the symmetric group $S_d$ consists of the hyperplanes $H_{ij} = \{ (x_1,\ldots, x_d) \in \RR^d~:~x_i-x_j=0 \}$, $1\leq i<j\leq d$. To each permutation $w \in S_d$ there corresponds a region $R_w$ of $\mathcal{B}_d$ given by $$R_w=\{(\lambda_1,\ldots,\lambda_d)\in \mathbb{R}^d:\lambda_{w(1)} > \lambda_{w(2)} > \cdots > \lambda_{w(d)}\}.$$ Hence the number of regions of $\mathcal{B}_d$ is $d!$. We write $R_{w,+}$ for the intersection of $R_w$ with $\RR_{\geq 0}^d$. It is easily seen that geometrically the closure of $R_{w,+}$ is a simplicial cone.
The intersection of the closures of the cones $R_{w,+}$, $w \in S_d$, and the standard $(d-1)$-simplex $\Delta_{d-1} =
\{ (\lambda_1, \ldots, \lambda_d) \in \RR^d ~|~\lambda_1+\cdots \lambda_d = 1, x_i \geq 0, 1 \leq i \leq d\}$ induces a simplicial decomposition of $\Delta_{d-1}$. This decomposition is called the *barycentric subdivision* of $\Delta_{d-1}$ and is denoted by $\sd(\Delta_{d-1})$. We are interested in a sequence $\sd^l(\Delta_{d-1})$, $1 \leq i \leq d$, of simplicial subdivisions of the simplex, which have the property that $\sd^1(\Delta_{d-1}) = \sd(\Delta_{d-1})$ and $\sd^{l-1}(\Delta_{d-1})$ is a refinement of $\sd^l(\Delta_{d-1})$.
For $1 \leq l \leq d$, we set $S_d^l$ to be the set of permutations $w \in S_d$ for which $w(1) > \cdots > w(l)$. We define the $l$-cone $R_w^l$ of a $w \in S_d^l$ to be $$R_w^l=\{(\lambda_1,\ldots,\lambda_d)\in \RR^d~:~\lambda_{w(1)},\ldots,\lambda_{w(l)} > \lambda_{w(l+1)} > \cdots >
\lambda_{w(d)}\}.$$ Clearly $R_w^l$ is a cone. We write $R_{w,+}^l$ for the intersection of $R_w^l$ with $\RR_{\geq 0}^d$. Again the closure of $R_{w,+}^l$ is a simplicial cone which is the union of all closures of the $R_{v,+}$ for $v \in S_d$ such that $v(i) = w(i)$ for $l+1 \leq i \leq n$. We call the simplicial decomposition induced by the collection of all $R_{w,+}^l$ for $w \in S_{d}^l$ on $\Delta_{d+1}$ the *$l$^th^ partial barycentric subdivision* of $\Delta_{d-1}$ and denote it by $\sd^l (\Delta_{d-1})$. Obviously, we have that $\sd^d(\Delta_{d-1}) = \Delta_{d-1}$, $\sd^{1}(\Delta_{d-1}) = \sd(\Delta_{d-1})$ and $\sd^{l-1}(\Delta_{d-1})$ is a refinement of $\sd^l(\Delta_{d-1})$. If $l > d$ then we set $\sd^l (\Delta_{d-1}) = \Delta_{d-1}$. For a $(d-1)-$ dimensional simplicial complex $\Delta$ on the vertex set $V=[n]$ its $l$^th^ partial barycentric subdivision is the complex $\sd^l(\Delta)$ which is the subdivision of $\Delta$ obtained by replacing each simplex by its $l$^th^ partial subdivision. Roughly speaking this means that we cone all $(k-1)$-faces of $\Delta$ over their barycenters for all $l \leq k$.
By construction the number of cones $R_w^l$, $w \in S_d^l$, is $\frac{d!}{(d-l)!} = d \cdot (d-1)\cdots (d-l+1)$. Next, we want to get a better understanding of the facial structure of $\sd^l(\Delta_{d-1})$.
We have already seen that the $(d-1)$-dimensional faces are in bijection with the permutations in $S_d^l$. We turn this description into a description by combinatorial objects that are more suitable for studying all faces of $\sd^l(\Delta_{d-1})$. We identify a permutation $w \in S_d^l$ with a formal chain $$w(1) , w(2) ,\ldots,w(l) > w(l+1) > \cdots > w_{d}$$ and this chain in turn with $A_0 \subset A_1 \subset \cdots \subset A_{d-l}$ for $A_i = \{w(1),\ldots, w(l), \ldots, w(l+i)\}$.
Using this chain description, a $2$-dim face of $\sd^l(\Delta_2)$ corresponding to $w = 1~2~3$ is either $\{1,2,3\}$ for $l = 3$ or $\{1,2\} \subset \{1,2,3 \}$ for $l = 2$ or $\{1\} \subset
\{1,2\} \subset \{1,2,3\}$ for $l = 1$.
More generally, the $(i-1)$-faces of $\sd^l(\Delta)$ are indexed by chains $A_0 \subset A_1 \subset \cdots \subset A_r$ for which:
----------------------------- -----------------------
- $0 \leq \# A_0 \leq l$, - $\#A_0 + r = i$,
- $l+1 \leq \# A_1$. - $A_r \in \Delta$.
----------------------------- -----------------------
At the beginning of Section \[se:transformation\] we will further reformulate this description in terms of yet another combinatorial objects.
Geometrically, the face of $\sd^l(\Delta_{d-1})$ corresponding to $A_0 \subset A_1 \subset \cdots \subset A_0$ is the set of points $(\lambda_1,\ldots,\lambda_{d}) \in \Delta_{d-1}$ for which
- We have $\lambda_i = \lambda_j$ if $i,j \in A_s \setminus A_{s-1}$ for some $1 \leq s \leq r$.
- We have $\lambda_i > \lambda_j$ if $i \in A_s$ and $j \in A_t$ for some $0 \leq s < t \leq r$.
- We have $\lambda_i = 0$ if $i \not\in A_r$.
In particular, a vertex $v$ of the $l$^th^ partial barycentric subdivision $\sd^{l}(\Delta)$ of a $(d-1)$-dimensional simplicial complex $\Delta$ either belongs to the vertex set of $\Delta$ or can be identified with an $(m-1)$-face $v= \{ v_{i_1}, v_{i_2}, \ldots , v_{i_m}\} \in \Delta$ of $\Delta$ for some $l\leq m \leq d$.
Algebraic aspects
-----------------
Let $\Delta$ be a $(d-1)$-dimensional simplicial complex on vertex set $[n]$. Let $k$ be a field and $R=k[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables. The *Stanley-Reisner ideal* $I_{\Delta}$ is the ideal of $R$ generated by the squarefree monomials $\prod_{i \in \mathcal{N}} x_i$ whose index set $\mathcal{N}$ is a non-face of $\Delta$. A non-face $\mathcal{N}\not\in \Delta$ is called a *minimal non-face* of $\Delta$ if no proper subset of $\mathcal{N}$ is a non-face of $\Delta$. It is easily seen that the generators in the unique minimal monomial generating set of the Stanley-Reisner ideal correspond to the minimal non-faces of $\Delta$. The quotient $R/I_{\Delta}$ is called the *Stanley-Reisner ring* or *face ring* and is denoted by $k[\Delta]$.
Let $\sd^l(\Delta)$ be the $l$^th^ partial barycentric subdivision of $\Delta$, where $0\leq l \leq d-1$. Similarly, let $I_{\sd^l(\Delta)}$ and $k[\sd^l(\Delta)]$ be the Stanley-Reisner ideal and face ring of $\sd^l(\Delta)$, respectively. Let $V_l=[n]\cup B_l$ be the set of vertices of $\sd^l(\Delta)$, where $B_l=\{b_1,\ldots,b_l\}$ is the set of barycenters of the faces of $\Delta$ whose dimension ranges between $l$ and $d-1$.
\[le:nonface\] Let $\Delta$ be a $(d-1)$-dimensional simplicial complex on ground set $[n]$. For $1 \leq l \leq d$ let $\mathcal{N}$ be a subset of the vertex set of $\sd^l(\Delta)$. In case $\mathcal{N}$ is a minimal non-face of $\sd^l(\Delta)$ then either $\mathcal{N}\subseteq [n]$ and $\mathcal{N}$ is a minimal non-face of $\Delta$ or $|\mathcal{N}| = 2$.
If $\mathcal{N}\subset [n]$ and a minimal non-face of $\Delta$, then we are done. Since two vertices of $\Delta$ are connected by an edge in $\sd^l(\Delta)$ if and only if they are connected by an edge in $\Delta$ it follows that any subset of $[n]$ that is a minimal non-face of $\sd^l(\Delta)$ must be a minimal non-face of $\Delta$. Now, suppose there exists at least one vertex $b \in \mathcal{N} \setminus [n]$ and $|\mathcal{N}|\neq 2$. If $|\mathcal{N}| =1 $ then $\mathcal{N} = \{ b\}$ but $b$ is a vertex and hence a face. This leads to a contradiction and we are left with the case $|\mathcal{N}| \geq 2$. Let $\mathcal{F} \in \sd^l(\Delta)$ be a facet such that $b \in \mathcal{F}$. Then there exists at least one vertex $v\in \mathcal{N} \backslash \mathcal{F}$, otherwise $\mathcal{N}$ is no more a non-face. But by construction the edge $\{ b, v\}$ is not an edge in $\sd^l(\Delta)$. Thus $\{b_j,v\}\subset \mathcal{N}$ from which $\{b_j,v\} = \mathcal{N}$ follows.
Now we have the following Proposition:
Let $\Delta$ be a $(d-1)$-dimensional simplicial complex on ground set $[n]$. Then for $1 \leq l \leq d-1$ the Stanley-Reisner ideal $I_{\sd^l(\Delta)}$ is generated by square free monomial ideals of degree at most $l+1$.
Suppose there exists some generator $u\in I_{\sd^l(\Delta)}$ with degree strictly larger than $l+1$. Then there exists a minimal non-face $\mathcal{N} \subset \sd^l(\Delta)$ such that $|\mathcal{N}|>l+1$. Thus by Lemma \[le:nonface\] it follows that $\mathcal{N} \subset [n]$. But in $\sd^l(\Delta)$, we have coned all faces $\mathcal{F} \subseteq [n]$ of dimension $\geq l$ over their barycenters. Thus there does not exist any non-face of dimension greater than $l$. Therefore, $|\mathcal{N}|\leq l+1,$ a contradiction.
$f$-vector and $h$-vector transformation {#se:transformation}
========================================
In this section we study the transformation maps sending the $f$- and $h$-vector of a simplicial complex $\Delta$ to the $f$- and $h$-vector of the $l$^th^ partial barycentric subdivision of $\Delta$.
We consider set systems $B=|B_0|B_1|...|B_{r}|$ such that $B_1, \ldots, B_r \neq \emptyset$ and $B_s \cap B_t = \emptyset$ for $0 \leq s < t
\leq r$. Despite the fact that $B_0$ can be empty we call such a system an ordered set partition. We write $R(j,i,l)$ for the number of such ordered set partitions $B=|B_0|B_1|...|B_{r}|$ for which
1. $B_0 \cup \cdots \cup B_r = [j]$
2. If $r \geq 1$ then $\# (B_0 \cup B_1) \geq l+1$
3. $\# B_0 + r = i$.
and call $R(j,i,l)$ the *restricted Stirling number* for the parameters $j$,$i$,$l$. For $l = 0$ by $\# (B_0 \cup B_1) \geq l+1 = 1$ it follows from $B_1 \neq \emptyset$ that $B_0 = \emptyset$. Hence $r = i$ and $|B_1|\cdots |B_i|$ is a usual ordered set partition of the $j$-element set $B_1 \cup \cdots \cup B_i$ into $i$ (non-empty) blocks. Thus $R(j,i,0)=i!S(j,i)$, where $S(i,j+1)$ is the Stirling number of second kind.
Recall that the $f$-vector $f^\Delta = (f_{-1}^\Delta, \ldots, f^\Delta_{d-1})$ of a $(d-1)$-dimensional simplicial complex is the vector with its $i$^th^ entry $f_i^\Delta$ counting the $i$-dimensional faces of $\Delta$. Using this notation, we have the following lemma.
\[le:fvector\] Let $\Delta$ be a $(d-1)-$dimensional simplicial complex with $f$-vector $f^\Delta=(f^\Delta_{-1},\ldots,f^\Delta_{d-1})$. Then $$f_{i-1}^{\sd^l(\Delta)}=\sum\limits_{j=0}^d f^\Delta_{j-1} \cdot R(j,i,l).$$
Let $A_0 \subset A_1 \subset \cdots \subset A_r$ be an $(i-1)$-dimensional face of $\sd^l(\Delta)$. Then by (C2) we have $i = \# A_0 + r$. We set $j = \# A_r$ and assume without loss of generality that $A_r = [j]$. We define $B_0 = A_0$ and $B_s = A_{s} \setminus A_{s-1}$ for $1 \leq s \leq r$. Then $i = \# B_0 + r$ and $j = \# (B_0 \cup \cdots \cup B_r)$. If $r \geq 1$ then by (C3) $\# A_1 = \# (B_0 \cup B_1) \geq l+1$. Hence $|B_0| \cdots |B_r|$ satisfies (P1)-(P3). Conversely, if $|B_0| \cdots |B_r|$ satisfies (P2)-(P3) and $(B_0 \cup \cdots \cup B_r) \in \Delta$ then one easily checks that for $A_s = B_0 \cup \cdots \cup B_s$, $0 \leq s \leq r$, the chain $A_0 \subset \cdots \subset A_r$ satisfies (C1)-(C4).
Thus for any $(j-1)$-dimensional face $F$ of $\Delta$ on gets $R(j,i,l)$ faces $A_0 \subset \cdots \subset A_r$ of dimension $i-1$ in $\sd^l(\Delta)$ with $F = A_r$.
Next we study the transformation of the $h$-vector. Recall that the $h$-vector of a $(d-1)$-dimensional simplicial complex $\Delta$ is the integer vector $h^\Delta = (h^\Delta_0,\ldots, h^\Delta_d)$ for which $\sum_{i=0}^d f^\Delta_{i-1}x^{d-i}
= \sum_{i=0}^d h^\Delta_i x^{d-i}$, where $x$ is some indeterminate. For a permutation $\sigma \in S_d$ we denoted by $$\D(\sigma)=\{i\in [d-1]~ |~ \sigma(i)>\sigma (i+1)\}$$ its decent set and write $\des(\sigma):= \# \D(\sigma)$ for its number of descents. Following [@BW] for $d \geq 1$ and integers $i$ and $j$ we denote by $A(d,i,j)$ the number of permutations $\sigma \in S_d$ such that $\sigma(1)=j$ and $\des (\sigma)=i$. In particular, $A(d,i,j)=0$ if $i\leq -1$ or $i\geq d$.
In the sequel, we define a refinement of the preceding statistics suitable for the study of our $h$-vector transformation. By definition of $S_d^l$ we have the following strictly increasing chain of subgroups: $$S_{d+1}^{d-1}\subset S_{d+1}^{d-2}\subset \cdots \subset S_{d+1}^{2}
\subset S_{d+1}^{1}=S_{d+1}.$$
We define the *$l$-descent set* $D^l(\sigma)$ of a permutation $\sigma \in S_d^l$ as follows:
A number $i\in [d-1]$ belongs to the $l$-descent set $D^l(\sigma)$ of $\sigma\in S_d^l$, if $i$ satisfies one of the following two conditions.
1. $i\in [l]$ and $\sigma(i)> \sigma(l+1)$,
2. $i\in [d-1]\setminus [l]$ and $\sigma(i)> \sigma(i+1).$
We write $\des^l(\sigma)=|D^l(\sigma)|$ for the *number of $l$-descents* of a permutation $\sigma \in S_d^l$. Note that for $l = 1$ condition (1) is equivalent to having a decent in position one and therefore $D^1(\sigma)$ is just the usual descent set of $\sigma$.
Let $\sigma_1,\sigma_2,\sigma_3 \in S_6^4$, such that, $\sigma_1 =(\underline{4,3,2,1},6,5)$, $\sigma_2 =(\underline{6,5,2,1},3,4)$ and $\sigma_3 =(\underline{6,4,3,2},5,1)$. Then
$
\begin{array}{ccc}
D^4(\sigma_1)=\{5\} &
D^4(\sigma_2)=\{1,2\} &
D^4(\sigma_3)=\{1,5\} \\
\des^4(\sigma_1)=1 &
\des^4(\sigma_2)=2 &
\des^4(\sigma_3)=2.
\end{array}
$
For all $d\geq 1,$ $1\leq l \leq d-1$ and all integers $i$ and $j$ we denote by $A(d,i,j,l)$ the number of permutations $\sigma \in S_d^l$ such that $\des^l(\sigma)=i$ and $\sigma(d+1)=j$. Note that $A(d,i,j,l)=0$ if $i\leq -1$ or $i\geq d$.
The following is our first main result. The case $l=1$ was treated in [@BW Thm. 1].
\[thm:main\] Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. Then $$h_j^{\sd^l(\Delta)}=\sum_{\mu=0}^d A(d+1,j,d+\mu-1,l)h_{\mu}^{\Delta}$$ for $1\leq l \leq d-1$ and $0\leq j \leq d.$
For all $0\leq j\leq d$, we have: $$\begin{aligned}
\nonumber h_j^{\sd^l(\Delta)} &=& \sum_{i=0}^{j}{d-i \choose j-i}(-1)^{j-i}f_{i-1}^{\sd^l(\Delta)} \\
\nonumber &=& \sum_{i=0}^j {d-i \choose j-i} (-1)^{j-i} \sum_{k=0}^d f_{k-1}^{\Delta}R(k,i,l) \\
\nonumber &=& \sum_{i=0}^j\sum_{k=0}^d {d-i \choose j-i} (-1)^{j-i}R(k,i,l)\sum_{\mu=0}^k {d-\mu \choose d-k}h_{\mu}^{\Delta}\\
\nonumber &=& \sum_{\mu=0}^d{(}\sum_{k=0}^d\sum_{i=0}^j (-1)^{j-i}{d-i \choose j-i}{d-\mu \choose d-k}R(k,i,l){)}h_{\mu}^{\Delta}.
\end{aligned}$$ Fix a permutation $\sigma \in S_d^l$ and let $\D^l(\sigma) =\{s_1,\ldots,s_k\}\subseteq [d]$ with $s_1<\cdots < s_k$ its descent set. We define $p_{\sigma}$ be $0$ if there is no $s_{p_{\sigma}}\in [l]$ for which $\sigma(s_{p_{\sigma}}) > \sigma(l+1)$ and to be the maximal number $p_\sigma \in [k]$ for which $s_{p_{\sigma}}\in [l]$ and $\sigma(s_{p_{\sigma}}) > \sigma(l+1)$. Thus by definition of an $l$-descent any set $S$ that can arise as the descent set of a permutation $\sigma \in S_d^l$ is of the form $S = \{1,\ldots, p, s_{p+1}, \ldots, s_k\}$ for some $0 \leq p_\sigma < l \leq s_{p}+1$. We want to count $$\left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)\subseteq S,p_{\sigma}=p \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\}.$$ First, we count the possibilities to choose $\sigma(s_{p+1}+1) , \ldots, \sigma(d+1)$. In this range a descent is just a usual descent and we have ${s_k \choose s_{p+1},s_{p+2}-s_{p+1},
\ldots,s_k-s_{k-1}}{d-\mu \choose d-t_k}$ possibilities. Having done this we are left with $s_{p+1}$ elements that we have to arrange accordingly. Let $U$ be this set of elements. Let us consider the options for $\sigma(l+1)$. In order to create no descent in the range of $l+1$ and $s_{p+1}-1$ there have to be at least $s_{p+1} - (l+1)$ elements larger than $\sigma(l+1)$. Hence there can be at most $l$ elements smaller than $\sigma(l+1)$. In order to create no descent between $\sigma(i)$ and $\sigma(l+1)$ for some $p < i < l+1$ there must be at least $l-p$ elements smaller than $\sigma(l+1)$. Hence $\sigma(l+1)$ can only range from the $(l+1-p)$^th^ element of $U$ to the $(l+1)$^st^ element of $U$. Let $l+1-p \leq i \leq l+1$ and assume $\sigma(l+1)$ is the $i$^th^ element of $U$. Then in order to fix the permutation $\sigma$ we have to fix $s_{p+1} -(l+1)$ element larger than $\sigma(l+1)$ that will become the images of $\sigma(l+2),\ldots, \sigma(s_{p+1})$ in increasing order. For that we have ${s_{p+1} -i \choose s_{p+1} - (l+1)}$ possibilities. Using $l+1 \leq s_{p+1}$ we obtain:
$$\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)\subseteq S,p_{\sigma}=p \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\}$$ $$=\sum_{i=l+1-p}^{l+1}{s_{p+1}-i \choose s_{p+1}-(l+1)}
{s_k \choose s_{p+1},s_{p+2}-s_{p+1},\ldots,s_k-s_{k-1}}
{d-\mu \choose d-t_k}$$ but, $$\nonumber \sum_{i=l+1-p}^{l+1}{s_{p+1}-i \choose s_{p+1}-(l+1)}$$ $$\begin{aligned}
\nonumber &=& {s_{p+1}-l-1+p \choose s_{p+1}-l-1}+
{s_{p+1}-l-2+p \choose s_{p+1}-l-1}+\cdots+
{s_{p+1}-l-1 \choose s_{p+1}-l-1}\\
\nonumber &=& {s_{p+1}-l-1+p+1 \choose s_{p+1}-l-1+1}=
{s_{p+1}-l+p \choose s_{p+1}-l}
\end{aligned}$$ so, $$\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)\subseteq S,p_{\sigma}=p \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\}$$ $$={s_{p+1}-l+p \choose s_{p+1}-l}
{s_k \choose s_{p+1},s_{p+2}-s_{p+1},\ldots,s_k-s_{k-1}}
{d-\mu \choose s-s_k}$$ therefore, $$\sum_{\{1\leq s_1<\cdots < s_k\leq d\}}\sum_{p=0}^l
\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)\subseteq S,p_{\sigma}=p \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\}$$ $$=\sum_{1\leq s_1< \cdots <s_k\leq d}\sum_{p=0}^l
{{s_{p+1}-l+p \choose s_{p+1}-l}
{s_k \choose s_{p+1},s_{p+2}-s_{p+1},\ldots,s_k-s_{k-1}}
{d-\mu \choose d-s_k}}$$ $$=\sum_{j=k}^d{d-\mu \choose d-j}\sum_{1\leq s_1< \cdots <s_{k-1}\leq j-1}
\sum_{p=0}^l {s_{p+1}-l+p \choose s_{p+1}-l}
{j \choose s_{p+1},s_{p+2}-s_{p+1},\ldots,j-s_{k-1}}$$
For a fixed sequence $s_1 < \cdots < s_k = j$ for which $s_1=1, \ldots, s_p = p$ and $s_{p+1} \geq l+1$ we set $B_0=[s_p]$ for $p \geq 1$ and $B_0=\emptyset$ for $p=0$. Also, we set $$B_{\omega}=[s_{p+\omega-1}+1,s_{p+\omega}]\text{ for }
1\leq \omega \leq k-p.$$ Then $B_0 \cup \cdots \cup B_r = [j]$ which implies (P1). If $r \geq 1$ then $\# (B_0 \cup B_1)=\# B_0 + \# B_1 \geq p+ (l+1-p)=l+1$ implying (P2). Finally $\# B_0 + r =p+(k-p)= k$ shows (P3).
For notational convenience we set $s_0=0$. So for $r=k-p$, $$\begin{gathered}
\label{eq:part}
\sum_{1\leq s_1< \cdots <s_{k-1}\leq j-1}\sum_{p=0}^l
{s_{p+1}-l+p \choose s_{p+1}-l}
{j \choose s_{p+1},s_{p+2}-s_{p+1},\ldots,j-s_{k-1}}
\end{gathered}$$ counts the number of $B=|B_0|B_1|...|B_{r}|$ of $[j]$ satisfying (P1) - (P3). Therefore equals $R(j,k,l)$.
Using this result we have, $$\sum_{S\subseteq [d],\# S =k}
\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)\subseteq S, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\}$$ $$=\sum_{j=k}^d {d-\mu \choose d-j} R(j,k,l)$$ therefore, $$\sum_{k=0}^d\sum_{i=0}^j (-1)^{j-i}
{d-i \choose j-i}
{d-\mu \choose d-k} R(k,i,l)$$ $$\begin{aligned}
%\nonumber to remove numbering (before each equation)
\nonumber &=& \sum_{i=0}^j
(-1)^{j-i}{d-i \choose j-i}\sum_{\{S\subseteq [d],\# S=i\}}
\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)\subseteq S, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\} \\
\nonumber &=& \sum_{\{S\subseteq [d],\# S\leq j\}}
(-1)^{j-\#S}{d-\#S \choose j-\#S}
\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)\subseteq S, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\} \\
\nonumber &=& \sum_{\{S\subseteq [d],\# S\leq j\}}
(-1)^{j-\#S}{d-\#S \choose j-\#S}\sum_{T\subseteq S}
\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)= T, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\} \\
\nonumber &=& \sum_{\{T\subseteq [d],\# T\leq j\}}\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)= T, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\} \sum_{\{S\supseteq T, \#S\leq j\}}(-1)^{j-\#S}{d-\#S \choose j-\#S}\\
\nonumber &=& \sum_{\{T\subseteq [d],\# T\leq j\}}\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)= T, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\}\sum_{i=\# T}^j(-1)^{j-i}{d-i \choose j-i}{d-\#T \choose i-\#T}.
\end{aligned}$$ But $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\nonumber \sum_{i=\# T}^j(-1)^{j-i}{d-i \choose j-i}{d-\#T \choose i-\#T} &=& {d-\#T \choose i-\#T}\sum_{i=\# T}^j(-1)^{j-i}{j-\#T \choose i-\#T} \\
\nonumber &=& \delta_{j,\#T}.
\end{aligned}$$ Hence\
$$\sum_{k=0}^d\sum_{i=0}^j (-1)^{j-i}{d-i \choose j-i}{d-\mu \choose d-k} R(k,i,l)$$ $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\nonumber &=& \sum_{\{T\subseteq [d],\# T= j\}}\# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\D^l(\sigma)= T, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\} \\
\nonumber &=& \# \left\{\sigma \in S_{d+1}^l\,\,\,\,\ \Big| \,\,\,\,\,\,\
\begin{array}{ll}
\des^l(\sigma)= j, \\
\sigma(d+1)=d+1-\mu
\end{array}
\right\}.
\end{aligned}$$ This completes the proof.
We note that since for a $(d-1)$-dimensional simplicial complex $\Delta$ and $l \leq d-1$ the subdivision operation $\sd^l(\bullet)$ is non-trivial in top-dimension it follows from Theorem 5.5. from [@D] that iterated application of $\sd^l(\bullet)$ will lead to a convergence phenomenon for the $f$-vector. More precisely, for a $(d-1)$-dimensional simplicial complex $\Delta$, set $\Delta^{(n)} := \underbrace{\sd^l(\cdots \sd^l}_n (\Delta)$ and $f^{(n)} (t) = \sum_{i=0}^d f_{i-1}^{\Delta^{(n)}}t^{d-i}$ then for $n \rightarrow \infty$ one root of $f^{(n)}(t)$ will go to $-\infty$ and the others converge to complex numbers independent of $\Delta$. This phenomenon was fist observed in [@BW Thm. 4.2] for the special case of classical barycentric subdivision $\sd^1(\bullet)$. In addition, in [@BW Thm. 3.1] it is shown that for $l=1$ the polynomial $f^{(1)}(t)$ has only real roots. Simple examples show that this is not the case for general $l$.
The Transformation Matrices
===========================
For a $(d-1)$-dimensional simplicial complex $\Delta$ we denote by $\mathfrak{H}_{d-1} = {(h_{ij}^{(d-1)})}_{0\leq i,j \leq d}
\in \RR^{(d+1) \times (d+1)}$ the matrix of the linear transformation that sends the $h$-vector of $\Delta$ to the $h$-vector of $\sd(\Delta)$ and $\mathfrak{H}^l_{d-1} = {(h_{ij}^{(d-1,l)})}_{0\leq i,j \leq d}
\in \RR^{(d+1) \times (d+1)}$ the matrix of the transformation of the $h$-vector of $\Delta$ to the $h$-vector of $\sd^l(\Delta)$. Thus $\mathfrak{H}^1_{d-1} = \mathfrak{H}_{d-1}$. By [@BW Thm. 1] we know $h_{ij}^{(d-1)}=A(d+1,i,j+1)$ and more generally by Theorem \[thm:main\] we know $h_{ij}^{(d-1,l)} = A(d+1,i,d+1-j,l)$.
As an illustration we present the matrices $\mathfrak{H}_d^l$ for $d=4$ and $l=3$ and $l=2$.
---------------------------------- ----------------------------------
$\mathfrak{H}_3^3=\left( $\mathfrak{H}_3^2=\left(
\begin{array}{ccccc} \begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\
1 & 2 & 1 & 1 & 1 \\ 5 & 5 & 5 & 2 & 1 \\
1 & 1 & 2 & 1 & 1 \\ 5 & 5 & 6 & 5 & 5 \\
1 & 1 & 1 & 2 & 1 \\ 1 & 2 & 3 & 5 & 5 \\
0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 \\
\end{array} \end{array}
\right) \right)
$ $
---------------------------------- ----------------------------------
The following lemma follows immediately from the definition of $A(d+1,i,j+1)$.
\[sumH\] The sum of all entries of $\mathfrak{H}^l_{d-1}$ is given by:\
$$\sum_{0\leq i,j \leq d}h_{ij}^{(d-1,l)}=\frac{(d+1)!}{l!},$$ and the sum of all entries of each column is given by:\
$$\sum_{0\leq j \leq d}h_{ij}^{(d-1,l)}=\frac{d!}{l!}, \quad 0 \leq i \leq d.$$
The next simple lemma gives an explicit formula for $\mathfrak{H}^{d-1}_{d-1}$ which will serve as the induction base for the proof of monotonocity of the $h$-vector under partial barycentric subdivision in Corollary \[Co:proof\].
\[le:entries\] Let $l=d-1$, then the entries of $\mathfrak{H}^{d-1}_{d-1}$ are given by:\
$$h_{ij}^{(d-1,d-1)}=\left\{
\begin{array}{ll}
0, & \hbox{$i=0,j\not=0$ or $i=d,j\not=d$;} \\
2, & \hbox{$i=j=1,\ldots,d-1$;} \\
1, & \hbox{otherwise.}
\end{array}
\right.$$\
and hence $$\mathfrak{H}^{d-1}_{d-1}=\left(
\begin{array}{cccccc}
1 & 0 & 0 & \cdots & 0 & 0 \\
1 & 2 & 1 & \cdots & 1 & 1\\
1 & 1 & 2 & \cdots & 1 & 1\\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
1 & 1 & 1 & \cdots & 2 & 1\\
0 & 0 & 0 & \cdots & 0 & 1 \\
\end{array}
\right)$$
We will prove it by describing the entries of an arbitrary column. Let $C_j=(A(d+1,i,d+1-j,d-1))_{0\leq i\leq d} \in \RR^{(d+1) \times 1}$ be the $j^{\text{th}}$ column of $\mathfrak{H}^{d-1}_{d-1}$. Then by definition the entries of $C_j$ count the permutations $\sigma\in S_{d+1}^{d-1}$ such that $\sigma(d+1)=d+1-j$ according to their number of $l$-descents. By fixing the last element, we are left with $d$ permutations $\{\sigma_0,\sigma_1,\ldots,\sigma_{d-1}\}\subset S_{d+1}^{d-1}$, and we arrange them in such a way that $\{\sigma_0(d),\sigma_1(d),\ldots,\sigma_{d-1}(d)\}$ are in descending order. We count the number of $l$-descents in the following way:
The number of $l$-descents of $\sigma_i$ in the first $d-1$ positions is $i$ by the way we have arranged $\sigma_0, \ldots, \sigma_d$. Moreover since $\sigma_i(d+1)=d+1-j$ there is a descent in position $d$ if and only if $i < j$. Therefore, the number of $l$-descents is $i+1$ if $i< j$ and $i$ if $i\geq j$. Now a simple count implies the assertion.
The examples above and the preceding lemma suggest some relations among the entries of $\mathfrak{H}^{l}_{d-1}$ that we verify in the next lemmas.
For $0\leq i,j\leq d$, $$A(d+1,i,d+1-j,l)=A(d+1,d-i,j+1,l).$$
Let us denote by $S_{d+1}^l(i,d+1-j)$ the set of permutations $\sigma \in S_{d+1}^l$ such that $\des^l(\sigma)=i$ and $\sigma(d+1)=d+1-j$. Thus $A(d+1,i,d+1-j,l)=\# S_{d+1}^l(i,d+1-j)$. To complete the proof it is enough to provide a bijection between $S_{d+1}^l(i,d+1-j)$ and $S_{d+1}^l(d-i,j+1)$. Let $$\varphi : S_{d+1}^l(i,d+1-j) \rightarrow S_{d+1}^l(d-i,j+1)$$ be the map that sends $\sigma =(\sigma(1),\ldots,\sigma(d+1)) \in S_{d+1}^l(i,d+1-j)$ to $$\varphi(\sigma):=(d+2-\sigma(l),\ldots,d+2-\sigma(1),d+2-\sigma(l+1),
\ldots,d+2-\sigma(d+1)).$$
Since $\sigma(1),\ldots,\sigma(l)$ are in descending order, we have that $d+2-\sigma(l),\ldots,d+2-\sigma(1)$ are also in descending order and hence $\varphi(\sigma)\in S_{d+1}^l$. By definition $\phi(\sigma)(d+1) = d+1 - \sigma(d+1) = d+1-j$. Thus to show that $\varphi(\sigma) \in S_{d+1}^l(d-i,j+1)$ it remains to verify that the number of $l$-descents of $\varphi(\sigma)$ is $d-i$.
We show that $m \in [d]$ is an $l$-descent of $\sigma$ if and only if $m$ is not an $l$-descent of $\varphi(\sigma)$. If $m \in[l]$ then $\sigma(j)>\sigma(l+1)$ implies $d+2-\sigma(j)<d+2-\sigma(l+1)$ and $\sigma(j)<\sigma(l+1)$ implies $d+2-\sigma(j)>d+2-\sigma(l+1)$. Analogously, if $m\in [d]\setminus [l]$ then $\sigma(m)>\sigma(m+1)$ implies $d+2-\sigma(m)<d+2-\sigma(m+1)$ and $\sigma(m)<\sigma(m+1)$ implies $d+2-\sigma(m)>d+2-\sigma(m+1)$.
Therefore, the number of $l$-descents of $\varphi(\sigma)$ is $d-i$. This completes the proof since $\varphi$ is clearly a bijection.
\[pr:ineq\] For $0\leq i,j \leq d$, $$\begin{aligned}
\label{eq:ineq}
A(d+1,i,d+1-j,l+1) & \leq & A(d+1,i,d+1-j,l).
\end{aligned}$$ In addition, for $d\geq 4,$ $0\leq j\leq d$ and $2\leq i \leq d-2$, inequality is strict.
For the sake of short notation, within the proof we say descent for corresponding $l$-descent and $``\,\,\hat{ }\,\,"$ means that the entry is missing in the permutation. We define a map $$\psi:S_{d+1}^{l+1}(i,d+1-j)\rightarrow S_{d+1}^l(i,d+1-j)$$ as follows:
Let $\sigma\in S_{d+1}^{l+1}(i,d+1-j)$ be a permutation for which $p$ is the number of descents in the first $l+1$ positions and $i-p$ descents in the remaining positions for some $0 \leq p \leq l$. Thus we can write $\sigma=(\sigma(1),\ldots,\sigma(p),\ldots,
\sigma(l+1),\sigma(l+2),\ldots,\sigma(d+1))$ such that $\sigma(p)> \sigma(l+2)$ and $\sigma(p+1)<\sigma(l+2)$, where $\sigma(1),\ldots,\sigma(l+1)$ are in descending order and $\sigma(d+1)=d+1-j$.\
We define $$\psi(\sigma)=(\sigma(1),\ldots,\hat{\sigma}(p+1),\ldots,\sigma(l+1),
\sigma(p+1),\ldots,\sigma(d));$$ i.e we change the position of $\sigma(p+1)$ from $p+1$ to $l+1$. It is easy to see that $\psi(\sigma)$ is a permutation for which $p$ is the number of descents in the first $l$ position and $i-p$ descents in the remaining position with $\psi(\sigma)(d)=d+1-j$. Therefore $\psi(\sigma)\in S_{d+1}^l(i,d+1-j)$. Clearly, $\psi$ is injective and hence we have $S_{d+1}^{l+1}(i,d+1-j)\subseteq S_{d+1}^l(i,d+1-j)$ which implies $A(d+1,i,d+1-j,l+1) \leq A(d+1,i,d+1-j,l)$.
Now assume $d\geq 4,$ $0\leq j\leq d$ and $2\leq i \leq d-2$. For the proof of the strict inequality in it suffices to find at least one element $\sigma\in S_{d+1}^l(i,d+1-j)$ that does not have a preimage under $\psi$ in $S_{d+1}^{l+1}(i,d+1-j)$. We consider two cases:
: We set $\sigma(d+1)=d+1-j$ and $\sigma(d)=d+1$. Now we are left with $d-1$ elements to be arranged with $i-1$ descents. Let $\rho_1,\ldots,\rho_{d-1}$ be the remaining elements of $[d+1]$ arranged in ascending order; i.e. $\rho_s < \rho_t$ for $s<t$. We have further two possibilities:\
- $i-1\leq l$. Then reordering the elements as $$\rho_{l+1},\ldots,\hat{\rho}_{l+2-i},\ldots,\rho_1,\rho_{l+2-i},\rho_{l+2}, \ldots,\rho_{d-1}$$ yields the required number of $l$-descents. Clearly, the formal preimage of $\sigma$ under $\psi$ has not belong to $S_{d+1}^{l+1}(i,d+1-j)$ since it has only one descent but $i\geq 2$.
- $i-1> l$. Then we reorder the elements in the first $l+1$ positions as $$\rho_{l+1},\ldots,\rho_2,\rho_1,$$ which contributes $l$ to the number of descents. The remaining $d-1-(l+1)$ elements can be arranged in such a way that they contribute $i-1-l$ to the number of descents. By this setting we cannot have a descent at $(l+1)^{\text{th}}$ and $(d-1)^{\text{st}}$ position, so at most we can have $d-2$ descents which coincides with the upper bound for $i$. Again, the formal preimage of $\sigma$ under $\psi$ does not belong to $S_{d+1}^{l+1}(i,d+1-j)$ since its number of descents are $i-l$.
: We set $\sigma(d+1):=d+1$. Thus we are left with the $d$ elements $[d]$ to be arranged yielding a permutation in $S_{d+1}^l(i,d+1-l)$. The same arrangement as in Case 1 with $d$ elements and $i$ descents will give us the required element $\sigma$.
As a consequence of Theorem \[thm:main\] and Proposition \[pr:ineq\] we can deduce a result on the growth of the $h$-vector under $l$^th^ partial barycentric subdivision.
\[Co:proof\] Let $\Delta$ be a $d$-dimensional simplicial complex such that $h_i^\Delta \geq 0$ for all $0 \leq i \leq d$. Then $h_i^\Delta \leq h_i^{\sd^l(\Delta)}$ for $0 \leq i \leq d$.
It is easy to see that $h_0^{\sd^l(\Delta)}=h_0^{\Delta}$ and $h_d^{\sd^l(\Delta)}=h_d^{\Delta}$, thus we are left with the case $1\leq i \leq d-1.$ Since by Theorem \[thm:main\] $h_i^{\sd^l(\Delta)}$ is a non-negative linear combination of the $h_j^\Delta$ it suffices to show that the entries of the submatrix ${(h_{ij}^{(d-1,l)})}_{1\leq i\leq d-1,\,\, 0\leq j \leq d}$ are non-zero. Again, by equation \[eq:ineq\] it is enough to consider the case $l=d-1$. By Lemma \[le:entries\] we complete the proof.
The consequence of the preceding corollary for the smaller class of Cohen-Macaulay simplicial complexes also follows from a very general result by Stanley [@St Theorem 4.10] using the fact that $l$^th^ partial barycentric subdivision is a quasi-geometric subdivision. Note that $h_i^\Delta \geq 0$ for Cohen-Macaulay simplicial complexes.
Let $\mathfrak{F}_{d-1}$ be the matrix of the transformation that sends $f$-vector of $\Delta$ to $f$-vector of $\sd(\Delta)$. We denote by $\mathfrak{F}_{d-1}^l$ the matrix of the transformation from the $f$-vector of $\Delta$ to the $f$-vector of $\sd^l(\Delta)$. Both matrices $\mathfrak{F}_{d-1}$ and $\mathfrak{F}^l_{d-1}$ are square matrices of order $d+1$, with $\mathfrak{F}_{d-1}^1=\mathfrak{F}_{d-1}$. By Theorem \[le:fvector\] the entries of $\mathfrak{F}^l_{d-1}={(f_{ij}^{(d-1,l)})}_{0\leq i,j \leq d}$ are given by $f_{ij}^{(d-1,l)}=R(j,i,l)$.
The following lemma shows that the matrices $\mathfrak{F}^l_{d-1}$ and $\mathfrak{H}^l_{d-1}$ are diagonalizable.
\[le:diagonalizble\] For $1\leq l\leq d-1$:
1. The matrices $\mathfrak{F}^l_{d-1}$ and $\mathfrak{H}^l_{d-1}$ are similar.
2. The matrices $\mathfrak{F}^l_{d-1}$ and $\mathfrak{H}^l_{d-1}$ are diagonalizable with eigenvalue $1$ of multiplicity $l+1$ and eigenvalues $\frac{(l+1)!}{l!},\ldots,\frac{d!}{l!}$ of multiplicity $1$.
- Since the transformation sending the $f$-vector of a simplicial complex to the $h$-vector of a simplicial complex is an invertible linear transformation, the first assertion follows.
- Clearly, $\mathfrak{F}^l_{d-1}$ is an upper triangular matrix with diagonal entries $$\underbrace{1,\ldots,1}_{(l+1)-\mbox{times}},
\frac{(l+1)!}{l!},\ldots,\frac{d!}{l!}.$$ Let $(\mathfrak{F}^l_{d-1})^\perp$ be the transpose of $\mathfrak{F}^l_{d-1}$. Then for $(\mathfrak{F}^l_{d-1})^\perp$, the first $(l+1)$ unit vectors are eigenvectors for the eigenvalue $1$. Also, The eigenvalues $\frac{(l+1)!}{l!},\ldots,\frac{d!}{l!}$ are pairwise different. This implies that $(\mathfrak{F}^l_{d-1})^\perp$ is diagonalizable. But then $\mathfrak{F}^l_{d-1}$ is diagonalizable.
\[Co:initialeigenvectors\] Let $\nu=(\nu_0,\ldots,\nu_{d})$ be an eigenvector of the matrix $\mathfrak{H}^l_{d-1}$ for the eigenvalue $\lambda$ such that $\lambda\not=\frac{d!}{l!}$. Then $\sum_{i=0}^d \nu_{i}=0.$
Since $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\nonumber \mathfrak{H}^l_{d-1}\nu &=& \lambda \nu \\
\Rightarrow \nonumber (1,\ldots,1)\mathfrak{H}^l_{d-1}\nu &=& (1,\ldots,1)\lambda \nu
\end{aligned}$$ But by Lemma \[sumH\], $(1,\ldots,1)\mathfrak{H}^l_{d-1}=\frac{d!}{l!}(1,\ldots,1)$. Therefore, either $\lambda=\frac{d!}{l!}$ or $\sum_{i=0}^d \nu_{i}=0$. Since $\lambda\not=\frac{d!}{l!}$ we are done.
Next we try to gain a better understanding of the eigenvectors of $\mathfrak{H}^l_{d-1}$.
\[le:Fd\] Let $d\geq 2$ and $\nu_1^{(1)},\ldots,\nu_1^{(l+1)},\nu_{l+1},\ldots,\nu_d$ be a basis of eigenvectors of the matrix $\mathfrak{F}^l_{d-1}$, where $\nu_1^{(1)},\ldots,\nu_1^{(l+1)}$ are eigenvectors for the eigenvalue $1$ and $\nu_{l+1},\ldots,\nu_d$ are eigenvectors for the eigenvalues $\{l+1,\ldots,d\}$, respectively. Then $(\nu_1^{(1)},0),\ldots,(\nu_1^{(l+1)},0),(\nu_{l+1},0),\ldots,(\nu_d,0)$ are eigenvectors of the matrix $\mathfrak{F}^l_d$ for the eigenvalues $\{\underbrace{1,\ldots,1}_{(l+1) times},l+1,\ldots,d\}.$
Since both $\mathfrak{F}^l_{d-1}$ and $\mathfrak{F}^l_{d}$ are upper triangular and $\mathfrak{F}^l_{d-1}$ is obtained by deleting $(d+2)$^nd^ column and row from $\mathfrak{F}^l_{d}$ the assertion follows.
Let $\hat{\mathfrak{H}}^l_{d-1}$ be a matrix obtained be deleting the first and last rows and columns of $\mathfrak{H}^l_{d-1}$. Thus $\hat{\mathfrak{H}}^l_{d-1}$ is a $d$ by $d$ square matrix.
\[le:innermatrixH\] The matrix $\hat{\mathfrak{H}}^l_{d-1}$ is diagonalizable.
By definition and \[thm:main\] the first row of $\mathfrak{H}^l_{d-1}$ is the first unit vector and the last row of $\mathfrak{H}^l_{d-1}$ is the $(d+1)$^st^ unit vector. Thus the characteristic polynomial of $\mathfrak{H}^l_{d-1}$ splits into $(1-t)^2$ times the characteristic polynomial of $\hat{\mathfrak{H}}^l_{d-1}$. Therefore, $\hat{\mathfrak{H}}^l_{d-1}$ has eigenvalues $$\underbrace{1,\ldots,1}_{(l-1)-\mbox{times}},\frac{(l+1)!}{l!},\ldots,\frac{d!}{l!}.$$ To show the matrix $\hat{\mathfrak{H}}^l_{d-1}$ is diagonalizable, it is enough to show that the eigenspace for the eigenvalue $1$ is of dimension $l-1$.
For this we again consider the full matrix $\mathfrak{H}^l_{d-1}$. Since $\mathfrak{H}^l_{d-1}$ is diagonalizable there is a basis $\omega_1^{(1)},\ldots,$ $\omega_1^{(l+1)},$ $\omega_{l+1},\ldots,\omega_{d}$ of $\RR^{d+1}$ consisting of eigenvectors of $\mathfrak{H}^l_{d-1}$. We can choose the numbering such that $\omega_1^{(i)},1\leq i\leq l+1$ are eigenvectors for the eigenvalue $1$ and $\omega_j$ is an eigenvector for the eigenvalues $\frac{j!}{l!},l+1 \leq j\leq d$, respectively.
Again, since the first and last row of $\mathfrak{H}^l_{d-1}$ are the first and $(d+1)$^st^ unit vector we can choose the eigenvectors of $\mathfrak{H}^l_{d-1}$ for the eigenvalue $\lambda=1$ as follows: $\omega_1^{(1)}$ and $\omega_1^{(2)}$ can be chosen such that $$\omega_1^{(1)}=(1,k_{11},\ldots,k_{1(d-1)},0) \text{ and } \omega_1^{(2)}=(0,k_{21},\ldots,k_{2(d-1)},1),$$ and $\omega_1^{(i)}$ can be chosen such that $\omega_1^{(i)}=(0,k_{i1},\ldots,k_{i(d-1)},0)$ for $3\leq i\leq l+1$. Clearly, this implies that deleting the leading and trailing $0$ form the $\omega_1^{(i)}$ for $3\leq i\leq l+1$ yields eigenvectors $\hat{\omega_1}^{(i)}=(k_{i1},\ldots,k_{i(d-1)})$ of $\hat{\mathfrak{H}}^l_{d-1}$ for the eigenvalue $\lambda=1$. Obviously, the set of vectors $\{\hat{\omega_1}^{(3)},\ldots,\hat{\omega_1}^{(l+1)}\}$ is linearly independent. Hence we have shown that the dimension of the eigenspace for the eigenvalue $1$ of $\hat{\mathfrak{H}}^l_{d-1}$ $l-1$.
The above lemma is a key ingredient in proving the following theorem.
\[th:eigenvector\] Let $d\geq 2$ and let $\omega_1^{(1)},\ldots,\omega_1^{(l+1)},\omega_{l+1},\ldots,\omega_{d}$ be a basis of eigenvectors of the matrix $\mathfrak{H}^l_{d-1}$, where $\omega_1^{(i)},1\leq i\leq l+1$ are eigenvectors for the eigenvalue $1$ and $\omega_j$ is an eigenvector for the eigenvalue $\frac{j!}{l!},l+1 \leq j\leq d$.
1. Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. If we expand $h$-vector of $\Delta$ in terms of eigenvectors of the matrix $\mathfrak{H}^l_{d-1}$, the coefficient of the eigenvector for the eigenvalue $\frac{d!}{l!}$ is non-zero.
2. The first and the last coordinate entry in $\omega_1^{(3)},\ldots,\omega_1^{(l+1)},\omega_{l+1},\ldots,\omega_{d}$ is zero.
3. The vectors $\omega_1^{(1)}$ and $\omega_1^{(2)}$ can be chosen such that $$\omega_1^{(1)}=(1,i_1,\ldots,i_{d-1},0)\text{ and }\omega_1^{(2)}=(0,j_1,\ldots,j_{d-1},1).$$
4. The vector $\omega_d$ can be chosen such that $\omega_d=(0,b_1,\ldots,b_{d-1},0)$ for strictly positive rational numbers $b_i$, $1\leq i\leq d-1$.
Let us expand the $f$-vector of $\Delta$ in terms of a basis of eigenvectors of the matrix $\mathfrak{F}^l_{d-1}$. Since $f^{\Delta}_{d-1}\not=0$ from Lemma \[le:Fd\] we deduce that the coefficient of the eigenvector for the highest eigenvalue is non-zero. Since $\mathfrak{F}^l_{d-1}$ and $\mathfrak{H}^l_{d-1}$ are similar so (1) follows.
Assertions (2) and (3) immediately follow from the proof of Lemma \[le:innermatrixH\].
For (4) consider the matrix $\hat{\mathfrak{H}}^l_{d-1}$ as defined above. It is easily seen that the entries of $\hat{\mathfrak{H}}^l_{d-1}$ are strictly positive numbers. Therefore, by the Perron-Frobenius Theorem [@HJ] it follows that there is an eigenvector $\hat{\omega}^l_d$ for the eigenvalue $\frac{d!}{l!}$ with strictly positive entries. Hence $(0,\hat{\omega}^l_d,0)$ is the required eigenvector.
Open Problems
=============
In this section we discuss some of the open problems related to the above work.
Corollary \[Co:initialeigenvectors\] describes properties of the eigenvectors of the matrix $\mathfrak{H}^l_{d-1}$ for the eigenvalue $\lambda$ such that $\lambda\not=\frac{d!}{l!}$. For the eigenvalue $\lambda=\frac{d!}{l!}$ we were able to deduce its non-negativity in Theorem \[th:eigenvector\] (4) but were not able to give more structural results or even provide an explicit description. By [@D] when applying $l$^th^ partial barycentric subdivision iteratively the limiting behavior of the $h$-vector is determined by this eigenvector, in a sense specified in [@D]. Hence some information can be read off from [@D] nevertheless complete information about that eigenvector would be desirable.
For example, for $d=4$ we have following eigenvectors, corresponding to the eigenvalues $\frac{4!}{3!}, \frac{4!}{2!}, \frac{4!}{1!}$, respectively. $$\left(
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{5}{3} \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{7}{2} \\
1 \\
0 \\
\end{array}
\right).$$ For $d=5$ we have the following eigenvectors, corresponding to the eigenvalues $\frac{5!}{4!}, \frac{5!}{3!}, \frac{5!}{2!}, \frac{5!}{1!}$, respectively. $$\left(
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{12}{7} \\
\frac{12}{7} \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{46}{11} \\
\frac{46}{11} \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{17}{2} \\
\frac{17}{2} \\
1 \\
0 \\
\end{array}
\right).$$ Similarly, for $d=6$ we have following eigenvectors, corresponding to the eigenvalues $\frac{6!}{5!},\frac{6!}{4!}, \frac{6!}{3!}, \frac{6!}{2!}, \frac{6!}{1!}$, respectively. $$\left(
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{7}{4} \\
\frac{7}{4} \\
\frac{7}{4} \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{1941}{437} \\
\frac{2146}{437} \\
\frac{1941}{437} \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{5431}{527} \\
\frac{8906}{527} \\
\frac{5431}{527} \\
1 \\
0 \\
\end{array}
\right),
\left(
\begin{array}{c}
0 \\
1 \\
\frac{586}{33} \\
\frac{5459}{132} \\
\frac{586}{33} \\
1 \\
0 \\
\end{array}
\right).$$ Thus the following problem appears to be interesting.
Give a description of eigenvectors of the matrices $\mathfrak{F}^l_{d-1}$ and $\mathfrak{H}^l_{d-1}$ for the eigenvalue $\frac{d!}{l!}$.
Let $V$ be a vertex set such that $\# V=d$ and let $2^V$ denote the simplex with vertex set $V$. Let $\Gamma$ be the first barycentric subdivision of $2^V$. The $h$-polynomial $h(\Gamma,x) = \sum_{i=0}^{d} h_ix^{d-i}$ of $\Gamma$ has the following combinatorial interpretation. $$\label{hvectorexc}
h(\Gamma,x) = \sum_{\sigma \in S_d}x^{\des(\sigma)} =\sum_{\sigma \in S_d}x^{\ex(\sigma)},$$ where $\ex(\sigma)$ denotes the number of [*excedances*]{} of $\sigma$, defined by $$\ex(\sigma)=\# \{i ~|~\sigma(i)>i\},$$ The first equality follows from [@St2 Theorem 3.13.1] (it is also a consequence of [@BW Thm 1] and Theorem \[thm:main\]), and the second is a consequence of [@St2 Proposition 1.4.3]. In [@St], the [*local $h$-polynomial*]{} $\ell_V(\Gamma,x)$ of $\Gamma$ has been defined and given in a similar passion of equation as follows: $$\label{localhvector}
\ell_V(\Gamma,x)=\sum_{\sigma\in D_d}x^{\ex(\sigma)},$$ where $D_d$ denotes the set of all derangements in $S_d$. We suggest the followings:
Give an interpretation of local $h$-polynomial for the $l$^th^ partial barycentric subdivision similar to in terms of a suitably defined $l$-excedance statistic on a newly defined set of $l$-derangements satisfying an analog of
Already the question of finding a statistic on $S_d^l$ fulfilling a statement analogous to seems to be hard and challenging. We note that in [@Ath] a theory of local $\gamma$-vectors of subdivisions was initiated.
\[problm 3\] Define an $l$-excedance statistic on $S_d^l$ such that the $l$-excedance and $l$-descent statistic on $S_d^l$ are equally distributed; i.e. satisfy an analog of .
For Problem \[problm 3\] we tried, different approaches. Despite not yielding a solution to the problem the following idea resulted in some interesting data. We define an injective map say $\chi:S_d^l \rightarrow S_d$ in the following way. Let $\sigma\in S_d^l$ such that $\sigma=(\sigma(1),\sigma(2),\ldots,\sigma(d)),$ with first $l$ elements are in descending order, then: $$\chi(\sigma)=\left\{
\begin{array}{ll}
(\sigma(l),\ldots,\sigma(1),\sigma(l+1),\ldots,\sigma(d)), & \hbox{if $\sigma(l+1)>\sigma(1)$} \\
& \hbox{or $\sigma(l)>\sigma(l+1)$;} \\
(\sigma(l_1),\ldots,\sigma(1),\sigma(l),\ldots,\sigma(l_1+1), & \\
\sigma(l+1),\ldots, \sigma(d)), & \hbox{if $\sigma(l_1)>\sigma(l+1)$}\\
& \hbox{and $\sigma(l_1+1)<\sigma(l+1)$}.\\
\end{array}
\right.$$ Now define the number of $l$-excedances $\ex^l(\sigma)$ of $\sigma\in S_d^l$ to be number of usual excedances of $\chi(\sigma)$, i.e. $$\ex^l(\sigma):=\#\{i~|~\chi(\omega)(i)>i\}.$$ We apply this definition for different values of $d$ and $l$. For a fixed $d$, the $l$-descent and $l$-excedance statistic are equally distributed on $S_d^l$ for $l=d-1$ and $l=d-2$. But for other values of $l$ the two statistics appear to be different. Nevertheless, the obtained data has some surprising and unexplained symmetry. For example, for $S_5^l$ we have following tables for the number of $l$-descents,
[cc|c|c|c|c]{} & &\
& & 4 & 3 & 2\
& & 1 & 1 & 1 &\
& & 1 & 6 & 16 &\
& & 1 & 6 & 26 &\
& & 1 & 6 & 16 &\
& & 1 & 1 & 1 &\
and the following table for the number of $l$-excedances.
[cc|c|c|c|c]{} & &\
& & 4 & 3 & 2\
& & 1 & 1 & 1 &\
& & 1 & 6 & 14 &\
& & 1 & 6 & 30 &\
& & 1 & 6 & 14 &\
& & 1 & 1 & 1 &\
Similarly, for $S_6^l$ the number of $l$-descents are shown in the following table,
[cc|c|c|c|c|c]{} & &\
& & 5 & 4 & 3 & 2\
& & 1 & 1 & 1 & 1 &\
& & 1 & 7 & 22 & 42 &\
& & 1 & 7 & 37 & 137 &\
& & 1 & 7 & 37 & 137 &\
& & 1 & 7 & 22 & 42 &\
& & 1 & 1 & 1 & 1 &\
and the number of $l$-excedances are shown in the following table.
[cc|c|c|c|c|c]{} & &\
& & 5 & 4 & 3 & 2\
& & 1 & 1 & 1 & 1 &\
& & 1 & 7 & 17 & 33 &\
& & 1 & 7 & 42 & 146 &\
& & 1 & 7 & 42 & 146 &\
& & 1 & 7 & 17 & 33 &\
& & 1 & 1 & 1 & 1 &\
We close by briefly mentioning an interesting problem relating to the $\gamma$-vector introduced by Gal [@G]. The behavior of the $\gamma$-vector under barycentric subdivision was studied in [@N]. In [@Ath] a theory of local $\gamma$-vectors are started and in [@AthSav Theorem 1.5] a nice interpretation of the local $\gamma$-vector was given for the classical barycentric subdivision. Again one can ask the same questions for the $\gamma$ and local $\gamma$-vector of the $l$^th^ partial barycentric subdivision.
[1]{} C.A. Athenasiadis, [*Flag subdivisions and $\gamma$-vectors*]{}, [http://arxiv.org/abs/1106.4520]{}. C.A. Athanasiadis, C. Savvidou, [*The local h-vector of the cluster subdivision of a simplex*]{}, Seminaire Lotharingien de Combinatoire 66 (2012), Article B66c, 21pp. F. Brenti, V. Welker, [*$f$-vectors of barycentric subdivisions*]{}, Math. Z. [**259**]{} (2008) 849-865. E. Delucchi, L. Sabalka, A. Pixton, f-polynomials of subdivisions of complexes Discrete Math. to appear, [http://arxiv.org/abs/1002.3201]{}. S.R. Gal, [*Real root conjecture fails for five- and higher-dimensional spheres*]{}, Discrete Comput. Geom. [**34**]{} (2005), 269–284. R. A. Horn, C. R. Johnson, [*Matrix Analysis*]{}, Cambridge University Press, Cambridge (1985). E. Nevo, T.K. Petersen, B. Tenner, [*The $\gamma$-vector of a barycentric subdivision*]{}, J. Combin. Theory Ser. A [**118**]{} (2011), 1364-1380. R. P. Stanley, [*Subdivision and local h-vectors*]{}, J. Amer. Math. Soc. [**5**]{} (1992) 805-851. R. P. Stanley, [*Enumerative Combinatorics*]{}, Vol. 1, Second edition, Cambridge University Press, 2012.
|
The nature of ordering in spin glasses below the transition temperature, $T_c$, remains a controversial issue. Two theories have been extensively discussed: the “droplet theory” proposed by Fisher and Huse[@fh] (see also Refs. ), and the replica symmetry breaking (RSB) theory of Parisi[@parisi; @mpv; @by]. An important difference between these theories concerns the number of large-scale, low energy excitations. In the RSB theory, which follows the exact solution of the infinite range SK model, there are excitations which involve turning over a finite fraction of the spins and which cost only a [*finite*]{} energy even in the thermodynamic limit. Furthermore, the surface of these excitations is argued[@qlink] to be space filling, i.e. the fractal dimension of their surface, $d_s$, is equal to the space dimension, $d$. By contrast, in the droplet theory, the lowest energy excitation which involves a given spin and which has linear spatial extent $L$ typically costs an energy of order $L^\theta$, where $\theta$ is a (positive) exponent. Hence, in the thermodynamic limit, excitations which flip a finite fraction of the spins cost an [*infinite*]{} energy. Also, the surface of these excitations is not space filling, [*i.e.*]{} $d_s < d$.
Recently we[@py1; @py2; @py3] investigated this issue by looking at how spin glass ground states in two and three dimensions change upon changing the boundary conditions. Extrapolating from the range of sizes studied to the thermodynamic limit, our results suggest that the low energy excitations have $d_s < d$. Similar results were found in two dimensions by Middleton[@midd]. In this paper, following a suggestion by Fisher[@dsf], we apply a perturbation to the ground states in the [*bulk*]{} rather than at the surface. The motivation for this is two-fold: (i) We can apply the same method both to models with short range interactions and to infinite range models, like the SK model, and so can verify that the method is able to distinguish between the RSB picture, which is believed to apply to infinite range models, and some other picture which may apply to short range models. (ii) It is possible that there are other low energy excitations which are not excited by changing the boundary conditions[@kawa; @hm]. We consider the short-range Ising spin glass in three and four dimensions, and, in addition, the SK and Viana-Bray[@vb] models. The latter is infinite range but with a finite average coordination number $z$, and is expected to show RSB behavior. All these models have a finite transition temperature. Our results for the SK and Viana-Bray models show clearly the validity of the RSB picture. However, for the short range models, our data is consistent with a picture suggested by Krzakala and Martin[@km] where there are extensive excitations with [*finite*]{} energy, i.e. their energy varies as $L^{\theta'}$ with $\theta' = 0$, but $d_s < d$. In three dimensions, this picture is difficult to differentiate from the droplet picture where the energy varies as $L^\theta$, because of the small value of $\theta$ ($\simeq 0.2$, obtained from the magnitude of the change of the ground state energy when the boundary conditions are changed from periodic to anti-periodic[@theta-3d]). It is easier to distinguish the two pictures in 4-D, even though the range of $L$ is less, because $\theta$ is much larger[@theta-4d] ($\simeq 0.7$). The Hamiltonian is given by $${\cal H} = -\sum_{\langle i,j \rangle} J_{ij} S_i S_j ,
\label{ham}$$ where, for the short range case, the sites $i$ lie on a simple cubic lattice in dimension $d=3$ or 4 with $N=L^d$ sites ($L \le 8$ in 3-D, $L \le 5$ in 4-D), $S_i=\pm
1$, and the $J_{ij}$ are nearest-neighbor interactions chosen from a Gaussian distribution with zero mean and standard deviation unity. Periodic boundary conditions are applied. For the SK model there are interactions between [*all*]{} pairs chosen from a Gaussian distribution of width $1/\sqrt{N-1}$, where $N \le 199$. For the Viana-Bray model each spin is connected with $z=6$ spins on average, chosen randomly, the width of the Gaussian distribution is unity, and the range of sizes is $N \le 399$. To determine the ground state we use a hybrid genetic algorithm introduced by Pal[@pal], as discussed elsewhere[@py2]. Let $S_i^{(0)}$ be the spin configuration in the ground state for a given set of bonds. Having found $S_i^{(0)}$, we then add a perturbation to the Hamiltonian designed to increase the energy of the ground state relative to the other states, and so possibly induce a change in the ground state. This perturbation, which depends upon a positive parameter $\epsilon$, changes the interactions $J_{ij}$ by an amount proportional to $S_i^{(0)} S_j^{(0)}$, i.e. $$\Delta {\cal H}(\epsilon) = \epsilon {1 \over N_b} \sum_{\langle i,j \rangle}
S_i^{(0)} S_j^{(0)} S_i S_j,$$ where $N_b$ is the number of bonds in the Hamiltonian. The energy of the ground state will thus increase exactly by an amount $ \Delta E^{(0)} = \epsilon .$ The energy of any other state, $\alpha$ say, will increase by the lesser amount $ \Delta E^{(\alpha)} = \epsilon\ {q_{l}}^{(0, \alpha)},$ where ${q_{l}}^{(0, \alpha)}$ is the “link overlap” between the states “0” and $\alpha$, defined by $${q_{l}}^{(0, \alpha)} = {1 \over N_b}\sum_{\langle i,j \rangle} S_i^{(0)} S_j^{(0)}
S_i^{(\alpha)} S_j^{(\alpha)} ,$$ in which the sum is over all the $N_b$ pairs where there are interactions. Note that the [*total*]{} energy of the states is changed by an amount of order unity.
The decrease in the energy [*difference*]{} between a low energy excited state and the ground state is given by $$\delta E^{(\alpha)} = \Delta E^{(0)} - \Delta E^{(\alpha)} =
\epsilon \ (1 - {q_{l}}^{(0, \alpha)}) .
\label{de}$$ If this exceeds the original difference in energy, $E^{(\alpha)} - E^{(0)}$, for at least one of the excited states, then the ground state will change due to the perturbation. We denote the new ground state spin configuration by $ \tilde{S}_i^{(0)}$, and indicate by ${q_{l}}$ and $q$, with no indices, the link- and spin-overlap between the new and old ground states.
Next we discuss the expected behavior of $q$ and ${q_{l}}$ for the various models. For the SK model, it is easy to derive the trivial relation, ${q_{l}}= q^2$ (for large $N$). Since RSB theory is expected to be correct, there are some excited states which cost a finite energy and which have an overlap $q$ less than unity. According to Eq. (\[de\]), these have a finite probability of becoming the new ground state. Hence the average value of $q$ and ${q_{l}}$ over many samples, denoted by $[\cdots]{_{\mathrm{av}}}$, should tend to a constant less than unity in the thermodynamic limit. This behavior is shown in the inset of Fig. \[q\_sk\]. For the Viana-Bray model, where there is no trivial connection between $q$ and ${q_{l}}$, we show in Fig. \[q\_sk\] data for $R = (1-[{q_{l}}]{_{\mathrm{av}}})/(1-[q]{_{\mathrm{av}}})$ for several values of $\epsilon$. This also appears to saturate. We plot this ratio rather than $[q]{_{\mathrm{av}}}$ or $[q_l]{_{\mathrm{av}}}$ for better comparison with the short range case below. For both models we took $\epsilon$ to be a multiple of the transition temperature (the mean field approximation to it, $T_c^{MF} = \sqrt{z}$, for the Viana-Bray model), so that a perturbation of comparable magnitude was applied in both cases.
What do we expect for the short range models? In the RSB theory, $1-[q]_{av}$ and $1-[q_l]_{av}$ (and hence the ratio $R$) should saturate to a finite value for large $L$. To derive the prediction of the droplet theory, suppose that the energy to create an excitation of linear dimension, $l$, has a characteristic scale of $l^{\theta'}$ (we use $\theta'$ rather than $\theta$ to allow for the possibility that this exponent is different from the one found by changing the boundary conditions). Let us assume that large clusters ($l \approx L$) dominate and ask for the probability that a large cluster is excited. The energy gained from the perturbation is $\epsilon (1 - {q_{l}}) \sim \epsilon / L^{(d - d_s)}$ since $1/L^{(d-d_s)}$ is the fraction of the system containing the surface (i.e. the broken bonds) of the cluster. Generally this will not be able to overcome the $L^{\theta'}$ energy cost to create the cluster. However, there is a distribution of cluster energies and if we make the plausible hypothesis that this distribution has a finite weight at the origin, then the probability that the cluster is excited is proportional to $1/L^{d - d_s + \theta'}$. In other words $$1 - [q]{_{\mathrm{av}}}\sim \epsilon / L^{\mu} \ \ {\mathrm where} \ \
\mu = \theta' + d - d_s .
\label{psis}$$ As discussed above, $1 - {q_{l}}$ is of order $1/L^{(d-d_s)}$ and so $$1 - [{q_{l}}]{_{\mathrm{av}}}\sim \epsilon / L^{{\mu_{l}}} \ \ {\mathrm where} \ \
{\mu_{l}}= \theta' + 2(d - d_s) .
\label{psil}$$ Similar expressions have been derived by Drossel et al.[@drossel] in another context. Eqs. (\[psis\]) and (\[psil\]) are expected to be valid only [*asymptotically*]{} in the limit $\epsilon \to 0$. In order to include data for a range of values of $\epsilon$ we note that the data is expected to scale as $$\begin{aligned}
1 - [q]{_{\mathrm{av}}}& = & F_q(\epsilon/L^\mu) , \nonumber \\
1 - [{q_{l}}]{_{\mathrm{av}}}& = & L^{-(d-d_s)}
F_{q_{l}}(\epsilon/L^\mu) ,
\label{scaling}\end{aligned}$$ where the scaling functions $F_q(x)$ and $F_{q_{l}}(x)$ both vary linearly for small $x$. Note that the above discussion applies also to a picture in which $\theta'=0$ and $d_s<d$.
A scaling plot of our results for $1 - [q]{_{\mathrm{av}}}$ in 3D is shown in Fig. \[q\_scaling\_3d\]. We consider a range of $\epsilon$ from $\sqrt{6}/4$ to $4\sqrt{6}$ (note that $T_c^{MF}= \sqrt{6}$) and find that the data collapse well onto the form expected in Eq. (\[scaling\]) with $ \mu = 0.44 \pm 0.02.$
It is also convenient to plot the ratio $R$, which represents the surface to volume ratio of the excited clusters. This has a rather weak dependence on $\epsilon$ and, as shown in Fig. \[q\_ql\_3d\], the data for [*each*]{} of the values of $\epsilon$ fits well the power law behavior $L^{-(d-d_s)}$, expected from Eqs. (\[psis\]) and (\[psil\]), with $d - d_s$ between 0.40 and 0.41 (the goodness of fit parameter, $Q$, is $0.07, 0.03, 0.85, 0.23, 0.10$, in order of increasing $\epsilon$). The inset to Fig. \[q\_ql\_3d\] shows that there are small deviations from the asymptotic behaviour, which can be accounted for by a scaling function with the same value of $\mu$ as in Fig. \[q\_scaling\_3d\] and with $$d - d_s = 0.42 \pm 0.02 \ \ (3D) .
\label{ds}$$ From this value of $\mu$ and Eqs. (\[psis\]) and (\[ds\]) we find $$\theta' = 0.02 \pm 0.03 \ \ (3D) .$$ In order to test the RSB prediction, we tried fits of the form $ R = a + b/L^c $, which give $a = 0.28 \pm 0.18, 0.01 \pm 0.14, 0.04 \pm 0.11$, and $-0.28 \pm 0.18$ ($Q=0.08,0.01,0.72$, and 0.52) for $\epsilon/\tau = 0.25, 0.5, 1$ and $2$. These are consistent with $a=0$ though a fairly small positive value, which would imply $d_s=d$, cannot be ruled out. For $\epsilon/\tau
=4$ the fit gives a small positive value, $0.18 \pm 0.07$ ($Q=0.79$), but this is likely too large a value of $\epsilon$ to be in the asymptotic regime for these sizes (see the inset of Fig. \[q\_ql\_3d\]). The form $R=a+b/L+c/L^2$ also fits reasonably well the data and gives $a$ between 0.41 and 0.48 ($Q=0.16,0.03,0.82,0.80,0.16$). However, for both forms the data are very far from the asymptotic limit $R\sim a$ for the sizes considered, unlike for the Viana-Bray model (compare the main parts of Figs. \[q\_sk\] and \[q\_ql\_3d\]). By contrast, the deviation from the asymptotic behavior $R\sim L^{-(d-d_s)}$ is quite small (see the inset of Fig. \[q\_ql\_3d\]).
In Fig. \[q\_all\_4d\] we show analogous results in 4-D. The calculations were performed for two different values $\epsilon = \sqrt{8}/4$ and $\sqrt{8}$ ($= T_c^{MF}$). The exponents are essentially the same for these two values of the perturbation and the fits give $ \mu = 0.26 \pm 0.04$, $d - d_s = 0.23 \pm 0.02$ , and so from Eq. (\[psis\]) we get our main results for 4D: $$\theta' = 0.03 \pm 0.05, \quad d - d_s = 0.23 \pm 0.02 \ \ (4D).$$ The data in Fig. \[q\_all\_4d\] is [*consistent*]{} with the scaling form in Eq. (\[scaling\]) but the data for the two values of $\epsilon$ are too widely separated to [*demonstrate*]{} scaling.
Interestingly, our results in both 3-D and 4-D are consistent with $\theta' =
0$, and, within the error bars, (which are purely statistical) incompatible with the relation $\theta' = \theta$, since $\theta \simeq 0.20$ in 3-D[@theta-3d; @py2] and $\theta \simeq 0.7$ in 4-D[@theta-4d]. In 3-D, $\theta - \theta'$ is small, but in 4-D this difference is larger and hence the conclusion that $\theta' \ne \theta$ is stronger. However, the conclusion that $d-d_s > 0$ is less strong in 4-D because our value for $d-d_s$ is quite small and the range of sizes is smaller than in 3-D. It would be interesting, in future work, to study the nature of these excitations to see how they differ from the excitation of energy $L^\theta$ (with $\theta > 0$) induced by boundary condition changes[@py2; @theta-3d; @theta-4d]. In particular, if their volume is space filling, one would expect a non-trivial order parameter distribution, $P(q)$, at finite temperatures. To conclude, an interpretation of our results for short range models which is natural, in that it fits the data with a minimum number of parameters and with small corrections to scaling, is that there are large-scale low energy excitations which cost a finite energy, and whose surface has fractal dimension less than $d$. This picture differs from the one suggested by Houdayer and Martin[@hm], in which $d_s=d$. Furthermore, the results for short range models appear quite different from those of the mean-field like Viana-Bray model for samples with a similar coordination number and a similar number of spins. Other scenarios, such as the droplet theory (with $\theta^\prime = \theta \ (> 0)$) or an RSB picture (where $\theta^\prime = 0, d - d_s = 0$), require larger corrections to scaling, but we cannot rule out the possibility of crossover to one of these behaviors at larger sizes. We would like to thank D. S. Fisher for suggesting this line of enquiry, and for many stimulating comments. We also acknowledge useful discussions and correspondence with G. Parisi, E. Marinari, O. Martin, M. Mézard and J.-P. Bouchaud. We are grateful to D. A. Huse, M. A. Moore and A. J. Bray for suggesting the scaling plot in Fig. \[q\_scaling\_3d\] and one of the referees for suggesting plotting the ratio $R$. This work was supported by the National Science Foundation under grant DMR 9713977. M.P. also is supported in part by a fellowship of Fondazione Angelo Della Riccia. The numerical calculations were supported by computer time from the National Partnership for Advanced Computational Infrastructure.
D. S. Fisher and D. A. Huse, J. Phys. A. [**20**]{} L997 (1987); D. A. Huse and D. S. Fisher, J. Phys. A. [**20**]{} L1005 (1987); D. S. Fisher and D. A. Huse, Phys. Rev. B [**38**]{} 386 (1988).
A. J. Bray and M. A. Moore, in [*Heidelberg Colloquium on Glassy Dynamics and Optimization*]{}, L. Van Hemmen and I. Morgenstern eds. (Springer-Verlag, Heidelberg, 1986).
W. L. McMillan, J. Phys. C [**17**]{} 3179 (1984).
C. M. Newman and D. L. Stein, Phys. Rev. B [**46**]{}, 973 (1992); Phys. Rev. Lett., [**76**]{} 515 (1996); Phys. Rev. E [**57**]{} 1356 (1998).
G. Parisi, Phys. Rev. Lett. [**43**]{}, 1754 (1979); J. Phys. A [**13**]{}, 1101, 1887, L115 (1980; Phys. Rev. Lett. [**50**]{}, 1946 (1983).
M. Mézard, G. Parisi and M. A. Virasoro, [*Spin Glass Theory and Beyond*]{} (World Scientific, Singapore, 1987).
K. Binder and A. P. Young, Rev. Mod. Phys. [**58**]{} 801 (1986).
E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. Ruiz-Lorenzo and F. Zuliani, J. Stat. Phys. [**98**]{}, 973 (2000).
M. Palassini and A. P. Young, Phys. Rev. [**B60**]{}, R9919, (1999).
M. Palassini and A. P. Young, Phys. Rev. Lett. [**83**]{}, 5216 (1999).
M. Palassini and A. P. Young, cond-mat/9910278.
A. A. Middleton, Phys. Rev. Lett. [**83**]{}, 1672 (1999).
D. S. Fisher, private communication.
N. Kawashima, J. Phys. Soc. Jpn. [**69**]{}, 987 (2000).
J. Houdayer and O. C. Martin, Europhys. Lett. [**49**]{}, 794 (2000).
L. Viana and A. J. Bray, J. Phys. C [**18**]{}, 3037 (1985).
A. K. Hartmann, Phys. Rev. E [**59**]{}, 84 (1999). A. J. Bray and M. A. Moore, J. Phys. C, [**17**]{}, L463 (1984); W. L. McMillan, Phys. Rev. B [**30**]{}, 476 (1984).
A. K. Hartmann, Phys. Rev. E [**60**]{}, 5135 (1999). K. Hukushima, Phys. Rev. E [**60**]{}, 3606 (1999).
F. Krzakala and O. C. Martin cond-mat/0002055.
K. F. Pal, Physica A [**223**]{}, 283 (1996); K. F. Pal, Physica A [ **233**]{}, 60 (1996).
B. Drossel, H. Bokil, M. A. Moore and A. J. Bray, Eur. Phys. J. [**13**]{}, 369 (2000).
|
---
abstract: 'Many problems in areas as diverse as recommendation systems, social network analysis, semantic search, and distributed root cause analysis can be modeled as pattern search on labeled graphs (also called “heterogeneous information networks” or HINs). Given a large graph and a query pattern with node and edge label constraints, a fundamental challenge is to find the top-$k$ matches according to a ranking function over edge and node weights. For users, it is difficult to select value $k$. We therefore propose the novel notion of an *any-$k$ ranking algorithm*: for a given time budget, return as many of the top-ranked results as possible. Then, given additional time, produce the next lower-ranked results quickly as well. It can be stopped anytime, but may have to continue until all results are returned. This paper focuses on acyclic patterns over arbitrary labeled graphs. We are interested in practical algorithms that effectively exploit (1) properties of heterogeneous networks, in particular selective constraints on labels, and (2) that the users often explore only a fraction of the top-ranked results. Our solution, [[KARPET]{}]{}, carefully integrates aggressive pruning that leverages the acyclic nature of the query, and incremental guided search. It enables us to prove strong non-trivial time and space guarantees, which is generally considered very hard for this type of graph search problem. Through experimental studies we show that [[KARPET]{}]{} achieves running times in the order of milliseconds for tree patterns on large networks with millions of nodes and edges.'
author:
- Xiaofeng Yang
- Deepak Ajwani
- Wolfgang Gatterbauer
- 'Patrick K. Nicholson'
- Mirek Riedewald
- Alessandra Sala
bibliography:
- 'refs.bib'
title: 'Any-k: Anytime Top-k Tree Pattern Retrieval in Labeled Graphs'
---
Introduction {#sec:intro}
============
Heterogeneous information networks (HIN) [@han2010mining], i.e., graphs with node and/or edge labels, have recently attracted a lot of attention for their ability to model many complex real-world relationships, thereby enabling rich queries. Often labels are used to represent types of nodes and their relationships:
\[ex:flickr1\] Consider a photo-sharing social network with three vertex type labels: user, photo, and group. Users are connected to the photos they upload, and photos are connected to groups when they are posted there. Finally, users can connect to groups by joining them. To maintain a vibrant community and alert users about potentially interesting photos, the social network might run queries of the type shown in Figure \[fig:FlickrExample\]: given *photo1* and two users, *user1* and *user2*, find alternative groups (matching nodes for *group2*) to post the photo in order to reach *user2* without spamming her directly. This is achieved by identifying a user belonging to both groups (*user3*), who can post the photo in the other group. There might be hundreds of matching triples (*group1*, *user3*, *group2*), and there would be many more if *user2* was not given in advance. Under these circumstances, the goal often is not to find *all* results, but only the *most important* ones. Importance can be determined based on node and edge weights, e.g., weights representing distances (or similarities). Then the query should return the lightest (or heaviest) pattern instances. For example, the weight of a group may be based on its number of members, the weight of a user on how active s/he is, and the weight of a link on the timestamp when it was established (to give preference to long-term relationships or more recent photo posts), or the sum of the PageRanks of its endpoints.
![Example query on a photo-sharing network: find the most important nodes of types ([user3]{}, [group1]{}, [group2]{}) for a given triple of specified nodes of types ([photo1]{}, [user1]{}, [user2]{}).[]{data-label="fig:FlickrExample"}](./figures/flickr.pdf){width="0.7\linewidth"}
These types of rich query semantics also appears in other contexts, e.g., root-cause analysis in distributed systems. The Vitrage service for OpenStack [@Vitrage] makes use of path and tree patterns to specify rules for automatic root cause deduction of alarms raised by virtual machines and hardware. Large OpenStack deployments—involving thousands of hosts and tens of thousands of virtual machines and hardware components—necessitate pattern matching algorithms to deduce the root cause of such patterns in near real-time.
We focus on efficient solutions for acyclic pattern queries on general labeled graphs. To this end, we propose the notion of *any-$k$* algorithms, a novel variant of top-$k$ algorithms. A top-$k$ algorithm exploits knowledge about the given $k$ to produce the top-$k$ lightest patterns faster than the “full enumeration” algorithm (which first produces all results and then ranks them by weight). In practice, it is difficult for users to know the value of $k$ upfront (“when will I have seen enough?”). An any-$k$ algorithm addresses this issue by not requiring a pre-set value for $k$. Instead, an any-$k$ algorithm
1. returns the top-ranked result as quickly as possible,
2. then returns the second-ranked result next, followed by the third-ranked, and so on,
3. until the user is satisfied and terminates the process.
In other words, the ranked enumeration can be stopped *anytime* and should then return as many top results as possible.
The queries we are interested in correspond to *subgraph isomorphism*, which is known to be hard in general. In particular, subgraph isomorphism on homogeneous graphs is already NP-complete in the size of the query (even for the path case as Hamiltonian path is a special case). And labeled graphs contain unlabeled graphs as a special case. On the other hand, labels provide more opportunities for achieving better performance *in practice* by exploiting heterogeneity where present. Note that a key reason for hardness of isomorphism lies in the “non-repetition constraint,” i.e., the same graph node cannot occur more than once in an answer. Without this constraint, pattern search would correspond to the easier *subgraph homomorphism* problem which can be solved in PTIME.
Our approach is based on three key insights: (1) Constraints on node or edge labels can dramatically reduce the number of matching results; (2) Mutually exclusive [type]{} labels “narrow the gap” in cardinality between the set of isomorphic subgraphs and the set of homomorphic subgraphs (which includes all isomorphic ones). The reason is that query pattern nodes of different types cannot be mapped to the same graph node, even when the algorithm is only searching for homomorphism. In the example photo-sharing network, users and photos cannot stand in for a group node. In the extreme, if all nodes in the query pattern have different types, then any solution for subgraph homomorphism also satisfies isomorphism. This suggests an approach that aggressively prunes for the homomorphism case and then filters based on node repetitions in the result patterns; and (3) In many real-world cases, output size is small relative to the combinatorial size of the pattern search space. Hence algorithm complexity bounds based on output size promise to deliver practically meaningful performance guarantees.
**Overview of the Solution.** Our approach combines three conceptually separate steps into a two-phase algorithm.
1\) The search space of possible homomorphic patterns is pruned to the provably smallest representation of the original graph. We use insights from the well-known Yannakakis algorithm [@DBLP:conf/vldb/Yannakakis81] for evaluating answers to acyclic conjunctive queries to create this representation in just one bottom-up and a subsequent top-down sweep through the query tree.
2\) We devise a novel any-$k$ algorithm for enumerating homomorphic tree patterns. It uses dynamic programming to perform a bottom-up cost calculation, followed by a top-down guided search.
3\) A final pruning step removes those homomorphic patterns that do not satisfy the isomorphism requirement.
We show how to combine the first two steps into just one bottom-up and one top-down phase. We then integrate the third step into the combined top-down phase. Our experiments show that even on graphs with millions of nodes and tens of millions of edges, we can return the top-ranked results in just a few milliseconds, whereas alternative approaches would take orders of magnitude longer. Our implementation can be downloaded from [@Code2018].
**Main contributions.** We devise [[KARPET]{}]{} (ernelization[^1] nd apid runing-based xploration for ree patterns), a novel and highly performant any-$k$ algorithm that can quickly identify top-ranked tree patterns in large graphs, then return the next lower-ranked ones when given extra time.
1\) [[KARPET]{}]{} is designed as an *anytime* ranking algorithm that enumerates homomorphic subtrees in order of total edge weight with strong theoretic guarantees: We show that our worst-case time complexity for returning all homomorphism results is identical to full enumeration. In addition, [[KARPET]{}]{} provides strong upper bound guarantees for the time to return the top-ranked homomorphism result, as well as the time between returning a homomorphism result and the next. For cases with “small gap” between homomorphism and isomorphism, i.e., when “sufficiently many” homomorphic patterns are also isomorphic patterns, these guarantees carry over to subgraph isomorphism.
2\) We propose fast and effective *local* pruning operations that exploit the heterogeneity of labeled graphs, proving that they also guarantee strong *global* pruning properties. Intuitively, for subgraph homomorphism, we show that inexpensive pruning based on 1-node neighborhoods efficiently removes all candidate nodes that are not part of any result pattern.
3\) In contrast to a lot of theoretical work on subgraph isomorphism algorithms, our algorithm is output-sensitive—its worst case complexity depends on the output size, which is smaller when the graph and the query are more heterogeneous, rather than being exponential in the size of the query pattern.
4\) We show how to speed up the search for top-ranked isomorphic answers by pushing the pruning for non-repeating nodes into the incremental result enumeration algorithm.
Any-k Algorithm {#sec:algorithm}
===============
We next present an approach for sub-graph *homomorphism*; this is a relaxation of sub-graph isomorphism in that we do not require the mapping ${\lambda}$ from query nodes to tree-pattern nodes to be bijective (in other words, a node can be repeated in the result pattern). Section \[sec:repetitions\] extends the approach for isomorphism.
[[KARPET]{}]{} consists of two phases: 1) a bottom-up sweep from leaves to the root of ${Q}$, and 2) a top-down depth-first traversal from root to leaves. The first phase prunes some of the spurious candidates and creates a “*candidate graph*” (discussed below) with “*minimum subtree weights*.” The second phase prunes the remaining spurious candidates and performs a search guided by the subtree weights. Here the term *spurious candidate* refers to a node or edge of the input graph that does not appear in any of the query results.
Bottom-Up Phase {#sec:bottomUp}
---------------
**Input**: query ${Q}$, node neighborhood index $N(v, \ell)$\
**Output**: ${\texttt{CandNode}\xspace}: u \mapsto [c \mapsto [u' \mapsto w_{\min}]]$\
${\texttt{CandEdge}\xspace}: (u, u') \mapsto [c \mapsto c']$\
\[alg:line:terminal\] $\forall c \in {V}.
{\varphi}(c) = {\psi}(u) :
{\texttt{CandNode}\xspace}(u).\textsc{Insert}(c \mapsto (\textrm{NIL} \mapsto 0))$
\[alg:line:traversal\]
\[alg:line:createcandidates\] ${\texttt{CandEdge}\xspace}(u, u').\textsc{Insert}(c \mapsto c')$
$C = \bigcap_{u' \textrm{ child nodes of } u} {\texttt{CandEdge}\xspace}(u, u').\textsc{Keys}$ \[alg:line:intersection\]
$C' = {\texttt{CandEdge}\xspace}(u, u').\texttt{Get}(c)$ $w_{i} \leftarrow \min_{c' \in C'}[w(c, c') + {\textsc{Weight}\xspace}(c')]$ ${\texttt{CandNode}\xspace}(u).\textsc{Insert}(c \mapsto (u_i \mapsto w_i))$ \[alg:line:minsubtree\]
[.31]{} 
[.31]{} 
[.31]{} 
The bottom-up phase traverses the query tree in any bottom-up order and constructs a “candidate graph” consisting of two index structures: (1) ${\texttt{CandNode}\xspace}(u)$ returns for query node $u$ a hash index that maps a node candidate $c$ to a list of minimum subtree weights, with one weight for each of $c$’s children. (2) ${\texttt{CandEdge}\xspace}(u, u')$ returns for each query edge between a node $u$ and its child $u'$ a hash index that maps a candidate node $c$ of $u$ to all adjacent candidates $c'$ of $u'$.
We illustrate Algorithm \[alg:pruning\] with Figures \[fig:bottomUp1\], \[fig:bottomUp2\], and \[fig:bottomUp3\]. It first inserts candidate nodes for each query leaf node $u$ into the corresponding candidates ${\texttt{CandNode}\xspace}(u)$, setting their weights to zero (line \[alg:line:terminal\]). Note that leaves do not have children, hence the NIL value in the expression. In Figure \[fig:bottomUp1\] there is a single candidate per leaf, but in practice it can be a larger subset of ${V}$ for each query leaf, depending on the node constraints. Then, for each query node $u$, the algorithm ($i$) finds possible candidate nodes, ($ii$) prunes them, and ($iii$) calculates the minimum subtree weights
In more detail: ($i$) for each query edge leading to a child $(u,u')$, it first finds all candidate edges $(c, c')$, storing the map ${\texttt{CandEdge}\xspace}: (u, u') \mapsto [c \mapsto c']$ (line \[alg:line:createcandidates\]). ($ii$) Then, the algorithm only keeps the list of candidates for each query node that are *reachable from candidate instances in all leaves of the query node* (line \[alg:line:intersection\]): In Figure \[fig:bottomUp3\], the list of candidates for query node *group1* is $\{c_1, c_2, c_3\}$. Notice how spurious candidates not reachable from the leaves, e.g., $e_1$ in *group2*, are not even accessed (compare with Figure \[fig:joinGraph\]). Similarly, while $d_1$ in *user3* is reachable from the left, it is not reachable from the right subtree and is thus automatically pruned as well. ($iii$) Then, the algorithm finds for each reachable node, the min weight along each query edge $(u, u')$ starting at $c$ (line \[alg:line:minsubtree\]). For example, in Figure \[fig:bottomUp3\], the left weight $5$ for $c_2$ is computed as the minimum of weights for following $(d_2, c_2)$, which is 5 as the sum of the weight of edge $(d_2, c_2)$ (= 2) plus the weight of $c_2$ (= 2+1), or for following $(d_2, c_3)$, which is 7 as the sum of the weight of edge $(d_2, c_2)$ (= 4) plus the weight of $c_3$ (= 2+1). Notice we use here ${\textsc{Weight}\xspace}(c)$ as short form for the sum of weights at a node $c$, which we get from [`CandNode`]{}. The two new created indices speed up finding adjacent edges in a subtree of the query pattern during top-down traversal.
Top-Down Phase
--------------
The second part of our algorithm performs top-down search, starting at the root node and proceeding downward to the leaves. This is essential for two reasons: First, the pre-computed subtree weights provide information to guide the search to the lightest patterns before exploring the heavier ones. Second, the top-down traversal implicitly prunes *all* remaining spurious candidates for sub-graph homomorphism, as we will prove in Section \[sec:analysis\]. Again, pruning actually happens implicitly by not reaching those candidates. To see the latter, consider *group1* candidate $c_1$ in Figure \[fig:bottomUp3\]. It is spurious, but could not be removed by the bottom-up sweep. However, it will never be accessed during top-down traversal, because $d_1$ was never recorded in [`CandNode`]{}by Algorithm \[alg:pruning\].
**Input**: Tree pattern ${Q}$, [`CandNode`]{}, [`CandEdge`]{}\
**Output**: All matches of ${Q}$, one-by-one in increasing order of weight
$\texttt{pq} \gets \textsc{PriorityQueue}()$ \[line:alg:initpq\] $Z \gets$ partial tree $(c)$ consisting of just one node $\texttt{pq}.\textsc{Insert}({\textsc{Weight}\xspace}(c), Z)$ \[line:update-cost\] $(\texttt{oldkey}, Z) \gets \texttt{pq}.\textsc{Pop-Minimum}$ return $Z$ $(u, u') \gets \textsc{NextPreorder}({Q},Z)$ \[line:alg:next\]Edge to expand pattern $Z' = Z.\textsc{Append}(c')$\[line:expansion\] $\texttt{newkey} \gets
\texttt{oldkey}
- {\texttt{CandNode}\xspace}(u).\texttt{Get}(c, u')
+
\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}
\hspace{21mm}\phantom{a}
w(c,c')
+ {\textsc{Weight}\xspace}(c')
$ \[line:newPrio\] `pq`.<span style="font-variant:small-caps;">Push</span>($\texttt{newkey}, Z')$
Algorithm \[alg:prioritized-search\] shows the pseudo-code for top-down guided search. Initially, all candidates $c$ in the query root $r$ are inserted into priority queue `pq` (line \[line:alg:initpq\]), with their priorities set to the sum of the candidate’s weights. In Figure \[fig:bottomUp3\], there is a single candidate, $d_2$, of weight $5+3=8$. Then the algorithm repeatedly pops the top element from `pq` and expands the partial pattern using pre-order traversal. Function $\textsc{NextPreorder}$ returns the edge, as the pair of parent and child node, along which the partial pattern will be expanded next (line \[line:alg:next\]). The priority value of each expanded partial match is defined as the sum of the pattern’s edge weights plus the sum of the weights of the unexplored subtrees. In the example, partial match $(d_2, c_2)$ is inserted into `pq` with priority 8 = 2 (edge weight) + (2+1) (weights of $c_2$) + 3 (weight of right subtree of $d_2$). Similarly, partial match $(d_2, c_3)$ is inserted with priority 4+(2+1)+3 = 10. Note that those values are computed incrementally during traversal (line \[line:newPrio\]). Consider expansion of $(d_2)$ to $(d_2, c_3)$. Priority of $d_2$ was 8, with weight 5 for the newly expanded subtree rooted at *group1*. After retrieving $c_3$ from [`CandEdge`]{}, priority of $(d_2, c_3)$ is computed as 8 (`old`) - 5 (newly expanded subtree) + 4 (weight of edge $(d_2, c_3)$ ) + 3 (priority of $c_3$) = 10 (line \[line:newPrio\] in Alg. \[alg:prioritized-search\]). Then $(d_2, c_2)$ is popped next, and expanded to partial match $(d_2, c_2, a)$ with priority 8 = 8 - 2 + 2 + (0+0). This pattern is then expanded next to $(d_2, c_2, a, b)$, $(d_2, c_2, a, b, e_2)$, and finally $(d_2, c_2, a, b, e_2, f)$—all with the same priority of 8. The latter is output as the minimal-weight solution. Only then will partial match $(d_2, c_3)$ with the higher priority value 10 be expanded analogously. Each expansion operation requires a pop operation from priority queue, visiting potential edges once.
Algorithm Analysis {#sec:analysis}
==================
All results in this section are for the relaxed version of the problem, based on sub-graph homomorphism instead of isomorphism. We discuss in Section \[sec:repetitions\] how to extend them to the isomorphism case. Proofs were omitted due to space constraints, but can be found in the extended version [@yang2018].
Minimality of Candidate Graph
-----------------------------
We show that during top-down search (Alg. \[alg:prioritized-search\]), no spurious candidate node will ever be accessed. A candidate node $c$ for a query node $q$ is “*spurious*” if there does not exist any homomorphic result pattern where $c$ is matched to $q$. Ensuring that no spurious nodes are accessed is crucial for proving strong upper bounds on the algorithm cost.
\[thm:spuriousNode\] If node candidate $c \in {\texttt{CandNode}\xspace}(q)$ for query node $q \in V_Q$ is accessed by Alg. \[alg:prioritized-search\], then there exists a homomorphic result pattern where ${\lambda}(q) = c$.
Each Pop, One Result—In Order {#sec:pq_pops}
-----------------------------
Next, we show a powerful result that is crucial in establishing important algorithm properties: During the top-down guided search, for each query result there is *at most* one push and *at most* one pop operation on priority queue `pq`. For this, we need the following lemmas.
\[lem:priorityMonotonicity\] The priority value of a partial pattern $P$ is always less than or equal to the priority of all its successors.
\[lem:sameWeightPQ\] Assume that Alg. \[alg:prioritized-search\] popped partial pattern $P=(c_1, c_2,\ldots, c_j)$, $j < |V_Q|$, of priority $w$ from `pq`. Then there exists a direct successor $(c_1, c_2,\ldots, c_j, c_{j+1})$ that has the same priority $w$.
Lemmas \[lem:priorityMonotonicity\] and \[lem:sameWeightPQ\] immediately imply:
\[cor:frontOptOk\] If the last pop operation on `pq` returned an incomplete pattern $P$, then one of the direct successors of $P$ will have priority equal to the minimum priority over all elements in `pq`.
\[ex:PETpriorityValues\] Consider the changes of `pq` for the example in Figure \[fig:bottomUp3\]. Initially it contains $[(d_2):8]$, the sole root node candidate with priority 5+3=8. This element is popped and expanded along edges $(d_2, c_2)$ and $(d_2, c_3)$. The priority of the former is 2 (weight of edge $(d_2, c_2)$) plus (2+1) (subtree weights of $c_2$) plus 3 (right subtree weight of $d_2$) = 8. It is identical to the initial priority of $d_2$, because edge $(d_2, c_2)$ is the one that determined the minimum left subtree weight of 5 in $d_2$. For $(d_2, c_3)$, priority is 10 due to the higher weight of edge $(d_2, c_3)$. After these two patterns are pushed, `pq` contains $[(d_2, c_2):8, (d_2, c_3):10]$. The next pop delivers $(d_2, c_2):8$, which is expanded to $(d_2, c_2, a):8$, followed by repeated pop and push operations on this pattern, every time obtaining the same priority of 8, until the top result $(d_2, c_2, a, b, e_2, f)$ of weight 8 is completed. Only then will expansion of $(d_2, c_3):10$ commence.
[**Front-element optimization.**]{} Based on Corollary \[cor:frontOptOk\], we next introduce an important optimization to Alg. \[alg:prioritized-search\]. Since the corollary guarantees that one of the direct successors of the partial pattern popped before will have a minimal priority value, we avoid the push-pop cycle for it and keep expanding it directly, only pushing the other direct successors. More precisely, assume the algorithm just popped partial match $P=(c_1, c_2,\ldots, c_i)$ of priority $w$ from `pq`. While expanding this pattern by one more node, it keeps in memory the first direct successor $P'=(c_1, c_2,\ldots, c_i, c'_{i+1})$ encountered that also has priority value $w$, pushing all other direct successors to `pq`. This way the algorithm still works on a min-priority element, but avoids the push-pop cycle for it. This seemingly minor optimization has strong implications as formalized in the following theorems.
\[thm:kPop\] Using front-element optimization, for any $k$, the $k$-th pop operation from `pq` produces the $k$-th lightest homomorphic result pattern, possibly requiring additional push operations, but no more pop operations until this result pattern is returned.
\[cor:numPushes\] No matter how many results are retrieved, Alg. \[alg:prioritized-search\] never performs more than $r_H$ push operations on `pq` *in total*. Here $r_H$ denotes the number of homomorphic subtrees in ${G}$.
This follows directly from Theorem \[thm:kPop\] and the following observation. Assume the algorithm continues to run until all query results are found. At that point it has removed all partial matches from `pq` and the queue is empty. Theorem \[thm:kPop\] implies that retrieving all results requires exactly $r_H$ pop operations. If the total number of push operations exceeded this, then the queue would not be empty. (And obviously, any execution of Alg. \[alg:prioritized-search\] that stops before returning all results will only have performed a subset of the push operations executed by the time all results are returned.)
Algorithm Cost {#sec:algCost}
--------------
To avoid notational clutter, we treat the size of the query pattern as a small constant and omit it from most formulas. (Note that pattern size is equal to the number of edges in ${E_Q}$, e.g., 5 in the photo-sharing network example.) It is straightforward to extend the formulas by including $|{E_Q}|$ as a variable.
[**Algorithm \[alg:pruning\].**]{} Theoretical worst case cost is $\mathrm{O}(|{E}|)$, i.e., linear in graph size: for each of the query pattern edges, in the worst case all graph edges are accessed. The time for constructing ${\texttt{CandEdge}\xspace}$ and ${\texttt{CandNode}\xspace}$ adds a constant overhead per edge processed. In practice, only a small fraction of ${E}$ will be accessed because of the label constraints. In particular, by using ${\texttt{GraphEdge}}$ in line \[alg:line:createcandidates\] in Alg. \[alg:pruning\], all neighbors of matching types (labels) are accessed in time linear in the number of these neighbors. Space cost is upper bounded by the combined size of ${\texttt{CandNode}\xspace}$ and ${\texttt{CandEdge}\xspace}$, i.e., cannot exceed $|{E_Q}|$ times input graph size.
[**Algorithm \[alg:prioritized-search\].**]{} The results from Section \[sec:pq\_pops\] lead to strong guarantees. Space complexity of Alg. \[alg:prioritized-search\] is equal to the maximum size of the priority queue. Corollary \[cor:numPushes\] immediately implies:
\[thm:spaceAlg2\] Space cost of Alg. \[alg:prioritized-search\] is upper bounded by $r_H$, the total result size for sub-graph homomorphism.
From a user’s point of view, the time it takes to produce the next lower-ranked result is crucial:
\[thm:interarrival\] The initial latency for Alg. \[alg:prioritized-search\] to return the top-ranked homomorphic match, and also the time between returning any two consecutive homomorphic matches, is $\mathrm{O}(\mathrm{outDegree} + \log r_H)$. Here $\mathrm{outDegree} \le r_H$ is greater of (1) the number of candidates in the root node and (2) the maximum cardinality of the set of adjacent node candidates $c'$ in ${\texttt{CandEdge}\xspace}: (u, u') \mapsto [c \mapsto \{(c', {w}(c, c'))\}]$ for any query graph edge $(u, u')$ and candidate $c$.
These strong results show that [[KARPET]{}]{} can effectively exploit selective label constraints. For instance, if there are a thousand homomorphic subgraphs in ${G}$, then Theorems \[thm:spaceAlg2\] and \[thm:interarrival\] guarantee that Alg. \[alg:prioritized-search\] will never store more than a thousand partial matches in memory and will perform at most a thousand (inexpensive) computation steps to deliver the next result to the user—no matter how big or connected the given graph!
We show next that the anytime property of [[KARPET]{}]{}, i.e., that it can deliver the top-ranked results quickly and then the next ones on request, incurs *no performance penalty* for producing *all* homomorphic matches:
\[thm:timeAlg2\] The lower bound for producing *all* homomorphic result patterns is $\Omega(r_H)$; sorting them costs $\mathrm{O}(r_H \log r_H)$. Alg. \[alg:prioritized-search\] has matching total time complexity $\mathrm{O}(r_H \log r_H)$.
Homomorphism to Isomorphism {#sec:repetitions}
===========================
[[KARPET]{}]{} as introduced in Section \[sec:algorithm\] returns homomorphic matches. To obtain the desired isomorphic matches, function ${\lambda}$ mapping query nodes to tree-pattern nodes has to be bijective. To guarantee this, one simply has to filter out all results where different query nodes are mapped to the same graph node. Instead of filtering on the final result, [[KARPET]{}]{} can perform early pruning by checking in line \[line:expansion\] in Alg. \[alg:prioritized-search\] if newly added node $c'$ already appears in partial match $Z$—discarding $Z'$ if it does. This modification has the following implications for the cost analysis results in Section \[sec:algCost\].
Since some of the items previously pushed to priority queue `pq` will now be discarded early, space consumption as well as computation cost of [[KARPET]{}]{} are lower than for finding all subgraph homomorphism results. However, worst-case complexity as established by Theorems \[thm:spaceAlg2\] and \[thm:timeAlg2\] remains the same. And the guarantees for the time between results (Theorem \[thm:interarrival\]) is weaker: In the worst case, e.g., when only the very first and the very last of the homomorphic matches represent isomorphic results, then time between consecutive results grows from $\mathrm{O}(\mathrm{outDegree} + \log r_H)$ to $\mathrm{O}(r_H \log r_H)$. Fortunately, as our experiments indicate, heterogeneity indeed results in a small gap between homomorphism and isomorphism, i.e., real-world performance is closer to $\mathrm{O}(\mathrm{outDegree} + \log r_H)$.
**Acknowledgments**. This work was supported in part by the National Institutes of Health (NIH) under award number R01 NS091421 and by the National Science Foundation (NSF) under award number CAREER III-1762268. The content is solely the responsibility of the authors and does not necessarily represent the official views of NIH or NSF. We would also like to thank the reviewers for their constructive feedback.
[^1]: *Kernelization* is a pre-processing technique that replaces the original input by a (usually) smaller representation called “kernel” in order to reduce the computation cost. Our approach enumerates solutions over a smaller pruned candidate graph.
|
---
abstract: 'G. Godefroy and the second author of this note proved in 1988 that in duals to Asplund spaces there always exists a projectional resolution of the identity. A few years later, Ch. Stegall succeeded to drop from the original proof a deep lemma of S. Simons. Here, we rewrite the condensed argument of Ch. Stegall in a more transparent and detailed way. We actually show that this technology of Ch. Stegall leads to a bit stronger/richer object —the so called projectional skeleton— recently constructed by W. Kubiś, via S. Simons’ lemma and with help of elementary submodels from logic.'
address:
- 'Faculty of Mathematics and Physics, Sokolovská 83, 18675 Praha 8, Czech Republic'
- 'Mathematical Institute of Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic'
author:
- Marek Cúth and Marián Fabian
title: 'Projections in duals to Asplund spaces made without Simons’ lemma'
---
In 1988, G. Godefroy and the second named author of this note constructed in [@fg] a projectional resolution of the identity in duals to general Asplund spaces; see, e.g., [@f Definitions 1.0.1 and 6.1.5]. A few years later, Ch. Stegall, presented a simplified variant of this construction by avoiding from the proof the use of S. Simons’ lemma [@f Lemma 8.1.3] or of any other substitute of it. He published it, in a rather condensed form in [@st]. In this note, we perform his argument in full details. Thus we believe that Ch. Stegall’s approach, so far overlooked by audience, will attract a broader attention. Ch. Stegall’s technology led us even to constructing a stronger/richer object, which includes the so called 1-[*projectional skeleton*]{}, introduced and studied recently by W. Kubiś, see [@kubis pages 765, 766], [@kkl pages 369, 370]. W. Kubiś’ construction was based on S. Simons’ lemma and was done by using elementary submodels from logic. Let $(X,\|\cdot\|)$ be any Banach space. If $V$ is a subspace of $X$ and $x^*
\in X^*$, then ${x^*}\!\!\upharpoonright_{V}$ means the restriction of $x^*$ to $V$; similarly, we put $M\!\!\upharpoonright_{V}:= \big\{{x^*}\!\!\upharpoonright_{V}:\ x^*\in M\}$ for any set $M$ in $X^*$. Alaoglu’s theorem asserts that the closed unit ball $B_{X^*}$ in $X^*$ provided with the weak$^*$ topology is a compact space. Let ${\C(B_{X^*})}$ denote the (Banach) space of all continuous functions on this compact space, endowed with the maximum norm $\|\cdot\|$. Consider the multivalued mapping $$\begin{aligned}
{\C(B_{X^*})}\ni f\lll \big\{x^*\in B_{X^*}:\ f(x^*)=\max f(B_{X^*})\big\}=:\partial(f)\, \subset B_{X^*};\end{aligned}$$ thus $\partial: {\C(B_{X^*})}\longrightarrow 2^{X^*}$. Clearly, for every $f\in {\C(B_{X^*})}$ the set $\partial(f)$ is non-empty and weak$^*$ compact. The mapping $\partial$ is also norm-to-weak$^*$ upper semicontinuous. To check this, consider any weak$^*$ open set $W$ in $X^*$. We have to show that $\{f\in\C(B_{X^*}):\ \partial(f)\subset W\}$ is an open set. Assume that this is not the case. Then there exists a sequence $f_0,f_1,f_2,\ldots$ of elements in $\C(B_{X^*})$ such that $\partial(f_0)\subset W$, that $\|f_n - f_0\| \to 0$ as $n\to\infty$, and that for every $n\in\N$ there is $x^*_n\in\partial(f_n)\setminus W$. Recalling that all the $x^*_n$’s belong to the (weak$^*$ compact) set $B_{X^*}$, the sequence $(x^*_n)$ has a weak$^*$ cluster point, $x^*\in B_{X^*}$, say. Also $$\begin{aligned}
\max f_0(B_{X^*})&\le& \max f_n(B_{X^*})+\|f_0-f_n\|\\
&=&f_n(x^*_n)-f_0(x^*_n)+\|f_0-f_n\|+f_0(x^*_n)
\le 2\|f_n-f_0\| + f_0(x^*_n)\end{aligned}$$ for every $n\in\N$. And, since $f_0$ is weak$^*$ continuous, $\big(f_0(x^*)\le\big)\ \max f_0(B_{X^*})\le f_0(x^*)$. We got that $x^*\in\partial(f_0)\ (\subset W)$. However, simultaneously, $x^*\not\in W$ as $x^*_n\not\in W$ for all $n\in \N$; a contradiction.
If the Banach space $(X,\|\cdot\|)$ is separable, the mapping $\partial: {\C(B_{X^*})}\longrightarrow 2^{X^*}$ has the following (crucial) property: For every $\xi\in B_{X^*}$ there is an $f\in \C(B_{X^*})$ such that $\partial(f)=\{\xi\}$. (This is the very point where the approach of Ch. Stegall starts.) Indeed, since the compact space $(B_{X^*},w^*)$ is then metrizable, $\xi$ is a $G_\delta$ point in it and Urysohn’s lemma [@e Theorem 1.5.10] provides easily a suitable function $f$. More specifically (and without using any topological tool), such an $f$ can be constructed by the formula $$\begin{aligned}
\label{zero}
B_{X^*}\ni x^*\longmapsto 1-\sum_{n=1}^\infty 2^{-n}\big|\la \xi,s_n\ra-\la x^*,s_n\ra\big| =:f(x^*)\end{aligned}$$ where $\{s_1,s_2,\ldots\}$ is a fixed countable dense subset of the closed unit ball $B_X$ in $X$. (By [@f Lemma 2.2.1(ii)] we know that the function $C(B_{X^*})\ni g\longmapsto \max g(B_{X^*})$ is Gateaux differentiable at $g:=f$.)
In what follows, let $(X,\|\cdot\|)$ be a Banach space of an arbitrary density. The observation in the latter paragraph leads to introducing the following family of functions from $\C(B_{X^*})$. Let $S$ be any nonempty set in $B_X$. By ${\ll}(S)$ we denote the family of all functions of the form $$\begin{aligned}
B_{X^*}\ni x^*\longmapsto 1-\sum_{n=1}^k 2^{-n}\big|a_n-\la x^*,s_n\ra\big|,\end{aligned}$$ where $k\in\N,\ a_1,a_2,\ldots,a_k$ are rational numbers in $[-1,1]$, and $s_1,s_2,\ldots,s_k$ are elements from $S$; note that $\#\ll (S)=\#S+\aleph_0$. It is easy to check that each element of ${\ll}(S)$ is weak$^*$ continuous and has maximum norm at most equal to $1$. Thus ${\ll}(S)$ is a subset of (the closed unit ball of) $\,\C(B_{X^*})$. We can easily check that $$\begin{aligned}
\label{17}
\overline{\ll(\overline{S})} = \overline{\ll (S)}\,;\end{aligned}$$ hence, the set in (\[17\]) is separable whenever $S$ is separable. Note that the $f$ defined in (\[zero\]) belongs to the (norm) closure of $\ll(\{s_1,s_2,\ldots\})\,$. From now on, assume that $(X,\|\cdot\|)$ is an Asplund space, which (equivalently) means that the dual space $X^*$ is weak$^*$ dentable [@ph Theorem 2.32]. Then, we are ready to apply the selection theorem of Jayne and Rogers [@jr Theorem 8], [@f Theorem 8.1.2] to our multivalued mapping $\partial$ (which is already known to be norm-to-weak$^*$ upper semicontinuous and weak$^*$ compact valued). Thus we get a sequence $\lambda_j: {\C(B_{X^*})}\longrightarrow X^*, \ j\in\N$, of norm-to-norm continuous mappings such that for every $f\in {\C(B_{X^*})}$ the limit $\lim_{j\to\infty} \lambda_j(f)=:\lambda_0(f)$ exists in the norm topology of $X^*$ and moreover $\lambda_0(f)\in\partial(f)$, that is, $f\big(\lambda_0(f)\big)=\max f(B_{X^*})$. Now, we define the multivalued mapping $${\C(B_{X^*})}\ni f\longmapsto \{\lambda_1(f),\lambda_2(f),\ldots\}
=:\Lambda(f)\subset X^*;$$ thus $\Lambda: {\C(B_{X^*})}\longrightarrow 2^{X^*}$. The continuity of the mappings $\lambda_j$’s and (\[17\]) then guarantee that $$\begin{aligned}
\label{18}
\overline{\Lambda\big(\overline{\ll(\overline{S})}\big)}
=\overline{\Lambda\big(\ll (S)\big)}\, .\end{aligned}$$
\[0\] Let $(X,\|\cdot\|)$ be any Asplund space and let $V$ be any subspace of it. Then $B_{V^*} \subset \overline{\Lambda\big({\ll(B_V)}\big)}\,\!\!\upharpoonright_V\,$ and $\dens V=\dens\ov{{\rm sp}\,\Lambda\big({\ll(B_V)}\big)}=\dens V^*$; in particular, $B_{X^*} \subset \overline{\Lambda\big({\ll(B_X)}\big)}$.
First, assume that $V$ is separable. Consider any $v^*\in B_{V^*}$. Find a countable dense subset $\{s_1,s_2,\ldots\}$ of $B_V$. Define $f:=1-\sum_{n=1}^\infty 2^{-n}\big|\la v^*,s_n\ra -\la\cdot,s_n\ra\big|$; thus $f\in\overline{\ll(B_V)}$. Then $f(x^*)=1$ for every $x^*\in B_{X^*}$, with ${x^*}\!\!\upharpoonright_{V}=v^*$, and $-1\le\ f(y^*)<1$ whenever $y^*\in B_{X^*}$ and ${y^*}\!\!\upharpoonright_V\neq v^*$. Thus $\partial(f)\!\!\upharpoonright_V=\{v^*\}$, and, since $\lambda_0(f)\in\partial(f)$, we have that $\lambda_0(f)\!\!\upharpoonright_V=v^*$. Recalling that $\|\lambda_j(f)-\lambda_0(f)\|\longrightarrow0$ as $j\to\infty$, we can conclude that $$\lambda_0(f)\in\overline {\Lambda(f)}
\subset \overline{\Lambda\big(\overline{\ll(B_V)}\,\big)}
=\overline{\Lambda(\ll(B_V))}$$ by (\[18\]). Therefore, $v^*=\lambda_0(f)\!\!\upharpoonright_{V}\in \overline{\Lambda\big(\ll(B_V)\big)}\,\!\!\upharpoonright_V \,$.
Second, assume that $V$ is non-separable (provided that $X$ is non-separable). Consider any $v^*\in B_{V^*}$. Find $\xi\in B_{X^*}$ so that $\xi\!\!\upharpoonright_V=v^*$. We shall construct countable sets $S_0\subset S_1\subset S_2\subset\cdots\subset B_V$ and separable subspaces $Z_0\subset Z_1\subset Z_2\subset\cdots\subset V$ as follows. Pick any countable subset $S_0$ of $B_V$ and any separable (rather infinite-dimensional) subspace $Z_0\subset V$. Let $m\in\N$ be given and assume that we have found $S_{m-1}$ and $Z_{m-1}$. Clearly, ${\Lambda\big(\ll(S_{m-1}})\,\big)$ is a countable subset of $X^*$. Find then a countable set $S_{m-1}\subset S_m\subset B_V$ such that $\overline{S_m}\supset B_X\cap{Z_{m-1}}$ and that $$\label{19}
\|\xi- {x^*}\| =\sup\big\{\la \xi,s\ra -
\la x^*,s\ra:\ s\in S_m\big\}\ \ \hbox{for every}\
x^*\in \Lambda\big(\ll(S_{m-1})\big)\,.$$ Put then $Z_m:=\overline{{\rm sp}\, (Z_{m-1}\cup S_m)}$. Do so for every $m\in\N$ and put finally $S:=S_0\cup S_1\cup S_2\cup\cdots$ and $Z\!:=\overline{Z_0\cup Z_1\cup Z_2\cup\cdots}\,$. Clearly, $S$ is a countable set, $Z$ is a separable subspace of $V$, and $\overline{S}=B_Z$. The “separable” case says that ${\xi}\!\!\upharpoonright_Z \in
\overline{\Lambda\big(\ll(B_{Z})\big)}\, \!\!\upharpoonright_Z \,$. By (\[18\]) we get that $\xi\!\!\upharpoonright_Z$ actually belongs to the set $\overline{\Lambda(\ll (S))}\!\!\upharpoonright_Z\,$. Pick $x^*\in \overline{\Lambda(\ll (S))}$ so that $\xi\!\!\upharpoonright_Z=x^*{}\!\!\upharpoonright_Z$. For every $i\in\N$ find $x^*_i\in\Lambda(\ll (S))$ so that $\|x^*-x^*_i\|<\frac1i$. A moment reflection reveals that for every $i\in\N$ there is $m_i\in\N$ so that $x^*_i\in\Lambda(\ll (S_{m_i-1}))$. Then, by (\[19\]), $$\begin{aligned}
\|\xi-x^*_i\|&=&\sup\big\{\la\xi,s\ra - \la x^*_i,s\ra:\ s\in S_{m_i}\big\}\\
&\le&\big\|\xi\!\!\upharpoonright_Z-x^*_i\!\!\upharpoonright_Z\!\big\| = \big\|x^*\!\!\upharpoonright_Z-x^*_i\!\!\upharpoonright_Z\big\|
\le \|x^*-x^*_i\|<\hbox{$\frac1i$}\end{aligned}$$ for every $i\in\N$. Therefore $\xi \in \overline{\Lambda(\ll (S))} \subset\overline{\Lambda(\ll(B_V))}\, ,$ and so, $v^*=\xi\!\!\upharpoonright_V\in \overline{\Lambda\big(\ll(B_V)\big)}\,\!\!\upharpoonright_V \,$.
Finally, consider any subspace $V$ of $X$. Choosing a dense subset $M$ of $B_V$, with $\#M=\dens V$, we have by (\[18\]) $$\begin{aligned}
\dens V&\le&\!\! \dens V^* \le \dens \overline{\Lambda\big(\ll(B_V)\big)}\,\!\!\upharpoonright_V
\le\dens \overline{\Lambda\big(\ll(B_V)\big)}
=\dens \overline{{\rm sp}\Lambda\big(\ll(B_V)\big)}\\
&=&\!\! \dens\ov{\Lambda(\ll(M))}
\le \#\Lambda\big(\ll(M)\big) = \# M\ =\dens V.\end{aligned}$$
\[7\] [*Proposition \[0\] is crucial for dropping S. Simons’ lemma from the original construction of a projectional resolution of the identity in duals to Asplund spaces; see [@fg] modulo [@f1]. Indeed, in [@fg], instead of the mapping $\partial:\C(B_{X^*})\longrightarrow 2^{B_{X^*}}$, there is considered the (so called duality) mapping $X\ni x\longmapsto \{x^*\in S_{X^*}\!: \, \langle x^*,x\rangle=\|x\|\}
=:J(x)$. This $J: X \longrightarrow 2^{S_{X^*}}$ is also norm-to-weak$^*$ upper semicontinuous and weak$^*$ compact valued. Hence, the Jayne-Rogers theorem [@f Theorem 8.1.2] yields norm-to-norm continuous mappings $D_1, D_2, \ldots$ (now) from $X$ into $X^*$ such that for every $x\in X$ the limit $\lim_{n\to\infty} D_n(x)=:D_0(x)$ exists in the norm topology of $X^*$ and moreover $\langle D_0(x),x\rangle=\|x\|$. Yet, there is no obvious guarantee that the set $D_0(X)$ is equal to, or at least norm-dense in the unit sphere $S_{X^*}$. Here, S. Simons’ lemma enters the argument and remedies the situation by showing that $D_0(X)$ is linearly dense in all of $X^*$.*]{}
\[1\] Let $(X,\|\cdot\|)$ be a non-separable Asplund space and let $Z\subset X$ be an infinite-dimensional subspace with [dens]{}$\, Z <\, $[dens]{}$\, X$. Then there exists an overspace $Z\subset V\subset X$, with [dens]{}$\, V=\,$[dens]{}$\, Z$, such that the restriction mapping $\ \overline{ {\rm sp}\,\Lambda\big(\ll(B_V)\big)}\ni x^*\longmapsto {x^*}\!\!\upharpoonright_V=:R(x^*)\in V^*$ is a (surjective) isometry.
Put $\aleph:=$ dens$\, Z$. By induction, we shall construct sets $S_0\subset S_1\subset S_2\subset\cdots \subset B_X$, all of cardinality $\aleph$, as follows. Let $S_0$ be a dense subset of $B_Z$, with $\# S_0=\aleph$. Let $m\in\N$ be fixed and assume that we have already found $S_{m-1}$. We already know that $\#\ll(S_{m-1})=\aleph$. Hence, by the norm-to-norm continuity of the mappings $\lambda_j$’s, we have that $\,\overline{{\rm sp}\, \Lambda\big(\ll(S_{m-1})\big)}$ is a subspace of $X^*$, with density $\aleph$; let $M$ be a dense set in it, with $\#M=\aleph$. Find a set $S_{m-1}\subset S_m\subset B_X$, with $\#S_m=\aleph$, so big that $\,$sp$\,S_{m-1}\cap B_X\subset
\overline{S_m}$ and that $\|\xi\|=\sup\,\la \xi,S_m\ra$ for every $\xi\in M$. Then, of course, $\|x^*\|=\sup\,\la x^*,S_m\ra$ for every $x^*\in{{\rm sp} \,\Lambda\big(\ll(S_{m-1})\big)}$. This finishes the induction (step).
Having constructed the $S_m$ for every $m\in\N$, put $S:=S_1\cup S_2\cup\cdots$ and $V:=\overline{{\rm sp}\, S}$; then $\#S=\aleph$ and $V$ is a subspace of $X$, with density $\aleph$. We shall show that this $V$, together with the corresponding $R$, serve for the conclusion of our proposition. Consider any $x^*\in
\overline{{\rm sp}\, \Lambda\big(\ll(B_V)\big)}$ and let $\ee>0$ be arbitrary. It is easy to verify that $\overline S=B_V$. By (\[18\]), $\overline{\Lambda(\ll(B_V))}=\overline{\Lambda(\ll (S))}$, and hence $\overline{{\rm sp}\,\Lambda(\ll(B_V))}=
\overline{{\rm sp}\,\Lambda(\ll (S))}\,$. Thus $x^*$ belongs to $\overline{{\rm sp}\, \Lambda\big(\ll (S)\big)}$. Find then $\xi\in {\rm sp}\, \Lambda\big(\ll (S)\big)$ so that $\|x^*-\xi\|<\ee$. We remark that $\xi$ belongs even to ${\rm sp}\, \Lambda\big(\ll (S_{m-1})\big)$ for some big $m\in\N$. Then $$\begin{aligned}
\|x^*\|-\ee &<&\|\xi\| = \sup\,\la \xi,S_m\ra \le \sup\,\la \xi,B_V\ra\\
& =& \|R(\xi)\| < \|R(x^*)\|+ \ee \le \|x^*\|+\ee.\end{aligned}$$ Thus $\|x^*\|=\|R(x^*)\|$. We proved that $R$ is an isometry. That $R$ is surjective follows immediately from Proposition \[0\].
\[2\] Let $V$ be a subspace of an Asplund space $(X,\|\cdot\|)$ such that the restriction mapping $\ \overline{ {\rm sp}\,\Lambda\big(\ll(B_V)\big)}\ni x^*\longmapsto {x^*}\!\!\upharpoonright_V=:R(x^*)\in V^*$ is a (surjective) isometry. Then the mapping $\ X^*\ni x^*\longmapsto
R^{-1}\big({x^*}\!\!\upharpoonright_V\big)=: P(x^*)$ is a linear norm-$1$ projection, $P(X^*)=\overline{{\rm sp}\,\Lambda\big(\ll(B_V)\big)}$, $\dens P(X^*)=\dens V$, and $\overline{V}^{\, w^*}=P^*(X^{**})$.
The first three statements concerning $P$ immediately follow from the definition of it. The “density” statement is contained in Proposition \[0\]. It remains to prove the last equality. That $V\subset P^*(X^{**})$ follows from the definition of $P$; hence $\overline{V}^{\,w^*}\subset P^*(X^{**})$. Assume there exists $x^{**}\in P^*(X^{**}) \setminus \overline{V}^{\,w^*}\!$. The Hahn-Banach separation theorem yields an $x^*\in X^*$ such that $\langle x^{**},x^*\rangle\neq0$ and ${x^*}\!\!\upharpoonright_{V}\equiv 0$. But $$\langle x^{**},x^*\rangle=\langle P^*(x^{**}),x^*\rangle=\langle x^{**},P(x^*)\ra = \big\langle x^{**},R^{-1}\big({x^*}\!\!\upharpoonright_{V}\big)\big\rangle =0;$$ a contradiction.
\[3\] Let $V_1,\, V_2$ be two subspaces of an Asplund space $(X,\|\cdot\|)$ such that $V_1\subset V_2$ and that the restriction mappings $\ \overline{ {\rm sp}\,\Lambda\big(\ll(B_{V_i})\big)}\ni x^*\longmapsto {x^*}\!\!\upharpoonright_{V_i}=:R_i(x^*)\in V_i^*,\ i=1,2$, are (surjective) isometries. Define $P_i: X^*\rightarrow X^*$ by $P_i(x^*)={R_i}^{-1}\big({x^*}\!\!\upharpoonright_{V_i}\big),\ x^*\in X^*,\
i=1,2$. Then $P_1\!\circ \!P_2 = P_1\ \,(=P_2\!\circ \!P_1)$, and $(P_2-P_1)(X^*)$ is isometrical with $(V_2/V_1)^*$.
\(i) From the definition of $P_i$’s we have immediately that $P_2\circ P_1= P_1$. Now, consider any $x^*\in X^*$ and any $x^{**}\in X^{**}$. Since $\overline{V_1}^{\,w^*}=P^*_1(X^{**})$ by Proposition \[2\], there is a net $(v_\tau)_{\tau\in T}$ in $V_1$ which weak$^*$ converges to $P^*_1(x^{**})$. Then, using the definition of $R_2$ and the inclusion $V_1\subset V_2$, we get $$\begin{aligned}
\la P^*_2\circ P^*_1(x^{**}), x^*\ra &=& \la P^*_1(x^{**}),P_2(x^*)\ra
=\lim_{\tau\in T}\,\la P_2(x^*),v_\tau\ra \\
&=& \lim_{\tau\in T}\, \la R_2{}^{-1}(x^*\!\!\upharpoonright_{V_2}),v_\tau\ra
=\lim_{\tau\in T}\,\la x^*,v_\tau\ra =\la P^*_1 (x^{**}),x^*\ra,\end{aligned}$$ and so $P^*_2\circ P^*_1 = P^*_1$, that is, $P_1\circ P_2=P_1 \,$. The “isometrical” statement can be shown as in the proof of [@f Proposition 6.1.9(iv)].
Now we are armed to construct a projectional resolution of the identity on the dual to every Asplund space. But we rather prefer to present a bit stronger/richer statement.
\[main\] Let $(X,\|\cdot\|)$ be a non-separable Asplund space. Then there exist a family $\vv$ of subspaces of $X$ and a family $\{Y_V\!:\ V\in\vv\}$ of subspaces of $X^*$ such that
1. $\bigcup\big\{V:\ V\in\vv \ {\it and}\ \dens V\!=\!\aleph\big\}=X$ and $\,\bigcup\big\{Y_V\!: V\in\vv \ {\it and}\ \dens V\!=\!\aleph\big\}$ $=X^*$ for every infinite cardinal $\aleph<\dens X$;
2. if $V_1,V_2\in \vv$, there is $V\in\vv$ such that $V\supset V_1\cup V_2$ and $\dens V=\max\{\dens V_1,\dens V_2\}$.
3. for every $V\in\vv$ the assignment $Y_V\!\ni\! x^*\longmapsto x^* \!\!\upharpoonright _V =:R_V(x^*)\in V^*$ is a surjective isometry, and hence the mapping $X^*\ni x^* \longmapsto R_V{}^{-1}(x^*\!\!\upharpoonright_V)$$=:P_V(x^*)$ is a norm-$1$ linear projection on $X^*$, with range $Y_V$, and $\dens P_V(X^*)=\dens V$;
4. $\ov{V}^{\,w^*} = {P_V}^*(X^{**})$ for every $V\in\vv$;
5. $\vv$ is complete in the following sense: if $\gamma$ is a limit ordinal, and $\{V_\alpha:\ 1\le\alpha<\gamma\}$ is an increasing long sequence of elements of $\,\vv$, then $V:=\ov{\bigcup_{1\le\alpha<\gamma}
V_\alpha}$ belongs to $\vv$ and $Y_V=\ov{\bigcup_{1\le\alpha<\gamma}
Y_{V_\alpha}}\,$;
6. if $\,V,U\in\vv$ and $V\subset U$, then $Y_V\subset Y_U$, $P_V\circ P_U=P_V\ \,(=P_U\circ P_V)$, and $(P_U-P_V)(X^*)$ is isometrical with $(U/V)^*$.
For every subspace $V$ of $X$ we put $Y_V:=\ov{{\rm sp}\,\Lambda(\ll(B_V))}$ and we consider the assignment $Y_V\ni x^*\longmapsto x^*\!\!\upharpoonright _V=:R_V(x^*)\in V^*$. Let $\vv$ consist of all subspaces $V$ of $X$ such that $R_V: Y_V\longrightarrow V^*$ is a surjective isometry.
\(i) and (ii) are guaranteed by Propositions \[1\] and \[0\].
\(iii) follows from the definition of $\vv$ via Propositions \[0\] and \[2\].
\(iv) follows from Proposition \[2\].
\(v) Assume the premise here holds. (\[18\]) and some elementary reasoning yields $$\begin{aligned}
\Lambda(\ll(B_V)) &\subset& \ov{\Lambda(\ll(B_V))}
=\ov{\Lambda\big(\ll(\ov{\hbox{$\bigcup_{\alpha<\gamma}$}B_{V_\alpha}}\big)\big)}
= \ov{\Lambda\big(\ll(\hbox{$\bigcup_{\alpha<\gamma}$}B_{V_\alpha}\big)\big)}\\
&=& \ov{\hbox{$\bigcup_{\alpha<\gamma}$}\Lambda\big(\ll(B_{V_\alpha})\big)}
\subset \ov{\hbox{$\bigcup_{\alpha<\gamma}$}\ov{{\rm sp}\Lambda\big(\ll(B_{V_\alpha})\big)}}
=\ov{\hbox{$\bigcup_{\alpha<\gamma}$} Y_{V_\alpha}}.\end{aligned}$$ Hence $$\begin{aligned}
\label{pet}
Y_V = \ov{{\rm sp}\Lambda\big(\ll(B_V)\big)} = \ov{\hbox{$\bigcup_{\alpha<\gamma}$} Y_{V_\alpha}}.\end{aligned}$$ It remains to show that $V\in\vv$, that is, that the mapping $R_V: Y_V\longrightarrow V^*$ is a surjective isometry. By Proposition \[0\] and (\[pet\]), $R_V$ is surjective. Further, fix any $x^*\in Y_V$ and let $\ee>0$ be arbitrary. Find $\xi\in \bigcup_{\alpha<\gamma}Y_\alpha$ such that $\|x^*-\xi\|<\ee$. Then $\xi$ belongs to $Y_\alpha$ for some $\alpha<\gamma$. Now, as $V_\alpha\in\vv$, we have $$\begin{aligned}
\|x^*\|-\ee &<& \|\xi\| =
\|R_{V_\alpha}(\xi)\| =\big\|\xi\!\!\upharpoonright_{V_\alpha}\big\|
\le \big\|\xi\!\!\upharpoonright_{V}\big\|\\ &<& \big\|x^*\!\!\upharpoonright_{V}\big\|
+\ee = \|R_V(x^*)\| + \ee \le\|x^*\| + \ee.\end{aligned}$$ Therefore, $\|x^*\|=\|R_V(x^*)\|$ and the mapping $R_V$ is shown to be an isometry. Thus $V\in\vv$.
\(vi) immediately follows from Proposition \[3\].
\[hlavni\] Let $(X,\|\cdot\|)$ be a non-separable Asplund space. Then $(X^*,\|\cdot\|)$ admits
\(i) [[@fg]]{} a projectional resolution of the identity, and also
\(ii) [[@kubis]]{} a $1$-projectional skeleton.
\(i) Let $\mu$ be the first ordinal with $\#\mu=\dens X$. Find a dense subset $\{x_\alpha:\ \omega<\alpha<\mu,\,$ $\alpha$ is non-limit$\}$ in $X$. By (i) find a separable element $V_\omega\in\vv$. Let $\gamma\in
(\omega,\mu]$ be any ordinal an assume that we already found $V_\alpha\in\vv$ for every $\omega\le\alpha<\gamma$. Assume first that $\gamma$ is non-limit. By (i) find a $V\in\vv$ such that $V\ni x_\gamma$ and $\dens V=\dens V_{\gamma-1}$. By (ii) find a $V_\gamma\in\vv$ such that $V_\gamma\supset V\cup V_{\gamma-1}$ and $\dens V_\gamma=V_{\gamma-1}$. Second, assume that $\gamma$ is a limit ordinal. Put then $V_\gamma=\ov{\bigcup_{\omega<\alpha<\gamma} V_\alpha}$. By (v), we have that $V_\gamma\in\vv$ and $Y_{V_\gamma}=\ov{\bigcup_{\omega\le\alpha<\gamma}
Y_{{V_\alpha}}}\,$. Now we can immediately verify that $\big\{ P_{V_\alpha}:\ \omega\le\alpha\le\mu\big\}$ is a projectional resolution of the identity on $(X^*,\|\cdot\|)$; see, e.g. [@f Definition 6.1.5].
\(ii) We note that the relation “$\subset$” on the family $\vv$ from Theorem \[main\] is a directed partial order, which is moreover $\sigma$-complete. Thus, the subfamily $\{P_V:\ V\in\vv\ {\rm and}\ \dens V=\aleph_0\}\,$ is a 1-[projectional skeleton]{} on $(X^*,\|\cdot\|)$ in the sense of the definition in [@kkl pages 369, 370].
**
1\. A 1-projectional skeleton in the dual to an Asplund space was originally constructed by W. Kubiś [@kubis]. He started from the existence of the so called [*projectional generator*]{} in the dual to an Asplund space [@f Proposition 8.2.1] and then he proceeded using elementary submodels from logic. It should be noted that the proof of [@f Proposition 8.2.1] is based on S. Simons’ lemma and on the Jayne-Rogers selection theorem [@jr Theorem 8].
2\. Let $\nn_1,\ \nn_2,\ \ldots$ be a sequence of equivalent norms on an Asplund space $X$. Let $\vv_1,\ \vv_2,\ \ldots$ be corresponding families found in Theorem \[main\] for these norms. It is not difficult to show that the family $\vv:=\bigcap_{i=1}^\infty\vv_i$ (is not only non-empty but that it even) satisfies all the conditions (i) – (vi) of Theorem \[main\]; see the proof of [@bm Proposition 1.1]. Then Corollary \[hlavni\] provides one projectional resolution of the identity and one $1$-projectional skeleton on $X^*$ which can be related to any of the norms $\nn_1,\ \nn_2,\ \ldots$
3\. Theorem \[main\] can be proved also with help of tools from [@fg] (where S. Simons’ lemma was used). Indeed, let the mappings $D_n: X\rightarrow X^*,\ n\in\N$, be from Remark \[7\] (they come from the Jayne-Rogers theorem) and define $D: X\rightarrow 2^{X^*}$ by $D(x)= \{D_1(x),D_2(x),\ldots\},\ x\in X$. For every subspace $W$ of $X$ put $Y_W:=\ov{{\rm sp}\,D(W)}$ and consider the assignment $Y_W\ni x^*\longmapsto x^*\!\!\upharpoonright _W=:R_W(x^*)\in W^*$. Let $\mathcal W$ consist of all subspaces $W$ of $X$ such that $R_W: Y_W\longrightarrow W^*$ is a surjective isometry. From [@fg pages 145, 146], where S. Simons’ lemma and other things were used, we know that $$\begin{aligned}
\label{fg}
\ov{{\rm sp}\, D(W)\!\!\upharpoonright_W}=W^*\quad \text{for every subspace}\quad W\subset X.\end{aligned}$$ (This is a weakened analogue of our Proposition \[0\]). By [@f1 Lemma 1], for every $Z\subset X$, with $\dens Z<\dens X$ there is a subspace $Z\subset W\subset X$, with $\dens W=\dens Z$, such that $R_W$ is an isometry from $Y_W$ into $W^*$. (This is an analogue of our Proposition \[1\].) Using this, we easily get that $\ov{{\rm sp}\, D(W)\!\!\upharpoonright_W}=\ov{{\rm sp}\, D(W)}\!\!\upharpoonright_W$, and by (\[fg\]), $R_W: Y_W\rightarrow W^*$ is a surjective isometry. Thus $W\in\mathcal W$. This way we get the properties (i) – (iv) and (vi) listed in Theorem \[main\]. (v) can be proved as in Theorem \[main\], now from the norm-to-norm continuity of the $D_n$’s.
4\. We use an opportunity to fix some inaccuracy in the book [@f]. In the proof of [@f Proposition 8.2.1], on the page 152, the equality (2) should read as $\overline{{\rm sp}\,\Phi(B_0)\!\!\upharpoonright_Y}=Y^*$ and the set $\Delta$ should be defined as $\overline{\Phi(B_0)\!\!\upharpoonright_Y}\cap B_{Y^*}\,$.
**Acknowledgment. We thank our colleague W. Kubiś for discussing the topic of this note.**
J.M. Borwein and W.B. Moors, [*Separable determination of integrability and minimality of the Clarke subdifferential mapping*]{}, Proc. Amer. Math. Soc., [**128**]{} (2000), 215–221.
R. Engelking, [*General topology*]{}, Warszawa 1977.
M.J. Fabian, *Gâteaux differentiability of convex functions and topology – weak Asplund spaces*, J. Wiley & Sons, 1997.
M. Fabian, *On projectional resolution of identity of the duals of certain Banach spaces*, Bull. Aust. Math. Soc., **35 (1987), 363–371.**
M. Fabian, G. Godefroy, *The dual of every Asplund space admits a projectional resolution of the identity*, Studia Math. [**91**]{} (1988), 141–151.
J.E. Jayne, C.A. Rogers, *Borel selectors for upper semicontinuous maps*, Acta Math. **155 (1985), 41–79.**
J. Kakol, W. Kubiś, M. López-Pellicer: *Descriptive Topology in Selected Topics of Functional Analysis Developments in Mathematics*, Vol. 24, Springer Science & Business Media, New York, 2011.
W. Kubiś, [*Banach spaces with projectional skeletons*]{} J. Math. Anal. Appl. **350 (2009) 758–-776.**
R.R. Phelps, *Convex functions, Monotone operators, and Differentiability*, Lecture Notes in Math. No. 1364 , 2nd Edition, Berlin, Springer Verlag 1993.
Ch. Stegall, *Spaces of Lipschitz functions on Banach spaces*, Functional Analysis Essen 1991, 265–278; Lect. Notes in Pure and Appl. Math. **150 New York, Dekker 1996.**
|
---
author:
- 'P. Noterdaeme'
- 'P. Petitjean'
- 'R. Srianand'
bibliography:
- 'hih2.bib'
title: 'The elusive $\rightarrow$H$_2$ transition in high-$z$ damped Lyman-$\alpha$ systems'
---
2[$\rm H_2$]{}
Introduction
============
The atomic to molecular hydrogen transition is a prerequisite process for star formation through the collapse of molecular clouds and therefore has important implications for the evolution of galaxies [e.g. @Kennicutt12]. The relative amount of dense molecular and diffuse atomic gas in nearby galaxies is found to be correlated with the hydrostatic pressure at the galactic mid-plane [@Blitz06], which is driven by the gravity of gas and stars. This is a natural consequence of thermal equilibrium of the gas, leading to multiple phases under an external pressure [e.g. @Wolfire95]. The transition between and H$_2$ can then be linked to a critical gas surface mass density above which star formation is triggered, inducing a Schmidt-Kennicutt relation [e.g. @Schaye01; @Altay11; @Lagos11; @Popping14].
The [**local**]{} abundance of H$_2$ in the interstellar medium (ISM) depends on the balance between its formation, primarily on the surface of dust grains (e.g. @Jura74b, but also in the gas phase though the H$^-$+H$\rightarrow$H$_2$+e$^-$ reaction, @Black87), and its dissociation by UV photons. Because the dissociation occurs through Lyman and Werner band line transitions [e.g. @Dalgarno70], self-shielding becomes very efficient when H$_2$ absorption lines from several rotational levels become saturated [e.g. @Draine96]. Dust grains also absorb Lyman and Werner band photons further contributing to decreasing the photo-dissociation rate. Theoretical microphysics models that include detailed treatment of the formation of H$_2$ onto dust grains and the dust- and self-shielding of H$_2$ show that the conversion from atomic to molecular occurs above a $N({\ion{H}{i}})$-threshold that increases with decreasing metallicity [e.g. @Krumholz09; @McKee10; @Gnedin11; @Sternberg14].
A sharp increase in the H$_2$ column densities has been first noticed above $\log N({\ion{H}{i}}) \sim 20.7$ in the local Galactic ISM by @Savage77. In turn, the first studies of the Magellanic clouds by @Tumlinson02 did not reveal any dependence of the H$_2$ content on the column density. This was explained by a high average UV radiation due to intense local star formation activity together with lower metallicities. At high redshift, H$_2$ is generally detected in about 10% of damped Lyman-$\alpha$ systems (DLAs) or less [@Petitjean00; @Ledoux03; @Noterdaeme08]. Physical conditions in these sub-solar metallicity systems indicate densities of the order of $n\sim50$ cm$^{-3}$ in the cold neutral medium, and ambient radiation field a few times the Draine field [e.g. @Srianand05; @Neeleman15]. @Noterdaeme08 noted that the presence of H$_2$ does not strongly depend on the total neutral hydrogen column density up to $\log N({\ion{H}{i}})\sim 21.5$. They concluded that large molecular hydrogen content, as predicted by @Schaye01 may be found at higher column densities.
Here, we investigate the H$_2$ content of high redshift -selected DLAs at the extreme column density end, a regime almost unprobed until now and only made reachable very recently thanks to very large DLA datasets [@Noterdaeme12c] and high-resolution spectroscopic follow-up.
The atomic to molecular hydrogen transition \[s:mol\]
=====================================================
--------------------------------------- -------------------------------------------
{width="0.43\hsize"} {width="0.43\hsize"}
--------------------------------------- -------------------------------------------
We have recently searched for 2 in four extremely strong DLAs (ESDLAs, defined as $\log N({\ion{H}{i}}) \ge 21.7$, @Noterdaeme14) using the Ultraviolet and Visual Echelle Spectrograph (UVES) on the Very Large Telescope (VLT). This brings the number of ESDLAs with H$_2$ searches (all with VLT/UVES) to seven. Details of and 2 measurements in these ESDLAs are summarised in Table \[table1\]. Combining this with other measurements we explore the 2 content as a function of $N$() in DLAs while refraining from drawing any conclusion on the [**overall**]{} H$_2$ detection rate. We can do so since the DLAs used for this study were selected only on the basis of their neutral hydrogen content. For this reason, we do not include recent H$_2$ detections obtained by directly targeting systems based on the presence of cold gas[^1].
Quasar [$z_{\rm abs}$]{} log $N$() log $N$(2) Ref.
--------------- ------------------- ---------------- ---------------- ------
HE0027$-$1836 2.402 21.75$\pm$0.10 17.43$\pm$0.02 $1$
QJ0154$+$1935 2.251 21.75$\pm$0.15 $\sim$ 18 $2$
Q0458$-$0203 2.040 21.70$\pm$0.10 $\le$ 14.60 $3$
QJ0816$+$1446 3.287 22.00$\pm$0.10 18.66$\pm$0.30 $4$
Q1157$+$0128 1.944 21.80$\pm$0.10 $\le$ 14.50 $1$
QJ1456$+$1609 3.352 21.70$\pm$0.10 17.10$\pm$0.09 $2$
QJ2140$-$0321 2.340 22.40$\pm$0.10 20.13$\pm$0.07 $2$
: 2 in ESDLAs with log $N$()$\ge$21.7 \[table1\]
In the left panel of Fig. \[f:h2hi\], we compare the total H$_2$ column density versus that of in our extended high-$z$ DLA sample (@Noterdaeme08 and the new ESDLAs) with values in the local Galactic ISM [@Savage77], in the SMC [@Welty12], and in DLAs associated with $\gamma$-ray burst afterglows (GRB-DLAs). In the overall population, we clearly see a bimodality in the distribution of $N$(H$_2$): most detections have $\log N($H$_2)$$>$17, far above the typical detection limits (a few times $10^{14}$[${\rm cm}^{-2}$]{}). In the following, we denote as “strong” (resp. “weak”) the systems with $\log N($H$_2)$$>$17 (resp. $<17$). The right panel shows the distribution of systems in each of these populations as a function of the column density[^2]. We find that H$_2$ is detected with column densities higher than 10$^{17}$ [${\rm cm}^{-2}$]{} in four (five if we include the possible 2 detection in the DLA towards J0154+1935) ESDLAs out of seven. This is significantly higher than the value seen in the the overall DLA population ($\sim 10$%, @Noterdaeme08 [@Balashev14] or possibly less, @Jorgenson14). The increase in the fraction of strong H$_2$ systems is significant but not as sharp as is seen in the Milky Way or in the Small Magellanic Cloud. In addition, the overall molecular fractions remain modest ($\sim 1\%$ or less).
To explain this, it must be noted that the multi-phase nature of the neutral gas is not equivalently probed by the different samples. The values corresponding to the Milky Way come from lines of sight towards nearby stars that are located only within $\sim$100 pc. These should therefore probe a single cloud that produces most of the total observed column density and in which the $N$() and $N$(2) can be directly related by microphysics. The situation is already different towards stars in the Magellanic clouds for which the observed column densities may include gas from different clouds or phases along the same line of sight. @Welty12 also argued that previous $N({\ion{H}{i}})$ determinations in the SMC were overestimated because they were derived from 21 cm emission, which averages structures in the ISM at scales smaller than the radio beams. Indeed, once the $N({\ion{H}{i}})$-values are more accurately determined using Ly-$\alpha-$absorption (i.e. along the same pencil-beam line of sight as used for H$_2$ measurements) higher molecular fractions are found in the SMC, revealing a clearer segregation between the strong and weak H$_2$ populations around $\log N({\ion{H}{i}}) \sim 21$ [^3]. In DLAs, the and H$_2$ column densities are measured through UV absorption along the same line of sight. However, a single quasar sight line likely samples multiple gas components having different physical conditions, as seen from the excitation of different species [e.g. @Srianand05; @Liszt15; @Noterdaeme15]. In addition, at a given redshift, different DLAs probe different galaxies with their own sets of physical conditions, which may contribute to smoothing the observation of any underlying transition. Recently, @Balashev15 have used chlorine to show that the [*local*]{} metallicity and molecular fraction in the H$_2$ components could be much higher than the line-of-sight averaged value, although this does not tell us whether the remaining is located in outer layers or in unrelated interloping clouds.
Our results show that a large amount of in ESDLAs could indeed be unrelated to H$_2$. This is also supported by the similar H$_2$ column densities seen in several much lower $N({\ion{H}{i}})$ systems. The large $N({\ion{H}{i}})$ probably results from a low impact parameter of the line of sight relative to the galactic centre [@Noterdaeme14] where the covering factor of H$_2$-bearing gas would be higher owing to higher ISM pressure [@Blitz06].
The situation could be similar along the lines of sight towards afterglows of long-duration $\gamma$ ray bursts (GRBs) where DLAs are often seen with $\log N({\ion{H}{i}}) > 22$ [e.g. @Jakobsson06]. As these GRBs are linked to the death of a massive star [@Bloom99], they are probably related to star forming regions that are typically denser and closer to the centre of the host galaxy than quasar-DLAs [@Pontzen10]. Because GRB-DLAs may be subject to a very intense UV radiation field [@Tumlinson07] one has to exercise caution when comparing them with quasar-DLAs. Nevertheless, although the sample is still small it appears that the detection rate is consistent with that seen in quasar-DLAs albeit with larger molecular fractions. This further supports the idea that most high column density lines of sight likely probe the central regions of a galaxy.
ESDLAs are very rare and huge surveys are needed to find them [@Noterdaeme09dla; @Noterdaeme12c]. However, one could question the fact that the to H$_2$ transition may induce a bias in the selection of the quasars against the detection of the corresponding systems.
![Metallicity versus column density. Data samples and corresponding symbols are as in Fig. \[f:xhhi\]. Filled (resp. unfilled) circles represent systems in which H$_2$ has been detected with $\log N($H$_2)\ge 17$ (resp. $\log N($H$_2)< 17$). The dotted (resp. dashed) line represents a constant $\log N({\ion{Zn}{ii}})=12.5$ (resp. $\log N({\ion{Zn}{ii}})=13.15$). \[f:xhhi\] ](xh_nhi.ps){width="0.90\hsize"}
The effect of ESDLAs on the colours of the background quasar.
=============================================================
In Fig. \[f:xhhi\], we identify the H$_2$-bearing DLAs in the $N({\ion{H}{i}})$-metallicity plane. Interestingly three quasar-DLAs and four GRB-DLAs are now known beyond the limit for significant dust obscuration proposed by @Boisse98 and long discussed in the literature [e.g. @Neeleman13]. Six of these DLAs show self-shielded 2. As observed by @Petitjean06 and as predicted by some models [e.g. @Krumholz09; @Sternberg14], the 2 detection rate is higher at high metallicity. However, the metallicity at which H$_2$ is found increases with decreasing $N({\ion{H}{i}})$. The presence of H$_2$ could be more closely related to the column density of dust grains [@Noterdaeme08]: using the column density of undepleted elements as a first-order proxy for that of dust [@Vladilo05], we can see that 10 of the 15 systems above a line of constant $\log N({\ion{Zn}{ii}})$=12.5 have $\log N$(H$_2)$$\ge$17, while this fraction is only 4/66 below.
Continuum absorption by dust and the absorption from lines in the Lyman series of and Lyman and Werner bands of H$_2$ can significantly affect the quasar transmitted flux in the different bands when column densities become very large. We quantify these effects by calculating the transmission for different and H$_2$ column densities and different reddening. For each absorption situation, the induced colour changes depend on the absorption redshift, the filter responses, and the input spectrum (quasar continuum plus [${\rm Ly}\,\alpha$]{} forest). For simplicity, we fixed ${\ensuremath{z_{\rm abs}}}=2.35$ (i.e. the redshift of our strongest ESDLA towards [J2140$-$0321]{}), used the filter responses of the Sloan Digital Sky Survey (SDSS, @York00), and considered a flat quasar spectrum. Our results are shown in Fig. \[f:col\]. We empirically checked that assuming a flat spectrum has little effect on the results. To this end, we introduced fake absorbers with known properties ($N({\ion{H}{i}})$, $N($H$_2)$, and dust) in [**real**]{} non-BAL quasar spectra (with emission redshift close to that of [J2140$-$0321]{}) and derived the colour changes. We find very good agreement ($\sim 0.01$ mag for $r,i,z$ and 0.05 mag in $g$) with our simple model [^4].
In the case of [J2140$-$0321]{}, we find that the damped [${\rm Ly}\,\alpha$]{} line alone severely affects the $g$-band (by 0.18 mag). Similarly, the strong H$_2$ absorption lines raise the $u$-band magnitude by $\sim$0.26 mag. The importance of these line absorption is similar to that of the continuum absorption owing to the presence of dust (for E(B-V) $\sim$ 0.05, estimated through SED profile fitting, see @Noterdaeme14) in these two bands. While the overall colour excesses estimated for [J2140$-$0321]{} are likely not large enough to push the quasar out of typical colour-selections, it is still significant. We note that a DLA with the same characteristics as those of the DLA associated with GRB080607 would in turn very likely escape current colour-based selections. In addition, even if such a spectrum were to be obtained, then the DLA would still be hard to recognise using automated detection procedures because of the little transmitted flux remaining between damped and H$_2$ absorption lines. This shows that alternative selection of quasars, based for example on NIR photometry [e.g. @Fynbo13; @Krogager15], are important in order to avoid biasing against intervening systems beyond the neutral to molecular transition. A promising technique for identifying these systems is to search for absorption [@Ledoux15].
![Colours of [J2140$-$0321]{} compared to those of other non-BAL quasars from SDSS-III at the same redshift. The arrows in the top right corner illustrate the estimates of colour excess due to , H$_2$, and dust (assuming SMC extinction law) at ${\ensuremath{z_{\rm abs}}}=2.34$. We note that because transmission is multiplicative, the corresponding magnitudes are additive and so are the colour vectors. The red arrow shows the estimated colour excess of [J2140$-$0321]{} (circled red) due to the intervening ESDLA. For comparison, the colour excess that would produce a DLA with same column densities and reddening as those associated with GRB080607 would reach the full ranges covered by both axes. \[f:col\] ](colours.ps){width="0.90\hsize"}
Conclusions \[s:conclusion\]
============================
We have extended the study of H$_2$ in DLAs to the very high column density end, allowing us to uncover a significant increase in the fraction of strong H$_2$ systems (that we define as having $\log N$(H$_2)>17$) at $\log N({\ion{H}{i}})>21.5$. While the high $N({\ion{H}{i}})$-threshold is qualitatively consistent with expectations from theoretical models describing H$_2$ microphysics, the mean molecular fraction in these systems remains relatively low. This can be explained by the quasar lines of sight having long path lengths through galaxies. In this picture, most of the is due to clouds unrelated to the molecular phase probed by H$_2$. The threshold for [*local*]{} ${\ion{H}{i}}$ to ${\rm H}_2$ conversion in high-$z$ DLAs could actually occur at $N({\ion{H}{i}})$ and metallicities similar to those in the Milky Way disc. The large H$_2$ column densities observed in EDLAs (with log $N$(H[i]{}) $>$ 21.7) could simply be due to the line of sight passing closer to the galaxy centre as shown by @Noterdaeme14, where the ISM pressure is higher and so is the probability of intercepting a molecular cloud. This is also consistent with the skewed $N({\ion{H}{i}})$-distributions observed in samples of absorbers selected for their high molecular content. The $N({\ion{H}{i}})$-distribution of $\sim$20 strong H$_2$ absorbers directly selected from the SDSS [@Balashev14] is indeed biased towards high $N({\ion{H}{i}})$ systems. Similarly, @Ledoux15 observe an excess of strong $N({\ion{H}{i}})$-systems among -selected absorbers, which appear to harbour high molecular content. We note that high molecular content is also found in some low $N({\ion{H}{i}})$ absorbers [e.g. @Srianand08; @Noterdaeme10co], which shows that the conversion from atomic to molecular hydrogen due to microphysics [which occurs on pc-scales, e.g. @Srianand13] does not require very high $N({\ion{H}{i}})$ [see also @Muzahid15].
We have investigated the impact on the quasar colours by the presence of systems beyond the neutral to molecular transition and showed that selection of quasars with NIR photometry would be important in order to avoid biasing against the detection of systems with high molecular content.
We thank the referee for careful reading of the manuscript and insightful remarks that helped improving the clarity of this paper.
[^1]: Either because of the detection of absorption [@Srianand08; @Noterdaeme10co], the presence of 21 cm absorption [@Srianand12], or direct evidence of H$_2$ lines [@Balashev14].
[^2]: For simplicity only the total H$_2$ column density is used for a given DLA. In spite of this, the total $N($H$_2$) is dominated in all cases by a few components that individually also have $\log N$(H$_2$) $>$ 17.
[^3]: We caution that part of the background stars were targeted for stellar studies (hence generally probing low extinction lines of sight), while others were specifically targeted to study the properties of dust, providing highly reddened lines of sight.
[^4]: This consistency check is not possible in the $u$-band because the corresponding wavelength range is not covered by the SDSS spectra.
|
---
abstract: 'One of the unsolved issues in the quantum gravity comes from the Wheeler-DeWitt equation, which is the second order functional derivative equation with non-linear term. In this paper, we introduce a method to solve the Wheeler-DeWitt equation with the static restriction introduced in this paper. Usually one treats the state functional of the spacetimes by the 3-dimensional metrics, which do not contain the timelike metrics. However we can expand this state to the state which has support on the space of the spacetime metrics with using additional constraint which requires the recovery of the 4-dimensional quantum gravity. Enlarging the support of the state functional of the spacetime metrics, we can simply solve the usual Wheeler-DeWitt equation with the additional constraint. Using this method we can solve some unsolved problems, such as the quantization of the black hole.'
author:
- Shintaro Sawayama
title: 'One method to solve Wheeler-DeWitt equation including black hole universe'
---
Introduction {#sec1}
============
The theory of the canonical quantum gravity is based on the Wheeler-DeWitt equation and traces back to 1967 [@De]. There are many methods treating the canonical quantum gravity. For example Hartle and Hawking [@Hart] considered a mini-superspace model. A loop approach to the canonical quantization was presented by Ashtekar and collaborators [@Rov][@As][@Thi]. Or there are string quantum cosmology [@AL]. In both schemes the difficulties with the Wheeler-DeWitt equation seams to remain. And only the homogeneous spacetimes can be quantized, while inhomogeneous spacetimes can not yet quantized completely. So we should try to remove or bypass the difficulties of the Wheeler-DeWitt equation.
In this paper, we show a technical method to solve the Wheeler-DeWitt equation, which we call up-to-down method. The up-to-down method consists of the following steps. First we add another dimension as an external time to the usual 4-dimensional metrics and create an enlarged Hilbert space which has support of the spacetime metrics, and then we reduce this quantum state to the physical 4-dimensional state, we can obtain the additional constraint to simplify the usual Wheeler-DeWitt equation i.e. we use static restriction in a quantum gravity. We show how this technique useful to the theories of canonical quantum gravity. The same method, however does not work for Klein-Gordon systems. The strength of our method in the quantum gravity is shown by considering of the Schwarzschild black hole. There is no idea of the statics in the quantum gravity. So our method is strong tool to solve the static Wheeler-DeWitt equation or usual Wheeler-DeWitt equation. The consistency of the additional constraint is checked by analyzing solution in the enlarged Hilbert space. In this paper, we consider the only vacuum spacetimes.
In section \[sec2\], we introduce the theory of the enlarged Hilbert space with one additional dimension. And show how we can solve the Hamiltonian constraint without fixing spacetime. This analysis is forward to the general discussion of the Wheeler-DeWitt equation. In subsection \[sec21\] we introduce what we call the up-to-down method. In subsection \[sec22\] we calculate general static solutions of the Wheeler-DeWitt equation.
In section \[sec3\], we analyze three mini-superspace models as an explicit test of the practicality of the up-to-down method introduced in this paper. In particular in \[sec31\], we consider a Friedmann universe model, and in subsection \[sec32\] we study an off-orthogonal metric space model, and in subsection \[sec33\], we quantize a Schwarzschild black hole. The black hole quantization is the main result achieved using the method introduced in this paper. Section \[sec4\] is devoted to a summary and discussion of the main results of our paper.
Up-to-down method and general solutions of the static Wheeler-DeWitt equation {#sec2}
=============================================================================
Up-to-down method {#sec21}
-----------------
In this section we explain in details what we call the up-to-down method. We analyze the auxiliary Hilbert space and the original Hilbert space and the projection.
We justify the definition of the projection given in this paper. Usually the Hilbert space has the support of the 3-dimensional spacelike metrics. However, we expand it here to have the support of the 4-dimensional spacetime metrics. We start by introducing the additional dimension which is an external euclidean time with positive signature, and thus create an artificial enlarged Hilbert space corresponding to this external time. We write such external dimension as $s$. Although there are many study of the higher dimensional spacetime, there is no physical meaning of $s$. However, there is mathematical meaning of the another time. This idea based on the Isham’s quantum category. So, we imagine many branes or many world interpretation and we quantize many world at the same time. The action may be written as $$\begin{aligned}
S=\int _{M\times s}{}^{(5)}RdMds. \label{f1}\end{aligned}$$ Where ${}^{(5)}R$ is the 5-dimensional Ricci scalar, built from the usual 4-dimensional metric and external time components. Rewriting the action by a 4+1 slicing of the 5-dimensional spacetime with lapse functionals given by the $s$ direction, we obtain the 4+1 Hamiltonian constraint and the diffeomorphism constraints as, $$\begin{aligned}
\hat{H}_S\equiv \hat{R}-\hat{K}^2+\hat{K}^{ab}\hat{K}_{ab} \label{f2}\\
\hat{H}_V^a\equiv \hat{\nabla} _b(\hat{K}^{ab}-\hat{K}\hat{g}^{ab}),\label{f3}\end{aligned}$$ where a hat means 4-dimensional and $a,b$ runs for $0,\cdots , 3$, e.g. the $\hat{K}_{ab}$ is extrinsic curvature defined by $\hat{\nabla}_a s_b$ and $\hat{K}$ is its trace, while $\hat{R}$ is the 4-dimensional Ricci scalar, and $\hat{\nabla} _a$ is the 4-dimensional covariant derivative.\
\
[*Definition 1.* ]{} The artificial enlarged functional space is defined by $\hat{H}_S|\Psi^{5} (g)\rangle =\hat{H}_V^a|\Psi^{5} (g)\rangle =0$, where $g$ is the 4-dimensional spacetime metrics $g_{\mu\nu}$ with ($\mu =0,\cdots ,3$). We insert the inner product usual sense i.e. $\langle \Psi^{5}(g)^{\dagger}|\Psi^{5}(g)\rangle$. We write this functional space as ${\cal H}_5$.\
Here, the definition of the canonical momentum $P$ is different from the usual one. Note in fact that the above state in ${\cal H}_5$ is not the usual 5-dimensional quantum gravity state, because the 4+1 slicing is along the $s$ direction. This is why we call this Hilbert space as artificial functional space. It is not defined by $\partial {\cal L}/(\partial dg/dt)$ but by $\partial {\cal L}/(\partial dg/ds)$, where ${\cal L}$ is the 5-dimensional Lagrangian.
In addition, we impose that 4-dimensional quantum gravity must be recovered from the above 5-dimensional action. The 3+1 Hamiltonian constraint and diffeomorphism constraint are, $$\begin{aligned}
H_S\equiv {\cal R}+K^2-K^{ij}K_{ij} \label{f4}\\
H_V^a\equiv D_j(K^{ij}-Kq^{ij}).\label{f5}\end{aligned}$$ where $i,j$ runs for $1,\cdots ,3$. Here $K_{ab}$ is the usual extrinsic curvature defined by $D_at_b$ and $K$ is its trace, while ${\cal R}$ is the 3-dimensional Ricci scalar, and $D_a$ is the 3-dimensional covariant derivative. Then we can define a subset of the auxiliary Hilbert space on which the wave functional satisfies the usual 4-dimensional constraints. In order to relate the 4 and 5 dimensional spaces we should define projections.\
\
[*Definition 2.*]{} The subset of ${\cal H}_5$ in which the five dimensional quantum state satisfies the extra constraints $H_S\Pi^1|\Psi ^5(g)\rangle=H_V^a\Pi^1|\Psi ^5(g)\rangle =0$ is called ${\cal H}_{5lim}$, where $P$ is the projection defined by $$\begin{aligned}
\Pi^1 :{\cal H}_5 \to {\cal F}_4 \ \ \
\{ \Pi^1 |\Psi^5(g)\rangle=|\Psi^5(g_{0\mu}={\rm const})\rangle \} ,\label{f6}\end{aligned}$$ where ${\cal F}_4$ is a functional space. And ${\cal H}_4$ is the usual four dimensional quantum gravity state with the restriction that $H_S|\Psi^4(q)\rangle=H_V^a|\Psi ^4(q)\rangle=0$. Here $q$ stands for the 3-dimensional metric $q_{ij}(i=1,\cdots ,3)$, and $\Pi^{2}$ is defined by $$\begin{aligned}
\Pi^{2}:{\cal H}_{5lim} \to {\cal H}_4 .\label{f7}\end{aligned}$$\
Although the measure of the projection may be zero, we can justify this projection in the next subsection.
From now we consider the recovery of the 4-dimensional vacuum Einstein gravity from the 5-dimensional wave functional. We consider 4-dimensional Ricci scalar in the 4+1 Hamiltonian constraint. The 4-dimensional Ricci scalar is decomposed as $$\begin{aligned}
\hat{R}=H_S+n_aH_V^a-\frac{1}{2}\dot{P}.\label{f8}\end{aligned}$$ Here $n_a$ is the orthonormal vector of the time and $P$ is the contraction of momentum. Then the modified Hamiltonian constraint for the 5-dimensional quantum state which contains the 4-dimensional Einstein gravity becomes, $$\begin{aligned}
\hat{H}_S\to -m\hat{H}_S:= -\hat{K}^2+\hat{K}^{ab}\hat{K}_{ab}-\frac{1}{2}\dot{P} \label{f9}\end{aligned}$$ There is the theoretical branch in using the Dirac constraint or Hamiltonian and diffeomorphism constraint. To use the Dirac constraint at this point create the additional constraint which restrict the state to the static.
Finally, the simplified Hamiltonian constraint in terms of the canonical representation becomes $$\begin{aligned}
m\hat{H}_S =(-g_{ab}g_{cd}+g_{ac}g_{bd})\hat{P}^{ab}\hat{P}^{cd}-\frac{1}{2}\dot{P}. \label{f10}\end{aligned}$$ The magic constant factor $-1$ for the term $g_{ab}g_{cd}$ is a consequence of the choice of dimensions for ${\cal H}_5,{\cal H}_4$. Here $\hat{P}^{ab}$ is the canonical momentum of the 4-dimensional metric $g_{ab}$, that is $\hat{P}^{ab}=\hat{K}^{ab}-g^{ab}\hat{K}$. And as we mentioned above, this canonical momentum is defined by the external time and not by the usual time.
We now give a more detailed definition of the artificial functional space as follows:\
\
[*Definition 3.*]{} The subset ${\cal H}_{5(4)}\subset {\cal H}_5$ is defined by the constraints, $ \hat{R}|\Psi ^5(g)\rangle =0$, and we write its elements as $|\Psi ^{5(4)}(g)\rangle$. We also define a projection $\Pi^3$ as $$\begin{aligned}
\Pi^3 : {\cal H}_{5(4)} \to {\cal H}_{4(5)} \ \ \ \{ P^*|\Psi ^{5(4)}(g)\rangle =|\Psi ^{5(4)}(g_{0\mu}={\rm const})\rangle
=: |\Psi ^{4(5)}(q)\rangle \} , \label{f11}\end{aligned}$$ where ${\cal H}_4$ is a subset of ${\cal H}_{4(5)}$. We can defien the inner product in the ${\cal H}_{4(5)}$ space like, $\langle \Psi^{4(5)}(q)^{\dagger}|\Psi^{4(5)}(q)\rangle$\
We notice the projection $\Pi^1$ and $P^2$ and $\Pi^3$ is all most all same. However, we use the different symbol because the projected functional space is different.\
[*Definition 4.*]{} In ${\cal H}_{4(5)}$ there is a subset whose state satisfy $H_S|\Psi^{4(5)}(q)\rangle=H_V|\Psi ^{4(5)}(q)\rangle=0$. We write such Hilbert space as ${\cal H}_{4(5)phys}$ and its elements as $|\Psi_{phys}^{4(5)}(q)\rangle$. If there are relations $\Pi^3|\Psi^{5(4)}(g)\rangle=|\Psi_{phys}^{4(5)}(q)\rangle$, we write such $|\Psi^{5(4)}(g)\rangle$ as $|\Psi^{5(4)}_{phys}(g)\rangle$ and we write such Hilbert space ${\cal H}_{5(4)phys}$.\
The ${\cal F}_4$ functional space or ${\cal H}_{5(4)}$ space does not need to be the $l^2$ norm spaces. As a consequence, we can obtain following theorem:\
\
We comment on the projections. Although there are three projection, these operation is same. However, the projecting space and projected space is different. So we use the number 1,2,3. The projection act only on the right hand side, it does not act left hand side. And usually it does not commute with the Hamiltonian constraint and the diffeomorphism constraint. Although the inverse projection like $\Pi^{-1}$ may be one to many projection, we can choose suitable inverse projection so that the measure of the projection does not become zero. Or we assume such projection exist. Although the measure of the projection may be zero, we assume there is at least one enlargement whose measure of the projection is not zero. Or we can only treat such subspace.\
[*Theorem 1.*]{} In this method, in the ${\cal H}_4$ additional constraint $m\hat{H}_S\Pi^3=0$ appears which we call static restriction, if there is no time evolution and the projection is defined by the definition 3.\
[*Sketch of the proof*]{}\
$$\begin{aligned}
\hat{H}_S\Pi^3=\hat{R}\Pi^3-\hat{K}^2\Pi^3 +\hat{K}^{ab}\hat{K}_{ab}\Pi^3 \nonumber \\
\to H_S\Pi^3 +n_aH_V^a\Pi^3-\frac{1}{2}\dot{P}\Pi^3-\hat{K}^2\Pi^3 +\hat{K}^{ab}\hat{K}_{ab}\Pi^3 \nonumber \\
\to -\frac{1}{2}\dot{P}\Pi^3-\hat{K}^2\Pi^3 +\hat{K}^{ab}\hat{K}_{ab}\Pi^3 \nonumber \\
\to (-q_{ij}g_{kl}+g_{ik}g_{jl})P^{ij}P^{kl}-\frac{1}{2}\dot{P}. \label{f12}\end{aligned}$$\
[*Lemma 1*]{}.If the term of the $\dot{P}$ became to zero acted on the state, the additional constraint become static restriction as, $$\begin{aligned}
{\cal S}=(-q_{ij}g_{kl}+g_{ik}g_{jl})P^{ij}P^{kl} \label{f13}\end{aligned}$$\
We explain this lemma. The static restriction $S$ is the special case of the additional constraint and the additional constraint usually does not commute with the Hamiltonian constraint and the diffeomorophis constraints. However, in the special mini-superspaces the static constraint and the Hamiltonian constraint does commute. There is another method of the up-to-down method, which uses $m\hat{H}_S$ in the enlarged functional space. Then there does not appear the additional constraint in the usual 4-dimensional constraints. This another method is useful to solve the usual static 4-dimensional Wheeler-DeWitt equation. We only consider static restriction or enlargement of the static restriction in this paper.
Now we also have the additional constraint equation coming from the assumption that the 5-dimensional vacuum Einstein gravity should reproduce 4-dimensional Einstein gravity in the classical limit. That is, $$\begin{aligned}
{}^{(4)}G_{ab}=\hat{K}\hat{K}_{ab}-\hat{K}_{a}^{\ c}\hat{K}_{bc}-2\nabla_a\beta_b=0, \label{f14}\end{aligned}$$ where, $$\begin{aligned}
\beta ^a:=s^b\nabla _bs^a-s^a\nabla _bs^b \label{f15}\end{aligned}$$ If we assume that l.h.s. of Eq. (\[f14\]) corresponds to the matter term, we can take its trace. And this additional constraint reduces to the sum of the 4+1 Hamiltonian constraint and the diffeomorphism constraint, that is, $$\begin{aligned}
8\pi T^a_a:=\hat{K}^2-\hat{K}_{ab}\hat{K}^{ab}
-2\nabla_a\beta^a = m\hat{H}_S-2s_a\hat{H}_V^a-\frac{1}{2}\dot{P} \approx m\hat{H}_S. \label{f16}\end{aligned}$$ In other words, the matter term $T_a^a$ has been promoted to the operator, it does not produce further constraints other than 5-dimensional modified Hamiltonian constraint. We don’t assume equation (\[f14\]), because it is too strong, determine the four independent metrics by the other metric components.\
\
[*Lemma2.*]{} The requirement to recover four dimensional gravity, $8\pi T_a^a |\Psi ^5(g)\rangle = 0$, is the same as $\hat{H}_S|\Psi ^5(g)\rangle = 0$. So $8\pi T_a^a \approx m\hat{H}_S \approx \hat{H}_S$.
We can summarize the procedure to solve the usual Wheeler-DeWitt equation.\
\
[*Steps to solve the Wheeler-DeWitt equation.*]{}\
There are 2 steps to solve the usual 3+1 Hamiltonian constraint.\
a1) Solve $m\hat{H}_S|\Psi^5(g)\rangle =0$ and obtain $|\Psi ^{5(4)}(g)\rangle$.\
a2) Project this state by $\Pi^3$.\
a3) Solve $H_S|\Psi ^{4(5)}(g)\rangle =0$ to obtain $|\Psi ^{4(5)}_{phys}(q)\rangle$.\
\
b1) Using the additional constraint $m\hat{H}_S\Pi^3|\Psi ^{4(5)}(q)\rangle =0$ in ${\cal H}_4$, solve $H_S|\Psi^{4(5)}(q)\rangle =0$ to obtain $|\Psi^{4(5)}_{phys}(q)\rangle$.\
b2) Then enlarge $|\Psi^{4(5)}_{phys}(q)\rangle \to |\Psi^{5(4)}_{phys}(g)\rangle$.\
\
The step a) is useful when we consider the Hamiltonian constraint in the general sense. The step b) is useful when we consider mini-superspace models and we use it in section \[sec3\]
General solution for the $|\Psi^{5(4)}(g)\rangle$ state and the projection theorem {#sec22}
----------------------------------------------------------------------------------
Let us consider $|\Psi^{5(4)}(g)\rangle$ as the state with support on the spacetime metrics which contain 4-dimensional gravity, and write the Hamiltonian and diffeomorphism constraints in the operator representation. Acting in the auxiliary Hilbert state as, $$\begin{aligned}
\sum (g_{ab}g_{cd}-g_{ac}g_{bd})
\frac{\delta}{\delta g_{ab}}\frac{\delta}{\delta g_{cd}}|\Psi^{5(4)}(g)\rangle =0 \label{f17}\end{aligned}$$ $$\begin{aligned}
(\hat{\nabla} _{a}\hat{P}^{ab})|\Psi\rangle
=\bigg[ \frac{\partial }{\partial x^{a}},\frac{\delta}{\delta g_{ab}}\bigg] |\Psi^{5(4)}(g)\rangle =0. \label{f18}\end{aligned}$$ We can easily find a solution to Eq. (\[f17\]), and such solution is the superposition of the two following states $$\begin{aligned}
|\Psi^{5(4)} (g)\rangle _d=\sum_{\mu}f_{(\mu )}[g_{\mu\mu}], \label{f19}\end{aligned}$$ and $$\begin{aligned}
|\Psi^{5(4)} (g)\rangle _{od}=\sum_{\mu \geq\nu}f_{(\mu\nu)}^{(1)}[g_{\mu\nu}]. \label{f20}\end{aligned}$$ Where $d$ stands for diagonal part, and $od$ stands for off-diagonal part. Here $f_{(\mu )}$ is an arbitrary functional of only the $g_{\mu\mu}$ and $f_{(\mu\nu)}^{(1)}$ is a first order functional of only the $g_{\mu\nu}$. This division of the state of the $|\Psi^{5(4)}(g)\rangle$ by $g_{\mu\mu}$ is important. The functional $f_{(\mu )}[g_{\mu\mu}]$ can be expressed as a function in the following way, $$\begin{aligned}
f_{(\mu )}[g_{\mu\mu}]=f_{(\mu )}(g_{\mu\mu},g_{\mu\mu ,\nu},...).\label{f21}\end{aligned}$$ The diffeomorphism constraint can be simply written as, $$\begin{aligned}
\bigg[ \frac{\partial }{\partial x^{\mu}},\frac{\delta}{\delta g_{\mu\mu}}\bigg]f_{(\mu )}[g_{\mu\mu}]
+\sum_{\mu\not=\nu}\bigg[ \frac{\partial }{\partial x^{\nu}},\frac{\delta}{\delta g_{\mu\nu}}\bigg]
f_{\mu\nu}^{(1)}[g_{\mu\nu}]=0
\ \ \ (\mu =0,1,2,3), \label{f22}\end{aligned}$$ One of the solutions of this equation is, $$\begin{aligned}
f_{(\mu )}(g_{\mu\mu})\subset f_{(\mu )}[g_{\mu\mu}] ,\label{f23}\end{aligned}$$ and $$\begin{aligned}
0 \subset f_{(\mu\nu )}^{(1)}[g_{\mu\nu}]\ \ \ (\mu\not=\nu ), \label{f24}\end{aligned}$$ These functionals are one of the solutions for the $|\Psi^{5(4)}(g)\rangle$ state. We note that the metric can be locally diagonalized.\
There are also solutions to the modified Hamiltonian constraint equation, that is the deterministic function as, $$\begin{aligned}
|\Psi^{5(4)} (g)\rangle =\sum_{\sigma}\prod _{\mu =0}^3g_{\mu\sigma (\mu )}, \label{f25}\end{aligned}$$ $\sigma$ stands for permutation. We note that Eq. (\[f25\]) does not involve the signature. In the derivation of the above formula, we did not assume any symmetry for the metric, in particular $g_{\mu\nu}=g_{\nu\mu}$ was not assumed. If we assume such a symmetry, we should add constant factors for the each element in Eq. (\[f25\]). Power series of $g_{\mu\sigma (\mu )} $ are also solutions, we may write $$\begin{aligned}
|\Psi^{5(4)} (g)\rangle =\sum_{\sigma}\prod _{\mu =0}^3g_{\mu\sigma (\mu )}^n, \label{f26}\end{aligned}$$ and also functionals of $g_{\mu\sigma (\mu )}$ is also solutions, that is $$\begin{aligned}
|\Psi^{5(4)} (g)\rangle =\sum_{\sigma}f\bigg[\prod _{\mu =0}^3g_{\mu\sigma (\mu )}\bigg] . \label{26}\end{aligned}$$ We can make the state (\[21\]) become a functional of the determinant of $g$ only, if we choose the coordinates in such a way that the metric becomes diagonal.\
[*Theorem 2.*]{} The projection of the above two states (\[19\]-\[20\]), (\[26\]) $\Pi^3|\Psi^{5(4)}(g)\rangle$ satisfy ${\cal S}\Pi^3|\Psi^{5(4)}(q)\rangle =0$.\
This fact is one of the motivations for introducing $\Pi^3$. Otherwise, this additional constraint does not appear in general solution, if it acts mini-superspaces, additional constraint appears.
Mini-superspaces {#sec3}
================
In this section we consider three mini-superspace models, i.e a Friedmann model, an anti-orthogonal model, and a Schwarzschild model. In subsection \[sec31\] we check whether the up-to-down method introduced in section \[sec2\] is really applicable. In subsection \[sec32\] we study the off-orthogonal metric model. One main progress is shown in subsection \[sec33\], which deals with a Schwarzschild black hole, and where it is shown that the black hole is quantized by up-to-down method. In this section we always ignore so called operator ordering.
Friedmann universe and the problem of time {#sec31}
------------------------------------------
Because the functional space ${\cal H}_{4(5)phys}$ may be empty, we should check that the up-to-down method is really consistent. In this paper we ignore the operator ordering for simplicity.
We simply assume that the metric corresponding to a Friedmann model with a cosmological constant $\Lambda$, i.e. $$\begin{aligned}
g_{ab}:= \begin{pmatrix}
b & 0 & 0 & 0 \\
0 & a & 0 & 0 \\
0 & 0 & a & 0 \\
0 & 0 & 0 & a
\end{pmatrix}.\label{f27}\end{aligned}$$ Here $a,b$ only depend on time $t$, and coordinates are chosen as $t,x,y,z$. The cosmological constant is included in the constraint of 4-dimensional gravity. The modified Hamiltonian constraint $m\hat{H}_S$ is explicitly written as, $$\begin{aligned}
m\hat{H}_S=6ab\frac{\partial}{\partial a}\frac{\partial}{\partial b}+a^2\frac{\partial^2}{\partial a^2} \\
=6\frac{\partial}{\partial \eta}\frac{\partial}{\partial \eta_b}+\frac{\partial^2}{\partial \eta^2}=0 ,\label{f28}\end{aligned}$$ where we have defined $\eta =\ln a$, $\eta _b=\ln b$. This constraint acts on $|\Psi^5(g)\rangle$, and solution of $m{\hat H}_S|\Psi^5(g)\rangle =0$ give $|\Psi^{5(4)}(g)\rangle$.
The usual 3+1 Hamiltonian constraint is $$\begin{aligned}
H_S
=\frac{9}{2}a^2\frac{\partial}{\partial a^2}+\Lambda \\
=\frac{9}{2}\frac{\partial}{\partial \eta^2}+\Lambda =0. \label{f29}\end{aligned}$$ In this Friedmann model we do not need additional constraint $m\hat{H}_SP^* \approx 0$. If the cosmological constant is zero, ${\cal H}_4$ and ${\cal H}_{4(5)}$ are the same Hilbert space. Here we enlarge the ${\cal H}_4$ state $|\Psi^4(g)\rangle $ to the ${\cal H}_{5(4)}$ state.
We can obtain the $|\Psi^4(q)\rangle$ state as, $$\begin{aligned}
|\Psi^4(\eta )\rangle =\exp (i\frac{\sqrt{2}}{3}\Lambda^{1/2} \eta) . \label{f30}\end{aligned}$$
The next step is to check the consistency of the up-to-down method. We can expand the state (21) to the state $|\Psi^{5(4)}(g)\rangle$ by the inverse projection $(\Pi^3)^-1$ as, $$\begin{aligned}
|\Psi^{5(4)}(\eta_b,\eta )\rangle =f(\eta_b) \exp (i\frac{\sqrt{2}}{3}\Lambda^{1/2} \eta) .\label{f31}\end{aligned}$$ Although the inverse projection can be defined another way, if we can find at least one enlargement, this method works well. Then we can solve the following constraint equation to derive $|\Psi^{5(4)}(g)\rangle$. $$\begin{aligned}
m\hat{H}_S|\Psi^5 (g)\rangle =\bigg( 6i\frac{\sqrt{2}}{3}\Lambda^{1/2} f'(\eta_b ) -6\frac{2}{9}\Lambda f(\eta_b) \bigg)
\exp (i\frac{\sqrt{2}}{3}\Lambda^{1/2} \eta) =0 \label{f32}\end{aligned}$$ From this equation we obtain $$\begin{aligned}
2i\sqrt{2}\Lambda^{1/2} f'(\eta_b ) -\frac{4}{3}\Lambda f(\eta_b) =0, \label{f33}\end{aligned}$$ or $$\begin{aligned}
f'=-i\frac{\sqrt{2}}{3}\Lambda^{1/2} f, \label{f34}\end{aligned}$$ whose solution is $$\begin{aligned}
f(\eta_b)=\exp \bigg( -i\frac{\sqrt{2}}{3}\Lambda ^{1/2}\eta_b \bigg) . \label{f35}\end{aligned}$$ From this result and Eq. (\[f30\]) we can obtain $|\Psi^{5(4)}(g)\rangle$ state as $$\begin{aligned}
|\Psi^{5(4)} (\eta_b ,\eta)\rangle =\exp \bigg( -i\frac{\sqrt{2}}{3}\Lambda ^{1/2}\eta_b \bigg)\exp (i\frac{\sqrt{2}}{3}\Lambda^{1/2} \eta) .\label{f36}\end{aligned}$$ This state is a solution to both the $m{\hat H}_S$ constraint and $H_S$ constraint, which shows that the up-to-down method is applicable at least to the simple Friedmann model. We comment on this enlargement. The norm of enlarged state is not important. Although, there are other enlargement, we choose multiplication enlargement. Because all the projection produces same state without constant factor. By this enlargement the measure of the projection does not become zero.
Quantization of the off-orthogonal metric space {#sec32}
-----------------------------------------------
As the next example we consider the quantization of a spacetime with anti-orthogonal metric i.e. $$\begin{aligned}
g_{ab}= \begin{pmatrix}
-c & 0 & 0 & 0 \\
0 & a & b & 0 \\
0 & b & a & 0 \\
0 & 0 & 0 & a
\end{pmatrix}.\label{f37}\end{aligned}$$ Here $a,b,c$ only depend on time $t$, and the coordinates are chosen as $t,x,y,z$. The modified 4+1 Hamiltonian constraint is now written as $$\begin{aligned}
m\hat{H}_S=-6ac\frac{\partial}{\partial a}\frac{\partial}{\partial c}
+(6a^2-2b^2)\frac{\partial^2}{\partial a^2}+2(b^2-a^2)\frac{\partial^2}{\partial b^2}=0, \label{f38}\end{aligned}$$ while the usual 3+1 Hamiltonian constraint is written as $$\begin{aligned}
H_S=-(5a^2-b^2)\frac{\partial}{\partial a^2}-(2b^2-a^2)\frac{\partial^2}{\partial b^2}=0.\label{f39}\end{aligned}$$ Since finding a solution to (\[39\]) is something difficult, we use additional constraint $$\begin{aligned}
{\cal S} =(6a^2-2b^2)\frac{\partial^2}{\partial a^2}+2(b^2-a^2)\frac{\partial^2}{\partial b^2}=0, \label{f40}\end{aligned}$$ to simplify it as $$\begin{aligned}
H_S=-\frac{(a^2+b^2)(2a^2-b^2)}{(a^2-b^2)}\frac{\partial^2}{\partial a^2}=0.\label{f41}\end{aligned}$$ Both constraints $H_S$ and $8\pi T_a^aP^*$ act on the state $|\Psi^{4(5)}(q)\rangle$ which belong to a subset of ${\cal H}_4$. The differential equation $H_S|\Psi^{4(5)}(q)\rangle$ can be solved easily and the result is, $$\begin{aligned}
|\Psi^{4(5)} (a,b)\rangle =C_1(b)a+C_2(b),\label{f42}\end{aligned}$$ here $C_1(b),C_2(b)$ are the arbitrary functions of $b$. The solution which also satisfy $m\hat{H}_SP^*\approx 0$ is now, $$\begin{aligned}
|\Psi^{4(5)} (a,b)\rangle =E_1ab+E_2a+E_3b+E_4.\label{f43}\end{aligned}$$ Here $E_1\cdots E_4$ are constants. We can finally expand the state (\[43\]) to the $|\Psi^{5(4)}(g)\rangle$ state, using the modified Hamiltonian constraint $m\hat{H}_S$, as $$\begin{aligned}
|\Psi ^{5(4)} (a,b,c)\rangle =F_1ab+F_2a+F_3bc+F_4c+F_5. \label{f44}\end{aligned}$$ This enlargement is also consistent with projection measure. The projected state is $|\Psi^{4(5)}$ state. In this example the static restriction and the Hamiltonian constraint does not commute. However, the obtained 4-dimensional state is state of the usual Hamiltonian constraint.
Quantization of the Schwarzschild black hole {#sec33}
--------------------------------------------
The most interesting result is the applicability of our up-to-down method to the model of a black hole spacetime. We consider the Schwarzschild black hole metric, i.e. $$\begin{aligned}
g_{ab}=\begin{pmatrix}
-f & 0 & 0 & 0 \\
0 & f^{-1} & 0 & \\
0 & 0 & A & 0 \\
0 & 0 & 0 & A\sin ^2\theta
\end{pmatrix}.\label{f45}\end{aligned}$$ Here we choose the usual coordinates $t,r,\theta,\phi$, and $f$ is the function that depends only on time $t$, and $A$ stands for the area i.e. $A\doteq r^2$. The derivative of its components is written as, $$\begin{aligned}
\frac{\delta}{\delta g_{tt}}=-\frac{\partial}{\partial f} \label{f46} \\
\frac{\delta}{\delta g_{rr}}=-f^2\frac{\partial}{\partial f} \label{f47} \\
\frac{\delta}{\delta g_{\theta\theta}}=\frac{\partial}{\partial A} \label{f48} \\
\frac{\delta}{\delta g_{\phi\phi}}
=\frac{1}{\sin^2\theta}\frac{\partial }{\partial A}+\frac{1}{A\sin 2\theta}\frac{\partial}{\partial \theta}.\label{f49}\end{aligned}$$ We make the simplifying assumption that the quantum state does not depend on $\theta$, because of spherical symmetry. Using the above formulas for the derivatives we can write the modified 4+1 Hamiltonian constraint as $$\begin{aligned}
m\hat{H}_S=-f^2\frac{\partial^2}{\partial f^2}+A^2\frac{\partial^2 }{\partial A^2}=0, \label{f50}\end{aligned}$$ while the usual 3+1 Hamiltonian constraint is $$\begin{aligned}
H_S=\frac{1}{2}\bigg(-f^2\frac{\partial^2 }{\partial f^2}
+Af\frac{\partial }{\partial f}\frac{\partial}{\partial A}
-2A^2\frac{\partial^2 }{\partial A^2}\bigg)+{\cal R}=0, \label{f51}\end{aligned}$$ where [R]{} is 3-dimensional Ricci scalar ${\cal R}=M/r^3=(1-f)/A$. The equation (\[f51\]) is a second order partial differential derivative with non-linear terms, and it is difficult to find explicit analytical solution. However, if we use the additional constraint $$\begin{aligned}
{\cal S}=-2fA\frac{\partial}{\partial f}\frac{\partial}{\partial A}+A^2\frac{\partial^2}{\partial A^2} \\
=A\frac{\partial}{\partial A}\bigg( -2f\frac{\partial}{\partial f}+A\frac{\partial}{\partial A}\bigg) =0, \label{f52}\end{aligned}$$ Then we can find the following relation between the two parameters $f$ and $A$, $$\begin{aligned}
f^{1/2}=cA,\label{f53}\end{aligned}$$ where $c$ is a non-zero complex constant. At this point we comment on the theoretical branch in Eq.(\[f52\]). And only if the above parameter relation holds, we can carry on simultaneous quantization. There are another way to quantize static black holes, i.e. Eq.(\[f51\]). We can find another way which is to use only the differential relation or to determine the form of $f$ in terms of $A$. Then there appears duality between $A$ and $f$. The singularity at the $A=0$ and $f=0$ become degenerate. And making it parameter relation, we can commute the static restriction and the Hamiltonian constraint. From Eq. (\[f53\]), we can deduce the analytic form of the mass operator, as $$\begin{aligned}
2\hat{M}=A^{1/2}-c^2A^{5/2}=r-c^2r^5, \label{f54}\end{aligned}$$ if we assume $f=1-2\hat{M}/r$. This is what we know from the additional constraint. Then the mass corresponding to the classical gravity can be defined by, $$\begin{aligned}
\langle M(c)\rangle :&=&\int_0^{\infty} 2\pi A^{1/2}\langle \Psi |\hat{M}|\Psi\rangle dA \nonumber \\
&=&\langle A\rangle -c^2\langle A^3\rangle .\label{f55}\end{aligned}$$ For the classical correspondence, we require the averaged mass is real and bounded from above. From the fact that averaged mass is real, we know $c$ is real or purely imaginary. Otherwise the relation of $A$ and $f$ seems to violating asymptotic flatness, if averaged value of the mass is constant (does not depend $f$ or $A$) and does not diverge, the quantum state is corresponding to the usual classical Schwarzschild black hole. The fact that the classical mass (\[f55\]) is constant is clear from the above formula.
Using the relation of $A$ and $f$ we can rewrite the usual 3+1 Hamiltonian constraint simply as second order ordinal differential equation, as $$\begin{aligned}
H_S=-\frac{7}{8}A^2\frac{\partial^2}{\partial A^2}+\frac{1-c^2A^2}{2A}=0. \label{f56}\end{aligned}$$ Here, we used $$\begin{aligned}
{\cal R}=\frac{\hat{M}}{r^3}=\frac{1-c^2A^2}{2A} \label{f57}\end{aligned}$$ This kind of constraint equation (\[f56\]) can always be solved numerically. And symbolically, we write its solution as $|\Psi^{4(5)}(c,A)\rangle =C_1F(c,A)$, where $C_1$ is constant factor for the later convenience.
If $A\to 0$, the 3+1 Hamiltonian constraint becomes $$\begin{aligned}
\lim_{A\to 0}H_S&=&-\frac{7}{8}A^2\frac{\partial^2}{\partial A^2}+\frac{1}{2A} \\
&\to &-\frac{7}{16}\frac{\partial^2}{\partial a}+1=0, \label{f58}\end{aligned}$$ where, $a=A^{-1/2}$. Such coordinate transformation can be done because of ignorance of operator ordering. So, if $A\to 0$, the constant $c$ disappears from the state.
In this limit the differential equation for $a$ can be solved analytically, and the solution is, $$\begin{aligned}
|\Psi (c,A\to 0)\rangle =E_1\exp \bigg( \frac{4}{\sqrt{7}}A^{-1/2}\bigg) +E_2\exp \bigg( -\frac{4}{\sqrt{7}}A^{-1/2}\bigg) ,\label{f59}\end{aligned}$$ where $E_1,E_2$ are constants. The state has no conical singularity at $A=0$, if $E_1=0$. Because the conical singularity and the coordinate singularity are degenerated now due to Eq.(50), we can say that in the quantisation of the black hole, the conical and the coordinate singularities are removed.
If we take a limit of $A\to \infty$, the Hamiltonian constraint becomes, $$\begin{aligned}
\lim_{A\to \infty}H_S&=&-\frac{7}{8}A^2\frac{\partial^2}{\partial A^2}-\frac{Ac^2}{2} \nonumber \\
&\to &-\frac{7}{16}\frac{\partial^2}{\partial a'^2}-c^2=0 \label{f60}\end{aligned}$$ Here, $a'=A^{1/2}=r$.
This solution is, $$\begin{aligned}
|\Psi (c,A\to \infty )\rangle =D_1\exp\bigg( i\frac{4}{\sqrt{7}}cA^{1/2}\bigg) +D_2\exp\bigg( -i\frac{4}{\sqrt{7}}cA^{1/2}\bigg) .\label{f61}\end{aligned}$$ Here, $D_1,D_2$ are constants. This function is different by $c$ is real or purely imaginary. To satisfy the requirement that the averaged mass has the upper bound, we calculate the averaged mass in the infinity. If $c$ is real, $$\begin{aligned}
\langle M(c)\rangle \approx \int_0^{C}2\pi A^{1/2}\langle \hat{M}\rangle dA+\int_C^{\infty}2\pi (|D_1|^2+|D_2|^2)(A-c^2A^3)dA \nonumber \\
+\int_C^{\infty}2\pi (A-c^2A^3)(D_1D_2^*e^{i8cA^{1/2}/\sqrt{7}}+D_1^*D_2e^{-i8cA^{1/2}/\sqrt{7}})dA \label{f62}\end{aligned}$$ Here, $C$ is some large number $(C\gg 1)$. In this case, the second term of the r.h.s. diverges to $-\infty$ and the third term of the r.h.s. does not converges. The first term of the right hand side is bounded from above, because we remove singularity. So if $c$ is real the contradiction is appeared because of the behaviour of the averaged mass. If $c$ is purely imaginary, the state becomes, as $$\begin{aligned}
|\Psi (c,A\to \infty )\rangle =D_1\exp\bigg( -\frac{4}{\sqrt{7}}|c|A^{1/2}\bigg) +D_2\exp\bigg( \frac{4}{\sqrt{7}}|c|A^{1/2}\bigg) . \label{f63}\end{aligned}$$ The constant $D_2$ is zero, because the norm of the state should converge. Then averaged value of the mass is, $$\begin{aligned}
\langle M(c)\rangle \approx \int_0^{C}2\pi A^{1/2}\langle \hat{M}\rangle dA+\int_C^{\infty}2\pi |D_1|^2(A+|c|^2A^3)\exp\bigg( -\frac{8}{\sqrt{7}}|c|A^{1/2}\bigg) dA. \label{f64}\end{aligned}$$ We know the second term of the r.h.s. converges because of exponential term and $\langle M(c)\rangle$ is bounded from above. If $c$ is purely imaginary, the averaged mass $\langle M(c)\rangle$ is always positive, so there does not appear any contradiction. From above discussions, we can say that $c$ is purely imaginary.
Expanding the $|\Psi ^{4(5)}(c,A)\rangle $ state to the $|\Psi^{5(4)}(c,f,A)\rangle $ state, we simply assume $C_1$ is a function of $f$. Although $f$ and $A$ are related in ${\cal H}_{4(5)}$, however, we assume that $f$ and $A$ are independent in ${\cal H}_{5(4)}$. Using the simplified 3+1 Hamiltonian constraint and the parameter relation in ${\cal H}_{4(5)}$, we can simplify $m\hat{H}_S$ as $$\begin{aligned}
m\hat{H}_S=-f^2\frac{\partial^2}{\partial f^2}+\frac{4}{7}\frac{c-cf}{f^{1/2}}=0. \label{f65}\end{aligned}$$ This constraint equation has always solutions which we symbolically write as $C_1(c,f)$. Then the states of the expanded Hilbert space are symbolically written as $ |\Psi^{5(4)}(c,f,A)\rangle =C_1(c,f) F(c,A) $. The existence of the expanded Hilbert space certificates the technique of up-to-down method.
Conclusion and discussions {#sec4}
==========================
The “up-to-down” method introduced in this paper looks as an interesting and powerful method to quantize important models in quantum gravity as a static solution. Otherwise the Hilbert space may not contain all ${\cal H}_4$ state in this method, the subset of physical quantum state is exist and this state is easy to find. We can easily find the static solutions of the Hamiltonian constraint by this method. From now on there are no static restriction in the quantum gravity. However, we find the static restriction in the quantum gravity. Usually static solution is easier to solve than to solve the dynamical solutions. We can say that it is same in the quantum gravity.
We studied in details three mini-superspace models i.e. a Friedmann model, an anti-orthogonal model, and a black hole model. In the Friedmann model we could check the up-to-down method is applicable. In the anti-orthogonal model we could derive only a trivial solution. However, the combination of the parameters of the expanded state is non-trivial. The major progress within the mini-superspace models seems to come from the black hole quantization. The up-to-down method allows to transform a two parameter partial differential equation into an ordinary second order differential equation.
And we comment on the enlargement. The three example of the mini-superspace are multiplication enlargement which are consistent of the measure of the projection. So we can say we can find at least one enlargement whose projection is non-zero.
We succeeded in the quantization of the Schwarzschild black holes. And we found the disappear of the conical and coordinate singularities at the quantum level. From the additional constraint, we could determine the explicit form of the mass operator and how to calculate the averaged mass.
The up-to-down method introduced in this paper is technically introduced, and the artificial Hilbert space is not the usual quantum gravity space. Although there may be a relation to the higher dimensional quantum gravity Hilbert space, we do not comment on it in this paper, because it is beyond the present work.
We would like to acknowledge A.Carlini and A.Hosoya for comments and discussions.
[99]{} B.S.DeWitt Phys. Rev. [**160**]{} 1113 (1967) J.J.Halliwell and J.B.Hartle Phys. Rev. [**D 43**]{} 1170 (1991)
C.Rovelli Quantum Gravity; Cambridge monographs on mathematical physics (2004) C.J.Isham and A.Ashtekar Class. Quant. Grav. [**9**]{} 1433 (1992) T.Thiemann gr-qc/0110034 (2001) A.Linde hep-th/0503195 (2005) S.Sawayama arXiv:0705.2916
|
---
author:
- 'The DAE$\delta$ALUS Collaboration$^*$'
title: |
A Study of Detector Configurations\
for the DUSEL CP Violation Searches\
Combining LBNE and DAE$\delta$ALUS\
---
------------------------------------------------------------------------
[*Abstract:*]{} This study presents comparative CP sensitivities for various sets of water Cerenkov and liquid argon detectors combined with various running scenarios associated with DAE$\delta$ALUS and LBNE neutrino beams at DUSEL. LBNE-only running scenarios show fairly small differences in sensitivity for the various detector combinations. On the other hand, the DAE$\delta$ALUS-only and DAE$\delta$ALUS-plus-LBNE running gives significantly better sensitivity for a detector combination that includes at least 200 kt of Gd-doped water Cerenkov detector, exceeding the sensitivity of a Project-X 10 year run. A 300 kt Gd-doped water Cerenkov detector yields the best sensitivity for combined running.
------------------------------------------------------------------------
This study examines the sensitivity of various design configurations proposed for DUSEL for the physics of CP violation. We rank-order ten configurations of beams and detectors. We consider three types of detector “units” which are then arranged in configurations consisting of sets of three units. The units are:
- **WCGd:**100 kt Gd-doped water Cerenkov detector with $\approx$20% high quantum efficiency PMT coverage and/or light concentrators to realize good efficiency for the $\sim$5 MeV Cerenkov light signal expected from neutron capture on Gd [@ntag];
- **WC:** 100 kt water Cerenkov detector modules with 15% high quantum efficiency PMT coverage;
- **LAr:** 17 kt of LAr.
We consider three types of neutrino sources, with 10 year running-periods:
- **LBNE alone** – which is 30$\times10^{20}$ protons on target (POT) in neutrino mode followed by 30$\times10^{20}$ POT in antineutrino mode. This is the standard 10-year run-plan prior to the startup of Project X [@Gina].
- **DAE$\delta$ALUS alone** – which is in antineutrino mode, following the plan described in ref. [@EOI].
- **DAE$\delta$ALUS + LBNE** – which is the standard plan for DAE$\delta$ALUS antineutrino running combined with LBNE running for the full 10 years in neutrino mode. These programs can proceed simultaneously.
The code used for this study is described in detail in ref. [@EOI]. The DAE$\delta$ALUS event rates do not depend on the mass hierarchy, but the LBNE rates do; for this study we use the normal hierarchy. For these comparisons, an input uncertainty of $\delta(\sin^{2} 2\theta_{13})=0.005$ from the upcoming reactor experiments, has been assumed [@EOI; @firstpaper; @huber].
The assumptions concerning beam fluxes for DAE$\delta$ALUS and LBNE have been described in ref. [@EOI]. The DAE$\delta$ALUS flux assumes a decay at rest (DAR) beam arising from the stopped pion decay chain produced by an incident proton beam of 800 MeV. The assumed locations and powers of the sources are: 1 MW at 1.5 km, 2 MW at 8 km, and 5 MW (when time averaged across the 2-phase run-plan) at 20 km. The LBNE flux files used in this discussion are [@MaryPrivate]:
- dusel120e250i002dr280dz1300km\_flux.txt (neutrino flux)
- dusel120e250ni002dr280dz1300km\_flux.txt (antineutrino flux)
These files are for an 120-GeV proton-on-target, on-axis, NuMI-like beam with a 280-m decay.
Refs. [@EOI], [@firstpaper] and [@Barger] describe the assumptions related to signal events and backgrounds in the water Cerenkov detector for both the DAE$\delta$ALUS and LBNE running. We assume a 67% reconstruction efficiency for DAE$\delta$ALUS inverse beta decay signal events in a WCGd. We assume a 15% reconstruction efficiency for LBNE charged current quasielastic signal events in either WC or WCGd. DAE$\delta$ALUS event backgrounds arise from beam-off sources, which can be measured during beam-off, and from intrinsic $\bar \nu_e$ in the beam, which is measured by the near accelerator. The systematics on these backgrounds arise from the statistical error on the measurements. LBNE event backgrounds arise from neutral current (NC) misidentification and intrinsic electron-flavor neutrinos in the beam. In the WC detector, we assume a 10% error on the LBNE NC and intrinsic backgrounds.
In this study, we also consider an LAr detector option. Because this target has no free protons, there is a low interaction rate for a DAR beam, and this detector does not contribute to the DAE$\delta$ALUS event sample. However, this detector can efficiently observe interactions at LBNE beam energies. To address the fact that the LAr detector has better signal to background for NC mis-identification, we scale the WC mis-identification background for each LAr unit. The assumption is that the overall efficiency for electron neutrinos or antineutrinos for LAr is 90%, which is 6 times higher than WC. The efficiency for accepting neutral current $\pi^0$ background is unchanged, so that this effectively gives a reduction factor for neutral current $\pi^0$ background of 6 (to 17% of the WC). These scale factors are consistent with the present assumptions of the LBNE collaboration [@private] and other studies [@T2KLAr; @T2KProp]. To be specific, for this study, we assume:
- 3$\times$WC = 1.0\*background,
- 2$\times$WC + 1$\times$LAr = 0.72\*background,
- 1$\times$WC + 2$\times$LAr = 0.44\*background,
- 3$\times$LAr = 0.17\*background,
where “background” refers to the mis-identification background for signal events produced by the LBNE beam.
As a simple benchmark to compare scenarios, we choose a point in\
$(\sin^{2} 2\theta_{13}, \delta_{CP})$ space and report the $1\sigma$ error on $\delta_{CP}$ for each configuration. We have chosen $(\sin^{2} 2\theta_{13}=0.05, \delta _{CP}=-90^{\circ})$. A comparison of ten scenarios is made in Table 1, ranked according to increasing sensitivity (1 is worst, 10 is best).
[|l|c| c|c|c|]{}Rank & Source & Configuration & 1$\sigma$ error& Comment\
1.& DAE$\delta$ALUS alone & 1$\times$WCGd & 34$^\circ$ & Worst\
2.&LBNE alone & 3$\times$WC & 25$^\circ$ &\
3.&LBNE alone & 2$\times$WC+1$\times$LAr & 24$^\circ$ &\
4.&LBNE alone & 1$\times$WC+2$\times$LAr & 23$^\circ$ &\
5.& DAE$\delta$ALUS alone & 2$\times$WCGd & 22$^\circ$ &\
6.& DAE$\delta$ALUS + LBNE & 1$\times$WCGd+2$\times$WC & 18$^\circ$ &\
7.& DAE$\delta$ALUS alone & 3$\times$WCGd & 17$^\circ$ &\
8.& DAE$\delta$ALUS + LBNE & 2$\times$WCGd+1$\times$WC & 15$^\circ$ &\
9.& DAE$\delta$ALUS + LBNE & 2$\times$WCGd+1$\times$LAr & 15$^\circ$ &\
10.& DAE$\delta$ALUS + LBNE & 3$\times$WCGd & 13$^\circ$ & Best\
In the case of running DAE$\delta$ALUS alone, only the number of units of WCGd are relevant, because the WC and LAr units are not sensitive to DAE$\delta$ALUS events. Table 1 shows that running DAE$\delta$ALUS alone with only 1 WCGd unit has poorest sensitivity (rank=1). As this configuration is statistics limited, adding units of WCGd immediately increases DAE$\delta$ALUS’ sensitivity. Two units of WCGd are already better than running LBNE alone (rank= 5) and 3 WCGd units is best for DAE$\delta$ALUS alone (rank=7). The 3 WCGd unit is also the best for the combined beams (rank=10).
The sensitivities for running LBNE alone are very similar for the various configurations (ranks = 2-4). This is simply a statement that the LAr sensitivity is designed to match the WC sensitivity by balancing target size against efficiency. Three configurations of WC and LAr units are provided. Note that WC and WCGd are equivalent in the case of LBNE. This is because the LBNE beam energy is $>100$ MeV; in this range neutron capture is no longer a relevant tag.
The best sensitivities arise when the two beam sources are combined with at least 200 kt of Gd-doped water Cerenkov detector. For a discussion of how the beams are complementary, resulting in this substantial improvement in sensitivity, see refs. [@EOI] and [@Agarwalla]. For the combined source running, the case of 1 WCGd module and 2 WC modules (rank=6) is significantly worse than the cases of 2 WCGd modules and 1 WC or 1 LAr (ranks =8 and 9). As discussed above, the best sensitivity arises from 300 kt of Gd-doped water Cerenkov detector (rank=10).
The conclusions of Table 1 follow for all values of $ \delta_{CP}$. To see this, we provide plots which show the comparisons across all values of $ \delta_{CP}$, for $\sin^2 2\theta_{13}=0.05$, in Figures 1 to 4.
Figure 5 shows the fraction of non-zero (and non-180$^\circ$) $\delta_{CP}$-space which can be sampled at 3$\sigma$. For simplicity, only the designs which ranked 6, 9 and 10 in Table 1 are shown. This allows comparison for the best cases for 100 kt, 200 kt and 300 kt WCGd. The Project X expectation is also shown. This assumes a 300 kt set of WC detectors with the LBNE conventional beam for $10^{22}$ POT in neutrino mode and $10^{22}$ POT in antineutrino mode. Both the 200 kt and 300 kt designs, paired with the combined DAE$\delta$ALUS-plus-LBNE-$\nu$-only neutrino source, exceed the Project X expectation [@EOI; @Agarwalla].
In conclusion, the reach of the combined LBNE and DAE$\delta$ALUS beams is outstanding, when at least 200 kt of Gd-doped water Cerenkov detector is included in the plan. The 300 kt design provides the best sensitivity.
[{width="5.5in"}]{}
[{width="5.5in"}]{}
[{width="5.5in"}]{}
[{width="5.5in"}]{}
[{width="5.5in"}]{}
[99]{}
The Super-Kamiokande Collaboration, arXiv:0811.0735.
G. Rameika, Private Communication, 2009.
The DAE$\delta$ALUS Collaboration, http://arxiv.org/abs/1006.0260.
J.M. Conrad and M.H. Shaevitz, Phys. Rev. Lett. 104:141802, 2010.
P. Huber, M. Lindner, T. Schwetz and W. Winter, JHEP [**0911**]{}, 044, 2009.
M. Bashai, Private Communication, 2009.
V. Barger, [*et al.*]{}, http://arxiv.org/abs/0705.4396, 2007.
From LBNE studies by B, Fleming and J. Kopp, private communication, 2010.
A. Meregaglia, Nuclear Physics B159, 101, 2006.
E. Kearns, [*et al.*]{}, IN2P3-00409176, v1, 2009.
S. Agarwall, [*et al.*]{}, http://arxiv.org/abs/1005.4055, 2010.
|
---
abstract: 'This paper is devoted to investigate the exact solutions of Bianchi type $I$ spacetime in the context of $f(R,T)$ gravity [@fRT1], where $f(R,T)$ is an arbitrary function of Ricci scalar $R$ and trace of the energy momentum tensor $T$. For this purpose, we find two exact solutions using the assumption of constant deceleration parameter and variation law of Hubble parameter. The obtained solutions correspond to two different models of the universe. The physical behavior of these models is also discussed.'
author:
- |
M. Farasat Shamir[^1]\
\
Department of Sciences & Humanities,\
National University of Computer & Emerging Sciences,\
Lahore Campus, Pakistan.
title: '**Bianchi Type $I$ Cosmology in $f(R,T)$ Gravity**'
---
[**Keywords:**]{} $f(R,T)$ gravity, Bianchi type $I$, deceleration parameter.\
[**PACS:**]{} 04.50.Kd.
Introduction
============
The most popular issue in the modern day cosmology is the current expansion of universe. It is now evident from observational and theoretical facts that our universe is in the phase of accelerated expansion [@acc1]-[@acc9]. The phenomenon of dark energy and dark matter is another topic of discussion [@de1]-[@de8]. It was Einstein who first gave the concept of dark energy and introduced the small positive cosmological constant. But after sometime, he remarked it as the biggest mistake in his life. However, it is now thought that the cosmological constant may be a suitable candidate for dark energy. Another proposal to justify the current expansion of universe comes from modified or alternative theories of gravity. $f(T)$ theory of gravity is one such example which has been recently developed. This theory is a generalized version of teleparallel gravity in which Weitzenböck connection is used instead of Levi-Civita connection. The interesting feature of the theory is that it may explain the current acceleration without involving dark energy. A considerable amount of work has been done in this theory so far [@ft]. Another interesting modified theory is $f(R)$ theory of gravity which involves a general function of Ricci scalar in standard Einstein-Hilbert Lagrangian. Some review articles [@rev] can be helpful to understand the theory.
Many authors have investigated $f(R)$ gravity in different contexts -[@f(R)gravity; @constrained; @by; @PPN; @parameters; @and; @stochastic; @background; @of; @gravitational; @waves]. Spherically symmetric solutions are most commonly studied solutions due to their closeness to the nature. Multam$\ddot{a}$ki and Vilja [@fr1] explored vacuum and perfect fluid solutions of spherically symmetric spacetime in metric version of this theory. They used the assumption of constant scalar curvature and found that the solutions corresponded to the already existing solutions in general relativity (GR). Noether symmetries have been used by Capozziello et al. [@fr2] to study spherically symmetric solutions in $f(R)$ gravity. Similarly many interesting results have been found using spherical symmetry in $f(R)$ gravity [@fr3]. Cylindrically symmetric vacuum and non-vacuum solutions have also been explored in this theory [@cylndr]. Sharif and Shamir [@me1] found plane symmetric solutions. The same authors [@me2] discussed the solutions of Bianchi types $I$ and $V$ cosmologies for vacuum and non-vacuum cases. Conserved quantities in $f(R)$ gravity via Noether symmetry approach have been recently calculated [@me3].
In a recent paper [@fRT1], Harko et al. proposed a new generalized theory known as $f(R,T)$ gravity. In this theory, gravitational Lagrangian involves an arbitrary function of the scalar curvature $R$ and the trace of the energy momentum tensor $T
$. Myrzakulov [@fRT2] discussed $f(R,T)$ gravity in which he explicitly presented point like Lagrangians. Sharif and Zubair [@fRT5] studied the laws of thermodynamics in this theory. The same authors [@fRT45] investigated holographic and agegraphic $f(R,T)$ models. Houndjo [@fRT4] reconstructed $f(R,T)$ gravity by taking $f(R,T)=f_1(R)+f_2(T)$ and it was proved that $f(R,T)$ gravity allowed transition of matter from dominated phase to an acceleration phase. Thus it is hoped that $f(R,T)$ gravity may explain the resent phase of cosmic acceleration of our universe. This theory can be used to explore many issues and may provide some satisfactory results.
The isotropic models are considered to be most suitable to study large scale structure of the universe. However, it is believed that the early universe may not have been exactly uniform. This prediction motivates us to describe the early stages of the universe with the models having anisotropic background. Thus, the existence of anisotropy in early phases of the universe is an interesting phenomenon to investigate. A Bianchi Type $I$ cosmological model, being the generalization of flat Friedmann-Robertson-Walker (FRW) model, is one of the simplest models of the anisotropic universe. Therefore, it seems interesting to explore Bianchi type models in the context of $f(R,T)$ gravity. Adhav [@fRT3] investigated the exact solutions of $f(R,T)$ field equations for locally rotationally symmetric Bianchi type $I$ spacetime. Reddy et al. [@fRT46] explored the solutions of Bianchi type $III$ spacetime using the law of variation of Hubble’s parameter. Bianchi type $III$ dark energy model in the presence of perfect fluid source has been reported [@fRT47]. Ahmed and Pradhan [@fRT48] studied Bianchi Type $V$ cosmology in this theory by involving the cosmological constant in the field equations. Naidu et al. [@fRT49] gave the solutions of Bianchi type $V$ bulk viscous string cosmological model.
In this paper, we are focussed to investigate the exact solutions of Bianchi type $I$ spacetime in the framework of $f(R,T)$ gravity. The plan of paper is as follows: In section **2**, we give some basics of $f(R,T)$ gravity. Section **3** provides the exact solutions for Bianchi type $I$ spacetime. Concluding remarks are given in the last section.
Some Basics of $f(R,T)$ Gravity
===============================
The $f(R,T)$ theory of gravity is the generalization or modification of GR. The action for this theory is given by [@fRT1] $$\label{frt1}
S=\int\sqrt{-g}\bigg(\frac{1}{16\pi{G}}f(R,T)+L_{m}\bigg)d^4x,$$ where $f(R,T)$ is an arbitrary function of the Ricci scalar $R$ and the trace $T$ of energy momentum tensor $T_{\mu\nu}$ while $L_{m}$ is the usual matter Lagrangian. It is worth mentioning that if we replace $f(R,T)$ with $f(R)$, we get the action for $f(R)$ gravity and replacement of $f(R,T)$ with $R$ leads to the action of GR. The energy momentum tensor $T_{\mu\nu}$ is defined as [@emt] $$\label{frt2}
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_m)}{\delta
g^{\mu\nu}}.$$ Here we assume that the dependance of matter Lagrangian is merely on the metric tensor $g_{\mu\nu}$ rather than its derivatives. In this case, we get $$\label{frt3}
T_{\mu\nu}=L_m g_{\mu\nu}-2\frac{\delta L_m}{\delta g^{\mu\nu}}.$$ The $f(R,T)$ gravity field equations are obtained by varying the action $S$ in Eq.(\[frt1\]) with respect to the metric tensor $g_{\mu\nu}$ $$\label{frt4}
f_R(R,T)R_{\mu\nu}-\frac{1}{2}f(R,T)g_{\mu\nu}-(\nabla_{\mu}
\nabla_{\nu}-g_{\mu\nu}\Box)f_R(R,T)=\kappa
T_{\mu\nu}-f_T(R,T)(T_{\mu\nu}+\Theta_{\mu\nu}),$$ where $\nabla_{\mu}$ denotes the covariant derivative and $$\Box\equiv\nabla^{\mu}\nabla_{\mu},~~ f_R(R,T)=\frac{\partial
f_R(R,T)}{\partial R},~~ f_T(R,T)=\frac{\partial
f_R(R,T)}{\partial
T},~~\Theta_{\mu\nu}=g^{\alpha\beta}\frac{\delta
T_{\alpha\beta}}{\delta g^{\mu\nu}}.$$ Contraction of (\[frt4\]) yields $$\label{frt04}
f_R(R,T)R+3\Box f_R(R,T)-2f(R,T)=\kappa T-f_T(R,T)(T+\Theta),$$ where $\Theta={\Theta_\mu}^\mu$. This is an important equation because it provides a relationship between Ricci scalar $R$ and the trace $T$ of energy momentum tensor. Using matter Lagrangian $L_m$, the standard matter energy-momentum tensor is derived as $$\label{frt5}
T_{\mu\nu}=(\rho + p)u_\mu u_\nu-pg_{\mu\nu},$$ where $u_\mu=\sqrt{g_{00}}(1,0,0,0)$ is the four-velocity in co-moving coordinates and $\rho$ and $p$ denote energy density and pressure of the fluid respectively. Perfect fluids problems involving energy density and pressure are not any easy task to deal with. Moreover, there does not exist any unique definition for matter Lagrangian. Thus we can assume the matter Lagrangian as $L_m=-p$ which gives $$\label{frt6}
\Theta_{\mu\nu}=-pg_{\mu\nu}-2T_{\mu\nu},$$ and consequently the field equations (\[frt4\]) take the form $$\label{frt7}
f_R(R,T)R_{\mu\nu}-\frac{1}{2}f(R,T)g_{\mu\nu}-(\nabla_{\mu}
\nabla_{\nu}-g_{\mu\nu}\Box)f_R(R,T)=\kappa
T_{\mu\nu}+f_T(R,T)(T_{\mu\nu}+pg_{\mu\nu}),$$ It is mentioned here that these field equations depend on the physical nature of matter field. Many theoretical models corresponding to different matter contributions for $f(R,T)$ gravity are possible. However, Harko et al. [@fRT1] gave three classes of these models $$f(R,T)= \left\lbrace
\begin{array}{c l}
{R+2f(T),}\\
{f_1(R)+f_2(T),}\\{f_1(R)+f_2(R)f_3(T).}
\end{array}
\right.$$\
In this paper we are focussed to the first class, i.e. $f(R,T)=R+2f(T)$. For this model the field equations become $$\label{frt8}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa
T_{\mu\nu}+2f'(T)T_{\mu\nu}+\bigg[f(T)+2pf'(T)\bigg]g_{\mu\nu},$$ where prime represents derivative with respect to $T$.
Exact Solutions of Bianchi Type $I$ Universe
============================================
In this section, we shall find exact solutions of Bianchi I spacetime in $f(R,T)$ gravity. For the sake of simplicity, we use natural system of units $(G=c=1)$ and $f(T)=\lambda T$, where $\lambda$ is an arbitrary constant. For Bianchi type $I$ spacetime, the line element is given by $$\label{6}
ds^{2}=dt^2-A^2(t)dx^2-B^2(t)dy^2-C^2(t)dz^2,$$ where $A,~B$ and $C$ are defined as cosmic scale factors. The Bianchi $I$ Ricci scalar turns out to be $$\label{7}
R=-2\bigg[\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}
+\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}\dot{C}}{BC}+\frac{\dot{C}\dot{A}}{CA}\bigg],$$ where dot denotes derivative with respect to $t$.
Using Eq.(\[frt8\]), we get four independent field equations, $$\begin{aligned}
\label{11}
\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}\dot{C}}{BC}+
\frac{\dot{C}\dot{A}}{CA}=(8\pi+3\lambda)\rho-\lambda
p,\\\label{12} \frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}
+\frac{\dot{B}\dot{C}}{BC}=\lambda
\rho-(8\pi+3\lambda)p,\\\label{13}
\frac{\ddot{C}}{C}+\frac{\ddot{A}}{A}
+\frac{\dot{C}\dot{A}}{AC}=\lambda
\rho-(8\pi+3\lambda)p,\\\label{14}
\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}
+\frac{\dot{A}\dot{B}}{AB}=\lambda \rho-(8\pi+3\lambda)p.\end{aligned}$$ These are four non-linear differential equations with five unknowns namely $A,~B,~C$, $\rho$ and $p$. Subtracting Eq.(\[13\]) from Eq.(\[12\]), Eq.(\[14\]) from Eq.(\[13\]) and Eq.(\[14\]) from Eq.(\[11\]), we get respectively $$\begin{aligned}
\label{015}
\frac{\ddot{A}}{A}-\frac{\ddot{B}}{B}
+\frac{\dot{C}}{C}\bigg(\frac{\dot{A}}{A}-\frac{\dot{B}}{B}\bigg)=0,\\\label{016}
\frac{\ddot{B}}{B}-\frac{\ddot{C}}{C}
+\frac{\dot{A}}{A}\bigg(\frac{\dot{B}}{B}-\frac{\dot{C}}{C}\bigg)=0,\\\label{017}
\frac{\ddot{A}}{A}-\frac{\ddot{C}}{C}
+\frac{\dot{B}}{B}\bigg(\frac{\dot{A}}{A}-\frac{\dot{C}}{C}\bigg)=0.\end{aligned}$$ These equations imply that $$\begin{aligned}
\label{15}
\frac{B}{A}=d_1\exp\bigg[{c_1\int\frac{dt}{a^3}}\bigg],\\\label{16}
\frac{C}{B}=d_2\exp\bigg[{c_2\int\frac{dt}{a^3}}\bigg],\\\label{17}
\frac{A}{C}=d_3\exp\bigg[{c_3\int\frac{dt}{a^3}}\bigg],\end{aligned}$$ where $c_1,~c_2,~c_3$ and $d_1,~d_2,~d_3$ are integration constants which satisfy the following relation $$\label{18}
c_1+c_2+c_3=0,\quad d_1d_2d_3=1.$$ Using Eqs.(\[15\])-(\[17\]), we can write the unknown metric functions in an explicit way $$\begin{aligned}
\label{19}
A=ap_1\exp\bigg[{q_1\int\frac{dt}{a^3}}\bigg],\\\label{20}
B=ap_2\exp\bigg[{q_2\int\frac{dt}{a^3}}\bigg],\\\label{21}
C=ap_3\exp\bigg[{q_3\int\frac{dt}{a^3}}\bigg],\end{aligned}$$ where $$\label{22}
p_1=({d_1}^{-2}{d_2}^{-1})^{\frac{1}{3}},\quad
p_2=(d_1{d_2}^{-1})^{\frac{1}{3}},\quad
p_3=(d_1{d_2}^2)^{\frac{1}{3}}$$ and $$\label{23}
q_1=-\frac{2c_1+c_2}{3},\quad q_2=\frac{c_1-c_2}{3},\quad
q_3=\frac{c_1+2c_2}{3}.$$ It is mentioned here that $p_1,~p_2,~p_3$ and $q_1,~q_2,~q_3$ also satisfy the relation $$\label{24}
p_1p_2p_3=1,\quad q_1+q_2+q_3=0.$$
Some Important Physical Parameters
----------------------------------
Now we present some important definitions of physical parameters. The average scale factor $a$ and volume scale factor $V$ are defined as $$\label{8}
a=\sqrt[3]{ABC}, \quad V=a^3=ABC.$$ The generalized mean Hubble parameter $H$ is given by $$\label{008}
H=\frac{1}{3}(H_1+H_2+H_3),$$ where $H_1=\frac{\dot{A}}{A},~H_2=\frac{\dot{B}}{B},~H_3=\frac{\dot{C}}{C}$ are defined as the directional Hubble parameters in the directions of $x,~y$ and $z$ axis respectively. The mean anisotropy parameter $A$ is $$\label{0000009}
A=\frac{1}{3}\sum^3_{i=1}\bigg(\frac{H_i-H}{H}\bigg)^2.$$ The expansion scalar $\theta$ and shear scalar $\sigma^2$ are defined as follows $$\begin{aligned}
\label{09}
\theta&=&u^\mu_{;\mu}=\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C},\\
\label{00009} \sigma^2&=&\frac{1}{2}\sigma_{\mu\nu}\sigma^{\mu\nu}
=\frac{1}{3}\bigg[\bigg(\frac{\dot{A}}{A}\bigg)^2+\bigg(\frac{\dot{B}}{B}\bigg)^2
+\bigg(\frac{\dot{C}}{C}\bigg)^2-\frac{\dot{A}\dot{B}}{AB}-\frac{\dot{B}\dot{C}}{BC}
-\frac{\dot{C}\dot{A}}{CA}\bigg],~~\end{aligned}$$ where $$\label{009}
\sigma_{\mu\nu}=\frac{1}{2}(u_{\mu;\alpha}h^\alpha_\nu+u_{\nu;\alpha}h^\alpha_\mu)
-\frac{1}{3}\theta h_{\mu\nu}$$ with $h_{\mu\nu}=g_{\mu\nu}-u_{\mu}u_{\nu}$ defined as the projection tensor. The deceleration parameter $q$ is the measure of the cosmic accelerated expansion of the universe. It is defined as $$\label{26}
q=-\frac{\ddot{a}a}{\dot{a}^2}.$$ The behavior of the universe models is determined by the sign of $q$. The positive value of deceleration parameter suggests a decelerating model while the negative value indicates inflation. Since there are four equations (\[11\])-(\[14\]) and five unknowns, so we need an additional constraint to solve them. Here we use a well-known relation [@15] between the average scale factor $a$ and Hubble parameter $H$ to solve the equations, $$\label{27}
H=la^{-n},$$ where $l$ and $n$ are positive constants. This is an important relation because it yields a constant value of deceleration parameter and consequently we obtain power law and exponential models of universe. Using Eqs.(\[008\]) and (\[27\]), we get $$\label{28}
\dot{a}=la^{1-n}$$ and the deceleration parameter becomes $$\label{29}
q=n-1.$$ Integrating Eq.(\[28\]), it follows that $$\label{30}
a=(nlt+k_1)^{\frac{1}{n}},\quad n\neq0,$$ and $$\label{31}
a=k_2\exp(lt),\quad ~~~n=0,$$ where $k_1$ and $k_2$ are constants of integration.
Singular Model of the Universe
------------------------------
Here we investigate the model of universe when $n\neq0$, i.e., $a=(nlt+k_1)^{\frac{1}{n}}$. In this case, the metric coefficients $A,~B$ and $C$ take the form $$\begin{aligned}
\label{35}
A&=&p_1(nlt+k_1)^{\frac{1}{n}}\exp\bigg[\frac{q_1(nlt+k_1)^
{\frac{n-3}{n}}}{l(n-3)}\bigg],\quad n\neq3,\\\label{36}
B&=&p_2(nlt+k_1)^{\frac{1}{n}}\exp\bigg[\frac{q_2(nlt+k_1)^
{\frac{n-3}{n}}}{l(n-3)}\bigg],\quad n\neq3,\\\label{37}
C&=&p_3(nlt+k_1)^{\frac{1}{n}}\exp\bigg[\frac{q_3(nlt+k_1)^
{\frac{n-3}{n}}}{l(n-3)}\bigg],\quad n\neq3.\end{aligned}$$ The directional Hubble parameters $H_i$ ($i=1,2,3$) turn out to be $$\label{38}
H_i=\frac{l}{nlt+k_1}+\frac{q_i}{(nlt+k_1)^{\frac{3}{n}}}.$$ The mean generalized Hubble parameter and volume scale factor are $$\label{39}
H=\frac{l}{nlt+k_1},\quad V=(nlt+k_1)^\frac{3}{n}.$$ The mean anisotropy parameter becomes $$\label{3929}
A=\frac{{q_1}^2+{q_2}^2+{q_3}^2}{3l^2(nlt+k_1)^{(6-2n)/n}}.$$ The expansion scalar and shear scalar for this model are given by $$\theta=\frac{3l}{nlt+k_1},\quad\sigma^2=\frac{{q_1}^2+{q_2}^2+{q_3}^2}{2(nlt+k_1)^{6/n}}.$$ Using Eqs. (\[11\])-(\[14\]), the energy density of the universe turns out to be $$\begin{aligned}
\nonumber
\rho=&&\frac{1}{12(\lambda+2\pi)(\lambda+4\pi)}\bigg[4(\lambda+3\pi)
\bigg\{\frac{3l^2}{(nlt+k_1)^2}+\frac{q_1q_2+q_2q_3+q_3q_1}{(nlt+k_1)^{\frac{6}{n}}}\bigg\}\\\label{ro1}
&-&\lambda
\bigg\{\frac{3l^2(1-n)}{(nlt+k_1)^2}+\frac{{q_1}^2+{q_2}^2+{q_3}^2}{(nlt+k_1)^{\frac{6}{n}}}\bigg\}\bigg]\end{aligned}$$ while the pressure of the universe becomes $$\begin{aligned}
\nonumber
p=&&\frac{-1}{12(\lambda+2\pi)(\lambda+4\pi)}\bigg[4\pi
\bigg\{\frac{3l^2}{(nlt+k_1)^2}+\frac{q_1q_2+q_2q_3+q_3q_1}{(nlt+k_1)^{\frac{6}{n}}}\bigg\}\\\label{p1}
&+&(3\lambda+8\pi)\bigg\{\frac{3l^2(1-n)}{(nlt+k_1)^2}+\frac{{q_1}^2+{q_2}^2+{q_3}^2}{(nlt+k_1)^{\frac{6}{n}}}\bigg\}\bigg].\end{aligned}$$ The plots of $\rho$, $p$ and equation of state parameter $w=p/\rho$ against time coordinate $t$ are shown in figure $1$ and $2$ respectively. It is evident from figure $2$ that $w\rightarrow \frac{1}{3}$ as $t\rightarrow\infty$. Thus the model corresponds to a radiation dominated universe as the time grows.
-- -- -- --
-- -- -- --
{width="2.8in" height="1.8in"}
Non-singular Model of the Universe
----------------------------------
For this model, $n=0$ and the average scale factor $a=k_2\exp(lt)$ turns the metric coefficients $A,~B$ and $C$ into $$\begin{aligned}
\label{43}
A&=&p_1k_2\exp(lt)\exp\bigg[-\frac{q_1\exp(-3lt)}{3l{k_2}^3}\bigg],\\\label{44}
B&=&p_2k_2\exp(lt)\exp\bigg[-\frac{q_2\exp(-3lt)}{3l{k_2}^3}\bigg],\\\label{37}
C&=&p_3k_2\exp(lt)\exp\bigg[-\frac{q_3\exp(-3lt)}{3l{k_2}^3}\bigg].\end{aligned}$$ The directional Hubble parameters $H_i$ become $$\label{44}
H_i=l+\frac{q_i}{{k_2}^3}\exp(-3lt) .$$ The mean generalized Hubble parameter and volume scale factor turn out to be $$\label{46}
H=l,\quad V={k_2}^3\exp(3lt).$$
-- -- -- --
-- -- -- --
The mean anisotropy parameter, expansion scalar and shear scalar are $$A=\frac{{q_1}^2+{q_2}^2+{q_3}^2}{3l^2{k_2}^6\exp(6lt)},~~
\theta=3l,\quad
\sigma^2=\frac{{q_1}^2+{q_2}^2+{q_3}^2}{2{k_2}^6\exp(6lt)}.$$ The energy density and pressure of the universe take the form $$\begin{aligned}
\nonumber
\rho=&&\frac{1}{12(\lambda+2\pi)(\lambda+4\pi)}\bigg[4(\lambda+3\pi)
\bigg\{3l^2+\frac{q_1q_2+q_2q_3+q_3q_1}{{k_2}^6\exp(6lt)}\bigg\}\\\label{ro2}
&-&\lambda\bigg\{3l^2+\frac{{q_1}^2+{q_2}^2+{q_3}^2}{{k_2}^6\exp(6lt)}\bigg\}\bigg],\end{aligned}$$ $$\begin{aligned}
\nonumber
p=&&\frac{-1}{12(\lambda+2\pi)(\lambda+4\pi)}\bigg[4\pi
\bigg\{3l^2+\frac{q_1q_2+q_2q_3+q_3q_1}{{k_2}^6\exp(6lt)}\bigg\}\\\label{p2}
&+&(3\lambda+8\pi)\bigg\{3l^2+\frac{{q_1}^2+{q_2}^2+{q_3}^2}{{k_2}^6\exp(6lt)}\bigg\}\bigg].\end{aligned}$$ For this model, the plots of $\rho$, $P$ and $w$ against time coordinate $t$ are shown in figure $3$ and $4$ respectively. It can be seen from figure $4$ that $w\rightarrow -1$ as $t\rightarrow\infty$ which indicates that the non-singular model corresponds to a vacuum fluid dominated universe.
{width="2.8in" height="1.8in"}
Concluding Remarks
==================
This paper is devoted to discuss the current phenomenon of accelerated expansion of universe in the framework of newly proposed $f(R,T)$ theory of gravity. For this purpose, we take $f(R,T)=R+2\lambda T$ and explore the exact solutions of Bianchi type $I$ cosmological model. We obtain two exact solutions using the assumption of constant value of deceleration parameter and the law of variation of Hubble parameter. The obtained solutions correspond to two different models of universe. The first solution forms a singular model with power law expansion while the second solution gives a non-singular model with exponential expansion of universe. The physical parameters for both of these models are discussed below.
The singular model of the universe corresponds to $n\neq0$ with average scale factor $a=(nlt+k_1)^{\frac{1}{n}}$. This model possesses a point singularity when $t\equiv t_s=-\frac{k_1}{nl}$. The volume scale factor and the metric coefficients $A,~B$ and $C$ vanish at this singularity point. The cosmological parameters $H_1,~H_2,~H_3,~H,~ \theta$, and $\sigma^2$ are all infinite at this point of singularity. If we choose $k_1=0$, figure $1$ suggests that energy density of the universe is zero at this time. The pressure approaches negative infinity as $t\rightarrow 0$. This strong negative pressure is an indication of dark energy. For this model, $w\rightarrow \frac{1}{3}$ as $t\rightarrow\infty$ which corresponds to a radiation dominated universe. The mean anisotropy parameter $A$ also becomes infinite at this point for $0<n<3$ and vanishes for $n>3$. Moreover, the isotropy condition, i.e., $\frac{\sigma^2}{\theta}\rightarrow 0$ as $t\rightarrow \infty$, is verified for this model. All these conclusive observations suggest that the universe starts its expansion with zero volume, strong negative pressure and energy density from $t=t_s$ and it will continue to expand for $0<n<3$.
Now we discuss the non-singular model of the universe corresponds to $n=0$. For this model the average scale factor is $a=k_2\exp(lt)$. The non-singularity is due to the exponential behavior of the model. The expansion scalar $\theta$ and mean generalized Hubble parameter $H$ are constant in this case. For finite values of $t$, the physical parameters $H_1,~H_2,~H_3,~\sigma^2$ and $A$ are all finite. The metric functions are defined for finite time and the isotropy condition is satisfied. There is an exponential increase in the volume as the time grows. However, energy density is approximately zero initially and becomes constant after some time. Pressure of the universe remains in the negative zone for this model which may be an indication of presence of dark energy in the universe. Figure $4$ suggests that $w\rightarrow -1$ as $t\rightarrow\infty$. Thus the exponential model corresponds to a vacuum fluid dominated universe. According to the observations [@nature], the expansion of the universe is accelerating when $w\approx-1$.
Therefore, it is hoped that the problematic issues such as dark energy and accelerated expansion of universe may be addressed using modified theories of gravity especially $f(R,
T)$ gravity. It would be interesting to explore more Bianchi type solutions in this context. Exact solutions of Bianchi type $V$ cosmological model in this theory are under process.\
\
**Acknowledgement**\
\
The author is thankful to National University of Computer and Emerging Sciences (NUCES) Lahore Campus for funding the PhD programme. The author is also grateful to the anonymous reviewer for valuable comments and suggestions to improve the paper.
[40]{}
Harko, T., Lobo, F.S.N., Nojiri, S. and Odintsov, S.D.: Phys. Rev. **D84**(2011)024020.
Riess, A.G. et al.: Astron. J. **116**(1998)1009; Riess, A.G. et al.: Astrophys. J. **607**(2004)665.
Perlmutter, S. et al.: Astrophys. J. **517**(1999)565.
Carmeli, M.: Commun. Theor. Phys. **5**(1996)159.
Seprgel, D. N. et al.: Astrophys. J. Suppl. **148**(2003)175; Spergel, D.N. et al.: Astrophys. J. Suppl. **170**(2007)377.
Allen, S. W. et al.: Mon. Not. R. Astron. Soc. **353**(2004)457.
Bennett, C. L. et al.: Astrophys. J. **148**(2003)1.
Hogan, J.: Nature **448**(2007)240.
Tegmark, M. et al.: Phys. Rev. **D69**(2004)103501
Abazajian, K. et al.: Astron. J. **129**(2005)1755.
Astier, P. et al.: Astron. Astrophys. **447**(2006)31.
Nojiri, S. and Odintsov, S.D.: Int. J. Geom. Meth. Mod. Phys. **4**(2007)115.
Turner, M.S., Huterer, D.: J. Phys. Soc. Jap. **76**(2007)111015; Frieman, J., Turner, M. and Huterer, D.: Ann. Rev. Astron. Astrophys. **46**(2008)385.
Li, M., Li, X.D., Wang, S. and Wang, Y.: Commun. Theor. Phys. **56**(2011)525.
Copeland, E.J., Sami, M. and Tsujikawa, S.: Int. J. Mod. Phys. **D15**(2006)1753.
Sahni, V. and Starobinsky, A.: Int. J. Mod. Phys. **D9** (2000) 373; Sahni, V.: Lect. Notes. Phys. **653** (2004) 141.
Carroll, S.M.: Living Rev. Rel. **4**(2001)1.
Weinberg, D.H.: New. Astron. Rev. **49**(2005)337.
Straumann, N.: Mod. Phys. Lett. **A21**(2006)1083.
Yang, R.J.: Europhys. Lett. **93**(2011)60001; Wei, H., Ma, X.P. and Qi, H.Y.: Phys. Lett. **B703**(2011)74; Wu, P.X. and Yu, H.W.: Eur. Phys. J. **C71**(2011)1552; Wu, P.X. and Yu, H.W.: Phys. Lett. **B703**(2011)223; Bamba, K., Geng, C.Q., Lee, C.C and Luo, L.W.: JCAP **1101**(2011)021; Li, B., Sotiriou, T.P. and Barrow, J.D.: Phys. Rev. **D83**(2011)104017.
Felice, A.D and Tsujikawa, S.: Living Rev. Rel. **13**(2010)3; Sotiriou, T.P. and Faraoni, V.: Rev. Mod. Phys. **82**(2010)451; Clifton, T., Ferreira, P.G., Padilla, A. and Skordis, C.: Phys. Rept. **513** (2012)1; Nojiri, S. and Odintsov, S.D.: Phys. Rept. **505**(2011)59. Bamba, K., Capozziello, S., Nojiri, S. and Odintsov, S.D.: Astrophys. Space Sci. **342**(2012)155.
Sharif, M. and Zubair, M.: Adv. High Energy Phys. **2013**(2013)790967.
Sharif, M. and Kausar, H.R.: JCAP **07**(2011)022.
Sharif, M. and Kausar, H.R.: Int. J. Mod. Phys. **D20**(2011)2239.
Sharif, M. and Kausar, H.R.: Astrophys. Space Sci. **331**(2011)281.
Sharif, M. and Kausar, H.R.: Mod. Phys. Lett. **A25**(2010)3299.
Bamba, K., Nojiri, S., Odintsov, S.D. and Saez-Gomez, D.: Phys. Lett. **B730**(2014)136.
Bamba, K., Makarenko, A.N., Myagky, A.N., Nojiri, S. and Odintsov, S.D.: JCAP **01**(2014)008.
Bamba, K., Nojiri, S. and Odintsov, S.D.: Phys. Lett. **B698**(2011)451.
Capozziello, S. and Vignolo, S.: Int. J. Geom. Meth. Mod. Phys. **8**(2011)167.
Capozziello, S., Darabi, F. and Vernieri, D.: Mod. Phys. Lett. **A26**(2011)65.
Capozziello, S., Laurentis, M.D., Odintsov, S.D. and Stabile, A.: Phys. Rev. **D83**(2011)064004.
Elizalde, E., Nojiri, S., Odintsov, S.D. and Saez-Gomez, D.: Eur. Phys. J. **C70**(2010)351.
Bamba, K., Geng, C., Nojiri, S. and Odintsov, S.D.: Mod. Phys. Lett. **A25**(2010)900.
Capozziello, S., Laurentis, M.D., Nojiri, S. and Odintsov, S.D.: Gen. Rel. Grav. **41**(2009)2313.
Multam$\ddot{a}$ki, T. and Vilja, I.: Phys. Rev. **D74**(2006)064022; Multam$\ddot{a}$ki, T. and Vilja, I.: Phys. Rev. **D76**(2007)064021.
Capozziello, S., Stabile, A. and Troisi, A.: Class. Quantum Grav. **24**(2007)2153.
Hollenstein, L. and Lobo, F.S.N.: Phys. Rev. **D78**(2008)124007; Sharif, M. and Kausar, H.R.: J. Phys. Soc. Jpn. **80**(2011)044004; Shojai, A. and Shojai, F.: Gen. Relativ. Gravit. **44**(2011)211.
Azadi, A., Momeni, D. and Nouri-Zonoz, M.: Phys. Lett. **B670**(2008)210; Momeni, D. and Gholizade, H.: Int. J. Mod. Phys. **D18**(2009)1719; Sharif, M. and Arif, S.: Astrophys. Space Sci. **342**(2012)237.
Sharif, M. and Shamir, M.F.: Mod. Phys. Lett. **A25**(2010)1281.
Sharif, M. and Shamir, M.F.: Class. Quantum Grav. **26**(2009)235020; Sharif, M. and Shamir, M.F.: Gen. Relativ. Gravit. **42**(2010)2643.
Shamir, M.F., Jhangeer, A. and Bhatti, A.A.: Chin. Phys. Lett. **29(8)**(2012)080402.
Myrzakulov, R.: Phys. Rev. **D84**(2011)024020.
Sharif, M. and Zubair, M.: JCAP **03**(2012)028.
Sharif, M. and Zubair, M.: J. Phys. Soc. Jpn. **82**(2013)064001.
Houndjo, M.J.S.: Int. J. Mod. Phys. **D21**(2012)1250003.
Adhav, K.S.: Astrophys. Space Sci. **339**(2012)365.
Reddy, D.R.K., Santikumar, R. and Naidu, R.L.: Astrophys. Space Sci. **342**(2012)249.
Reddy, D.R.K., Santikumar, R. and Pradeepkumar, T.V.: Int. J. Theor. Phys. **52**(2013)239.
Naidu, R.L., Reddy, D.R.K., Ramprasad, T. and Ramana, K.V.: Astrophys. Space Sci. **348**(2013)247.
Ahmed, N. and Pradhan, A.: Int. J. Theor. Phys. **53**(2014)289.
Landau, L.D. and Lifshitz E.M.: *The Classical Theory of Fileds* (Butterworth-Heinemann, 2002).
Berman, M.S.: Nuov. Cim. **B74**(1983)182; Shamir, M.F. and Bhatti, A.A.: Can. J. Phys. **90**(2012)193; Singh, C.P. and Kumar, S.: Int. J. Theor. Phys. **47**(2008)3171-3179; Singh, C.P., Zeyauddin, M. and Ram, S.: Int. J. Theor. Phys. **47**(2008)3162-3170.
Hogan, J.: Nature **448**(2007)240.
[^1]: [email protected]
|
---
abstract: |
We study the roots of the generalised Hermite polynomials $H_{m,n}$ when both $m$ and $n$ are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular lattice, as was numerically observed by Clarkson. We describe the elliptic region and the deformed lattice in terms of elliptic integrals and their degenerations.
Keywords: Generalised Hermite polynomials; roots asymptotics; Painleve IV; Boutroux Curves; Tritronquee solution.
address:
- 'Grupo de Física Matemática e Departamento da Matemática da Universidade de Lisboa, Campo Grande Edifício C6, 1749-016 Lisboa, Portugal.'
- 'SISSA, via Bonomea 265, 34136 Trieste, Italy.'
author:
- Davide Masoero and Pieter Roffelsen
bibliography:
- 'referencesunited.bib'
title: Roots of generalised Hermite polynomials when both parameters are large
---
[ ]{}
Introduction
============
For $m,n\in\mathbb{N}$, the generalised Hermite polynomial $H_{m,n}$ is the polynomial of degree $m\times n$ defined by the determinantal formula $$H_{m,n}(z)=\gamma_{m,n}\begin{vmatrix}
h_m(z) & h_{m+1}(z) & \ldots & h_{m+n-1}(z)\\
h_m^{(1)}(z) & h_{m+1}^{(1)}(z) & \ldots & h_{m+n-1}^{(1)}(z)\\
\vdots &\vdots & \ddots & \vdots\\
h_m^{(n-1)}(z) & h_{m+1}^{(n-1)}(z) & \ldots & h_{m+n-1}^{(n-1)}(z)\\
\end{vmatrix}$$ where $h_k^{(l)}(z)$ denotes the $l$-th derivative of the $k$-th Hermite polynomial $$h_k(z)=(-1)^ke^{z^2}\frac{\partial^k}{\partial z^k}\left[e^{-z^2}\right]$$ and $\gamma_{m,n}\in\mathbb{C}^*$ is a, for the purpose of this paper, irrelevant constant multiplier.
The goal of the present paper is to study the asymptotic distribution of the roots of $H_{m,n}$ as $m+n \to \infty$. We call the asymptotics unrestricted since, in contrast to our previous paper [@maro18], we do not require any of the two parameters to remain bounded.
The most striking property of the generalised Hermite polynomials is that they yield families of rational solutions to the fourth Painlevé equation $$\label{eq:piv}
{\omega}_{zz}=\displaystyle\frac{1}{2{\omega}}{\omega}_z^2+ \frac{3}{2} {\omega}^3 + 4 z{\omega}^2+2(z^2+1-2\theta_\infty){\omega}-\frac{8\theta_0^2}{{\omega}}
,\quad \theta:=(\theta_0,\theta_\infty) \in \mathbb{C}^2.$$ For example, the functions $\omega_{m,n}^{(\text{I})}=\frac{d}{dz}\log\frac{H_{m+1,n}}{H_{m,n}}$ solve the above equation with parameters $\theta_0=\tfrac{1}{2}n, \theta_\infty=m+\tfrac{1}{2}n+1$ [@noumiyamada]. This establishes an explicit relation among poles of rational solutions of Painleve IV and roots of generalised Hermite polynomials; hence the problem of our interest can be restated as the study of the asymptotic distribution of singularities of the Hermite-family of rational solutions of the Painleve IV equation [@maro18].
We mentioned in our previous paper [@maro18] that the asymptotic analysis of rational solutions of Painlevé equations (and more generally special solutions) has recently been the object of intense study, see e.g. [@costin12; @piwkb]. This is even more the case at the time of writing since in the meanwhile many new results have appeared in the literature [@bothner18; @buckingham18; @vanass18]. There is a clear reason for this: most often when a problem in applied or pure mathematics is solved by means of a solution of a Painlevé equation (see e.g. [@dubrovin2009]), this is singled-out by some special (i.e. non generic) asymptotic expansion, which reflects itself in a special associated Riemann-Hilbert problem amenable to a thorough analysis to a degree not attainable for generic solutions [^1].
The above is clearly the case for generalised Hermite polynomials. In fact in [@maro18] we showed that in the case of the generalised Hermite polynomials the isomonodromic deformation method for Painleve IV simplifies dramatically. To be more precise, we proved the following theorem, which characterises the roots of $H_{m,n}$ by means of an eigenvalue problem for a specific class of anharmonic oscillators.
\[thm:inversemonodromy\] For $m,n\in\mathbb{N}$, the point $a \in {\mathbb{ C }}$ is a root of $H_{m,n}$ if and only if there exists $b \in {\mathbb{ C }}$ such that the anharmonic oscillator $$\begin{aligned}
\label{eq:anharmonic}
&\psi''(\lambda)=({\lambda}^2+2 a {\lambda}+ a^2 - (2m+n) -\frac{b}{{\lambda}}+\frac{n^2-1}{4 \lambda^2})\psi(\lambda),
\end{aligned}$$ satisfies the following two properties:
1. **Apparent Singularity Condition.** The resonant singularity at ${\lambda}=0$ is apparent or equivalently the monodromy around the singularity is scalar. In a formula, $$\psi(e^{2 \pi i} {\lambda})=(-1)^{n+1} \psi({\lambda}) \, , \quad \forall \psi \mbox{ solution of \eqref{eq:anharmonic}.}$$
2. **Quantisation Condition.** There exists a non-zero solution of which solves the following boundary value problem $$\lim_{{\lambda}\to +\infty} \psi({\lambda})=\lim_{{\lambda}\to 0^+} \psi({\lambda})=0 \; .$$
Studying the asymptotic solution of the above inverse problem, we obtain our main result, which we name the *bulk asymptotics*:
1. We determine the region of the complex plane which asymptotically gets filled densely by the roots. This is a quadrilateral domain that we call the elliptic region following Buckingham [@buckingham18], who had already described it.
2. We describe the bulk asymptotics of the roots, that is, we obtain an asymptotic description of the roots uniformly on any fixed compact subset of the interior of the elliptic region. In particular we show that the roots asymptotically organise themselves on a deformed lattice, thus confirming Clarkson’s numerical findings [@clarkson2003piv; @clarksonpiv].
The paper is organised as follows. In Section \[sec:results\] we state our main results, and we announce our results concerning the critical asymptotics, that is, the asymptotics of roots approaching the four corners of the elliptic region (we postpone a detailed analysis of this to a forthcoming publication, in preparation). In Section \[sec:WKB\] we develop a complex WKB method for equation , Section \[section:elliptic\] is devoted to the study of the elliptic region and in Section \[section:bulkasymptotic\] we prove the main theorems concerning the bulk asymptotics. Finally in the appendix we collect some results from the theory of elliptic integrals and of Stokes complexes which are used in the main body.
Before we begin our paper, in Figure \[fig:intro\] we show the reader a pictorial description of our results, which is worth a thousand lemmas.
![Asymptotically the roots lie on the vertices of a deformed lattice. The asymptotic prediction is stunningly precise even for moderate $m,n$. In the picture, the elliptic region with in purple the deformed lattice, and in blue the roots of $H_{m,n}(z)$ with $(m,n)=(22,16)$. []{data-label="fig:intro"}](m22n16){width="10cm"}
Acknowledgements {#acknowledgements .unnumbered}
----------------
D.M. is partially supported by the FCT Project PTDC/MAT-PUR/ 30234/2017 ‘Irregular connections on algebraic curves and Quantum Field Theory’ and by the FCT Investigator grant IF/00069/2015 ‘A mathematical framework for the ODE/IM correspondence’.
Results {#sec:results}
=======
We introduce a large parameter $E=2m+n$, and the scaled parameters $\alpha, \beta, \nu$, $$\label{eq:scaledparameters}
E=2m+n,\quad \alpha=E^{-\frac12}a,\quad \beta=E^{-\frac32}b, \quad \nu=\frac{n}{E}.$$ Without loss in generality we assume that $m\geq n$, since $H_{n,m}(z)=H_{m,n}(iz)$, hence $\nu\in[0,\frac13]$. We further assume that $\nu \in (0,\frac13]$. The case $\nu=0$ requires a different approach and was dealt with in our previous paper [@maro18].
By scaling the independent variable ${\lambda}\to E^{\frac12}{\lambda}$, the problem laying before us is the rigorous study of the no-logarithm and quantisation condition on the anharmonic oscillator $$\begin{aligned}
\label{eq:scaled}
&\psi''({\lambda})=\left(E^2V(\lambda;\alpha,\beta,\nu)-\frac{1}{4 {\lambda}^2} \right)\psi({\lambda}),\\ \label{eq:scaledpotential}
&V(\lambda;\alpha,\beta,\nu)=\lambda^2+2\alpha \lambda+\alpha^2-1-\beta\lambda^{-1}+\frac{\nu^2}{4}\lambda^{-2},\end{aligned}$$ in the $E\rightarrow +\infty$ limit. Our approach to the inverse monodromy problem is based on the complex WKB analysis of equation . The latter builds on the approximation of solutions by means of the (multivalued) WKB functions $$\label{eq:wkb_functions}
\psi=V^{-\frac14}e^{\pm E \int^{\lambda}\sqrt{V(\mu)}d\mu},$$ where $V$ is the function appearing in . Notice that in the above formula we have neglected the term $-\frac{1}{4 {\lambda}^2}$. This is called the Langer modification and it is necessary to obtain a correct approximation when regular singularities are present; we will discussed it further when proving our results.
The Elliptic Region
-------------------
In this subsection we introduce the compact region $K_a$ in the complex $\alpha$-plane which asymptotically gets filled with the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ as $E\rightarrow \infty$. This region is defined as the projection onto the $\alpha$-plane of a compact set $K\subseteq \{(\alpha,\beta)\in\mathbb{C}^2\}$ which asymptotically gets filled with the solutions $(\alpha,\beta)$ of the no-logarithm and quantisation condition on .
(1.75,0) arc (0:180:1.75cm); (-1.75,0) arc (-180:0:1.75cm);
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\lambda_1$]{};
at (-1.5,0) ; at (-1.5,0) [$\lambda_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\lambda_3$]{};
at (3,0) ; at (3,0) [$\lambda_4$]{};
(-3,0) – (-1.5,0); (1.5,0) – (3,0); (0,0) circle (1.5cm);
(-3,0) – (-4.5,1.5); (-4.5,1.5) – (-4.75,1.75);
(3,0) – (4.5,1.5); (4.5,1.5) – (4.75,1.75);
(-3,0) – (-4.5,-1.5); (-4.5,-1.5) – (-4.75,-1.75);
(3,0) – (4.5,-1.5); (4.5,-1.5) – (4.75,-1.75);
plot \[smooth cycle,tension=0.6\] coordinates [(0.9,0) (1.5,0.5) (3, 0.5) (3.6,0) (3,-0.5) (1.5,-0.5) ]{};
at (2.35,0.75) [$\boldsymbol{\gamma_1}$]{}; at (0.05,1.95) [$\boldsymbol{\gamma_2}$]{};
Before defining $K$, we first introduce the elliptic curve underlying the elliptic integrals in the WKB functions . Consider the affine curve $$\Gamma_{\alpha,\beta} = \lbrace P= ({\lambda},y) \in {\mathbb{ C }}^2, y^2= \lambda^4+2\alpha \lambda^3+(\alpha^2-1){\lambda}^2-
\beta\lambda+\frac{\nu^2}{4}={\lambda}^2 V({\lambda})\rbrace \; .$$ This curve can be compactified by adding two points at infinity, $\infty_{\pm}$, in accordance with the rule $\lim_{P \to \infty_{\pm}} \frac{y}{{\lambda}^2} =\pm1$. In this way we obtain a -possibly singular- elliptic curve, which we denote by $\widehat{\Gamma}=\widehat{\Gamma}(\alpha,\beta)$, with corresponding projection $({\lambda},y)\mapsto {\lambda}$ onto the Riemann sphere. The pull-back of the multivalued differential $\sqrt{V({\lambda})}d{\lambda}$ is the meromorphic differential $\omega=\frac{y}{{\lambda}}d{\lambda}$ on $\hat{\Gamma}$.
In order to introduce $K$, we first define the Stokes complex associated with the potential $V(\lambda)$. In the WKB analysis of the Stokes lines of $V(\lambda)$ play an important role. These are defined as the levels sets $\Re\int_{\lambda^*}^\lambda \sqrt{V({\lambda})}d{\lambda}=0$ in $\mathbb{P}^1$, where $\lambda^*$ any zero of $V(\lambda)$. The Stokes complex $\mathcal{C}=\mathcal{C}(\alpha,\beta)\subseteq \mathbb{P}^1$ of $V(\lambda)$ is defined as the union of all its Stokes lines and zeros. For example, in Figure \[fig:approximate\] the Stokes complex of $V(\lambda)$ is depicted with $(\alpha,\beta)=(0,0)$.
\[defi:RK\] We define $R$ as the set of $(\alpha,\beta)\in\mathbb{C}^2$ such that the Stokes complex $\mathcal{C}(\alpha,\beta)$ is homeomorphic to the Stokes complex at $(0,0)$, denote its closure by $K=\overline{R}$ and define $K_a$ as the projection of $K$ onto the $\alpha$-plane. We call $K_a$ the elliptic region.
[r]{}[0.3]{}
{width="27.00000%"}
The region $K_a$ is a quadrilateral domain, invariant under complex conjugation and reflection in the origin: It is a simply connected region whose boundary is a Jordan curve composed of four analytic pieces, which we call edges, meeting at four corners, as in Figure \[fig:marked\]. The interior of $K_a$ corresponds to regular elliptic curves, while edges and corners are made up of singular elliptic curves: the edges correspond to the coalescence of a pair of branching points, and the corners correspond to the coalescence of three branching points.
Call $e_k, 1\leq k\leq 4$, the edges of $K_a$ and $c_k, 1\leq k\leq 4$, the corners of $K_a$, see Figure \[fig:marked\], so that $$\partial K_a=e_1\sqcup e_2\sqcup e_3\sqcup e_4 \sqcup\{c_1,c_2,c_3,c_4\}.$$ For $1\leq k\leq 4$, the corner $c_k$ of $K_a$ equals the unique root of $$\label{eq:defiCintro}
C(\alpha):=\alpha^8-6(3\nu^2+1)\alpha^4+8(1-9\nu^2)\alpha^2-3(9\nu^4+6\nu^2+1)$$ in the $k$-th quadrant of the complex $\alpha$-plane. For $1\leq k\leq 4$, the edge $e_k$ is a smooth curve in the half-plane $\{\tfrac{1}{2}\pi(k-2)<\arg{\alpha}<\tfrac{1}{2}\pi k\}$ with end-points $c_{k-1}$ and $c_k$, where $c_{-1}:=c_4$. We have the following implicit parametrisation of $\partial K_a$: let $x=x(\alpha)$ be the unique algebraic function which solves the quartic $$3 x^4+4\alpha x^3+(\alpha^2-1)x^2-\frac{\nu^2}{4}=0$$ analytically in the complex $\alpha$-plane with $x(\alpha)\sim\frac{\nu}{2}\alpha^{-1}$ as $\alpha\rightarrow \infty$ and branch-cuts the diagonals $[c_1,c_3]$ and $[c_2,c_4]$. On the same cut plane there exists a unique algebraic function $y=y(\alpha)$ which solves $$y^2=\alpha^2+6 x\alpha+6x^3-1$$ with $y(\alpha)\sim \alpha$ as $\alpha\rightarrow \infty$. We set $$\label{eq:parametrisation}
\psi(\alpha)=\tfrac{1}{2}\Re\left[\alpha y+\tfrac{1}{2}(1-\nu)\log(p_1)-\log(p_2)+\nu \log(p_3)\right],$$ where $$p_1=1-2 x\alpha-2x^2,\quad p_2=2x+\alpha+y,\quad p_3=\frac{x(\alpha^2+5 x\alpha+4x^2-1) +\frac{1}{2}\nu y}{x^2}.$$ Then $\psi$ is a univalued harmonic function on the cut plane and its level set $\{\phi(\alpha)=0\}$ consists of the boundary $\partial K_a$ plus four additional lines which emanate from the corners and go to infinity along the asymptotic directions $e^{\frac{\pi}{4}(2k-1)}\infty$, $1\leq k\leq 4$, see Figure \[fig:parametrisation\]. Buckingham [@buckingham18 Sections 1.1 and 3.2] gives a slightly different but equivalent parametrisation of the boundary, which also includes the four additional lines emanating from the corners.
To formulate our asymptotic results, we need to introduce a pair of functions on $K$ and $K_a$. For any $ (\alpha,\beta) \in K$ we define the basis of cycles $\{\gamma_1,\gamma_2\}$ as in Figure \[fig:approximate\]. By Definition \[defi:RK\] we have $\Re \oint_{\gamma_1} \omega= \Re \oint_{\gamma_2} \omega=0$, thus yielding a real mapping from $K$ (resp. $K_a$) to ${\mathbb{ R }}^2$: $$\label{def:S}
\mathcal{S}: K\rightarrow \mathbb{R}^2, (\alpha,\beta)\mapsto
(-is_1(\alpha,\beta),-is_2(\alpha,\beta)), \qquad \mathcal{S}_a=\mathcal{S} \circ \Pi_a^{-1}$$ with
\[eq:defs1s2\] $$\begin{aligned}
\label{eq:s1}
s_1(\alpha,\beta)&=\int_{\gamma_1} \omega+\frac{i \pi (1-\nu)}{2},\\ \label{eq:s2}
s_2(\alpha,\beta)&=\int_{\gamma_2} \omega
\end{aligned}$$
and $\Pi_a:K\rightarrow K_a$ the projection onto $K_a$, which we prove to be invertible.
It turns out that $K$ (resp. $K_a$) are homeomorphic under $\mathcal{S}$ (resp. $\mathcal{S}_a$) to the rectangle $$\label{eq:defiQ}
Q:=\left[-\tfrac{1}{2}(1-\nu)\pi,+\tfrac{1}{2}(1-\nu)\pi\right]\times \left[-\nu\pi,+\nu\pi\right],$$ and the above homeomorphisms map the interior of $K$ (resp. $K_a$) $C^{\infty}$-diffeomorphically onto the interior of $Q$. Moreover the image of the edges $e_k$ are the edges $\hat{e}_k$ of the rectangle and the image of the corners $c_k$ are the corners $\hat{c}_k$ of the rectangle, as shown in Figure \[fig:marked\].
For $1\leq k\leq 4$, the edges $e_k$ and $e_{k+1}$ meet at the corner $c_k$ with interior angle equal to $\frac{2}{5}\pi$. In particular $\mathcal{S}_a$ is not conformal.
(imageKa) at (0,0) [![Edges and corners of respectively $K_a$ and $Q$.[]{data-label="fig:marked"}](Ka.pdf "fig:"){width="35.00000%"}]{}; (imageQ) at (0.45,0) [![Edges and corners of respectively $K_a$ and $Q$.[]{data-label="fig:marked"}](Q.pdf "fig:"){width="35.00000%"}]{};
at (0.96,0.875) [$\widehat{c}_1$]{}; at (0.40,0.875) [$c_1$]{}; at (0.60,0.875) [$\widehat{c}_2$]{}; at (0.04,0.875) [$c_2$]{}; at (0.605,0.134) [$\widehat{c}_3$]{}; at (0.04,0.12) [$c_3$]{}; at (0.963,0.137) [$\widehat{c}_4$]{}; at (0.40,0.12) [$c_4$]{};
at (0.97,0.66) [$\widehat{e}_1$]{}; at (0.40,0.65) [$e_1$]{};
at (0.7,0.88) [$\widehat{e}_2$]{}; at (0.14,0.875) [$e_2$]{};
at (0.6,0.35) [$\widehat{e}_3$]{}; at (0.042,0.34) [$e_3$]{};
at (0.86,0.118) [$\widehat{e}_4$]{}; at (0.3,0.12) [$e_4$]{};
at (0.21875,0.5) [$0$]{}; at (0.78125,0.5) [$0$]{};
Bulk Asymptotics {#section:bulk}
----------------
For sake of simplicity we state our result when $m,n\to \infty$ with the ratio $\frac{m}{n}$ fixed. We thus choose $p\geq q$, $p,q$ either equal or co-prime, and fix the ratio $\frac{m}{n}=\frac{p}{q}$. Hence the numbers $m,n$ take values in the sequences $m=tq,n=tp, t \in {\mathbb{ N }}^*$. Correspondingly $\nu=\frac{p}{2q+p}\in(0,\frac13]$ is fixed and the large parameter $E$ belongs to the sequence $(2q+p)t, t \in {\mathbb{ N }}^*$.
Given an integer number $m \in {\mathbb{ N }}^*$ we denote $I_m=\lbrace -m+1,-m+3,\dots,m-1\rbrace \subseteq {\mathbb{ Z }}$. $\forall (j,k) \in I_m \times I_n$ we let $(\alpha_{j,k},\beta_{j,k}) \in K$ be the unique solution of $\mathcal{S}(\alpha,\beta)=(\frac{\pi j}{E},\frac{\pi k}{E})$.
A filling fraction is a real number $\sigma \in (0,1)$. We let $I_m^{\sigma}=I_m \cap [\sigma(-m+1), \sigma(m-1)]$ and define $Q^{\sigma} \subset Q$ as the closed rectangle $[-\frac{\pi \lfloor \sigma (m-1) \rfloor}{E},
\frac{\pi \lfloor \sigma (m-1) \rfloor}{E}]
\times[-\frac{\pi \lfloor \sigma (n-1) \rfloor}{E},
\frac{\pi \lfloor \sigma (n-1) \rfloor}{E}]$, which in the large $E$ limit converges to $\sigma\cdot Q$.
Finally we define $K^{\sigma}=\mathcal{S}^{-1}(Q^{\sigma})$ and $K_a^{\sigma}=\Pi_a(K^{\sigma})$ as the projection of $K^{\sigma}$ on the $\alpha$-plane.
Our main result is the following theorem which provides an asymptotic formula for the roots of the generalised Hermite polynomials as well as an estimate of the error.
\[thm:bulkintro\] Fix a filling fraction $\sigma \in (0,1)$. Then there exists an $R_{\sigma}>0$ such that for $E$ large enough the following hold true:
1. Each disc with center $E^{\frac12}\alpha_{j,k}$ and radius $R_{\sigma}E^{-\frac32}$, $(j,k) \in I_m^{\sigma} \times I_n^{\sigma} $, contains a unique root of the generalised Hermite polynomial $H_{m,n}$.
2. Let ${\mathcal{ K }}^{\sigma}=\lbrace a \in {\mathbb{ C }}, E^{-\frac12}a \in K_a^{\sigma}\rbrace$, then the $\epsilon$-neighbourhood of ${\mathcal{ K }}^{\sigma}$ with radius $R_{\sigma}E^{-\frac{3}{2}}$ contains exactly $\lfloor \sigma m \rfloor \times \lfloor \sigma n \rfloor$ roots of the generalised Hermite polynomials.
Clarkson [@clarkson2003piv; @clarksonspecialpol] observed numerically that the zeros of generalised Hermite polynomial $H_{m,n}$ seem to organise themselves on the intersection points of a deformed grid of $m$ vertical and $n$ horizontal lines, see the top pictures in Figure \[fig:grid\]. The mapping $\mathcal{S}_a$ rectifies this deformed grid. Namely, after Theorem \[thm:bulkintro\], the images under $\mathcal{S}_a$ of the rescaled roots of $H_{m,n}$ asymptotically organise themselves along the intersection points of the true orthogonal grid made of the respective equally spaced $m$ vertical and $n$ horizontal lines $$\label{eq:gridlines}
l_v^{(j)}= \{(x,y) \in Q, x=\frac{\pi j}{E}\}\quad (j \in I_m), \quad l_h^{(k)}=\lbrace (x,y) \in Q, y=\frac{\pi k}{E}\}\quad (k \in I_n),$$ see Figure \[fig:grid\].
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![On the top row the elliptic region $K_a$ with in purple the inverse images under $\mathcal{S}_a$ of the grid lines and the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ in blue, and on the bottom row the rectangle $Q$ with in purple the grid lines and the images of the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ under $\mathcal{S}_a$ in blue for the respective values $(m,n)=(2,2)$, $(7,5)$, $(14,9)$.[]{data-label="fig:grid"}](m2n2 "fig:"){width="4.5cm"} ![On the top row the elliptic region $K_a$ with in purple the inverse images under $\mathcal{S}_a$ of the grid lines and the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ in blue, and on the bottom row the rectangle $Q$ with in purple the grid lines and the images of the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ under $\mathcal{S}_a$ in blue for the respective values $(m,n)=(2,2)$, $(7,5)$, $(14,9)$.[]{data-label="fig:grid"}](m7n5 "fig:"){width="4.5cm"} ![On the top row the elliptic region $K_a$ with in purple the inverse images under $\mathcal{S}_a$ of the grid lines and the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ in blue, and on the bottom row the rectangle $Q$ with in purple the grid lines and the images of the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ under $\mathcal{S}_a$ in blue for the respective values $(m,n)=(2,2)$, $(7,5)$, $(14,9)$.[]{data-label="fig:grid"}](m14n9 "fig:"){width="4.5cm"}
![On the top row the elliptic region $K_a$ with in purple the inverse images under $\mathcal{S}_a$ of the grid lines and the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ in blue, and on the bottom row the rectangle $Q$ with in purple the grid lines and the images of the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ under $\mathcal{S}_a$ in blue for the respective values $(m,n)=(2,2)$, $(7,5)$, $(14,9)$.[]{data-label="fig:grid"}](m2n2Q "fig:"){width="4.5cm"} ![On the top row the elliptic region $K_a$ with in purple the inverse images under $\mathcal{S}_a$ of the grid lines and the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ in blue, and on the bottom row the rectangle $Q$ with in purple the grid lines and the images of the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ under $\mathcal{S}_a$ in blue for the respective values $(m,n)=(2,2)$, $(7,5)$, $(14,9)$.[]{data-label="fig:grid"}](m7n5Q "fig:"){width="4.5cm"} ![On the top row the elliptic region $K_a$ with in purple the inverse images under $\mathcal{S}_a$ of the grid lines and the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ in blue, and on the bottom row the rectangle $Q$ with in purple the grid lines and the images of the roots of $H_{m,n}(E^\frac{1}{2}\alpha)$ under $\mathcal{S}_a$ in blue for the respective values $(m,n)=(2,2)$, $(7,5)$, $(14,9)$.[]{data-label="fig:grid"}](m14n9Q "fig:"){width="4.5cm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The deformed lattice is actually a regular lattice for small $\alpha$’s. Indeed we have the following immediate corollary of Theorem \[thm:bulkintro\].
\[cor:smallalphaintro\] Fix $N_0$ and suppose $|j|,|k| \leq N_0 $. The roots lie on the regular lattice generated by the vectors $\frac{\mathcal{K}\left(\frac{1-\nu}{1+\nu}\right)}{E \sqrt{1+\nu}}$ and $i \frac{\mathcal{K}\left(\frac{2\nu}{1+\nu}\right)}{E \sqrt{1+\nu}}$, where $\mathcal{K}(m)$ denotes the standard complete elliptic integral of the first kind with respect to the parameter $m=k^2$.
More precisely, the following asymptotic formula holds $$\label{eq:smalljkintro}
\alpha_{j,k}= \frac{1}{\sqrt{1+\nu}}\left(\mathcal{K}\left(\frac{1-\nu}{1+\nu}\right)\frac{j}{E}+
\mathcal{K}\left(\frac{2\nu}{1+\nu}\right)\frac{k}{E}i\right)+ O\big(E^{-2}\big)$$ Note that the above equation describes a square lattice when $\nu=\frac{1}{3}$.
#### **Announcement of a result concerning the critical behaviour** {#announcement-of-a-result-concerning-the-critical-behaviour .unnumbered}
We announce here the formula describing the critical asymptotics of roots of generalised Hermite polynomials. Global error estimates, and full proofs will be provided in a forthcoming publication [@maroprep].
Theorem \[thm:bulkintro\] does not cover the asymptotics of roots approaching the corner of the elliptic region. These are called critical asymptotics, and, according to our result, are described by means of the Tritronquee solution $y_T$ of the Painlevé I equation $$\label{eq:PI}
y''(z)=6y^2(z)-z \; .$$ Remarkably, also the critical asymptotics for the poles of rational solutions of the Painlevé II equation are described in terms of the Tritronquee solution, as proven in [@buckinghamcr].
Let us be more precise. The Tritronquee solution, which was discovered by Boutroux [@boutroux], can be defined as the unique solution of equation which does not have poles in the closed sector $|\arg{z}|\leq \frac{4 \pi}{5}$ [@costin12]. It has however an infinite number of poles [@piwkb], which a fortiori lie in the sector $|\arg{z}|> \frac{4 \pi}{5}$. Let $c_1$ be the upper right corner of the elliptic region, and $p$ be a pole of the Tritronquee solution $y_T$. For $E$ large enough, there is a unique root $\alpha_p$ of $H_{m,n}$ with the following asymptotic behaviour $$\label{eq:tritronqueeas}
\alpha_p=c_1-4\kappa p E^{-\frac{2}{5}}+\mathcal{O}(E^{-\frac{3}{5}}) \mbox{ as } E\rightarrow \infty \; .$$ Here $\kappa$ is a constant, independent of $p$, defined by the equation $$\kappa^5=\frac{2 c_1^3(2+c_1^2)^3}{(c_1^4-3\nu^2-1)(c_1^4+4 c_1^2+3\nu^2+1)},
\quad -\frac{3}{4}\pi-\frac{\pi}{10}\leq \arg \kappa<-\frac{3}{4}\pi+\frac{\pi}{10}.$$ The critical asymptotics for the other corners are given by the above formulas, upon substituting $c_1$ for the corresponding $c_k,k\neq 1$, and by choosing the appropriate solution of the quintic equation.
Note on the Literature
----------------------
The problem of describing the asymptotic location of roots of generalised Hermite polynomials when both parameters are large were raised by Peter Clarkson in [@clarksonpiv]. The slightly more general problem of describing the asymptotics of generalised Hermite polynomials was the object of much recent interest, see e.g. [@maro18; @buckingham18; @vanass18]. In our previous paper [@maro18] we dealt with the asymptotic location of roots in case when one of the two parameters stays bounded. Buckingham studied the asymptotics of generalised Hermite polynomials when both parameters are large in the complement of the elliptic region, that is, in the region where no roots are present [@buckingham18]. We present here the first correct (a previous attempt is carefully analysed in [@buckingham18; @vanass18], and shown to be erroneous) asymptotic description of roots of generalised Hermite polynomials when both parameters are large.
WKB asymptotics {#sec:WKB}
===============
In the present section we study the inverse monodromy problem of equation , which characterises roots of generalised Hermite polynomials in the rescaled variables $\alpha,\beta$, in the large $E$ limit. We do this by developing a suitable complex WKB method. We remark that, in the present section and unless otherwise stated, $m,n,E=2m+n$ are arbitrary positive real numbers, since most results hold true irrespective of their integer nature.
We consider the complex WKB method for an equation of the kind $$\label{eq:schrlan}
\psi''(x)=(E^2 V({\lambda})+r({\lambda}))\psi({\lambda}) \;.$$ Specifying to $V=V({\lambda};\alpha,\beta,\nu)$ and $r=-\frac{1}{4 {\lambda}^2}$, we obtain the main object of our analysis, namely equation (\[eq:scaled\]).
The WKB asymptotic is based on the approximation of solutions by means of the multivalued functions $$\begin{aligned}
\label{eq:wkb}
& J({\lambda},{\lambda}')=e^{- E S({\lambda},{\lambda}')-\frac{1}{4} L({\lambda},{\lambda}')}, \\ \nonumber
& S({\lambda},{\lambda}')=\int^{{\lambda}}_{{\lambda}'}\sqrt{V(\mu)} d\mu, \quad L({\lambda},{\lambda}')=\int_{{\lambda}'}^{\lambda}\frac{V'(\mu)}{V(\mu)} d \mu\end{aligned}$$
In order to measure the difference between a WKB approximation $J({\lambda},{\lambda}')$ and a solution $\psi({\lambda},{\lambda}_0,{\lambda}')$, we introduce the ratio $z({\lambda},{\lambda}_0):=z({\lambda},{\lambda}_0,{\lambda}')=\frac{\psi({\lambda},{\lambda}_0,{\lambda}')}{J({\lambda},{\lambda}')}$ and impose the boundary condition $z({\lambda}_0,{\lambda}_0)=1$ (which implies that $z$ does not depend on ${\lambda}'$). The ratio (after a few algebraic steps, see [@erdelyi10 §Section 4] for the details) is shown to satisfy the following Volterra integral equation $$\begin{aligned}
\nonumber
& z({\lambda},{\lambda}_0)=1-\frac{1}{E}\int^{{\lambda}}_{{\lambda}_0} K({\lambda},{\lambda}') F({\lambda}') d{\lambda}' \, , \quad B({\lambda},{\lambda}')= \frac{e^{ 2 E S({\lambda},{\lambda}')}-1}{2} \\
\label{eq:volterra}
& F({\lambda})=\frac{\phi({\lambda})}{ V^{\frac12}({\lambda})} \, , \quad
\phi({\lambda})=-r({\lambda})+\frac{ -4 V''({\lambda}) V({\lambda})+5V'^2({\lambda})}{16 V({\lambda})^2}\end{aligned}$$ We begin the analysis of the integral equation by studying the singularity of the forcing form $F({\lambda}) d{\lambda}$.
The only singularities, on the compact Riemann surface $\widehat{\Gamma}$, of the form $F({\lambda}) d{\lambda}$ are the zeros of $V$, namely the branching points.
The form is manifestly regular outside ${\lambda}=0,\infty$ and $V({\lambda})=0$: in a neighbourhood of $\infty$, $\rho({\lambda})=O(\lambda^{-3})$; in a neighbourhood of $0$, $\rho({\lambda})=O(1)$; while at a zero of $V$, $\rho$ is is not integrable.
We notice that formula is the so-called Langer modified WKB approximation. In fact, the standard WKB approximation is obtained by studying approximate solutions of the form $\widetilde{J}({\lambda},{\lambda}')=e^{-\widetilde{S}({\lambda},{\lambda}')}$ with $\widetilde{S}({\lambda},{\lambda}')=\int_{{\lambda}'}^{\lambda}{\sqrt{Q(\mu)}}d\mu$, $Q=E^2 V+r$. This choice leads to an integral equation with the forcing term $$\widetilde{F}({\lambda}) d{\lambda}=\frac{ -4 Q''({\lambda}) Q({\lambda})+5Q'^2({\lambda})}
{16 Q^{\frac52}({\lambda})} d{\lambda}\; .$$
The latter 1-form coincides asymptotically with $E^{-1}F({\lambda}) d{\lambda}$ in all of $\widehat{\Gamma}$ but for a neighbourhood of the Fuchsian singularity ${\lambda}=0$, where $F({\lambda})d{\lambda}$ is integrable while $\widetilde{F}({\lambda}) d{\lambda}$ is not. For such a reason, the standard WKB approximation fails at that point, while the Langer modified WKB provides the correct result.
Let ${\lambda}_0$ be a regular point of $F({\lambda}) d{\lambda}$. We say that $\gamma:[0,1] \to \overline{{\mathbb{ C }}}, \gamma(0)={\lambda}_0$ is an admissible curve if $\gamma$ is a rectifiable curve such that $\Re S(\gamma(t),{\lambda}_0) $ is monotone not increasing on $[0,1]$.
\[pro:continuation\] If $\gamma$ is admissible, then for large enough $E$ there exists a unique solution $\psi({\lambda},{\lambda}_0,{\lambda}')$ of such that $$\label{eq:wkbasymptotic}
{\left| \frac{\psi(\gamma(t),{\lambda}_0,{\lambda}')}{J(\gamma(t),{\lambda}')}-1 \right|} \leq e^{\frac{\rho(\gamma(t))}{E}}-1$$ with $$\rho(\gamma(t))= \int_0^t|F(s) \dot{\gamma}(s)|ds \; .$$
Details can be found in [@piwkb]. The integral equation is of the form $z=1+E^{-1}B[1]$, where $B$ is a continuous linear (integral) operator on the space $L^{\infty}([0,1])$, and $1$ the constant function $1$. We construct its unique solution by means of the Neumann series $z=\sum_{l=0}^{\infty}E^{-l}B^l[1]$ where $B^l[1]$ is the $l$-th power (or iterate) of $B$ applied to the constant function $1$.
We want to estimate the operator norm $\|{B}^l\|_{\infty}$ of $K^l$. By definition of admissible curve we have that $|B({\lambda},{\lambda}')|\leq 1$, from which it follows that $\|{B}\|_{\infty}\leq \rho(\gamma(1))$. Since by iterating $B$ $l$-times we are integrating on a simplex of dimension $l$, which has volume $\frac{1}{l!}$ and thus $\|{B^l}\|_{\infty}\leq \frac{\|{B}\|_{\infty}^l}{l!}=\frac{\rho^l(\gamma(1))}{l!}$. The thesis follows from the latter estimate.
In the case of the potential $V=V({\lambda},\alpha,\beta,\nu)$, we have that $S({\lambda},{\lambda}')=\pm E\frac{{\lambda}^2}{2}+ O({\lambda}), E >0$ as ${\lambda}\to \infty$. It follows that the lines of steepest descent (or ascent) for the function $S({\lambda},{\lambda}')$ are asymptotic to one of the following four rays, the rays with argument $0,\frac{\pi}{2},-\frac{\pi}{2}$, or $\pi$. This corresponds to the fact that there are four Stokes sectors, each with a unique subdominant solution. Alternatively, we can think of the complex plane with four adjoined points at infinity, namely $\pm\infty,\pm i \infty$, and of the unique solution subdominant at one of each adjoined points. In order to solve the inverse monodromy problem characterising the roots of the generalised Hermite polynomials we have to understand the linear relations among these four solutions, and among them and the solution subdominant at ${\lambda}=0$. Such linear relations can be computed in WKB approximation and the error can be estimated by means of a number of those error terms $\rho$’s described in the previous proposition.
Let us consider the complex plane cut along the negative real semi-axis. Denote by $\chi_+,\psi_0,\psi_{\pm1}$ the unique (up to a normalising constant) solutions of the anharmonic oscillator such that $\lim_{{\lambda}\to 0}\chi_+({\lambda})=\lim_{{\lambda}\to + \infty}\psi_0({\lambda})=\lim_{{\lambda}\to \pm i \infty}\psi_{\pm 1}({\lambda})=0$.
We define $\Gamma_{\pm}$ as the set of admissible paths such that $\lim_{t\to 0+}\gamma(t)=0, \lim_{t \to 1^-}\gamma(t)=\pm i \infty$, and $\Gamma_L,\Gamma_R$ the set of admissible paths such that $\lim_{t\to 0^+}\gamma(t)=-\infty, \lim_{t \to 1^-}\gamma(t)= i \infty$ passing ${\lambda}=0$ respectively from the left or from the right (namely $\int_0^1 d\arg{\gamma(t)}=\pm \pi$ provided that $\gamma$ belongs to $\gamma_R$ or $\gamma_L)$. Finally, we let $$\rho_{\pm}=\inf_{\gamma \in \Gamma_{\pm}} \rho(\gamma(1)) \; ,$$ and $$\rho_{L,R}=\inf_{\gamma \in \Gamma_{L,R}} \rho(\gamma(1)) \; \; .$$
In principle one can define the minimal errors $\rho(\infty,\pm i \infty)$ and $\rho(0,\infty)$ of all admissible paths joining $\infty$ and $\pm i \infty$, and of all admissible paths joining $0$ and $+\infty$. However this is unnecessary for two opposite reasons: The error $\rho(\infty,\pm i \infty)=0$ because these asymptotic directions belong to neighbouring Stokes sectors, see [@piwkb]; The error $\rho(\infty,\pm i \infty):=\infty$ because, due to the topology of the Stokes complex, there are no admissible paths connecting $0$ and $\infty$ if $(\alpha,\beta)\in K$.
\[def:W\] Fixed $\nu$, we notice that the errors $\rho_{\pm}$, and $\rho_L,\rho_R$ are functions of $\alpha,\beta$. We say that a compact subset $D \subseteq {\mathbb{ C }}^2 \ni
(\alpha,\beta)$ has the $W$ property, if the above functions (i.e. the $\rho$’s) are bounded. We often denote such domains by $D_W$.
By construction, any compact subset $K'$ of $R$ has the $W$ property. By lower semi-continuity, any such $K'$ admits an epsilon neighbourhood with the $W$ property.
We now prove the WKB estimate for the eigenvalue conditions and for the trivial-monodromy condition. For sake of clarity, we analyse them in three distinct theorems (the first about the quantisation condition, the other two about the trivial monodromy condition).
\[thm:wkbquantisation\] Fix $\nu \in (0,\frac 13]$, not necessarily rational, let $\alpha,\beta$ belong to a domain $D$ on which $\rho_R,\rho_{\pm}$ are uniformly bounded.
On the cut plane ${\mathbb{ C }}\setminus{{\mathbb{ R }}_-}$, let the solutions $\chi_+,\psi_0$ of equation be uniquely determined up to multiplicative pre-factors as the solutions subdominant at $0,+\infty$. Denote their Wronskian by $Wr[\chi_+,\psi_0]$.
Then there exist $C_0,E_0 >0$ such that, for all the $E \geq E_0$ the following estimate holds (after a suitable normalisation of $\chi_+,\psi_0$): $$\label{eq:quantisationwkb}
\left| (Wr[\chi_+,\psi_0]+1) e^{E\oint_{\gamma_1}\sqrt{V}}+1\right|\leq \frac{C_0}{E} \qquad \forall(\alpha,\beta) \in D.$$
\[thm:wkbmonodromy\] Let $D_W$ a domain with the $W$ property. Fix $\nu\in (0,\frac{1}{3}]$ rational and restrict $E$ to integer values such that $E \nu=n \in {\mathbb{ N }}$.
On the plane cut along the positive imaginary semi-axis, let $\psi_{1}^L,\psi_1^R$ be the solutions of equation uniquely determined up to multiplicative pre-factors as the solutions subdominant at $+i\infty$ respectively to the left and to the right of the branch cut. Denote their Wronskian by $Wr[\psi_1^R,\psi_1^L]$.
Then there exist $C_0,E_0 >0$ such that, for all $E \geq E_0$ the following hold true:
1. the monodromy of equation is trivial if and only if $Wr[\psi_1^R,\psi_1^L]=0$;
2. we have the estimate (after suitable normalisation of $\psi_1^{R,L}$) $$\label{eq:monodromywkb}
\left| (Wr[\psi_1^R,\psi_1^L]+1) e^{-E\oint_{\gamma_2}\sqrt{V}+i\pi n}+1\right|\leq \frac{C_0}{E} \qquad \forall (\alpha,\beta) \in D_W.$$
Below we compute the full monodromy matrix in WKB approximation, even though the above theorem suffices to compute asymptotically the roots of the generalised Hermite polynomials. We do provide the complete monodromy matrix because, to our knowledge, this has not appeared in the scientific literature at the time of writing.
\[thm:wkbmonodromy2\] Fix $\nu \in (0,\frac 13]$, not necessarily rational, and a domain $D_W$ with the $W$ property.
On the plane cut along the negative real semi-axis, let $\psi_{\pm1}$ be the solutions of equation uniquely determined up to multiplicative pre-factors as the solutions subdominant at $\pm i \infty$.
Then there exist $C_0,E_0 >0$ such that for all $E \geq E_0$ the following hold true:
1. the pair $\lbrace \psi_{1},\psi_{-1} \rbrace$ is a basis of solutions;
2. letting ${\mathcal{ M }}$ denote the monodromy of equation with respect to the basis $\lbrace \psi_{1},\psi_{-1} \rbrace$, we have $$\| {\mathcal{ M }}\cdot{\mathcal{ M }}^{-1}_{W}-{\mathbb{ I }}_2\|\leq \frac{C_0}{E} \qquad \forall (\alpha,\beta) \in D_W, \mbox{ with}$$ $$\label{eq:monodromyMwkb}
{\mathcal{ M }}_{W}=\begin{pmatrix}
-2 \cos{\pi n}-e^{E\oint_{\gamma_2}\sqrt{V}} & \kappa(\alpha,\beta)\big(1+e^{ i \pi n } e^{E\oint_{\gamma_2}\sqrt{V}}\big)\\
-\kappa^{-1}(\alpha,\beta)\big(1+e^{ -i \pi n } e^{E\oint_{\gamma_2}\sqrt{V}}\big)& e^{E\oint_{\gamma_2}\sqrt{V}}
\end{pmatrix}, \quad {\mathbb{ I }}_2=\begin{pmatrix}
1 & 0\\
0& 1
\end{pmatrix},$$ where $k(\alpha,\beta)$ is a non-zero (immaterial) normalising constant that can be chosen in such a way that the above estimate hold.
After the estimates (\[eq:quantisationwkb\],\[eq:monodromywkb\]), in WKB approximation the quantisation and monodromy conditions read: $$\begin{aligned}
\label{eq:wkbcondition1}
& E\oint_{\gamma_1}\sqrt{V} = (2k+1) \pi, k \in {\mathbb{ Z }} \\ \label{eq:wkbcondition2}
& E\oint_{\gamma_2}\sqrt{V} = 2l \pi , l \in {\mathbb{ Z }} \mbox{ if } n \mbox { odd }, \; l \in {\mathbb{ Z }}+\frac12\mbox{ if } n \mbox { even }
\end{aligned}$$ Notice that in this approximation the quantisation and monodromy conditions are on an equal footing despite of their different definition. This is however consistent with the following fact: that there is a dual inverse monodromy problem which interchanges the roles of $m,n$, see [@maro18].
Finally notice that the trace and determinant of the monodromy matrix ${\mathcal{ M }}$, are exact in the WKB approximation, as indeed the trace and the determinant of ${\mathcal{ M }}_{W}$ are, respectively, $-2 \cos \pi n$ and $1$.
plot \[smooth,tension=0.7\] coordinates [(1.5,0) (3,1) (4.5, 0) (6,-1) (7.5,0) ]{};
plot \[smooth,tension=0.7\] coordinates [(1.5,0) (3,-1) (4.5, 0) (6,1) (7.5,0) ]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-6,0); at (-6,0) [$\lambda_1$]{};
at (-3,0) ; at (-3,0) [$\lambda_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\lambda_+$]{};
at (3,0) ; at (3,0) [$\lambda_3$]{};
at (4.5,0) ; at (4.5,0) [$\lambda_R$]{};
at (6,0) ; at (6,0) [$\lambda_4$]{};
at (7.5,0) ; at (7.5,0) [$\lambda_0$]{}; at (-7.5,0) [$\color{white}\lambda_0$]{};
at (3,1) [$\beta$]{}; at (3,-1) [$\alpha$]{};
at (6,1) [$\overline{\beta}$]{}; at (6,-1) [$\overline{\alpha}$]{};
at (-4.5,0) [$\lambda_L$]{}; at (-4.5,0) ;
(-6,0) – (-3,0); (3,0) – (6,0);
plot \[smooth,tension=0.9\] coordinates [(4.5,0) (3.5,-1.7) (0, -1) (-3.5,1) (-4.5,0) ]{};
(3.375,0) ellipse (1.125cm and 0.5cm);
(0,0) circle (0.6cm);
(1.5,0) arc (-180:0:1.5cm and 1.1cm);
(-4.5,0) arc (-180:0:3cm and 1.1cm);
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-6,0); at (-6,0) [$\lambda_1$]{};
at (-3,0) ; at (-3,0) [$\lambda_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\lambda_+$]{};
at (3,0) ; at (3,0) [$\lambda_3$]{};
at (4.5,0) ; at (4.5,0) [$\lambda_R$]{};
at (6,0) ; at (6,0) [$\lambda_4$]{};
at (7.5,0) [$\color{white}\lambda_0$]{}; at (-7.5,0) [$\color{white}\lambda_0$]{};
at (3.375,0.5) [$\beta_R$]{}; at (3,-1.1) [$\alpha_R$]{};
at (-1.5,-1.1) [$\alpha_L$]{}; at (1.2,-1.45) [$\beta_L$]{};
at (0,0.6) [$\sigma$]{};
at (-4.5,0) ; at (-4.5,0) [$\lambda_L$]{};
(-6,0) – (-3,0); (3,0) – (6,0);
plot \[smooth,tension=0.8\] coordinates [(-4.5,0) (-3,0.5) (-1.5, 0) (0,-0.5) (1.5,0) ]{};
plot \[smooth,tension=0.8\] coordinates [(-4.5,0) (-3,-0.5) (-1.5, 0) (0,0.5) (1.5,0) ]{};
(3,0) ellipse (1.5cm and 0.5cm);
(-1.5,0) ellipse (3cm and 1.4cm);
(0,0) ellipse (4.5cm and 2.2cm);
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-6,0); at (-6,0) [$\lambda_1$]{};
at (-3,0) ; at (-3,0) [$\lambda_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\lambda_+$]{};
at (3,0) ; at (3,0) [$\lambda_3$]{};
at (4.5,0) ; at (4.5,0) [$\lambda_R$]{};
at (6,0) ; at (6,0) [$\lambda_4$]{};
at (-4.5,0) ; at (-4.5,0) [$\lambda_L$]{};
at (7.5,0) [$\color{white}\lambda_0$]{}; at (-7.5,0) [$\color{white}\lambda_0$]{};
at (3,0.5) [$\beta$]{}; at (3,-0.5) [$\alpha$]{};
at (-3,0.5) [$\overline{\delta}$]{}; at (-3,-0.5) [$\gamma$]{};
at (-1.5,1.4) [$\delta$]{}; at (-1.5,-1.4) [$\overline{\gamma}$]{};
at (0,2.2) [$\epsilon$]{}; at (0,-2.2) [$\vartheta$]{};
(-6,0) – (-3,0); (3,0) – (6,0);
We choose points ${\lambda}_+,{\lambda}_L,{\lambda}_R,{\lambda}_0$ as in Figures \[fig:paths\].
Proof of Theorem \[thm:wkbquantisation\]. We normalise the solutions $\chi_+,\psi_{\pm1},\psi_0$, subdominant at ${\lambda}=0,{\lambda}=\pm i \infty,{\lambda}=\infty$, by the requirements $\chi_+({\lambda}_+)=\psi_{\pm1}({\lambda}_R)=\psi_0({\lambda}_0)=1$.
Since $\rho_R < \infty$, the solutions $\lbrace \psi_{1},\psi_{-1} \rbrace$ form a basis. In fact, by Proposition \[pro:continuation\], $$\lim_{{\lambda}\to -i\infty}\left|\psi_1({\lambda})e^{\int_{{\lambda}_R}^{{\lambda}}\sqrt{V(\mu)}+\frac{V'(\mu)}{V(\mu)}}-1\right|
\leq E^{-1}\rho_R \to 0 \, ,$$ where $\sqrt{V}$ is chosen in such a way that $\lim_{{\lambda}\to \pm i\infty}\Re \int_{{\lambda}_R}^{{\lambda}}\sqrt{V(\mu)}=\pm \infty$. Hence $\psi_1$ is eventually dominant at $-i\infty$, while $\psi_{-1}$ is by definition subdominant at $-i\infty$.
We write $$\label{eq:ABAB}
\chi_+=A\psi_1+B\psi_{-1} \,,\quad \psi_0=\overline{A} \psi_1+ \overline B\psi_{-1} \, ,$$ so that $$\label{eq:wrABAB}
Wr[\chi_+,\psi_0]= Wr[\psi_{1},\psi_{-1}] B \overline{A} \big( \frac{A \overline{B}}{B \overline{A}} -1 \big) \; .$$ To compute $A,B,\overline{A},\overline{B}$ in the WKB approximation, we first notice that $$\begin{aligned}
A=\lim_{{\lambda}\to -i\infty}\frac{\chi_+({\lambda})}{\psi_1({\lambda})}, \,
\overline{A}=\lim_{{\lambda}\to -i\infty}\frac{\psi_0({\lambda})}{\psi_1({\lambda})}, \,
B=\lim_{{\lambda}\to i\infty}\frac{\chi_+({\lambda})}{\psi_{-1}({\lambda})}, \,
\overline{B}=\lim_{{\lambda}\to i\infty}\frac{\psi_0({\lambda})}{\psi_{-1}({\lambda})}.\end{aligned}$$ The above limits can be computed using the WKB approximations of $\chi_+,\psi_{\pm1},\psi_0$ via Proposition \[pro:continuation\] - see [@piwkb] for a guided introduction to these kind of computations: $$\begin{aligned}
\nonumber
|Ae^{-\int_\alpha\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|&\leq
E^{-1}\big( \rho_- + \rho_{R}\big),\\ \nonumber
|\overline{A}e^{-\int_{\overline{\alpha}}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|&\leq
E^{-1}\rho_{R}, \\ \nonumber
|B e^{-\int_\beta\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|&\leq
E^{-1}\big( \rho_+ + \rho_{R}\big),\\ \label{eq:AB}
|\overline{B}e^{-\int_{\overline{\beta}}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1| &\leq
E^{-1} \rho_{R}.
\end{aligned}$$ Here the paths $\alpha,\overline{\alpha},\beta,\overline{\beta}$ are chosen as in the Figure \[fig:path1\]. We notice that $\oint_{\alpha-\beta+\overline{\beta}-\overline{\alpha}}\sqrt{V}=-\oint_{\gamma_1}\sqrt{V}$ and $\oint_{\alpha-\beta+\overline{\beta}-\overline{\alpha}}\frac{V'}{V}=\oint_{\gamma_1}\frac{V'}{V}$. Combining formula (\[eq:wrABAB\]) with the estimates (\[eq:AB\]), and latter homological computations, we obtain the thesis.
We now prove Theorem \[thm:wkbmonodromy\]. We let $\chi_+,\psi_{-1},\psi_{1}^R,\psi_1^L$ be the solutions subdominant at $0,-i\infty,+i\infty$ to the right (left) of the cut, normalised by the requirements $\chi_+({\lambda}+)=\psi_{-1}({\lambda}_R)=\psi_1^R({\lambda}_R)=\psi_1^L({\lambda}_L)=1$. We begin by proving part (1). It follows from $\rho(0,i\infty)<\infty$ that $\psi_1^{R,L}$ and $\chi_+$ are linearly independent if $E$ is big enough (by reasoning similarly as in the beginning of the proof of Theorem \[thm:wkbquantisation\]). It follows that $\psi_1^{R}(e^{2i\pi}{\lambda})=(-1)^{n+1}\psi_1^{R}(e^{2i\pi}{\lambda})$ if and only if ${\lambda}=0$ is an apparent singularity. Moreover $\psi_1^{R}(e^{2i\pi}{\lambda})=(-1)^{n+1}\psi_1^{R}(e^{2i\pi}{\lambda})$ if and only if $\psi_1^{R}$ is subdominant at $+i\infty$ to the left of the branch-cut, which is equivalent to $Wr[\psi_1^R,\psi_1^L]=0$.
We now prove part (2). We write $$\label{eq:CDCD}
\psi_1^{R,L}=A^{R,L}\chi_+ + B^{R,L}\psi_{-1}$$ so that $$\label{eq:wrCDCD}
Wr[\psi_+^R,\psi_1^L]= Wr[\chi_+,\psi_{-1}] A^LB^R \big( \frac{A^R B^L}{A^L B^R} -1 \big) \; .$$ To compute $A^{R,L},B^{R,L}$ in the WKB approximation, we reason as above. Noticing that $$\begin{aligned}
A^{R,L}=\lim_{{\lambda}\to -i\infty}\frac{\psi^{R,L}({\lambda})}{\chi_+({\lambda})}, \,
B^{R,L}=\lim_{{\lambda}\to 0}\frac{\psi^{R,L}({\lambda})}{\psi_{-1}({\lambda})},\end{aligned}$$ we obtain, via Proposition \[pro:continuation\], the WKB estimates $$\begin{aligned}
\nonumber
|A^{R,L}e^{-\int_{\alpha_{R,L}}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|&\leq
E^{-1}\big( \rho_- + \rho_{R,L}\big),\\
|B^{R,L} e^{-\int_{\beta_{R,L}}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|&\leq
E^{-1}\big( \rho_+ + \rho_-\big). \label{eq:ABrl}\end{aligned}$$ Here the paths $\alpha_R,\alpha_L,\beta_L,\beta_R,\sigma$ are as in Figure \[fig:path2\]. Combining equations (\[eq:wrCDCD\]) and (\[eq:ABrl\]), and noticing that $\alpha_R-\alpha_L+\beta_L-\beta_R=\gamma_2-\sigma$, we obtain the thesis.
We now prove Theorem \[thm:wkbmonodromy2\]. To this aim we fix the following paths $\alpha,\overline{\alpha},
\gamma,\overline{\gamma},\delta,\overline{\delta},{\varepsilon},\vartheta$ on $\widehat{\Gamma}$ as in Figure \[fig:path3\]. In the plane cut along the *positive* real semi axis, we define $\psi_{\pm1}^{L}$ as the solution subdominant at $\pm i \infty$ with the normalisation $\psi_{\pm1}^{L}({\lambda}_L)=1$. We use this alternative pair of solutions to find a convenient LU factorisation of the monodromy matrix:
1. We denote by ${\mathcal{ M }}_U$ the matrix of change of basis among the solutions $\psi_{\pm1}^R$ and $\psi_{\pm1}^L$, $(\psi^L_{1},\psi^L_{-1})^T=
{\mathcal{ M }}_U (\psi_{1}^R,\psi_{-1}^R)^T$, when $\psi_{\pm1}^R$ are analytically continued to ${\lambda}_L$ along an arc passing above ${\lambda}=0$.
2. We denote by ${\mathcal{ M }}_D$ the matrix of change of basis, $(\psi^R_{1},\psi^R_{-1})^T=
{\mathcal{ M }}_D (\psi_{1}^L,\psi^L_{-1})^T$, when $\psi_{\pm1}^L$ are analytically continued to ${\lambda}_R$ along an arc passing below ${\lambda}=0$.
3. We thus have ${\mathcal{ M }}={\mathcal{ M }}_D\cdot{\mathcal{ M }}_U$, where, as it turns out, ${\mathcal{ M }}_U$ is lower triangular while ${\mathcal{ M }}_D$ is upper triangular.
We analyse $M_U$ first. We notice that ${\mathcal{ M }}_U$ is equivalently defined as the matrix of change of basis among $\psi_{\pm1}^R$ and $\psi_{\pm1}^L$, when we work on the complex plane cut along the negative imaginary semi axis, and we define $\psi_{-1}^R$ as the solution subdominant at $-i\infty$ right to the cut (and such that $\psi_{-1}^R({\lambda}_R)=1)$), and $\psi_{-1}^L$ as the solution subdominant at $-i\infty$ left to the cut (and such that $\psi_{-1}^L({\lambda}_L)=1)$). With such a geometry, the solutions $\psi_{1}^L$ and $\psi_{1}^R$ coincide but for a normalising factor, which can be easily computed by WKB approximation: $$\label{eq:psi+up}
\psi_1^L= F \psi_{1}^R \,,\quad |F e^{-\int_{{\varepsilon}}\sqrt{V}+\frac{V'}{4V}}-1|\leq \frac{\rho_L+\rho_R}{E} \; .$$
On the contrary, $\psi_{-1}^L,\psi_{-1}^R$ are not in general proportional. To overcome such a difficulty, we utilise the solution $\chi_+$. Writing $$\label{eq:chi+up}
\chi_+=A\psi_{1}^R+B \psi_{-1}^R=C\psi_{1}^L+D \psi_{-1}^L$$ we have that $$\label{eq:MUdef}
{\mathcal{ M }}_U=\begin{pmatrix}
F & 0\\
D^{-1}(A-C F)& D^{-1}B
\end{pmatrix}$$ The WKB approximation of the terms $A,B$ was computed in above. The same sort of computations show that $$\begin{aligned}
\nonumber
& |Ce^{-\int_{\gamma}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|\leq
E^{-1}\big( \rho_- + \rho_{L}\big),\\ \label{eq:CD}
& |De^{-\int_{\delta}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|\leq
E^{-1}\big( \rho_+ + \rho_{L}\big).\end{aligned}$$
We can repeat the very same arguments to compute ${\mathcal{ M }}_D$, after having introduced a branch-cut on the positive imaginary semi-axis. We have that $$\label{eq:psi-down}
\psi_{-1}^R= T \psi_{-1}^L \,,\quad |T e^{-\int_{\theta}\sqrt{V}+\frac{V'}{4V}}-1|\leq \frac{\rho_L+\rho_R}{E} \; .$$ Defining $$\label{eq:chi+down}
\chi_+=\overline{C}\psi_{1}^L+\overline{D} \psi_{-1}^L \; ,$$ we have $$\label{eq:MDdef}
{\mathcal{ M }}_D=\begin{pmatrix}
A^{-1}\overline{C} & A^{-1}(\overline{D}-B T)\\
0 & T
\end{pmatrix}$$ where WKB approximations of $\overline{C},\overline{D}$ are given by $$\begin{aligned}
\nonumber
& |\overline{C}e^{-\oint_{\overline{\gamma}}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|\leq
E^{-1}\big( \rho_- + \rho_{L}\big),\\ \label{eq:CDbarra}
& |\overline{D}e^{-\oint_{\overline{\delta}}\big(\sqrt{V}+\frac{V'}{4 V} \big)} -1|\leq
E^{-1}\big( \rho_+ + \rho_{L}\big).\end{aligned}$$ Using (\[eq:MUdef\],\[eq:MDdef\]) we obtain $$\label{eq:M}
{\mathcal{ M }}={\mathcal{ M }}_D\cdot{\mathcal{ M }}_U= \begin{pmatrix}
A^{-1}\overline{C}F+ D^{-1}(A-C F) & A^{-1}(\overline{D}-B T)D^{-1}B\\
D^{-1}(A-C F) T & D^{-1}BT
\end{pmatrix}$$ Substituting $A,B,C,D,\overline{C},\overline{D},F,T$ with their WKB approximations (\[eq:AB\],\[eq:CD\],\[eq:CDbarra\],\[eq:psi+up\],\[eq:psi-down\]), and using some elementary homological calculations (e.g. $\epsilon+\bar{\gamma}-\alpha=\gamma_2$), we obtain that the WKB approximation ${\mathcal{ M }}_{W}$ of ${\mathcal{ M }}$ is $$\label{eq:Mwkb}
{\mathcal{ M }}_{W}=\begin{pmatrix}
-2 \cos{\pi n}-\exp{E\oint_{\gamma_2}\sqrt{V}} & \kappa (1+e^{ i \pi n } \exp {E\oint_{\gamma_2}\sqrt{V}})\\
-\kappa^{-1}(1+e^{ -i \pi n } \exp {E\oint_{\gamma_2}\sqrt{V}})& \exp{E\oint_{\gamma_2}\sqrt{V}}
\end{pmatrix}$$ where $\kappa=-e^{\int_{\alpha-\beta}\sqrt{V}+\frac{V'}{4V}}e^{-i \pi n}$ .
Finally the estimate $\| {\mathcal{ M }}\cdot{\mathcal{ M }}^{-1}_{W}-{\mathbb{ I }}_2\|\leq \frac{C_0}{E} $ can be proven by using the factorisation ${\mathcal{ M }}={\mathcal{ M }}_D {\mathcal{ M }}_U$. Indeed, if we let ${\mathcal{ M }}_{U(W)},{\mathcal{ M }}_{D(W)}$ denote the WKB approximation of ${\mathcal{ M }}_{U,D}$ when which are obtained substituting $A,B,C,D,\overline{C},\overline{D},F,T$ with their WKB approximations, then it is straightforward to show that $\| {\mathcal{ M }}_{D}\cdot{\mathcal{ M }}^{-1}_{D (W)}-{\mathbb{ I }}_2\|\leq \frac{C_0}{E}$ and $\|{\mathcal{ M }}_{U}\cdot{\mathcal{ M }}^{-1}_{U (W)}-{\mathbb{ I }}_2\|\leq \frac{C_0}{E} $.
The Elliptic Region {#section:elliptic}
===================
This section is devoted to the study of the regions $R$, $K$ and $K_a$, defined in Definition \[defi:RK\], as well as the projection $\Pi_a:K\rightarrow K_a$ and mapping $\mathcal{S}:K\rightarrow Q$ introduced in equation , which we reintroduce here. For convenience of the reader we have collected some explicit formulas (that we will need in the sequel) for complete elliptic integrals and their Jacobian in the Appendix \[append:computation\].
The regions $R$, $K$ and $K_a$ are defined entirely in terms of the Stokes geometry of the potential $$\label{eq:potential}
V(\lambda;\alpha,\beta)=\lambda^2+2\alpha \lambda+\alpha^2-1-\beta\lambda^{-1}+\frac{\nu^2}{4}\lambda^{-2}.$$ We therefore quickly review some basic notions of WKB theory concerning Stokes complexes, see [@fedoryuk; @strebel; @jenkins] for more details.
Firstly we recall the notion of turning points.
\[def:turn\] A zero of multiplicity $n$ of is called a turning point of degree $n$ of the potential, for $n\geq 1$. The turning points and $\lambda=0$ are called the critical points of the potential. Other points are called generic.
Let $\lambda_0$ be a generic point and consider, for any choice of sign, the so called action integral $$S(\lambda_0,\lambda)=\int_{\lambda_0}^{\lambda}\sqrt{V(\mu;\alpha,\beta)}d\mu$$ defined on the universal covering of the $\lambda$-plane minus critical points. Let $i_{\lambda_0}$ be the level curve $\Re S(\lambda_0,\lambda)=0$ through $\lambda_0$ on the universal covering and consider its projection $\widetilde{i}_{\lambda_0}$ onto the $\lambda$ plane minus critical points. Clearly $\widetilde{i}_{\lambda_0}$ cannot self-intersect and is hence either diffeomorphic to a circle or a line. We call $\widetilde{i}_{\lambda_0}$ the level curve through $\lambda_0$.
Note that different level curves cannot intersect and thus the set of level curves forms a complete partition of the $\lambda$-plane minus critical points. We have the following important dichotomy.
\[lem:stokes\_curves\] Let $\lambda_0$ be a generic point of the potential and $\widetilde{i}_{\lambda_0}$ its corresponding level curve. If $\widetilde{i}_{\lambda_0}$ is diffeomorphic to a circle, then it is homotopic to a simple encircling of $\lambda=0$ in the $\lambda$-plane minus critical points. Otherwise $\widetilde{i}_{\lambda_0}$ is diffeomorphic to a line. Let $x\mapsto \gamma_{\lambda_0}(x)$ be a diffeomorphism of $\mathbb{R}$ onto $\widetilde{i}_{\lambda_0}$, then, for $\epsilon\in \{\pm 1\}$, we have the following dichotomy: either
1. $\lim_{x\rightarrow \epsilon \infty}=\infty$ and the curve is asymptotic to one of the four rays $e^{\frac{1}{4}(2k-1)\pi i}\mathbb{R}_+$, $k\in\mathbb{Z}_4$, in the $\lambda$-plane.
2. or $\lim_{x\rightarrow \epsilon \infty}=\lambda_*$ with $\lambda_*$ a turning point of the potential.
See Strebel [@strebel].
In alignment with the above lemma, we make the following definition.
\[defi:dichotomy\] Considering the dichotomy in Lemma \[lem:stokes\_curves\], when $\widetilde{i}_{\lambda_0}$ is diffeomorphic to a line, we call, in case
1. the asymptotic direction $\infty_k:=e^{\frac{1}{4}(2k-1)\pi i}\infty$ an endpoint of $\widetilde{i}_{\lambda_0}$,
2. the corresponding turning point $\lambda_*$ an endpoint of $\widetilde{i}_{\lambda_0}$.
We call $\widetilde{i}_{\lambda_0}$ a Stokes line if at least one endpoint is a turning point.
We define $\mathbb{C}_\infty$ as the complex plane with the addition of four marked points at infinity $\infty_k:=e^{\frac{1}{4}(2k-1)\pi i}\infty$, $1\leq k \leq 4$, with the unique topology making the following map a homeomorphism, $$\begin{aligned}
&L:\mathbb{C}_\infty\rightarrow \mathbb{D}\cup\{e^{\frac{1}{4}(2k-1)\pi i}:1\leq k\leq 4\},\\
&L(\rho e^{i\phi})=\frac{2}{\pi} \arctan(\rho)e^{i\phi}\quad (\rho\in \mathbb{R}_{\geq 0}, \phi \in \mathbb{R}),\\
&L(\infty_k)=e^{\frac{1}{4}(2k-1)\pi i}\quad (1\leq k\leq 4),\end{aligned}$$ where $\mathbb{D}$ denotes the open unit disc. We denote $\mathbb{C}_\infty^*=\mathbb{C}_\infty\setminus\{0\}$.
\[def:stokescomplex\] The Stokes complex ${\mathcal{ C }}={\mathcal{ C }}(\alpha,\beta)\subseteq \mathbb{C}_\infty^*$ of the potential is the union of the marked points at infinity and the Stokes lines and turning points of the potential. The corresponding internal Stokes complex is defined as the union of the turning points and those Stokes lines with only turning points as endpoints.\
We call two Stokes complexes isomorphic if there exists a topological homeomorphism between them which respects the markings. Such an isomorphism always has an extension to an automorphism of $\mathbb{C}_\infty^*$.
From a topological point of view the Stokes complex ${\mathcal{ C }}$ is an embedded graph into $\mathbb{C}_\infty^*$ with vertices equal to the turning points and the four points at infinity, with edges given by the Stokes lines.
In the following proposition we summarise some basic facts concerning the Stokes complexes under consideration.
\[pro:basic\] Let $\mathcal{C}$ be the Stokes complex of the potential , then
- For any turning point $\lambda^*$, say of degree $n$, there are precisely $n+2$ Stokes lines emanating from it, counting Stokes lines with all end points equal to $\lambda_*$ double.
- $\mathcal{C}$ is connected;
- The internal Stokes complex contains precisely one Jordan curve, the interior of which contains $\lambda=0$;
- If two different Stokes lines have the same endpoints, then they are homotopically inequivalent in $\mathbb{C}_\infty^*$.
See [@fedoryuk; @strebel].
As an example, let us consider the Stokes complex corresponding to the potential with $(\alpha,\beta)=(0,0)$. In Figure \[fig:complex00\] its depicted as an embedded graph in $\mathbb{C}_\infty^*$, and in Figure \[fig:approximate\] it is depicted in $\mathbb{C}$, with real turning points $$\begin{aligned}
&\lambda_1=-\sqrt{\tfrac{1}{2}(1+\sqrt{1-\nu^2})},\quad \lambda_2=-\sqrt{\tfrac{1}{2}(1-\sqrt{1-\nu^2})},
\label{eq:turning}\\ &\lambda_3=+\sqrt{\tfrac{1}{2}(1-\sqrt{1-\nu^2})},\quad \lambda_4=+\sqrt{\tfrac{1}{2}(1+\sqrt{1-\nu^2})}.\nonumber\end{aligned}$$
= \[circle, minimum width=4pt, fill, inner sep=0pt\];
(0,0) ellipse (6.3cm and 2.15cm); at (-3,0); at (-3,0) [$\lambda_1$]{};
at (-1.5,0) ; at (-1.5,0) [$\lambda_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\lambda_3$]{};
at (3,0) ; at (3,0) [$\lambda_4$]{};
(-3,0) – (-1.5,0); (1.5,0) – (3,0); (0,0) circle (1.5cm);
(-3,0) – (-4.5,1.5); at (-4.5,1.5) ; at (-4.5,1.5) [$\infty_2$]{};
(3,0) – (4.5,1.5); at (4.5,1.5) ; at (4.5,1.5) [$\infty_1$]{};
(-3,0) – (-4.5,-1.5); at (-4.5,-1.5) ; at (-4.5,-1.5) [$\infty_3$]{};
(3,0) – (4.5,-1.5); at (4.5,-1.5) ; at (4.5,-1.5) [$\infty_4$]{};
For convenience of the reader, we reintroduce the regions $R$, $K$ and $K_a$ in the following definition.
\[defi:K\] We denote by $R\subseteq {\mathbb{ C }}^2$ the set of all $(\alpha,\beta)\in {\mathbb{ C }}^2$ for which the corresponding Stokes complex ${\mathcal{ C }}(\alpha,\beta)$ is isomorphic to ${\mathcal{ C }}(0,0)$. Furthermore, we define
- $K$ as the closure of $R$;
- $R_a$ and $R_b$ as the projection of $R$ onto respectively the $\alpha$-plane and $\beta$-plane, with corresponding projections $\Pi_a$ and $\Pi_b$;
- $K_a$ and $K_b$ as the projection of $K$ onto respectively the $\alpha$-plane and $\beta$-plane.
We call $K_a$ the elliptic region.
We proceed in discussing the mapping $\mathcal{S}$ introduced in . Strictly speaking, the formulas defining $\mathcal{S}$ are only unambiguously defined on the region $R$ and it will require a bit work to show that the formulae have a well-defined continuous extension to the closure $K=\overline{R}$. Firstly, note that $s_1$ and $s_2$ are uniquely defined by for $(\alpha,\beta)\in R$ and indeed, due the the Stokes geometry of the potential on $R$, $$\label{eq:res1s2}
\Re s_1=0,\quad \Re s_2=0.$$ We define the mapping $$\label{def:SR}
\mathcal{S}: R\rightarrow \mathbb{R}^2, (\alpha,\beta)\mapsto (-is_1(\alpha,\beta),-is_2(\alpha,\beta)) \; ,$$ and notice that set $R$ is invariant under negation and complex conjugation, and that $(0,0)
\in R$.
\[lem:sym\] The set $R$ is invariant under complex conjugation $(\alpha,\beta)\mapsto (\overline{\alpha},\overline{\beta})$ and reflection $(\alpha,\beta)\mapsto (-\alpha,-\beta)$. Furthermore, for $(\alpha,\beta)\in R$, $$\begin{aligned}
\mathcal{S}(\overline{\alpha},\overline{\beta})&=\left(\mathcal{S}_1(\alpha,\beta),-\mathcal{S}_2(\alpha,\beta)\right),\\
\mathcal{S}(-\alpha,-\beta)&=\left(-\mathcal{S}_1(\alpha,\beta),-\mathcal{S}_2(\alpha,\beta)\right),
\end{aligned}$$ and in particular $\mathcal{S}(0,0)=(0,0)$.
This follows from the symmetries $$V(\lambda;\overline{\alpha},\overline{\beta})=\overline{V(\overline{\lambda};\alpha,\beta)},\quad
V(\lambda;-\alpha,-\beta)=V(-\lambda;\alpha,\beta).\qedhere$$
The remainder of the present section consists of two subsections. In Subsection \[subsection:1\] we study the sets $R,K$ and the map $\mathcal{S}$. In particular we prove the following
\[pro:extension\] The set $R$ is a smooth $2$-dimensional regular submanifold of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$. The mapping $\mathcal{S}$ has a unique continuous extension $\mathcal{S}:K\rightarrow Q$ which is a homeomorphism and maps $R$ diffeomorphically onto $Q^\circ$.
In Subsection \[subsection:2\] we prove the following
\[pro:projection\] The projection $\Pi_a:K\rightarrow K_a$ is a homeomorphism which maps $R$ diffeomorphically onto $R_a$.
Moreover we study the domain $\Pi_a(K)=K_a \subset {\mathbb{ C }}$: we prove that it is a quadrilateral domain and we prove the explicit description of its boundary as given in Section \[sec:results\].
$K$ and the map $\mathcal{S}$ {#subsection:1}
-----------------------------
We start this subsection by estimating of the range of ${\mathcal{ S }}$.
\[lem:range\] The image of $R$ under $\mathcal{S}$ is contained in the open rectangle $$Q^{\circ}:=\left(-\tfrac{1}{2}(1-\nu)\pi,+\tfrac{1}{2}(1-\nu)\pi\right)\times \left(-\nu\pi,+\nu\pi\right).$$
The meromorphic differential $\omega$ has four poles, $0_{\pm}:= ({\lambda}=0,y=\pm \frac{\nu}{2})$ and $\infty_{\pm}$, and the corresponding residues are easily computed, $$\operatorname*{Res}_{P=0_\pm}\omega=\pm \frac{\nu}{2}, \quad \operatorname*{Res}_{P=\infty_{\pm}}\omega=\mp \frac12 \; .$$ Consider the oriented contours $\delta_i$, $1\leq i\leq 6$, defined as in Figure \[fig:contours\], where the two blue lines are the Stokes line between $\lambda_1$ and $\lambda_2$ and the Stokes line between $\lambda_3$ and $\lambda_4$ acting as branch cuts, such that all the contours lie in the same sheet of the elliptic curve $\widehat{\Gamma}$ where $\omega\sim \tfrac{1}{2}\nu \lambda^{-1}d\lambda$. We define $$\label{eq:defir}
r_i=-i\int_{\delta_i}{\omega}\quad (1\leq i\leq 6),$$ all of which are real and positive due to the Stokes geometry of the potential. Furthermore, $r_1=r_6$ and $r_3=r_4$.
Since $0_+$ and $\infty_-$ lie in the same sheet as the above contours, the residue theorem yields $$\begin{aligned}
r_2+r_5&=2\pi\operatorname*{Res}_{\lambda=0_+}{\omega}=\pi\nu,\\
r_1+r_2+r_3+r_4+r_5+r_6&=2\pi \operatorname*{Res}_{\lambda=\infty_-}{\omega} =\pi.
\end{aligned}$$ It follows in particular that $$\label{eq:r4r6}
r_4+r_6=\tfrac{1}{2}(1-\nu)\pi.$$ Note that $s_2(\alpha,\beta)=(r_5-r_2)i$ and since $r_2+r_5=\pi\nu$ with $r_2,r_5>0$, we have $$-\pi\nu<-is_2(\alpha,\beta)< +\pi\nu.$$ Similarly $s_1(\alpha,\beta)=-2r_4i+\tfrac{1}{2}(1-\nu)\pi i$ and it follows from equation that $0<r_4<\tfrac{1}{2}(1-\nu)\pi$, so $$-\tfrac{1}{2}(1-\nu)\pi<-is_1(\alpha,\beta)< +\tfrac{1}{2}(1-\nu)\pi,$$ which finishes the proof of the lemma.
plot \[smooth cycle,tension=0.8\] coordinates [(1.5,0) (2.25,-0.5) (3, 0) (2.25,0.5) ]{};
plot \[smooth cycle,tension=0.8\] coordinates [(-3,0) (-2.25,-0.5) (-1.5, 0) (-2.25,0.5) ]{};
plot \[smooth,tension=0.6\] coordinates [(-1.5,0) ( [1.59\*cos(-168.75)]{} , [1.59\*sin(-168.75)]{} ) ( [1.63\*cos(-157.5)]{} , [1.63\*sin(-157.5)]{} ) ( [1.65\*cos(-146.25)]{} , [1.65\*sin(-146.25)]{} ) ( [1.65\*cos(-135)]{} , [1.65\*sin(-135)]{} ) ( [1.65\*cos(-123.75)]{} , [1.65\*sin(-123.75)]{} ) ( [1.65\*cos(-112.5)]{} , [1.65\*sin(-112.5)]{} ) ( [1.65\*cos(-101.25)]{} , [1.65\*sin(-101.25)]{} ) (0,-1.65) ( [1.65\*cos(-78.75)]{} , [1.65\*sin(-78.75)]{} ) ( [1.65\*cos(-67.5)]{} , [1.65\*sin(-67.5)]{} ) ( [1.65\*cos(-56.25)]{} , [1.65\*sin(-56.25)]{} ) ( [1.65\*cos(-45)]{} , [1.65\*sin(-45)]{} ) ( [1.65\*cos(-33.75)]{} , [1.65\*sin(-33.75)]{} ) ( [1.63\*cos(-22.5)]{} , [1.63\*sin(-22.5)]{} ) ( [1.59\*cos(-11.25)]{} , [1.59\*sin(-11.25)]{} ) (1.5,0) ]{};
plot \[smooth,tension=0.6\] coordinates [(1.5,0) ( [1.59\*cos(-168.75+180)]{} , [1.59\*sin(-168.75+180)]{} ) ( [1.63\*cos(-157.5+180)]{} , [1.63\*sin(-157.5+180)]{} ) ( [1.65\*cos(-146.25+180)]{} , [1.65\*sin(-146.25+180)]{} ) ( [1.65\*cos(-135+180)]{} , [1.65\*sin(-135+180)]{} ) ( [1.65\*cos(-123.75+180)]{} , [1.65\*sin(-123.75+180)]{} ) ( [1.65\*cos(-112.5+180)]{} , [1.65\*sin(-112.5+180)]{} ) ( [1.65\*cos(-101.25+180)]{} , [1.65\*sin(-101.25+180)]{} ) (0,1.65) ( [1.65\*cos(-78.75+180)]{} , [1.65\*sin(-78.75+180)]{} ) ( [1.65\*cos(-67.5+180)]{} , [1.65\*sin(-67.5+180)]{} ) ( [1.65\*cos(-56.25+180)]{} , [1.65\*sin(-56.25+180)]{} ) ( [1.65\*cos(-45+180)]{} , [1.65\*sin(-45+180)]{} ) ( [1.65\*cos(-33.75+180)]{} , [1.65\*sin(-33.75+180)]{} ) ( [1.63\*cos(-22.5+180)]{} , [1.63\*sin(-22.5+180)]{} ) ( [1.59\*cos(-11.25+180)]{} , [1.59\*sin(-11.25+180)]{} ) (-1.5,0) ]{};
at (-2.25,-0.75) [$\delta_1$]{}; at (0,-1.9) [$\delta_2$]{}; at (2.25,-0.75) [$\delta_3$]{}; at (2.25,0.75) [$\delta_4$]{}; at (0,1.9) [$\delta_5$]{}; at (-2.25,0.75) [$\delta_6$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\lambda_1$]{};
at (-1.5,0) ; at (-1.5,0) [$\lambda_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\lambda_3$]{};
at (3,0) ; at (3,0) [$\lambda_4$]{};
(-3,0) – (-1.5,0); (1.5,0) – (3,0); (0,0) circle (1.5cm);
(-3,0) – (-4.5,1.5); (-4.5,1.5) – (-4.75,1.75);
(3,0) – (4.5,1.5); (4.5,1.5) – (4.75,1.75);
(-3,0) – (-4.5,-1.5); (-4.5,-1.5) – (-4.75,-1.75);
(3,0) – (4.5,-1.5); (4.5,-1.5) – (4.75,-1.75);
at (-4.5,0) [[II]{}]{}; at (4.5,0) [[V]{}]{}; at (2.25,1.6) [[I]{}]{}; at (2.25,-1.6) [[IV]{}]{}; at (0,-0.75) [[III]{}]{};
To proceed further with our analysis we introduce the notion of Boutroux curves [@boutroux; @piwkb; @bertola2011; @bertola2009] tailored to our setting.
\[def:boutroux\] We call the elliptic curve $\widehat{\Gamma}$ a Boutroux curve if $\Re\oint_\gamma\omega=0$ for any closed cycle $\gamma$ in $\widehat{\Gamma}$. We denote by $\Omega$ the set of all $(\alpha,\beta)\in\mathbb{C}^2$ such that $\widehat{\Gamma}(\alpha,\beta)$ is a Boutroux curve.
By definition of the region $R$, we have $R\subseteq \Omega$. In proving Lemma \[lem:deformation\] and Proposition \[pro:extension\], we have to deal with deformations of Boutroux curves for which we have collected the necessary results in Appendix \[append:boutroux\].
Firstly we focus our attention on Lemma \[lem:deformation\]. We make use of the factorisation $$\lambda^2V=(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_3)(\lambda-\lambda_4),$$ where the turning points are defined, up to permutations, by the following algebraic constraints
\[eq:constraint\] $$\begin{aligned}
&\lambda_1\lambda_2\lambda_3\lambda_4=\tfrac{1}{4}\nu^2,\label{eq:constraint1}\\
&\lambda_1\lambda_2+\lambda_1\lambda_3+\lambda_1\lambda_4+\lambda_2\lambda_3+
\lambda_2\lambda_4+\lambda_3\lambda_4=\tfrac{1}{2}(\lambda_1^2+\lambda_2^2+\lambda_3^2+\lambda_4^2)-2,\label{eq:constraint2}\\
&
\tfrac{1}{2}(\lambda_1+\lambda_2+\lambda_3+\lambda_4)=-\alpha, \label{eq:alphaoflambdas}\\ \label{eq:betaoflambdas}
&\tfrac{1}{4}\nu^2 (\lambda_1^{-1}+\lambda_2^{-1}+\lambda_3^{-1}+\lambda_4^{-1})= \beta.\end{aligned}$$
When $(\alpha,\beta)\in R$, we may unambiguously, i.e. not just up to permutation, define the turning points via the Stokes complex and we will always do this in alignment with Figure \[fig:complex00\].
Having introduced the notion of Boutroux curves we proceed with the proof of Proposition \[pro:extension\] - actually with a stronger version of it which we call Theorem \[thm:Sextension\] - together with an explicit description of $K\setminus R$, see Corollary \[cor:edgesbeta\].
Our analysis is divided in several preparatory lemmas. In Lemma \[lem:deformation\] we show that $R$ is a smooth submanifold of ${\mathbb{ C }}^2$, in Lemma \[lem:injective\] we show that ${\mathcal{ S }}$ is injective, in Lemma \[lem:compact\] we show that $K$ is compact, in Lemmas \[lem:deltaR\],\[lem:partition\] we show that $K\setminus R$ is made of singular elliptic curves, and eventually in Lemma \[lem:extension\] we prove that ${\mathcal{ S }}$, as defined in $R$, admits a unique extension to $K$, which maps $K$ to $Q$.
\[lem:deformation\] The set $R$ is a smooth $2$-dimensional regular submanifold of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$ and the mapping $\mathcal{S}$ is locally diffeomorphic[^2] on $R$. Furthermore, for any given $(\alpha^*,\beta^*)\in R$, there exist simply connected open sets $U\subseteq\mathbb{C}$ and $V\subseteq\mathbb{C}$, containing respectively $\alpha^*$ and $\beta^*$, and a diffeomorphism $B:U\rightarrow V$, such that $$R\cap(U\times V)=\{(\alpha,B(\alpha)):\alpha\in U\}.$$
As $(\alpha^*,\beta^*)\in R$, we know that the turning points of $V(\lambda;\alpha^*,\beta^*)$ are all simple. Hence we can take small open discs $U_0\subseteq \mathbb{C}$ and $V_0\subseteq \mathbb{C}$, centred respectively at $\alpha^*$ and $\beta^*$, such that on $U_0\times V_0$ the turning points of $V(\lambda;\alpha,\beta)$ do not coalesce. On this set the turning points $\lambda_i=\lambda_i(\alpha,\beta)$ are analytic in $(\alpha,\beta)$, $1\leq i \leq 4$, and the defining equations of $s_1(\alpha,\beta)$ and $s_2(\alpha,\beta)$ have unique analytic continuation to $U_0\times V_0$, the result of which we denote by $s_1(\alpha,\beta)$ and $s_2(\alpha,\beta)$ as well. The idea of the proof is to make use of the fact that $R$ is locally defined by equations .
Using the standard decomposition into real and imaginary part, $$\alpha=\alpha_R+i\alpha_I,\quad \beta=\beta_R+i\beta_I,\quad s_i=s_i^R+is_i^I\quad (i=1,2),$$ the analytic mapping $$(s_1,s_2):U_0\times V_0\rightarrow \mathbb{C}^2,(\alpha,\beta)\mapsto (s_1(\alpha,\beta),s_2(\alpha,\beta))$$ can be rewritten as a real smooth mapping $$H:(\alpha_R,\alpha_I,\beta_R,\beta_I)\mapsto (s_1^R,s_1^I,s_2^R,s_2^I)$$ on an open environment of $(\alpha_R^*,\alpha_I^*,\beta_R^*,\beta_I^*)$.
Due to equation , we know that $(s_1,s_2)$ is locally biholomorphic near $(\alpha^*,\beta^*)$ and thus $H$ is locally diffeomorphic near $(\alpha_R^*,\alpha_I^*,\beta_R^*,\beta_I^*)$. To prove the thesis, we proceed in showing that the mapping $(\beta_R,\beta_I)\mapsto (s_1^R,s_2^R)$ is a local diffeomorphism near $(\beta_R^*,\beta_I^*)$, for $(\alpha_R,\alpha_I)=(\alpha_R^*,\alpha_I^*)$ fixed.
To this end we apply the implicit function theorem to show that $\{\Re{\alpha},\Im{\alpha}\}$ are good local coordinates on the zero set $$\Re{s_1(\alpha,\beta)}=0,\quad \Re{s_2(\alpha,\beta)}=0.$$ This requires that the Jacobian determinant $$\delta=\delta(\alpha,\beta)=\begin{vmatrix}
\frac{\partial s_1^R}{\partial \beta_R} & \frac{\partial s_1^R}{\partial \beta_I}\\
\frac{\partial s_2^R}{\partial \beta_R} & \frac{\partial s_2^R}{\partial \beta_I}
\end{vmatrix}$$ does not vanish at $(\alpha,\beta)=(\alpha^*,\beta^*)$. By the Cauchy-Riemann equations, we have $\delta=\Im{\left[\frac{\partial s_1}{\partial \beta}\cdot \overline{\frac{\partial s_2}{\partial \beta}}\right]}$. Now, note that $$\frac{\partial s_1}{\partial \beta}=-\frac{1}{2}\int_{\gamma_1}\widetilde{\omega},\quad
\frac{\partial s_2}{\partial \beta}=-\frac{1}{2}\int_{\gamma_2}\widetilde{\omega},$$ where $\widetilde{\omega}$ is the holomorphic differential form $\widetilde{\omega}=\frac{d\lambda}{y}$. As $\gamma_1$ and $\gamma_2$ are homologically independent, $(p_1,p_2):=(-2 \frac{\partial s_1}{\partial \beta},-2\frac{\partial s_2}{\partial \beta})$ form a pair of $\mathbb{R}$-linearly independent periods of the elliptic curve $\widehat{\Gamma}(\alpha,\beta)$. Therefore $p_1\overline{p}_2\notin \mathbb{R}$ and thus $\delta(\alpha,\beta)\neq 0$ on $U_0\times V_0$ and in particular at $(\alpha,\beta)=(\alpha^*,\beta^*)$. We note that this can also be proven using the explicit formulae in Appendix \[append:computation\].
By the implicit function theorem, and the fact that $(s_1,s_2)$ is a local biholomorphism, there exist simply connected open sets $U\subseteq U_0$ and $V\subseteq V_0$ with $\alpha^*\in U$ and $\beta^*\in V$, and a diffeomorphism $B:U\rightarrow V$ such that $$\{(\alpha,B(\alpha)):\alpha\in U\}=\{(\alpha,\beta)\in U\times V: \Re{s_1(\alpha,\beta)}=\Re{s_2(\alpha,\beta)}=0\}$$ and hence in particular $B(\alpha^*)=\beta^*$. Furthermore, note that $$\{(\alpha,B(\alpha)):\alpha\in U\}=\Omega\cap(U\times V).$$
Applying Lemma \[lem:deformboutroux\], it follows that the Stokes complex $\mathcal{C}(\alpha,B(\alpha))$ is homeomorphic to $\mathcal{C}(\alpha^*,\beta^*)$ and hence $(\alpha,B(\alpha))\in R$ for all $\alpha\in U$. Therefore $$\label{eq:ROmega}
R\cap(U\times V)=\Omega\cap(U\times V)=\{(\alpha,B(\alpha)):\alpha\in U\}.$$
By (i), we may choose a simply connected open environment $W\subseteq \{(\Re{\alpha},\Im{\alpha},\Re{\beta},
\Im{\beta})\in\mathbb{R}^4\}$ of $(\alpha^*,\beta^*)$, with $W\subseteq U\times V$, such that $H|_W:W\rightarrow \mathbb{R}^4$ is a diffeomorphism onto its open image $H(W)\subseteq \mathbb{R}^4$. So $(W,H|_W)$ is a local chart of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$ and $$H|_W(R\cap W)=H(W)\cap \{(\sigma_1^R,\sigma_1^I,\sigma_2^R,\sigma_2^I)\in\mathbb{R}^4:\sigma_1^R=\sigma_2^R=0\}.$$ We conclude that $R$ is a smooth 2-dimensional regular submanifold of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$. Furthermore, $(R\cap W,(s_1^I,s_2^I))$ is a local chart of $R$, and as $\mathcal{S}=(s_1^I,s_2^I)$, we find that $\mathcal{S}$ is locally diffeomorphic on $R$.
As a corollary of the previous lemma we obtain that $R_a\subseteq \mathbb{C}$ and $R_b\subseteq \mathbb{C}$ are both open. Furthermore, the turning points $\lambda_i=\lambda_i(\alpha,\beta)$, $1\leq i\leq 4$, are smooth on $R$.
Next we turn our attention to Proposition \[pro:extension\]. To prove it, we need a number of preparatory lemmas.
\[lem:injective\] The mapping $\mathcal{S}$ is injective.
We just give a rough sketch of the proof and refer the interested reader to [@bertola2009; @jenkins] for a complete treatment of similar arguments. Suppose $\mathcal{S}(\alpha,\beta)=\mathcal{S}(\alpha',\beta')$ and let respectively $\lambda_1,\ldots,\lambda_4$ and $\lambda_1',\ldots,\lambda_4'$ be corresponding turning points. Furthermore, let us denote corresponding differentials by $\omega$ and $\omega'$.
Note that the Stokes complex of the potential $V(\lambda,\alpha,\beta)$ naturally cuts the complex $\lambda$-plane into five disjoint open connected regions $I,\ldots,V$ as in Figure \[fig:contours\]. By choosing the sign correctly, the action integral $$S(\lambda)=\int_{\lambda_1}^{\lambda}\omega$$ defines a uniformisation of $I$ (resp. homeomorphism from $\overline{I}$) onto the open (resp. closed) left-half plane, mapping the turning points $\lambda_1,\lambda_2,\lambda_3,\lambda_4$ to the respective marked points $0$, $r_6i$, $(r_5+r_6)i$ and $(r_4+r_5+r_6)i$, where $r_4,r_5,r_6$ as defined in Lemma \[lem:range\].
Similarly, the Stokes complex of the potential $V(\lambda,\alpha',\beta')$ cuts the complex plane into five pieces $I',\ldots, V'$ and $$S'(\lambda)=\int_{\lambda_1'}^{\lambda}\omega'$$ defines a uniformisation of $I'$ (resp. homeomorphism from $\overline{I'}$) onto the open (resp. closed) left-half plane, which maps the turning points $\lambda_1',\lambda_2',\lambda_3',\lambda_4'$ to the same respective marked points $0$, $r_6i$, $(r_5+r_6)i$ and $(r_4+r_5+r_6)i$, as $\mathcal{S}(\alpha,\beta)=\mathcal{S}(\alpha',\beta')$.
Thus $\chi:\overline{I}\rightarrow \overline{I'}$, defined by $$\chi:=S'|_{\overline{I'}}^{-1}\circ S|_{\overline{I}},$$ is a homeomorphism which maps $I$ conformally and bijectively onto $I'$, mapping turning point $\lambda_k$ to $\lambda_k'$ for $1\leq k\leq 4$.
Without going into further detail, one may prove that $\chi$ can be uniquely extended to an automorphism of $\mathbb{P}^1$, mapping each region $J\in \{I,\ldots V\}$ to its corresponding region $J'$ and in particular $$\chi(0)=0,\quad\chi(\infty)=\infty,\quad\chi(\lambda_i)=\lambda_i'\quad (1\leq i\leq 4).$$ It follows that $\chi$ is a dilation. There are only two dilations which preserve equations and , namely $\chi(\lambda)\equiv +\lambda$ and $\chi(\lambda)\equiv -\lambda$. Since by construction $\chi$ leaves the asymptotic direction $e^{-\frac{3}{4}\pi i}\infty$ invariant, we must have $\chi(\lambda)\equiv +\lambda$. In particular $(\alpha',\beta')=(\alpha,\beta)$ by equations and , and we conclude that $\mathcal{S}$ is indeed injective.
\[lem:compact\] The region $K=\overline{R}$ is compact.
Suppose $K$ is not compact, then $R$ is unbounded. Take a sequence $(\alpha_n,\beta_n)_{n>1}$ in $R$ such that $|\alpha_n|+|\beta_n|\rightarrow \infty$ as $n\rightarrow \infty$. Let us write $\widetilde{\beta}_n=\frac{\beta_n}{\alpha_n^3}$ for $n\geq 1$. By replacing $(\alpha_n,\beta_n)_{n>1}$ by an appropriate subsequence if necessary, we may assume that we are in one of the following four scenarios:
1. $\alpha_n\rightarrow \alpha^*\in\mathbb{C}$ and $|\beta_n|\rightarrow \infty$,
2. $|\alpha_n|\rightarrow \infty$ and $|\widetilde{\beta}_n|\rightarrow \infty$,
3. $|\alpha_n|\rightarrow \infty$ and $\widetilde{\beta}_n\rightarrow {\widetilde{\beta}}^*\in\mathbb{C}^*$, or
4. $|\alpha_n|\rightarrow \infty$ and $\widetilde{\beta}_n\rightarrow 0$
as $n\rightarrow \infty$.
Each of the four cases leads to a contradiction in a similar fashion and we therefore limit our discussion to one of them, case (iii). Setting $\lambda=\alpha_n\mu$, we have $$\label{eq:differential}
\sqrt{V(\lambda;\alpha_n,\beta_n)}d\lambda=\alpha_n^2\sqrt{\mu^2+2\mu+1-\widetilde{\beta}_n\mu^{-1}-
\alpha_n^{-2}+\alpha_n^{-4}\frac{\nu^2}{4}\mu^{-2}}d\mu.$$ Let us write the turning points of ${V(\lambda;\alpha_n,\beta_n)}$ by $\lambda_j^n=\lambda_j(\alpha_n,\beta_n)$ and define $\mu_j^n:=\alpha_n^{-1}\lambda_j^n$ for $1\leq j\leq 4$ and $n\geq 1$. Then, by replacing $(\alpha_n,\beta_n)_{n>1}$ by an appropriate subsequence if necessary, we may assume that there exists a permutation $\sigma\in S_4$ such that $$\mu_{\sigma(4)}^n\rightarrow 0,\quad \mu_{\sigma(j)}^n\rightarrow u_j\in\mathbb{C}^*\quad (1\leq j\leq 3)$$ as $n\rightarrow \infty$, where $\{u_1,u_2,u_3\}$ are the roots of $\mu^3+2\mu^2+\mu-{\widetilde{\beta}}^*$.
However, $\mu=0$ is a simple pole of the differential . Recalling definition of $r_4,r_6\in\mathbb{R}_+$ in the proof of Lemma \[lem:range\], we hence obtain, as $n\rightarrow \infty$ and thus $\mu_{\sigma(4)}^n\rightarrow 0$, that either $r_4\rightarrow +\infty$ or $r_6\rightarrow +\infty$, contradicting equation .
We define the border of $R$ by $\delta R=\overline{R}\setminus R$ in $\mathbb{R}^4$, not to be confused with the topological boundary $\partial R$ of $R$, so that $K=R\sqcup \delta R$. In order to extend the mapping $\mathcal{S}$ continuously to $K$, we have understand what happens with the turning points and cycles $\gamma_1,\gamma_2$ upon approaching the border. We thus proceed with studying the border of $R$.
Let us denote the discriminant of the polynomial $$\lambda^4+2\alpha\lambda^3+(\alpha^2-1)\lambda^2-\beta \lambda+\tfrac{1}{4}\nu^2$$ by $$\begin{aligned}
\Delta(\alpha,\beta)=&-27 \beta^4+4\alpha(3-\alpha^2)\beta^3+2\left(3\nu^2(5\alpha^2-6)+2(\alpha^2-1)^2\right)\beta^2\\
&+4\nu^2 \alpha\left(6\nu^2-(1-\alpha^2)(10-\alpha^2)\right)\beta+\nu^2\left(4\nu^4+\nu^2(\alpha^4-20\alpha^2-8)+4(1-\alpha^2)^3\right).\end{aligned}$$ As a first step, in the following lemma we prove that the border of $R$ is characterised by the merging of turning points.
\[lem:deltaR\] We have $\delta R\subseteq \Omega\cap \{\Delta(\alpha,\beta)=0\}$.
Firstly we show that $\delta R\subseteq \Omega$. Let $(\alpha^*,\beta^*)\in \delta R$, then there exists a sequence $(\alpha_n,\beta_n)_{n\geq 1}$ in $R$ such that $(\alpha_n,\beta_n)\rightarrow (\alpha^*,\beta^*)$ as $n\rightarrow \infty$. Take any Jordan curve $\gamma$ in $\mathbb{C}^*$ minus turning points of $V(\lambda;\alpha^*,\beta^*)$, which encircles an even number of turning points counting multiplicity, then $\sqrt{V(\lambda)}$ can be chosen single-valued along $\gamma$ and $$\Re\oint_{\gamma}\sqrt{V(\lambda;\alpha^*,\beta^*)}d\lambda=\lim_{n\rightarrow \infty}
\Re\oint_{\gamma}\sqrt{V(\lambda;\alpha_n,\beta_n)}d\lambda=0.$$ It follows that $(\alpha^*,\beta^*)\in \Omega$ and hence $\delta R\subseteq \Omega$.
Next, let $(\alpha^*,\beta^*)\in \delta R$ and suppose that $\Delta(\alpha^*,\beta^*)\neq 0$. The proof of equation can be used line for line to show that there exists an open environment $W\subseteq \{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$ of $(\alpha^*,\beta^*)$ such that $$W\cap \Omega=\{(\alpha,\beta)\in W:\mathcal{C}(\alpha,\beta)\sim \mathcal{C}(\alpha^*,\beta^*)\}.$$ As $W\cap \Omega\cap R=W\cap R\neq \emptyset$, it follows that $\mathcal{C}(\alpha^*,\beta^*)\sim \mathcal{C}(0,0)$ and hence $(\alpha^*,\beta^*)\in R$. But $R\cap \delta R=\emptyset$ and we have arrived at a contradiction. We conclude that $\Delta(\alpha^*,\beta^*)=0$.
By the above lemma, at points on the border $\delta R$ two or more of the turning points of $V(\lambda)$ have merged. We classify points on the border by the isomorphism class of their Stokes complex. To this end, we introduce the isomorphism classes $E_1,\ldots, E_4$ and $C_1,\ldots, C_4$ in Figure \[fig:stokes\_complexes\]. For convenience, we have included the isomorphism class $G_0$ of the Stokes complex at $(\alpha,\beta)=(0,0)$ in the figure.
The possible mergers of turning points near the border are heavily constrained, since two turning points can only merge if they are connected by a Stokes line, due to Lemma \[lem:double\], and Lemma \[lem:triple\] gives a similar constraint for the merging of three turning points. This in turn constraints the possible Stokes complexes attainable at the border and the following lemma shows that the isomorphism classes introduced in Figure \[fig:stokes\_complexes\] indeed suffice to cover the entire border.
[ c | c | c ]{}
at (2.5,1.25) [$C_2$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (3,0); at (3,0) [$\eta_1$]{};
at (0,0) ; at (0,0) [0]{};
at (-0.75,-0.75) ; at (-0.78,-0.75) [$\eta_2$]{};
(3,0) – (4.75,1.75);
(3,0) – (4.75,-1.75);
plot \[smooth,tension=0.6\] coordinates [(3,0) (2,-0.1) (0.6,-1.4) (-0.75,-0.75) ]{};
plot \[smooth,tension=0.9\] coordinates [(-0.75,-0.75) (-0.5,0.6) (0.85,0.85) (0.5,-0.5) (-0.75,-0.75) ]{};
plot \[smooth,tension=0.8\] coordinates [(-0.75,-0.75) (-1,-0.5) (-1,0.8) (-1.75,1.75) ]{}; plot \[smooth,tension=0.8\] coordinates [(-0.75,-0.75) (-1.75,-1.75) ]{};
&
at (-2.45,1.125) [$E_2$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\mu_1$]{};
at (0,-1) ; at (0,-1) [$\mu_3$]{};
at (0,0) ; at (0,0) [0]{};
at (3,0) ; at (3,0) [$\mu_2$]{};
plot \[smooth,tension=0.6\] coordinates [(-3,0) (-2.5,-0.2) (-1,-1.3) (0,-1) ]{}; plot \[smooth,tension=0.6\] coordinates [(0,-1) (1,0) (0,1) (-1,0) (0,-1) ]{};
plot \[smooth,tension=0.6\] coordinates [(3,0) (2.5,-0.2) (1,-1.3) (0,-1) ]{};
(-3,0) – (-4.75,1.75);
(3,0) – (4.75,1.75);
(-3,0) – (-4.75,-1.75);
(3,0) – (4.75,-1.75);
plot \[smooth cycle,tension=0.7\] coordinates [(4.3,0) (3,-0.7) (2,-1.5) (1,-1.5) (0,-1) (1,0) (2,0.3) (3,0.7) ]{};
at (2.35,1) [$\boldsymbol{\gamma_1}$]{};
&
at (-2.5,1.25) [$C_1$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\eta_1$]{};
at (0,0) ; at (0,0) [0]{};
at (0.75,-0.75) ; at (0.78,-0.75) [$\eta_2$]{};
(-3,0) – (-4.75,1.75);
(-3,0) – (-4.75,-1.75);
plot \[smooth,tension=0.6\] coordinates [(-3,0) (-2,-0.1) (-0.6,-1.4) (0.75,-0.75) ]{};
plot \[smooth,tension=0.9\] coordinates [(0.75,-0.75) (0.5,0.6) (-0.85,0.85) (-0.5,-0.5) (0.75,-0.75) ]{};
plot \[smooth,tension=0.8\] coordinates [(0.75,-0.75) (1,-0.5) (1,0.8) (1.75,1.75) ]{}; plot \[smooth,tension=0.8\] coordinates [(0.75,-0.75) (1.75,-1.75) ]{};
\
at (2.5,1.25) [$E_3$]{}; (1.75,0) arc (0:180:1.625cm and 1.75cm); (-1.5,0) arc (-180:0:1.625cm and 1.75cm);
= \[circle, minimum width=4pt, fill, inner sep=0pt\];
at (-1.5,0) ; at (-1.5,0) [$\mu_3$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\mu_2$]{};
at (3,0) ; at (3,0) [$\mu_1$]{};
(1.5,0) – (3,0); (0,0) circle (1.5cm);
(-1.5,0) – (-3.25,1.75);
(3,0) – (4.75,1.75);
(-1.5,0) – (-3.25,-1.75);
(3,0) – (4.75,-1.75);
at (0.05,2.1) [$\boldsymbol{\gamma_2}$]{};
&
at (-2.5,1.25) [$G_0$]{}; (1.75,0) arc (0:180:1.75cm); (-1.75,0) arc (-180:0:1.75cm);
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\lambda_1$]{};
at (-1.5,0) ; at (-1.5,0) [$\lambda_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\lambda_3$]{};
at (3,0) ; at (3,0) [$\lambda_4$]{};
(-3,0) – (-1.5,0); (1.5,0) – (3,0); (0,0) circle (1.5cm);
(-3,0) – (-4.75,1.75);
(3,0) – (4.75,1.75);
(-3,0) – (-4.75,-1.75);
(3,0) – (4.75,-1.75);
(2.25,0) ellipse (1.85cm and 0.7cm) ;
at (2.35,1.0) [$\boldsymbol{\gamma_1}$]{}; at (0.05,2.1) [$\boldsymbol{\gamma_2}$]{};
&
at (-2.5,1.25) [$E_1$]{}; (1.5,0) arc (0:180:1.625cm and 1.75cm); (-1.75,0) arc (-180:0:1.625cm and 1.75cm);
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\mu_1$]{};
at (-1.5,0) ; at (-1.5,0) [$\mu_2$]{};
at (0,0) ; at (0,0) [0]{};
at (1.5,0) ; at (1.5,0) [$\mu_3$]{};
(-3,0) – (-1.5,0); (0,0) circle (1.5cm);
(-3,0) – (-4.75,1.75);
(1.5,0) – (3.25,1.75);
(-3,0) – (-4.75,-1.75);
(1.5,0) – (3.25,-1.75);
at (0.05,2.1) [$\boldsymbol{\gamma_2}$]{};
\
at (2.5,1.25) [$C_3$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (3,0); at (3,0) [$\eta_1$]{};
at (0,0) ; at (0,0) [0]{};
at (-0.75,0.75) ; at (-0.78,0.75) [$\eta_2$]{};
(3,0) – (4.75,1.75);
(3,0) – (4.75,-1.75);
plot \[smooth,tension=0.6\] coordinates [(3,0) (2,0.1) (0.6,1.4) (-0.75,0.75) ]{};
plot \[smooth,tension=0.9\] coordinates [(-0.75,0.75) (-0.5,-0.5) (0.85,-0.85) (0.5,0.6) (-0.75,0.75) ]{};
plot \[smooth,tension=0.8\] coordinates [(-0.75,0.75) (-1,0.5) (-1,-0.8) (-1.75,-1.75) ]{}; plot \[smooth,tension=0.8\] coordinates [(-0.75,0.75) (-1.75,1.75) ]{};
&
at (-2.5,1.2) [$E_4$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\mu_1$]{};
at (0,1) ; at (0,1) [$\mu_3$]{};
at (0,0) ; at (0,0) [0]{};
at (3,0) ; at (3,0) [$\mu_2$]{};
plot \[smooth,tension=0.6\] coordinates [(-3,0) (-2.5,0.2) (-1,1.3) (0,1) ]{}; plot \[smooth,tension=0.6\] coordinates [(0,1) (1,0) (0,-1) (-1,0) (0,1) ]{};
plot \[smooth,tension=0.6\] coordinates [(3,0) (2.5,0.2) (1,1.3) (0,1) ]{};
(-3,0) – (-4.75,1.75);
(3,0) – (4.75,1.75);
(-3,0) – (-4.75,-1.75);
(3,0) – (4.75,-1.75);
plot \[smooth cycle,tension=0.7\] coordinates [(4.3,0) (3,0.7) (2,1.5) (1,1.5) (0,1) (1,0) (2,-0.3) (3,-0.7) ]{};
at (2.35,1.75) [$\boldsymbol{\gamma_1}$]{};
&
at (-2.5,1.25) [$C_4$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (-3,0); at (-3,0) [$\eta_1$]{};
at (0,0) ; at (0,0) [0]{};
at (0.75,0.75) ; at (0.78,0.75) [$\eta_2$]{};
(-3,0) – (-4.75,1.75);
(-3,0) – (-4.75,-1.75);
plot \[smooth,tension=0.6\] coordinates [(-3,0) (-2,0.1) (-0.6,1.4) (0.75,0.75) ]{};
plot \[smooth,tension=0.9\] coordinates [(0.75,0.75) (0.5,-0.5) (-0.85,-0.85) (-0.5,0.6) (0.75,0.75) ]{};
plot \[smooth,tension=0.8\] coordinates [(0.75,0.75) (1,0.5) (1,-0.8) (1.75,-1.75) ]{}; plot \[smooth,tension=0.8\] coordinates [(0.75,0.75) (1.75,1.75) ]{};
\[lem:partition\] For any point $(\alpha,\beta)\in \delta R$ on the border the associated Stokes complex $\mathcal{C}(\alpha,\beta)$ falls in one of the classes $E_k,1\leq k\leq 4$ or $C_k,1\leq k\leq 4$, introduced in Figure \[fig:stokes\_complexes\].
Let $(\alpha^*,\beta^*)\in \delta R$, then there exists a sequence $(\alpha_n,\beta_n)_{n\geq 1}$ in $R$ such that $(\alpha_n,\beta_n)\rightarrow (\alpha^*,\beta^*)$ as $n\rightarrow \infty$. Let $\{\mu_1,\ldots,\mu_m\}$ be the turning points of $V(\lambda;\alpha^*,\beta^*)$, discarding multiplicity, so that $2\leq m\leq 4$. By Lemma \[lem:deltaR\], we know that $m=2$ or $m=3$. Let us denote the turning points of $V(\lambda;\alpha_n,\beta_n)$ by $\lambda_j^n=\lambda_j(\alpha_n,\beta_n)$ for $1\leq j\leq 4$ and $n\geq 1$, then we may assume, by replacing $(\alpha_n,\beta_n)_{n\geq 1}$ by an appropriate subsequence if necessary, that there exists a surjective mapping $\sigma:\{1,2,3,4\}\rightarrow \{1,2,\ldots,m\}$ such that $\lambda_j^n\rightarrow \mu_{\sigma(j)}$ as $n\rightarrow \infty$ for $1\leq j\leq 4$.
Let us first consider the case $m=3$ and label the turning points of $V(\lambda;\alpha^*,\beta^*)$ such that $\mu_1$ and $\mu_2$ are simple and $\mu_3$ is the double turning point. Applying Lemma \[lem:double\], either
1. $\lambda_1^n,\lambda_2^n\rightarrow \mu_3$ as $n\rightarrow \infty$,
2. $\lambda_2^n,\lambda_3^n\rightarrow \mu_3$ as $n\rightarrow \infty$, or
3. $\lambda_3^n,\lambda_4^n\rightarrow \mu_3$ as $n\rightarrow \infty$.
In case (i), it follows from Lemma \[lem:deformationlines\] that
- $\mu_3$ has one emanating Stokes line asymptotic to $e^{+\frac{3}{4}\pi i}\mathbb{R}_+$ and another asymptotic to $e^{-\frac{3}{4}\pi i}\mathbb{R}_+$,
- $\mu_1$ or $\mu_2$ has one emanating Stokes line asymptotic to $e^{+\frac{1}{4}\pi i}\mathbb{R}_+$ and another asymptotic to $e^{-\frac{1}{4}\pi i}\mathbb{R}_+$.
It is now easy to see that $\mathcal{C}(\alpha^*,\beta^*)\in E_3$, using Proposition \[pro:basic\], Lemma \[lem:boutrouxclas\] and the fact that $(\alpha^*,\beta^*)\in \Omega$ by Lemma \[lem:deltaR\].
In case (iii) it follows analogously that $\mathcal{C}(\alpha^*,\beta^*)\in E_1$.
In case (ii), it follows from Lemma \[lem:deformationlines\] that we may, if necessary, renumber $\{\mu_1,\mu_2\}$ such that
- $\mu_1$ has one emanating Stokes line asymptotic to $e^{+\frac{3}{4}\pi i}\mathbb{R}_+$ and another asymptotic to $e^{-\frac{3}{4}\pi i}\mathbb{R}_+$,
- $\mu_2$ has one emanating Stokes line asymptotic to $e^{+\frac{1}{4}\pi i}\mathbb{R}_+$ and another asymptotic to $e^{-\frac{1}{4}\pi i}\mathbb{R}_+$.
It is now easy to see that either $\mathcal{C}(\alpha^*,\beta^*)\in E_2$ or $\mathcal{C}(\alpha^*,\beta^*)\in E_4$, using Proposition \[pro:basic\] and Lemma \[lem:boutrouxclas\].
Next, let us consider $m=2$ and label the turning points of $V(\lambda;\alpha^*,\beta^*)$ so that $\mu_1$ is simple and $\mu_2$ is the triple turning point. By Lemma \[lem:triple\], either
1. $\lambda_1^n,\lambda_2^n,\lambda_3^n\rightarrow \mu_2$ as $n\rightarrow \infty$, or
2. $\lambda_2^n,\lambda_3^n,\lambda_4^n\rightarrow \mu_2$ as $n\rightarrow \infty$.
Similarly as above, it follows that in case (1) that either $\mathcal{C}(\alpha^*,\beta^*)\in C_2$ or $\mathcal{C}(\alpha^*,\beta^*)\in C_3$, and in case (2) that either $\mathcal{C}(\alpha^*,\beta^*)\in C_1$ or $\mathcal{C}(\alpha^*,\beta^*)\in C_4$.
We define $$\widetilde{e}_k=\{(\alpha,\beta)\in \delta R:\mathcal{C}(\alpha,\beta)\in E_k\},
\quad \widetilde{c}_k=\{(\alpha,\beta)\in \delta R:\mathcal{C}(\alpha,\beta)\in C_k\}\quad (1\leq k\leq 4),$$ then Lemma \[lem:partition\] implies the following partition of the border, $$\label{eq:disjoint}
\delta R=\widetilde{e}_1\sqcup \widetilde{e}_2\sqcup \widetilde{e}_3\sqcup
\widetilde{e}_4\sqcup \widetilde{c}_1\sqcup \widetilde{c}_2\sqcup \widetilde{c}_3\sqcup \widetilde{c}_4.$$
Let us denote by $l_+$ and $l_-$ respectively the upper and lower Stokes line with respect to $0$, as depicted in Figure \[fig:complex00\], with endpoints $\lambda_2$ and $\lambda_3$. Then we have the following corollary of Lemma \[lem:partition\] which characterises the separate parts of the border in the disjoint union .
\[cor:merging\] Near the edges $\widetilde{e}_k$, $1\leq k\leq 4$, two of the turning points of $V(\lambda)$ merge,
- here $\lambda_3$ and $\lambda_4$ merge,
- here $\lambda_2$ and $\lambda_3$ merge via the vanishing of $l_-$,
- here $\lambda_2$ and $\lambda_3$ merge via the vanishing of $l_+$,
- here $\lambda_1$ and $\lambda_2$ merge.
Similarly, the corners $\widetilde{c}_k$ are characterised by the merging of three turning points,
- here $\lambda_2$, $\lambda_3$ and $\lambda_4$ merge via the vanishing of $l_-$,
- here $\lambda_1$, $\lambda_2$ and $\lambda_3$ merge via the vanishing of $l_-$,
- here $\lambda_1$, $\lambda_2$ and $\lambda_3$ merge via the vanishing of $l_+$,
- here $\lambda_2$, $\lambda_3$ and $\lambda_4$ merge via the vanishing of $l_+$.
Let $(\alpha^*,\beta^*)\in \widetilde{e}_1$ and let $(\alpha_n,\beta_n)_{n\geq 1}$ be a sequence in $R$ which converges to $(\alpha^*,\beta^*)$. Let $\lambda_k^n=\lambda_k(\alpha_n,\beta_n)$ denote the turning points of $V(\lambda;\alpha_n,\beta_n)$ for $1\leq k\leq 4$ and $n\geq 1$, and $\mu_1,\mu_2,\mu_3$ denote the turning points of $V(\lambda;\alpha^*,\beta^*)$ as in Figure \[fig:stokes\_complexes\] for the class $E_1$.
We have to show that $\lambda_k^n\rightarrow \mu_{\sigma_0(k)}$ for $1\leq k\leq 4$, where $\sigma_0(k)=k$ for $1\leq k\leq 3$ and $\sigma_0(4)=3$. Suppose this is not the case, then there exists a subsequence $(\alpha_{n_m},\beta_{n_m})_{m\geq 1}$ together with a surjective mapping $\sigma:\{1,2,3,4\}\rightarrow \{1,2,3\}$ not equal to $\sigma_0$, such that $\lambda_k^{m_n}\rightarrow \mu_{\sigma(k)}$ as $n\rightarrow \infty$ for $1\leq k\leq 4$. But, by applying the argument in Lemma \[lem:partition\] to the subsequence $(\alpha_{n_m},\beta_{n_m})_{m\geq 1}$, it then follows that the Stokes complex $\mathcal{C}(\alpha^*,\beta^*)$ must fall in one of the classes $E_2,E_3$ or $E_4$, which contradicts that $(\alpha^*,\beta^*)\in \widetilde{e}_1$. We conclude that $\lambda_k^n\rightarrow \mu_{\sigma_0(k)}$ for $1\leq k\leq 4$ so that $\widetilde{e}_1$ is indeed characterised by the merging of the turning points $\lambda_3$ and $\lambda_4$. The other characterisations are shown analogously.
Now that we know which turning points merge on the different parts of the border, it is clear how to extend $\mathcal{S}$ continuously to $K$, as detailed in the proof of the following lemma.
\[lem:extension\] The mapping $\mathcal{S}$ on $R$ has a unique continuous extension to $K$, $$\mathcal{S}:K\rightarrow Q,\quad Q=\left[-\tfrac{1}{2}(1-\nu)\pi,+\tfrac{1}{2}(1-\nu)\pi\right]\times \left[-\nu\pi,+\nu\pi\right],$$ with $$\begin{aligned}
\mathcal{S}(\widetilde{e}_k)\subseteq \widehat{e}_k,\quad \mathcal{S}(\widetilde{c}_k)\subseteq \{\widehat{c}_k\}\quad (1\leq k \leq 4),
\end{aligned}$$ where the open edges and corners $\widehat{e}_k, \widehat{c}_k$, $1\leq k\leq 4$ of $Q$ are defined as in Figure \[fig:marked\].\
Furthermore, for $1\leq k\leq 4$, the ‘edge’ $\widetilde{e}_k$ is a smooth $1$-dimensional regular submanifold of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$ and the restricted mapping $$\label{eq:edgediffeo}
\mathcal{S}|_{\widetilde{e}_k}:\widetilde{e}_k\rightarrow \widehat{e}_k$$ is locally diffeomorphic.
We first discuss the extension of $\mathcal{S}$ to $R\cup \widetilde{e}_1$. Let $(\alpha,\beta)\in \widetilde{e}_1$ and take any sequence $(\alpha_n,\beta_n)_{n\geq 1}$ in $R$ with $(\alpha_n,\beta_n)\rightarrow (\alpha^*,\beta^*)$ as $n\rightarrow \infty$. We label the turning points $\{\mu_1,\mu_2,\mu_3\}$ of $V(\lambda;\alpha,\beta)$ such that $\mu_3$ is the double turning point, $\mu_2$ is the unique simple turning point adjacent to $\mu_3$ and $\mu_1$ is the remaining simple turning point, in accordance with Figure \[fig:stokes\_complexes\].
Let us denote the turning points of $V(\lambda;\alpha_n,\beta_n)$ by $\lambda_j^n=\lambda_j(\alpha_n,\beta_n)$ for $1\leq j\leq 4$ and $n\geq 1$. Due to Corollary \[cor:merging\], $$\lambda_1^n\rightarrow \mu_1,\quad \lambda_2^n\rightarrow \mu_2,\quad \lambda_3^n\rightarrow \mu_3,\quad \lambda_4^n\rightarrow \mu_3$$ as $n\rightarrow \infty$ and thus $$s_1(\alpha_n,\beta_n)\rightarrow +\tfrac{1}{2}(1-\nu)\pi i,\qquad s_2(\alpha_n,\beta_n)\rightarrow
s_2^{e_1}(\alpha,\beta)\quad (n\rightarrow \infty),$$ where $$\label{eq:defiS2}
s_2^{e_1}(\alpha,\beta)=\int_{\gamma_2}\omega$$ with $\gamma_2$ as defined in Figure \[fig:stokes\_complexes\] on the class $E_1$. In conclusion, $$\label{eq:limitSn}
\mathcal{S}(\alpha_n,\beta_n)\rightarrow (+\tfrac{1}{2}(1-\nu)\pi, -is_2^{e_1}(\alpha,\beta))\quad (n\rightarrow \infty)$$ and we therefore set $$\mathcal{S}(\alpha,\beta)=(+\tfrac{1}{2}(1-\nu)\pi, -is_2^{e_1}(\alpha,\beta)).$$ It is easy to see that $\mathcal{S}(\alpha,\beta)\in \widehat{e}_1$, following the lines in the proof of Lemma \[lem:range\].
To prove continuity of $\mathcal{S}$ on $R\cup \widetilde{e}_1$, it remains to be shown that $s_2^{e_1}$ is continuous on $\widetilde{e}_1$. We proceed in showing that $\widetilde{e}_1$ is a regular smooth submanifold of $\mathbb{R}^4$ and that $s_2^{e_1}$ is locally diffeomorphic on $\widetilde{e}_1$, which in particular implies the required continuity.
Let $(\alpha^*,\beta^*)\in \widetilde{e}_1$, then $\Delta(\alpha^*,\beta^*)=0$. Take a simply connected open environment $U$ of $\alpha^*$, a simply connected open environment $V$ of $\beta^*$ and a biholomorphism $B_\Delta:U\rightarrow V$ such that $$W:=\{\Delta(\alpha,\beta)=0\}\cap (U\times V) = \{(\alpha,B_\Delta(\alpha)):\alpha\in U\}.$$ Note that $W$ is a regular complex submanifold of $\mathbb{C}^2$ and $\widetilde{e}_1\cap (U\times V) \subseteq W$, which allows us to study $\widetilde{e}_1$ locally within $W$. To this end, we extend $s_2^{e_1}$ to a analytic mapping on $W$ and show that it is biholomorphic at $(\alpha^*,\beta^*)$.
Let $\{\mu_1^*,\mu_2^*,\mu_3^*\}$ be the turning points of $V(\lambda;\alpha^*,\beta^*)$, labelled in accordance with Figure \[fig:stokes\_complexes\] as above. Let $D_j\subseteq \mathbb{C}^*$ with $\mu_j^*\in D_j$, $1\leq j\leq 3$, be mutually disjoint open discs. We define the following analytic function on $D_1\times D_2\times D_3$, $$\widetilde{S}_2(\mu_1,\mu_2,\mu_3)=\oint_{\gamma_2}{\frac{\sqrt{(\mu-\mu_1)(\mu-\mu_2)}(\mu-\mu_3)}{\mu}d\mu},$$ with branch and contour chosen consistently with $s_2^{e_1}$ in equation .\
For $1\leq j\leq 3$, there exists a unique analytic function $\mu_j: W\rightarrow \mathbb{C}^*$ with $\mu_j(\alpha^*,\beta^*)=\mu_j^*$, $1\leq j\leq 3$, such that $\mu_j(\alpha,\beta)$ is a turning point of $V(\lambda;\alpha,\beta)$ for $(\alpha,\beta)\in W$. We extend $s_2^{e_1}$, defined in , analytically to $W$, by setting $$s_2^{e_1}(\alpha,\beta)=s_2^{e_1}(\alpha,B_\Delta(\alpha))=\widetilde{S}_2(\mu_1(\alpha,B_\Delta(\alpha)),\mu_2(\alpha,B_\Delta(\alpha)),\mu_3(\alpha,B_\Delta(\alpha))),$$ for $(\alpha,\beta)\in W$.
Now note that $$\Omega\cap W=\{(\alpha,\beta)\in W:\Re{s_2^{e_1}(\alpha,\beta)}=0\}.$$ We wish to show that $s_2^{e_1}:W\rightarrow \mathbb{C}$ is locally biholomorphic at $(\alpha^*,\beta^*)$, i.e. $$\label{eq:derivative}
\frac{\partial}{\partial \alpha}s_2^{e_1}(\alpha,B_\Delta(\alpha))$$ does not vanish at $\alpha=\alpha^*$.
In order to compute , we introduce, inspired by and , $$\begin{aligned}
\widetilde{S}_3(\mu_1,\mu_2,\mu_3)&=\mu_1\mu_2\mu_3^2-\tfrac{1}{4}\nu^2,\\
\widetilde{S}_4(\mu_1,\mu_2,\mu_3)&=\mu_1\mu_2+2\mu_1\mu_3+2\mu_2\mu_3-\tfrac{1}{2}(\mu_1^2+\mu_2^2)+2,
\end{aligned}$$ and compute the Jacobian $$|J_{(\widetilde{S}_2,\widetilde{S}_3,\widetilde{S}_4)}(\mu)|=
\pm 2 \mu_3(\mu_2-\mu_1)(\mu_3-\mu_1)^{\frac{3}{2}}(\mu_3-\mu_2)^{\frac{3}{2}},
\quad J_{(\widetilde{S}_2,\widetilde{S}_3,\widetilde{S}_4)}(\mu):=
\left(\frac{\partial \widetilde{S}_{m+1}}{\partial\mu_n}\right)_{1\leq m,n\leq 3}.$$ It follows that $M:=J_{(\widetilde{S}_2,\widetilde{S}_3,\widetilde{S}_4)}(\mu^*)$ is invertible. By the chain rule, we have $$\begin{pmatrix}
\frac{\partial \widetilde{S}_2}{\partial\alpha} \\
\frac{\partial \widetilde{S}_3}{\partial\alpha} \\
\frac{\partial \widetilde{S}_4}{\partial\alpha}
\end{pmatrix}=
J_{(\widetilde{S}_2,\widetilde{S}_3,\widetilde{S}_4)}(\mu)\cdot \begin{pmatrix}
\frac{\partial \mu_1}{\partial\alpha}\\
\frac{\partial \mu_2}{\partial\alpha}\\
\frac{\partial \mu_3}{\partial\alpha}
\end{pmatrix}$$ which, when specialised to $(\alpha,\beta)=(\alpha^*,\beta^*)$, gives $$\begin{pmatrix}
\frac{\partial}{\partial \alpha}s_2^{e_1}(\alpha,B(\alpha))|_{\alpha=\alpha^*} \\
0\\
0
\end{pmatrix}=
M\cdot \begin{pmatrix}
\frac{\mu_1^*}{\mu_3^*-\mu_1^*}\\
\frac{\mu_2^*}{\mu_3^*-\mu_2^*}\\
\frac{\mu_3^*(\mu_1^*+\mu_2^*-2\mu_3^*)}{2(\mu_3^*-\mu_1^*)(\mu_3^*-\mu_2^*)}
\end{pmatrix}$$ Since $M$ is invertible, it thus follows that does not vanishes at $\alpha=\alpha^*$ and hence $s_2^{e_1}(\alpha,\beta)$ is a local biholomorphism at $(\alpha,\beta)=(\alpha^*,\beta^*)$. Therefore, there exists a simply connected open environment $W_0\subseteq W$ of $(\alpha^*,\beta^*)$ such that $$(F_1,F_2):W_0\rightarrow \mathbb{R}^2,(\alpha,\beta)\mapsto (\Re \mathcal{S}_2(\alpha,\beta),\Im \mathcal{S}_2(\alpha,\beta))$$ is a diffeomorphism onto its open image $F(W_0)\subseteq \mathbb{R}^2$ with $$F(W_0)\cap(\{0\}\times \mathbb{R})=\{0\}\times I,$$ where $I\subseteq \mathbb{R}$ is an open interval. We apply Lemma \[lem:deformboutroux\] with $$T:=\{(\alpha,\beta)\in W_0:\Re \mathcal{S}_2(\alpha,\beta)=0\},$$ giving $$T=W_0\cap \Omega =W_0\cap \widetilde{e}_1.$$ Let us in particular note that, by choosing an open environment $Z\subseteq \{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$ of $(\alpha^*,\beta^*)$ with $Z\cap \{\Delta(\alpha,\beta)=0\}\subseteq W_0$, we have $$\label{eq:edge_isolation}
Z\cap \{\Delta(\alpha,\beta)=0\}\cap \Omega=Z\cap \widetilde{e}_1.$$
We conclude that $\widetilde{e}_1$ is a smooth $1$-dimensional regular submanifold of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$. Furthermore $F_2|_T$ maps $T$ diffeomorphically onto $I$. As $\mathcal{S}|_T=F_2|_T$, it follows that $\mathcal{S}|_{\widetilde{e}_1}:\widetilde{e}_1\rightarrow \widehat{e}_1$ is locally diffeomorphic at $(\alpha,\beta)=(\alpha^*,\beta^*)$ and in particular continuous. Combined with , this shows that the extension of $\mathcal{S}$ to $R\cup \widetilde{e}_1$ is indeed continuous.
Next, we will extend $\mathcal{S}$ to $R\cup \widetilde{e}_1\cup \widetilde{e}_2$ in much the same way. For $(\alpha,\beta)\in \widetilde{e}_2$, we define $$\mathcal{S}(\alpha,\beta)=(-is_1^{e_2}(\alpha,\beta),+\nu \pi),$$ where $$\label{eq:defiS1}
s_1^{e_2}(\alpha,\beta)=\int_{\gamma_1}\omega+\frac{i \pi (1-\nu)}{2}$$ with $\gamma_1$ as defined in Figure \[fig:stokes\_complexes\] on the class $E_2$, and the branch chosen consistently with the definition of $s_1$. Analogously to the above, it is shown that $\widetilde{e}_2$ is a smooth $1$-dimensional regular submanifold of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$, and that $\mathcal{S}|_{\widetilde{e}_2}\widetilde{e}_2\rightarrow \widehat{e}_2$ is locally diffeomorphic.
Continuity of the extension $\mathcal{S}$ to $R\cup \widetilde{e}_1\cup \widetilde{e}_2$ on $\widetilde{e}_2$ is proven as it was on $\widetilde{e}_1$, noting that $\overline{\widetilde{e}_1}\cap \widetilde{e}_2=\emptyset$ by equation .
Similarly, it is shown that $\widetilde{e}_3$ and $\widetilde{e}_4$ are smooth $1$-dimensional regular submanifolds, we have $\overline{\widetilde{e}_i}\cap \widetilde{e}_j=\emptyset$ for $1\leq i,j\leq 4$ with $i\neq j$, and we extend $\mathcal{S}$ to $R\cup \widetilde{e}_1\cup \widetilde{e}_2\cup \widetilde{e}_3\cup \widetilde{e}_4$, such that $\mathcal{S}(\widetilde{e}_i)\subseteq \widehat{e}_i$ and $\mathcal{S}|_{\widetilde{e}_i}\widetilde{e}_i\rightarrow \widehat{e}_i$ is locally diffeomorphic for $1\leq i\leq 4$.
Finally, we define $\mathcal{S}(\alpha,\beta)=\widehat{c}_i$ if $(\alpha,\beta)\in \widetilde{c}_i$ for $1\leq i\leq 4$, so that $\mathcal{S}:K\rightarrow Q$, and it remains to be shown that $\mathcal{S}$ is continuous at such points.
Let $(\alpha,\beta)\in c_1$ and let us denote the simple and triple turning point of $V(\lambda;\alpha,\beta)$ by respectively $\eta_1$ and $\eta_2$, in accordance with Figure \[fig:stokes\_complexes\]. Suppose $(\alpha_n,\beta_n)_{n\geq 1}$ is a sequence in $R$ such that $(\alpha_n,\beta_n)\rightarrow (\alpha,\beta)$ as $n\rightarrow \infty$ and let us denote the turning points of $V(\lambda;\alpha_n,\beta_n)$ by $\lambda_j^n=\lambda_j(\alpha_n,\beta_n)$ for $1\leq j\leq 4$ and $n\geq 1$. It follows from Lemmas \[lem:deformationlines\] and \[lem:triple\] that $$\lambda_1^n\rightarrow \eta_1,\quad \lambda_j^n\rightarrow \eta_2\quad (2\leq j\leq 4)$$ and hence $$\mathcal{S}(\alpha_n,\beta_n)=(-is_1(\alpha_n,\beta_n),-is_2(\alpha_n,\beta_n))\rightarrow \widehat{c}_1$$ as $n\rightarrow \infty$.
Similarly, if $(\alpha_n,\beta_n)_{n\geq 1}$ is a sequence in $\widetilde{e}_1$ which converges to $(\alpha,\beta)$, we denote the turning points of $V(\lambda;\alpha_n,\beta_n)$ by $\mu_j^n$, $1\leq j\leq 3$ in accordance with Figure \[fig:stokes\_complexes\], and Lemma \[lem:triple\] yields $$\mu_1^n\rightarrow \eta_1,\quad \mu_j^n\rightarrow \eta_2\quad (2\leq j\leq 3)$$ and thus $$\mathcal{S}(\alpha_n,\beta_n)=(+\tfrac{1}{2}(1-\nu)\pi,-is_2^{e_1}(\alpha_n,\beta_n))\rightarrow \widehat{c}_1$$ as $n\rightarrow \infty$.
We handle sequences in $\widetilde{e}_2$ converging to $(\alpha^*,\beta^*)$ similarly and finally note that the sets $\widetilde{c}_i$,$1,\leq i\leq 4$ are discrete and $(\alpha,\beta)\notin \overline{\widetilde{e}_i}$ for $3\leq i \leq 4$. It follows that $\mathcal{S}$ is continuous at $(\alpha,\beta)\in \widetilde{c}_1$. Continuity at points in $\widetilde{c}_j$ for $2\leq j\leq 4$ are proven analogously and it follows that $\mathcal{S}$ is globally continuous.
We now have all the ingredients to prove the following theorem, which generalises Proposition \[pro:extension\].
\[thm:Sextension\] The continuous extension $\mathcal{S}:K\rightarrow Q$ defined in Lemma \[lem:extension\] is a homeomorphism and maps the interior $R$ of $K$ diffeomorphically onto the open rectangle $Q^\circ$.\
For $1\leq i\leq 4$, the extension $\mathcal{S}$ maps the ‘edge’ $\widehat{e}_i$ diffeomorphically [^3] onto the open edge $\widehat{e}_i$ of $Q$.\
Furthermore, for $1\leq i\leq 4$, the set $\widetilde{c}_i$ is a singleton and $\mathcal{S}$ maps $\widetilde{c}_i\equiv\{\widetilde{c}_i\}$ to the corner $\widehat{c}_i$ of $Q$.
We know that the restriction of $\mathcal{S}$ to $R$ is an injective smooth mapping from $R$ to $Q^\circ$, and hence, by the domain invariance theorem, $\widehat{R}=\mathcal{S}(R)$ is an open subset of $Q^\circ$ and $\mathcal{S}$ maps $R$ diffeomorphically onto $\widehat{R}$. By construction of $\mathcal{S}$ in Lemma \[lem:extension\], we have $\mathcal{S}(\delta R)\subseteq \partial Q$. Since, by Lemma \[lem:compact\], $K$ is compact, $\partial \widehat{R}\subseteq\mathcal{S}(\delta R)$ and hence $\partial \widehat{R}\subseteq \partial Q$.
There exists only one non-empty open subset of $Q^\circ$ whose boundary is contained in $\partial Q^\circ$, namely $Q^\circ$ itself. We conclude that $\widehat{R}=Q^\circ$ and in particular $\mathcal{S}$ maps $R$ diffeomorphically onto the open rectangle $Q^\circ$.
It follows that $\mathcal{S}(\delta R)=\partial \widehat{R}=\partial Q$ and we have $\mathcal{S}(K)=Q$, i.e. $\mathcal{S}$ is surjective.
Let $1\leq i\leq 4$, then, by construction of $\mathcal{S}$ in Lemma \[lem:extension\], we have $\mathcal{S}^{-1}(\widehat{e}_i)\subseteq \widetilde{e}_i$, so the restricted mapping $\mathcal{S}|_{\widetilde{e}_i}:\widetilde{e}_i\rightarrow \widehat{e}_i$ is surjective. Analogously to the proof of Lemma \[lem:injective\] it is shown that the restricted mapping $\mathcal{S}|_{e_i}$ is injective. Since, by Lemma \[lem:extension\], the restricted mapping $\mathcal{S}|_{e_i}:e_i\rightarrow \widehat{e}_i$ is locally diffeomorphic it follows that $\mathcal{S}$ maps $\widetilde{e}_i$ diffeomorphically onto $\widehat{e}_i$. We conclude that $\mathcal{S}$ maps $R\cup \widetilde{e}_1\cup
\widetilde{e}_2\cup \widetilde{e}_3 \cup \widetilde{e}_4$ bijectively onto $Q^{\circ}\cup \widehat{e}_1\cup \widehat{e}_2\cup \widehat{e}_3 \cup \widehat{e}_4$.
Let $1\leq i\leq 4$, then, by construction of $\mathcal{S}$ in Lemma \[lem:extension\], we have $\mathcal{S}^{-1}(\widehat{c}_i)\subseteq \widetilde{c}_i$. Hence the surjectivity of $\mathcal{S}$ implies that $\widetilde{c}_i$ is non-empty. On the other hand, it is easy to see that $\widetilde{c}_i$ has at most one element, applying the same line of argument as in the proof of Lemma \[lem:injective\]. So $\widetilde{c}_i$ is a singleton and $\mathcal{S}$ maps $\widetilde{c}_i\equiv\{\widetilde{c}_i\}$ to the corner $\widehat{c}_i$ of $Q$.
All together we obtain that $\mathcal{S}$ is a continuous bijection. Since $K$ is compact, $\mathcal{S}$ sends closed sets to closed sets and therefore $\mathcal{S}$ is a homeomorphism.
\[cor:edgesbeta\] Let $1\leq k\leq 4$, then the corner $\widetilde{c}_k=\{\widetilde{c}_k\}$ of $K$ is the unique point $(\alpha,\beta)\in\mathbb{C}^2$ such that the Stokes complex $\mathcal{C}(\alpha,\beta)$ of $V(\lambda)$ lies in the isomorphism class $C_k$, introduced in Figure \[fig:stokes\_complexes\]. The edge $\widetilde{e}_k$ equals the set of all $(\alpha,\beta)\in\mathbb{C}^2$ such that the Stokes complex $\mathcal{C}(\alpha,\beta)$ lies in the isomorphism class $E_k$, also introduced in Figure \[fig:stokes\_complexes\].
Let us consider the edge $\widetilde{e}_1$. Note that equation defines a mapping $$s_2^{e_1}:\{(\alpha,\beta)\in\mathbb{C}^2:\mathcal{C}(\alpha,\beta)\in E_i\}\rightarrow i(-\nu \pi,+\nu \pi)$$ which is easily proven to be injective by the same line of argument as the proof of Lemma \[lem:injective\]. But $\widetilde{e}_1$ is a subset of its domain and $s_2^{e_1}=i(-\nu \pi,+\nu \pi)$ by Theorem \[thm:Sextension\]. It follows that $\widetilde{e}_1$ equals the domain of $s_2^{e_1}$. The other characterisations are proven similarly.
The Projection $\Pi_a$ and Region $K_a$ {#subsection:2}
---------------------------------------
In this subsection we derive explicit parametrisations of the boundary $\partial K_a$ and border $\delta R$, proving in particular the parametrisation of $\partial K_a$ in Section \[sec:results\], see equation . Along the way we establish the following
\[lem:crucial\] The projection $\Pi_a$ maps the border $\delta R$ homeomorphically onto $\partial R_a$.
Proposition \[pro:projection\] follows from the above lemma using standard topological arguments.
We already know that $R_a$ is open, $K_a$ is compact and $\overline{R}_a=K_a$. Due to Theorem \[thm:Sextension\], we know that $\delta R$ is homeomorphic to the Jordan curve $\partial Q$. It thus follows from Lemma \[lem:crucial\] that $\partial R_a=\partial K_a$ is a Jordan curve and $R_a$ equals its interior since $R_a$ is open and connected.
Consider the mapping $g=\Pi_a\circ \mathcal{S}^{-1}:Q\rightarrow K_a$. It follows from Lemma \[lem:deformation\] and Theorem \[thm:Sextension\] that $g|_{Q^{\circ}}:Q^{\circ}\rightarrow R_a$ is a local diffeomorphism. Furthermore, $g$ maps the jordan curve $\partial Q$ homeomorphically onto the Jordan curve $\partial R_a$. It follows from the latter and compactness of $K$ that $g|_{Q^{\circ}}:Q^{\circ}\rightarrow R_a$ is proper. Therefore $g^{-1}(\alpha)$ is finite for any $\alpha\in R_a$, and since $g|_{Q^{\circ}}$ is locally diffeomorphic, it follows that $g|_{Q^{\circ}}:Q^{\circ}\rightarrow R_a$ is a covering map. Since $Q^{\circ}$ and $R_a$ are simply connected, $g|_{Q^{\circ}}$ must be injective and thus $g$ is a continuous bijection which maps $Q^{\circ}$ diffeomorphically onto $R_a$. As $K$ is compact, $g$ maps closed sets to closed sets and thus $g$ is a homeomorphism. The proposition now follows since $\Pi_a=g\circ \mathcal{S}$.
We proceed with deriving explicit parametrisations for $\partial K_a$ and $\delta R$. Let us denote $$e_k=\Pi_a(\widetilde{e}_k),\quad c_k=\Pi_a(\widetilde{c}_k)\quad (1\leq k\leq 4),$$ so that $$\partial R_a\subseteq \Pi_a(\delta R)=e_1\cup e_2\cup e_3\cup e_4 \cup\{c_1,c_2,c_3,c_4\},$$ since $K$ is compact.
We know that the edges and corners of $K$ parametrise singular Boutroux curves, that is $$\delta R\subseteq \Omega \cap \{\Delta(\alpha,\beta)=0\}.$$ Thus the algebraic surface $\{\Delta(\alpha,\beta)=0\}$ plays an important role in our analysis of them. We first study the corners of $K$ which correspond to branching points of this algebraic surface. Then we derive aformentioned explicit parametrisations.
Let us first consider the corners of $K$. They correspond the merging of three turning points and are thus branching points of the algebraic surface $\{\Delta(\alpha,\beta)=0\}$. Therefore the $c_k$, $1\leq k\leq 4$, are roots of the discriminant of $\Delta(\alpha,\beta)$ w.r.t. $\beta$, and thus zeros of $$\label{eq:defiC}
C(\alpha):=\alpha^8-6(3\nu^2+1)\alpha^4+8(1-9\nu^2)\alpha^2-3(9\nu^4+6\nu^2+1).$$ Direct computation gives that the (finite) branching locus of $\{\Delta(\alpha,\beta)=0\}$ is given by the discrete set $$\label{eq:branchinglocus}
\{(\alpha,\beta)\in\mathbb{C}^2:C(\alpha)=0,\Delta(\alpha,\beta)=0\}=\{(\alpha,\beta)\in\mathbb{C}^2:C(\alpha)=0,\beta=
f(\alpha)\},$$ where $$f(\alpha):=-\frac{\alpha^2(\alpha^2-2)+3\nu^2+1}{6\alpha}.$$ We therefore obtain the following relation, $$\label{eq:cornerequation}
\widetilde{c}_k=(c_k,f(c_k))\quad (1\leq k\leq 4).$$ It remains to be determined to which root of $C(\alpha)$ the corner $\widetilde{c}_k$ corresponds, for $1\leq k\leq 4$. We have the following lemma concerning the roots of $C(\alpha)$.
\[lem:zerospolynomial\] The polynomial $C(\alpha)$ has precisely two real roots and two purely imaginary roots.
Application of Descartes’ rule of signs on $C(\pm i \alpha)$ immediately gives that $C(\alpha)$ has precisely two purely imaginary roots. Note that $$C'(\alpha)=8\alpha(\alpha^2+2)(\alpha^2-(1-3\nu^2))(\alpha^2-(1+3\nu^2))$$ and direct computation gives that $C(\alpha)<0$ for each of the five real roots of $C'(\alpha)$. Since $C(\alpha)\sim 8\alpha^8$ as $\alpha\rightarrow \infty$, it follows that $C(\alpha)$ has precisely two real roots.
By Lemma \[lem:zerospolynomial\] and the fact that $C(\alpha)$ is real and symmetric, we may define $u_k$ as the unique root of $C(\alpha)$ in the $k$-th quadrant of the complex plane $\{\alpha\in\mathbb{C}\}$ for $1\leq k\leq 4$, so that $$\label{eq:defiu}
u_2=-\overline{u}_1,\quad u_3=-u_1,\quad u_4=\overline{u}_1.$$ We also define $v_k$ as the unique root of $C(\alpha)$ within $i^{k-1}\mathbb{R}_+$ so that $$v_3=-v_1,\quad v_4=-v_2,$$ see Figure \[fig:complexl1\]. Due to the symmetries studied in Lemma \[lem:sym\] we have $$c_2=-\overline{c_1},\quad c_3=-c_1,\quad c_4=\overline{c_1},\label{eq:csym}$$ and therefore clearly $$\label{eq:corners}
\{c_k:1\leq k\leq 4\}=\{u_k:1\leq k\leq 4\}.$$ We make the final identification $c_k=u_k$, $1\leq k\leq 4$, while deriving the parametrisation of $\partial K_a$.
We turn our attention to $\partial K_a$ and for convenience of the reader, first re-introduce the function $\psi(\alpha)$, given in equation , which we proof to parametrise it. Consider the quartic equation $$\label{eq:defix}
3 X^4+4\alpha X^3+(\alpha^2-1)X^2-\frac{\nu^2}{4}=0.$$ Its branching points are given by the zeros of $C(\alpha)$. Let $x=x(\alpha)$ be the unique algebraic function which solves the quartic analytically in the complex $\alpha$-plane with $x(\alpha)\sim\frac{\nu}{2}\alpha^{-1}$ as $\alpha\rightarrow \infty$ and branch-cuts the diagonals $[u_1,u_3]$ and $[u_2,u_4]$ plus $[v_1,v_3]$ and $[v_2,v_4]$. Since equation is of fourth order in $X$, we can compute all its branches explicitly. In particular, for $\alpha\in (v_1,\infty)$ all the branches of are real and only one is positive, namely $x(\alpha)$. Since $X=0$ is never a root of , it follows that $x(\alpha)$ does not coincide with any other branch of at $\alpha =v_1$, so $x(\alpha)$ is single-valued at $\alpha=v_1$. Similarly it follows that $x(\alpha)$ is single-valued at $v_k$ for $2\leq k\leq 4$ and thus $x(\alpha)$ is analytic on the $\alpha$-plane merely cut along $[u_1,u_3]$ and $[u_2,u_4]$.
An analogous argument shows that there exists a unique algebraic function $y=y(\alpha)$ which solves $$\label{eq:ybranch}
y^2=\alpha^2+6 x\alpha+6x^3-1$$ on the same cut $\alpha$-plane with $y(\alpha)\sim \alpha$ as $\alpha\rightarrow \infty$. We set $$\label{eq:parametrisation2}
\psi(\alpha)=\tfrac{1}{2}\Re\left[\alpha y+\tfrac{1}{2}(1-\nu)\log(p_1)-\log(p_2)+\nu \log(p_3)\right],$$ where $$p_1=1-2 x\alpha-2x^2,\quad p_2=2x+\alpha+y,\quad p_3=\frac{x(\alpha^2+5 x\alpha+4x^2-1) +\frac{1}{2}\nu y}{x^2}.$$ In the following proposition we justify the parametrisation of $\partial K_a$ given in Section \[sec:results\].
\[pro:boundary\] Considering the function $\psi(\alpha)$ defined in , the following hold true.
1. The function $\psi(\alpha)$ is a harmonic function on the $\alpha$-plane cut along $[u_1,u_3]$ and $[u_2,u_4]$, and $\psi(\alpha)\rightarrow 0$ as $\alpha$ approaches any of the branch points $u_k$,$1\leq k \leq 4$ within the cut plane.
2. The zero set $\{\phi(\alpha)=0\}$ takes the form depicted in Figure \[fig:parametrisation2\], namely it consists of the four roots $u_k$,$1\leq k \leq 4$ of $C(\alpha)$, and eight (mutually disjoint) level curves, $\epsilon_{k}$, $l_{k}$, $1\leq k\leq 4$. From each $u_k$ emanate three level curves, $\epsilon_{k},\epsilon_{k+1}$ and $l_k$, with $l_k$ going to infinity asymptotic to $e^{\frac{\pi i}{4}(2k-1)}\infty$, for $1\leq k\leq 4$.
3. For $1\leq k\leq 4$, the internal radial angle between $\epsilon_{k}$ and $\epsilon_{k+1}$ at $u_k$ equals $\frac{2}{5}\pi$.
4. Define $$\label{eq:defiB}
B_\Delta(\alpha)=x(-2+4 x^2+6 x\alpha+2\alpha^2),$$ where $x=x(\alpha)$ is the algebraic function introduced in , then $B_\Delta$ is an analytic function on the cut $\alpha$-plane with a well-defined limiting value $B_\Delta(\alpha)=f(\alpha)$ at the branching points $\alpha=u_k$, $1\leq k\leq 4$, where $f$ is defined in , and for $1\leq k\leq 4$, the corner $\widetilde{c}_k$ and edge $\widetilde{e}_k$ of $K$ are parametrised by
$$\begin{aligned}
c_k&=u_k,& \widetilde{c}_k&=(u_k,B_\Delta(u_k)),\label{eq:paracorner}\\
e_k&=\epsilon_k,& \widetilde{e}_k&=\{(\alpha,B_\Delta(\alpha)):\alpha \in \epsilon_k\}.\label{eq:paraedge}
\end{aligned}$$
If $\nu=\frac{1}{3}$, then $\psi(e^{\frac{\pi i}{4}}\alpha)=-\psi(\alpha)$ and $l_k$ simply equals the straight diagonal line $\{u_k+te^{\frac{\pi i}{4}(2k-1)}:t\in \mathbb{R}_+\}$ for $1\leq k\leq 4$.
(image) at (0,0) [![The level set $\{\psi(\alpha)=0\}$ and zeros of the polynomial $C(\alpha)$, defined in , with the branch cuts $[u_1,u_3]$ and $[u_2,u_4]$ in dashed red.[]{data-label="fig:parametrisation2"}](parametrisation2 "fig:"){width="50.00000%"}]{};
at (0.5,0.5) [$0$]{}; at (0.654,0.654) [$u_1$]{}; at (0.346,0.654) [$u_2$]{}; at (0.346,0.346) [$u_3$]{}; at (0.654,0.346) [$u_4$]{};
at (0.915,0.5) [$v_1$]{}; at (0.5,0.915) [$v_2$]{}; at (0.085,0.5) [$v_3$]{}; at (0.5,0.085) [$v_4$]{}; (0.36,0.36) – (0.64,0.64); (0.36,0.64) – (0.64,0.36);
at (0.64,0.5) [$\epsilon_1$]{}; at (0.5,0.64) [$\epsilon_2$]{}; at (0.36,0.5) [$\epsilon_3$]{}; at (0.5,0.36) [$\epsilon_4$]{};
at (0.85,0.85) [$l_1$]{}; at (0.15,0.85) [$l_2$]{}; at (0.15,0.15) [$l_3$]{}; at (0.85,0.15) [$l_4$]{};
Before proving Proposition \[pro:boundary\], let us show how it implies Lemma \[lem:crucial\] and thus Proposition \[pro:projection\].
Due to Proposition \[pro:boundary\], we know that $$\Pi_a(\delta R)=e_1\sqcup e_2\sqcup e_3\sqcup e_4 \sqcup\{c_1,c_2,c_3,c_4\}$$ is a Jordan curve in the $\alpha$ plane. Let us denote its interior by $I$. We know that $R_a$ is connected, since $R$ is connected, and that $\partial R_a\subseteq \Pi_a(\delta R)$, since $K$ is compact. But there exists only one open non-empty connected set $U$ with $\partial U\subseteq \Pi_a(\delta R)$, namely $U=I$. We conclude that $R_a=I$ equals the interior of the Jordan curve $\Pi_a(\delta R)$.
It follows in particular that $\Pi_a(\delta R)=\partial R_a$, thus $\Pi_a|_{\delta R}:\delta R\rightarrow \partial R_a$ is surjective. Furthermore it follows directly from part (4) of Proposition \[pro:boundary\] that $\Pi_a|_{\delta R}:\delta R\rightarrow \partial R_a$ has a continuous inverse, namely $B_\Delta$. The lemma follows.
Let us first recall Lemma \[lem:deltaR\], which states that $$\delta R\subseteq \Omega\cap \{\Delta(\alpha,\beta)=0\}.$$ Namely, on the border $\delta R$ two or three turning points have coalesced, so that the resulting Boutroux curve $\widehat{\Gamma}$ is singular. To prove the proposition, we first describe a method to compute $\Omega\cap \{\Delta(\alpha,\beta)=0\}$, and then we will specialise it to the subset of our interest, namely $\delta R$.
Suppose $(\alpha,\beta)\in\mathbb{C}^2$ is such that $\Delta(\alpha,\beta)=0$, then there exist a unique turning point $\lambda=X$ of $V(\lambda;\alpha,\beta)$ which is not simple. Consider for the moment the generic case in which $X$ is double and thus $V(\lambda)$ has two remaining simple turning points, say $x_{1,2}$. Then $$V(\lambda;\alpha,\beta)=\lambda^{-2}(\lambda-x_1)(\lambda-x_2)(\lambda-X)^2,$$ yielding the following helpful relations,
$$\begin{aligned}
&3 X^4+4\alpha X^3+(\alpha^2-1)X^2-\frac{\nu^2}{4}=0,\label{eq:sing1}\\
&X(-2+4 X^2+6 X\alpha+2\alpha^2)=\beta,\label{eq:sing2}\\
&x_1+x_2=-2(X+\alpha),\label{eq:sing3}\\
&(x_2-x_1)^2=4(1-2 X\alpha-2 X^2).\label{eq:sing4}
\end{aligned}$$
Here we recognise equations and in (\[eq:sing1\],\[eq:sing2\]).
Now clearly $\widehat{\Gamma}(\alpha,\beta)$ is a Boutroux curve if and only if $$\Re \int_{x_*}^X\frac{\sqrt{(\lambda-x_1)(\lambda-x_2)}(\lambda-X)}{\lambda}d\lambda=0,$$ for any choice of $x_*\in\{x_1,x_2\}$ and choice of contour. This integral can be evaluated explicitly, yielding $$\label{eq:parametrisationgen}
\Psi=\tfrac{1}{2}\Re\left[\alpha y+\tfrac{1}{2}(1-\nu)\log(P_1)-\log(P_2)+\nu \log(P_3)\right],$$ with $$P_1=\tfrac{1}{4}(x_2-x_1)^2,\quad P_2=X-\tfrac{1}{2}(x_1+x_2)+Y,\quad P_3=\frac{X(x_1x_2-\tfrac{1}{2}(x_1+x_2)X) +\frac{1}{2}\nu Y}{X^2},$$ where $Y$ is a branch of $$\label{eq:Ybranch}
Y^2=(X-x_1)(X-x_2)=\alpha^2+6 X\alpha+6X^3-1.$$ Firstly note that $\Psi$ is invariant under interchanging of $x_1$ and $x_2$ and we may thus eliminate them from the formula using equations (\[eq:sing3\],\[eq:sing4\]), yielding equation with $x\rightarrow X$ and $y\rightarrow Y$.
In summary, we can compute $\Omega\cap \{\Delta(\alpha,\beta)=0\}$ as follows. Take a local branch $X=X(\alpha)$ of and choose the correct branch $Y=Y(\alpha)$ of realising equation , define the algebraic function $\beta_\Delta(\alpha_0)$ by the left-hand side of , then $\Psi$ is (locally) a harmonic function in $\alpha$ and for any $\alpha_0$ in its zero set we have $$(\alpha_0,\beta_\Delta(\alpha_0))\in \Omega\cap \{\Delta(\alpha,\beta)=0\},$$ yielding a local parametrisation of $\Omega\cap \{\Delta(\alpha,\beta)=0\}$. By considering each of the four branches $X$ of and following the above procedure we can in principle completely compute $\Omega\cap \{\Delta(\alpha,\beta)=0\}$. To prove the proposition this however will not be necessary.
In the case of our interest the relevant branch of is given by $X=x$, introduced in equation , and the corresponding correct branch of is given by $Y=y$ introduced in . Let us first prove part (1) of the proposition. For $\alpha$ on the plane cut along $[u_1,u_3]$ and $[u_2,u_4]$ we have $$V(\lambda;\alpha,B_\Delta(\alpha))=\lambda^{-2}(\lambda-x_1)(\lambda-x_2)(\lambda-X)^2,$$ for some up to permutation unique $x_1$ and $x_2$. Note that they cannot coincide on the cut plane and we may thus specify each uniquely by choosing $x_{1,2}=x_{1,2}(\alpha)$ analytically on the cut plane with $$\label{eq:x1x2asymptotic}
x_1=-\alpha-\sqrt{1-\nu}+\mathcal{O}(\alpha^{-1}),\quad x_2=-\alpha+\sqrt{1-\nu}+\mathcal{O}(\alpha^{-1})$$ as $\alpha\rightarrow \infty$, due to equations (\[eq:sing3\],\[eq:sing4\]).
Since $x_1\neq x_2$ on the cut plane it follows from equation , that the term $p_1$ in formula does not vanish and thus $\Re\log(p_1)$ is finite and harmonic on the cut plane. Similarly, if $p_2=0$, then $Y^2=(X-\tfrac{1}{2}(x_1+x_2))^2$ which implies $x_1=x_2$. It follows that $p_2$ does not vanish and thus $\Re\log(p_2)$ is finite and harmonic on the cut plane. The same argument, using the identity $x_1x_2X^2=\frac{1}{4}\nu^2$, works for $p_3$ and we conclude that $\psi(\alpha)$ is a harmonic function on the entire cut plane.
The branching points of $X(\alpha)$ are characterised by the coalescing of one of the two simple turning points $\{x_1,x_2\}$ with the double turning point $X(\alpha)$: at $\alpha=u_1,u_4$ the turning point $x_2$ coalesces with $X$ and at $\alpha=u_2,u_3$ the turning point $x_1$ coalesces with $X$.
The local behaviour of $\psi(\alpha)$ near the branching points is easily computed. Firstly, let us note that $\psi$ has the following symmetries, $$\label{eq:psisym}
\psi(-\alpha)=\psi(\alpha),\quad \psi(\overline{\alpha})=\overline{\psi(\alpha)}$$ and we thus merely have to compute the behaviour near the branching point $\alpha =u_1$ where $x_2$ and $X$ coalesce. By direct computation we have the corresponding Puiseux series $$\begin{aligned}
x_1&=\eta_1+\mathcal{O}(\alpha-u_1),\\
x_2&=\eta_2-2r(\alpha-u_1)^{\frac{1}{2}}+\mathcal{O}(\alpha-u_1),\\
X&=\eta_2+r(\alpha-u_1)^{\frac{1}{2}}+\mathcal{O}(\alpha-u_1),
\end{aligned}$$ as $\alpha\rightarrow u_1$, where $$\eta_1=-3\eta_2-2u_1,\quad \eta_2=\frac{1+3\nu^2-u_1^4}{4u_1(2+u_1^2)},$$ for a unique root $r$ of $r^2=-\tfrac{1}{3}\eta_2$. Therefore $$\begin{aligned}
\psi(\alpha)&=-\frac{4}{15}\Re\left[ \frac{(x_2-x_1)^{\frac{1}{2}}}{x_2}(X-x_2)^{\frac{5}{2}}\right](1+\mathcal{O}(\alpha-u_1)^{\frac{1}{2}})\nonumber\\
&=
-\frac{4}{15}3^{\frac{5}{4}}\Re\left[\kappa(\alpha-u_1)^{\frac{5}{4}}\right](1+\mathcal{O}(\alpha-u_1)^{\frac{1}{2}}) \label{eq:puiseux}
\end{aligned}$$ where the branch of $z^{\frac{5}{4}}$ is taken real and positive on $\mathbb{R}_+$ with branch cut $\{\arg z=-\frac{3\pi i}{4}\}$, for a unique $\kappa=\kappa(\nu)$ which satisfies $$\label{eq:kappa}
\kappa^4=-(\eta_2-\eta_1)^2\eta_2.$$ In particular $\psi$ vanishes near $u_1$ and due to symmetries we know that $\psi$ vanishes near the other branching points as well, finishing the proof of part (1) of the proposition.
We now turn our attention to the zero set $\{\psi(\alpha)=0\}$. We first compute it locally near the branching point $u_1$. To this end, note that $\kappa=\kappa(\nu)$ in equation is an algebraic function in $\nu$. It is the unique branch of equation satisfying $\arg{\kappa(\frac{1}{3})}=\frac{3}{16}\pi$. By direct computation it can be shown that $\arg{\kappa}:(0,\frac{1}{3}]\rightarrow (\frac{1}{6}\pi,\frac{3}{16}\pi]$.
The right-hand side of can be expanded into a complete Puiseux series and since the argument of $\kappa$ is bounded by $\frac{1}{20}\pi<\arg{\kappa}<\frac{9}{20}\pi$, it follows that there are three level curves of $\{\psi(\alpha)=0\}$ emanating radially from $\alpha=u_1$ with angles $$\label{eq:angles}
\frac{4}{5}\left(\frac{1}{2}\pi-\arg{\kappa}+m \pi i\right)\quad (m\in\{-1,0,+1\}).$$ Due to the symmetries , it follows that each branching point $u_k$ has precisely three level curves emanating from it. We call them, going around $u_k$ in anti-clockwise direction starting from the branch-cut, $\epsilon_k,l_k^*,\epsilon_{k+1}^*$. Note in particular that the radial angle at $u_k$: between $\epsilon_{k}$ and $l_k^*$ equals $\frac{4}{5}\pi$, between $l_k^*$ and $\epsilon_{k+1}^*$ equals $\frac{4}{5}\pi$ and between $\epsilon_{k+1}^*$ and $\epsilon_{k}$ equals $\frac{2}{5}\pi$, for $1\leq k\leq 4$, due to equation .
Clearly $$\psi(\alpha)=\tfrac{1}{2}\Re[\alpha^2]+\mathcal{O}(\log|\alpha|)\quad (\alpha\rightarrow \infty)$$ and it is relatively straightforward to show that there are precisely four level curves $\{\psi(\alpha)=0\}$ emanating from $\alpha=\infty$ along the asymptotic directions $e^{\frac{\pi i}{4}(2k-1)}\infty$, $1\leq k\leq 4$. We call these level curves $l_k$, $1\leq k\leq 4$, in accordance with Figure \[fig:parametrisation2\].
The final piece of information to deduce the correctness of Figure \[fig:parametrisation2\] is that $\psi(\alpha)$ is strictly monotonic along the branch cuts: for $k=1,2$ the lower (upper) branch of $\psi$ along $[0,u_{k}]$ is strictly increasing (decreasing) as $\alpha$ traverses along it in the direction of $u_{k}$ whereas for $k=3,4$ the lower (upper) branch of $\psi$ along $[0,u_{k}]$ is strictly decreasing (increasing) as $\alpha$ traverses along it in the direction of $u_{k}$. Indeed this implies that both the upper and lower branches of $\psi$ along the branch cuts are nonzero, except for at the branching points $u_k,1\leq k \leq 4$. Thus the $\epsilon_k,\epsilon_k^*,l_k,l_k^*$ make up all the level curves of $\{\psi(\alpha)=0\}$, and since $\psi$ is harmonic, it is easy to deduce that we must have $l_k=l_k^*$ and $\epsilon_k=\epsilon_k^*$ for $1\leq k\leq 4$, where $\epsilon_1^*:=\epsilon_5^*$, yielding part (2) of the proposition. Furthermore, since $\epsilon_k=\epsilon_k^*$, it follows that the internal radial angle between $\epsilon_{k}$ and $\epsilon_{k+1}$ at $u_k$ equals $\frac{2}{5}\pi$, for $1\leq k\leq 4$, establishing part (3).
Due to Lemma \[lem:deformboutroux\], we know that the isomorphism class of the Stokes complex $\mathcal{C}(\alpha,B_{\Delta}(\alpha))$ is constant along each of the level curves $\epsilon_k,l_k$, $1\leq k\leq 4$. To prove part (4), we have to determine the isomorphism class on each level curve. We first compute the isomorphism classes on the $l_k$, $1\leq k\leq 4$, after which the isomorphism classes at the branching points $u_k$ and curves $\epsilon_k$, $1\leq k\leq 4$, are easily deduced.
We proceed in computing the isomorphism class on $l_1$. Firstly, due to Lemma \[lem:boutrouxclas\], we know that the inner Stokes complex is connected along $l_1$. By equations , setting $\lambda=-\alpha+t$ with $t=\mathcal{O}(1)$ and $x_k=-\alpha+\widetilde{x}_k$ for $k=1,2$, we have $$V(\lambda)=\widetilde{V}(t)(1-\mathcal{O}(\alpha^{-1})),\quad \widetilde{V}(t):=(t-\widetilde{x}_1)(t-\widetilde{x}_2),$$ as $\alpha\rightarrow e^{\frac{\pi}{4}i}\infty$ along $l_1$. Furthermore note that $\widetilde{X}=X+\alpha$ and $\widetilde{0}=0+\alpha$ are asymptotic to $e^{\frac{\pi}{4}i}\infty$ in the same limit. The Stokes complex of the leading order potential $\widetilde{V}(t)$ is depicted in Figure \[fig:complexl1\] under $\widetilde{L}_1$.
Similarly, setting $\lambda=|\alpha|^{-1}s$ with $s=\mathcal{O}(1)$ and writing $\widehat{X}=|\alpha|^{-1} X$, we have $$V(\lambda)=\widehat{V}(s)(1-\mathcal{O}(\alpha^{-1})),\quad \widehat{V}(s):=i\frac{(s-\widehat{X})^2}{s^2},$$ as $\alpha\rightarrow e^{\frac{\pi}{4}i}\infty$ along $l_1$. Furthermore $\widehat{x}_{1,2}=|\alpha|x_{1,2}$ are asymptotic to $e^{-\frac{3\pi}{4}i}\infty$ in the same limit. The Stokes complex of the leading order potential $\widehat{V}(s)$ is depicted in Figure \[fig:complexl1\] under $\widehat{L}_1$.
There is only one isomorphism class consistent with above two limiting behaviours, namely the class $L_1$ defined in Figure \[fig:complexl1\]. Thus on $l_1$ the Stokes complex falls in the class $L_1$.
[| c | c | c |]{}
at (0,2.5) [$L_1$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (1,0); at (1,0) [$x_2$]{};
at (-1,0) ; at (-1,0) [$x_1$]{};
(-1,0) – (1,0);
(1,0) – (4,3); (1,0) – (2,-1); (-1,0) – (-2,1); (-1,0) – (-2,-1);
at (3,2) ; at (2.95,2.05) [$X$]{};
at (2.3,2.7) ; at (2.3,2.7) [$0$]{};
plot \[smooth,tension=1.0\] coordinates [(3,2) (2.85,3.25) (1.7, 3.3) (1.75, 2.15) (3,2) ]{};
&
at (0,2.6) [$\widetilde{L}_1$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\]; at (1,0); at (1,0) [$\hspace{0.1cm}\widetilde{x}_2$]{};
at (-1,0) ; at (-1,0) [$\widetilde{x}_1\hspace{0.1cm}$]{};
(-1,0) – (1,0);
(1,0) – (2,1); (1,0) – (2,-1); (-1,0) – (-2,1); (-1,0) – (-2,-1);
&
at (2.8,4.5) [$\widehat{L}_1$]{};
= \[circle, minimum width=4pt, fill, inner sep=0pt\];
(2,1) – (4,3);
at (3,2) ; at (2.95,2.05) [$\widehat{X}$]{};
at (2.3,2.7) ; at (2.3,2.7) [$0$]{};
plot \[smooth,tension=1.0\] coordinates [(3,2) (2.85,3.25) (1.7, 3.3) (1.75, 2.15) (3,2) ]{};
\
As $\alpha\rightarrow u_1$ along $l_1$, we know that $x_2$ merges with $X$, and thus the resulting Stokes complex of $V(\lambda)$ at $(\alpha,\beta)=(u_1,B_\Delta(u_1))$ is given by $C_1$, defined in Figure \[fig:stokes\_complexes\]. We conclude that $c_1=u_1$ and hence $\widetilde{c}_1= (u_1,B_\Delta(u_1))$ by equation . Using the symmetries we obtain equation for all $1\leq k\leq 4$.
Next we consider the Stokes complex of $V(\lambda;\alpha,B_\Delta(\alpha))$ along $\epsilon_2$. Again we know it’s inner Stokes complex must be connected due to Lemma \[lem:boutrouxclas\]. Furthermore, we have
- as $\alpha\rightarrow u_1=c_1$ along $\epsilon_2$, the turning point $x_2$ merges with $X$ and the resulting Stokes complex falls in the class $C_1$;
- as $\alpha\rightarrow u_2=c_2$ along $\epsilon_2$, the turning point $x_1$ merges with $X$ and the resulting Stokes complex falls in the class $C_2$;
- The isomorphism class of the Stokes complex is invariant under reflection in the imaginary axes (and interchanging of marked points $\infty_1\leftrightarrow \infty_2$ and $\infty_3\leftrightarrow \infty_4$).
Indeed the third easily follows from the symmetries given in equation . Clearly, there is only one isomorphism class of Stokes complexes which satisfies these three conditions, namely $E_2$ defined in Figure \[fig:stokes\_complexes\], with $\mu_{1,2}=x_{1,2}$ and $\mu_3=X$. Thus the Stokes complex along $\epsilon_2$ falls in the class $E_2$ and we have $$\label{eq:parasub}
\epsilon_2\subseteq e_2,\quad\{(\alpha,B_\Delta(\alpha)):\alpha \in \epsilon_2\}\subseteq\widetilde{e}_2.$$ Since $\widetilde{e}_2$ is a smooth non-self intersecting curve with end-points $\widetilde{c}_1$ and $\widetilde{c}_2$, and the same holds true for $\{(\alpha,B_\Delta(\alpha)):\alpha \in \epsilon_2\}$, equation implies $\{(\alpha,B_\Delta(\alpha)):\alpha \in \epsilon_2\}=\widetilde{e}_2$ and thus $\epsilon_2=e_2$. Analogously we prove equation for the remaining $k\in\{1,3,4\}$ and with part (4) thus established, the proposition is proven.
We define $$\mathcal{S}_a=\mathcal{S}\circ \Pi_a^{-1}:K_a\rightarrow Q,$$ which is a homeomorphism that maps $R_a$ diffeomorphically onto $Q^\circ$. We may now justifiably call $e_k$,$1\leq k\leq 4$ and $c_k$,$1\leq k\leq 4$ the edges and corners of $K_a$. Note that $\mathcal{S}_a$ maps $c_k$ to $\widehat{c}_k$ and $e_k$ onto $\widehat{e}_k$, see Figure \[fig:marked\].
Bulk Asymptotic {#section:bulkasymptotic}
===============
In the present section we use the WKB analysis of the equation to provide an asymptotic formula for the roots of the generalised Hermite polynomials. The reader should recall, see e.g. the Section \[sec:results\], the definition of the domains $K$, $K_a$ (i.e. the projection of $K$ on the $\alpha$-plane, $K_a=\Pi_a(K)$), the function ${\mathcal{ S }}$ and the rectangle $Q=\left[-\tfrac{1}{2}(1-\nu)\pi,\tfrac{1}{2}(1-\nu)\pi\right]\times \left[-\nu\pi,\nu\pi\right]$, which is the image of $K$ under ${\mathcal{ S }}$.
In order to state and prove our results, for sake of simplicity we choose $p\geq q$, $p,q$ either equal or co-prime, and thus fix a ratio $\frac{m}{n}=\frac{p}{q}$. Hence, in the whole section, the numbers $m,n$ take values in the sequences $m=xq,n=xp, x \in {\mathbb{ N }}$. Correspondingly $\nu=\frac{p}{2q+p}\in(0,\frac13]$ is fixed and the large parameter $E$ belongs to the sequence $(2q+p)x$. However, all results are essentially unchanged if we let $\nu$ vary on $[\nu_0,\frac13]$, for some fixed $\nu_0 >0$.
Given an integer number $m \in {\mathbb{ N }}^*$ we let $I_m=\lbrace -m+1,-m+3,\dots,m-1\rbrace \subseteq {\mathbb{ Z }}$.
For each $j \in I_m, k \in I_n$, we let $(\alpha_{j,k},\beta_{j,k}) \in K$ be the unique solution of $S(\alpha,\beta)=(\frac{\pi j}{E},\frac{\pi k}{E})$.
A filling fraction is a real number $\sigma \in (0,1)$. We let $I_m^{\sigma}=I_m \cap [\sigma(-m+1), \sigma(m-1)]$ and define $Q^{\sigma} \subset Q$ as the closed rectangle $[-\frac{\pi \lfloor \sigma (m-1) \rfloor}{E},
\frac{\pi \lfloor \sigma (m-1) \rfloor}{E}]
\times[-\frac{\pi \lfloor \sigma (n-1) \rfloor}{E},
\frac{\pi \lfloor \sigma (n-1) \rfloor}{E}]$, which in the large $E$ limit converges to $\sigma\cdot Q$.
Finally we define $K^{\sigma}=\mathcal{S}^{-1}(Q^{\sigma})$ and $K_a^{\sigma}=\Pi_a(K^{\sigma})$ as the projection of $K^{\sigma}$ on the $\alpha$-plane.
Our main result is the following asymptotic characterisation of the solutions $(\alpha,\beta)$ of the inverse monodromy problem characterising the roots of generalised Hermite polynomials.
\[thm:bulkbeta\] Fix $\sigma \in (0,1)$. Then there exists an $R_{\sigma}>0$ such that for $E$ large enough the following hold true:
1. In each ball of centre $(\alpha_{j,k},\beta_{j,k})$, $(j,k) \in I_m^{\sigma} \times I_n^{\sigma}$ and radius $R_{\sigma}E^{-2}$ there exists a unique point $(\alpha,\beta)$ such that the anharmonic oscillator satisfies the inverse monodromy problem characterising the roots of generalised Hermite polynomials.
2. In the $\epsilon$ neighbourhood of $K^{\sigma}$ with radius $R_{\sigma}E^{-2}$, there are exactly $\lfloor \sigma m \rfloor \times \lfloor \sigma n \rfloor$ points $(\alpha,\beta)$ such that the anharmonic oscillator (\[eq:scaled\]) satisfies the inverse monodromy problem characterising the roots of generalised Hermite polynomials.
The proofs of the Theorem \[thm:bulkintro\] and of the Corollary \[cor:smallalphaintro\] stated in the Results Section follows directly from the above Theorem.
A point $a \in {\mathbb{ C }}$ is a root of $H_{m,n}$ if and only if there exists $\beta$ such that $(E^{-\frac{1}{2}}a,\beta)$ provides a solution to the inverse monodromy problem for the scaled oscillator (\[eq:scaled\]). Since the projection $\Pi_a:K^{\sigma} \to K^{\sigma}_a$ is a local diffeomorphism (Proposition \[pro:projection\]), the thesis follows from Theorem \[thm:bulkbeta\].
By the above Theorem, we need to solve $S(\alpha_{j,k},\beta_{j,k})=(\frac{\pi j}{E},\frac{\pi k}{E})$ for $j,k$ bounded. Since $j,k$ are bounded, up to a uniform (and immaterial) $O(E^{-2})$ error we can solve the above equation using the first order Taylor expansion of ${\mathcal{ S }}$ at $(\alpha,\beta)=(0,0)$ (recall that ${\mathcal{ S }}(0,0)=(0,0)$). More precisely, if we let $(J)_{ij}$ be the Jacobian of ${\mathcal{ S }}$ at $(\alpha,\beta)=0$, we have that $\alpha_{j,k}$ is, up to the $O(E^{-2})$ error, the first component of the solution of the linear equation $J.(\alpha,\beta)=(\frac{\pi j}{E},\frac{\pi k}{E})$. That is $$\begin{aligned}
\alpha_{j,k}= (\det J)^{-1} \frac{\pi i}{E} \big( J_{22} j- J_{12} k \big)+O\big(E^{-2}\big) \; .
\end{aligned}$$ The thesis follows from the above formula when substituting the actual values of $(J)_{ij}$ obtained by computing the relevant elliptic integrals.
In order to prove Theorem \[thm:bulkbeta\] we will use the multidimensional Rouché theorem, which we state below
Let $D,E$ be bounded domains in ${\mathbb{ C }}^n $, $ \overline{D} \subset E$ , and let $f(z) , g(z)$ be holomorphic maps $ E \to {\mathbb{ C }}^n$ such that
- $f(z) \neq 0 ,\; \forall z \in \partial D$,
- ${\left| g(z) \right|} < {\left| f(z) \right|}, \;\forall z \in \partial D$,
then $w(z)=f(z)+g(z)$ and $f(z)$ have the same number (counted with multiplicities) of zeroes inside $D$. Here ${\left| f(z) \right|}$ is any norm on ${\mathbb{ C }}^n$.
The proof is essentially an improved version of the proof of the analogous result for poles of the Tritronquee solution of Painleve I [@piwkb2].
For sake of definiteness, we use on ${\mathbb{ C }}^2$ the euclidean norm and we denote it by $\| \,\|$.
We denote by $\widetilde{K}^{\sigma}$ the epsilon neighbourhood of $K^{\sigma}$ where the radius epsilon is $R_{\sigma}E^{-2}$, for some $R_{\sigma}>0$ to be determined.
Since $\widetilde{K}^{\sigma}$ is shrinking to a compact subset of the interior of $K$, namely $K^{\sigma}$, it follows that $\widetilde{K}^{\sigma}$ eventually possesses the $W$ property as per Definition \[def:W\]. Therefore by Theorem \[thm:wkbquantisation\] and Theorem \[thm:wkbmonodromy\], for $E$ large enough the solutions of the inverse monodromy problem, restricted to $\widetilde{K}^\sigma$, coincide with the zeros of the function $$w:\widetilde{K}^{\sigma} \to {\mathbb{ C }}^2\, , \; w=(Wr[\chi_+,\psi_0],Wr[\psi_1^R,\psi_1^L]) \; .$$ Here the solutions $\chi_+,\psi_0,\psi_1^R,\psi_1^L$ of equation are defined in Theorem \[thm:wkbquantisation\] and Theorem \[thm:wkbmonodromy\].
Moreover since $|e^{E\oint_{\gamma_i}\sqrt{V}}|=1$ for all $(\alpha,\beta) \in K^{\sigma}$, for any fixed $R_{\sigma}$ we can find an $E_{\sigma}$ such that for all $E\geq E_{\sigma}$, $$\label{eq:boundexpbulk}
|e^{\oint_{\gamma_i}\sqrt{V}}|\leq 2\quad \forall (\alpha,\beta) \in \widetilde{K}^{\sigma}.$$ We conclude, after Theorem \[thm:wkbquantisation\] and Theorem \[thm:wkbmonodromy\], that there exist $C_\sigma,E_{\sigma}$ such that for all $E\geq E_{\sigma}$, the following estimates hold, $$\begin{aligned}
\label{eq:quantproof}
\left| Wr[\chi_+,\psi_0]+1 + e^{E\oint_{\gamma_1}\sqrt{V}}\right|&\leq \frac{C_{\sigma}}{E}
\quad \forall (\alpha,\beta) \in \widetilde{K}^{\sigma}, \\
\label{eq:monodproof}
\left| Wr[\psi_1^R,\psi_1^L]+1 + e^{E\oint_{\gamma_2}\sqrt{V}-i\pi n}\right|&\leq \frac{C_{\sigma}}{E} \quad \forall (\alpha,\beta)
\in \widetilde{K}^{\sigma}.\end{aligned}$$ We define $$f:\widetilde{K}^{\sigma} \to {\mathbb{ C }}^2 , \,
f=-(e^{E\oint_{\gamma_1}\sqrt{V}}+1,e^{E\oint_{\gamma_2}\sqrt{V}-i\pi n}+1)=-
(e^{i E s_1(\alpha,\beta)- i \pi m}+1,e^{i E s_2(\alpha,\beta)-i\pi n}+1)$$ and notice that its zeros coincide with the points $(\alpha_{j,k},\beta_{j,k}), (j,k) \in I_m^{\sigma} \times I_n^{\sigma}$.
In order to prove the thesis by means of the Rouché theorem, it is sufficient to find an $R_{\sigma}$ such that
1. $\forall (j,k) \in I_m^{\sigma} \times I_n^{\sigma}$, $\|f\|> \sqrt{2} E^{-1}C_{\sigma}$ on $\Sigma^{(j,k)}_{R_{\sigma}}$, where $\Sigma^{(j,k)}_{R_{\sigma}}$ is the sphere of centre $(\alpha_{j,k},\beta_{j,k})$, $j \in I_m^{\sigma}, k \in I_n^{\sigma}$ and radius $R_{\sigma}E^{-2}$;
2. $\|f\|> \sqrt{2} E^{-1}C_{\sigma}$ on the boundary of $\widetilde{K}^{\sigma}$.
We do this by proving the corresponding estimates on the images of these sets under $\mathcal{S}$.
We choose the co-ordinates $(\hat{x},\hat{y})$ on $Q^{\sigma}$ and we embed $Q^{\sigma}$ in ${\mathbb{ C }}^2$ by allowing $\hat{x},\hat{y}$ to be complex numbers. We define the function $\hat{f}=-(e^{i E \hat{x} - i\pi m},e^{iE \hat{y}-i \pi n})$ and notice that $\hat{f}=f \circ {\mathcal{ S }}$.
We begin by proving $(1)$. The following statement follows from a simple computation: if $Y>0$ is big enough then for each $Y'>Y$ and each $(j,k)\in I_m \times I_n$, on the spherical shell $\widehat{B}(j,k)_{Y,Y'} \subset {\mathbb{ C }}^2$ of centre $(\frac{\pi j}{E},\frac{\pi k}{E})$, internal radius radius $YE^{-2}$ and external radius $Y'E^{-2}$, the estimate $\|\hat{f}\|>\sqrt{2} E^{-1}C_{\sigma}$ is satisfied for $E$ big enough. Let us then choose a $Y>0$ for which the above property holds and consider $S^{-1}(\widehat{B}(j,k)_{Y,Y'})$ for $(j,k) \in I_m^{\sigma} \times
I_n^{\sigma}$ and some $Y'>Y$. Denoting $J(j,k)$ the Jacobian of ${\mathcal{ S }}$ at $(\alpha_{j,k},\beta_{j,k})$ and $A(j,k)$ its inverse (recall that $S$ restricted to $K^{\sigma}$ is a diffeomorphism), we have that -up to a negligible $O(E^{-4})$ contribution- the counter-image of a sphere of radius $E^{-2} Y$ is just the ellipsoid with semi-axis $\sqrt{{\lambda}_i(j,k)} Y E^{-2},i=1,2$ where ${\lambda}_i(j,k)>0$’s are eigenvalues of the matrix $A^{\dagger}(j,k)A(j,k)$. Since the compact set $K^{\sigma}$ is a subset of the open domain $R=K \setminus \partial K$ where ${\mathcal{ S }}$ is a diffeomorphism, the eigenvalues ${\lambda}_i$’s are uniformly bounded. It means that there is a $R_{\sigma}$ and a $Y'>Y$ such that $ \Sigma^{(j,k)}_{R_{\sigma}}\subset {\mathcal{ S }}^{-1}(\widehat{B}(j,k)_{Y,Y'})$ for each $(j,k) \in I_{m}^{\sigma} \times I_n^{\sigma}$. Hence $\|f\|>\sqrt{2} E^{-1}C_{\sigma}$ for each of such $\Sigma^{(j,k)}_{R_{\sigma}}$, and thus the thesis is proven.
Part $(2)$ can be proven similarly. For $Y,E$ big enough, in the boundary of the ${\varepsilon}$ neighbourhood of $Q^{\sigma}$ with radius $E^{-2}Y$ the estimate $\|\hat{f}\|>\sqrt{2} E^{-1}C_{\sigma}$ is satisfied. By the same reasoning as above, we can find a $R_{\sigma}$ such that the estimate $\|f\|>\sqrt{2} E^{-1}C_{\sigma}$ is satisfied on the boundary of $\widetilde{K}^{\sigma}$.
The Theorems \[thm:bulkbeta\],\[thm:bulkintro\],\[cor:smallalphaintro\] hold true unchanged if we let $\nu$ vary on the interval $[\nu_0,\frac13]$ for some fixed $\nu_0>0$. By this we mean that the constant $R_{\sigma}$ in the theorems above can be chosen to be independent on $\nu$.
Elliptic Integrals {#append:computation}
==================
In this appendix we collect a number of explicit formulae for the elliptic integrals under consideration in the paper, which can be derived using standard elliptic function theory. The formulae are explicitly given in terms of the zeros of $V(\lambda;\alpha,\beta)$. For $(\alpha,\beta)$ close to $(0,0)$ the zeros $\lambda_k=\lambda_k(\alpha,\beta)$ of $V(\lambda;\alpha,\beta)$ do not coalesce and are analytic in $(\alpha,\beta)$. They are determined by equations , up to permutation, and we fix them unambiguously by the initial conditions at $(\alpha,\beta)=(0,0)$.
Firstly, we have the following explicit formulas for $s_1$ and $s_2$, $$\begin{aligned}
s_1&=+\frac{2 i}{\sqrt{(\lambda_4-\lambda_1)(\lambda_3-\lambda_2)}}F(\lambda_1,\lambda_2,\lambda_3,\lambda_4)+\frac{1}{2} i\pi(1-\nu),\\
s_2&=-\frac{2 }{\sqrt{(\lambda_4-\lambda_3)(\lambda_2-\lambda_1)}}F(\lambda_4,\lambda_1,\lambda_2,\lambda_3)+i\pi\nu,\end{aligned}$$ with $$\begin{aligned}
F(\lambda_1,\lambda_2,\lambda_3,\lambda_4)=&-\tfrac{1}{4}(\lambda_4-\lambda_2)(\lambda_3-\lambda_2)(3\lambda_1-\lambda_2+\lambda_3+\lambda_4)\mathcal{K}(m)\\
&+\tfrac{1}{4}(\lambda_4-\lambda_1)(\lambda_3-\lambda_2)(\lambda_1+\lambda_2+\lambda_3+\lambda_4)\mathcal{E}(m)\\
&+(\lambda_4-\lambda_2)\Pi(n_1,m)+2\lambda_1\lambda_3(\lambda_4-\lambda_2) \Pi(n_2,m),\end{aligned}$$ where $\mathcal{K}(m)$, $\mathcal{E}(m)$ and $\Pi(n,m)$ denote the standard complete elliptic integrals of the respective first, second and third kind with parameter $m=k^2$ and elliptic characteristic $n$, in the above formula equal to $$m=\frac{(\lambda_2-\lambda_1)(\lambda_4-\lambda_3)}{(\lambda_3-\lambda_1)(\lambda_4-\lambda_2)},\quad n_1=-\frac{\lambda_4-\lambda_3}{\lambda_3-\lambda_2},\quad n_2=-\frac{(\lambda_4-\lambda_3)\lambda_2}{(\lambda_3-\lambda_2)\lambda_4}.$$ The formula for $s_1$ holds for $(\alpha,\beta)$ close to $(0,0)$ with all branches chosen principal, and is globally correct (on an open environment of $R$) via appropriate analytic continuation. The formula for $s_2$ holds for $(\alpha,\beta)$ close to $(0,0)$, with all branches chosen principal except the one for $\Pi(n_2,m)$, namely $$\Pi(n_2,m)=\begin{cases}
\Pi^{(p)}(n_2,m) &\text{if $\Im{n_2}> 0$,}\\
\Pi^{(p)}(n_2,m)+\frac{i\pi}{\sqrt{(n_2-1)(1-m/n_2)}} &\text{if $\Im{n_2}\leq 0$,}\\
\end{cases}$$ where $\Pi^{(p)}(n_2,m)$ denotes the principal branch, so that $\Pi(n_2,m)$ is analytic in $n_2$ on an open environment of $(1,\infty)$ for $m\in \mathbb{C}\setminus [1,\infty)$. It is globally correct (on an open environment of $R$) via appropriate analytic continuation.
Considering the Jacobian $$\label{eq:jacobianexplicit}
J_{(s_1,s_2)}(\alpha,\beta)=\begin{pmatrix}
\frac{\partial s_1}{\partial \alpha}& \frac{\partial s_1}{\partial \beta}\\
\frac{\partial s_2}{\partial \alpha}& \frac{\partial s_2}{\partial \beta}
\end{pmatrix}$$ we have the following explicit expressions $$\begin{aligned}
\frac{\partial s_1}{\partial \alpha}&=+\frac{2i}{\sqrt{(\lambda_4-\lambda_1)(\lambda_3-\lambda_2)}}\left[(\lambda_3-\lambda_2)(\lambda_4-\lambda_1)\mathcal{E}(m_{11})-(\lambda_1\lambda_2+\lambda_3\lambda_4)\mathcal{K}(m_{11})\right],\\
\frac{\partial s_2}{\partial \alpha}&=-
\frac{2}{\sqrt{(\lambda_4-\lambda_3)(\lambda_2-\lambda_1)}}\left[(\lambda_2-\lambda_1)(\lambda_4-\lambda_3)\mathcal{E}(m_{21})-(\lambda_2\lambda_3+\lambda_1\lambda_4)\mathcal{K}(m_{21})\right],\\
\frac{\partial s_1}{\partial \beta}&=-\frac{2i}{\sqrt{(\lambda_3-\lambda_1)(\lambda_4-\lambda_2)}}\mathcal{K}(m_{12}),\\
\frac{\partial s_2}{\partial \beta}&=+\frac{2}{\sqrt{(\lambda_3-\lambda_1)(\lambda_4-\lambda_2)}}\mathcal{K}(m_{22}),\end{aligned}$$ with parameters $$m_{12}=\frac{(\lambda_2-\lambda_1)(\lambda_4-\lambda_3)}{(\lambda_3-\lambda_1)(\lambda_4-\lambda_2)},\quad m_{22}+m_{12}=1,\quad m_{11}+m_{22}^{-1}=1,\quad m_{11}m_{21}=1,$$ which holds near $(\alpha,\beta)=(0,0)$ with all branches taken principally, and holds globally via appropriate analytic continuation. In particular $$\label{eq:jacobiandet}
|J_{(s_1,s_2)}(\alpha,\beta)|\equiv 2\pi i$$ due to Legendre’s identity.
Boutroux Curves {#append:boutroux}
===============
We have the following equivalent characterisations of Boutroux curves.
\[lem:boutrouxclas\] Let $(\alpha,\beta)\in \mathbb{C}^2$, then the following are equivalent:
1. $\widehat{\Gamma}(\alpha,\beta)$ is a Boutroux curve,
2. each external vertex of the Stokes complex $\mathcal{C}(\alpha,\beta)$ has valency one,
3. the inner Stokes complex corresponding to $V(\lambda;\alpha,\beta)$ is connected.
Let us note that (iii) trivially implies (i) and it is easy to see that (ii) implies (iii) using the fact that the Stokes complex is connected, see Proposition \[pro:basic\]. It remains to be shown that (i) implies (ii).
Suppose $\widehat{\Gamma}(\alpha,\beta)$ is a Boutroux curve. By Proposition \[pro:basic\], each external vertex has valency at least one. Suppose the valency of any particular external vertex, say $e^{\frac{1}{4}(2k-1)\pi i}$, is greater than one. Then there exist distinct Stokes lines $l_1$ and $l_2$, both asymptotic to $e^{\frac{1}{4}(2k-1)\pi i}\mathbb{R}_+$, such that $$\label{eq:realint}
\Re\int_{\Lambda_1}^{\Lambda_2}\sqrt{V(\lambda)}d\lambda=0,$$ for any choice of $\Lambda_1\in l_1$, $\Lambda_2\in l_2$ and choice of connecting contour. Let $C,R>0$ be such that $$|\sqrt{V(\lambda)}-\lambda|\leq C,$$ for $|\lambda|>R$. Then, for any $\Lambda_1\in l_1$ and $\Lambda_2\in l_2$ with $|\Lambda_1|,|\Lambda_2|>R$, equation with a choice of contour lying in $\{|\lambda|>R\}$, implies $$\label{eq:inequality}
|\Re(\Lambda_2^2-\Lambda_1^2)|\leq 2C|\Lambda_2-\Lambda_1|.$$ We choose $\Lambda_j=re^{i\theta_i(r)}\in l_j$ for $r\gg0$, with $\theta_j(r)\sim \tfrac{1}{4}(2k-1)\pi$ as $r\rightarrow +\infty$, for $j\in 1,2$. Note that $\theta_1(r)-\theta_2(r)\notin 2\pi \mathbb{Z}$ for any $r$. Now inequality translates to $$r\leq 2 C \frac{|e^{i(\theta_2(r)-\theta_1(r))}-1|}{|\cos{(2\theta_2(r))}-\cos{(2\theta_1(r))}|},$$ but the right-hand side converges to $C$ as $r\rightarrow +\infty$ and we have arrived at a contradiction. We infer that each external vertex must has valency one, which completes the proof of the lemma.
We proceed with discussing deformations of Stokes complexes, and in particular Boutroux curves. Firstly we consider generic deformations of Stokes lines in the following lemma.
\[lem:deformationlines\] Let $(\alpha^*,\beta^*)\in \mathbb{C}^2$ and consider the Stokes complex of $V(\lambda;\alpha^*,\beta^*)$. Let $\mu^*$ be a turning point of $V(\lambda;\alpha^*,\beta^*)$ and let $l^*$ be a Stokes line with $\mu^*$ as one of its endpoints. Let $\gamma_0:\mathbb{R}_{\geq 0}\rightarrow \mathbb{C}$ be a homeomorphism onto $l^*\cup\{\mu^*\}$ such that its restriction to $\mathbb{R}_+$ is a diffeomorphism onto $l^*$. Then, for any $0<T<\infty$, there exists
- a simply connected open environment $W \subseteq \mathbb{C}^2$ of $(\alpha^*,\beta^*)$,
- an analytic function $\mu(\alpha,\beta)$ on $W$ with $\mu(\alpha^*,\beta^*)=\mu^*$, and
- a smooth mapping $\gamma:W\times [0,T)\rightarrow \mathbb{C},(\alpha,\beta,t)\mapsto \gamma_{(\alpha,\beta)}(t)$,
such that $\gamma_{(\alpha^*,\beta^*)}([0,T))=\gamma_0([0,T))$, and, for all $(\alpha,\beta)\in W$,
- $\mu(\alpha,\beta)$ is a turning point of $V(\lambda;\alpha,\beta)$,
- $\gamma_{(\alpha,\beta)}(0)=\mu(\alpha,\beta)$, and
- $\gamma_{(\alpha,\beta)}:[0,T)\rightarrow \mathbb{C}$ is, when restricted to $(0,T)$, a diffeomorphism onto part of a unique Stokes line $l=l(\alpha,\beta)$ of $V(\lambda;\alpha,\beta)$.
Furthermore, if $l^*$ is asymptotic to $e^{\frac{1}{4}(2k-1)\pi i}\mathbb{R}_+$ for some $k\in \mathbb{Z}_4$, then the above also holds for $T=+\infty$, so that $\gamma_{(\alpha,\beta)}:[0,\infty)\rightarrow \mathbb{C}$ is, when restricted to $(0,\infty)$, a diffeomorphism onto a unique Stokes line $l=l(\alpha,\beta)$ of $V(\lambda;\alpha,\beta)$ asymptotic to $e^{\frac{1}{4}(2k-1)\pi i}\mathbb{R}_+$, for all $(\alpha,\beta)\in W$.
The proof is a straightforward but technical exercise in analysis. We leave it to the interested reader.
\[lem:deformboutroux\] Let $T$ be a both locally and globally simply connected metric space, together with a continuous mapping $$(\alpha,\beta):T\rightarrow \mathbb{C}^2,t\mapsto (\alpha(t),\beta(t))$$ such that $(\alpha(t),\beta(t))\in \Omega$ for all $t\in T$. Further suppose that turning points of $V(\lambda;\alpha(t),\beta(t))$ do not merge or split on $T$, i.e. there exists an $m\in \{2,3,4\}$ and continuous functions $\mu_j:T\rightarrow \mathbb{C}$, for $1\leq j\leq m$, such that $\{\mu_j(t): 1\leq j\leq m\}$ are the turning points of $V(\lambda;\alpha(t),\beta(t))$ and $|\{\mu_j(t): 1\leq j\leq m\}|=m$ for all $t\in T$.\
Then the isomorphism class of the Stokes complex $\mathcal{C}(\alpha(t),\beta(t))$ is constant on $T$.
Note that it is enough to show that, for every $t^*\in T$, there exists an open environment $B\subseteq T$ of $t^*$ such that the Stokes complex $\mathcal{C}(\alpha(t),\beta(t))$ is isomorphic to $\mathcal{C}(\alpha(t^*),\beta(t^*))$ for $t\in B$.
Let $t^*\in T$ and write $(\alpha^*,\beta^*)=T(t^*)$. We denote $V^*(\lambda)=V(\lambda;\alpha^*,\beta^*)$ and $V_t(\lambda)=V(\lambda;\alpha(t),\beta(t))$ for $t\in T$. Let us pick one of the turning points, say $\mu_j(t)$, $1\leq j\leq m$. It is convenient to number the Stokes lines of $V_t(\lambda)$ emanating from $\mu_j(t)$ uniquely. Let $r_j\geq 1$ be the degree of the turning point and $l_1^*,\ldots,l_{r_j+2}^*$ be the Stokes lines of $V^*(\lambda)$ emanating from $\mu_j^*$[^4]. We may choose a unique $\theta_1^*\in [0,\frac{1}{r_j+2}2\pi)$, write $\theta_s^*=\theta_1^*+\frac{s-1}{r_j+2}2\pi$, and order the Stokes lines such that $l_s^*$ emanates from $\mu_j^*$ with angle $\theta_s^*$ for $1\leq s\leq r_j+2$. It is now easy to see that there exists a unique continuous mapping $\theta_1: T\rightarrow \mathbb{R}$ with $\theta_1(t^*)=\theta_1^*$, so that, for every $t\in T$ and $1\leq s\leq r_j+2$, there exists a unique Stokes line $l_s(t)$ of $V_t(\lambda)$ which emanates from $\mu_j(t)$ with angle $\theta_s=\theta_1(t)+\frac{s-1}{r_j+2}2\pi$. We call $l_s(t)$ the continuous extension to $T$ of $l_s^*$ with respect to the turning point $\mu_j^*$ and angle $\theta_j^*$. Note that every Stokes line of $V^*(\lambda)$ has a unique continuous extension to $T$ with respect to every turning point from which it emanates for every of the angles by which it does so. In particular, a priori it is not excluded that an internal Stoke line might have two different continuous extensions to $T$.
Let $1\leq s\leq r_j+2$. Suppose $l_s^*$ is asymptotic to $e^{\frac{1}{4}(2k-1)\pi i}\mathbb{R}_+$ for a $k\in\mathbb{Z}_4$, then we may apply Lemma \[lem:deformationlines\] to find an open environment $B_{j,s}\subseteq T$ of $t^*$ such that the Stokes line $l_s(t)$ is asymptotic to $e^{\frac{1}{4}(2k-1)\pi i}\mathbb{R}_+$ for all $t\in B_{j,s}$.
Suppose instead that $l_s^*$ is an internal Stokes line and has endpoints $\{\mu_j^*,\mu_k^*\}$. It might be that $\mu_j^*=\mu_k^*$, in which case there is an $1\leq s'\leq r_j+2$ with $s'\neq s$ such that $l_s^*=l_{s'}^*$. Regardless of this, let $\theta_\diamond^*\in [0,2\pi)$ by the angle by which $l_s^*$ emanates from $\mu_k^*$ and denote by $l_{\diamond}(t)$ the unique continuous extension of $l_s^*$ to $T$ with respect to $\mu_k^*$ and angle $\theta_\diamond^*$. We wish to show that there exists an open environment $B_{j,s}\subseteq T$ of $t^*$ such that $l_s(t)=l_\diamond(t)$, and thus $l_s(t)$ is an internal Stokes line with endpoints $\{\mu_j(t),\mu_k(t)\}$, for all $t\in B_{j,s}$.
Assume, for the sake of contradiction, that such a set does not exist. Then there exists a sequence $(t_n)_{n\geq 1}$ in $T$ with $t_n\rightarrow t^*$ as $n\rightarrow \infty$ such that $l_s(t_n)\neq l_\diamond(t_n)$ for all $n\geq 1$. Fix a point $\Lambda\in l_s(t^*)=l_\diamond(t^*)=l_s^*$. Using Lemma \[lem:deformationlines\], it is easy to see that there exists a sequence $(\Lambda_n)_{n\geq 1}$ in $\mathbb{C}$ with $\Lambda_n\rightarrow \Lambda$ as $n\rightarrow \infty$ such that $\Lambda_n\in l_s(t_n)$ for $n\geq 1$. Similarly there exists a sequence $(\Lambda_n^\diamond)_{n\geq 1}$ in $\mathbb{C}$ with $\Lambda_n^\diamond\rightarrow \Lambda$ as $n\rightarrow \infty$ such that $\Lambda_n^\diamond\in l_\diamond(t_n)$ for $n\geq 1$. In fact, it is not difficult to see that we may choose these sequences such that $\Lambda_n^\diamond-\Lambda_n$ is approximately orthogonal to the Stokes line $l_s^*$ through $\Lambda$ for large $n$, namely $$\label{eq:orthogonal}
\Lambda_n^\diamond-\Lambda_n=\epsilon_n |\Lambda_n^\diamond-\Lambda_n| \overline{\sqrt{V^*(\Lambda)}}+o(\Lambda_n^\diamond-\Lambda_n)$$ as $n\rightarrow \infty$, for some irrelevant choice of signs $\epsilon_n\in\{\pm 1\}$, $n\geq 1$.
Since $(\alpha(t),\beta(t))\in \Omega$ for $t\in T$, we know, by Lemma \[lem:boutrouxclas\], that the internal Stokes complex of $V_{t_n}(\lambda)$ is connected and in particular $$\label{eq:vanish}
\Re\int_{\Lambda_n}^{\Lambda_n^\diamond}{\sqrt{V_{t_n}(\lambda)}d\lambda}=0,$$ for every choice of contour avoiding critical points and $n\geq 1$. For large $n$, say $n\geq N\gg0$, we may choose as contour the line segment between $\Lambda_n$ and $\Lambda_n^\diamond$ and find a $c$, independent of $n$, such that $$\left|\int_{\Lambda_n}^{\Lambda_n^\diamond}{\sqrt{V_{t_n}(\lambda)}d\lambda}-(\Lambda_n^\diamond-\Lambda_n)\sqrt{V_{t_n}(\Lambda)}\right|\leq c|\Lambda_n^\diamond-\Lambda_n|^2$$ for $n\geq N$, so that equation yields $$|\Re(\Lambda_n^\diamond-\Lambda_n)\sqrt{V_{t_n}(\Lambda)}|\leq c|\Lambda_n^\diamond-\Lambda_n|^2$$ for $n\geq N$. Therefore, using , we obtain $$|\Re \overline{\sqrt{V_{0}(\Lambda)}}\sqrt{V_{t_n}(\Lambda)}|=o(1)$$ as $n\rightarrow \infty$. However, the left-hand side converges to $|V_{0}(\Lambda)|$ as $n\rightarrow \infty$ and therefore $V_{0}(\Lambda)=0$. Clearly $\Lambda$ is not a turning point of $V^*(\lambda)$ and we have arrived at a contradiction. We conclude that there exists an open environment $B_{j,s}\subseteq T$ of $t^*$ such that $l_s(t)=l_\diamond(t)$, and thus $l_s(t)$ is an internal Stokes line with endpoints $\{\mu_j(t),\mu_k(t)\}$, for all $t\in B_{j,s}$.
Now $$B=\bigcap_{1\leq j\leq m,1\leq s\leq r_j+2} {B_{j,s}}$$ is an open environment of $t^*$ and the Stokes complex $\mathcal{C}(\alpha(t),\beta(t))$ is isomorphic to $\mathcal{C}(\alpha^*,\beta^*)$ for $t\in B$.
\[lem:double\] Let $(\alpha^*,\beta^*)\in \Omega$ be such that $V(\lambda;\alpha^*,\beta^*)$ has one double turning point and two simple turning points, say $\mu_1^*,\mu_2^*\in\mathbb{C}^*$. Choose a simply connected open environment $W\subseteq \mathbb{C}^2$ of $(\alpha^*,\beta^*)$ such that, for $j\in \{1,2\}$, there exists a unique analytic function $\mu_j:W\rightarrow \mathbb{C}^*$ with $\mu_j(\alpha^*,\beta^*)=\mu_j^*$, such that $\mu_j(\alpha,\beta)$ is a simple turning point of $V(\lambda;\alpha,\beta)$ for $(\alpha,\beta)\in W$.\
Then there exists an open environment $W_0\subseteq W$ of $(\alpha^*,\beta^*)$, such that, for all $(\alpha,\beta)\in W_0\cap \Omega$, if $\Delta(\alpha,\beta)\neq 0$, then the two turning points of $V(\lambda;\alpha,\beta)$, not equal to $\mu_1$ or $\mu_2$, are connected by a unique Stokes line which is homotopic to the straight line segment between the two in $\mathbb{C}^*$ minus critical points.
Note that, for $(\alpha,\beta)\in \Omega$, considering the Stokes $\mathcal{C}(\alpha,\beta)$, the sum of the valencies of the vertices equals twice the number of edges minus $4$. Using this identity in conjunction with Lemma \[lem:deformationlines\], the lemma follows quite directly. We leave the details to the reader.
The following lemma is proven similarly to the above.
\[lem:triple\] Let $(\alpha^*,\beta^*)\in \Omega$ be such that $V(\lambda;\alpha^*,\beta^*)$ has one triple turning point and one simple turning point, say $\mu_1^*\in\mathbb{C}^*$. Choose a simply connected open environment $W\subseteq \mathbb{C}^2$ of $(\alpha^*,\beta^*)$ such that there exists a unique analytic function $\mu_1:W\rightarrow \mathbb{C}^*$ with $\mu_1(\alpha^*,\beta^*)=\mu_1^*$, such that $\mu_1(\alpha,\beta)$ is a simple turning point of $V(\lambda;\alpha,\beta)$ for $(\alpha,\beta)\in W$.\
Then there exists an open environment $W_0\subseteq W$ of $(\alpha^*,\beta^*)$, such that, for all $(\alpha,\beta)\in W_0\cap \Omega$ not equal to $(\alpha^*,\beta^*)$,
- if $\Delta(\alpha,\beta)\neq 0$, then the three simple turning points $\{\mu_2,\mu_3,\mu_4\}$ of $V(\lambda;\alpha,\beta)$, not equal to $\mu_1(\alpha,\beta)$, can be labelled such that $\mu_2$ and $\mu_3$ are connected by a Stokes line which is homotopic to the straight line segment between the two in $\mathbb{C}^*$ minus critical points, and the same for the pair $\{\mu_3,\mu_4\}$;
- if $\Delta(\alpha,\beta)=0$, then the two turning points $\{\mu_2,\mu_3\}$, one simple and one double, of $V(\lambda;\alpha,\beta)$, not equal to $\mu_1(\alpha,\beta)$, are connected by a Stokes line which is homotopic to the straight line segment between the two in $\mathbb{C}^*$ minus critical points.
[^1]: On the other side, the structure of the Painlevé equations can be properly understood only when studying the general solution, see [@fokas; @guzzetti12; @iorgov15].
[^2]: In this paper, diffeomorphic always means $C^\infty$-diffeomorphic.
[^3]: diffeomorphically with respect to the geometric structure on $\widetilde{e}_i$ as a smooth regular submanifold of $\{(\Re{\alpha},\Im{\alpha},\Re{\beta},\Im{\beta})\in\mathbb{R}^4\}$, see Lemma \[lem:extension\].
[^4]: Here we allow for duplicity of a Stokes line if all its endpoints equal $\mu_j(t)$.
|
---
abstract: 'In this paper, a nonconforming finite element method has been proposed and analyzed for the von Kármán equations that describe bending of thin elastic plates. Optimal order error estimates in broken energy and $H^1$ norms are derived under minimal regularity assumptions. Numerical results that justify the theoretical results are presented.'
author:
- 'Gouranga Mallik[^1] and Neela Nataraj[^2]'
bibliography:
- 'vKeBib.bib'
title: A Nonconforming Finite Element Approximation for the von Karman Equations
---
von Kármán equations, Morley element, plate bending, non-linear\
35J61, 65N12, 65N30
Introduction {#intro}
============
Let $\Omega \subset\mathbb{R}^2$ be a polygonal domain with boundary $\partial \Omega$. Consider the von Kármán equations for the deflection of very thin elastic plates that are modeled by a non-linear system of fourth-order partial differential equations with two unknown functions defined by: for given $f\in L^{2}(\Omega)$, seek the vertical displacement $u$ and the Airy stress function $v$ such that $$\label{vke}
\left.
\begin{array}{l l}
\Delta^2 u &=[u,v]+f \\
\Delta^2 v &=-{\frac{1}{2}}[u,u]
\end{array}
\right\} \text{in } \Omega$$ with clamped boundary conditions $$\label{vkb}
u=\frac{\partial u}{\partial \nu} = v = \frac{\partial v}{\partial \nu} = 0 \text{ on } \partial\Omega,$$ where the biharmonic operator $\Delta^2$ and the von Kármán bracket $[\cdot,\cdot]$ are defined by $$\Delta^2\varphi:=\varphi_{xxxx}+2\varphi_{xxyy}+\varphi_{yyyy} \mbox{ and }
[\eta,\chi]:=\eta_{xx}\chi_{yy}+\eta_{yy}\chi_{xx}-2\eta_{xy}\chi_{xy}={{\rm cof}}(D^2\eta):D^2\chi,$$ ${{\rm cof}}(D^2\eta)$ denotes the co-factor matrix of $D^2\eta$ and $\nu$ denotes the unit outward normal to the boundary $\partial\Omega$ of $\Omega$.
Depending on the thickness to length ratio, several plate models have been studied in literature; the most important ones being linear models like Kirchhoff and Reissner-Mindlin plates for [*thin*]{} and [*moderately thick*]{} plates respectively; and non-linear von Kármán plate model for [*very thin*]{} plates. Many practical applications deal with the Kirchhoff model for [*thin*]{} plates in which the transverse shear deformation is negligible. On the other hand, the Reissner-Mindlin plate model for [*moderately thick*]{} plates takes into consideration the shear deformation. The displacements of [*very thin*]{} plates are so large that a non-linear model is essential to consider the membrane action. The assumptions made in the von Kármán model are similar to those of Kirchhoff model except for the linearization of the strain tensor, which in fact, leads to the non-linearity in the model.
For the theoretical study as regards the existence of solutions, regularity and bifurcation phenomena of von Kármán equations, see [@CiarletPlates; @Knightly; @Fife; @Berger; @BergerFife; @BlumRannacher] and the references therein. Due to the importance of the problem in application areas, several numerical approaches have also been attempted in the past. The major challenges are posed by the non-linearity and the higher order nature of the equations. The convergence analysis and error bounds for conforming finite element methods are analyzed in [@Brezzi]. The papers [@Miyoshi; @Reinhart] and [@Quarteroni] investigate and analyze the Hellan-Hermann-Miyoshi mixed finite element method and a stress-hybrid method, respectively for the von Kármán equations. In these papers, the authors simultaneously approximate the unknown functions and their derivatives. The papers [@Brezzi; @Miyoshi; @Quarteroni] deal with the approximation and error bounds for isolated solutions, thereby not discussing the difficulties arising from the non-uniqueness of the solution and the bifurcation phenomena.
Over the last few decades, the finite element methodology has developed in various directions. For higher-order problems, nonconforming methods and discontinuous Galerkin methods are gaining popularity as they have a clear advantage over conforming finite elements with respect to simplicity in implementation. In this paper, an attempt has been made to study the von Kármán equations using nonconforming Morley finite elements. The Morley finite element method has been proposed and analyzed for the biharmonic equation in [@MingXu] and for the Monge-Ampère equation in [@MNeilan]. In [@XLR], a two level additive Schwarz method for a non-linear biharmonic equation using Morley elements is discussed under the assumption of smallness of data. The $C^0$ interior penalty method, a variant of the discontinuous Galerkin method has been used to analyze the Monge-Ampère equation in [@Gudi].
The solutions $u,v$ of clamped von Kármán equations defined on a polygonal domain belong to $H^2_0(\Omega)\cap H^{2+\alpha}(\Omega)$[@BlumRannacher], where $\alpha\in ({\frac{1}{2}}, 1]$ referred to as the index of elliptic regularity is determined by the interior angles of $\Omega$. Note that when $\Omega$ is convex, $\alpha=1$. This paper discusses a nonconforming finite element discretization of - and develops $a~priori$ error estimates for the displacement and Airy stress functions in polygonal domains with possible corner singularities. To highlight the contributions of this work, we have
- obtained an approximation of an isolated solution pair $(u,v)$ of - using nonconforming Morley elements;
- developed optimal order error estimates in broken energy and $H^1$ norms under realistic regularity assumptions;
- performed numerical experiments that justify the theoretical results.
The advantages of the method are that the nonconforming Morley elements which are based on piecewise quadratic polynomials are simpler to use and have lesser number of degrees of freedom in comparison with the conforming Argyris finite elements with 21 degrees of freedom in a triangle or the Bogner-Fox-Schmit finite elements with 16 degrees of freedom in a rectangle. Moreover, the method is easier to implement than mixed/hybrid finite element methods.
The difficulties due to non-conformity of the space increases the technicalities in the proofs of error estimates. Moreover, one loses the symmetry property with respect to all the variables in the discrete formulation for nonconforming case. An important aid in the proofs is a companion conforming operator, also known in the literature as the enriching operator which maps the elements in the nonconforming finite element space to that of the conforming space. Also, as proved in [@HuShi] for the biharmonic problem, it is true that when Morley finite elements are used for the von Kármán equations, the $L^2$ error estimates cannot be further improved. This is evident from the results of the numerical experiments presented in Section \[sec:num\].
The paper is organized as follows. Section \[intro\] is introductory and Section \[sec:weakformulation\] introduces the weak formulation for the problem. This is followed by description of nonconforming finite element formulation in Section \[sec:ncfem\]. Section \[sec:ee\] deals with the existence of the discrete solution and the error estimates in broken energy and $H^1$ norms. The results of the numerical experiments are presented in Section \[sec:num\]. Conclusions and perspectives are discussed in Section \[conclusions\]. The analysis of a more generalized form of - is dealt with in Appendix A.
Throughout the paper, standard notations on Lebesgue and Sobolev spaces and their norms are employed. We denote the standard $L^2$ scalar or vector inner product by $(\cdot,\cdot)$ and the standard norm on $H^{s}(\Omega)$, for $s>0$ by $\|\cdot\|_{s}$. The positive constants $C$ appearing in the inequalities denote generic constants which may depend on the domain $\Omega$ but not on the mesh-size.
Weak formulation {#sec:weakformulation}
================
The weak formulation corresponding to - is: given $f\in{L^2(\Omega)}$, find $u,v\in \: V:={H^2_0(\Omega)}$ such that
\[wform\] $$\begin{aligned}
& a(u,\varphi_1)+ b(u,v,\varphi_1)+b(v,u,\varphi_1)=l(\varphi_1) {\quad\forall\,}\varphi_1\in V\label{wforma}\\
& a(v,\varphi_2)-b(u,u,\varphi_2) =0 {\quad\forall\,}\varphi_2 \in V\label{wformb}
\end{aligned}$$
where ${\quad\forall\,}\eta,\chi,\varphi\in V$, $$\begin{aligned}
&a(\eta,\chi):={\int_\Omega}D^2 \eta:D^2\chi{\;\textit{dx}}, \; \; b(\eta,\chi,\varphi):={\frac{1}{2}}{\int_\Omega}{{\rm cof}}(D^2\eta)D\chi\cdot D\varphi{\;\textit{dx}}\; \; {\mbox {and}} \; \;
l(\varphi):=(f,\varphi).
\end{aligned}$$ Note that $b(\cdot,\cdot,\cdot)$ is derived using the divergence-free rows property [@Evans; @MNeilan]. Since the Hessian matrix $D^2\eta$ is symmetric, ${{\rm cof}}(D^2\eta)$ is symmetric. Consequently, $b(\cdot,\cdot,\cdot)$ is symmetric with respect to the second and third variables, that is, $b(\eta,\xi,\varphi)=b(\eta,\varphi,\xi)$. Moreover, since $[\cdot,\cdot]$ is symmetric, $b(\cdot,\cdot,\cdot)$ is symmetric with respect to all the variables in the weak formulation.
An equivalent vector form of the weak formulation which will be also used in the analysis is defined as: for $F=(f,0)$ with $f\in{L^2(\Omega)}$, seek $\Psi=(u,v)\in {\mathcal V}:=V\times V$ such that $$\label{vform}
A(\Psi,\Phi)+B(\Psi,\Psi,\Phi)=L(\Phi) {\quad\forall\,}\Phi \in {\mathcal V}$$ where${\quad\forall\,}\, \Xi=(\xi_{1},\xi_{2}),\Theta=(\theta_{1},\theta_{2})$ and $ \Phi=(\varphi_{1},\varphi_{2})\in \mathcal{V}$, $$\begin{aligned}
&A(\Theta,\Phi):=a(\theta_1,\varphi_1)+a(\theta_2,\varphi_2),\label{defnA}\\
&B(\Xi,\Theta,\Phi):=b(\xi_{1},\theta_{2},\varphi_{1})+b(\xi_{2},\theta_{1},\varphi_{1})-b(\xi_{1},\theta_{1},\varphi_{2}) \; \; \mbox{and}\label{defnB} \\
&L(\Phi):=(f,\varphi_1).\label{defnL}
\end{aligned}$$
It is easy to verify that the bilinear forms $A(\cdot,\cdot)$ and $B(\cdot,\cdot,\cdot)$ satisfy the following continuity and coercivity properties. That is, there exist constants $C$ such that $$\begin{aligned}
{A}(\Theta,\Phi)&\leq& C{\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_2 \: {\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_2 \quad \forall \Theta,\,\Phi\in \mathcal{V},\\
{A}(\Theta,\Theta)& \geq& C{\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_2^2 \quad \forall \Theta \in \mathcal{V}, \\
B(\Xi, \Theta, \Phi) & \leq & C {\ensuremath{\left| \! \left| \! \left|}}\Xi{\ensuremath{\right| \! \right| \! \right|}}_2 \: {\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_2 \: {\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_2 \quad \forall \Xi, \,\Theta, \, \Phi \in \mathcal{V},\end{aligned}$$ where the product norm ${\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_2:=\sqrt{A(\Phi,\Phi)}{\quad\forall\,}\Phi\in{\mathcal V}$. In the sequel, the product norm defined on $(H^s(\Omega))^2$ and $(L^2(\Omega))^2$ are denoted by ${\ensuremath{\left| \! \left| \! \left|}}\cdot{\ensuremath{\right| \! \right| \! \right|}}_s$ and ${\ensuremath{\left| \! \left| \! \left|}}\cdot{\ensuremath{\right| \! \right| \! \right|}}$, respectively.
For the results on existence of solution of the weak formulation, we refer to [@Berger; @BergerBook; @Knightly; @CiarletPlates]. More precisely, the weak solution $\Psi=(u,v)$ of - can be characterized as the solution of the operator equation $I\Psi=T\Psi$ defined on ${\mathcal V}$ where $T$ is a compact operator on ${\mathcal V}$ and $I$ is an identity operator on ${\mathcal V}$. In [@Knightly], it has been proved that there exists at least one solution of the operator equation. Also, the uniqueness of solution under the assumption on smallness of the data function $f$ has been derived.
In this paper, we follow [@Brezzi] and assume that the solution $\Psi=(u,v)$ is isolated. That is, the linearized problem defined by: for given $G=(g_1,g_2)\in( L^2(\Omega) )^2 \subset {\mathcal V}'$, seek $\Theta=(\theta_1,\theta_2)\in \mathcal{V}$ such that $$\label{vforml}
{\mathcal A}(\Theta,\Phi)=(G,\Phi) {\quad\forall\,}\Phi \in \mathcal{V}$$ where ${\mathcal A}(\Theta,\Phi):=A(\Theta,\Phi)+B(\Psi,\Theta,\Phi)+B(\Theta,\Psi,\Phi)$ is well posed and satisfies the $a~priori$ bounds $$\label{apriorilin23}
{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\Theta}{\ensuremath{\right| \! \right| \! \right|}}_2\leq C{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}, \quad {\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\Theta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}\leq C{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}$$ where $\alpha$ is the index of elliptic regularity.
Nonconforming Finite Element Method (NCFEM) {#sec:ncfem}
===========================================
In the first subsection, the Morley element is defined and some preliminaries are introduced. In the second subsection, nonconforming finite element formulation for von Kármán equations and the corresponding linearized problem are presented. Some properties and auxiliary results necessary for the analysis are discussed in the third subsection.
The Morley Element
------------------
Let $\mathcal T_h$ be a regular, quasi-uniform triangulation [@Brenner; @Ciarlet] of $\bar\Omega$ into closed triangles. Set $h_T={\rm diam}(T){\quad\forall\,}T\in \mathcal{T}_h$ and $h=\max_{T\in\mathcal{T}_h}h_T$. For $T\in \mathcal{T}_h$ with vertices $a_i=(x_i,y_i),\: i=1,2,3$, let $m_4, m_5$ and $m_6$ denote the midpoints of the edges opposite to the vertices $a_1, a_2$ and $ a_3$ respectively (see Figure \[fig:MorleyElement\]). We denote the set of vertices (resp. edges) of $\mathcal{T}_h$ by $\mathfrak{V}_h$ (resp. $\mathfrak{E}_h$). For $ e\in\mathfrak{E}_h$, let $h_e={\rm diam}(e)$.
\(m) at (0,0) ;
(m.corner 1) circle (2pt); (-62,-31)[$a_1$]{} (52,-31)[$a_2$]{} (-7,62)[$a_3$]{}
(m.corner 2) circle (2pt); (m.corner 3) circle (2pt); (m.side 1) – ($(m.side 1)!0.5!90:(m.corner 1)$); (8,8)[$m_4$]{} (m.side 2) – ($(m.side 2)!0.5!90:(m.corner 2)$); (-22,8)[$m_5$]{} (m.side 3) – ($(m.side 3)!0.5!90:(m.corner 3)$); (-7,-25)[$m_6$]{}
[@Ciarlet] The [*Morley finite element*]{} is a triplet $(T,P_T,\Phi_T)$ where
- $T$ is a triangle
- $P_T=P_2(T)$ is the space of all quadratic polynomials on $T$ and
- $\Phi_T=\{\phi_i\}_{i=1}^6$ are the degrees of freedom defined by: $$\phi_i(v)=v(a_i),\; i=1,2,3\text{ and }\phi_i(v)=\frac{\partial v}{\partial\nu}(m_i),\; i=4,5,6.$$
The nonconforming [*Morley element space*]{} associated with the triangulation $\mathcal{T}_h$ is defined by $$\begin{aligned}
V_h:=\Big\{&\varphi \in {L^2(\Omega)}\, :\, \varphi|_{T}\in P_2(T){\quad\forall\,}T\in \mathcal{T}_h,\,
\varphi\text{ is continuous at the vertices $\{ a_i\}_{i=1}^3$ of the triangle }\\
&\text{and the normal derivatives of } \varphi\text{ at the midpoint of the edges $\{ m_i\}_{i=4}^6$ are continuous, }\\
&\varphi =0\text{ at the vertices on }\partial\Omega,\;
\frac{\partial \varphi }{\partial \nu} =0\text{ at the midpoint of the edges on }\partial\Omega\Big\}.
\end{aligned}$$
For $\varphi \in V_h$ and $\Phi=(\varphi_1,\varphi_2) \in {\mathcal V}_h:=V_h\times V_h$, the mesh dependent semi-norms which are equivalent to the norms denoted as $|\varphi|_{2,h}$ and ${\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}$, respectively, are defined by: $$|\varphi|_{2,h}^2:=\sum_{T\in\mathcal{T}_h}|\varphi|_{2,T}^2, \quad
{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}^2:=|\varphi_{1}|_{2,h}^2+|\varphi_{2}|_{2,h}^2.$$ Also, for a non-negative integer $m,\:1\leq p<\infty$ and $\varphi\in W^{m,p}(\Omega;{\mathcal T}_h)$ $$|\varphi|_{m,p,h}^2:=\sum_{T\in\mathcal{T}_h}|\varphi|_{m,p,T}^2,\quad
\|\varphi\|_{m,p,h}^2:=\sum_{T\in\mathcal{T}_h}\|\varphi\|_{m,p,T}^2,$$ and for $p=\infty$ $$|\varphi|_{m,\infty,h}:=\max_{T\in\mathcal{T}_h}|\varphi|_{m,\infty,T},\quad
\|\varphi\|_{m,\infty,h}:=\max_{T\in\mathcal{T}_h}\|\varphi\|_{m,\infty,T},$$ where $|\cdot|_{m,p,T}$ and $\|\cdot\|_{m,p,T}$ denote the usual semi-norm and norm in the Banach space $W^{m,p}(T)$ and $W^{m,p}(\Omega;{\mathcal T}_h)$ denotes the broken Sobolev space with respect to the mesh ${\mathcal T}_h$. For $\Phi=(\varphi_{1},\varphi_{2})$ with $\varphi_1,\varphi_2\in W^{m,p}(\Omega;{\mathcal T}_h)$, define $\displaystyle
{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{m,p,h}^2:=|\varphi_{1}|_{m,p,h}^2+|\varphi_{2}|_{m,p,h}^2.\ $ When $p=2$, the notation is abbreviated as $|\cdot|_{m,h}$ and $\|\cdot\|_{m,h}$.
Nonconforming Finite Element Formulation
----------------------------------------
The NCFEM formulation corresponding to - can be stated as: for $f\in{L^2(\Omega)}$, seek $(u_h,v_h)\in \mathcal{V}_h$ such that
\[wformd\] $$\begin{aligned}
&a_h(u_h,\varphi_1)+b_h(u_h,v_h,\varphi_1)+b_h(v_h,u_h,\varphi_1)=l_h(\varphi_1) {\quad\forall\,}\varphi_1\in V_h \label{wformda}\\
& a_h(v_h,\varphi_2)-b_h(u_h,u_h,\varphi_2) =0 {\quad\forall\,}\varphi_2 \in V_h \label{wformdb}
\end{aligned}$$
where ${\quad\forall\,}\eta,\chi,\varphi\in V_h$, $$\begin{aligned}
& a_h(\eta,\chi):={\sum_{T\in\mathcal{T}_h}\int_T}D^2 \eta:D^2\chi{\;\textit{dx}}, \; \; b_h(\eta,\chi,\varphi):={\frac{1}{2}}{\sum_{T\in\mathcal{T}_h}\int_T}{{\rm cof}}(D^2\eta)D\chi\cdot D\varphi{\;\textit{dx}}\; \mbox{ and } \\
& l_h(\varphi):={\sum_{T\in\mathcal{T}_h}\int_T}f\varphi{\;\textit{dx}}.
\end{aligned}$$ As in the continuous formulation, the discrete form $b_h(\cdot,\cdot,\cdot)$ is symmetric with respect to the second and third variables. However, unlike in the conforming case [@Brezzi], $b_h(\cdot,\cdot, \cdot)$ is [*not*]{} symmetric with respect to the first and second variables or the first and third variables. The equivalent vector form corresponding to - is given by: seek $\Psi_h=(u_h,v_h)\in \mathcal{V}_h$ such that $$\label{vformd}
A_h(\Psi_h,\Phi)+B_h(\Psi_h,\Psi_h,\Phi)=L_h(\Phi) {\quad\forall\,}\Phi \in \mathcal{V}_h$$ where${\quad\forall\,}\, \Xi=(\xi_{1},\xi_{2}),\Theta=(\theta_{1},\theta_{2})$ and $ \Phi=(\varphi_{1},\varphi_{2})\in \mathcal{V}_h,$ $$\begin{aligned}
&A_h(\Theta,\Phi):=a_h(\theta_1,\varphi_1)+a_h(\theta_2,\varphi_2)\label{defnAh}, \\
&B_h(\Xi,\Theta,\Phi):=b_h(\xi_{1},\theta_{2},\varphi_{1})+b_h(\xi_{2},\theta_{1},\varphi_{1})-b_h(\xi_{1},\theta_{1},\varphi_{2})\label{defnBh} \; \; \mbox{and}\\
&L_h(\Phi):={\sum_{T\in\mathcal{T}_h}\int_T}f\varphi_1{\;\textit{dx}}.\label{defnLh}
\end{aligned}$$
The nonconforming finite element formulation corresponding to reads as: for given $G\in{(L^2(\Omega))^2}$, find $\Theta_h \in {{\mathcal V}}_h$ such that $$\label{vformld}
{\mathcal A}_h(\Theta_h,\Phi)=(G,\Phi) {\quad\forall\,}\Phi \in \mathcal{V}_h$$ where ${\mathcal A}_h(\Theta_h,\Phi):=A_h(\Theta_h,\Phi)+B_h(\Psi,\Theta_h,\Phi)+B_h(\Theta_h,\Psi,\Phi)$ and $ A_h(\cdot,\cdot),\,B_h(\cdot,\cdot,\cdot)$ are defined in and , respectively.
Auxiliary Results
-----------------
In this subsection, some auxiliary results which are essential for the analysis are stated.
[*(Integral average)*]{} [@Braess]\[defnia\] The projection $P_e:L^2(T) {\longrightarrow}P_0(e)$ defined by $ \displaystyle P_e \varphi=\frac{1}{h_e}\int_e \varphi{\;\textit{ds}}$, satisfies $$\label{ia}
\|\varphi - P_e \varphi \|_{0,e} \le C h_T^{1/2} |\varphi|_{1,T} \qquad \forall \varphi \in H^1(T).$$
[(Interpolant)]{}[@Ciarlet; @ScbSungZhang; @LasLes]\[interpolant\] Let $\Pi_h:V{\longrightarrow}V_h$ be the Morley interpolation operator defined by: $$\begin{aligned}
&(\Pi_h\varphi)(p)=\varphi(p){\quad\forall\,}p\in\mathfrak{V}_h,\\
&\int_e\frac{\partial \Pi_h \varphi}{\partial \nu}{\;\textit{ds}}=\int_e\frac{\partial \varphi}{\partial \nu}{\;\textit{ds}}{\quad\forall\,}e\in\mathfrak{E}_h.\end{aligned}$$ Then for $\varphi\in H^{2+\alpha}(\Omega)$, $\alpha\in(0,1]$, it holds: $$\begin{aligned}
&\|\varphi-\Pi_h\varphi\|_{m,p,h} \leq C h^{1+\alpha-m+\frac{2}{p}}\|\varphi\|_{2+\alpha},\qquad 0\leq m\leq 2, 1\leq p< \infty.
$$
For simplicity of notation, the interpolant of $\Phi \in {\mathcal V}$ is denoted by $\Pi_h \Phi$ and belongs to ${\mathcal V}_h$.
[ (Enrichment function)]{}\[enrich1\][@ScbSungZhang] Let $V_c$ be chosen as [*Hsieh-Clough-Tocher*]{} macro element space [@ScbSungZhang; @Ciarlet] which is a conforming relative of the Morley finite element space $V_h$. For any $\varphi \in V_h$, there exists ${E}_h\varphi \in V_c \subset V$ such that $$\begin{aligned}
\sum_{T \in \mathcal{T}_h} \left( h_T^{-4}|\varphi - E_h\varphi|^2_{0,T} +
h_T^{-2}|\varphi - E_h\varphi |^2_{1,T}
\right) + |E_h \varphi |^2_{2,h}
&\leq C|\varphi|^2_{2,h}.
\end{aligned}$$
Again, for $\Phi \in {\mathcal V}_h$, the enrichment function corresponding to $\Phi$ denoted by $E_h \Phi$, belongs to ${\mathcal V}$.
In the next lemma, we establish an imbedding result. A similar result has been proved in [@XLR Lemma 3.1] for the case of [*convex*]{} polygonal domains. However, for the sake of completeness, we provide a detailed proof for the case of polygonal domains. Note that only the edge estimation in is different from the proof in [@XLR].
(An imbedding result)\[imdedding\] For $\varphi\in V_h$, it holds: $$|\varphi|_{1,4,h}\leq C|\varphi|_{2,h}.$$
The tangential and normal derivative of $\varphi\in V_h$ are continuous at the midpoint of each edges of $T\in{\mathcal T}_h$. That is $\varphi_x,\,\varphi_y\in S_h$ where $S_h$ is the nonconforming Crouzeix-Raviart finite element space defined by $$\begin{aligned}
S_h:=\Big\{& w \in {L^2(\Omega)}\, :\, w|_{T}\in P_1(T){\quad\forall\,}T\in \mathcal{T}_h,\, w \text{ is continuous at the midpoints of}\\
&\text{ the triangle edges and }
w =0\text{ at the midpoint of the edges on }\partial\Omega\Big\}.
\end{aligned}$$ It is enough to prove $
|w|_{0,4,h}\leq |w|_{1,h}{\quad\forall\,}w\in S_h.
$
Consider the auxiliary problem: given $\theta \in H^{-1}(\Omega)$, seek $\xi$ such that $$\begin{aligned}
\label{auxlap}
-\Delta \xi&=\theta \text{ in } \Omega,\quad\xi=0\text{ on } \partial\Omega.\end{aligned}$$ The solution satisfies the following $a~priori$ bounds $$\label{apb}
\|\xi\|_1\leq C\|\theta\|_{-1},\; \|\xi\|_{1+\gamma}\leq C\|\theta\|,$$ where $\gamma\in ({\frac{1}{2}},1]$ denotes the elliptic regularity of the problem . Let $I_h\xi\in S_h$ be an interpolant which satisfies the estimate [@B; @Brenner] $$\label{CRIh}
|\xi-I_h\xi|_{0,h}+h|\xi-I_h\xi|_{1,h}\leq Ch^{1+\gamma}\|\xi\|_{1+\gamma}.$$ A multiplication of with $w$ and a use of Green’s formula leads to $$\label{intbyparts}
(\theta,w)=(-\Delta \xi,w)=\sum_{T\in{\mathcal T}_h}(\nabla \xi,\nabla w)-\sum_{T\in {\mathcal T}_h} \int_{\partial T}\frac{\partial\xi}{\partial\nu} w{\;\textit{ds}}$$ The boundary term can be estimated as follows: $$\begin{aligned}
\sum_{T\in {\mathcal T}_h} \int_{\partial T}\frac{\partial\xi}{\partial\nu} w{\;\textit{ds}}&=\sum_{T\in {\mathcal T}_h}\sum_{e\subset \partial T} \int_{e}\frac{\partial\xi}{\partial\nu} (w-P_e w){\;\textit{ds}}.\end{aligned}$$ Since $\displaystyle \int_e(w-P_e w){\;\textit{ds}}=0 \; \:\forall e\in\mathfrak{E}_h$ and $\displaystyle \frac{\partial }{\partial \nu} I_h \xi$ is a constant over each edge, we obtain $$\begin{aligned}
\sum_{T\in {\mathcal T}_h}\sum_{e\subset \partial T} \int_{e}\frac{\partial\xi}{\partial\nu} (w-P_e w){\;\textit{ds}}&=
\sum_{T\in {\mathcal T}_h}\sum_{e\subset \partial T} \int_{e}\frac{\partial}{\partial\nu} (\xi-I_h\xi)(w-P_e w){\;\textit{ds}}\\
&\leq\sum_{T\in {\mathcal T}_h}\sum_{e\subset \partial T}\|\xi-I_h\xi\|_{1,e}\|w-P_e w\|_{0,e}.\end{aligned}$$ A use of trace theorem, Lemma \[defnia\] and leads to the estimate $$\Big{|}-\sum_{T\in {\mathcal T}_h} \int_{\partial T}\frac{\partial\xi}{\partial\nu} w{\;\textit{ds}}\Big{|}\leq C h^\gamma\|\xi\|_{1+\gamma}|w|_{1,h}.$$ Therefore, the [*a priori*]{} bounds in yields $$\begin{aligned}
\label{bdw}
(\theta,w)\leq (|\xi|_1+Ch^\gamma\|\xi\|_{1+\gamma})|w|_{1,h}\leq C(\|\theta\|_{-1}+h^\gamma\|\theta\|)|w|_{1,h}.\end{aligned}$$ A choice of $\theta=w^3$ in leads to $$\label{w04h}
|w|_{0,4,h}^4\leq C(\|w^3\|_{-1}+h^\gamma \|w^3\|)|w|_{1,h}.$$ A use of inverse inequality yields $$\label{w3l}
\|w^3\|=\|w\|_{L^6(\Omega)}^3\leq Ch^{-\frac{1}{4}}\|w\|_{L^4(\Omega)}^3.$$ Also, [Hölder’s ]{}inequality and the imbedding result $L^4(\Omega)\hookrightarrow H_0^1(\Omega)$ lead to $$\label{w3dual}
(w^3,\xi)=\|w\|_{L^4(\Omega)}^3\|\xi\|_{L^4(\Omega)}\leq C\|w\|_{L^4(\Omega)}^3|\xi|_{1} \Longrightarrow \|w^3\|_{-1}\leq C\|w\|_{L^4(\Omega)}^3.$$ Hence, a use of and in leads to the required result $$|w|_{0,4,h}\leq C(1+h^{\gamma-\frac{1}{4}})|w|_{1,h}\leq C|w|_{1,h}.$$
The next lemma follows from [@ScbSungZhang Lemmas 4.2 & 4.3].
(Bounds for $A_h(\cdot,\cdot)$)\[enrichreg\] (i) Let $\boldsymbol{\chi}\in (H^{2+\alpha}(\Omega))^2$ and $\Phi\in{\mathcal V}_h$. Then, it holds $$A_h(\boldsymbol{\chi},E_h\Phi-\Phi)\leq C h^\alpha{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\chi}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}.$$ (ii) Further, for $\boldsymbol{\chi}\in (H^{2+\alpha}(\Omega))^2$ and $\Phi\in (H^{2}_0(\Omega))^2\cap (H^{2+\alpha}(\Omega))^2 $, it holds $$A_h(\boldsymbol{\chi},\Pi_h\Phi-\Phi)\leq C h^{2\alpha}{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\chi}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}.$$
A use of the definition of $B_h(\cdot, \cdot,\cdot)$, generalized [Hölder’s ]{}inequality and Lemma \[imdedding\] leads to a bound given by $$\label{boundBh1}
B_h(\Xi,\Theta,\Phi) \leq C_b{\ensuremath{\left| \! \left| \! \left|}}\Xi{\ensuremath{\right| \! \right| \! \right|}}_{2,h} {\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_{2,h} {\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h},$$ where $C_b$ is a positive constant independent of $h$.
\[boundBh2\] (A bound for $B_h(\cdot,\cdot,\cdot)$) For $\Xi\in (H^{2+\alpha}(\Omega))^2$ and $\Theta,\Phi\in {\mathcal V}+{\mathcal V}_h$, there holds $$B_h(\Xi,\Theta,\Phi)\leq C{\ensuremath{\left| \! \left| \! \left|}}\Xi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha} {\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_ {1,4,h}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq C{\ensuremath{\left| \! \left| \! \left|}}\Xi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha} {\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_ {2,h}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{1,h}.$$
Consider $$\label{defnbh2}
b_h(\eta,\chi,\varphi)={\frac{1}{2}}{\sum_{T\in\mathcal{T}_h}\int_T}\left((\eta_{yy}\chi_{x}-\eta_{xy}\chi_{y})\varphi_x+(\eta_{xx}\chi_{y}-\eta_{xy}\chi_{x}
)\varphi_y\right){\;\textit{dx}}.$$ For $\eta \in H^{2+\alpha}(\Omega)$, a use of generalized [Hölder’s ]{}inequality and the imbedding result $H^{2+\alpha}(\Omega)\hookrightarrow W^{2,4}(\Omega)$ leads to an estimate of the first term on the right hand side of as $$\begin{aligned}
{\frac{1}{2}}\Big{|}{\sum_{T\in\mathcal{T}_h}\int_T}\eta_{yy}\chi_x\varphi_x {\;\textit{dx}}\Big{|}
&\leq \left({\sum_{T\in\mathcal{T}_h}}|\eta|_{2,4,T}^4\right)^{\frac{1}{4}}\left({\sum_{T\in\mathcal{T}_h}}|\chi|_{1,4,T}^4\right)^{\frac{1}{4}}
\left({\sum_{T\in\mathcal{T}_h}}|\varphi|_{1,2,T}^2\right)^{\frac{1}{2}}\\
&\leq \|\eta\|_{W^{2,4}(\Omega)}\,|\chi|_{1,4,h}\,|\varphi|_{1,h}
\leq C\|\eta\|_{2+\alpha}\,|\chi|_{1,4,h}\,|\varphi|_{1,h}.\end{aligned}$$ Similar bounds hold true for the remaining three terms in . Hence the required result follows using the definition of $B_h(\cdot,\cdot,\cdot)$ and Lemma \[imdedding\].
\[boundBh3\] Using a proof similar to that of Lemma \[boundBh2\], it can be deduced that for $\Xi\in (H^{2+\alpha}(\Omega))^2$ and $\Theta,\Phi\in {\mathcal V}+{\mathcal V}_h$, there holds $$B_h(\Xi,\Theta,\Phi)\leq C{\ensuremath{\left| \! \left| \! \left|}}\Xi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha} {\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_ {1,h}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{1,4,h}\leq C{\ensuremath{\left| \! \left| \! \left|}}\Xi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha} {\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_ {1,h}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}.$$
Using the definition of $b_h(\cdot,\cdot, \cdot)$, an integration by parts and a use of , the following lemma holds true.
(An intermediate result)\[bhflip12\] For $\eta\in (V\cap H^{2+\alpha}(\Omega))+V_h$ and $\chi,\varphi\in V_h$, it holds $$b_h(\eta,\chi,\varphi)=b_h(\chi,\eta,\varphi)+{\frac{1}{2}}{\sum_{T\in\mathcal{T}_h}\int_{\partial T}}\left(\eta_x\chi_y-\eta_y\chi_x \right)\nabla\varphi\cdot\tau{\;\textit{ds}}$$ where $\tau$ is the unit tangent to the boundary $\partial T$ of the triangle $T$. Moreover, $$\forall \eta,\chi,\varphi\in V_h,\quad {\frac{1}{2}}{\sum_{T\in\mathcal{T}_h}\int_{\partial T}}\left(\eta_x\chi_y-\eta_y\chi_x \right)\nabla\varphi\cdot\tau{\;\textit{ds}}\leq C h|\eta|_{2,h}\,|\chi|_{1,\infty,h}\,|\varphi|_{2,h}.$$
\[flipbound\] A use of Lemma \[bhflip12\], Remark \[boundBh3\] and imbedding result $H^{2+\alpha}(\Omega)\hookrightarrow W^{1,\infty}(\Omega)$ leads to:\
for $\Xi_h,\Phi_h\in{\mathcal V}_h$ and $\boldsymbol{\xi}\in(H^{2+\alpha}(\Omega))^2$, $$|B_h(\Xi_h,\boldsymbol{\xi},\Phi_h)|\leq |B_h(\boldsymbol{\xi},\Xi_h,\Phi_h)|+Ch{\ensuremath{\left| \! \left| \! \left|}}\Xi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\xi}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Phi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}.$$
The next lemma which will be used to establish the well posedness of the linearized problem , follows easily under the assumption that $\Psi$ is an isolated solution of .
(Well posedness of dual problem)\[ctsdual\] If $\Psi$ is an isolated solution of , then the dual problem defined by: given $Q\in (H^{-1}(\Omega))^2$, find $\boldsymbol{\zeta}\in {\mathcal V}$ such that $$\label{ctsaux}
{\mathcal A}(\Phi,\boldsymbol\zeta)=(Q,\Phi) {\quad\forall\,}\Phi\in{\mathcal V}$$ is well posed and satisfies the $a~priori$ bounds: $$\label{apriori23}
{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_2\leq C{\ensuremath{\left| \! \left| \! \left|}}Q{\ensuremath{\right| \! \right| \! \right|}}_{-1}, \quad {\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}\leq C{\ensuremath{\left| \! \left| \! \left|}}Q{\ensuremath{\right| \! \right| \! \right|}}_{-1},$$ where $\alpha$ denotes the elliptic regularity index and $\displaystyle{\ensuremath{\left| \! \left| \! \left|}}Q{\ensuremath{\right| \! \right| \! \right|}}_{-1}:=\sup_{\boldsymbol{\varphi}\in (H^1_0(\Omega))^2}\frac{(Q,\boldsymbol{\varphi})}{{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\varphi}{\ensuremath{\right| \! \right| \! \right|}}_1}$.
Since the Morley finite element space ${\mathcal V}_h$ is not a subspace of ${\mathcal V}$ and the discrete form $b_h(\cdot, \cdot, \cdot)$ is non-symmetric with respect to first and second or first and third variables, we encounter additional difficulties in establishing the well posedness of the discrete problem in comparison to the conforming case.
(Well posedness of discrete linearized problem)\[wellposeld\] If $\Psi$ is an isolated solution of , then for sufficiently small $h$, the discrete linearized problem is well-posed.
The space $V_h$ being finite dimensional, uniqueness of solution of implies existence of solution. Uniqueness follows if an $a~priori$ bound for the solution of can be established. That is, we aim to prove that $${\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq C{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}$$ for sufficiently small $h$. For $\Phi\in{\mathcal V}_h$, using Lemma \[boundBh2\] and Remark \[flipbound\], the following Gårding’s type inequality holds true: $$\begin{aligned}
{\mathcal A}_h(\Phi,\Phi)&= A_h(\Phi,\Phi)+B_h(\Psi,\Phi,\Phi)+B_h(\Phi,\Psi,\Phi)\notag\\
&\geq{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}^2-C{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{1,h}-Ch{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}.\label{garding}\end{aligned}$$ Substitute $\Phi=\Theta_h$ in and use to obtain $$\label{Thetabdd}
{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq C(h{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}).$$ Note that $$\label{Ehuse}
{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq {\ensuremath{\left| \! \left| \! \left|}}\Theta_h-E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq Ch{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+{\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1}.$$ Now we estimate ${\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1}$. Choose $Q=-\Delta E_h\Theta_h$ and $\Phi=E_h\Theta_h$ in and use to obtain $$\begin{aligned}
{\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1}^2&={\mathcal A}(E_h\Theta_h,\boldsymbol{\zeta})={\mathcal A}_h(E_h\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+{\mathcal A}_h(E_h\Theta_h,\Pi_h\boldsymbol{\zeta})\notag\\
&={\mathcal A}_h(E_h\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+{\mathcal A}_h(E_h\Theta_h-\Theta_h,\Pi_h\boldsymbol{\zeta})+(G,\Pi_h\boldsymbol{\zeta})\notag\\
&=A_h(E_h\Theta_h-\Theta_h,\boldsymbol{\zeta})+A_h(\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+B_h(\Psi,E_h\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+B_h(E_h\Theta_h,\Psi,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})\notag\\
&\quad+B_h(\Psi,E_h\Theta_h-\Theta_h,\Pi_h\boldsymbol{\zeta})+B_h(E_h\Theta_h-\Theta_h,\Psi,\Pi_h\boldsymbol{\zeta})+(G,\Pi_h\boldsymbol{\zeta})\label{Ehwellbd}\end{aligned}$$ A use of Lemmas \[interpolant\], \[enrich1\], \[enrichreg\], , , Remarks \[boundBh3\] and \[flipbound\] leads to $$\begin{aligned}
{\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1}^2&\leq C\left(h^{\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}+h^{\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_2{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}+h{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_2+{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_2\right)\\
&\leq C\left(h^\alpha{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}\right){\ensuremath{\left| \! \left| \! \left|}}-\Delta E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{-1}\leq C\left(h^{\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}\right){\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1}.\end{aligned}$$ Therefore, $$\label{Ehestimate}
{\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq C(h^{\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}).$$ Now, - yield $$\begin{aligned}
&{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq C_{*}h^{\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+C{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}.
$$ That is, ${\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq C{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}$ for a choice of $h\leq h_1=(\frac{1}{2C_*})^{\frac{1}{\alpha}}$ with $\alpha\in ({\frac{1}{2}},1]$.
\[wellposelddual\] If $\Psi$ is an isolated solution of , then for sufficiently small $h$, the discrete linearized dual problem: given $\mathcal{G}\in {(L^2(\Omega))^2}$, find $\boldsymbol{\zeta_h}\in{\mathcal V}_h$ such that $$\label{vformlddual}
{\mathcal A}_h(\Phi,\boldsymbol\zeta_h)=(\mathcal{G},\Phi) {\quad\forall\,}\Phi\in{\mathcal V}_h$$ is well posed. The proof is similar to that of Theorem \[wellposeld\] and hence is skipped.
Existence, Uniqueness and Error Estimates {#sec:ee}
=========================================
In view of Theorem \[wellposeld\] and Remark \[wellposelddual\], the bilinear form ${\mathcal A}_h(\cdot,\cdot): {\mathcal V}_h \times {\mathcal V}_h \rightarrow {\mathbb R}$ defined by $$\label{nonsing}
{\mathcal A}_h(\Theta,\Phi)= A_h(\Theta,\Phi)+B_h(\Psi,\Theta,\Phi)+B_h(\Theta,\Psi,\Phi)$$ is nonsingular on ${\mathcal V}_h\times{\mathcal V}_h$.
The next lemma establishes that the perturbed bilinear form $\tilde{{\mathcal A}}_h(\cdot,\cdot)$, constructed using $\Pi_h\Psi$ is also nonsingular. Though a similar result is proved in [@Brezzi] for the conforming case, we provide a proof here for the sake of completeness.
[(Nonsingularity of perturbed bilinear form)]{}\[nonsingular\] Let $\Pi_h\Psi$ be the interpolation of $\Psi$ as defined in Lemma \[interpolant\]. Then, for sufficiently small $h$, the perturbed bilinear form defined by $$\label{defnAht}
\tilde {\mathcal A}_h(\Theta,\Phi)= A_h(\Theta,\Phi)+B_h(\Pi_h\Psi,\Theta,\Phi)+B_h(\Theta,\Pi_h\Psi,\Phi)$$ is nonsingular on ${\mathcal V}_h\times{\mathcal V}_h$, if is nonsingular on ${\mathcal V}_h\times{\mathcal V}_h$.
The bilinear form ${\mathcal A}_h:{\mathcal V}_h\times{\mathcal V}_h{\longrightarrow}{\mathbb R}$ is bounded and satisfies $$\begin{aligned}
&\sup_{{\ensuremath{\left| \! \left| \! \left|}}\Theta {\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1} {\mathcal A}_h(\Theta,\Phi)\geq \beta{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}, \quad \sup_{{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1}{\mathcal A}_h(\Theta,\Phi)\geq \beta{\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_{2,h},\end{aligned}$$ where $\beta >0$ is a constant. For $\tilde{\Psi} \in {\mathcal V}+{\mathcal V}_h$, a use of the above properties of ${\mathcal A}_h(\cdot,\cdot)$ and continuity of $B_h(\cdot,\cdot,\cdot)$ (see ) yields $$\begin{aligned}
&\sup_{{\ensuremath{\left| \! \left| \! \left|}}\Phi {\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1} A_h(\Theta,\Phi)+B_h(\Psi-\tilde{\Psi},\Theta,\Phi)+B_h(\Theta,\Psi-\tilde{\Psi},\Phi)\\
&\geq \sup_{{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1} {\mathcal A}_h(\Theta,\Phi)-\sup_{{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1} \left(B_h(\tilde{\Psi},\Theta,\Phi)+B_h(\Theta,\tilde{\Psi},\Phi)\right)\\
&\geq \beta{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}-2C_b{\ensuremath{\left| \! \left| \! \left|}}{\tilde \Psi}{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_{2,h}
\geq \frac{\beta}{2}{\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_{2,h},\end{aligned}$$ provided $\displaystyle{\ensuremath{\left| \! \left| \! \left|}}{\tilde \Psi} {\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq\frac{\beta}{4C_b}$. Such a choice is justified for sufficiently small $h\leq h_2$ (say), by setting $\tilde{\Psi}=\Psi-\Pi_h\Psi$ and using Lemma \[interpolant\]. Similarly, $ \displaystyle \sup_{{\ensuremath{\left| \! \left| \! \left|}}\Theta {\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1} \tilde{{\mathcal A}}_h(\Theta ,\Phi)\geq \frac{\beta}{2}{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\quad\forall\,}\Phi\in{\mathcal V}_h.$ Hence the required result.
Existence and Local Uniqueness Results
--------------------------------------
Consider the nonlinear operator $\mu:{\mathcal V}_h{\longrightarrow}{\mathcal V}_h$ defined by $$\label{defnmu}
\tilde{{\mathcal A}}_h(\mu(\Theta),\Phi)=L_h(\Phi)+B_h(\Pi_h\Psi,\Theta,\Phi)+B_h(\Theta,\Pi_h\Psi,\Phi)-B_h(\Theta,\Theta,\Phi){\quad\forall\,}\Phi\in{\mathcal V}_h.$$ A use of Lemma \[nonsingular\] leads to the fact that the mapping $\mu$ is well-defined and continuous. Also, any fixed point of $\mu$ is a solution of and vice-versa. Hence, in order to show the existence of a solution to , we will prove that the mapping $\mu$ has a fixed point. As a first step to this, define ${\mathbb B}_R(\Pi_h\Psi):=\left\{\Phi \in{\mathcal V}_h: {\ensuremath{\left| \! \left| \! \left|}}\Phi-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R \right\}$.
(Mapping of ball to ball) \[mapball2ball\] For a sufficiently small choice of $h$, there exists a positive constant $R(h)$ such that for any $\Theta \in {\mathcal V}_h$, $${\ensuremath{\left| \! \left| \! \left|}}\Theta-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R(h)\Rightarrow {\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta)-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R(h).$$ That is, $\mu$ maps the ball ${\mathbb B}_{R(h)}(\Pi_h\Psi)$ to itself.
Since the bilinear form $ \tilde{{\mathcal A}}_h(\cdot, \cdot)$ is nonsingular, from Lemma \[nonsingular\], there exists ${\bar\Phi}\in{\mathcal V}_h$ such that ${\ensuremath{\left| \! \left| \! \left|}}{\bar\Phi}{\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1$ and $$\begin{aligned}
&\frac{\beta}{4}{\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta)-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \tilde{\mathcal A}_h(\mu(\Theta)-\Pi_h\Psi, {\bar\Phi})\end{aligned}$$ Let $E_h{\bar\Phi}$ be an enrichment of ${\bar\Phi}$ (see Lemma \[enrich1\]). A use of , and yields $$\begin{aligned}
&\tilde{\mathcal A}_h(\mu(\Theta)-\Pi_h\Psi, {\bar\Phi})
=\tilde{\mathcal A}_h(\mu(\Theta), {\bar\Phi})-\tilde{\mathcal A}_h(\Pi_h\Psi, {\bar\Phi})\notag\\
&\quad=L_h({\bar\Phi})+B_h(\Pi_h\Psi,\Theta,{\bar\Phi})+B_h(\Theta,\Pi_h\Psi,{\bar\Phi})-B_h(\Theta,\Theta,{\bar\Phi})-A_h(\Pi_h\Psi,{\bar\Phi})-2B_h(\Pi_h\Psi,\Pi_h\Psi,{\bar\Phi})\notag\\
&\quad=L_h({\bar\Phi}-E_h{\bar\Phi}) +\left(A_h(\Psi, E_h{\bar\Phi})-A_h(\Pi_h\Psi,{\bar\Phi})\right)+\left(B_h(\Psi,\Psi, E_h{\bar\Phi})-B_h(\Pi_h\Psi,\Pi_h\Psi,{\bar\Phi})\right)\notag\\
&\qquad+B_h(\Pi_h\Psi-\Theta,\Theta-\Pi_h\Psi,{\bar\Phi})=:T_1+T_2+T_3+T_4. \label{Aht_est}\end{aligned}$$ Now we estimate $\{T_i\}_{i=1}^4$. $T_1$ can be estimated using Lemma \[enrich1\] and the continuity of $L_h$. Using Lemma \[enrichreg\], continuity of $A_h(\cdot,\cdot)$ and Lemma \[interpolant\], we obtain $$\begin{aligned}
T_2 &\leq |A_h(\Psi, E_h{\bar\Phi})-A_h(\Pi_h\Psi,{\bar\Phi})|\leq |A_h(\Psi,E_h{\bar\Phi}-{\bar\Phi})|+|A_h(\Psi-\Pi_h\Psi,{\bar\Phi})|\leq Ch^\alpha {\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}.\end{aligned}$$ A use of Lemmas \[boundBh2\], \[enrich1\], \[interpolant\] and leads to $$\begin{aligned}
T_3 &\leq | B_h(\Psi,\Psi,E_h{\bar\Phi})-B_h(\Pi_h\Psi,\Pi_h\Psi,{\bar\Phi})|\\
&\leq |B_h(\Psi,\Psi,E_h{\bar\Phi}-{\bar\Phi})-B_h(\Pi_h\Psi-\Psi,\Pi_h\Psi,\Phi)-B_h(\Psi,\Pi_h\Psi-\Psi,{\bar\Phi})|\\
&\leq Ch{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}\,{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_2\,{\ensuremath{\left| \! \left| \! \left|}}E_h{\bar\Phi}{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+Ch^{\alpha} {\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}\,{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_2\,{\ensuremath{\left| \! \left| \! \left|}}{\bar\Phi}{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\\
&\leq Ch^\alpha{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_2{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}.\end{aligned}$$ Finally, $T_4$ is estimated using as $$\begin{aligned}
T_4&\leq|B_h(\Pi_h\Psi-\Theta,\Theta-\Pi_h\Psi,{\bar\Phi})|\leq C{\ensuremath{\left| \! \left| \! \left|}}\Theta-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}^2.\end{aligned}$$ A substitution of the estimates derived for $T_1, T_2, T_3$ and $T_4$ in and an appropriate grouping of the terms yields $$\begin{aligned}
\label{muestimate}
&{\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta)-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq C_1\left(h^{\alpha}+{\ensuremath{\left| \! \left| \! \left|}}\Theta-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}^2\right) \end{aligned}$$ for some positive constants $C_1$ independent of $h$ but dependent on ${\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}$. A choice of $h\leq h_3$, where $h_3=\left(\frac{1}{4C_1^2}\right)^\frac{1}{\alpha}$, yields $4C_1^2 h^\alpha\leq 1$. Since ${\ensuremath{\left| \! \left| \! \left|}}\Theta-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R(h)$, for $h\leq h_3$, a choice of $R(h):=2C_1 h^\alpha$ leads to $$\begin{aligned}
&{\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta)-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq C_1 h^\alpha\left(1+4C_1^2 h^\alpha\right)\leq R(h)\end{aligned}$$ This completes the proof.
(Existence)\[exitenceuniqueness\] For sufficiently small $h$, there exists a solution $\Psi_h$ of the discrete problem that satisfies ${\ensuremath{\left| \! \left| \! \left|}}\Psi_h-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R(h)$, for some positive constant $R(h)$ depending on $h$.
Lemma \[mapball2ball\] leads to the fact that $\mu$ maps the ball ${\mathbb B}_{R(h)}(\Pi_h\Psi)$ to itself. Therefore, an application of Schauder fixed point theorem [@Kesavan] yields that the mapping $\mu$ has a fixed point, say $\Psi_h$. Hence, $\Psi_h$ is an approximate solution of which satisfies ${\ensuremath{\left| \! \left| \! \left|}}\Psi_h-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R(h)$.
(Contraction result)\[contractionthm\] For $\Theta_1,\Theta_2\in {\mathbb B}_{R(h)}(\Pi_h\Psi)$ with $R(h)$ as defined in Theorem \[mapball2ball\], the following contraction result holds true: $$\label{contractioneqn}
{\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta_1)-\mu(\Theta_2){\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq Ch^{\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Theta_1-\Theta_2{\ensuremath{\right| \! \right| \! \right|}}_{2,h},$$ for some positive constant $C$ independent of $h$.
For $\Theta_1,\Theta_2\in {\mathbb B}_{R(h)}(\Pi_h\Psi)$, let $\mu(\Theta_i), i=1,2$ be the solutions of: $$\begin{aligned}
\label{mueqn1}
\tilde{\mathcal A}_h(\mu(\Theta_i),\Phi)&=L_h(\Phi)+B_h(\Pi_h\Psi,\Theta_i,\Phi)+B_h(\Theta_i,\Pi_h\Psi,\Phi)-B_h(\Theta_i,\Theta_i,\Phi){\quad\forall\,}\Phi\in{\mathcal V}_h.
$$ The nonsingularity of $\tilde{\mathcal A}_h(\cdot,\cdot)$ yields a $\bar{\Phi}$ with ${\ensuremath{\left| \! \left| \! \left|}}\bar\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1$. With and , we obtain $$\begin{aligned}
&\frac{\beta}{4}{\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta_1)-\mu(\Theta_2){\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \tilde{\mathcal A}_h(\mu(\Theta_1)-\mu(\Theta_2),\bar{\Phi})\notag\\
&=B_h(\Pi_h\Psi,\Theta_1-\Theta_2,\bar\Phi)+B_h(\Theta_1-\Theta_2,\Pi_h\Psi,\bar\Phi)+B_h(\Theta_2,\Theta_2,\bar\Phi)-B_h(\Theta_1,\Theta_1,\bar\Phi)\\
&=B_h(\Theta_2-\Theta_1,\Theta_1-\Pi_h\Psi,\bar{\Phi})+B_h(\Theta_2-\Pi_h\Psi,\Theta_2-\Theta_1,\bar{\Phi})\\
&\leq C{\ensuremath{\left| \! \left| \! \left|}}\Theta_2-\Theta_1{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\left({\ensuremath{\left| \! \left| \! \left|}}\Theta_1-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+
{\ensuremath{\left| \! \left| \! \left|}}\Theta_2-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\right).\end{aligned}$$ Since $\Theta_1,\Theta_2\in {\mathbb B}_{R(h)}(\Pi_h\Psi)$, for a choice of $R(h)$ as in the proof of Theorem \[mapball2ball\], for sufficiently small $h$, we obtain $${\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta_1)-\mu(\Theta_2){\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq Ch^{\alpha} {\ensuremath{\left| \! \left| \! \left|}}\Theta_2-\Theta_1{\ensuremath{\right| \! \right| \! \right|}}_{2,h},$$ for some positive constant $C$ independent of $h$. This completes the proof.
(Local uniqueness) Let $\Psi$ be an isolated solution of . For sufficiently small choice of $h$, Theorem \[contractionthm\] establishes the local uniqueness of the solution of .
Error Estimates
---------------
In this subsection, the error estimates in the broken energy and $H^1$ norms are established.
[(Energy norm estimate)]{}\[eetimate\] Let $\Psi$ and $\Psi_h$ be the solutions of and respectively. Under the assumption that $\Psi$ is an isolated solution, for sufficiently small $h$, it holds $${\ensuremath{\left| \! \left| \! \left|}}\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq C h^{\alpha},$$ where $\alpha\in ({\frac{1}{2}},1]$ is the index of elliptic regularity.
A use of triangle inequality yields $$\label{newnn}
{\ensuremath{\left| \! \left| \! \left|}}\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq {\ensuremath{\left| \! \left| \! \left|}}\Psi-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+{\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}.$$ For sufficiently small $h$, Theorem \[exitenceuniqueness\] leads to $$\label{pihsoln}
{\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq Ch^\alpha.$$ Now, Lemma \[interpolant\] , and establish the required estimate.
[($H^1$ estimate)]{}\[h1estimate\] Let $\Psi$ and $\Psi_h$ be the solutions of and respectively. Assume that $\Psi$ is an isolated solution. Then, for sufficiently small $h$, it holds $${\ensuremath{\left| \! \left| \! \left|}}\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq C h^{2\alpha},$$ where $\alpha\in ({\frac{1}{2}},1]$ is the index of elliptic regularity.
A use of triangle inequality yields $$\label{trienq}
{\ensuremath{\left| \! \left| \! \left|}}\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq {\ensuremath{\left| \! \left| \! \left|}}\Psi-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq {\ensuremath{\left| \! \left| \! \left|}}\Psi-\Pi_h\Psi {\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\rho}-E_h\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}E_h\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{1},$$ where $\boldsymbol{\rho}=\Pi_h\Psi-\Psi_h$. A choice of $Q=-\Delta E_h\boldsymbol{\rho}$ and $\Phi=E_h\boldsymbol{\rho}$ in the dual problem and a use of , leads to $$\begin{aligned}
\left(\nabla E_h\boldsymbol{\rho},\nabla E_h\boldsymbol{\rho}\right) &= {\mathcal A}_h({E_h\boldsymbol{\rho},\boldsymbol \zeta})={\mathcal A}_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})+{\mathcal A}_h(\boldsymbol{\rho},{\boldsymbol \zeta})\nonumber\\
&= A_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})+B_h(\Psi,E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})+B_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},\Psi,{\boldsymbol \zeta})\nonumber\\
&\quad+A_h(\Pi_h\Psi-\Psi,{\boldsymbol\zeta})+A_h(\Psi-\Psi_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+A_h(\Psi,\Pi_h\boldsymbol{\zeta}-\boldsymbol{\zeta})+L_h(\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})\nonumber\\
&\quad
+\left(B_h(\Psi,\Pi_h\Psi-\Psi_h,{\boldsymbol \zeta})+B_h(\Pi_h\Psi-\Psi_h,\Psi,{\boldsymbol \zeta})-B_h(\Psi,\Psi,{\boldsymbol \zeta})+B_h(\Psi_h,\Psi_h,\Pi_h{\boldsymbol \zeta})\right)\nonumber\\
&=:\sum_{i=1}^{8} T_i.\end{aligned}$$
$T_1$ is estimated using Lemma \[enrichreg\] and . $T_4$ and $T_6$ are estimated using Lemma \[enrichreg\]. $T_5$ is estimated using continuity of $A_h(\cdot,\cdot)$, Lemma \[interpolant\] and Theorem \[eetimate\]. The term $T_7$ is estimated using continuity of $L_h$ and Lemma \[interpolant\]. $T_2$ is estimated using Remark \[boundBh3\], Lemma \[enrich1\] and as $$\begin{aligned}
\label{T2}
T_2&\leq |B_h(\Psi,E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})|
\leq C{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha} {\ensuremath{\left| \! \left| \! \left|}}E_h\boldsymbol{\rho}-\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{1,h}{\ensuremath{\left| \! \left| \! \left|}}{\boldsymbol \zeta}{\ensuremath{\right| \! \right| \! \right|}}_2\leq C h^{1+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}{\boldsymbol \zeta}{\ensuremath{\right| \! \right| \! \right|}}_2.\end{aligned}$$ $T_3$ is estimated using Remark \[flipbound\], Lemma \[enrich1\], and as $$T_3\leq|B_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},\Psi,{\boldsymbol \zeta})|\leq |B_h(\Psi,E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})|+Ch{\ensuremath{\left| \! \left| \! \left|}}E_h\boldsymbol{\rho}-\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{2,h}{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}{\boldsymbol \zeta}{\ensuremath{\right| \! \right| \! \right|}}_2\leq Ch^{1+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}{\ensuremath{\left| \! \left| \! \left|}}{\boldsymbol \zeta}{\ensuremath{\right| \! \right| \! \right|}}_2.$$ Finally, a use of Remarks \[boundBh3\] , \[flipbound\], Lemmas \[interpolant\], \[boundBh2\], Theorem \[eetimate\] and yields an estimate for $T_8$ as $$\begin{aligned}
T_8& =B_h(\Psi,\Pi_h\Psi-\Psi_h,{\boldsymbol \zeta})+B_h(\Pi_h\Psi-\Psi_h,\Psi,{\boldsymbol \zeta})-B_h(\Psi,\Psi,{\boldsymbol \zeta})+B_h(\Psi_h,\Psi_h,\Pi_h{\boldsymbol \zeta})\\
&=B_h(\Psi,\Pi_h\Psi-\Psi,{\boldsymbol \zeta})+B_h(\Pi_h\Psi-\Psi,\Psi,{\boldsymbol \zeta})\\
&\quad+B_h(\Psi,\Psi-\Psi_h,{\boldsymbol \zeta})+B_h(\Psi-\Psi_h,\Psi,{\boldsymbol \zeta})-B_h(\Psi,\Psi,{\boldsymbol \zeta})+B_h(\Psi_h,\Psi_h,\Pi_h{\boldsymbol \zeta})\\
&=B_h(\Psi,\Pi_h\Psi-\Psi,{\boldsymbol \zeta})+B_h(\Pi_h\Psi-\Psi,\Psi,{\boldsymbol \zeta})\\
&\quad+B_h(\Psi-\Psi_h,\Psi-\Psi_h,{\boldsymbol \zeta})+B_h(\Psi_h-\Psi,\Psi_h,\Pi_h{\boldsymbol \zeta}-{\boldsymbol \zeta})+B_h(\Psi,\Psi_h,\Pi_h{\boldsymbol \zeta}-{\boldsymbol \zeta})\\
&\leq Ch^{2\alpha}({\ensuremath{\left| \! \left| \! \left|}}{\boldsymbol \zeta}{\ensuremath{\right| \! \right| \! \right|}}_2 +{\ensuremath{\left| \! \left| \! \left|}}{\boldsymbol \zeta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}).\end{aligned}$$ A combination of the estimates $T_1$ to $T_8$ and $a~priori$ bounds for the linearized dual problem yields $$\begin{aligned}
\label{gradEh}
&(\nabla E_h\boldsymbol{\rho},\nabla E_h\boldsymbol{\rho})\leq Ch^{2\alpha}{\ensuremath{\left| \! \left| \! \left|}}-\Delta E_h\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{-1}\leq Ch^{2\alpha}{\ensuremath{\left| \! \left| \! \left|}}E_h\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{1} \Longrightarrow {\ensuremath{\left| \! \left| \! \left|}}E_h\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{1}\leq C h^{2\alpha}.\end{aligned}$$ A use of Lemmas \[interpolant\], \[enrich1\], and the last statement of in completes the proof.
Convergence of the Newton’s Method
----------------------------------
In this subsection, we define a working procedure to find an approximation for the discrete solution $\Psi_h$. The discrete solution $\Psi_h$ of is characterized by the fixed point of . This depends on the unknown $\Pi_h\Psi$ and hence the approximate solution for is computed using Newton’s method in implementation. The iterates of the Newton’s method are defined by $$\label{NewtonIterate}
A_h(\Psi_h^{n},\Phi)+B_h(\Psi_h^{n-1},\Psi_h^{n},\Phi)+B_h(\Psi_h^{n},\Psi_h^{n-1},\Phi)=B_h(\Psi_h^{n-1},\Psi_h^{n-1},\Phi)+L_h(\Phi){\quad\forall\,}\Phi\in {\mathcal V}_h.$$ Now we establish that these iterates in fact converge quadratically to the solution of .
(Convergence of Newton’s method)\[NewtonThm\] Let $\Psi$ be an isolated solution of and let $\Psi_h$ solve . There exists $\rho> 0$, independent of $h$, such that for any initial guess $\Psi_h^0$ which satisfies $\displaystyle {\ensuremath{\left| \! \left| \! \left|}}\Psi_h^0-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \rho,\:\, {\ensuremath{\left| \! \left| \! \left|}}\Psi_h^n-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h} \leq\frac{\rho}{2^n}$ holds true. That is, the iterates of the Newton’s method defined in are well defined and converge quadratically to $\Psi_h$.
From Lemma \[nonsingular\], there exists $\delta>0$ such that for each $Z_h\in{\mathcal V}_h$ satisfying ${\ensuremath{\left| \! \left| \! \left|}}Z_h-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \delta$, the form $$\label{NewtonNonsingular}
A_h(\Theta,\Phi)+B_h(Z_h,\Theta,\Phi)+B_h(\Theta,Z_h,\Phi)$$ is non singular in ${\mathcal V}_h\times{\mathcal V}_h$. From , for sufficiently small $h$, ${\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq Ch^\alpha$. Thus $h$ can be chosen sufficiently small so that ${\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \frac{\delta}{2}$. Define $$\label{defnrho}
\rho:=\min\left\{\frac{\delta}{2},\frac{\beta}{16C_b}\right\}$$ where $\beta$ and $ C_b$ are respectively the coercivity constant of ${\mathcal A}_h(\cdot,\cdot)$ and boundedness constant of $B_h(\cdot,\cdot,\cdot)$ (see ). Assume that the initial guess $\Psi_h^0$ satisfies ${\ensuremath{\left| \! \left| \! \left|}}\Psi_h-\Psi_h^0{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \rho$. Then $${\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h^0{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq {\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}+{\ensuremath{\left| \! \left| \! \left|}}\Psi_h-\Psi_h^0{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \delta$$ Since is nonsingular, the first iterate $\Psi_h^1$ of the Newton’s method in is well defined for the initial guess $\Psi_h^0$. Using the nonsingularity of , there exists $\bar\Phi\in{\mathcal V}_h$ such that ${\ensuremath{\left| \! \left| \! \left|}}\bar\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1$ which satisfies $$\frac{\beta}{8}{\ensuremath{\left| \! \left| \! \left|}}\Psi_h^1-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq A_h(\Psi_h^1-\Psi_h,\bar\Phi)+B_h(\Psi_h^0,\Psi_h^1-\Psi_h,\bar\Phi)+B_h(\Psi_h^1-\Psi_h,\Psi_h^0,\bar\Phi).$$ A use of , , yields $$\begin{aligned}
&A_h(\Psi_h^1-\Psi_h,\bar\Phi)+B_h(\Psi_h^0,\Psi_h^1-\Psi_h,\bar\Phi)+B_h(\Psi_h^1-\Psi_h,\Psi_h^0,\bar\Phi)\notag\\
&=B_h(\Psi_h^0,\Psi_h^0,\bar \Phi)+L_h(\bar \Phi)-A_h(\Psi_h,\bar \Phi)-B_h(\Psi_h^0,\Psi_h,\bar \Phi)-B_h(\Psi_h,\Psi_h^0,\bar \Phi)\notag\\
&=B_h(\Psi_h^0,\Psi_h^0,\bar \Phi)+B_h(\Psi_h,\Psi_h,\bar \Phi)-B_h(\Psi_h^0,\Psi_h,\bar \Phi)-B_h(\Psi_h,\Psi_h^0,\bar \Phi)\notag\\
&=B_h(\Psi_h^0-\Psi_h,\Psi_h^0-\Psi_h,\bar \Phi)\leq C_b{\ensuremath{\left| \! \left| \! \left|}}\Psi_h^0-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}^2.\label{induction0}\end{aligned}$$ Hence, ${\ensuremath{\left| \! \left| \! \left|}}\Psi_h^1-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \frac{8C_b}{\beta}{\ensuremath{\left| \! \left| \! \left|}}\Psi_h^0-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}^2$. Since ${\ensuremath{\left| \! \left| \! \left|}}\Psi_h^0-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}\leq \rho\leq \frac{\beta}{16C_b}$, we obtain $$\label{induction0b}
{\ensuremath{\left| \! \left| \! \left|}}\Psi_h^1-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq{\frac{1}{2}}{\ensuremath{\left| \! \left| \! \left|}}\Psi_h^0-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \frac{\rho}{2}.$$ Since ${\ensuremath{\left| \! \left| \! \left|}}\Psi_h^1-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \rho$, the form is nonsingular for $Z_h=\Psi_h^1$. Continuing the process, we obtain $${\ensuremath{\left| \! \left| \! \left|}}\Psi_h^n-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \frac{\rho}{2^n}.$$ Moreover, proceeding as in the proof of the estimate , it can be shown that $${\ensuremath{\left| \! \left| \! \left|}}\Psi_h^{n+1}-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \left(8C_b/\beta\right){\ensuremath{\left| \! \left| \! \left|}}\Psi_h^n-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{2,h}^2.$$ This establishes that the Newton’s method converges quadratically to $\Psi_h$. This completes the proof.
[(Local uniqueness)]{} The local uniqueness of solution of also follows from Theorem \[NewtonThm\]. We observe that the definition of $\rho$ in does not depend on $h$. From Theorem \[NewtonThm\], it is clear that for any initial guess $\Psi_h^0$ which lies in the ball of radius $\rho$ with center at $\Psi_h$, the sequence generated by will converge uniquely to $\Psi_h$. In particular, if we choose the initial guess $\Psi_h^0=\Pi_h\Psi$, then the sequence generated by the iterates of the Newton’s method will also converge to $\Psi_h$ which shows the local uniqueness of the solution $\Psi_h$.
Numerical Experiments {#sec:num}
=====================
In this section, two numerical experiments that justify the theoretical results are presented. The implementations have been carried out in MATLAB. The results illustrate the order of convergence obtained for the numerical solution of - computed using the Morley finite element scheme. For a detailed description of construction of basis functions for the Morley element, see Ming & Xu [@MingXu]. We implement the Newton’s method defined in to solve the discrete problem .
Example 1 {#example1}
---------
In the first example, we choose the right hand side load functions such that the exact solution is given by $$\begin{aligned}
u(x,y)&=x^2(1-x)^2y^2(1-y)^2;\quad v(x,y)=\sin^2(\pi x)\sin^2(\pi y)
\end{aligned}$$ on the unit square. The initial triangulation is chosen as shown in Figure \[fig:MT\_h\](a). In the uniform red-refinement process, each triangle $T$ is divided into four similar triangles [@AlbertyCarstensenFunken] as in Figure \[fig:MT\_h\](b).
Let the mesh parameter at the $N$-th level be denoted by $h_N$ and the computational error by $e_N$. The experimental order of convergence at the $N$-th level is defined by $$\alpha_N:=log(e_{N-1}/e_N)/log(h_{N-1}/h_{N})=log(e_{N-1}/e_N)/log(2).$$
Tables \[table:OC\_NCFEM\_uh\] and \[table:OC\_NCFEM\_vh\] show the errors and experimental convergence rates for the variables $u_h$ and $v_h$. In Figures \[fig:convrate\_uhM\]-\[fig:convrate\_vhM\], the convergence history of the errors in broken energy, $H^1$ and $L^2$ norms are illustrated. The computational order of convergences in broken $H^2,\; H^1$ norms are quasi-optimal and verify the theoretical results obtained in Theorems \[eetimate\] and \[h1estimate\] for $\alpha=1$. The order of convergence with respect to $L^2$ norm is sub-optimal justifying the results in [@HuShi] that using a lower order finite element method, the order of convergence in $L^2$ norm cannot be improved than that of the $H^1$ norm.
\# unknowns $|u-u_h|_{2,h}$ Order $|u-u_h|_{1,h}$ Order $\|u-u_h\|_{L^2}$ Order
------------- ----------------- -------- ----------------- -------- ------------------- --------
25 0.874685E-1 - 0.102155E-1 - 0.386068E-2 -
113 0.405787E-1 1.1080 0.257318E-2 1.9891 0.919743E-3 2.0695
481 0.209921E-1 0.9508 0.732470E-3 1.8127 0.248134E-3 1.8901
1985 0.106209E-1 0.9829 0.191118E-3 1.9383 0.636227E-4 1.9635
8065 0.532754E-2 0.9953 0.483404E-4 1.9831 0.160158E-4 1.9900
32513 0.266595E-2 0.9988 0.121213E-4 1.9956 0.401107E-5 1.9974
: Errors and convergence rates of $u_h$ in broken $H^2,H^1$ and $L^2$ norms[]{data-label="table:OC_NCFEM_uh"}
\# unknowns $|v-v_h|_{2,h}$ Order $|v-v_h|_{1,h}$ Order $\|v-v_h\|_{L^2}$ Order
------------- ----------------- -------- ----------------- -------- ------------------- --------
25 19.245671 - 2.140613E-0 - 0.770876E-0 -
113 9.5043699 1.0178 0.569979E-0 1.9090 0.177898E-0 2.1154
481 5.0549209 0.9109 0.161737E-0 1.8172 0.482777E-1 1.8816
1985 2.5758939 0.9726 0.421546E-1 1.9398 0.123930E-1 1.9618
8065 1.2944929 0.9926 0.106618E-1 1.9832 0.312076E-2 1.9895
32513 0.6480848 0.9981 0.267351E-2 1.9956 0.781643E-3 1.9973
: Errors and convergence rates of $v_h$ in broken $H^2,H^1$ and $L^2$ norms[]{data-label="table:OC_NCFEM_vh"}
![Convergence history of displacement for Example 1[]{data-label="fig:convrate_uhM"}](uh_OC_SqDomainT7){height="3in" width="5in"}
![Convergence history of Airy stress for Example 1[]{data-label="fig:convrate_vhM"}](vh_OC_SqDomainT7){height="3in" width="5in"}
Example 2 {#example2}
---------
Consider the L-shaped domain $\Omega=(-1,1)^2 \setminus([0,1)\times(-1,0])$ (see Figure \[fig:Lshape\]). Choose the right hand functions such that the exact singular solution [@Grisvard] in polar coordinates is given by $$\begin{aligned}
u(r,\theta)&=(r^2 cos^2\theta-1)^2 (r^2 sin^2\theta-1)^2 r^{1+\alpha}g_{\alpha,\omega}(\theta);\quad v(r,\theta)=u(r,\theta),\end{aligned}$$ where $\omega:=\frac{3\pi}{2}$ and $\alpha:= 0.5444837367$ is a non-characteristic root of $\sin^2(\alpha\omega) = \alpha^2\sin^2(\omega)$ with $$\begin{aligned}
g_{\alpha,\omega}(\theta)=&\left(\frac{1}{\alpha-1}\sin\big{(}(\alpha-1)\omega\big{)}-\frac{1}{\alpha+1}\sin\big{(}(\alpha+1)\omega\big{)}\right)\Big{(}\cos\big{(}(\alpha-1)\theta\big{)}-\cos\big{(}(\alpha+1)\theta\big{)}\Big{)}\\
&-\left(\frac{1}{\alpha-1}\sin\big{(}(\alpha-1)\theta\big{)}-\frac{1}{\alpha+1}\sin\big{(}(\alpha+1)\theta\big{)}\right)\Big{(}\cos\big{(}(\alpha-1)\omega\big{)}-\cos\big{(}(\alpha+1)\omega\big{)}\Big{)}.
\end{aligned}$$ Tables \[table:OC\_Lshape\_uh\] and \[table:OC\_Lshape\_vh\] show the errors and experimental convergence rates for the variables $u_h$ and $v_h$. The domain being non-convex, we do not obtain linear and quadratic order of convergences in broken energy and $H^1$ norms for displacement and Airy stress functions.
(0,0)–(1,0)–(1,1)–(2,1)–(2,2)–(0,2)–(0,0); (2.5,0)–(3.5,0)–(3.5,1)–(4.5,1)–(4.5,2)–(2.5,2)–(2.5,0); (2.5,0)–(4.5,2); (2.5,1)–(3.5,2); (2.5,1)–(3.5,1)–(3.5,2); (2.5,1)–(3.5,0); (2.5,2)–(3.5,1); (3.5,2)–(4.5,1);
\# unknowns $|u-u_h|_{2,h}$ Order $|u-u_h|_{1,h}$ Order $\|u-u_h\|_{L^2}$ Order
------------- ----------------- -------- ----------------- -------- ------------------- --------
17 29.209171 - 6.363539E-0 - 2.769499E-0 -
81 14.130192 1.0476 1.682747E-0 1.9190 0.693436E-0 1.9977
353 7.5651300 0.9013 0.491659E-0 1.7750 0.200814E-0 1.7879
1473 3.9620126 0.9331 0.146551E-0 1.7462 0.583024E-1 1.7842
6017 2.0841141 0.9267 0.487106E-1 1.5891 0.179703E-1 1.6979
24321 1.1252534 0.8891 0.187772E-1 1.3752 0.613474E-2 1.5505
: Errors and the experimental convergence rates for $u_h$ in broken $H^2, H^1$ and $L^2$ norms for L-shaped domain[]{data-label="table:OC_Lshape_uh"}
\# unknowns $|v-v_h|_{2,h}$ Order $|v-v_h|_{1,h}$ Order $\|v-v_h\|_{L^2}$ Order
------------- ----------------- -------- ----------------- -------- ------------------- --------
17 24.759835 - 4.932699E-0 - 2.069151E-0 -
81 15.293270 0.6951 1.779132E-0 1.4712 0.727981E-0 1.5070
353 7.8509322 0.9619 0.483823E-0 1.8786 0.199644E-0 1.8664
1473 4.0531269 0.9538 0.137278E-0 1.8173 0.557622E-1 1.8400
6017 2.1219988 0.9336 0.439086E-1 1.6445 0.165699E-1 1.7507
24321 1.1421938 0.8936 0.165883E-1 1.4043 0.545066E-2 1.6040
: Errors and the experimental convergence rates for $v_h$ in broken $H^2, H^1$ and $L^2$ norms for L-shaped domain[]{data-label="table:OC_Lshape_vh"}
Conclusions & Perspectives {#conclusions}
==========================
In this work, an attempt has been made to obtain approximate solutions for the clamped von Kármán equations defined on polygonal domains using nonconforming Morley elements. Error estimates in broken energy and $H^1$ norms are established for sufficiently small discretization parameters. Numerical results that substantiate the theoretical results are obtained. A future area of interest would be derivation of reliable $a~posteriori$ error estimates that drive the adaptive mesh refinements.
[**Acknowledgments:** ]{} The authors would like to sincerely thank Professors S. C. Brenner and Li-yeng Sung for their suggestions on extension of the results to non-convex polygonal domains and to Dr. Thirupathi Gudi for his comments. The first author would also like to thank National Board for Higher Mathematics (NBHM) for the financial support towards the research work.\
Appendix
========
We consider one of the variants of von Kármán equations which is important in practical applications and give a brief sketch of the extension of the analysis. Consider the following form of von Kármán equations: $$\label{vkes}
\left.
\begin{array}{l l}
\Delta^2 u &=[u,v]-\frac{p}{D}\Delta u+f \\
\Delta^2 v &=-{\frac{1}{2}}[u,u]
\end{array}
\right\} \text{in } \Omega$$ with clamped boundary conditions $$\label{vkbs}
u=\frac{\partial u}{\partial \nu} = v = \frac{\partial v}{\partial \nu} = 0 \text{ on } \partial\Omega,$$ where $p$ is a real parameter known as the bifurcation parameter and $D$ denotes the flexural rigidity of the plate. The weak formulation of - reads as: given $F=(f,0)$, find $\Psi\in{\mathcal V}$ such that $$\label{vforms}
A(\Psi,\Phi)+B(\Psi,\Psi,\Phi)+{\mathfrak C}(\Psi,\Phi)=L(\Phi) {\quad\forall\,}\Phi \in \mathcal{V}$$ where $A(\cdot,\cdot),\;B(\cdot,\cdot,\cdot),\; L(\cdot)$ are defined in - respectively, and ${\mathfrak C}(\cdot,\cdot)$ is defined as $${\mathfrak C}(\Theta,\Phi)=-\frac{p}{D}\int_{\Omega}\nabla \theta_1\cdot\nabla\varphi_1{\;\textit{dx}}{\quad\forall\,}\Theta=(\theta_1,\theta_2) \text{ and } \Phi=(\varphi_1,\varphi_2)\in {\mathcal V}.$$ The corresponding nonconforming finite element formulation is given by: find $\Psi_h\in{\mathcal V}_h$ such that $$\label{vformsd}
A_h(\Psi_h,\Phi)+B_h(\Psi_h,\Psi_h,\Phi)+{\mathfrak C}_h(\Psi_h,\Phi)=L_h(\Phi) {\quad\forall\,}\Phi \in \mathcal{V}_h$$ where $A_h(\cdot,\cdot),\;B_h(\cdot,\cdot,\cdot),\; L_h(\cdot)$ are defined in - respectively, and ${\mathfrak C}_h(\cdot,\cdot)$ is defined as $${\mathfrak C}_h(\Theta,\Phi)=-\frac{p}{D}\sum_{T\in{\mathcal T}_h}\int_{T}\nabla \theta_1\cdot\nabla\varphi_1{\;\textit{dx}}{\quad\forall\,}\Theta=(\theta_1,\theta_2) \text{ and } \Phi=(\varphi_1,\varphi_2)\in {\mathcal V}_h.$$ For the newly introduced bilinear form ${\mathfrak C}(\cdot,\cdot)$, the following boundedness properties hold true: $$\begin{aligned}
{\mathfrak C}(\Theta,\Phi)&\leq C{\ensuremath{\left| \! \left| \! \left|}}\Theta{\ensuremath{\right| \! \right| \! \right|}}_1{\ensuremath{\left| \! \left| \! \left|}}\Phi{\ensuremath{\right| \! \right| \! \right|}}_1{\quad\forall\,}\Theta,\Phi\in{\mathcal V}\label{Cbcont}\\
{\mathfrak C}_h(\Theta_h,\Phi_h)&\leq C{\ensuremath{\left| \! \left| \! \left|}}\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}{\ensuremath{\left| \! \left| \! \left|}}\Phi_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}{\quad\forall\,}\Theta_h,\Phi_h\in{\mathcal V}_h\label{Cbnconf}.\end{aligned}$$ For the modified problem , the linearized problem (see ) is defined by: for given $G\in{(L^2(\Omega))^2}$, find $\Theta\in{\mathcal V}$ such that $$\label{vformsl}
{\mathcal A}(\Theta,\Phi)=(G,\Phi) {\quad\forall\,}\Phi \in \mathcal{V}$$ where $$\label{defnCh}
{\mathcal A}(\Theta,\Phi):=A(\Theta,\Phi)+B(\Psi,\Theta,\Phi)+B(\Theta,\Psi,\Phi)+{\mathfrak C}(\Theta,\Psi).$$ The dual problem is stated as: given $Q\in (H^{-1}(\Omega))^2$, find $\boldsymbol{\zeta}\in{\mathcal V}$ such that $$\label{vformsldual}
{\mathcal A}(\Phi,\boldsymbol{\zeta})=(Q,\Phi) {\quad\forall\,}\Phi \in \mathcal{V}.$$ It can be observed that if $\Psi$ is an isolated solution of , then and are well posed and satisfy the $a~priori$ bounds $$\label{apriorislin23}
{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\Theta}{\ensuremath{\right| \! \right| \! \right|}}_2\leq C{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}},\:\: {\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\Theta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}\leq C{\ensuremath{\left| \! \left| \! \left|}}G{\ensuremath{\right| \! \right| \! \right|}}\text{ and } {\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_2\leq C{\ensuremath{\left| \! \left| \! \left|}}Q{\ensuremath{\right| \! \right| \! \right|}}_{-1}, \:\: {\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}\leq C{\ensuremath{\left| \! \left| \! \left|}}Q{\ensuremath{\right| \! \right| \! \right|}}_{-1},$$ where $\alpha$ is the index of elliptic regularity. The discrete linearized problem is defined as: find $\Theta_h\in{\mathcal V}_h$ such that $$\label{vformsld}
{\mathcal A}_h(\Theta_h,\Phi)=(G,\Phi) {\quad\forall\,}\Phi \in \mathcal{V}_h$$ where $$\label{defnAhs}
{\mathcal A}_h(\Theta_h,\Phi):=A_h(\Theta_h,\Phi)+B_h(\Psi,\Theta_h,\Phi)+B_h(\Theta_h,\Psi,\Phi)+{\mathfrak C}_h(\Theta_h,\Phi).$$
With this background, Theorem \[wellposeld\], Lemma \[nonsingular\] and Theorems \[mapball2ball\]-\[NewtonThm\] can be modified for the new formulation, leading to the applicability of the analysis to a more general form of the von Kármán equations. We will sketch the proofs of the important results.
(Well posedness of discrete linearized problem)\[wellposesld\] If $\Psi$ is an isolated solution of , then for sufficiently small $h$, the discrete linearized problem is well-posed.
[*Outline of the proof.*]{} Following the proof of Theorem \[wellposeld\], we easily arrive at using . To estimate ${\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_1$ in this case, choose $Q=-\Delta E_h\Theta_h$ and $\Phi=E_h\Theta_h$ in and use to obtain $$\begin{aligned}
{\ensuremath{\left| \! \left| \! \left|}}E_h\Theta_h{\ensuremath{\right| \! \right| \! \right|}}_{1}^2
&=A_h(E_h\Theta_h-\Theta_h,\boldsymbol{\zeta})+A_h(\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+B_h(\Psi,E_h\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+B_h(E_h\Theta_h,\Psi,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})\\
&\quad+B_h(\Psi,E_h\Theta_h-\Theta_h,\Pi_h\boldsymbol{\zeta})+B_h(E_h\Theta_h-\Theta_h,\Psi,\Pi_h\boldsymbol{\zeta})+(G,\Pi_h\boldsymbol{\zeta})\\
&\quad+\left({\mathfrak C}_h(E_h\Theta-\Theta_h,\boldsymbol{\zeta})+{\mathfrak C}_h(\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})\right).\end{aligned}$$ The last term can be estimated using , Lemmas \[interpolant\] and \[enrich1\] as $$|{\mathfrak C}_h(E_h\Theta-\Theta_h,\boldsymbol{\zeta})+{\mathfrak C}_h(\Theta_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})|\leq Ch{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_2.$$ The remaining terms are estimated as in Theorem \[wellposeld\] and result follows.
The next lemma follows as in Lemma \[nonsingular\] using and hence the proof is skipped.
[(Nonsingularity of perturbed bilinear form)]{}\[nonsingulars\] Let $\Pi_h\Psi$ be the interpolation of $\Psi$ as defined in Lemma \[interpolant\]. Then, for sufficiently small $h$, the perturbed bilinear form defined by $$\tilde {\mathcal A}_h(\Theta,\Phi)= A_h(\Theta,\Phi)+B_h(\Pi_h\Psi,\Theta,\Phi)+B_h(\Theta,\Pi_h\Psi,\Phi)+{\mathfrak C}(\Theta,\Phi)$$ is nonsingular on ${\mathcal V}_h\times{\mathcal V}_h$, if is nonsingular on ${\mathcal V}_h\times{\mathcal V}_h$.
(Mapping of ball to ball) \[mapball2balls\] For a sufficiently small choice of $h$, there exists a positive constant $R(h)$ such that for any $\Theta \in {\mathcal V}_h$, $${\ensuremath{\left| \! \left| \! \left|}}\Theta-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R(h)\Rightarrow {\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta)-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq R(h).$$ That is, $\mu$ maps the ball ${\mathbb B}_{R(h)}(\Pi_h\Psi)$ to itself.
[*Outline of the proof.*]{} Proceeding as in the proof of Theorem \[mapball2ball\], using nonsingularity of $\tilde{{\mathcal A}}_h(\cdot,\cdot)$ and Lemma \[nonsingulars\], there exists $\bar\Phi\in{\mathcal V}_h$ such that ${\ensuremath{\left| \! \left| \! \left|}}\bar\Phi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}=1$ and $$\begin{aligned}
&\frac{\beta}{4}{\ensuremath{\left| \! \left| \! \left|}}\mu(\Theta)-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq \tilde{\mathcal A}_h(\mu(\Theta)-\Pi_h\Psi, {\bar\Phi})\notag\\
&=L_h({\bar\Phi}-E_h{\bar\Phi}) +\left(A_h(\Psi, E_h{\bar\Phi})-A_h(\Pi_h\Psi,{\bar\Phi})\right)+\left(B_h(\Psi,\Psi, E_h{\bar\Phi})-B_h(\Pi_h\Psi,\Pi_h\Psi,{\bar\Phi})\right)\notag\\
&\quad+B_h(\Pi_h\Psi-\Theta,\Theta-\Pi_h\Psi,{\bar\Phi})\notag+\left({\mathfrak C}_h(\Psi,E_h\bar\Phi)-{\mathfrak C}_h(\Pi_h\Psi,\bar\Phi)\right)=:\sum_{i=1}^{5} T_i.\end{aligned}$$ The terms $T_1$ to $T_4$ can be estimated as in the proof of Theorem \[mapball2ball\]. The last term $T_5$ is estimated using , Lemmas \[interpolant\] and \[enrich1\] as: $$|{\mathfrak C}_h(\Psi,E_h\bar\Phi)-{\mathfrak C}_h(\Pi_h\Psi,\bar\Phi)|\leq |{\mathfrak C}_h(\Psi,E_h\bar\Phi-\bar\Phi)|+|{\mathfrak C}_h(\Psi-\Pi_h\Psi,\bar\Phi)|\leq Ch{\ensuremath{\left| \! \left| \! \left|}}\Psi{\ensuremath{\right| \! \right| \! \right|}}_2.$$ The remaining proof follows exactly same as the proof of Theorem \[mapball2ball\].\
The existence of solution $\Psi_h$ of follows using Theorem \[mapball2balls\] and satisfies the estimate $$\label{solnee}
{\ensuremath{\left| \! \left| \! \left|}}\Psi_h-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{2,h}\leq Ch^\alpha.$$ A contraction result similar to Theorem \[contractionthm\] also holds true in this case. The energy estimate follows exactly as in the proof of Theorem \[eetimate\].
[($H^1$ estimate)]{}\[h1estimate\_s\] Let $\Psi$ and $\Psi_h$ be the solutions of and respectively. Assume that $\Psi$ is an isolated solution. Then, for sufficiently small $h$, it holds $${\ensuremath{\left| \! \left| \! \left|}}\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq C h^{2\alpha},$$ where $\alpha\in ({\frac{1}{2}},1]$ is the index of elliptic regularity.
[*Outline of the proof.*]{} A use of triangle inequality yields $$\label{trienqs}
{\ensuremath{\left| \! \left| \! \left|}}\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq {\ensuremath{\left| \! \left| \! \left|}}\Psi-\Pi_h\Psi{\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}\Pi_h\Psi-\Psi_h{\ensuremath{\right| \! \right| \! \right|}}_{1,h}\leq {\ensuremath{\left| \! \left| \! \left|}}\Psi-\Pi_h\Psi {\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\rho}-E_h\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{1,h}+{\ensuremath{\left| \! \left| \! \left|}}E_h\boldsymbol{\rho}{\ensuremath{\right| \! \right| \! \right|}}_{1},$$ where $\boldsymbol{\rho}=\Pi_h\Psi-\Psi_h$. A choice of $Q=-\Delta E_h\boldsymbol{\rho}$ and $\Phi=E_h\boldsymbol{\rho}$ in the dual problem leads to $$\begin{aligned}
\left(\nabla E_h\boldsymbol{\rho},\nabla E_h\boldsymbol{\rho}\right) &= {\mathcal A}_h({E_h\boldsymbol{\rho},\boldsymbol \zeta})={\mathcal A}_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})+{\mathcal A}_h(\boldsymbol{\rho},{\boldsymbol \zeta})\nonumber\\
&= A_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})+B_h(\Psi,E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})+B_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},\Psi,{\boldsymbol \zeta})\nonumber+{\mathfrak C}_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})\\
&\quad+A_h(\Pi_h\Psi-\Psi,{\boldsymbol\zeta})+A_h(\Psi-\Psi_h,\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})+A_h(\Psi,\Pi_h\boldsymbol{\zeta}-\boldsymbol{\zeta})+L_h(\boldsymbol{\zeta}-\Pi_h\boldsymbol{\zeta})\nonumber\\
&\quad
+\left(B_h(\Psi,\Pi_h\Psi-\Psi_h,{\boldsymbol \zeta})+B_h(\Pi_h\Psi-\Psi_h,\Psi,{\boldsymbol \zeta})-B_h(\Psi,\Psi,{\boldsymbol \zeta})+B_h(\Psi_h,\Psi_h,\Pi_h{\boldsymbol \zeta})\right)\nonumber\\
&\quad+
\left({\mathfrak C}_h(\Pi_h\Psi-\Psi,\boldsymbol{\zeta})+{\mathfrak C}_h(\Psi_h,\Pi_h\boldsymbol{\zeta}-\boldsymbol{\zeta})\right)\end{aligned}$$ Combining all the terms related to ${\mathfrak C}_h$ and using , and Lemmas \[interpolant\], \[enrich1\], we obtain the estimate $${\mathfrak C}_h(E_h\boldsymbol{\rho}-\boldsymbol{\rho},{\boldsymbol \zeta})+{\mathfrak C}_h(\Pi_h\Psi-\Psi,\boldsymbol{\zeta})+{\mathfrak C}_h(\Psi_h,\Pi_h\boldsymbol{\zeta}-\boldsymbol{\zeta})\leq Ch^{1+\alpha}{\ensuremath{\left| \! \left| \! \left|}}\boldsymbol{\zeta}{\ensuremath{\right| \! \right| \! \right|}}_{2+\alpha}.$$ Estimating the remaining terms as in the proof of Theorem \[h1estimate\], the result follows.\
The Newton’s iterates in this case are defined by $$\label{NewtonIterate_s}
A_h(\Psi_h^{n},\Phi)+B_h(\Psi_h^{n-1},\Psi_h^{n},\Phi)+B_h(\Psi_h^{n},\Psi_h^{n-1},\Phi)+{\mathfrak C}_h(\Psi_h^n,\Phi)=B_h(\Psi_h^{n-1},\Psi_h^{n-1},\Phi)+L_h(\Phi){\quad\forall\,}\Phi\in {\mathcal V}_h.$$ The quadratic convergence result follows by a similar proof as in Theorem \[NewtonThm\].\
Example 3 {#example3}
---------
In this example, we perform numerical experiments for the problem - with $p/D=10$, over a unit square domain. Choose the right hand side load functions such that the exact solution is given by $$\begin{aligned}
u(x,y)&=x^2(1-x)^2y^2(1-y)^2,\quad v(x,y)=\sin^2(\pi x)\sin^2(\pi y).
\end{aligned}$$ We consider the same initial triangulation and its uniform refinement process as in Example \[example1\]. Tables \[table:OC\_NCFEM\_uhs\] and \[table:OC\_NCFEM\_vhs\] show the errors and experimental convergence rates for the variables $u_h$ and $v_h$. The computational order of convergences in broken $H^2,\; H^1$ norms are quasi-optimal and verify the theoretical results. Also, the order of convergence with respect to $L^2$ norm is sub-optimal justifying the results in [@HuShi].
\# unknowns $|u-u_h|_{2,h}$ Order $|u-u_h|_{1,h}$ Order $\|u-u_h\|_{L^2}$ Order
------------- ----------------- -------- ----------------- -------- ------------------- --------
25 0.101724E-0 - 0.129574E-1 - 0.469669E-2 -
113 0.391714E-1 1.3767 0.275863E-2 2.2317 0.957470E-3 2.2943
481 0.195023E-1 1.0061 0.767382E-3 1.8459 0.252196E-3 1.9246
1985 0.974844E-2 1.0004 0.198544E-3 1.9504 0.641987E-4 1.9739
8065 0.487399E-2 1.0000 0.500990E-4 1.9866 0.161298E-4 1.9928
32513 0.243697E-2 1.0000 0.125546E-4 1.9965 0.403763E-5 1.9981
: Errors and Convergence rates of $u_h$ in broken $H^2,H^1$ and $L^2$ norms[]{data-label="table:OC_NCFEM_uhs"}
\# unknowns $|v-v_h|_{2,h}$ Order $|v-v_h|_{1,h}$ Order $\|v-v_h\|_{L^2}$ Order
------------- ----------------- -------- ----------------- -------- ------------------- --------
25 19.245650 - 2.140609E-0 - 0.770875E-0 -
113 9.5043692 1.0178 0.569978E-0 1.9090 0.177898E-0 2.1154
481 5.0549208 0.9109 0.161737E-0 1.8172 0.482777E-1 1.8816
1985 2.5758938 0.9726 0.421546E-1 1.9398 0.123930E-1 1.9618
8065 1.2944929 0.9926 0.106618E-1 1.9832 0.312076E-2 1.9895
32513 0.6480848 0.9981 0.267351E-2 1.9956 0.781643E-3 1.9973
: Errors and Convergence rates of $v_h$ in broken $H^2,H^1$ and $L^2$ norms[]{data-label="table:OC_NCFEM_vhs"}
[^1]: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. Email. [email protected]
[^2]: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. Email. [email protected]
|
---
abstract: 'We explore heavy-element nucleosynthesis in the explosion of massive stars which are triggered by a quark–hadron phase transition during the early post-bounce phase of core-collapse supernovae. The present study is based on general-relativistic radiation hydrodynamics simulations with three-flavor Boltzmann neutrino transport in spherical symmetry, which utilize a quark-hadron hybrid equation of state based on the MIT bag model for strange quark matter. The quark–hadron phase transition inside the stellar core forms a shock wave propagating toward the surface of the proto-neutron star. This shock wave results in an explosion and ejects neutron-rich matter from the outer accreted layers of the proto-neutron star. Later, during the cooling phase, the proto-neutron star develops a proton-rich neutrino-driven wind. We present a detailed analysis of the nucleosynthesis outcome in both neutron-rich and proton-rich ejecta and compare our integrated nucleosynthesis with observations of the solar system and metal-poor stars. For our standard scenario we find that a “weak" $r$-process occurs and elements up to the second peak ($A \sim 130$) are successfully synthesized. Furthermore, uncertainties in the explosion dynamics could barely allow to obtain the strong $r$-process which produces heavier isotopes including the third peak ($A \sim 195$) and actinide elements.'
author:
- |
Nobuya Nishimura, Tobias Fischer, Friedrich-Karl Thielemann,\
Carla Fröhlich, Matthias Hempel, Roger Käppeli, Gabriel Martínez-Pinedo,\
Thomas Rauscher, Irina Sagert, and Christian Winteler
title: |
Nucleosynthesis in core-collapse supernova explosions\
triggered by a quark-hadron phase transition
---
Introduction
============
Nucleosynthesis from core-collapse supernovae is certainly responsible for the production of intermediate mass elements including the so-called alpha elements and a certain fraction of iron isotopes and their neighbors (the Fe-group nuclei). The lighter alpha elements from oxygen to silicon have contributions from hydrostatic burning which takes place during stellar evolutions, while the heavier ones and the Fe-group isotopes originate from explosive burning [@WoosleyWeaver:1995; @Thielemann:etal:1996; @Woosley:etal:2002; @Nomoto:etal:2006; @WoosleyHeger:2007; @HegerWoosley:2010; @Thielemann:etal:2011]. A major open question is related to the source of heavy elements beyond iron.
There have been strong expectations that the innermost layer of ejecta, close to the forming neutron star, remains neutron-rich and is a possible site for the $r$-process. For many years the late neutrino-driven wind (NDW), following the actual explosion, seemed an adequate $r$-process site in core-collapse supernovae that formed a neutron star [@QianWoosley:1996; @Takahashi:etal:1994; @Woosley:etal:1994]. Such investigations were followed up by many parametrized calculations that explored the sensitivity of the most relevant $r$-process parameter, the neutron-to-seed ratio, via variations of entropy $S$, electron fraction (or number of proton per nucleon) $Y_e$ and expansion timescale $\tau$ [@Hoffman:etal:1997; @MeyerBrown:1997; @Freiburghaus:etal:1999; @Farouqi:etal:2010]. However, steady state wind models [@Thompson:etal:2001; @Wanajo:2006] showed that it is very hard to attain the required entropies. These results have been recently confirmed by fully hydrodynamical simulations [@Arcones:etal:2007] that in addition showed that the presence of a reverse shock does not have any major impact on the neutron-to-seed ratio.
Furthermore, recent investigations noticed that the early NDW turns matter proton-rich, producing specific Fe-group isotopes and in the subsequent $\nu p$-process nuclei with a mass number up to $A\sim 80$ – $90$ [@Liebendoerfer:etal:2003; @Pruet:etal:2005; @Froehlich:etal:2006a; @Froehlich:etal:2006b; @Pruet:etal:2006; @Wanajo:2006]. While it was initially hoped that there exists a chance that the late NDW still turns neutron-rich (after its initial, early proton-rich phase) existing core-collapse calculations seem to indicate that the wind becomes even more proton-rich in the long-term evolution [@Fischer:etal:2010; @Huedepohl:etal:2010; @Fischer:2012], although very recent investigations [@MartinezPinedo:2012; @Roberts:2012] related to the spectral evolution of $\nu_e$ and $\bar\nu_e$ in medium might change this somewhat.
It was realized that the best chance to eject neutron-rich matter, is to utilize material stemming from the initial collapse and compression, where electron captures turned it neutron-rich early in the explosion, before neutrino interactions have the chance to convert it into proton-rich matter in the NDW. Electron-capture supernovae, which explode without a long phase of accretion onto the proto-neutron star, apparently provide such conditions [@Wanajo:etal:2011]. However, the $Y_{e}$’s obtained under such conditions do not support a strong $r$-process, which successfully reproduces the platinum peak of $r$-elements around $A=195$.
An alternative explosive scenario as a possible site for the $r$-process has been suggested by @Jaikumar:2007, where after a supernova explosion a quark–hadron phase transition inside the neutron star can eject very neutron-rich material from the neutron star surface. This scenario has been referred to as a *quark-nova*. In the present paper, we follow a different approach and investigate core-collapse supernova explosions that are triggered by a quark-hadron phase transition during the early post-bounce phase [for details, see @Sagert:etal:2009; @Fischer:etal:2011] and their nucleosynthesis features. Under such conditions, zones which are initially neutron-rich, can be (promptly) ejected without experiencing the strong effects of the neutrino flux which comes from the central proto-neutron star.
In the following sections, we discuss the supernova explosion model and the nuclear physics inputs utilized for the nucleosynthesis calculations in Section 2, the conditions experienced and the resulting ejecta in Section 3, and give a detailed analysis of the ejecta composition in Section 4, followed by conclusions, also in comparison to alternative $r$-process sites, in Section 5.
Methodology and nuclear physics input
=====================================
We explore nucleosynthesis in core-collapse supernovae which are triggered by a deconfinement phase transition during the early post-bounce phase. The explosion models have been discussed in detail by @Fischer:etal:2011. In this section, we summarize the main features of the explosion models and the input physics of the nuclear reaction network utilized for nucleosynthesis.
Hydrodynamics and Equation of State
-----------------------------------
A self-consistent supernova explosion model, which we adopt to investigate nucleosynthesis in the present work, has been carried out by using the AGILE-BOLTZTRAN code. This numerical code is based on general-relativistic radiation hydrodynamics in spherical symmetry with an adaptive grid and employs a three-flavor Boltzmann neutrino transport [@Liebendoerfer:etal:2004] with detailed microphysics, e.g., a realistic nuclear equation of state (EOS), weak processes, and (other) nuclear reactions. For the current study, the standard weak processes were considered as listed in Table 1 of @Fischer:etal:2011.
The baryonic EOS in supernovae needs in general to be known for the following three intrinsically different regimes:
1. At temperatures below $6$ GK ($\simeq 0.5$ MeV), the baryon EOS is dominated by heavy nuclei and their abundances, determined by individual nuclear reactions which are not necessarily in equilibrium. In @Fischer:etal:2010, we included a nuclear reaction network, consisting of 20 nuclei, which permits to simulate a large domain of the progenitor up to the helium–layer and is utilized to predict the amount of energy generation by explosive nuclear burning.
2. At temperatures above $6$ GK ($\simeq 0.5$ MeV), the nuclear statistical equilibrium (NSE) can be applied, where the baryonic EOS from @LattimerSwesty:1991 and @Shen:etal:1998 are commonly used in supernova simulation studies.
3. Above the nuclear matter density, the state of matter is highly uncertain. There exists the possibility of a deconfinement phase transition. Therefore, we extended the hadronic EOS by @Shen:etal:1998 at high densities and temperatures, making use of a quark EOS based on the bag model for strange quark matter. For the first-order phase transition between hadronic and quark phases, we applied Gibbs conditions leading to a mixed phase during the transition. This results in a continuous phase transformation. Details of the quark–hadron hybrid EOS are discussed in @Fischer:etal:2011.
Besides the contributions from electrons and positrons as well as photons, Coulomb corrections to the EOS are added by the method of @TimmesArnett:1999.
For the current nucleosynthesis predictions, we select the explosion calculations of the $10.8$ $M_\odot$ progenitor model, where a quark–hadron hybrid EOS was used with an early phase transition to quark matter close to normal nuclear matter density [labeled EOS2, second line of Table 2 in @Fischer:etal:2011]. We chose this model because it has an explosion energy consistent with the expected order of magnitude of $10^{51}$ erg [see the second line of Table 3 in @Fischer:etal:2011]. The maximum gravitational mass for this EOS is $1.5026$ $M_\odot$. Though it is in agreement with the highest precisely known mass of a compact star, the Hulse–Taylor pulsar of $1.44$ $M_\odot$, recent mass limits for the physical EOS are based on the millisecond pulsars J1903+0327 with $M = 1.667 \pm 0.021$ $M_\odot$ [@Freire:etal:2011] and J1614–2230 with a high mass of $M = 1.97 \pm 0.04$ $M_\odot$ [@Demorest:etal:2010]. The inclusion of corrections from the strong interaction coupling constant can stiffen the quark EOS and lead to higher maximum masses [see, e.g., @Schertler00; @Alford:etal:2005; @Sagert:etal:2010; @Sagert:etal:2011; @Weissenborn:etal:2011].
The physical conditions for a possible quark-hadron phase transition in the proto-neutron star are highly uncertain. Therefore, @Fischer:etal:2011 also constructed a quark-hadron hybrid EOS which includes corrections from the strong interaction coupling constant. They showed that such an EOS, with a maximum mass of 1.67 M$_\odot$ and a phase transition to quark matter close to nuclear saturation density [see EOS3 in table 2 and 3 in @Fischer:etal:2011], leads to a qualitatively similar explosion scenario in spherical symmetry as obtained in the explored models with several EOSs which have lower maximum masses. Note, the different post-bounce times for the onset of deconfinement, and hence for the onset of the explosion, lead to different proto-neutron star structures. In general, a delayed phase transition at higher densities translates to a longer mass accretion phase (for the same progenitor). It results in a more massive protoneutron star with a steeper density gradient at its surface and a lower $Y_e$ at the onset of the explosion triggered by the phase transition. An earlier phase transition and less steep density gradient would result in a slower expansion and a slightly higher $Y_e$.
This has important consequences for the nucleosynthesis of the ejected material since expansion timescale and enclosed mass, and also neutrino luminosities and spectra, depend on these aspects [for a detailed discussion, see @Fischer:etal:2011] and the behavior of EOS1 and EOS2 in their Figure 17.
@Weissenborn:etal:2011 recently showed that it is possible to obtain a quark–hadron hybrid EOS which allows for both, a maximum mass larger than 2 $M_\odot$ and a low critical density for the appearance of quark matter. Whether such quark–hadron hybrid EOS will result in a similar dynamical evolution as discussed in @Fischer:etal:2011 will be examined in future simulations.
Explosion Scenario
------------------
The supernova post-bounce evolution is characterized by mass accretion causing a continuous rise of the central density. Once the central density exceeds the critical density for the onset of deconfinement, the quark–hadron phase transition takes place, leading to the appearance of a quark–hadron mixed phase. Thereby, quark matter appears in the supernova core where the highest densities are experienced. The timescale for the appearance of quark matter is given by the timescale for the central density to rise. This depends on the progenitor model, which determines mass accretion rates, and the hadronic EOS. Note that the EOS in the mixed phase is significantly softer than in hadronic and pure quark phases. It is a consequence of the assumed first-order phase transition. Mass accretion from the outer layers of the progenitor onto the central supernova core leads to a continuous rise of the central enclosed mass. When the critical mass (given by the hybrid EOS) of the configuration is obtained, it becomes gravitationally unstable and the supernova core begins to contract. The contraction proceeds into a collapse which rises the density and converts the hadronic core into quark matter at around the center. A massive pure quark core forms at the center, where the EOS stiffens and the collapse halts, and an accretion shock forms. The shock wave propagates out of the high-density supernova core, remaining an accretion front with no matter outflow. Once it reaches the outer layers of the central core, where the density drops over several orders of magnitude, the accretion front accelerates and turns into a dynamic shock with matter outflow. This moment determines the onset of an explosion, also for supernova models which would otherwise not explode in spherical symmetry, based on explosion mechanisms discussed so far. Finally, at distances on the order of $100$ km the expanding shock wave merges with the standing shock from the initial core bounce at nuclear densities, which was unaffected by the dynamics occurring in the supernova core.
Zone No. $M_{\#}$ ($10^{-2} M_{\odot}$) $\Delta\overline{M_\#}$ ($M_{\odot}$) $Y_{e, \rm{NSE}}$ $t_{\rm{ej}}$ (s)
---------- -------------------------------- --------------------------------------- --------------------------------- -------------------
001–014 $0.000$–$0.208$ [$1.496\times10^{-4}$]{} $ 0.20$ $\cdots$
: Summary of mass zones and their properties
\[tab-properties\]
When the (second) shock reaches the neutrino spheres, an additional millisecond neutrino burst is released. It appears in all flavors, however dominated by $\bar\nu_e$ and $\nu_{\mu/\tau}$, in contrast to the $\nu_e$-deleptonization burst related to the early bounce shock propagation across the neutrino spheres between $200$ and $500$ ms after core bounce. This second neutrino burst is of particular interest for water-Cherenkov neutrino detectors, which are more sensitive to $\bar\nu_e$ than to $\nu_e$. @Dasgupta:etal:2010 demonstrated that the currently operating generation of water-Cherenkov neutrino detectors (e.g., Super-Kamiokande and IceCube detectors) can resolve such millisecond neutrino burst of the explosion models by @Fischer:etal:2011.
The matter considered for nucleosynthesis studies of heavy elements (see Section 3) belongs originally to the inner parts of the silicon and sulfur layers of the $10.8$ $M_\odot$ progenitor from @Woosley:etal:2002, with temperature and electron fraction of $3$ GK and $Y_e \simeq 0.5$, respectively, at $800$ – $1000$ km from the center on the pre-collapse phase. During the collapse and the explosion, material contracts and is heated by the shock to temperatures exceeding $100$ GK and hence completely dissociates into free nucleons. At high densities weak processes, mainly electron captures, establish a very low proton-to-baryon ratio $Y_e \simeq 0.1$ during the core-collapse. The further evolution is discussed in detail in Section 3.
Nuclear Reaction Network
------------------------
The nuclear reaction network utilized for the following nucleosynthesis simulations of the ejecta is an extension of previous ones, which have already been described in detail [see @Nishimura:etal:2006; @Fujimoto:etal:2008]. The network includes more than $4000$ nuclei from neutrons and protons up to fermium with atomic number $Z = 100$ [for detail, see Table 1 in @Nishimura:etal:2006] and includes proton-rich isotopes as well as neutron-rich ones far from stability. It includes two- and three-body reactions, decay channels, and electron as well as positron capture [for details, see network A in @Fujimoto:etal:2007] and screening effects for all relevant charged-particle reactions. Experimentally determined masses [@AudiWapstra:1995] and reaction rates are adopted if available. Otherwise, theoretical predictions for nuclear masses, reaction rates, and beta-decays are applied, based on the finite range droplet mass model [@Moller:etal:1995]. Spontaneous and beta-delayed fission processes [@StaudtKlapdor-Kleingrothaus:1992] are taken into account. We adopt the empirical formula for fission fragments by @KodamaTakahashi:1975.
![ Initial distribution of mass and $Y_e$ as a function of mass zone number. Thick lines and dashed lines relate to Lagrangian mass coordinate and electron fraction, respectively. Masses are measured from the surface of the proto-neutron star (starting at 0 for zone 001 of Table \[tab-properties\]) and the electron fractions are adopted at the time when the temperature decreases down to $T=9\times 10^9$ K for ejected matter and the end of the hydrodynamic simulation for inner non-ejected zones). The plot shows only the mass zone range 001 to 090 for enclosed mass and $Y_e$. For the mass zones 90 to 120, $Y_e$ stays essentially constant at $\simeq 0.5$ and the enclosed mass continues to be proportional to mass zone numbers.[]{data-label="fig-init"}](./f1.eps){width="\hsize"}
We also employ neutrino interactions with matter in order to include dominant weak interactions affecting the evolution of the overall proton/nucleon ratio $Y_{e}$. For (anti-)electron neutrino captures by nucleons, we adopt reaction rates derived by @QianWoosley:1996 but ignore any reactions with heavy isotopes because the amount of neutrino capture is negligible for the early phase of nucleosynthesis. The neutrino fluxes, resulting from the detailed neutrino-radiation hydrodynamics calculation described in the previous subsection, are utilized to determine the actual rates as a function of time. These reaction rates depend on the distance from the proto-neutron star and the mean energy and luminosity of neutrinos emitted from the proto-neutron star. It depends on the structure and the evolution, which is sensitive to the EOS, and the precise evolution history of ejected matter including the early phase of the core bounce. Thus, this is different from the treatment of other nuclear reaction rates, which are determined only by local thermodynamic conditions and density and temperature.
Nucleosynthesis in the ejecta
=============================
Dynamic Evolution of Mass Zones
-------------------------------
In order to calculate the nucleosynthesis evolution of ejected matter within a postprocessing approach, the dynamic evolution is required in radial Lagrangian mass zones. For this reason, the evolution determined with the radiation hydrodynamics code AGILE-BOLTZTRAN [see @MezzacappaBruenn:1993a; @MezzacappaBruenn:1993b; @MezzacappaBruenn:1993c; @Liebendoerfer:etal:2001a; @Liebendoerfer:etal:2001b] which is based on an adaptive grid, was mapped on a Lagrangian grid of 120 mass zones. This provides the Lagrangian evolution of physical quantities, such as density, temperature, electron fraction, and velocity of the ejected material, and in addition the neutrino fluxes experienced as a function of time.
The mass zones which are ejected in the explosion are classified in three different categories, related to their ejection process and thermodynamic quantities. As listed in Table [\[tab-properties\]]{}, zones 001 to 120, given with the ejection timescale after bounce ($t_{\rm{ej}}$) and the final $Y_{e,\rm{NSE}}$ which is $Y_e$ at the end of NSE (below $T=9$ GK), cover the material from the surface of the inner core at 1.48000 $M_{\odot}$ to layers with a corresponding mass of 1.49482 $M_{\odot}$. They are also shown with respect to their mass as well as $Y_e$-distribution in Figure \[fig-init\].
![Top: radial trajectories of mass elements as a function of time after bounce. The colors indicate the properties of these mass elements: black, red, green and blue lines refer to matter which is either (black) not ejected, (red) part of the neutrino-driven wind, (green) initially stalled matter which gets boosted by the wind and (blue) matter which experiences a prompt ejection. Bottom: evolution of $Y_e$ as a function of time after the core bounce. The deconfinement phase transition happens about 0.4 s after this initial bounce, causing the explosion and ejection of matter. The colors are the same as in the top panel.[]{data-label="fig-properties"}](./f2.eps){width="\hsize"}
These zones coincide with the matter discussed at the end of Section 2.2, where at high densities a $Y_e$ decrease down to even $0.02$ has been noticed. The evolution of these mass zones in time is displayed in Figure [\[fig-properties\]]{}. Zones 001 to 014 are not ejected within $0.5$ s after the core bounce. These zones preserve the original low-$Y_e$ obtained during collapse and shock wave propagation. As they are not ejected, we ignore them in the further nucleosynthesis discussion, plus all matter originating from regions at smaller radii. In Figure [\[fig-properties\]]{}, they are displayed in black. Zones 015 to 019 are ejected in the so-called NDW, shown in red. Their $Y_e$ is strongly affected by neutrino interactions, turning this matter proton-rich. Zones 020 to 050 (displayed in green) have stalled from infall after shock formation and are ejected thereafter due to neutrino heating and dynamic effects [@Fischer:etal:2011]. The adjacent zones 051 to 120 are ejected in a prompt way, due to the shock wave originating from the deconfinement phase transition (displayed in blue). We clearly see the division of matter which is ejected in a prompt fashion (blue), matter which is coasting and falling in again, but gets reaccelerated outward by neutrino energy deposition (green), matter which falls back onto the neutron star, but becomes part of the NDW ejecta (red), and finally matter which stays on the neutron star and will never be ejected (black).
![Evolution of temperature (top), density (middle), and entropy (bottom) of mass zones as a function of time after bounce. []{data-label="fig-postprocess"}](./f3.eps){width="\hsize"}
![Radius of neutrino spheres for electron neutrinos and electron anti-neutrinos (top). Luminosities (middle) and mean energies (bottom) experienced by mass zone 018, which belongs to the neutrino-driven wind with proton-rich ejecta, as a function of time after bounce. The difference between the mean energy of anti-neutrinos and those of neutrinos is of the order $3$ MeV, i.e., less than $4\Delta$, where $\Delta$ is the neutron–proton mass difference.[]{data-label="fig-neutrinos"}](./f4.eps){width="\hsize"}
As shown in the bottom part of Figure [\[fig-properties\]]{}, the blue and green zones are neutronized during the collapse via electron capture to various degrees, depending on the maximum density attained. Thus the main feature is that a strong compression during infall leads to high densities (at still low entropies, i.e. highly degenerate matter) with large electron Fermi energies which endorse electron captures and a strong neutronization of matter with a small $Y_e$. Their $Y_e$-values range from $0.35$ to $0.5$. The prompt or quasi-prompt ejection does not change this value (with minor effects on the innermost zones, being partially affected by the NDW). The effect of neutrino and anti-neutrino exposure from the core leads also to an increase of $Y_e$, as discussed before. Therefore, the major point in favor of an $r$-process is a fast expansion of that material having two aspects: (1) timescales are shorter than needed for attaining weak equilibrium and (2) matter moves fast to larger radii where the effect of neutrinos vanishes ($1/r^2$). The mass zones in red experience a similar effect in their early evolution during collapse, but the later evolution leads to values of $Y_e$ exceeding $0.5$. This is similar to recent studies of the NDW [see the Introduction and @Fischer:etal:2010], where similar neutrino and anti-neutrino spectra and flux intensities favor proton-rich matter due to the neutron–proton mass difference, resulting in different energies available for the neutrino/anti-neutrino captures. As the neutrino luminosity is still high in the ejection phase, we expect $\nu p$-process nucleosynthesis. Finally non-ejected mass zones (black) can initially also experience interaction with the neutrino flux and turn proton-rich while still at larger radii and small(er) densities. Once they settle on the surface of the neutron star at high densities, capture of degenerate electrons dominates over the neutrino effects, and they turn neutron-rich again.
![ $Y_e$ and entropy $S$, which set the conditions for explosive nucleosynthesis at the time of matter ejection. The innermost ejected zones are proton-rich due the effect of the neutrino-driven wind, which also heats matter efficiently, leading to high entropies. The outer mass zones, ejected in a more prompt fashion keep their original (slightly neutron-rich) $Y_e$ from the infall/compression phase. []{data-label="fig-entYe"}](./f5.eps){width="\hsize"}
The thermodynamic conditions (density, temperature, and entropy), which are responsible for the nucleosynthesis results, are shown in Figure [\[fig-postprocess\]]{} for the ejected mass zones. These figures indicate a temperature, density, and entropy maximum when the quark–hadron phase transition occurs, which causes a second core bounce (about $0.4$ s after the first bounce at nuclear densities; see Figure \[fig-properties\]) and an outgoing shock front forms. The expansion follows a close to constant entropy, i.e. is adiabatic, once matter is ejected (after $0.4$ s for the prompt ejection and after $1.5$ s for the delayed ejection). The matter which initially fell back onto the proto-neutron star and is finally ejected by the NDW, experiences heating and an entropy rise due to this energy deposition by neutrinos.
Neutrinos from the Proto-neutron Star
-------------------------------------
The properties of the neutrino and anti-neutrino flux (luminosities, average energies and neutrino sphere radii) can be found in Figure
\[fig-neutrinos\]. It is clearly seen that the initial bounce ($0$ s) at nuclear densities leads to a neutrino burst due to electron captures, while the second shock wave caused by the quark-hadron phase transition (at $0.4$ s) also produces antineutrinos. From that point on in time the neutrino and anti-neutrino luminosities are comparable (slightly smaller for anti-neutrinos). The average energies are larger for anti-neutrinos than for neutrinos, but the difference remains less than 4 MeV. Neutrino and anti-neutrino captures determine the neutron/proton ratio due to the reactions $$\bar\nu_e + p \rightarrow n + e^+ \ \ \ \ \ \nu_e + n \rightarrow p + e^-.$$
![ Neutron/seed ratio and remaining mass fractions of $^4$He ($\alpha$-particles) after charged-particle freeze-out. Both properties result from the original $Y_e$ and entropy $S$ in these mass zones. In turn they determine the fraction of heavy elements and whether those experience further neutron capture after charged-particle freeze-out, which is the key to the pattern of heavy nuclei and the maximum mass number attained. In the inner part (green lines) an effect is seen, which results from an intermediate fallback before final ejection. These mass zones experience first (due to the shock from the deconfinement phase transition) a maximum temperature and density, expand afterwards close to adiabatically, heat up during the intermediate fallback, and then expand freely. The line indicated with ”no-boost” gives the initial neutron/seed ratio after the first expansion. The alpha-fraction in the inner zones results from the reheating phase. The initial expansion at low $Y_e$’s would result in vanishing alpha-fractions.[]{data-label="fig-Xalnseed"}](./f6.eps){width="\hsize"}
Based on the neutron/proton mass difference of $\Delta = 1.293$ MeV, [@Froehlich:etal:2006a] could show (see their Equation (4)), that with the use of Equations (64a) and (64b) in @QianWoosley:1996, $\dot Y_e > 0$ in the case that the difference between the mean antineutrino and neutrino energies fulfills $\epsilon_{\bar{\nu}} - \epsilon_\nu < 4\;\Delta$, with $\epsilon = \langle E^2\rangle/\langle E \rangle$ and where $\langle E \rangle$ is the mean energy and $\langle E^2 \rangle$ is the square value of the root-mean-square (rms) energy. Therefore $Y_e > 0.5$ is obtained if the timescale for neutrino/anti-neutrino captures is shorter than the dynamic timescale. Thus, for all conditions discussed here, where neutrino and anti-neutrino captures are responsible for the $n$/$p$ ratio, proton-rich conditions are attained, i.e., $Y_e > 0.5$. That is exactly what is seen in Figures \[fig-properties\] and \[fig-neutrinos\] for mass zones which experience essential neutrino fluxes (weighted by $1/r^2$) at radii of about 100 km. It is related to the spectral evolution of $\nu_e$ and $\bar\nu_e$ after the onset of explosion and has been discussed in detail in @Fischer:2012. For matter at larger radii (about 1000 km), the timescale for this process is too long and minor $Y_e$ changes occur, i.e., the initial $Y_e$ from the collapse phase is retained. Matter at smaller radii, on top of the neutron star, experiences high densities and electron Fermi energies, where electron captures dominate which make matter neutron-rich.
Figure \[fig-entYe\] underlines this effect due to neutrino interactions or electron capture. All outer mass zones keep their original $Y_e$, which is due to electron capture at high densities during the collapse, and ranges from about $0.48$ (further out) to $0.32$ (for the inner quasi-prompt ejected matter). Material which fell in initially onto the surface of the neutron star and is then ejected via the NDW, has been turned proton-rich by neutrino and anti-neutrino captures with values up to $Y_e=0.55$. The NDW also leads to energy deposition and an entropy increase to maximum values of about $85$ $k_B$ per baryon. The entropy in the outer ejected regions is of the order $30$–$50$ $k_B$ per baryon, caused by shock heating during the passage of the ejection shock wave.
Nucleosynthesis Results
-----------------------
In the following we show final nucleosynthesis results for a number of typical mass zones. In order to get a rough idea about the results of explosive nucleosynthesis, one can utilize either maximum densities and temperatures prior to an adiabatic expansion or the entropies attained in the expanding matter (in radiation-dominated regimes $S\propto T^3/\rho$).
{width="0.7\hsize"}
Comparing entries in Figure 5 of [@Thielemann:etal:1990] and Fig.3 of [@Thielemann:etal:1996] leads to the conclusion that (1) these are typical conditions for an alpha-rich freeze-out from explosive Si-burning and (2) one would expect remaining alpha mass-fractions after charged-particle freeze–out of the order 20% – 100%. One should consider, however, that those calculations were performed for hydrodynamic (i.e., free fall) expansion timescales, which can differ from the actual simulation, and a value of $Y_e=0.4988$, i.e. matter neither neutron nor proton-rich. Therefore we expect the following changes: (1) higher/lower entropies within the given variety will lead to higher/lower remaining alpha-fractions; (2) higher $\dot{Y}_e$, i.e., more proton-rich matter causes an alpha-rich charged-particle freeze-out with remaining free protons, which can thereafter lead to a $\nu p$-process, if a sufficient flux of electron anti-neutrinos is still present; (3) smaller $Y_e$’s, i.e., more neutron-rich matter permits to bypass the slower triple-alpha reaction via the faster $\alpha\alpha n$-reaction, in order to produce heavier nuclei and a reduction in the remaining alpha-fraction is expected. In addition, free neutrons are remaining after the charged-particle freeze-out. With respect to this latter aspect, we expect also the additional behavior: higher entropies and lower $Y_e$’s lead to a larger neutron/seed ratio, seed nuclei being the heaviest nuclei formed after charged-particle freeze-out, and permit therefore more neutron captures on these seed-nuclei. Dependent on the neutron to seed ratio, this could lead to light, medium or strong $r$-processing, producing nuclei in the first, second or third $r$-process peaks, around A=80, 130 or 195, depending on the actual n/seed ratio attained.
![**Top:** The comparison of results with (solid) and without (dashed) the effect of reheating and second expansion (boost) integrated over the whole range of mass zones 20-50. **Bottom:** The ratios of the two cases, based on the full hydrodynamic evolution with reheating (boost) and the neglection of reheating around $1.5$ s.The reheating leads to photodisintegrations, a strong appearance of $^{4}\rm{He}$ and a reshaping of the heavy element distribution.[]{data-label="fig-boost"}](./f8.eps){width="\hsize"}
![ Resulting nucleosynthesis for the entire mass zones in the neutrino-driven wind, experiencing a $\nu$p-process. The result is shown for two options: (a) including the anti-neutrino captures which turn protons into neutrons in the late phase of nucleosynthesis and permit to overcome the $^{64}$Ge beta-decay bottle neck via an (n,p)-reaction (filled red circles), (b) neglecting this effect (open black circles). It can be seen that the inclusion of anti-neutrino reactions enhances abundances of nuclei heavier than $A = 64$ and permits the production of nuclei up to $A = 80 - 90$.[]{data-label="fig-nup"}](./f9.eps){width="\hsize"}
In Figure \[fig-Xalnseed\] the properties of mass zones 020 to 120 are summarized, those with a $Y_e < 0.5$ which experience neutron-rich conditions to a varying degree. Properties (1), i.e., the degree of alpha-rich freeze-out and (3), the resulting neutron/seed ratio, are displayed. We see a complex dependence of these properties on entropy $S$, $Y_e$, expansion timescale $\tau$, plus further complications from reheating and re-expansion which do not fit to a simple expansion interpretation. For similar $Y_e$’s, the remaining alpha-fraction increases with entropy, as expected, when looking at the behavior of mass zones 080 to 120. Then, following mass zones further in, the decrease in $Y_e$ dominates, which permits to pass the alpha-to-carbon bottle-neck more efficiently, via the $\alpha\alpha n$-reaction, and the remaining alpha-fraction vanishes. As known from moderately neutron-rich NDW simulations, none of the entropies encountered here leads to sizable neutron/seed ratios. For such low entropies, only a strongly neutron-rich initial composition permits large(r) neutron/seed ratios. This is what can be noticed when following mass zones from 070 down to 040, where the lowest $Y_e$’s are encountered. Mass zones 020 to 040 experience a more complicated history, initial expansion after shock passage, later partial fallback, and then final ejection and further expansion. Here, the neutron/seed ratio after the first charged-particle freezeout is the one of importance for the production of heavy elements.
![ Overview of dominating conditions, i.e. abundances of a few key nuclei after charged-particle freeze-out, as a function of radial Lagrangian mass, with variations from (a) typical explosive Si-burning products and an alpha-rich freeze-out in the outer zones to (b) an increasing remaining neutron abundance after charged-particle freeze-out, permitting a weak $r$-process, down to (c) proton-rich neutrino-driven wind ejecta, permitting the onset of an $\nu$p-process.[]{data-label="fig-overview"}](./f10.eps){width="\hsize"}
The reheating leads to partial photodisintegration of heavy elements, the production of alphas, and the buildup toward nuclei with mass numbers around $A=90$–$110$ during the final expansion. The neutron/seed ratio at this second charged-particle freezeout (boost) is rather a measure for local rearrangements of matter.
The maximum neutron/seed ratio of slightly above 10 obtained over the complete range of ejected mass zones, does not support conditions to produce the third $r$-process peak, in fact only a small production of the second peak is expected. The effect of the “boost", i.e. second reheating and expansion in mass zones 020–040, rearranges/reshapes the abundance distribution via photodisintegrations and captures, but does not alter this conclusion.
With this background we have a look at the final abundances of representative mass zones, displayed in Figure \[fig-finab\]. We see in the outer layers remaining alpha-fractions beyond 60%. These are regions, which experience entropies of $S = 33$–$50 k_b$/baryon and $Y_e$’s close to 0.5 and see vanishing neutron/seed ratios. Thus, we expect essentially the production of the Fe-group up to $A=50$–$70$. This can be observed in the last two subfigures of Figure \[fig-finab\] (zones 080 and 120). Mass zones 60 and 70, which experience the highest entropies and moderately decreased $Y_e$’s, can move matter up to and (slightly) beyond $A=90$ (for mass zone 70 still with a large fraction of matter in the Fe-group), mostly due to a more neutron-rich (lower $Y_e$) charged-particle freeze-out and not due to further neutron processing (see neutron/seed ratio in Figure \[fig-Xalnseed\]). One can also see a variation in the final carbon-fraction, underlining how effective the bridging of the alpha-to-carbon bottle neck of reactions is in comparison to subsequent capture reactions to heavier nuclei. At smaller radii (mass zones 020 to 051 and $Y_e$’s as small as 0.33), also the $A = 130$ peak starts to be populated. This is due to the neutron/seed ratio of up to 10 attained after charged-particle freezeout. For the mass zones 20–45, abundances are shown after the first expansion (dashed) and after reheating/boost (solid), which is characterized by some photo-disintegration of heavy nuclei, the appearance of $^4$He, and further processing of nuclei beyond the Fe-group (see also Figure \[fig-boost\]). Summarizing the results of all mass zones with $Y_e < 0.5$, we note that none of these zones produce matter beyond $A \sim 130$, nor show the 130 peak dominating abundances.
Deeper mass zones, ejected via the NDW turns proton-rich and they experience entropies as high as $85 k_B$ per baryon. These are conditions where we expect a $\nu p$-process [@Froehlich:etal:2006b; @Pruet:etal:2006; @Wanajo:2006]. This can be seen in the first three panels of Figure \[fig-finab\], dashed lines show abundances without the inclusion of the $\nu p$-process, solid lines show the final results after $\nu p$-processing. While in terms of total mass fraction the production of nuclei beyond the Fe-group is not too impressive, Figure [\[fig-nup\]]{} displays this more prominently, where the overproduction ratio over solar is plotted. In the proton-rich environment anti-neutrino captures on free protons produce neutrons and permit to overcome the $rp$-process waiting point $^{64}$Ge via an $(n,p)$-reaction, winning against a slower $\beta^+$-decay. In this way, nuclei up to $A = 80 - 90$ can be produced on the proton-rich side of stability. The comparison of the black and red bullets indicates the strength of this process.
![Integrated abundance distributions as a function of atomic mass number $A$ for all ejecta in comparison to solar r-abundances [*(top)*]{} (normalized to the A=100 region) and separated by ejection process[*(bottom)*]{}. We also indicate the effect of uncertainties in the $Y_e$ determination, given in terms of percentage $P_{cor}$, discussed in section 5.[]{data-label="fig-integ"}](./f11.eps){width="\hsize"}
Survey and integrated composition
=================================
After having discussed the individual composition, ejected from different positions in the exploding model, we want to give a final survey of the conditions attained in all mass zones of explosive Si-burning which are affected by the explosion mechanism (here the deconfinement phase transition). The discussion of nucleosynthesis in layers further out is only affected by the energy in the shock wave, as has been discussed extensively in the literature [@Thielemann:etal:1996], and will not be repeated here. Finally, we also discuss the overall features of the integrated yields (of these inner mass zones, close to the explosion mechanism). The survey is displayed in Figure [\[fig-overview\]]{} which features abundances after charged-particle freeze-out, and thus the setting for the final nucleosynthesis features, if free neutrons or protons are remaining in sizable fractions.
![Density evolution as a function of time. Shown is the standard case with the original explosion simulation according to EOS2 and exponential density expansions fitted to this case ($f=1$) as well as slower expansions with longer expansion timescales by a factor 2, 5, and 10.[]{data-label="fig-dens"}](./f12.eps){width="\hsize"}
Essentially, all mass zones have total entropies in excess of $S=30 k_B$ per baryon. In the outer mass zones with a $Y_e$ close to 0.5 this produces high alpha-fractions plus dominantly $^{56}$Ni. Moving somewhat further in with slightly increasing entropies, noticeable amounts of $^{64}$Ge are produced as well, with the $^{64}$Ge/$^{56}$Ni being a measure of entropy. Decreasing $Y_e$, i.e., having more neutron-rich conditions, changes the initially pure alpha-rich charged-particle freezeout to an alpha-rich freeze-out with sizable abundances up to $A=90$. A further decrease in $Y_e$ causes remaining free neutrons, which permit an additional sequence of neutron captures. The decrease of $Y_e$ down to $0.33$, leads to a charged-particle freeze-out with vanishing alpha-fractions but a sufficient amount of free neutrons, which permit later to produce nucleosynthesis ejecta in the mass A=130 peak. The mass zones which are affected by a reheating boost are characterized by photo-disintegrations and a second charged-particle freeze-out with remaining alpha-fractions, Finally the innermost proton-rich zones, with higher entropies of about $S=85 k_B$ per baryon and experiencing a continuous neutrino-flux, show an alpha-rich and proton-rich freeze-out with $Y_e$’s up to $0.55$. This causes a $\nu p$-process after charged-particle freeze-out , permitted by neutron production via anti-neutrino captures on free protons. This process will produce nuclei up to $A=90$, but on the proton-rich side of stability, especially Sr, Y, and Zr isotopes.
After this survey of the conditions at charged-particle freeze-out, combined with the final ejecta composition as a function of radial mass coordinate as discussed in the previous section, we want to give an integrated presentation for these inner mass zones which experience conditions for the possible formation of nuclei beyond the Fe-group. Figure \[fig-integ\] (top) shows the composition for this range in mass numbers in comparison to solar $r$-abundances. As we do not yet know the frequency of such events, i.e. which range of the initial mass function of stellar masses leads to these types of explosions, we show a scaling normalized to the $A \sim 100$ mass region (where the dominant abundances are obtained). The bottom part of Figure \[fig-integ\] shows the individual contributions to the overall abundances by the different mass zones as presented in Table \[tab-properties\] and Figure \[fig-init\], and shown in different colors in Figure \[fig-properties\] and \[fig-postprocess\] as well as \[fig-entYe\], \[fig-Xalnseed\], and \[fig-finab\]. The high end of the abundance distribution in the prompt (blue, dashed) as well as the delayed (boosted) ejecta (green, solid) are due to the most neutron-rich conditions with $Y_e$ close to $0.33$. The outermost ejected mass zones with $Y_e$ closer to $0.5$ contribute to the Fe-group and matter up to $A = 80$ (lower range of mass numbers of blue dashed line). The NDW (red line) with slightly proton-rich ejecta ( $Y_e = 0.55$) produces significant abundances only up to $A = 64$, if the same normalization for the relevant mass zones is used (see, however, also Figure \[fig-nup\]).
![The corresponding temperature evolutions with adiabatic index 4/3.[]{data-label="fig-temp"}](./f13.eps){width="\hsize"}
It is clearly visible that, first of all, objects like these supernovae, exploding by a mechanism based on EOSs with a low-density quark–hadron phase transition, do only experience a weak $r$-process in ejected mass zones which were neutronized during collapse. There is no matter produced in the third $r$-process peak. If normalizing the abundance curve at $A=100$, in order to avoid an overproduction of this mass region, also only a small contribution (less than 10% is expected to the second $r$-process peak, i.e., $A=130$). The major production affects the atomic mass range from $A=80$ to $115$, curiously also reproducing a minimum at $A=97$–$99$. Thus, the type of events discussed here, contribute to the whole mass region beyond the Fe-group up to $A=115$ in a significant way, accompanied by minor contributions to the second $r$-process peak at $A = 130$.
Model Uncertainties
===================
Equation of State Uncertainties
-------------------------------
The major uncertainty in the present calculations is expected from the uncertain properties of the EOS, especially the quark–hadron phase transition which is at the origin of the explosion mechanism discussed here. This was already reviewed extensively in Section 2.1. The analysis done by @Fischer:etal:2011 with variations in the properties of the phase transition, led to different post-bounce times for the onset of the explosion and different neutron star structures. Later transitions at higher densities cause more massive progenitors with steeper density gradients at the surface. The standard case of the present paper corresponds to EOS2 in Figure 17 of @Fischer:etal:2011, i.e., a late(r) explosion with steep(er) density gradient and thus a fast expansion. Variations toward the properties of EOS1, with an earlier explosion, would cause slower expansions. Therefore, we performed test (nucleosynthesis) calculations with slower expansions for a specific mass zone (here zone 51, see also Figures \[fig-finab\]) which provided among the best $r$-process conditions in our standard case. Figs.[\[fig-dens\]]{} and [\[fig-temp\]]{} show variations with exponential density expansions fitted to the standard case ($f=1$) and slower expansions with longer expansion times scales by a factor of two, five, and 10 (choosing the temperature according to adiabatic expansions with the same entropy).
![ Final abundances for mass zone \# 51 with different expansion speeds. The slower expansions ($f=2, 10$) lead to smaller remaining alpha fractions (as expected), but also to a slightly stronger (weak) $r$-process, indicated by the abundances in the $A=130$ peak.[]{data-label="fig-fabund"}](./f14.eps){width="\hsize"}
In typical NDW environments for given $S$ and $Y_e$, a fast(er) expansion leads to a strong(er) alpha-rich freeze-out, thus more alpha particles and less seeds and a higher neutron/seed ratio, promoting the options for a stronger $r$-process. However, in the mass zones of interest here, the $Y_e$ is quite low and the remaining alpha-fraction close to negligible. This feature was discussed extensively in Section 3.3 with respect to Figures \[fig-Xalnseed\] and \[fig-finab\]. We see the effect in Figure \[fig-fabund\], that the fastest expansion ($f=1$) shows the highest remaining alpha-fraction, but only of the order $10^{-4}$. This behavior lowers the initial seed abundance after charged-particle freeze-out by the same amount, which is negligible. This causes essentially the same seed abundances and neutron/seed ratio for all calculations with different expansion speeds ($f=1$ – $10$) and cancels this dominant effect in typical neutrino-driven wind environments. In fact, the slowest expansion leads to the highest abundances at $A=130$. Here, a second-order effect seems to take over, the different density and temperature conditions for slower expansions lead to an apparently slightly more neutron-rich $r$-process path, encountering smaller beta-decay half-lives which permit to proceed faster to heavier nuclei. This behavior, as shown in Figure \[fig-fabund\], indicates a slightly stronger (weak) $r$-process for the slower expansions. However, the effect is not strong, only clearly visible for $f=10$, which is larger than the uncertainty encountered in our EOS variations.
$Y_e$ Uncertainties
-------------------
One can argue, that there might exist some uncertainty in weak interactions which determine $Y_e$, especially due to recent developments of including charged-current rates that are consistent with the EOS [for details, see @MartinezPinedo:2012; @Roberts:2012]. These very recent preliminary investigations include medium effects for the (electron) neutrino and antineutrino capture reactions, which lead to changed reaction $Q$-values by adding nucleon interaction potential differences for neutrons and protons, enhancing as a result the difference between average anti-neutrino and neutrino energies. This causes a change in $Y_e$. All our earlier discussions had the main emphasis that the influence of the neutrino flux from the proto-neutron star would cause an increase in $Y_e$. The preliminary analysis of such effects by @MartinezPinedo:2012 and @Roberts:2012, which were not yet included in the present calculations, shows that the $Y_e$ increase is weakened or even moderately neutron-rich conditions ($Y_e \lesssim 0.5$) can result.
For this reason, we also repeated the present nucleosynthesis calculations with variations in the initial $Y_e$ for the mass zones experiencing prompt and delayed explosions, according to the following recipe: $$Y_{e,\; \rm{cor}} = 0.5 + (Y_e - 0.5) \times \left(1 + \frac{p_{\rm{cor}}}{100}\right) \ ,$$ where the ${Y_{e,\; \rm{cor}}}$’s are the corrected ones and $p_{\rm{cor}}$ denotes the percentage of uncertainty in deviations of $Y_e$ from the symmetric value $0.5$, which enlarges these deviation from $0.5$. This lowers the initially only moderately neutron-rich $Y_e$’s in regions which already produced $A=130$ nuclei. The nucleosynthesis results are presented in Figures \[fig-integ\] and \[fig-honda\] and show the options of obtaining a full or weak $r$-process. However, we expect that any uncertainties beyond $20$% are unrealistic for the explosion model we adopt in the current work.
As is obvious from the discussion above, that only a “weak" $r$-process can be supported by the nucleosynthesis conditions found in the explosion mechanism discussed and presented here, one might wonder whether such conditions support abundance features found in “weak $r$-process" low-metallicity stars as observed by @Honda:etal:2006. For this reason, we also show such a comparison in Figure \[fig-honda\]. What can be seen is that these observations also show sizable $r$-process features above the $A=130$ peak, although weaker than in solar $r$-element abundances. If such abundance distributions are the result of a single nucleosynthesis pollution, also the “weak" $r$-process found in the present paper can only marginally explain such features. One could argue, however, that such observed abundance features are a combination of at least two pollutions, one (low level) solar $r$-contribution plus another “weak" $r$-contribution extending only up to $A=130$.
Multi-Dimensional Effects
-------------------------
Nucleosynthesis uncertainties due to multi-dimensional effects, i.e., convective turnover, have not been considered here, but we want to discuss their possible impact. Note that convection and the presence of unstable fluid motion are typically associated with compositional mixing, also of the ejected material. Their development, e.g., after the shock passage across a low-density environment, can leave peculiar features such as discussed in [@WanajoJanka:2012] at the example of a $8.8 M_\odot$ star explosion. In order to give an estimate for our nucleosynthesis results obtained in the presence of possible convection, we evaluated the Ledoux criterion [@WilsonMayle:1988]. It is related to entropy per baryon and lepton-number gradients. We find that regions where these gradients become largely negative, i.e. regions where convection may develop, correspond to either fall-back of material or the shock propagation in $Y_e\simeq0.5$ material. Thus, matter with such uncertainties is either not ejected (fallback) or mixing occurs in regions with very similar $Y_e$. Hence, we expect the impact of possible convection on the here discussed nucleosynthesis scenario to be of minor importance, in particular the ejected material which attains low-$Y_e$ is not affected. Independent of this expected unimportance of convection for nucleosynthesis results, we should keep in mind that matter is strongly accelerated and thus ejected very fast. Even if instabilities exist, they should not lead to significant mixing on such small timescales after the onset of the explosion and affect the nucleosynthesis.
![Integrated abundance distributions as a function of atomic charge number $Z$ in comparison to abundance features in “weak $r$-process” low metallicity stars as observed by @Honda:etal:2006. We also indicate the effect of uncertainties in the $Y_e$ determination, given in terms of percentage $P_{\rm{cor}}$[]{data-label="fig-honda"}](./f15.eps){width="\hsize"}
Nevertheless, this nucleosynthesis site inhabits another possibility of convection due to the explosion mechanism related to the quark–hadron phase transition. It takes place on timescales on the order of several 100 ms, depending on details of the quark–hadron EOS, in the absence of any other, e.g., neutrino-driven explosion. On these timescales, large-scale convection may develop before the transition/explosion is initiated. Exploring its impact on the entire explosion and nucleosynthesis scenario will require multi-dimensional simulations, especially related to the point in time when this explosion starts (possibly affected by convection and related neutrino losses and/or uncertainties in the EOS, which both determine the $Y_e$ at the onset of the explosion). This is beyond the scope of the present article and will require future investigations.
Discussion and Summary
======================
Supernova nucleosynthesis is well understood for the outer ejected mass zones, which can be well approximated by a shock wave with appropriate energy passing through the layers of the progenitor [@WoosleyWeaver:1995; @Thielemann:etal:1996; @Woosley:etal:2002; @Nomoto:etal:2006; @WoosleyHeger:2007; @HegerWoosley:2010; @Thielemann:etal:2011]. What remains uncertain is the composition of the innermost ejecta, directly linked to the explosion mechanism, i.e., the collapse and explosion phase. In the present paper, we analyzed these mass zones of core-collapse supernova explosions triggered by a quark–hadron phase transition during the early post-bounce phase [@Sagert:etal:2009; @Fischer:etal:2011]. A number of aspects are important for understanding these results. The very innermost ejecta are strongly affected by the NDW. Recent investigations noticed that this NDW turns matter proton-rich, producing specific Fe-group isotopes and in the subsequent $\nu p$-process nuclei with masses up to $A=80-90$ [@Liebendoerfer:etal:2003; @Pruet:etal:2005; @Froehlich:etal:2006a; @Froehlich:etal:2006b; @Pruet:etal:2006; @Wanajo:2006]. Even in the long-term evolution proton-rich conditions prevail [@Fischer:etal:2010; @Huedepohl:etal:2010]. Thus, there seems to exist no chance to produce $r$-process matter in these innermost regions, despite many interesting parameter studies for NDW ejecta in terms of entropy $S$, electron fraction $Y_e$, and expansion timescale $\tau$ [@Hoffman:etal:1997; @MeyerBrown:1997; @Freiburghaus:etal:1999; @Farouqi:etal:2010] or hydrodynamic studies, partially with parameter variations [@Arcones:etal:2007; @Kuroda:etal:2008; @PanovJanka:2009; @Roberts:etal:2010; @ArconesMontes:2011]. Very recently, @MartinezPinedo:2012 and @Roberts:2012, explored charged-current weak processes (i.e., electron-neutrino and anti-neutrino captures on neutrons and protons), consistent with the EOSs in neutrino-driven supernova explosion. They find modifications of the reaction $Q$-values due to medium effects, which increase spectral differences between $\nu_e$ and $\bar\nu_e$. This results in slightly neutron-rich ejecta. Nevertheless, since the entropy per baryon is still low a strong $r$-process is unlikely to occur.
In order to obtain $r$-process conditions, a better chance to eject neutron-rich matter is provided, when neutron-rich matter stems from the initial collapse and compression, where electron captures made it neutron-rich, early in an explosion, before neutrino interactions have the chance to turn it proton-rich in the NDW. Core-collapse supernovae, exploding via the quark–hadron phase transition (the focus of the present study), or electron-capture supernovae [@Wanajo:etal:2011], which explode without a long phase of accretion onto the proto-neutron star, both lead to a rather prompt ejection of prior compressed and neutronized matter. However, the $Y_e$ obtained under such conditions does not support a full $r$-process. This kind of outcome also characterizes the conditions we find in the prompt and quasi-prompt ejecta of the present study, which did not experience a strong neutrino flux. However, the $Y_e$-values attained are not smaller than $0.33$. Whether uncertainties in the input or explosion physics can change this down to values close to $0.23$, necessary for obtaining the third $r$-process peak, remains to be shown in an upcoming article where we will include charged-current reaction rates that are consistent with the EOS following @MartinezPinedo:2012 and @Roberts:2012.
A related, but different, phenomenon has been discussed in @Jaikumar:2007 [, and references therein]. In their scenario, the quark–hadron phase transition does not happen shortly after core collapse and is not the cause of the supernova explosion. Instead a regular neutrino-driven supernova explosion occurs first and leads to a deleptonized neutron star with low-$Y_e$ and a steep density gradient at the surface. Their scenario of a “quark-novae", occurring via a quark–hadron phase transition in an existing neutron star, is reported to produce $r$-process elements via the prompt ejection of extremely neutron-rich material from the neutron star surface. In this sense, quark-novae differ from the scenario explored in the current study. They result in a more compact object already in $\beta$-equilibrium, such that the expansion timescale of the ejected material is much faster, compared to the scenario discussed in the present paper. However, in @Jaikumar:2007 the authors apply a very simplified description for the quark–hadron phase transition and ignore neutrino transport in general, as well as weak processes in the hadronic part of their neutron star. The latter may be essential for producing neutrinos which may change $Y_e$ during shock passage and hence lead to different nucleosynthesis results than what has been discussed in @Jaikumar:2007. It remains to be shown if the obtained favorable conditions for the $r$-process will remain, applying more sophisticated input physics.
Given our results described above, we conclude that supernova explosions triggered via the quark–hadron phase transition during the early post-bounce phase can contribute to a weak $r$-process, consistent with observations of low-metallicity stars [@Honda:etal:2006] and with LEPP abundances [@Travaglio:etal:2004]. Such conditions apparently do not occur in regular core-collapse supernovae. However, strong $r$-process conditions, which also produce the third $r$-process peak and the actinides, have in simulations only materialized in neutron star mergers [@Freiburghaus:etal:1999b; @Goriely:etal:2011], fast rotating core-collapse supernovae with strong magnetic fields and jet ejecta [@Cameron:2003; @Nishimura:etal:2006; @Fujimoto:etal:2008] or accretion disks around black holes [@Surman:etal:2008; @WanajoJanka:2012]. For the explosion model discussed here, investigation of uncertainties in the hydrodynamics and the nuclear physics input showed that a strong $r$-process to produce up to the third peak elements is rarely possible to obtain.
Neutron star mergers have been shown to be powerful sources of $r$-process matter, in fact ejecting a factor of $100$ – $1000$ more $r$-process material than required on average from core-collapse supernovae, if those would have to explain solar $r$-process abundances. This would actually support the large scatter of Eu/Fe in comparison to, e.g., O, Mg, Si, S, and Ca/Fe, where the latter are clearly produced in supernovae. The only problem is that it might be hard to explain the early appearance of $r$-process matter for metallicities at and below $\rm{[Fe/H]}=-3$. Neutron star mergers will appear after the first supernovae have already produced Fe and studies by @Argast:etal:2004 showed that one expects $r$-process products only for metallicities $\rm{[Fe/H]} = -3 \sim -2$. Some recent studies, which include the fact that our Galaxy is possibly the result from smaller merging subsystems (with different star formation rates) have been expected to show a way out of this dilemma. If this cannot solved, we need another strong $r$-process source already at low metallicities, and possibly jets from rotating core collapses with strong magnetic fields could be the solution [see, e.g, @Nishimura:etal:2006; @Fujimoto:etal:2007; @Winteler:2012].
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to the anonymous referee for the valuable comments and suggestions to improve our manuscript. N.N. acknowledges M. Hashimoto and S. Fujimoto for early development of the nuclear reaction network which is extended and used in this work and also thanks for A. Arcones to critical and useful comments. The project was funded by the Swiss National Science Foundation grant Nos. PP00P2-124879/1 and 200020-122287, the Helmholtz Research School for Quark Matter Studies, and the Helmholtz International Center (HIC) for FAIR. N.N. is supported by the National Astronomical Observatory of Japan under “FY2012 Visiting Fellowship Program". T.F is supported by the Swiss National Science Foundation under project No. PBBSP2-133378 and G.M.P is partly supported by the Sonderforschungsbereich 634, the ExtreMe Matter Institute EMMI and HIC for FAIR. C.F. acknowledges support from the DOE Topical Collaboration “Neutrinos and Nucleosynthesis in Hot and Dense Matter" under contract DE-FG02-10ER41677. M.L. and M.H. acknowledge support from the High Performance and High Productivity Computing (HP2C) project. M.H. is supported by the Swiss National Science Foundation (SNF) under project No. 200020-132816/1 and is also grateful for participating in the ENSAR/THEXO project. T.R. is supported by the European Commission under the FP7 ENSAR/THEXO project. The authors are additionally supported by CompStar, a research networking program of the European Science Foundation and EuroGENESIS, a collaborative research program of the ESF. F.-K.T. is an Alexander von Humboldt awardee.
[99]{}
Alford, M., Braby, M., Paris, M. & Reddy, R. 2005, , 629, 969
Argast, D., Samland, M., Thielemann, F.-K. & Qian, Y.-Z. 2004, , 416, 997
Audi, G. & Wapstra, A. H. 1995, , 595, 409
Arcones, A., Janka, H.-Th., & Scheck, L. 2007, , 467, 1227
Arcones, A., & Montes, F. 2011, , 731, 5
Cameron, A. G. W. 2003, , 587, 327
Cole, A. L., Anderson, T. S., Zegers, R. G. T., et al. 2012, , 86, 015809
Dasgupta, B., Fischer, T., Horiuchi, S., et al. 2010, , 81, 103005
Demorest, P. B., Pennucci, T., Ransom, S. M., et al. 2010, , 467, 1081
Farouqi, K., Kratz, K.-L., Pfeiffer, et al. 2010, , 712, 1359
Fischer, T., Whitehouse, S. C., Mezzacappa, A., et al. 2010, , 517, A80
Fischer, T., Sagert, I., Pagliara, G., et al. 2011, , 194, 28
Fischer, T., Mart[í]{}nez-Pinedo, G., Hempel, M., et al. 2012, , 85, 083003
Freiburghaus, C., Rembges, J.-F., Rauscher, T., et al. 1999a, , 516, 381
Freiburghaus, C., Rosswog, S., & Thielemann, F.-K. 1999b, , 525, L121
Freire, P. C. C., Bassa, C. G., [Wex]{}, N., et al. 2011, , 412, 2763
Fröhlich, C., Hauser, P., Liebendörfer, M., et al. 2006a, , 637, 415
Fröhlich, C., Martínez-Pinedo, G., Liebendörfer, M., et al. 2006b, , 96, 142502
Fujimoto, S.-i., Hashimoto, M.-a., Kotake, K., & Yamada, S. 2007, , 656, 382
Fujimoto, S.-i., Nishimura, N. & Hashimoto, M.-a. 2008, , 680, 1350
Goriely, S., Bauswein, A., & Janka, H.-T. 2011, , 738, L32
Heger, A.., & [Woosley]{}, S.E. 2010, , 724, 341
Hoffman, R. D., Woosley, S.E., & Qian, Y.-Z. 1997, , 482, 951
Honda, S., Aoki, W., Ishimaru, Y., & Wanajo, S. 2006, , 643, 1180
H[ü]{}depohl, L., M[ü]{}ller, B., Janka, H.-T., et al. 2010, , 104, 251101
Janka, H.-T., Langanke, K., Marek, A., et al. 2007, , 442, 38
Jaikumar, P., Meyer, B. S., Otsuki, K., Ouyed, R. 2007, , 471, 227
Kodama, T. & Takahashi, K. 1975, , 239, 489
Kuroda, T., Wanajo, S., & Nomoto, K. 2008, , 672, 1068
Langanke, K., & Mart[í]{}nez-Pinedo, G. 2003, Reviews of Modern Physics, 75, 819
Lattimer, J. M. & Swesty, D. F. 1991, , 535, 331
Liebendörfer, M., Mezzacappa, A. & Thielemann, F.-K., 2001a, , 63, 104003
Liebendörfer, M., Rosswog, S., Thielemann, F.-K., et al. 2001b, , 63, 103004
Liebendörfer, M., Mezzacappa, A., Messer, O. E. B., et al. 2003, , 719, 144
Liebendörfer, M., Messer, O. E. B., Mezzacappa, A., et al. 2004, , 150, 263
Martínez-Pinedo, G., Fischer, T., Lohs, A., Huther, L. 2012, arXiv:1205.2793
Meyer, B.S., & Brown, J. 1997, , 112, 199
Mezzacappa, A., & Bruenn, S. W. 1993a, , 405, 637
Mezzacappa, A., & Bruenn, S. W. 1993b, , 405, 669
Mezzacappa, A., & Bruenn, S. W. 1993c, , 410, 740
Möller, P., Nix, J. R., Myers, W. D., & Swiatecki, W. J. 1995, At. Data Nucl. Data Tables, 59, 185
Nishimura, S., Kotake, K., Hashimoto, M., et al. 2006, , 642, 410
, K., Tominaga, N., Umeda, H., et al. 2006, , 777, 424
Panov, I. V., & Janka, H.-T. 2009, , 494, 829
Pruet, J., Woosley, S. E., Buras, R., et al. 2005, , 623, 325
Pruet, J., Hoffman, R. D., Woosley, S. E., et al. 2006, , 644, 1028
Qian, Y.-Z. & Woosley, S. E. 1996, , 471, 331
Roberts, L. F., Woosley, S. E., & Hoffman, R. D., 2010, , 722, 954
Roberts, L. F. & Reddy, S., 2012, arXiv:1205.4066
, I., [Fischer]{}, T., [Hempel]{}, M., et al. 2009, , 102, 081101
Sagert, I., Fischer, T., Hempel, M., et al. 2010, J. Phys. G: Nucl. Part. Phys., 37, 094064
Sagert, I., Tolos, L., Chatterjee, D., et al. 2010, arXiv:1111.6058
Schertler, K., Greiner, C., Schaffner-Bielich, J. & Thoma, M. H. 2000, , 677, 463
Shen, H., Toki, H., Oyamatsu, K. & Sumiyoshi, K. 1998, , 637, 435
Staudt, A. & Klapdor-Kleingrothaus, H. 1992, , 549, 254
Surman, R., McLaughlin, G. C., Ruffert, M., et al. 2008, , 679, L117
Takahashi, K., Witti, J., & Janka, H.-T. 1994, , 286, 857
Thielemann, F.-K., Hashimoto, M., & Nomoto, K. 1990, , 349, 222
Thielemann, F.-K., Nomoto, K., & Hashimoto, M. 1996, , 460, 408
Thielemann, F.-K., Hirschi, R., Liebendörfer, M., & Diehl, R. 2011, in Lecture Notes in Physics, Berlin Springer Verlag, Vol. 812, Lecture Notes in Physics, Berlin Springer Verlag, ed. R. Diehl, D. H. Hartmann, & N. Prantzos, 153-232
Thompson, T. A., Burrows, A., & Meyer, B. S. 2001, , 562, 887
Timmes, F. X., & [Arnett]{}, D., 1999, , 125, 277
Travaglio, C., Gallino, R., Arnone, E., et al. 2004, , 601, 864
Wanajo, S. 2006, , 647, 1323
Wanajo, S., & Janka, H.-T. 2011, , 746, 180
Wanajo, S., Janka, H.-T., & Müller, B. 2011, , 726, L15
Weissenborn, S., Sagert, I., Pagliara, G., et al. 2011, , 740, L14
Wilson, J. R., & Mayle, R. W., 1988, , 163, 63
Winteler, C.; Käppeli, R.; Perego, A. et al., 2012, , 750, 22
Woosley, S. E., Wilson, J. R., Mathews, G. J., et al. 1994, , 433, 229
Woosley, S. E., & Weaver, T. E. 1995, , 101, 181
Woosley, S. E., Heger, A., & Weaver, T. E. 2002, Rev. Mod. Phys., 74, 1015
Woosley, S. E., & [Heger]{}, A. 2007, , 442, 269
|
---
abstract: 'We perform local measurements of both the shear rate and the particle fraction to study viscous resuspension in non-Brownian suspensions. A suspension of PMMA spherical particles dispersed in a lighter Newtonian fluid (*Triton X100*) is sheared in a vertical Couette cell. The vertical profiles of the particle volume fraction are measured for Shields numbers ranging from $10^{-3}$ to $1$, and the variation in the particle normal stress in the vorticity direction of the particle fraction is deduced.'
author:
- 'Enzo d’Ambrosio, Frédéric Blanc'
- Elisabeth Lemaire
bibliography:
- 'biblio\_resuspension.bib'
title: 'Viscous resuspension of non-Brownian particles: determination of the concentration profiles and particle normal stresses.'
---
non-Brownian suspensions, viscous resuspension, particle normal stresses
Introduction
============
Understanding the flow properties of concentrated suspensions is a real challenge in the development of many industrial products (e.g., solid propellant rocket motors and fresh concrete) and in the description of various environmental flows (e.g., torrential lava, mud flows, and submarine slides). Among other transport properties, shear-induced particle migration has received increasing attention in recent decades. Particle migration can be due to inertial effects [@segre1962behaviour] but also occurs at low Reynolds numbers, for instance, in a Poiseuille flow in which the particles tend to migrate towards the centre of the channel [@koh1994experimental; @hampton1997migration; @butler1999imaging; @snook2016dynamics], in wide-gap Couette flow towards the outer cylinder [@graham1991note; @abbott1991experimental; @chow1994shear; @sarabian2019fully] and outward in cone-and-plate geometry [@chow1995particle]. Another typical example of shear-induced migration is viscous resuspension whereby an initially settled layer of negatively buoyant particles expands vertically when a shear flow is applied. Viscous resuspension has been observed for the first time by [@gadala1979rheology] and later explained by [@leighton1986viscous] and [@acrivos1993shear], who demonstrated that the height of the resuspended particle layer results from the balance between a downward gravitational flux and an upward shear-induced diffusion flux. The authors studied the resuspension of various particles (different sizes and densities) in two different liquids (different viscosities and densities) sheared in a cylindrical Couette device. They measured the height of the resuspended layer of particles, $h_s$, as a function of the shear rate and showed that the difference between $h_s$ and $h_0$ (i.e., the initial sediment height) normalized by $h_0$ was a function of only the Shields number defined as the ratio between viscous and buoyancy forces: $$\dfrac{h_s-h_0}{h_0}=f(A) \mbox{ with } A=\dfrac{9}{2}\dfrac{\eta_0 \dot{\gamma}}{\Delta \rho g h_0}
\label{A}$$ Their experimental results were found to be in very good agreement with the diffusive flux model developed by [@leighton1986viscous]. Later, [@zarraga2000characterization] revisited the results of [@acrivos1993shear] to determine the particle normal stress in the vorticity direction, $\Sigma_{33}^p$, from the height of the resuspended layer of particles by writing the Cauchy momentum balance in the vertical direction: $$\dfrac{\partial \Sigma_{33}^p}{\partial z}=\Delta \rho g \phi
\label{Cauchy}$$ Then, a relation between $\Sigma_{33}^p$ and the particle volume fraction at the bottom, $\phi_0$, is obtained by the integration of Eq. \[Cauchy\] from the interface between the suspended layer and the clear liquid at the bottom together with the equation of particle number conservation. The relationship between particle normal stress and shear-induced migration (or resuspension) has been the subject of several studies and is still an active area of investigation [@nott1994pressure; @mills1995rheology; @morris1998pressure; @morris1999curvilinear; @deboeuf2009particle; @lhuillier2009migration; @nott2011suspension; @ovarlez2013migration]. The suspension balance model proposed by [@morris1999curvilinear] and refined by [@lhuillier2009migration] and [@nott2011suspension] offers a promising framework for modelling shear-induced particle migration, but it suffers from a relative lack of experimental data on particle normal stresses. In addition to the above-cited work of [@zarraga2000characterization], who used the viscous resuspension experiment of [@acrivos1993shear] to deduce $\Sigma_{33}^p$ for particle volume fractions ranging from $0.3$ to $0.5$, [@deboeuf2009particle] determined $\Sigma_{33}^p$ for particle volume fractions ranging from $0.3$ to $0.5$ through the measurement of the pore pressure in a cylindrical Couette flow. [@boyer2011unifying] used a pressure-imposed shear cell to measure $\Sigma_{22}^p$ in the range $\phi \in [0.4,0.585]$, and [@dbouk2013normal] determined $\Sigma_{22}^p$ in the range $\phi \in [0.3,0.47]$ through the measurement of both the total stress $\Sigma_{22}$ and the pore pressure. See [@guazzelli2018rheology] for a review. All of these studies show a linear relationship between the particle normal stress components and the shear rate, but recently, [@saint2019x] performed X-ray radiography experiments on viscous resuspension that revealed a non-linear relationship with the shear rate.
In this paper, we present the experimental results of viscous resuspension in a Couette device in which the local particle volume fraction and the local shear rate are measured by optical imaging. $\Sigma_{33}^p$ is obtained by integrating Eq.\[Cauchy\] from the interface between the clear fluid and the resuspended layer to any height $z$ below the interface. These experiments present the dual advantage that $\Sigma_{33}^p$ can be determined for a wide range of particle fractions and that the local shear rate can be measured to accurately test the scaling of particle normal stresses with shear rate.
Materials and Methods
=====================
Suspension and device
---------------------
PMMA spheres (Arkema BS572), $2a=268\pm 25\,\mu m$ in diameter and $1.19\, 10^{3}\pm 10\, kg/m^3$ in density, are used. The particles are dispersed in *Triton X 100* to which a small amount of a fluorescent dye (*Nile Blue A*, REF) is added. This mixture is Newtonian with a viscosity of $\eta_0=0.34
\pm0.02\, Pa.s$ and of density $1.06\,10^3\pm 10\, kg/m^3$ at $T=23^oC$. The characteristic settling velocity of the particles is then $V_S= 2/9\, \Delta \rho g/ \eta\approx 20\,\mu m/s$. The liquid and the particles are chosen to have roughly the same refractive index, $ 1.49$, and accurate index matching is achieved by tuning the temperature of the chamber that contains the rheometer.
The resuspension experiments are conducted in a Couette cell made of PMMA mounted on a controlled-stress rheometer (Mars II, Thermofisher) (see figure \[fig:schema\](a)). The rotor has a radius $R_1 = 19\, mm$, and the stator has a radius $R_2 = 24\, mm$. Thus, the gap is much larger than the particle diameter ($(R_2-R_1)/a \approx 37$) but is small enough for the shear stress variation to be weak: $\Sigma_{12}(R_1)/\Sigma_{12}(R_2)=R_2^2/R_1^2\approx1.6$. Thus, radial shear-induced particle migration is expected to be weak.
![a) Sketch of the experimental device. b) View from above. The vertical laser sheet is shifted by an offset of length $y_0$ from the radial position. $x$ is the horizontal position in the laser sheet, and $z$ is the vertical position. $z=0$ is set by the mercury/suspension interface.[]{data-label="fig:schema"}](schema.pdf){width="60.00000%"}
The bottom of the Couette cell is filled with mercury to prevent the particles from migrating out of the gap (under the cup) and to maximize slip at the suspension/bottom interface to have a shear rate as homogeneous as possible inside the gap. The suspension is poured into the rheometer cell and illuminated by a thin vertical laser sheet (thickness $\approx 50\, \mu m$) offset by $y_0=16.2\, mm$ from the radial plane (see figure \[fig:schema\](b)). A camera (IDS, nominal frequency 33 Hz, full resolution $4104 \times 2174 \, px^2$) is placed at $90^o$ to the enlightened plane. The accurate matching of the refractive index, the thinness of the laser sheet and the resolution of the camera allow the recording of high-quality images with a resolution of $30\,px$ per particle.
### Experimental procedure
In this paper, we will focus on the steady state of resuspension obtained for various angular velocities of the rotor, $\Omega$: $0.3,\ 0.5,\ 1,\ 2,\ 5,\ 10,\ 20,\ 30,\ 40$ and $60$ rounds per minute ($rpm$). For all these angular velocity values, the Reynolds number ($\mathcal{R}e=\rho \Omega R_1(R_2-R_1)/\eta$) is less than $1$, and the Péclet number ($\mathcal{P}e=6\pi \eta a^3\dot{\gamma}/k_B T$) is very large ($\mathcal{P}e >10^{8} $).
To reach the steady state, the suspension is first sheared with an angular velocity of the rotor equal to $5\,rpm$ for one hour. Then, the speed is set to the desired value for a period until the steady state is reached; the steady stated is considered attained when the torque applied by the rheometer becomes constant. The time duration necessary to achieve the steady state is approximately a few hours. Figure \[fig:cartograhie\] shows the viscous resuspension observed for a few rotor angular velocity values. As $\Omega$ increases, the resuspended height increases and the bulk particle concentration decreases. For the slowest rotation speeds (see figure \[fig:cartograhie\]), particle layering appears near the walls. This structuring of the suspension is clearly observed by averaging images (see figure \[fig:glissement\](b)).
![Left: typical images recorded for different rotor rotation speeds. A photo of the settled layer ($\Omega=0$) is also presented (height $21.3\, mm$). The rotor is on the left of each frame, the stator is on the right, and the mercury/suspension interface corresponds to the bottom of each frame. Centre: the mapping of the particle volume fraction averaged over 10000 images. Right: the orthoradial velocity normalized by the rotor velocity $\Omega R_1$ and averaged over 100 velocity fields.[]{data-label="fig:cartograhie"}](cartographie.pdf){width="100.00000%"}
Velocity and concentration fields
---------------------------------
### Settled layer
Figure \[fig:cartograhie\] shows an image of the suspension in the settled state. The sediment height is $h_0=21.3\, mm\approx 4(R_2-R_1)$. The volume fraction of the sediment was measured in a graduated cylinder approximately $1\, cm$ in diameter. A given mass of particles is poured into the vessel containing the suspending liquid (Triton X100), and the sediment height is carefully measured. We took three measurements and obtained $\phi_0=0.574\pm0.003$.
### Concentration fields
To determine the concentration field in the $(x,z)$ vertical laser plane, each image is binarized with a local threshold. The particles are detected through a watershed segmentation process [@vincent1991watersheds]. The position of the barycentre of each segmented zone gives the position of each particle centre in the $(x,z)$ plane sampled with rectangular cells $(i,j)$ of size $\delta x=(R_2-R_1)/8$ and $\delta z=2a$. In each cell, the number of particle centres, $N_{ij}$, is measured. The particle density $n_{ij}=N_{ij}/(\delta x \delta z)$ is reconstructed in the $(r,z)$ plane, making the change of variable $r=\sqrt{y_0^2+x^2}$. Due to the non-zero thickness of the laser sheet and of the slight polydispersity of the particles, $n_{ij}$ is not the absolute particle density, and to compute the true particle volume fraction, we use the particle volume conservation from the sediment to the resuspended state: $$\phi(r,z)=\beta n(r,z) \mbox{ with }\beta= \dfrac{\phi_0 \pi ( R_2^2-R_1^2 )h_0}{\int_0^h \int_{R_1}^{R_2} n_{ij} 2\pi r\, \mathrm{d}r \mathrm{d}z }$$
To compute the mean particle volume fraction, this procedure is repeated over $10000$ decorrelated images. Note that the acquisition time can be as long as $100\,hrs$ for the lowest rotation speed of the rotor. Examples of the concentration field are given in figure \[fig:cartograhie\], which deserves a few comments:
- Near the walls, the particle fraction is lower than in the bulk of the suspension, which should stem from the layering of the particles near the walls [@Suzuki2008Study; @Yeo2010Ordering; @blanc2013microstructure; @gallier2016effect; @deboeuf2018imaging].
- Outside of the structured zones, no or very weak radial particle migration is observed: the maximum difference in the particle volume fraction is evaluated to be less than $2\%$.
- Along the vertical direction, a concentration gradient is observed as expected in the case of resuspension flows with a sharp interface separating the suspension and the pure fluid [@acrivos1994measurement].
### Velocity fields
The aim of the present study is to investigate resuspension and to link it to particle normal stresses. Because $\Sigma_{33}^p$ is a function of the shear rate, it is essential that the shear rate is known as precisely as possible. For this purpose, we measured the velocity field in the gap. The shift in the laser sheet out of the radial plane allows particle image velocimetry (PIV) measurements [@manneville2004high] in the $(x,z)$ plane. Under the assumption that the radial component of the velocity is zero or much smaller than the orthoradial component, $v_{\theta}$ can be deduced from a simple projection of $v_{x}$ along the orthoradial direction (see figure \[fig:schema\](b)): $$v_\theta(x,z)=v_x(x,z) \dfrac{\sqrt{x^2+y_0^2}}{y_0}$$ The velocity field $\boldsymbol{v}(v_x(x,z),v_z(x,z))$ is computed using the open source software DPIVSOFT[^1] [@meunier2003analysis]. Each image is divided into correlation windows of size $128 \times 128\, px^2$. Each correlation window contains approximately $10$ particles that are the PIV tracers. The cross correlation of the corresponding windows from two successive images yields the mean velocity of the particles in the window. The in-plane loss of pairs error is decreased by translating the correlation windows in a second run [@westerweel1997fundamentals], thus reducing the correlation windows size to $64 \times 64\, px^2$. The same procedure performed on all the windows gives the velocity field, which is averaged over $100$ images.
The mapping of the $\theta$-component of the velocity field in the plane $(x,z)$ is then obtained and used to reconstruct the velocity field in the $(r,z)$ plane. Velocity maps are shown in figure \[fig:cartograhie\], in which the velocity normalized by the velocity of the rotor is represented for several values of $\Omega$. Note that the PIV measurements also enable estimation of the z-component of the velocity, particularly to check that there is no significant secondary flow. For all the experiments, we checked that the vertical velocity was less than $1\%$ of $v_\theta$ and did not present any peculiar spatial correlation. It is clear from figure \[fig:cartograhie\] that there is a significant apparent wall slip, especially for the low angular velocities, i.e., the large particle volume fractions. The wall slip phenomenon in concentrated non-Brownian suspensions is well known [@jana1995apparent; @ahuja2009slip; @blanc2011local; @korhonen2015apparent] and can be at the origin of the very large discrepancy between the macroscopic expected shear rate, $\dot{\gamma}_N=2\Omega \dfrac{R_1^2 R_2^2}{R_2^2-R_1^2} \dfrac{1}{r^2}$, and the true local shear rate that can be deduced from the PIV measurements: $$\dot{\gamma}(r,z)=r \dfrac{\partial (v_\theta(r,z)/r)}{\partial r}
\label{gamma_dot}$$
Figure \[fig:glissement\] displays the ratio of the measured shear rate to the nominal shear rate calculated at the middle of the gap, $\dot{\gamma}_N(r = (R_1+R_2)/2)$, as a function of $\phi$ for all the values of $\Omega$. A few comments on this figure are needed. First, it is observed that all the data collapse on a unique curve regardless of the angular velocity of the rotor. This finding is consistent with the results of [@jana1995apparent] and the fact that the difference between $\dot{\gamma}$ and $\dot{\gamma}_N$ is mainly due to apparent wall slip that arises from particle layering near the cylinders [@blanc2013microstructure]. Particle layering is clearly seen in figure \[fig:glissement\](inset), which is an averaged image obtained for $\Omega=2\,rpm$. Second, wall slip becomes negligible when the particle volume fraction is small enough $\phi\approx0.2$. In contrast, for higher concentrations, the local shear rate can substantially deviate from $\dot{\gamma}_N$; for the smallest values of $\Omega$ (the largest values of $\phi$), the true shear rate can be as small as one-fourth of the apparent macroscopic shear rate, making it necessary to measure the velocity field in the gap.
![Ratio of the local shear rate to the nominal shear rate vs. the local volume fraction. Each colour corresponds to a given value of $\Omega$. Each point was obtained by averaging $\dot{\gamma}$ and $\phi$ over the central third of the gap for a given height $z$. $\dot{\gamma}_N$ is calculated at the middle of the gap $r=(R_1 + R_2)/2$. Insert: a zoomed-in image of an averaged image over 10000 decorrelated frames obtained for $\Omega = 2\, rpm$. Red line: spatial variation in the particle volume fraction (a.u.) []{data-label="fig:glissement"}](gp_sur_gpapp_phi.pdf){width="60.00000%"}
Results
-------
### Concentration profiles
Figure \[fig:concentration\_profile\](a) shows two concentration profiles averaged over the central third of the gap at low ($\Omega=0.5\, rpm$) and high ($\Omega=20\, rpm$) angular velocities. It is observed that the concentration is almost constant in the resuspended layer and drops to zero quite sharply, even for the highest angular velocity. This sharp interface between the resuspended layer and the clear fluid was already predicted by [@acrivos1993shear] when interpreting their experiments in light of a diffusive flux model. Figure \[fig:concentration\_profile\](a) also shows the profiles predicted by [@acrivos1993shear]. The agreement is quite good even though the resuspension height that we measured at low angular velocity is slightly larger than that obtained by [@acrivos1993shear] and marginally smaller at high angular velocity. This trend is seen in figure \[fig:concentration\_profile\], where, as in [@acrivos1993shear], we observe a power-law dependence of the sediment expansion with the Shields number (\[A\]) but with an exponent slightly lower than $1/3$ [@zarraga2000characterization].
Finally, it should be noted that near the bottom of the Couette cell, the particle concentration tends to decrease. This finding may be related to a problem of particle detection near the interface with mercury, which reflects light and may downgrade the image quality in its vicinity. In the next section, in which the concentration will be used to evaluate $\Sigma_{33}^p$, we will not consider this zone in which we are not absolutely confident in the particle concentration measurement.
![a) Examples of the vertical concentration profiles obtained by averaging $\phi(r)$ over the central third of the gap. The corresponding Shields numbers are computed using the local shear rate (averaged over the central third), and the results are compared to the predictions of [@acrivos1993shear] (red lines). b) Relative expansion of the particle layer versus the Shields number. $h_s$ is arbitrary defined as the height in which $\phi=0.1$, and the results are compared with the correlation proposed by [@acrivos1993shear].[]{data-label="fig:concentration_profile"}](concentration_profile.pdf){width="60.00000%"}
### Determination of $\Sigma_{33}^p$
To determine $\Sigma_{33}^p$, Eq.\[Cauchy\] is integrated from the interface between the resuspended layer and the clear fluid to the height $z(\phi)$:
$$\Sigma_{33}^p(r,z)=-\int_{z(\phi)}^{z(\phi=0)}\Delta \rho g \phi(r,z) dz$$
Thus, for each point $(r,z)$, the third particle normal stress, the local shear rate and the particle volume fraction are computed. Figure \[fig:sigma33\] shows the variation of $-\Sigma_{33}^p$ normalized by $\eta_0 \dot{\gamma}$ as a function of $\phi$. To avoid boundary effects, we discarded the measurements taken for $z<h_s/4$ and $r<(R_2-R_1)/3$ or $r>2(R_2-R_1)/3$.
![Third particle normal stress normalized by the product of the fluid viscosity and the local shear rate versus the local particle volume fraction. The agreement with the correlation proposed by [@zarraga2000characterization] where $\phi_0=0.574$ is very good (red line). In contrast, it is not possible to represent the experimental results by the correlation obtained by [@boyer2011unifying] for $\Sigma_{22}^p$ together with $\lambda_3=1/2$ (blue line).[]{data-label="fig:sigma33"}](sigma33p_phi.pdf){width="60.00000%"}
In figure \[fig:sigma33\], we observe that the data almost collapse on a single curve for a wide range of $\phi$ between $0.2$ and $0.6$ with variation in $\Sigma_{33}^p/\eta_0 \dot{\gamma}$ over more than five decades. We restricted the data to particle volume fractions greater than $0.2$ because, as shown in figure \[fig:concentration\_profile\], below this value, the concentration profile is very sharp, which makes it difficult to measure the concentration. The data are somewhat scattered, especially for the largest values of $\phi$. This finding may have different origins. First, it can stem from experimental issues because, as observed in figure \[fig:concentration\_profile\], for the lowest angular velocity values (i.e., the larger particle volume fractions), the profile is nearly flat, which means that $\Sigma_{33}^p$ given by the integral of $\phi(z)$ varies greatly with $\phi$. Thus, even a small error in $\phi$ is likely to cause a large error in the computation of $\Sigma_{33}^p$. In addition to these experimental issues, the shear-thinning character displayed by most non-Brownian suspensions [@acrivos1994measurement; @lobry2019shear; @vazquez2016shear; @vazquez2017investigating; @tanner2018bootstrap] can also be at the origin of the scattering observed for the largest concentrations. Indeed, it is expected that particle normal stresses vary with the shear stress (and not the shear rate) [@boyer2011unifying]. Thus, as the angular velocity increases, the viscosity, $\eta_S$, decreases and the ratio $\Sigma_{33}^p/\eta_S$ increases for a given $\phi$, which should improve the collapse of the data.
The red and black lines in figure \[fig:sigma33\] represent the correlation proposed by [@zarraga2000characterization]:
$$\Sigma_{33}^p=-\eta_0 \dot{\gamma} \dfrac{\phi^3}{\left(1-\dfrac{\phi}{\phi_0}\right)^3}$$
The black curve is obtained with the original value of $\phi_0$ proposed by [@zarraga2000characterization] ($\phi_0=0.62$), while the red curve has been obtained for $\phi_0=0.574$: the value of the particle volume fraction inside the settled layer that we measured. We observe a very good agreement between the experimental data and the correlation from [@zarraga2000characterization]. Furthermore, [@zarraga2000characterization] established the correlation for a particle volume fraction ranging from $0.3$ to $0.5$, while our results show that this correlation can be expanded to a wider range of concentrations. The blue curve is obtained by using the correlation obtained by [@boyer2011unifying] for $\Sigma_{22}^p$ (with $\phi_0=0.574$): $$\Sigma_{22}^p=-\eta_0 \dot{\gamma}\dfrac{\phi^2}{\left(\phi_0-\phi\right)^2}$$ and assuming that $\lambda_2=\Sigma_{22}^p/\Sigma_{11}^p\approx 1$ and $\lambda_3=\Sigma_{33}^p/\Sigma_{11}^p=0.5$, as suggested by [@morris1999curvilinear].
The agreement between the blue curve and our data is not satisfactory. In our opinion, this discrepancy does not call into question the results obtained by [@boyer2011unifying] but rather the lack of variability in $\lambda_{3}$ with $\phi$. This last result has already been noted by [@gallier2014rheology] and was previously suggested by [@morris1999curvilinear] themselves.
In conclusion, with the aim of studying viscous resuspension, we conducted local measurements of both the shear rate and the particle volume fraction and deduced the variation of $\Sigma_{33}^p/\eta_0\dot{\gamma}$ with $\phi$. Our results confirm the correlation proposed by [@zarraga2000characterization] and extend it over a wider range of particle volume fractions.
We are grateful to L. Lobry, F. Peters and B. Saint-Michel for fruitful discussions, and D. Gilbert for the 3D sketch of the experimental device.
[^1]: Available on the web (https://www.irphe.fr/meunier/)
|
---
author:
- |
David Eppstein Michael T. Goodrich Nodari Sitchinava\
Department of Computer Science\
University of California, Irvine\
[ {eppstein, goodrich, nodari}(at)ics.uci.edu]{}
bibliography:
- 'localize.bib'
title: Guard Placement For Wireless Localization
---
Introduction
============
Natural Angle-Guard Placements
==============================
An Upper Bound For Arbitrary Polygons
=====================================
Lower Bounds
============
Polygon Classes that Require Fewer than $n-2$ Guards
====================================================
In this section we consider classes of polygons for which the general upper bound of $n-2$ guards for arbitrary polygons can be considerably improved.
Polygons with a Sublinear Number of Guards
------------------------------------------
We now present a class of polygons for which a square-root number of guards is sufficient to solve the sculpture garden problem, providing an upper bound within a constant factor of the lower bound of Theorem \[thm:sqrt-lower\].
Conciseness Trade-offs
======================
Conclusion and Open Problems
============================
In this paper, we introduced the sculpture garden problem for placing angle guards in such a way as to define a polygon $P$ and prove when points are inside $P$. We presented a number of results concerning the kinds and number of guards needed to define various polygons. We provided the $n-2$ upper and $\frac n2$ lower bounds for general polygons, as well as conjectured the existence of some polygons which require as many as $n-2$ angle guards. We also provided several classes of polygons which require substantially fewer guards than the general upper bound. We feel this paper begins an interesting new branch of work on polygon guarding problems and hope that it will inspire future work in this direction. In particular, we leave the following open problems:
1. Is there a simple polygon that requires $n-2$ angle guards to define it (our conjecture is “yes”)?
2. Our results also apply to the inverse sculpture garden problem, so that mobile devices outside a polygon can prove they are outside. What are the best upper and lower bounds for a generalization of the sculpture garden problem so that devices inside or outside the polygon can prove their respective locations?
3. Establish tight upper and lower bounds for solving the sculpture garden problem for orthogonal polygons.
4. Is the problem of finding the minimum number of angle guards for a particular polygon NP-hard?
Acknowledgment {#acknowledgment .unnumbered}
--------------
We would like to thank Matthew Dickerson for helpful comments regarding several of the proofs contained in this paper.
|
---
abstract: 'There is a big difference between quasi-galois closed in the eprint (arXiv:0907.0842) and pseudo-galois in the sense of Suslin-Voevodsky. It is nontrivial.'
address: 'School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, People’s Republic of China'
author:
- 'Feng-Wen An'
title: 'a short note on quasi-galois closed and pseudo-galois covers'
---
Let $X$ and $Y$ be integral schemes of finite types over $Spec(\mathbb{Z}).$ Let $\varphi :X\rightarrow Y$ be a morphism of finite type.
We define quasi-galois closed covers in **Definition 1.1** of [@An]. If $\varphi $ is a finite morphism, Suslin and Voevodsky define pseudo-galois in **Definition 5.5** of [@VS].
**Remark.** There is a big difference between quasi-galois closed and pseudo-galois. This is nontrivial. For quasi-galois closed, the function field $k\left( X\right) $ is not necessarily algebraic over $k\left(
Y\right) .$ At the same time, for pseudo-galois, $k\left( X\right) $ must be an algebraic extension over $k\left( Y\right) .$
**Example.** Let $X=Spec(\mathbb{
Z}[t])$ and $Y=Spec(\mathbb{Z})$ and let $\varphi :X\rightarrow Y$ be the morphism induced by the inclusion. Here $t$ is a variable over $\mathbb{Q}.$ Then $X$ is quasi-galois closed over $Y.$ It is clear that $X$ is not pseudo-galois over $Y$.
Many thanks for an anonymous referee’s comments.
[9]{} An, F-W. Automorphism groups of quasi-galois closed arithmetic schemes. eprint arXiv:0907.0842.
Suslin, A; Voevodsky, V. Singular homology of abstract algebraic varieties. Invent. Math. 123 (1996), 61-94.
|
---
abstract: 'The number of scientific fields that regularly collect data that are spatio-temporal continues to grow. An intuitive feature of this type of data is that measurements taken on experimental units near each other in time and space tend to be similar. As such, many methods developed to accommodate spatio-temporal dependent structures attempt to borrow strength among units close in space and time, which constitutes an implicit space-time grouping. We develop a class of dependent random partition models that explicitly models this spatio-temporal clustering by way of a dependent random partition model. We first detail how temporal dependence is incorporated so that partitions evolve gently over time. Then conditional and marginal properties of the joint model are derived. We then demonstrate how space can be integrated. Computation strategies are detailed and we illustrate the methodology through simulations and an application.'
author:
- |
Garritt L. Page\
Brigham Young University, Provo, USA\
BCAM - Basque Center of Applied Mathematics, Bilbao, Spain\
and\
Fernando A. Quintana [^1]\
Pontificia Universidad Católica de Chile, Santiago, Chile\
and\
David B. Dahl\
Brigham Young University, Provo, USA.
bibliography:
- '../reference.bib'
title: '**Spatio-Temporal Random Partition Models**'
---
\#1
1
[1]{}
0
[1]{}
[**Spatio-Temporal Random Partition Models**]{}
[*Keywords:*]{} correlated partitions, spatio-temporal clustering, hierarchical Bayes modeling, Bayesian nonparametrics, time dependent partitions.
Introduction {#Intro}
============
We introduce a method to directly model spatio-temporal dependence in a sequence of random partitions. Our approach is motivated by the practical problem of modeling a prior distribution for a sequence of random partitions that exhibit substantial overlap over time, and where cluster formation may also be spatially influenced. Traditionally, dependencies in random partitions (i.e., the clustering of units) have been obtained as a by-product of dependent random measures in Bayesian nonparametric (BNP). We will argue, however, that when partitions are the inferential objects of principal interest, then the partition should be modeled *directly* rather than relying on *induced* random partition models such as those originating from temporal, or spatio-temporal dependent BNP models. But first, we review the literature on dependent BNP methods.
BNP methods that incorporate time include @Caron:2007, @nieto-barajas:2012, @isadora, @gutierrez:2016, @jo2017 and [@caron:2017]. Those accommodating space include @sDP, , @duan:2007, , and @GelfandDiggleFuentesGuttorp:2010. The BNP literature is more sparse for combined space-time methods, with being the first to construct a spatio-temporal BNP model for areal data by adding an AR(1)-like temporal transition structure to the spatial Dirichlet process of [@sDP]. consider a model for functional magnetic resonance imaging data and model temporal dependence in the error term and spatial dependence through a hierarchical Dirichlet process mixture model on voxel-specific coefficients (whose clustering induces spatial dependence in the partition). [@Savitsky:2016] apply a spatio-temporal BNP model to the American Community Survey with varying spatial resolution. [@cassese2018] construct a space-time species sampling model that permits the identification of disease outbreaks.
![Lagged ARI values using the method of [@caron:2017] based on concentration parameter $M=0.5$, discount parameter set to zero, and 10,000 Monte Carlo samples. The temporal dependence parameter is $\alpha \in [0,1]$. []{data-label="caron2"}](caron3.pdf)
A common aspect of all these methods is that temporal, spatial, or spatio-temporal dependence is accommodated in the sequence of random measures by way of the atoms or weights of the stick-breaking representation [@Sethuraman:1994]. The induced random partitions, however, exhibit only weak dependence even when a sequence of random probability measures is highly correlated. To illustrate this point, we conducted a small Monte Carlo simulation where a sequence of partitions were generated with 10 time points and 20 units using the method of @caron:2017. To measure similarity of partitions at different time points, we use the lagged adjusted rand index (ARI). Figure \[caron2\] shows these values averaged over 10,000 Monte Carlo samples. Notice that as $\alpha$ increases, the partitions from time period $t$ to $t+1$ only become slightly more similar, such that the dependance between partitions is, at best, only weak. Further, the dependence is not temporally intuitive as it does not decay as a function of lag. This behavior is not unique to [@caron:2017]’s approach, as noticed the same type of behavior when using a linear dependent Dirichlet process mixture model. In fact, all BNP methods that model a sequence of random probability measures will induce a random partition model with similar weak-correlation behavior. This behavior is analogous to trying to induce dependence among random variables from distributions with correlated parameters. There is no guarantee that correlated parameters would produce strong correlations among the random variables themselves.
[@paci:2018]’s motivation is more similar to ours as their principal interest is spatially referenced partitions over time. However, their approach is based on a mixture of experts model whose weights depend on space and time. As such, their method retains the same properties as the BNP methods.
Our approach is to consider the sequence of partitions indexed by time as the object of principal interest and propose a method that models it directly. This will provide more control over how “smoothly” partitions evolve over time. Perhaps the work closest to ours (in the sense of explicitly modeling a sequence of partitions) can be found in [@carlosetal]. Their modeling approach for temporally-referenced sequence of partitions differs from ours in that they do not focus on smooth evolution of spatial partitions over time.
The rest of the article is organized as follows. Section \[model\] details our approach to modeling partitions temporally and spatially. In Section \[model\] we also provide a few theoretical results and some computational strategies. In Section \[simulation.studies\] we detail a number of simulation studies that illustrate the method and highlight its utility. Then we consider a PM$_{10}$ data set that is publicly available. Section \[conclusions\] contains some concluding remarks.
Joint Model for a Sequence of Partitions {#model}
========================================
We begin with some notation. Let $i=1, \ldots, m$ denote the $m$ experimental units at time $t$ for $t=1,\ldots, T$. Let $\rho_t = \{S_{1t}, \ldots, S_{k_tt}\}$ denote a partition of the $m$ experimental units at time $t = 1, \ldots, T$ into $k_t$ clusters. An alternative partition notation is based on $m$ cluster labels at time $t$ denoted by $\bm{c}_t = \{c_{1t}, \ldots, c_{mt}\}$ where $c_{it} = j$ implies that $i \in S_{jt}$. Notice the one-to-one correspondence between $\rho_t$ and $\bm{c}_t$. Finally, any quantity with a “$\star$” superscript will be cluster-specific. For example, we will use $\mu^{\star}_{jt}$ to denote the mean of cluster $j$ at time $t$ so that $\mu_{it} = \mu^{\star}_{jt}$ if $c_{it} = j$.
Temporal Modeling for Sequences of Partitions {#depedentPartitions}
---------------------------------------------
We first describe our approach to correlating partitions over time and subsequently, in the next subsection, detail the inclusion of space. Introducing temporal dependence in a collection of partitions requires formulating a joint probability model for $\{\rho_1, \ldots, \rho_T\}$. Generically, we will denote this joint model with $\text{Pr}(\rho_t, \ldots, \rho_T)$. Temporal dependence among the $\rho_t$’s implies that the cluster configurations found in $\rho_{t-1}, \rho_{t-2}, \ldots, \rho_1$ could impact the cluster configuration in $\rho_t$. However, we assume that the probability model for the sequence of partitions has a Markovian structure. That is, the conditional distribution of $\rho_t$ given $\rho_{t-1}, \rho_{t-2}, \ldots, \rho_1$ only depends on $\rho_{t-1}$. Thus, we construct $\text{Pr}(\rho_t, \ldots, \rho_T)$ as $$\begin{aligned}
\label{joint.model}
\text{Pr}(\rho_1, \ldots, \rho_T) = \text{Pr}(\rho_T | \rho_{T-1}) \text{Pr}(\rho_{T-1} | \rho_{T-2})\cdots \text{Pr}(\rho_2 | \rho_1)\text{Pr}(\rho_1).\end{aligned}$$ Here $\text{Pr}(\rho_1)$ is an exchangeable partition probability function (EPPF) that describes how the $m$ experimental units at time period 1 are grouped into $k_1$ distinct groups with frequencies $n_{11}, \ldots, n_{1k_1}$. One characteristic of an exchangeable EPPF that will prove useful in what follows is sample size consistency (or what refer to as the addition rule). This property dictates that marginalizing the last of $m + 1$ elements leads to the same model as if we only had $m$ elements. A commonly encountered EPPF is that induced by a Dirichlet process (DP). This particular EPPF is sometimes referred to as a Chinese restaurant process (CRP) and corresponds to a special case from the family of product partition models (PPM). For more details see . Because we employ the CRP-type EPPF in what follows, we provide its form here $$\begin{aligned}
\label{crp}
\text{Pr}(\rho | M) = \frac{M^k}{\prod_{i=1}^{n}(M+i-1)} \prod_{i=1}^k (|S_i| - 1)!,\end{aligned}$$ where $k$ is the number of clusters in $\rho$ and $M$ is a concentration parameter controlling the number of clusters. We will denote this random partition distribution as $CRP(M)$.
Although conceptually straightforward, is silent regarding how $\rho_{t-1}$ influences the form of $\rho_{t}$. To make this explicit, we introduce an auxiliary variable that guides how similar $\rho_t$ is to $\rho_{t-1}$. Now, if two partitions are highly correlated, then the cluster configurations between them will change very little and as a result only a few of the $m$ experimental units will change cluster assignment. Conversely, two partitions that exhibit low correlation will likely be comprised of very different cluster configurations. The auxiliary variable we introduce identifies which of the experimental units at time $t-1$ will be considered for possible cluster reallocation at time $t$. Specifically, let $\gamma_{i t}$ denote the following $$\begin{aligned}
\gamma_{i t} =
\left\{
\begin{array}{c l}
1 & \mbox{if unit $i$ is {\it not} reallocated when moving from time $t-1$ to $t$} \\
0 & \mbox{otherwise}.
\end{array}
\right.\end{aligned}$$ By construction we set $\gamma_{i1} = 0$ for all $i$ (i.e., all experimental units are allocated to clusters during the first time period). We then assume that $\gamma_{i t} \stackrel{ind}{\sim} Ber(\alpha_t)$. Note that each of the $\alpha_t \in [0,1]$ acts as a temporal dependence parameter. Specifically, we will interpret $\alpha_t =1$ as implying that $\rho_t = \rho_{t-1}$ with probability 1. Conversely, when $\alpha_t=0$, then $\rho_t$ is independent of $\rho_{t-1}$. For notational convenience we introduce $\bm{\gamma}_{t} = (\gamma_{1 t}, \gamma_{2 t}, \ldots, \gamma_{m t})$ which is an $m$-tuple comprised of zeros and ones. The augmented joint model changes to $$\begin{gathered}
\label{joint.joint.model}
\text{Pr}(\bm{\gamma}_1,\rho_1, \ldots, \bm{\gamma}_T, \rho_T) = \text{Pr}(\rho_T | \bm{\gamma}_T, \rho_{T-1}) \text{Pr}(\bm{\gamma}_T) \times \\
\text{Pr}(\rho_{T-1} |\bm{\gamma}_{T-1}, \rho_{T-2})\text{Pr}(\bm{\gamma}_{T-1})\cdots \text{Pr}(\rho_2 |\bm{\gamma}_2, \rho_1)\text{Pr}(\bm{\gamma}_2) \text{Pr}(\rho_1).\end{gathered}$$
In Section \[toy.example\] of the online Supplementary Material, we provide a toy example that illustrates how our construction produces intuitive conditional partition distributions. In addition to exhibiting intuitive behavior conditionally, it would be appealing if marginally each of the $\rho_t$ follow the parent EPPF (i.e., the probability model assumed for $\rho_1$), so that the joint probability model for partitions would become stationary. The following proposition establishes this result which is a consequence of the fact that conditioning on $\bm{\gamma}_t$ provides a “reduced” EPPF.
\[proposition1\] Let $\rho_1 \sim EPPF$ and $\bm{\gamma}_1 = 0$. If a joint model for $\rho_1 \ldots, \rho_T$ is constructed as described above by introducing $\bm{\gamma}_t$ for $t = 2, \ldots, T$, then we have that marginally $\rho_{1},\ldots,\rho_{T}$ are identically distributed with law coming from the EPPF used to model $\rho_1$. Specifically, letting $\rho_{-t} = (\rho_1, \ldots, \rho_{t-1}, \rho_{t+1}, \ldots, \rho_T)$ and $\bm{\gamma} = (\gamma_1, \ldots, \gamma_T)$, we have that for all $\lambda \in P$, $$\begin{aligned}
\textup{Pr}(\rho_t = \lambda) = \sum_{\rho_{-t} \in P^{\otimes}} \sum_{\bm{\gamma} \in \Gamma^{\otimes}} \textup{Pr}(\bm{\gamma}_1, \rho_1, \ldots, \rho_t=\lambda, \ldots, \bm{\gamma}_T, \rho_T) = \textup{Pr}(\rho_1=\lambda),\end{aligned}$$ where $P^{\otimes} = P \times P \times \ldots \times P$, $P$ a collection of all partitions of $m$ units and $\Gamma^{\otimes} = \Gamma \times \Gamma \times \ldots \times \Gamma$, $\Gamma$ a collection of all possible binary vectors of size $m$.
See the Appendix.
In what follows we will use $tRPM(\bm{\alpha}, M)$ to denote our temporal random partition model parameterized by $\alpha_1, \ldots, \alpha_T$ and EPPF .
We briefly mention that introducing $\gamma_{it}$ is similar in spirit to the approach taken by [@Caron:2007; @caron:2017]. However, they use $\bm{\gamma}_t$ to identify a partial partition at time $t$ that informs how [*all*]{} the observational units will be reallocated at time $t+1$. While this difference may seem benign at first glance, it has drastic ramifications on the type of dependence that exists among the actual sequence of partitions. To see this, similar to what was done in the simulation described in the Introduction, we generate 10,000 sequences of partitions based on our construction an provide the average lagged ARI values in Figure \[ours2\]. Notice now that the similarity of the partitions behaves in an intuitive way as a function of lag. Mainly, that as lags increase the similarity between partitions decreases. Further, $\alpha$ has a clear impact on the dependence between partitions with large $\alpha$ values resulting in strong dependence. Observe also that the range of ARI values achieved by this construction can be substantially higher than what was described earlier in the discussion leading to Figure \[caron2\].
![Lagged ARI values using concentration parameter $M=0.5$ based on 10,000 Monte Carlo samples using our $tRPM$. Our method shows strong temporal dependence, whereas little is seen in Figure \[caron2\] for [@caron:2017].[]{data-label="ours2"}](ours3.pdf)
Spatio-Temporal Model for a Sequence of Partitions
--------------------------------------------------
Before studying how our joint partition model can be employed in Bayesian modeling, we next describe our approach to incorporating space in the partition model. One possible way of adding a spatial component in the joint model would be to make the auxiliary variables $\bm{\gamma}_{it}$ spatially referenced. However, sample size consistency would be lost and as a result the marginal property in Proposition \[proposition1\] would not hold. An alternative approach that we adopt is to include spatial information directly in the EPPF. If the spatially referenced EPPF employed preserves sample size consistency, then Proposition \[proposition1\] still holds. To this end, we consider the spatial product partition model (sPPM) developed in . As a way of introducing the sPPM, let $\bm{s}_i$ denote the spatial coordinates of the $i$th item (note that these coordinates do not change over time) and let $\bm{s}^{\star}_{jt}$ be the subset of spatial coordinates that belong to the $j$th cluster at time $t$. Then we express the EPPF of the $t$th partition with the following product form $$\begin{aligned}
\label{sPPM}
\text{Pr}(\rho_t | \nu_0, M) \propto \prod_{j=1}^{k_t} c(S_{jt}|M) g(\bm{s}^{\star}_{jt}|\nu_0).\end{aligned}$$ Here $c(\cdot|M) \ge 0$ is called the cohesion and is a set function that produces cluster weights [*a priori*]{}. We consider the cohesion $c(S_{jt}|M) = M \times (|S_{jt}| - 1)!$ as it has connections with the CRP making this version of the sPPM a type of spatially re-weighted CRP. The similarity function $g(\cdot|\nu_0)$ is a set function parametrized by $\nu_0$ that measures the compactness of the spatial coordinates in $\bm{s}^{\star}_{jt}$ producing higher values if the spatial coordinates in $\bm{s}^{\star}_{jt}$ are less alike. Not all similarity functions preserve sample size consistency so to ensure this, after standardizing spatial locations, we employ $$\begin{aligned}
\label{similarity.function}
g(\bm{s}^{\star}_{jt}|\nu_0) = \int \prod_{i \in S_{jt}} N(\bm{s}_i | \bm{m},\bm{V}) NIW(\bm{m},\bm{V}| \bm{0}, 1, \nu_0, \bm{I})d\bm{m}d\bm{V},\end{aligned}$$ where $N(\cdot | \bm{m},\bm{V})$ denotes a bivariate normal density and $NIW(\cdot,\cdot| \bm{0}, 1, \nu_0, \bm{I})$ a normal-inverse-Wishart density with mean $\bm{0}$, scale equal to 1, inverse scale matrix equal to $\bm{I}$, and $\nu_0$ being the user-supplied degrees of freedom. Note that larger values of $\nu_0$ increase spatial influence on partition probabilities. For more details on why this formulation preserves sample size consistency, see [@PPMxMullerQuintanaRosner] and . For more information regarding the impact of $\nu_0$ on product form of the partition model, see . We will denote the random partition distribution defined in and using $sPPM(\nu_0, M)$.
We mention briefly that it would be very straightforward to build a partition model based on space and time by extending the sPPM so similarity function $g$ is a function of both space and time. Although this ensures that partitions will be influenced by space and time, the desire for partitions to evolve over time would be lost. In this setting, each spatial location by time point combination would be treated as an observational unit and would create clusters that transect time, which is something we wanted to avoid in our formulation.
We will use $stRPM(\bm{\alpha}, \nu_0, M)$ to denote our spatio-temporal random partition model parameterized by $\alpha_1, \ldots, \alpha_T$ and EPPF detailed in and .
Hierarchical Data Model {#hierarchical.model}
-----------------------
Once a partition model is specified, there is tremendous flexibility regarding how to model space/time (global or cluster-specific) at different levels of a hierarchical model (at the data level or process level or both). Since we are interested to see how including space/time in the partition model impacts clustering and model fits, in the simulations of the next section, we consider a hierarchical model where space/time only appears in the partition model. In particular, using cluster label notation, we will employ the following hierarchical model $$\label{dat.gen.mech}
\begin{aligned}
Y_{it} | \bm{\mu}^{\star}_{t}, \bm{\sigma}_t^{2\star}, \bm{c}_{t} & \stackrel{ind}{\sim} N(\mu^{\star}_{c_{it}t}, \sigma_{c_{it}t}^{2\star}), \ i = 1, \ldots, m \ \mbox{and} \ t=1, \ldots, T, \\
(\mu_{jt}^{\star}, \sigma^{\star}_{jt})|\theta_t, \tau^2 & \stackrel{ind}{\sim} N(\theta_t, \tau^2) \times UN(0,A_{\sigma}), \ j = 1, \dots, k_t ,\\
(\theta_t, \tau) & \stackrel{iid}{\sim} N(\phi_0, \lambda^2) \times UN(0, A_{\tau}), \ t = 1, \ldots, T,\\
\{\bm{c}_{t}, \ldots, \bm{c}_{T}\} & \sim \mbox{{\it joint RPM}},
\end{aligned}$$ where $Y_{it}$ denotes the response measured on the $i$th unit at time $t$, $\mbox{{\it joint RPM}}$ denotes some joint random partition model, and $UN$ denotes a uniform distribution. The remaining assumptions (e.g., independence across clusters and exchangeability within each cluster) are commonly employed. Notice that in this model three entities are in some sense “competing” when determining cluster membership, namely: a) time, b) space, and c) response. This competition, however, is carried out in a probabilistic and coherent fashion.
Computation {#PostComputation}
-----------
As the posterior distribution implied by the model in is not available in closed form, we build an algorithm that permits sampling from it. The construction of $\text{Pr}(\rho_1, \ldots, \rho_T)$ naturally leads one to consider a Gibbs sampler. In the Gibbs sampler, $\bm{\gamma}_t$ will need to be updated in addition to $\rho_t$ (by way of $\bm{c}_t$). But the Markovian assumption reduces some of the cost as we only need to consider $\rho_{t-1}$ and $\rho_{t+1}$ when updating $\rho_t$. Even though each update of $\rho_t$ and $\bm{\gamma}_t$ for $t=1,\ldots, T$ needs to be checked for compatibility (i.e. proposed moves do not violate the prior construction), it is fairly straightforward to adapt standard algorithms, e.g. Algorithm 8 of [@neal:2000], with care to make sure that only experimental units with $\gamma_{it} = 0$ are considered when updating cluster labels at time $t$. In what follows we assume that the in is the $stRPM(\bm{\alpha}, \nu_0, M)$ described earlier.
The MCMC algorithm we employ depends on deriving the complete conditionals for $\rho_t$ and $\gamma_t$. Before describing them, we introduce some needed notation. Let $N_{0t} = \sum_{j = 1}^m I[\gamma_{jt} = 0]$ denote the number of units to be reallocated when moving from time $t-1$ to $t$ (note that $N_{0t}\sim{\rm Bin}(m,1-\alpha_t)$) and denote with $\rho_{t}^{-N_{0t}}$ the “reduced” partition that remains after removing the $N_{0t}$ units that are to be reallocated at time $t$ as indicated by $\bm{\gamma}_t$. A key result needed to derive the full conditionals of $\gamma_{it}$ and $c_{it}$ is provided in the following proposition.
Based on the construction of a joint probability model as described in Section \[depedentPartitions\] and $\rho_1 \sim EPPF$, then we have $$\begin{aligned}
\textup{Pr}(\rho_t | \bm{\gamma}_t, \rho_{t-1}) = \left\{
\begin{array}{l l}
\textup{Pr}(\rho_t)/\textup{Pr}(\rho^{-N_{0t}}_t) & \rho_t \asymp \rho_{t-1}\\[3pt]
0 & otherwise,
\end{array}
\right.\end{aligned}$$ where $\rho_t \asymp \rho_{t-1}$ indicates that $\rho_t$ is compatible with $\rho_{t-1}$.
See the Appendix.
When updating $\gamma_{it}$ in a Gibbs sampler, one can think of removing $\gamma_{it}$ from $\bm{\gamma}_t$, and then reinsert it as either a 0 or 1. To this end, let $N^{(-i)}_{0t} = \sum_{j\ne i} I[\gamma_{it} = 0]$ denote the case when $\gamma_{it}$ is reinserted as a 1 and $N^{(+i)}_{0t} = N^{(-i)}_{0t} + 1$ denote the case when $\gamma_{it}$ is reinserted as a 0. Now, the full conditional for $\gamma_{it} = 1$, denoted by $\text{Pr}(\gamma_{it} = 1 | -)$, is $$\begin{aligned}
\text{Pr}(\gamma_{it} = 1 | -) & \propto \text{Pr}(\rho_t | \bm{\gamma}_t, \rho_{t-1}) \text{Pr}(\bm{\gamma}_t)\text{I}[\rho_t \asymp \rho_{t-1}], \\
& \propto \frac{\text{Pr}(\rho_t)}{\text{Pr}(\rho^{-N^{(+i)}_{0t}}_t)} \alpha_t^{\gamma_{it}}\text{I}[\rho_t \asymp \rho_{t-1}].\end{aligned}$$ Here $\text{I}[\cdot]$ denotes an indicator function. The resulting normalized full conditional for $\gamma_{it}$ is $$\begin{aligned}
\label{full.conditional.gamma1}
\text{Pr}(\gamma_{it} = 1 | -) = \displaystyle\frac{\alpha_t \text{Pr}(\rho^{-N^{(-i)}_{0t}}_t)}{\alpha_t \text{Pr}(\rho^{-N^{(-i)}_{0t}}_t) + (1-\alpha_t) \text{Pr}(\rho^{-N^{(+i)}_{0t}}_t)}\text{I}[\rho_t \asymp \rho_{t-1}].\end{aligned}$$ For a given EPPF that has a closed form (e.g., CRP), it is straightforward to compute $\text{Pr}(\rho^{-N^{(-i)}_{0t}}_t)$ and $\text{Pr}(\rho^{-N^{(+i)}_{0t}}_t)$. If, however, the EPPF does not have a closed form (e.g., sPPM), then note that can be re-expressed as $$\begin{aligned}
\text{Pr}(\gamma_{it} = 1 | -) = \displaystyle\frac{\alpha_t\text{I}[\rho_t \asymp \rho_{t-1}]}{\alpha_t + (1-\alpha_t)\text{Pr}(\rho^{-N^{(+i)}_{0t}}_t)/\text{Pr}(\rho^{-N^{(-i)}_{0t}}_t)}.\end{aligned}$$ The quantity $\text{Pr}(\rho^{-N^{(+i)}_{0t}}_t)/\text{Pr}(\rho^{-N^{(-i)}_{0t}}_t)$ is a commonly encountered expression in MCMC methods employed in random partition modeling. See for example Neal’s Algorithm 8 [@neal:2000]. Those same methods can be employed to calculate the desired probabilities.
The full conditional for $c_{it}=h$ corresponding to $\gamma_{it} = 0$ is the following $$\begin{aligned}
\text{Pr}(c_{it} = h | -) & \propto N(Y_{it} | \mu^{\star}_{c_{it} = h, t}, \sigma^{2\star}_{c_{it} = h, t})\text{Pr}(c_{1t}, \ldots, c_{it} = h, \ldots, c_{mt})\text{I}[\rho_t \asymp \rho_{t-1}].
$$ The case that unit $it$ forms a new cluster must also be considered so that $$\begin{aligned}
\text{Pr}(c_{it} = h | -) \propto
\left\{
\begin{array}{cl}
N(Y_{it} | \mu^{\star}_{c_{it} = h, t}, \sigma^{2\star}_{c_{it} = h, t})\text{Pr}(c_{it} = h)\text{I}[\rho_t \asymp \rho_{t-1}] & \mbox{$h = 1, \ldots, k_t^{-i}$, } \\
N(Y_{it} | \mu^{\star}_{new_h, t}, \sigma^{2\star}_{new_h, t})\text{Pr}(c_{it} = h)\text{I}[\rho_t \asymp \rho_{t-1}] & \mbox{$h = k_t^{-i}+1$,}
\end{array}
\right.\end{aligned}$$ where $\text{Pr}(c_{it} = h) = \text{Pr}(c_{1t}, \ldots, c_{it} = h, \ldots, c_{mt})$, $\mu^{\star}_{new_h, t}$ and $\sigma^{2\star}_{new_h, t}$ are auxiliary parameters drawn from the prior as in [@neal:2000]’s Algorithm 8 (with one auxiliary parameter) and $k_t^{-i}$ are the number of clusters at time $t$ when the $i$th unit has been removed. Details of computation procedures associated with the sPPM can be found in . Given $\rho_t$ and $\bm{\gamma}_t$, the full conditionals of the remaining parameters in model follow standard techniques. A sample can be drawn from the posterior distribution implied by model by iterating through the complete conditionals for $\bm{\gamma}_t$ and $\rho_t$ and those of other model parameters.
Simulation Studies {#simulation.studies}
==================
In this section we describe three simulation studies that explore the performance of our proposal. The first simulation study is focused on the temporal dependence among estimated partitions, the second on the temporal dependence that the joint partition model induces among the $\bm{Y}_{i} = (Y_{i1}, \ldots, Y_{iT})$, and the third on the impact that including space in the partition model has on model fit.
Simulation 1: Temporal Dependence in Estimated Partitions
---------------------------------------------------------
The purpose of the first simulation is to study the accuracy of partition estimates (i.e., $\hat{\rho}_t$) and how much they change over time. As such, in this study we do not consider spatial clustering. We do however, explore accuracy in estimating $\mu_{it}$ and $\alpha_t$. To this end, we considered model as a data generating mechanism to create one hundred datasets with fifty observations at five time points. For the [*joint RPM*]{} in model we used $tRPM(\bm{\alpha})$ with $\alpha_t=\alpha$ for all $t$ and generate synthetic datasets under $\alpha \in \{0, 0.1, 0.25, 0.5, 0.75, 0.9, 0.999\}$. For all $i$ and $t$, we set $\sigma_{c_{it}t}^{2\star} = \sigma^2 = 1$, $\tau^2 = 25$, and $\theta_t = 0$.
To each synthetic data set we fit model using the MCMC algorithm detailed in Section \[PostComputation\] by collecting 10,000 iterates and discarding the first 5,000 as burn-in and thinning by 5 (resulting in 1,000 MCMC samples) after setting $A_{\sigma} = 5$ and $A_{\tau} = 10$. All partition point estimates were estimated using the method developed in the [salso]{} R package (@dahl2019) with the Binder loss function (@binder:1978, ). To measure similarity between partitions, we employed the adjusted Rand index () and we used WAIC () to measure model fit.
Table \[sampleARI\] displays the lagged 1 and 4 adjusted Rand index (ARI) as a function of $\alpha$. As expected, for both lags the ARI increases as $\alpha$ increases. Also as expected lagged 4 ARI increases less as a function of $\alpha$ compared to the lagged 1 ARI. Note that on average the lagged 1 ARI for $\alpha \in \{ 0.1, 0.25\}$ is smaller than that for $\alpha = 0$. This is because the variability associated with lagged 1 ARI when $\alpha=0$ is much larger than when $\alpha > 0$ producing a few lagged ARI values that are large. The median of the lagged ARI values increase as a function of $\alpha$ monotonically.
----------------------- ----------------------------------- ----------------------------------- ------------- ------------- ------ ------
(r)[4-5]{} (r)[6-7]{} $ARI(\hat{\rho}_1, \hat{\rho}_2)$ $ARI(\hat{\rho}_1, \hat{\rho}_5)$ $\alpha$ $\mu_{i t}$ tRPM CRP
$\alpha=0.0$ 0.192 (0.03) 0.182 (0.03) 0.00 (0.00) 0.94 (0.01) 1711 1709
$\alpha=0.1$ 0.122 (0.02) 0.151 (0.02) 0.97 (0.02) 0.94 (0.01) 1689 1694
$\alpha=0.25$ 0.180 (0.02) 0.130 (0.02) 0.95 (0.02) 0.94 (0.01) 1645 1669
$\alpha=0.5$ 0.434 (0.02) 0.132 (0.02) 0.88 (0.03) 0.94 (0.01) 1627 1723
$\alpha=0.75$ 0.714 (0.02) 0.254 (0.02) 0.89 (0.03) 0.96 (0.01) 1576 1636
$\alpha=0.9$ 0.874 (0.01) 0.546 (0.02) 0.91 (0.03) 0.93 (0.01) 1501 1710
$\alpha=0.9999$ 0.980 (0.00) 0.941 (0.01) 0.49 (0.05) 0.93 (0.01) 1502 1611
----------------------- ----------------------------------- ----------------------------------- ------------- ------------- ------ ------
: Adjusted Rand index when comparing $\hat{\rho}_1$ to $\hat{\rho}_2$ and $\hat{\rho}_1$ to $\hat{\rho}_5$. Note that $ARI(\cdot, \cdot)$ denotes the adjusted Rand index as a function of two partitions. Coverage rates for $\alpha$ and $\mu_{it}$ and model fit metrics for $tRPM(\alpha, M)$ and $CRP(M)$. These values are averaged over the 100 generated data sets. The values in parenthesis are Monte Carlo standard errors. Note that smaller values of WAIC indicate better fit.
\[sampleARI\]
To study the ability to recover $\mu_{it}$ and $\alpha$, 95% credible intervals for each were computed and coverage was estimated. Results are provided in Table \[sampleARI\]. Notice that coverage for $\alpha$ is low when the true $\alpha$ is at or near the boundary (e.g., $\alpha \in \{0, 0.9999\})$ which is to be expected. The coverage associated with $\mu_{it}$ is close to the nominal rate regardless of the value of $\alpha$. Therefore, temporal dependence in the partition model does not adversely impact the ability to estimate individual means.
Lastly, to compare model fit when using $tRPM(\alpha, M)$ as the RPM in model relative to $\rho_t \stackrel{iid}{\sim} CRP(M)$, we calculated the WAIC for each data set when fitting model under both RPMs. Results are provided in Table \[sampleARI\] where each entry is an average WAIC value over all 100 datasets. Notice that, when the independent partitions were used to generate data (i.e., $\alpha = 0$), modeling partitions independently produces slightly better model fit as would be expected. But even if relatively weak temporal dependence exists among the sequence of partitions, there are gains in modeling the sequence of partitions with $tRPM(\alpha, M)$, with gains becoming substantial as $\alpha$ increases.
The upshot from this simulation study is that lagged partition estimates when employing $tRPM(\alpha, M)$ display intuitive behavior in that similarity between partition estimates decreases as lag increases. In addition, employing the $tRPM(\alpha, M)$ partition model does not negatively impact parameter estimation and produces improved model fits when dependence is present in the sequence of partitions and a minimal cost in model fit when it is not.
![Lagged auto-correlations among the $(Y_{i1}, \ldots, Y_{iT})$ when modeling $\mu^{\star}_{jt}$ with an AR(1) type structure.[]{data-label="MarginalCorrelations"}](MarginalCorrelationPlots4.pdf)
Simulation 2: Induced Correlation at the Response Level
-------------------------------------------------------
A potential benefit of developing a joint model for partitions is the ability to accommodate temporal dependence that may exist between $Y_{it}$ and $Y_{it+1}$. To study this, we conducted a small Monte Carlo simulation study that is comprised of sampling repeatedly from the $tRPM(\alpha, M)$ using the computational approach of Section \[PostComputation\]. Once the partition is generated, the temporal dependence among the $\bm{Y}_{i}$ depends on specific model choices for $\mu_{jt}^{\star}$. Here we use $\mu^{\star}_{jt} \sim N(\phi_1\mu_{jt-1}^{\star}, \tau^2(1-\phi_1^2))$ for $t > 2$, $j=1, \ldots, k_t$, and $|\phi_1| \le 1$. For $t=1$ we use $\mu^{\star}_{j1} \sim N(0, \tau^2)$ and if $k_{t+1} > k_t$ new $\mu^{\star}_{jt+1}$ values are drawn from $N(0, \tau^2)$. Now setting $m=25$, $T=10$, $\tau=10$, and $\sigma=1$, 100 data sets were generated for $\phi_1 \in \{0, 0.25, 0.5, 075, 0.9, 1\}$. For each data set generated, the lagged auto-correlations among $\bm{Y}_i$ were computed for $i=1,\ldots, m$. The results found in Figure \[MarginalCorrelations\] are the lagged auto-correlations averaged over the $m$ units for $\alpha \in \{0, 0.25, 0.5, 0.75, 0.9\}$.
As can be seen in Figure \[MarginalCorrelations\], when partitions are independent (i.e., $\alpha = 0$), no correlation propagates to the data level. The same can be said if atoms are $iid$ (i.e., $\phi_1 = 0$). As the temporal dependance among $\mu_{jt}^{\star}$ increases (i.e., $\phi_1$ increases), there is stronger temporal dependence among $Y_{i1}, \ldots, Y_{iT}$. Notice further that this dependence persists longer in time as $\alpha$ increases as one would expect.
Simulation 3: Dependence in Estimated Partitions {#simstudy3}
------------------------------------------------
We now discuss our final simulation study, where we investigated the performance of our procedure when both space and time are considered. To do so, we created synthetic data sets that contain spatio-temporal structure. Each employs a $15 \times 15 $ regular grid with spatial locations coming from the unit interval. In addition, either 5 or 10 time points were considered resulting in 1,125 or 2,250 total observations. Response values were generated in two ways. The first employs a Gaussian process with a separable spatio-temporal exponential covariance function. We set the spatial scale to 0.3, the temporal scale to 2 and the sill to 1.75 (see for more details). Note that no “true” partition exists for this data generating mechanism. However, we study it to explore performance of our method when spatial structure exists among observations but was not induced through partitioning. The second method of generating response values essentially employs model as a data generating mechanism. Spatio-temporal partitions were generated using together with conditional cluster label probabilities of @PPMxMullerQuintanaRosner [pg. 265] and setting $\alpha_t=\alpha$ for all $t$ with $\alpha \in \{0, 0.5, 0.9\}$ (note that for $\alpha = 0$ no temporal dependence exists among partitions). In the similarity function we considered $\nu_0 \in \{2,20\}$ where $\nu_0 = 2$ corresponds to light weight on spatial proximity and $\nu_0 = 20$ moderate weight. Finally, we set $\tau^2=1$ and $\sigma^{2\star}_{c_{it}t} = \sigma^2 = 0.04$ for all $i$ and $t$ resulting in smaller with-in cluster variability relative to between-cluster variability.
To determine the impact that each component of our spatio-temporal partition model has on model fit, we fit the hierarchical model to each synthetic data set using a variety of random partition models which are listed below. As a competitor, we consider a linear dependent Dirichlet process [@MacEachern:2000; @deIorio:2009], indexing the random probability measure through the mean function of the atoms by space and time. To ensure sufficient flexibility, B-spline basis functions for both spatial coordinates were employed. The details of each model considered are
1. Model 1: $(\rho_1, \ldots, \rho_T) \sim stRPM(\bm{\alpha}, \nu_0, M)$
2. Model 2: $\rho_t \stackrel{iid}{\sim} sPPM(\nu_0, M)$ for $t = 1, \ldots, T$.
3. Model 3: $(\rho_1, \ldots, \rho_T) \sim tRPM(\bm{\alpha}, M)$
4. Model 4: $\rho_t \stackrel{iid}{\sim} CRP(M)$ for $t = 1, \ldots, T$.
5. Model 5: linear dependent Dirichlet process mixture model (DDPM).
Additionally, for each model that employs the sPPM, we considered both $\nu_0 = 2$ (models 1a, 2a) and $\nu_0 = 20$ (models 1b, 2b). For each data generating scenario, 100 data sets were created and each of the models listed was fit by collecting 1,000 MCMC samples after discarding the first 5,000 as burn-in and thinning by 5 after setting $A_{\sigma} = 1$ and $A_{\tau} = 2$. Model fits were compared using WAIC. Results can be found in Figures \[waicGP\] and \[waicDRP\].
![Results from simulation study when observations were generated using a spatio-temporal Gaussian process. Boxplots display the 100 WAIC values that correspond to model fit for each synthetic data set. Note that smaller WAIC values indicate a better fit. []{data-label="waicGP"}](gpWAIC4.pdf)
![Results from simulation study for the scenario in which partition structure is included in data generation process. Boxplots display the 100 WAIC values that correspond to model fit for each synthetic data generating scenario. Note that smaller indicates better fit.[]{data-label="waicDRP"}](drp5WAIC4.pdf)
The primary purpose of Figure \[waicGP\] is to compare model fit from the spatio-temporal partition model we develop to that from the linear DDPM (model 5). It appears that all methods are competitive to the linear DDPM, which is particularly true with 10 time points. Thus, our dependent partition model accommodates temporal dependence more efficiently relative to the linear DDPM under this data generating scenario. Note that regardless of the number of time points, model 1b ($stRPM(\alpha, \nu_0, M)$ with $\nu_0 = 20$) appears to perform best. Surprisingly, $tRPM(\alpha, M)$ (model 4) is quite competitive, particularly with 10 time points. The conclusion here is that employing $stRPM(\alpha, \nu_0, M)$ to model partitions appears to accommodate spatio-temporal dependence even if there is no underlying partition structure.
From Figure \[waicDRP\] we see that when partitions are generated independently, there is very little lost by employing the dependent joint model in terms of model fit (see top left panel for model 3 and 4). However, as spatial and/or temporal structure is introduced in the partition model, there are clear gains in terms of model fit when employing $tRPM(\bm{\alpha}, M)$ and/or $stRPM(\bm{\alpha}, \nu_0,M)$. From this simulation it seems that employing the $tRPM(\alpha, M)$ regardless of the strength of temporal dependence among partitions is reasonable as there is minimal cost in terms of model fit even when partitions are generated independently. Finally, it appears that $stRPM(\bm{\alpha}, \nu_0,M)$ performed best.
Application {#applications}
-----------
In this section we apply our method to a real-world data set coming from the field of environmental science. A second application in educational measurement is provided in Section \[SIMCE.application\] of the online Supplementary Material. As mentioned previously, once a partition model is specified there is quite a bit of flexibility regarding how (or if) temporal dependence is incorporated in other parts of a hierarchical model. To illustrate this, we incorporate temporal dependence in three places of the hierarchical model we construct.
As part of preliminary exploratory data analysis (not shown), we examined serial dependence for each experimental (monitoring station), and concluded that they all exhibited a particular type of temporal dependence. Because of this, we introduce a unit-specific temporal dependence parameter $|\eta_{1i}| \le 1$ and model observations from a single unit over time ($Y_{1i}, \ldots, Y_{iT}$) with an AR(1) structure. In addition, motivated by a desire for parsimony, we employed a Laplace prior for $\eta_{1i}$. Finally, to permit the temporal dependence in the partition model to propagate through the hierarchical model, we model $\theta_t$ with an AR(1) structure. The full hierarchical model is detailed in . $$\label{FullModel}
\begin{aligned}
Y_{it} | Y_{i t-1}, \bm{\mu}^{\star}_{t},\bm{\sigma}^{2\star}_t, \bm{\eta}, \bm{c}_{t} & \stackrel{ind}{\sim} N(\mu^{\star}_{c_{it}t} + \eta_{1i}Y_{i t-1},\sigma_{c_{it}t}^{2\star}(1-\eta_{1i}^2)), \\
Y_{i1} & \stackrel{ind}{\sim} N(\mu^{\star}_{c_{i1}1}, \sigma_{c_{i1}1}^{2\star}),\\
\xi_i = \mbox{Logit}(0.5(\eta_{1i} + 1)) & \stackrel{iid}{\sim} Laplace(a,b), \\
(\mu_{jt}^{\star}, \sigma^{\star}_{jt}) & \stackrel{ind}{\sim} N(\theta_t, \tau^2) \times UN(0,A_{\sigma}), \\
\theta_t | \theta_{t-1} & \stackrel{ind}{\sim} N(\phi_0 + \phi_1\theta_{t-1}, \lambda^2(1-\phi_1^2)), \\
(\theta_1, \tau) & \sim N(\phi_0, \lambda^2) \times UN(0,A_{\tau}),\\
(\phi_0, \phi_1, \lambda) & \sim N(0, s^2) \times UN(-1,1) \times UN(0,A_{\lambda}), \\
\{\bm{c}_{t}, \ldots, \bm{c}_{T}\} & \sim stRPM(\bm{\alpha}, \nu_0, M), \ \mbox{with $\alpha_t \stackrel{iid}{\sim} Beta(a_{\alpha}, b_{\alpha})$},
\end{aligned}$$ where all Roman letters correspond to parameters that are user supplied. Notice that there are a number of special cases embedded in our hierarchical model. For example, $\eta_{i1} = 0$ for all $i$ results in conditionally independent observations. Further, $\phi_1 = 0$ results in independent atoms and $\alpha_t = 0$ for all $t$ in independent partitions over time. Note that model used in the simulation studies is a special case of ($\phi_1 = 0$ and $\eta_{i1} = 0$ for all $i$). $A_{\sigma}$ may influence partition formation. If this value is selected to be too large, then all observational units could plausibly be allocated to one cluster. If it is too small then many spurious clusters could potentially be formed. Therefore, this parameter must be selected thoughtfully. Our approach is to set $A_{\sigma}$ to about half the sample standard deviation computed using all observations.
Rural Background PM$_{10}$ Data Application {#pm10.application}
-------------------------------------------
The rural background PM$_{10}$ data is taken from the European air quality database. These data are comprised of the daily measurements of particulate matter with a diameter less than 10 $\mu$m from rural background stations in Germany and are publicly available in the [gstat]{} package (@gstat:2016) found on CRAN in [R]{} (@R:2018). We focus on average monthly PM$_{10}$ measures from the year 2005. Of the 69 stations, 9 were removed because of missing values. We fit the hierarchical model to these data and consider all the possible special cases (i.e., $\eta_{1i} = 0$ or not, $\phi_1 = 0$ or not, $\alpha_t = 0$ or not, with and without space). This resulted in 16 total models that were fit by collecting 1,000 MCMC iterates after discarding the first 10,000 as burn-in and thinning by 10. The prior values employed were $A_{\sigma} = A_{\tau} = 5$, $s^2 = 100$, $a = 0$, $b=1$, $a_{\alpha} = b_{\alpha} = 1$, and $\nu_0 = 5$. The WAIC and log pseudo marginal likelihood (LPML) for each model are presented in Table \[PM10\].
Notice that among all the model fits, employing a variant of $tRPM(\bm{\alpha}, M)$ (i.e., rows with “Yes” in the “In Partition” column) generally improves model fit. The best performing model in terms of WAIC and LPML includes spatio-temporal dependence in the partition model, temporal dependence among the atoms, and temporal dependence in the likelihood. To see how the different models impact how partitions evolve over time, we provide Figure \[LaggedARI\]. This figure displays the lagged ARI values for each of the 16 models. Notice that when partitions are modeled independently (first or third rows of Figure \[LaggedARI\]) then partitions evolve over time quite erratically in the sense that the cluster configuration can change dramatically from one time point to the next. However, when employing $tRPM(\bm{\alpha}, M)$ (second row of Figure \[LaggedARI\]) the partitions seemed to evolve much more “smoothly” as there is less drastic changes in cluster configuration. Finally, it appears that employing the $stRPM(\bm{\alpha}, \nu_0, M)$ (fourth row of Figure \[LaggedARI\]) not only produces partitions that evolve “smoothly” over time, but the temporal dependence seems to decay quicker than when employing $tRPM(\bm{\alpha}, M)$ only. In fact the model that produces the best model fit metrics (right most plot of the bottom row) seems to produce partitions that change quite gently over time as desired.
-------------------------------------------- ------------ ------- ------- ------ --------------- --------------
(r)[1-3]{} (r)[4-5]{} (r)[6-7]{} Partition Likelihood Atoms LPML WAIC LPML WAIC
No No No -1973 3464 -1904 3655
No No Yes -1973 3467 -1899 3653
No Yes No -1762 3071 -1562 3125
No Yes Yes -1770 3070 -1560 3170
Yes No No -1639 3226 -1613 3003
Yes No Yes -1618 3120 -1579 3015
Yes Yes No -1758 3153 -3240 3014
Yes Yes Yes -1590 3016 [**-1535**]{} [**2911**]{}
-------------------------------------------- ------------ ------- ------- ------ --------------- --------------
: PM$_{10}$ data: Results of model fitting. The bold font identifies best model fits in terms of LPML and WAIC. Higher values for LPML indicate better fit while lower values for WAIC indicate better fit.
\[PM10\]
![PM$_{10}$ data. Each figure is a summary of the lagged $ARI$ values corresponding to the 16 models in Table \[PM10\].[]{data-label="LaggedARI"}](LaggedARI.pdf)
Conclusions
===========
We developed a joint probability model for a sequence of partitions that explicitly considers temporal dependence among the partitions. Further we showed that our methodology is capable of accommodating partitions that evolve slowly over time in that the adjusted Rand index between estimated partitions decays as the lag in time increases. Further, we showed that in the absence of temporal dependence between partitions, the cost in terms of model fit is minimal.
Even though our main focus is constructing a dependent probability model for a sequence of random partitions, our method, when coupled with a simple hierarchical model, could provide an alternative approach to general space-time modeling that completely avoids inverting matrices. This could result in computation gains compared to employing computationally intense non-separable covariance functions. In addition, assumptions associated with stationarity and/or isotropy can be avoided.
The predictive nature of the spatio-temporal prior on a sequence of random partitions we have presented has a (first-order) Markovian structure. Various extensions can be considered, such as adding higher order dependence across time or dependence in baseline covariates. All of these cases would build on our constructive definition, as extra refinements of the basic idea of carrying smooth transitions on time and space. The Markovian approach can also be used for predictive inference, although that was not our main motivation for the models implemented here, and therefore we have not explored this avenue.
Proof of Proposition 2.1
========================
For clarity, here we introduce notation that highlights the dependence of partitions on sample size. For example, $\rho_{t,m} = (S_{1,t},\ldots,S_{k_t(m), t })$ and $[m]=\{1,\ldots,m\}$. By assumption $\text{Pr}(\rho_{1,m})$ is specified by means of an EPPF which we now construct. Denote $\mathbb{N}^*=\cup_{k=0}^{\infty}\mathbb{N}^k$, and identify any $\bm{n}=(n_1,\ldots,n_k)\in\mathbb{N}^*$ with the infinite sequence $(n_1,\ldots,n_k,0,0,\ldots)$. Given $\bm{n}\in\mathbb{N}^*$, let $k(\bm{n})$ denote the number of non-zero entries in $\bm{n}$ and denote by $\bm{n}^{j+}$ the result of incrementing $\bm{n}$’s $j$th component (i.e., $n_j$) by 1, with $1\le j\le k(\bm{n})+1$. An EPPF is then any function $r:\mathbb{N}^* \longrightarrow [0,1]$ that is symmetric in its arguments and where $$\label{eq:EPPF}
r(1)=1\qquad\mbox{and}\qquad r(\bm{n})=\sum_{j=1}^{k(\bm{n})+1}r(\bm{n}^{j+})
\qquad \mbox{for all $\bm{n}\in\mathbb{N}^*$.}$$ Condition implies that a EPPF is sample size consistent, i.e., marginalizing the $(n+1)$st element leads to the model for $n$ elements. The EPPF also implies exchangeability of configurations in the sense that a EPPF is invariant under permutations of the elements that keep the cluster sizes unaltered. We also note that any valid EPPF defines a predictive rule of the form $$\label{eq:PPF}
r_j(\bm{n})=\frac{r(\bm{n}^{j+})}{r(\bm{n})},\qquad\mbox{for $1\le j\le k(\bm{n})+1$,}$$ where it is assumed that $r(\bm{n})>0$ and $r_j(\bm{n})$ represents the probability of a new element joining the $j$th already existing cluster, for $1\le j\le k(\bm{n})$, or starting a new one (the $k(\bm{n})+1$). The one-step rule can also be extended to predictions of two or more elements by simply iterating the one-step rule as many times as needed. Now, given an EPPF $r$, we have that $$\label{eq:priorrho1}
\text{Pr}(\rho_{1,m}=(S_{1,1},\ldots,S_{k_1(m),1}))=r(n_{1,1},\ldots,n_{k_1(m),1}).$$
To prove the result, it suffices to show that it holds for $\rho_{2,m}$ and then by induction the result holds generally. Denote by $[\Gamma]=\{i\in\{1,\ldots,m\}: \,\gamma_{i2}=0\}$ the (random) set of elements removed from $\rho_{1,m}$. Then, $\rho_{1,m}^{-N_{02}}$ is a partition of the elements of $[m]-[\Gamma]$ (where as before $N_{02} = \sum_{j = 1}^m I[\gamma_{j2} = 0]$). By exchangeability and the fact that an EPPF is sample size consistent, we have that for any partition $S^-_1,\ldots,S^-_{k([m]-[\Gamma])}$ of $[m]-[\Gamma]$: $$\begin{aligned}
\text{Pr}(\rho_{2,m}^{-N_{02}}=(S^-_1,\ldots,S^-_{k([m]-[\Gamma])})\mid [\Gamma]) & = \text{Pr}(\rho_{1,m}^{-N_{02}}=(S^-_1,\ldots,S^-_{k([m]-[\Gamma])})\mid [\Gamma])\\
& =r(|S^-_1|,\ldots,|S^-_{k([m]-[\Gamma])}|),\end{aligned}$$ where $|S_j|$ is the number of elements in $S_j$. In addition, and again by exchangeability and sample size consistency, the predictive rule starting from $[m]-[\Gamma]$ (or from any subset of $[m]$ for that matter) depends only on the sizes of the subsets in that partition. Thus, conditioning on all reallocation configurations and initial partition after subject removal we have: $$\begin{aligned}
\begin{split}
\text{Pr}(\rho_{2,m}=(S_1,\ldots,S_k))&= \sum_{[\Gamma]} \sum_{\rho_{2,m}^{-N_{02}}} \text{Pr}(\rho_{2,m}=(S_1,\ldots,S_k)\mid[\Gamma],\,\rho_{2,m}^{-N_{02}}) \times \\
& \qquad \qquad \qquad \text{Pr}(\rho_{2,m}^{-N_{02}}\mid [\Gamma])\text{Pr}([\Gamma]),
\end{split} \\
\begin{split}
& =\sum_{[\Gamma]} \sum_{\rho_{1,m}^{-N_{02}}} \text{Pr}(\rho_{1,m}=(S_1,\ldots,S_k)\mid[\Gamma],\,\rho_{1,m}^{-N_{02}}) \times \\
& \qquad \qquad \qquad \text{Pr}(\rho_{1,m}^{-N_{02}}\mid [\Gamma]) \text{Pr}([\Gamma]),
\end{split} \\
& =\text{Pr}(\rho_{1,m}=(S_1,\ldots,S_k)),\end{aligned}$$ where the second to last equality follows from the constructive description given earlier and the properties of the EPPF. The result then follows.
Proof of Proposition 2.2
========================
Let $P_{C_t} = \{\rho_t \in P : \rho_t \asymp \rho_{t-1}\}$ denote the collection of all partitions of the elements of $[m]$ at time $t$ that are compatible with $\rho_{t-1}$ based on $\bm{\gamma}_t$. Then by construction, $\text{Pr}(\rho_t | \bm{\gamma}_t, \rho_{t-1})$ is a random partition distribution whose support is $P_{C_t}$ so that $$\begin{aligned}
\text{Pr}(\rho_t = \lambda | \bm{\gamma}_t, \rho_{t-1}) = \displaystyle \frac{\text{Pr}(\rho_t = \lambda) I[\lambda \in P_{C_t}]}{\sum_{\lambda} \text{Pr}(\rho_t = \lambda)I[\lambda \in P_{C_t}]}.\end{aligned}$$ It only remains to show that $\sum_{\lambda \in P_{C_t}} \text{Pr}(\rho_t = \lambda) = \text{Pr}(\rho_t^{-N_{0t}})$ which is more easily seen employing cluster label notation. Let $c_{\gamma_t} = \{ c_{it} : \gamma_{it = 0}\}$. By iteratively invoking the sample size consistency property we have that $$\begin{aligned}
\text{Pr}(\rho_t^{-N_{0t}}) & = \sum_{c_{\gamma_t}} \text{Pr}(\rho_t = \{c_{1t}, \ldots, c_{mt}\}) \\
& = \sum_{\lambda \in P_{C_t}} \text{Pr}(\rho_t = \lambda ),\end{aligned}$$ where the last equality holds since summing over $c_{\gamma_t}$ is based only on cluster labels that are not fixed from time point $t-1$ to $t$ which results in summing over all possible compatible partitions (i.e., $\lambda \in P_{C_t}$).
[**SUPPLEMENTARY MATERIAL**]{}
Title:
: Supplementary Material. This file contains details of our model applied to an additional application in the field of education.
R-package for the $stRPM$ routine:
: An R-package titled [drpm]{} contains code used to fit model described in .
[^1]: Partially supported by grant FONDECYT 1180034 and by Iniciativa Científica Milenio - Minecon Núcleo Milenio MiDaS
|
---
author:
- Shiyin Shen
- 'M. Argudo-Fernández'
- Li Chen
- Xiaoyan Chen
- Shuai Feng
- Jinliang Hou
- Yonghui Hou
- Peng Jiang
- Yipeng Jing
- Xu Kong
- Ali Luo
- Zhijian Luo
- Zhengyi Shao
- Tinggui Wang
- Wenting Wang
- Yuefei Wang
- Hong Wu
- 'Xue-Bing Wu'
- Haifeng Yang
- Ming Yang
- Fangting Yuan
- Hailong Yuan
- Haotong Zhang
- Jiannan Zhang
- Yong Zhang
- Jing Zhong
title: A sample of galaxy pairs identified from the LAMOST spectral survey and the Sloan Digital Sky Survey
---
Introduction {#sect:intro}
============
In the standard hierarchical structure formation model, galaxies are built up through merging processes. Numerical simulations show that the galaxy mergers can trigger the star burst, feed the central super-massive black hole and transform the galaxy morphology[@Springel05]. These different physical processes take places at different stages of galaxy merging. At early stage, as two galaxies approaching, they start to have interactions on each other out to a distance of about 100 kpc. After the first passage, the galaxies start to show strong tidal tails and undergo star bursts. After few times of passages, the galaxies quickly evolve into final coalescence. The whole time scale of the merging progress takes bout $1-2$ Gyr[@Torrey12].
In observation, the processes of galaxy merging have been probed by statistical studies of galaxy pairs as function of their separation, stellar mass, morphology, mass ratio and many other parameters[e.g. @Nikolic04; @Ellison08; @Ellison13]. In such studies, a large and unbiased sample of galaxy pairs is crucial. By far, the largest low redshift galaxy pair sample is identified from the main galaxy(MG) sample of the Sloan Digital Sky Survey[SDSS, @York00], which is a spectroscopic survey of a magnitude-limited sample down to $r<17.77$[@EDR]. The spectroscopic completeness of the MG sample in SDSS is quite high [$\sim
90\%$, @Hogg04]. Based on the spectroscopic MGs in the final data release of the SDSS legacy survey[Data Release Seven, DR7 @DR7] , the number of galaxy pairs is over 10,000[e.g. @Ellison11]. Despite of this large number, the galaxy pair sample identified from the spectroscopic MGs is far from complete. The incompleteness of the galaxy pairs is mainly caused by the fiber collision effect in SDSS, which is a minimum separation of 55 arcsec between any two fibers for any given spectroscopic plate. As a result, the completeness of the galaxy pairs identified from the spectroscopic MGs is estimated to be only about 35 percent[@Patton08]. In other words, the SDSS missed galaxies caused by the fiber collision have a very high probability being in galaxy pairs. Therefore, spectroscopic targeting of the SDSS missed galaxies is an efficient way to identify new galaxy pairs. Only if all these SDSS missed galaxies could be targeted by a new spectroscopic survey, a complete and unbiased galaxy pair sample could be finally made. More importantly, such a sample would be a benchmark on the studies of small-scale environmental effect of low redshift galaxies.
In this manuscript, we introduce the project of observing the SDSS missed MGs with the Guo Shou Jing Telescope (also named as the Large Sky Area Multi-Object Fiber Spectroscopic Telescope - LAMOST)[^1] and present its early result: a new sample of galaxy pairs. This paper is organized as follows. In section 2, we introduce the project of the spectroscopic observation of the SDSS missed MGs with LAMOST. In section 3, we present a new galaxy pair sample using new redshifts from LAMOST survey. Finally, we make short discussions and give summary in Section 4.
LAMOST survey: complementary galaxy sample {#sect:Obs}
==========================================
LAMOST is a special quasi-meridian reflecting Schmidt telescope located at Xinglong Station of National Astronomical Observatory of China. The design of LAMOST provides an effective aperture about 4 meters, a diameter of $\sim 5^\circ$ field of view and a spectroscopy system with $\sim 4000$ fibers[@Wang96; @Su04; @Cui12]. After about one year of pilot survey [@pilot], the LAMOST regular spectral survey has started from September of 2012 and would last for 5 five years. An overview of the LAMOST spectral survey can be seen in @Zhao12. The LAMOST regular survey mainly focuses on Galactic stars, but also includes a significant fraction of extragalactic objects[e.g. @Huo13; @Shi14]. One of the sample of the extragalactic sources is the SDSS missed MGs , and is named as the complementary galaxy sample in the LAMOST survey[@LDR1].
the complementary galaxy sample
-------------------------------
![The SDSS spectroscopic MGs and complementary galaxy sample. Top: the $r$ band Petrosian magnitude distribution. The solid and dotted histograms show the distributions of the complementary galaxies and SDSS spectroscopic MGs respectively, while the dashed histogram shows the 3,456 complementary galaxies with spectra in LAMOST DR2(Section 2.2). Bottom: the angular separation to the nearest SDSS MG(complementary galaxy: solid, SDSS spectroscopic MG: dotted). All the histograms have been normalized to unit area. []{data-label="Comp"}](fig1.eps){width="120mm"}
The complementary galaxy sample is constructed from the catalog archive server of the SDSS legacy survey, where all the galaxies with $r$ band Petrosian magnitude (Galactic reddening corrected) brighter than $r=17.77$ and not yet with spectroscopic redshifts are selected. The footprint of the complementary galaxies in the LAMOST survey is restricted in the north Galactic cap region ($ -10 < \delta < 60$ deg and $b> 0$ deg). [^2] After that, we remove a small fraction of galaxy targets that might be contaminated by nearby bright stars using the spherical polygon masks in the NYU value added galaxy catalog [@VAGC]. The final number of the complementary galaxy sample in the input catalog of the LAMOST spectral survey is 66,263.
In SDSS DR7, the number of the MGs that have been targeted with spectroscopy in north Galactic cap is 639,428. That is to say, the fraction of SDSS missed MGs(66,263 of 705,691) is about 10 percent. In Fig. \[Comp\], we show the $r$ band Petrosian magnitude distributions of the complementary galaxy sample and the SDSS spectroscopic MGs respectively. As can be seen, the complementary galaxies have similar magnitude distribution as the SDSS spectroscopic MGs, but are slightly biased toward faint galaxies. Most of the SDSS missed MGs are due to the fiber collision effect, therefor they are expected to be biased toward the high density region. To show this effect quantitatively, we match each complementary galaxy to the global MG sample and obtain the projected distance $\theta_{\rm min}$(in unit of arcsec) to its nearest neighbor. For comparison, we also calculate $\theta_{\rm
min}$ for each SDSS spectroscopic MG. The two distributions of $\theta_{\rm min}$ are plotted in the bottom panel of Fig. \[Comp\]. Strong biases of two $\theta_{\rm min}$ distributions at 55 arcsec are clearly seen. This result not only shows that the complementary galaxies are biased to the high density environment, but also implies that the galaxy pair sample identified from the SDSS spectroscopic sample alone is far from complete[@Patton08].
To quantify the incompleteness of the galaxy pairs in SDSS, we select photometric galaxy pairs in the SDSS MG sample, which are defined as the galaxies and their nearest neighbors inside the radii of 100 arsec. We match photometric pairs from the SDSS spectroscopic MGs and from all the MGs respectively. The fraction of the galaxy pairs of these two samples gives us a estimation of the completeness of the galaxy pairs in SDSS spectroscopic MGs. We show the resulted completeness as function of the pair angular separation in the top panel of Fig. \[fcomp\]. Again, we see that a strong incompleteness jumps at the angular separation $\theta < 55$ arcsec, where the completeness is only about 30 percent and even decreases with the decreasing of $\theta$. In the bottom panel, we show the completeness of the galaxy pairs as function of the redshift. Here, we have defined the galaxy pairs as these with projected separation ${r_{\rm p}}< 100
{\,h_{70}^{-1} {\rm kpc}}$(see Section 3) and assumed that the separation follows a random distribution. In this case, the galaxy pairs in SDSS spectroscopic MGs decreases with increasing of redshift and reaches a plateau $\sim 30$ percent at $z>0.09$ where the 55 arcsec limit corresponds to a projected distance 100${\,h_{70}^{-1} {\rm kpc}}$. Since the peak redshift of the SDSS MGs is at $z\sim0.1$, the global completeness of the galaxy pairs in SDSS spectroscopic MG sample is less than 40 percent.
![The completeness of galaxy pairs in SDSS spectroscopic MGs. The top panel shows the completeness as function of the angular separation of the pair members. The bottom panel shows the completeness of the galaxy pairs (defined as these with projected distance${r_{\rm p}}< 100{\,h_{70}^{-1} {\rm kpc}}$) as function of redshift. []{data-label="fcomp"}](fig2.eps){width="120mm"}
LAMOST observation
------------------
The complementary galaxies are mixed together with other LAMOST targets(most of them are Galactic stars) and then compiled into the LAMOST survey plates. In each plate, the number density of the complementary galaxies is very low, which are therefore assigned fibers with higher priority than stars. In the LAMOST survey, the input sources are tiled into three different types of plates, the bright(B), medium(M) and faint(F) plates, which are designed to reach the average signal-to-noise ratio ($S/N\sim10$) for the objects down to the magnitude limits $r<16.5,r<17.8$ and $r<18.5$ respectively. Most of the complementary galaxies have their magnitudes in the range $16.5<r<17.8$ (Fig. \[Comp\]) and therefor are mainly tiled into the M plates. On the other hand, due to the limited number of the dark nights, the observing time allocated for the M plates is quite few. Because of that, only a small fraction of the complementary galaxy sample have been targeted yet.
The spectroscopic data used in this study is from the LAMOST Data Release 2 (DR2) , i.e the data till June 2014 . In LAMOST DR2, there are 3,456 complementary galaxies that have been targeted with spectroscopy and with spectra released. The magnitude distribution of these 3,456 LAMOST targeted galaxies is plotted as the dashed histogram in the top panel of Fig. \[Comp\]. Compared with the input of the 66,263 complementary galaxies, the LAMOST targets are evidently biased toward bright galaxies. There are two reasons for this bias. First, some of the bright complementary galaxies($r<16.5$) are compiled into the target list of LAMOST B plates, which far outnumbers the M plates. The other reason for the bias against the faint galaxies is the failure of the spectroscopies because of their low $S/N$(see next section).
redshift measurements
---------------------
![The signal-to-noise ratio($S/N$) of the LAMOST spectra of the complementary galaxies. The top panel shows the magnitude as function of $S/N$, while the bottom panel shows the histograms of $S/N$. Top: the small dots represent all 3,456 complementary galaxies. The squares with error bars show the median and 16/84 percentiles of the $S/N$ distribution in magnitude bins. Bottom: the solid histogram shows all the 3,456 complementary galaxies, while the dashed and dotted histograms show the sub-samples of galaxies with redshifts measured from the PCA algorithm and form the LAMOST 1-D pipeline respectively. The fractions of galaxies with redshift measured(right $y$-axis) for the spectra at different $S/N$ are shown by the dotted line(PCA algorithm) and dashed line(1-D pipeline).[]{data-label="SNR"}](fig3.eps){width="120mm"}
In the catalog of LAMOST DR2, only 1,951 of the 3456 complementary galaxies have redshifts measured from the LAMOST 1-D pipeline and published in the LAMOST catalog. The failure of the redshift measurements is mainly due to the low $S/N$ of the LAMOST spectra of the faint galaxies($r>16.5$). We show the mean $S/N$ in the $r$-band wavelength range against the $r$-band magnitude of the 3,456 LAMOST targeted galaxies in the top panel of Fig. \[SNR\]. The distribution of the $S/N$ is shown in the bottom panel. As can be seen, when $r>16.5$, the median $S/N$ of the LAMOST spectra becomes lower than 10, which makes the number of the spectra with LAMOST catalog $z$ decreases significantly (see dotted histogram in the bottom panel of Fig. \[SNR\]).
To further improve the successful rate of the redshift measurement, we developed an independent PCA redshift measurement algorithm following the pipeline used for the ‘CMASS’ galaxies in the Baryon Oscillation Spectroscopic Survey of the SDSS III[@Bolton12]. The detail of the pipeline will be presented in an upcoming paper. Here, we outline the basic routines in Appendix A. With this algorithm, we obtained 2,796 redshift measurements. In Fig. \[SNR\], the dashed line shows the fraction of the spectra with PCA redshifts measured. It is clearly that our new PCA algorithm makes a significant improvement on the redshift measurement of the LAMOST spectra, especially at the low $S/N$ end.
![Number histogram of the redshift differences (in terms of recessional velocity) between the LAMOST and SDSS measurements. The solid histogram shows the differences between the SDSS redshifts and our PCA redshift measurements, while the dotted histogram represents the differences between the SDSS redshifts and and LAMOST 1-D redshifts. []{data-label="LSz"}](fig4.eps){width="120mm"}
To quantify the redshift measurements from the LAMOST 1-D pipeline and our PCA algorithm, we compare their redshifts with other independent measurements. After the SDSS legacy survey, some of the SDSS missed MGs had been targeted by the BOSS spectroscopy in SDSS III[@BOSS] [^3]. We matched the 3,456 complementary galaxies with SDSS DR12 spectroscopic catalog and obtained 1,056 matches. For these 1,056 galaxies, the LAMOST catalog lists 604 redshifts while our PCA algorithm provides 923 redshift measurements. In Fig. \[LSz\], we show the histograms of the differences of the redshifts (in terms of recessional velocity difference $\Delta V$) of these galaxies with both LAMOST and SDSS redshifts. The LAOMST redshifts from 1-D pipeline and PCA algorithm both show good consistences with SDSS values. For the LAMOST catalog $z$, the standard deviation of $\Delta V$ is about 80${\,\rm{km\,s^{-1}}}$. For the PCA $z$, the scatter of $\Delta V$ is even smaller, $\sim 58{\,\rm{km\,s^{-1}}}$. Given the better consistence with the SDSS redshifts of the PCA redshift measurements, we use the PCA redshifts for these galaxies with both PCA redshifts and catalog redshifs. For the 1,056 galaxies with SDSS redshifts, we take their redshifts from SDSS catalog. As we will show in the next section, the criterion of the velocity difference we adopted to identify galaxy pairs is ${|\Delta V|}< 500
{\,\rm{km\,s^{-1}}}$. Therefore, the scatter between the SDSS and LAMOST redshift measurements have few impacts on the pair identification.
the galaxy pair sample
======================
![Basic statistical properties of the 1,102 galaxy pairs identified from the LAMOST complementary galaxies and SDSS MGs(solid histograms). The statistical properties of the SDSS selected galaxy pairs are plotted as the dotted histograms in each panel for comparison (normalized to 1,102). The top left, top right, bottom left and bottom right panels show the histograms of the angular separation, redshift, projected distance and recessional velocity differences respectively. In the bottom right panel, the dotted curve shows a Gaussian distribution function with a standard deviation $218 {\,\rm{km\,s^{-1}}}$.[]{data-label="hBasic"}](fig5.eps){width="120mm"}
In this section, we combine the redshifts of the complementary galaxies with the SDSS spectroscopic MGs to identify new galaxy pairs. For the 3,456 complementary galaxies, we have obtained 3,137 redshifts. Among them, 1,056 redshifts come from SDSS DR12, 1906 from PCA algorithm, 175 from LAMOST 1-D pipeline.
In observation, a galaxy pair is typically defined from the projected distance ${r_{\rm p}}$ and recessional velocity difference ${|\Delta V|}$ of two neighboring galaxies. However, there is no consensus on the critical values of ${r_{\rm p}}$ and ${|\Delta V|}$. For example, both based on SDSS DR7, @Liu11 defined an AGN pair sample with ${|\Delta V|}< 600 {\,\rm{km\,s^{-1}}}$ and ${r_{\rm p}}< 100{\,h_{70}^{-1} {\rm kpc}}$, whereas @Patton11 searched galaxy pairs using ${|\Delta V|}< 1,000{\,\rm{km\,s^{-1}}}$ and ${r_{\rm p}}< 80{\,h_{70}^{-1} {\rm kpc}}$[see also @Scudder12; @Maria15]. For the projected distance ${r_{\rm p}}$, there are evidences that galaxies show interactions on their neighbors at ${r_{\rm p}}>
80{\,h_{70}^{-1} {\rm kpc}}$ [@Scudder12]. In this study, we set a critical value ${r_{\rm p}}< 100 {\,h_{70}^{-1} {\rm kpc}}$. For $\Delta V$, a large critical value (for example, ${|\Delta V|}<1,000 {\,\rm{km\,s^{-1}}}$) might introduce a significant fraction of contamination from the high density environment(e.g. galaxy groups and clusters). In this study, we select galaxy pairs using ${|\Delta V|}< 500 {\,\rm{km\,s^{-1}}}$ (see Appendix B for more discussions).
We match the redshifts of 3,137 complementary galaxies with the spectroscopic MGs in SDSS DR7 using the criteria ${r_{\rm p}}< 100 {\,h_{70}^{-1} {\rm kpc}}$ and ${|\Delta V|}< 500 {\,\rm{km\,s^{-1}}}$ and obtain 1,141 galaxy pair candidates. In a few cases, a complementary galaxy may have more than one SDSS spectroscopic MGs matched. In this case, we choose the galaxy with the smallest ${r_{\rm p}}$ as the matched pair member and mark this pair being in a multiple system. We will come to the multiple systems in Section 3.1. Moreover, to have a better quality control to the pair sample, we make visual inspections on the SDSS images for all the pair candidates. In a few cases, the imperfect SDSS pipeline de-blends big galaxies into several small children. This effect results in 39 fake pairs. Thus, our final sample include 1,102 galaxy pairs.
We show the histograms of the ${r_{\rm p}}$ and ${|\Delta V|}$ of the final 1,102 galaxy pairs in the bottom two panels of Fig. \[hBasic\], where the distributions of the angular separation $\Theta$ (in acrsec) between the pair members and their average redshifts are shown in the top two panels. To have a better understanding of the statistical properties of the new pair sample, we show the distributions of the SDSS only pairs as the dotted histograms in each panel of Fig. \[hBasic\] for comparison. The SDSS only pairs are selected from the SDSS DR7 group catalog of @Yang08 using the same criteria above. The number of the SDSS pairs is 16,973. Because of its large number, we have not made visual inspections on this sample. However, according to the fact that there are only 39 of 1,141 LAMOST-SDSS pairs are fake, we expect the impact of the fake pairs to its statistical properties should be quite small. For convenience, we abbreviate the LAMOST-SDSS pairs as the LS pairs and the SDSS only pairs as SS pairs below.
As can be seen from Fig. \[hBasic\], except the distribution of ${|\Delta V|}$, the LS pairs show significant difference from the SS pairs. The angular separation of the SS pairs is clearly biased toward large values($\Theta > 55$ arcsec, top left panel). This, as we mentioned, is because of the 55 arcsec fiber collision effect in SDSS. The SS pairs are also biased toward lower redshifts(top right panel), which again is because of the fiber collision effect. Although the low $z$ selection effect compensates the high $\Theta$ bias, as a combination, the SS pairs are still biased toward large separation (${r_{\rm p}}$) ones (bottom left panel). Compared with the SS pair sample, our new LS pairs only increase the number of pairs by a few percent ($\sim 7\%$). However, for the close pairs with more significant interaction$({r_{\rm p}}< 30{\,h_{70}^{-1} {\rm kpc}})$, the number of the LS pairs enlarges the SS pairs by more than 14 percent(358 versus 2527). Therefor, the LS pair sample could be a useful supplement to the current SS pair sample, especially for the close pairs, which are valuable in the statistical studies of galaxy interaction and merging .
For the velocity difference (bottom right panel), the standard deviation of the LS and SS pairs is 218 and 198 ${\,\rm{km\,s^{-1}}}$ respectively. Considering the fact that the dispersion of the redshift differences between the LAMOST and SDSS measurements could be as large as $\sim 80{\,\rm{km\,s^{-1}}}$(Fig. \[LSz\]), these two results are consistent with each other. In this panel, we also plot a Gaussian distribution with standard deviation $\sigma=218{\,\rm{km\,s^{-1}}}$ for comparison. As we can see, the distribution of ${|\Delta V|}$ of LS pairs deviates from the Gaussian function, especially at the tails (see Appendix B for a more detailed discussion).
The catalog of the LS pairs are listed in Table 1. Besides basic parameters(e.g. Ra, Dec, redshift) for each pair member, we also list two extra flags(${\rm{M\_Flag}}$ and ${\rm{O\_Flag}}$) for each pair, which characterizes the multiplicity and the image overlapping of the pair members respectively. In the following two sub-sections, we make brief descriptions on these two flags.
---- ------------------- ------------------- --------- ------------------- ------------------- --------- ------------------ ------------------
ID $\alpha_1$\[deg\] $\delta_1$\[deg\] $z_1$ $\alpha_2$\[deg\] $\delta_2$\[deg\] $z_2$ ${\rm{M\_Flag}}$ ${\rm{O\_Flag}}$
1 112.23653 36.91987 0.06006 112.24269 36.91610 0.05988 0 0
2 114.54001 28.13682 0.07852 114.54659 28.12799 0.07942 0 0
3 116.04450 23.99016 0.07545 116.05284 23.99939 0.07533 0 0
4 116.46624 26.47179 0.12311 116.46654 26.47719 0.12373 0 0
5 119.86285 23.97182 0.09218 119.85204 23.98273 0.09318 0 0
6 120.09847 39.83033 0.01320 120.17009 39.87050 0.01326 1 0
7 120.40500 15.70968 0.01545 120.36157 15.74989 0.01637 1 0
8 120.83544 23.96435 0.05784 120.84409 23.96933 0.05725 0 0
9 120.89831 28.54465 0.14178 120.89262 28.55000 0.14096 0 0
10 121.02204 31.44029 0.07302 121.03594 31.43697 0.07303 0 0
11 121.37741 22.13299 0.13852 121.38057 22.12452 0.14003 0 0
12 123.44543 8.38181 0.11409 123.44723 8.38925 0.11324 0 0
13 123.98402 8.28538 0.14295 123.97318 8.28571 0.14438 0 0
14 124.05299 3.85966 0.08668 124.05083 3.85807 0.08777 0 1
15 124.46973 7.57755 0.12465 124.46754 7.57477 0.12484 0 0
---- ------------------- ------------------- --------- ------------------- ------------------- --------- ------------------ ------------------
: The catalog of galaxy pairs identified from the LAMOST complementary galaxies and SDSS MGs. For each pair, $\alpha_1,\delta_1,z_1$ are the Right Accession, Declination, redshift of the LAMOST complementary galaxy, while $\alpha_2,\delta_2,z_2$ are those of the SDSS galaxy. ${\rm{M\_Flag}}$ and ${\rm{O\_Flag}}$ are the multiplicity and overlapping flags (see Section 3.1 and 3.2 for detail). The table is sorted in ascending order of $\alpha_1$. The complete table is available on-line.[]{data-label="Tab_pair"}
multiple system
---------------
![Distribution of the the $g-r$ color of the pair members. The solid and dotted histograms (normalized to unit area) show the galaxies in the pairs with ${\rm{M\_Flag}}=0$ (dual) and ${\rm{M\_Flag}}=1$ (multiple) respectively.[]{data-label="Mflag"}](fig6.eps){width="120mm"}
Our galaxy pair sample is defined from two simple observational criteria (${r_{\rm p}}<100 {\,h_{70}^{-1} {\rm kpc}}$ and ${|\Delta V|}< 500 {\,\rm{km\,s^{-1}}}$) . Besides the pair members, the other neighboring galaxies have not been taken into consideration. However, to study the galaxy interaction and galaxy merging using galaxy pairs, it is better to refine a sample of $physical$ pairs, where the interaction between two pair members overtakes the effects from other neighbors[e.g. @Maria15]. For this purpose, we mark the galaxy pairs in multiple system with a multiple flag ‘${\rm{M\_Flag}}$’. Our aim is to remind that these pairs may not be suitable for studying galaxy interactions by only considering their members.
We define a galaxy pair in multiple system when either of its member has another main galaxy neighbor($r<17.77$) that also satisfies the pair definition(${r_{\rm p}}< 100 {\,h_{70}^{-1} {\rm kpc}}$ and ${|\Delta V|}< 500{\,\rm{km\,s^{-1}}}$) . Within current data, 197 of the 1,102 LS pairs are found in multiple systems. These pairs are marked with flag ${\rm{M\_Flag}}= 1$ in Table 1 and noted as ‘multiple’ pairs below. For comparison, the pairs with ${\rm{M\_Flag}}= 0$ are noted as ‘dual’ pairs.
To check the possible distinction between the multiple and dual pairs, we compare the $g-r$ color(taking from the SDSS model magnitudes and with $K$-correction applied) distributions of their members in Fig. \[Mflag\]. The galaxies in multiple pairs are averagely redder than in dual ones. In specific, the red galaxy fraction($g-r>0.7$) in the dual and multiple pairs are 0.52 and 0.66 respectively. This color bias is caused by the fact that the multiple pairs are biased toward high density environment (i.e. galaxy groups and clusters).
We remind that the ${\rm{M\_Flag}}$ set for the multiple systems is quite a preliminary parameter. Since the pairs are selected from a magnitude-limited sample, both of the pair sample and the multiplicity flag have strong redshift dependence. For example, a triplet system with $M_{{\rm r},i}=-20,-21,-22$ mag would be identified as a triplet, a pair and a single galaxy in the SDSS MG sample ($r<17.77$) at redshift $z=0.05,0.1,0.15$ respectively. On the other hand, the multiplicity flag is also not complete, which is because of the high incompleteness of the SDSS spectroscopic MGs at small scales and also because of the small fraction of the complementary galaxies been targeted by LAMOST yet. For the dual pairs with ${\rm{M\_Flag}}=0$, they are still possibly in multiple systems, i.e. having another companion galaxy but without redshift measured yet.[^4] That is to say, the fraction of the pairs in multiple system (197 of 1102) is actually a lower limit. Given the significant fraction of galaxy pairs in multiple system, the studies of galaxy interactions with galaxy pairs should take the multiplicity of the galaxy systems into consideration(Shen et al. 2015, in preparation).
overlapping pairs
-----------------
![Distribution of the $g-r$ color of the pair members. The solid and dotted histograms(normalized to unit area) show the galaxies in the pairs with ${\rm{O\_Flag}}=0$ and ${\rm{O\_Flag}}=1$ respectively.[]{data-label="Oflag"}](fig7.eps){width="120mm"}
The luminosity/mass ratio of the pair members plays an important role in galaxy merging[@Jiang14]. However, photometry of a very close galaxy pair is a nontrivial task[@Simard11]. During the visual inspection of the galaxy pairs, we noticed some of the galaxy pairs whose images of their members overlap each other significantly. We mark such pairs($N=216$) with a flag ${\rm{O\_Flag}}= 1$. For these overlapping pairs, the uncertainties of their photometry might introduce uncertainties and possible biases in the estimation of the stellar mass and mass ratio.
We show the $g-r$ color distribution of the overlapping pairs as the dotted histogram in Fig. \[Oflag\], where the distribution of the other pairs is shown as the solid histogram for comparison. As can be seen, although the overlapping pairs have much closer projected distance, their colors are biased toward redder values. The fraction of red galaxies($g-r>0.70$) in the overlapping pairs is 62 percent, which is significantly higher than the fraction 0.52 for the non-overlapping pairs. Unless there are big uncertainties in the photometry of these overlapping pairs, it is hard to explain the color bias we see in Fig. \[Oflag\]. Actually, @Patton11 have already shown that the poor photometry of SDSS official pipeline is largely responsible for the suspicious and large fraction of of extreme red galaxies(e.g. $g-r>0.9$) in the very close galaxy pairs. Therefore, to get a better estimation of the mass ratio of these overlapping pairs, more detailed photometry is required[e.g. @Simard11]. Here, we set this flag for a caution and leave the detailed photometry for a future work.
Conclusion
==========
In this paper, we presented the project of the spectroscopic survey of the SDSS missed MGs($r<17.77$) using LAMOST. The SDSS missed MGs are named as the complementary galaxy sample in the LAMOST survey. In the first two years of the LAMOST survey, due to the limited survey time of the medium(M) plates, only a small fraction (3,456 of 66,263) of the complementary galaxies had obtained LAMOST spectra. The majority of these spectra have quite low $S/N$, which are mostly due to the poor seeing condition. We developed a PCA algorithm to improve the redshift measurements of these low $S/N$ spectra. Together with the SDSS DR12 match and LAMOST 1-D pipeline results, we finally obtained 3,137 redshifts of the 3,456 complementary galaxies.
Considering the fact that the SDSS missed galaxies are mainly caused by the fiber collision effect, the spectroscopy of the complementary galaxies has a great potential in identifying new galaxy pairs. We present such a catalog of galaxy pairs identified from the first two years data of the LAMOST survey. From the redshifts of 3,137 complementary galaxies, we obtained a sample of 1,102 galaxy pairs after a careful visual inspection. Compared with the galaxy pairs selected from SDSS data only, our pair sample includes a larger fraction of close pairs(${r_{\rm p}}< 30{\,h_{70}^{-1} {\rm kpc}}$). Because of such advantages, our sample increased the current SDSS close pairs sample by about an amount of $\sim15$ percent. In common with other studies, our pairs are selected using two simple observational criteria(${r_{\rm p}}$ and ${|\Delta V|}$) . Whether they are physical bounding systems have not been taken into consideration. We find that at least $\sim20$ percent of the pairs are actually located in multiple systems((${\rm{M\_Flag}}=1$) ). Moreover, during the visual inspection, we find, for about 20 percent the pairs, the images of their members overlap each other. Therefor, the photometry and stellar mass estimation of the galaxies in these overlapping pairs(${\rm{O\_Flag}}=1$) should be in caution.
With the acceleration of the LAMOST survey on the complementary galaxies in the current and future season(from Sep. of 2014), we expect that the completeness of the galaxy pairs will be further improved. Once the LAMOST survey had finished the spectroscopies of the complementary galaxy sample, the great progresses on the studies of galaxy pair, galaxy interactions, galaxy merging and the small-scale environmental effects of galaxies would be expected.
Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.
This work is supported by the “973 Program” 2014 CB845705, Strategic Priority Research Program “The Emergence of Cosmological Structures” of the Chinese Academy of Sciences (CAS; grant XDB09030200) and the National Natural Science Foundation of China (NSFC) with the Project Number of 11573050 and 11433003.
[30]{}
natexlab\#1[\#1]{}
, K. N., [Adelman-McCarthy]{}, J. K., [Ag[ü]{}eros]{}, M. A., [Allam]{}, S. S., [Allende Prieto]{}, C., [An]{}, D., [Anderson]{}, K. S. J., [Anderson]{}, S. F., [Annis]{}, J., [Bahcall]{}, N. A., & et al. 2009, , 182, 543
, M., [Verley]{}, S., [Bergond]{}, G., [Duarte Puertas]{}, S., [Ramos Carmona]{}, E., [Sabater]{}, J., [Fern[á]{}ndez Lorenzo]{}, M., [Espada]{}, D., [Sulentic]{}, J., [Ruiz]{}, J. E., & [Leon]{}, S. 2015, ArXiv e-prints
, M. R., [Schlegel]{}, D. J., [Strauss]{}, M. A., [Brinkmann]{}, J., [Finkbeiner]{}, D., [Fukugita]{}, M., [Gunn]{}, J. E., [Hogg]{}, D. W., [Ivezi[ć]{}]{}, [Ž]{}., [Knapp]{}, G. R., [Lupton]{}, R. H., [Munn]{}, J. A., [Schneider]{}, D. P., [Tegmark]{}, M., & [Zehavi]{}, I. 2005, , 129, 2562
, A. S., [Schlegel]{}, D. J., [Aubourg]{}, [É]{}., [Bailey]{}, S., [Bhardwaj]{}, V., [Brownstein]{}, J. R., [Burles]{}, S., [Chen]{}, Y.-M., [Dawson]{}, K., [Eisenstein]{}, D. J., [Gunn]{}, J. E., [Knapp]{}, G. R., [Loomis]{}, C. P., [Lupton]{}, R. H., [Maraston]{}, C., [Muna]{}, D., [Myers]{}, A. D., [Olmstead]{}, M. D., [Padmanabhan]{}, N., [P[â]{}ris]{}, I., [Percival]{}, W. J., [Petitjean]{}, P., [Rockosi]{}, C. M., [Ross]{}, N. P., [Schneider]{}, D. P., [Shu]{}, Y., [Strauss]{}, M. A., [Thomas]{}, D., [Tremonti]{}, C. A., [Wake]{}, D. A., [Weaver]{}, B. A., & [Wood-Vasey]{}, W. M. 2012, , 144, 144
, X.-Q. e. a. 2012, Research in Astronomy and Astrophysics, 12, 1197
, K. S., [Schlegel]{}, D. J., [Ahn]{}, C. P., [Anderson]{}, S. F., [Aubourg]{}, [É]{}., [Bailey]{}, S., [Barkhouser]{}, R. H., [Bautista]{}, J. E., [Beifiori]{}, A., [Berlind]{}, A. A., [Bhardwaj]{}, V., [Bizyaev]{}, D., [Blake]{}, C. H., [Blanton]{}, M. R., [Blomqvist]{}, M., [Bolton]{}, A. S., [Borde]{}, A., [Bovy]{}, J., [Brandt]{}, W. N., [Brewington]{}, H., [Brinkmann]{}, J., [Brown]{}, P. J., [Brownstein]{}, J. R., [Bundy]{}, K., [Busca]{}, N. G., [Carithers]{}, W., [Carnero]{}, A. R., [Carr]{}, M. A., [Chen]{}, Y., [Comparat]{}, J., [Connolly]{}, N., [Cope]{}, F., [Croft]{}, R. A. C., [Cuesta]{}, A. J., [da Costa]{}, L. N., [Davenport]{}, J. R. A., [Delubac]{}, T., [de Putter]{}, R., [Dhital]{}, S., [Ealet]{}, A., [Ebelke]{}, G. L., [Eisenstein]{}, D. J., [Escoffier]{}, S., [Fan]{}, X., [Filiz Ak]{}, N., [Finley]{}, H., [Font-Ribera]{}, A., [G[é]{}nova-Santos]{}, R., [Gunn]{}, J. E., [Guo]{}, H., [Haggard]{}, D., [Hall]{}, P. B., [Hamilton]{}, J.-C., [Harris]{}, B., [Harris]{}, D. W., [Ho]{}, S., [Hogg]{}, D. W., [Holder]{}, D., [Honscheid]{}, K., [Huehnerhoff]{}, J., [Jordan]{}, B., [Jordan]{}, W. P., [Kauffmann]{}, G., [Kazin]{}, E. A., [Kirkby]{}, D., [Klaene]{}, M. A., [Kneib]{}, J.-P., [Le Goff]{}, J.-M., [Lee]{}, K.-G., [Long]{}, D. C., [Loomis]{}, C. P., [Lundgren]{}, B., [Lupton]{}, R. H., [Maia]{}, M. A. G., [Makler]{}, M., [Malanushenko]{}, E., [Malanushenko]{}, V., [Mandelbaum]{}, R., [Manera]{}, M., [Maraston]{}, C., [Margala]{}, D., [Masters]{}, K. L., [McBride]{}, C. K., [McDonald]{}, P., [McGreer]{}, I. D., [McMahon]{}, R. G., [Mena]{}, O., [Miralda-Escud[é]{}]{}, J., [Montero-Dorta]{}, A. D., [Montesano]{}, F., [Muna]{}, D., [Myers]{}, A. D., [Naugle]{}, T., [Nichol]{}, R. C., [Noterdaeme]{}, P., [Nuza]{}, S. E., [Olmstead]{}, M. D., [Oravetz]{}, A., [Oravetz]{}, D. J., [Owen]{}, R., [Padmanabhan]{}, N., [Palanque-Delabrouille]{}, N., [Pan]{}, K., [Parejko]{}, J. K., [P[â]{}ris]{}, I., [Percival]{}, W. J., [P[é]{}rez-Fournon]{}, I., [P[é]{}rez-R[à]{}fols]{}, I., [Petitjean]{}, P., [Pfaffenberger]{}, R., [Pforr]{}, J., [Pieri]{}, M. M., [Prada]{}, F., [Price-Whelan]{}, A. M., [Raddick]{}, M. J., [Rebolo]{}, R., [Rich]{}, J., [Richards]{}, G. T., [Rockosi]{}, C. M., [Roe]{}, N. A., [Ross]{}, A. J., [Ross]{}, N. P., [Rossi]{}, G., [Rubi[ñ]{}o-Martin]{}, J. A., [Samushia]{}, L., [S[á]{}nchez]{}, A. G., [Sayres]{}, C., [Schmidt]{}, S. J., [Schneider]{}, D. P., [Sc[ó]{}ccola]{}, C. G., [Seo]{}, H.-J., [Shelden]{}, A., [Sheldon]{}, E., [Shen]{}, Y., [Shu]{}, Y., [Slosar]{}, A., [Smee]{}, S. A., [Snedden]{}, S. A., [Stauffer]{}, F., [Steele]{}, O., [Strauss]{}, M. A., [Streblyanska]{}, A., [Suzuki]{}, N., [Swanson]{}, M. E. C., [Tal]{}, T., [Tanaka]{}, M., [Thomas]{}, D., [Tinker]{}, J. L., [Tojeiro]{}, R., [Tremonti]{}, C. A., [Vargas Maga[ñ]{}a]{}, M., [Verde]{}, L., [Viel]{}, M., [Wake]{}, D. A., [Watson]{}, M., [Weaver]{}, B. A., [Weinberg]{}, D. H., [Weiner]{}, B. J., [West]{}, A. A., [White]{}, M., [Wood-Vasey]{}, W. M., [Yeche]{}, C., [Zehavi]{}, I., [Zhao]{}, G.-B., & [Zheng]{}, Z. 2013, , 145, 10
, A. & [Geller]{}, M. J. 1996, , 467, 19
, S. L., [Mendel]{}, J. T., [Patton]{}, D. R., & [Scudder]{}, J. M. 2013, , 435, 3627
, S. L., [Patton]{}, D. R., [Mendel]{}, J. T., & [Scudder]{}, J. M. 2011, , 418, 2043
, S. L., [Patton]{}, D. R., [Simard]{}, L., & [McConnachie]{}, A. W. 2008, , 135, 1877
, D. W., [Blanton]{}, M. R., [Brinchmann]{}, J., [Eisenstein]{}, D. J., [Schlegel]{}, D. J., [Gunn]{}, J. E., [McKay]{}, T. A., [Rix]{}, H.-W., [Bahcall]{}, N. A., [Brinkmann]{}, J., & [Meiksin]{}, A. 2004, , 601, L29
, Z.-Y. e. a. 2013, , 145, 159
, C. Y., [Jing]{}, Y. P., & [Han]{}, J. 2014, , 790, 7
, X., [Shen]{}, Y., [Strauss]{}, M. A., & [Hao]{}, L. 2011, , 737, 101
, A.-L. & et al. 2012, Research in Astronomy and Astrophysics, 12, 1243
, A.-L. & et al. 2015, Research in Astronomy and Astrophysics, 15, 1095
, B., [Cullen]{}, H., & [Alexander]{}, P. 2004, , 355, 874
, D. R. & [Atfield]{}, J. E. 2008, , 685, 235
, D. R., [Ellison]{}, S. L., [Simard]{}, L., [McConnachie]{}, A. W., & [Mendel]{}, J. T. 2011, , 412, 591
, J. M., [Ellison]{}, S. L., [Torrey]{}, P., [Patton]{}, D. R., & [Mendel]{}, J. T. 2012, , 426, 549
, R. K. 1996, , 279, 1310
, Z.-X., [Luo]{}, A.-L., [Comte]{}, G., [Chen]{}, X.-Y., [Wei]{}, P., [Zhao]{}, Y.-H., [Wu]{}, F.-C., [Zhang]{}, Y.-X., [Shen]{}, S.-Y., [Yang]{}, M., [Wu]{}, H., [Wu]{}, X.-B., [Zhang]{}, H.-T., [Lei]{}, Y.-J., [Zhang]{}, J.-N., [Wang]{}, T.-G., [Jin]{}, G., & [Zhang]{}, Y. 2014, Research in Astronomy and Astrophysics, 14, 1234
, L., [Mendel]{}, J. T., [Patton]{}, D. R., [Ellison]{}, S. L., & [McConnachie]{}, A. W. 2011, , 196, 11
, V., [Di Matteo]{}, T., & [Hernquist]{}, L. 2005, , 361, 776
, C., [Lupton]{}, R. H., [Bernardi]{}, M., [Blanton]{}, M. R., [Burles]{}, S., [Castander]{}, F. J., [Connolly]{}, A. J., [Eisenstein]{}, D. J., [Frieman]{}, J. A., [Hennessy]{}, G. S., [Hindsley]{}, R. B., [Ivezi[ć]{}]{}, [Ž]{}., [Kent]{}, S., [Kunszt]{}, P. Z., [Lee]{}, B. C., [Meiksin]{}, A., [Munn]{}, J. A., [Newberg]{}, H. J., [Nichol]{}, R. C., [Nicinski]{}, T., [Pier]{}, J. R., [Richards]{}, G. T., [Richmond]{}, M. W., [Schlegel]{}, D. J., [Smith]{}, J. A., & [Strauss]{}, M. A. 2002, , 123, 485
, D.-Q. & [Cui]{}, X.-Q. 2004, , 4, 1
, P., [Cox]{}, T. J., [Kewley]{}, L., & [Hernquist]{}, L. 2012, , 746, 108
, S.-G., [Su]{}, D.-Q., [Chu]{}, Y.-Q., [Cui]{}, X., & [Wang]{}, Y.-N. 1996, , 35, 5155
, X., [Mo]{}, H. J., & [van den Bosch]{}, F. C. 2008, , 676, 248
, D. G., [Adelman]{}, J., [Anderson]{}, Jr., J. E., [Anderson]{}, S. F., [Annis]{}, J., [Bahcall]{}, N. A., [Bakken]{}, J. A., [Barkhouser]{}, R., [Bastian]{}, S., [Berman]{}, E., [Boroski]{}, W. N., [Bracker]{}, S., [Briegel]{}, C., [Briggs]{}, J. W., [Brinkmann]{}, J., [Brunner]{}, R., [Burles]{}, S., [Carey]{}, L., [Carr]{}, M. A., [Castander]{}, F. J., [Chen]{}, B., [Colestock]{}, P. L., [Connolly]{}, A. J., [Crocker]{}, J. H., & [Csabai]{}, I. 2000, Astro. J., 120, 1579
, G., [Zhao]{}, Y.-H., [Chu]{}, Y.-Q., [Jing]{}, Y.-P., & [Deng]{}, L.-C. 2012, Research in Astronomy and Astrophysics, 12, 723
\[lastpage\]
redshift measurements with PCA eigen-templates
==============================================
We developed an independent algorithm to measure the redshifts of the LAMOST complementary galaxies using PCA eigen-templates and chi-squared fitting.
The eigen-templates are derived from the 1,056 complementary galaxies with DR12 redshifts. In specific, we first divide 1,056 galaxies into 20 $g-r$ color bins with similar numbers. Then, using the SDSS DR12 redshifts, we stack their LAMOST spectra and build the LAMOST composite spectra for galaxies in each $g-r$ bin. We make a PCA decomposition for these 20 composite spectra and find that the first 4 eigenvectors can recover most of the spectroscopic features.
We explore the redshift of each galaxy in the range $0.005<z<0.5$ by taking trial values that are moved with steps of each spectrum pixels. For each trial redshift, we fit the observed spectrum with the error-weighted least-square linear combination of the four ‘eigen-spectra‘ and a four-order poly-nominal. The poly-nominal is introduced to compensate the calibration of the LAMOST spectrum. The reduced $\chi^2$ value(the resulted $\chi^2$ divided by the number of fitting pixels) for each trial redshift define a $\chi^2(z)$ curve in the probed redshift range. The best redshift estimation is then defined by the minimal of the $\chi^2(z)$ curve. The error is evaluated at the location where the $\chi^2$ is increased by one at each side of the minimum values. During the fitting, besides the pixels with ANDMASK set, we also have masked the pixels at the wavelengths where the sky-subtraction residuals are 3 times more than the average noise.
The pair-wise peculiar velocity distribution of galaxy pairs
============================================================
The distribution of the recessional velocity difference between the galaxy pair members, also known as the pair-wise peculiar velocity distribution function(PVDF), plays a very important role in the large scale structure studies. On the non-linear scales($r<1$ Mpc), both observations and theoretical models suggest an exponential PVDF, which can be explained and approximated by a weighted integral of Gaussian distributions of subunits[e.g. @Diaferio96; @Sheth96]. However, the PVDF of the pairs in our analysis is on very small scale, even smaller than the viral radius of galaxies ($r<100$ kpc), which has not drawn much attention in related studies.
We show the pair-wise peculiar velocity($\Delta V$) distribution of the SDSS selected pairs in Fig. \[dV\]. The SDSS pairs are selected with criteria ${r_{\rm p}}<100$ kpc and ${|\Delta V|}<
800{\,\rm{km\,s^{-1}}}$. As we can see, the $\Delta V$ distribution shows long tails out to $\pm 800 {\,\rm{km\,s^{-1}}}$ and can not be fitted well by a Gaussian function. We first fit the observed $\Delta V$ distribution with the usually adopted exponential profile,
$$f(\Delta V)=\frac{1}{\sqrt{2}\sigma} {\rm exp} \left(-\frac{\sqrt{2}\Delta
V}{\sigma} \right)\,,$$
where $\sigma$ characterizes the pair-wise peculiar velocity dispersion. The best fitting of the exponential profile has $\sigma=324{\,\rm{km\,s^{-1}}}$ and is shown as the blue line in Fig. \[dV\]. Although better than the Gaussian profile fitting, the exponential profile still can not fit well the fine structure of the observed $\Delta V$ distribution, especially at ${|\Delta V|}> 200{\,\rm{km\,s^{-1}}}$ ranges.
Alternatively, as a preliminary test, we fit the observed $\Delta V$ distribution with a multi-component model. The observed $\Delta V$ distribution might be contributed by different components and each component corresponds to different physical circumstances. The ideal case of a galaxy pair is that the pair members form a gravitational bound system. In this case, $\Delta V$ represents the orbital velocity of the pair members. A more common case of the observed galaxy pair is that the galaxy pairs have other close neighbors. That is to say, the pair members and their neighbors together locate in a bigger galaxy system, e.g. galaxy groups and clusters. In this case, the pair members orbit individually in the gravitational potential of the host halo and $\Delta V$ represents its velocity dispersion. Finally, $\Delta V$ may be dominated by the difference of the Hubble flow. In this case, the radial distance of the members ($\sim$ Mpc) is much larger than the tangential distance ($<100$ kpc). That is to say, the ‘pair’ members actually have few physical interactions on each other and such pairs could be considered as contaminations from projection. Motivated by above scenario, we fit the observed $\Delta V$ distribution with three components, one narrow Gaussian profile(orbital velocities of ideal pairs), one broad Gaussian profile(velocity dispersion of host halos) and a flat component(projection contamination). The best fitting of this three-component model is shown as the solid line in Fig. \[dV\]. As we can see, this model makes a pretty good fitting to the observed $\Delta V$ distribution. The $\Delta V$ profiles of the three components are shown as the dotted lines in Fig. \[dV\]. The narrow Gaussian component has a standard deviation $\sim 110{\,\rm{km\,s^{-1}}}$ and contributes about 40 percent of the galaxy pairs with ${|\Delta V|}<800{\,\rm{km\,s^{-1}}}$. The broad component has a deviation $\sim 300{\,\rm{km\,s^{-1}}}$ and makes a half contribution. The constant component contributes the last 10 percent.
In Section 3.2, we select the LAMOST-SDSS galaxy pairs with criteria ${r_{\rm p}}<100$ kpc and ${|\Delta V|}< 500{\,\rm{km\,s^{-1}}}$. Our motivation of choosing a smaller critical value of ${|\Delta V|}$ is to reduce the contribution from galaxy groups and clusters, i.e. the broad component in Fig. \[dV\], while include all possible circumstances of ideal pairs. A more detailed study on the $\Delta V$ distribution of the galaxy pairs in different environments is in preparation.
![Recessional velocity difference $\Delta V$ distribution of the galaxy pairs in SDSS. The best exponential model fitting is shown as the blue line. The best three-component model fitting is shown as the red line, where the contributions of each component are shown as the dotted lines.[]{data-label="dV"}](figB1.eps){width="120mm"}
[^1]: http://www.lamost.org
[^2]: For the south Galactic region, since the SDSS MGs are only located in three stripes, the LAMOST survey includes another independent galaxy spectroscopic survey project, which aims to get the redshifts of of a magnitude-limited sample down to $r<18$.
[^3]: These galaxies typically have photo flag BRIGHT$\_$GAL in SDSS III.
[^4]: The pair with ID=52 in Table 1 is actually an isolated and compact galaxy triplet, which is found to be a triplet candidate during our visual inspection and then spectroscopically confirmed by a follow-up observation with the 2.16m telescope in the Xinglong Station(Feng et al. , 2015) .
|
---
abstract: 'Caldeira-Leggett model of reservoir is generalized to a reservoir modeled by a continuum of real Klein-Gordon fields, instead of harmonic oscillators. A quantum Langevin type dissipative equation is obtained for the scalar field. The susceptibility of the medium is defined in terms of the reservoir Green’s function and the coupling function satisfying causality condition. The connection between the coupling function and the susceptibility of the medium is found to be a Hankel transform from which the coupling function can be determined in terms of the susceptibility of the medium. Noise currents and their fluctuation-dissipation relation are obtained. In a homogeneous medium or reservoir, explicit form of the quantum scalar field, and its large-time limit, are found.'
author:
- 'A. Refaei'
- 'F. Kheirandish'
title: 'Dissipative noninteracting scalar field theory: A covariant formulation'
---
Introduction
============
Dissipative scalar field theories appear in a variety of problems in physics, for example in Casimir physics, a scalar field is the fluctuating field interacting linearly with some external matter fields defined inside or over some specific surfaces. In this case dissipation appears in the equation of motion of the scalar field trough the susceptibility of the medium. The scalar field or the main field, satisfies Dirichlet or Neumann boundary conditions on these surfaces causing fluctuating-induced forces among matter fields [@Casimir]. Another interesting example is the reheating of the universe after an inflationary epoch has passed [@Guth; @Brand].
A similar situation appears in the electromagnetic field quantization in the presence of metals or dielectrics where due to the specified geometry the electromagnetic field can be basically considered as two independent scalar fields. In this case one uses the Huttner-Barnett model [@Huttner; @Kheir] to quantize the electromagnetic field in the presence of matter fields modelled by continuum sets of harmonic oscillators modeling electric and magnetic properties of the medium. For a covariant approach to quantize the electromagnetic field in the presence of matter fields see [@Amoo].
In 1983, Caldeira and Leggett proposed a model to incorporate dissipation into the equation of motion based on a microscopic theory [@Caldeira]. They modeled the matter field or reservoir as a combination of non interacting harmonic oscillators with different mass and frequencies which were coupled to the main system linearly, knowing as system-reservoir model. In this model, by writing Heisenberg equations for both system and reservoir degrees of freedom and eliminating the reservoir degrees of freedom one finds a Langevin type equation for the main system including a noise field naturally induced from the vacuum fluctuations of the matter or reservoir system. This method has been applied to a variety of problems for example to the polaron motion in a crystal or the dynamics of Josephson junction [@Weiss] with successful experimental predictions [@Voss; @Martinis]. In Caldeira-Legget and Huttner-Barnett model of the matter fields, the Lagrangians associated with these matter fields are not Lorentz-Invariant and so the total Lagrangian density is not lorentz invariant.
In the present article, inspired by Caldeira-Leggett model, we present a covariant formulation of a non-interacting dissipative quantum scalar field by modeling the medium or reservoir as a continuum of Klein-Gordon fields instead of harmonic oscillators and find a quantum Langevin type dissipative equation for the scalar field. We follow a canonical approach to quantize the field-reservoir model though a path integral quantization of the system is also possible and may be more effective while considering interacting quantum field theories. The susceptibility of the medium is defined in terms of the reservoir Green function and coupling function satisfying causality condition. The connection between the coupling function and the susceptibility of the medium is found to be a Hankel transform. In the special case where the reservoir is assumed to be homogeneous, the explicit form of the quantum scalar field is obtained. Knowing the explicit form of the quantum field, one can proceed and construct Fock’s space or find important quantities like the two-point function or Feynman Green function. Including finite temperature corrections is also straightforward since the field is expressed in terms of the creation and annihilation operators of the medium which can be easily thermalized. A more realistic problem appears when one considers an interacting field theory where a path integral approach seems to be more effective, at least from perturbative point of view.
Lagrangian
==========
Throughout the article, without losing any generality, we assume that the fields are defined in $1+1$-dimensional space-time and use natural units $\hbar=c=1$, for notational convenience. For a real scalar field $\varphi(x,t)$, interacting with an environment defined by real scalar fields $Y_\omega (x,t)$, we assume the following covariant Lagrangian density for the field-environment system $$\begin{aligned}
\label{L}
\mathcal{L} &=& {{\frac{1}{2}}}\,{{\partial}}_\mu\,\varphi (x,t)\,{{\partial}}^\mu\,\varphi (x,t)-{{\frac{1}{2}}}\,m^2 \varphi^2 (x,t)\nonumber\\
&& + \,{{\frac{1}{2}}}\,\int_0^{\infty}d\omega\,[{{\partial}}_\mu\,Y_\omega (x,t) \,{{\partial}}^\mu\,Y_\omega (x,t)-\omega^2 Y^2_\omega (x,t)]\nonumber\\
&& +\int_0^{\infty}d\omega\,f(\omega)\,\varphi(x,t)\,Y_\omega(x,t),\end{aligned}$$ where $f(\omega)$ is a coupling function which is assumed to be homogeneous here, that is independent on space-time coordinates. In a nonhomogeneous medium it depends on space-time variables and should be considered as a scalar classical field to keep the total Lagrangian covariant. The difference between the Lagrangian density in Eq.(\[L\]) and in previous models [@Huttner; @Kheir], is the modified matter Lagrangian density, which is now assumed to be a continuum of $1+1$-dimensional real Klein-Gordon fields instead of a continuum of harmonic fields. Note that the index of continuity is the frequency parameter $\omega\in(0,\infty)$. From Euler-Lagrange equations, we find the classical equations of motion for the fields as $$\label{Efi}
(\partial^2+m^2)\,\varphi(x,t)=\int_0^\infty d\omega\,f(\omega)\,Y_\omega(x,t),$$ and $$\label{EY}
(\partial^2+\omega^2)\,Y_\omega(x,t)=f(\omega)\,\varphi(x,t),$$ where $\partial^2={{\partial}}^2_t-{{\partial}}^2_x$. These are coupled integro-differential equations. In the next section, we quantize the model via canonical quantization approach and find similar coupled integro-differential equations for the quantum fields which we are interested in.
Canonical quantization
======================
From the Lagrangian density (\[L\]), the conjugate momenta corresponding to the fields $\varphi$ and $Y_\omega$ are defined by $$\label{momfi}
\pi (x,t) = \frac{{{\partial}}\mathcal{L}}{{{\partial}}\dot{\varphi}}=\dot{\varphi}(x,t),$$ $$\label{momYi}
\Pi_\omega (x,t) = \frac{\delta \mathcal{L}}{\delta \dot{Y}_\omega}=\dot{Y}_\omega (x,t).$$ To quantize the theory canonically, the following equal-time commutation relations are be imposed on the fields and their conjugate momenta $$\label{comfi}
[\hat{\varphi}(x,t),\,\hat{\pi}(x',t)]=i\,\delta(x-x'),$$ $$\label{comYi}
[\hat{Y}_\omega (x,t),\,\hat{\Pi}_{\omega'} (x',t)=i\,\delta(\omega-\omega')\delta(x-x'),$$ and the other commutation relations are identically zero. Using the canonical momenta Eqs.(\[momfi\],\[momYi\]) and the Lagrangian density Eq.(\[L\]), we find Hamiltonian density from the definition $$\hat{\mathcal{H}} = \hat{\pi}\dot{\hat{\varphi}}+\int_0^\infty d\omega\,\hat{\Pi}_\omega \dot{\hat{Y}}_\omega-\hat{\mathcal{L}}.$$ The Hamiltonian of the field-environment system is defined by $$\begin{aligned}
\label{H}
\hat{H} &=& \int\limits_{-\infty}^{\infty} dx\,\hat{\mathcal{H}},\nonumber\\
&=& \int\limits_{-\infty}^{\infty} dx\, \bigg\{{{\frac{1}{2}}}(\hat{\pi}^2+({{\partial}}_x\hat{\varphi})^2+m^2\hat{\varphi}^2)\nonumber\\
&& + \,{{\frac{1}{2}}}\int_0^\infty d\omega\,(\Pi^2_\omega+({{\partial}}_x \hat{Y}_\omega)^2+\omega^2 \hat{Y}^2_\omega) \nonumber\\
&& - \int_0^\infty d\omega\,f(\omega)\,\hat{\varphi}(x,t)\,\hat{Y}_\omega (x,t)\bigg\}.\end{aligned}$$ Having the Hamiltonian and using the Heisenberg equations $$\begin{aligned}
i\,\frac{{{\partial}}\hat{\varphi}(x,t)}{{{\partial}}t} &=& [\hat{\varphi}(x,t),\hat{H}], \\
i\,\frac{{{\partial}}\hat{Y}_\omega(x,t)}{{{\partial}}t} &=& [\hat{Y}_\omega(x,t),\hat{H}],\end{aligned}$$ we obtain the equations of motion for the field $\hat{\varphi}$ and matter fields $\hat{Y}_\omega$, respectively as $$\label{Qfi}
(\partial^2+m^2)\,\hat{\varphi}(x,t)=\int_0^\infty d\omega\,f(\omega)\,\hat{Y}_\omega(x,t),$$ $$\label{QY}
(\partial^2+\omega^2)\,\hat{Y}_\omega(x,t)=f(\omega)\,\hat{\varphi}(x,t),$$ which are similar to their classical counterparts Eqs.(\[Efi\],\[EY\]). The formal solution of Eq.(\[QY\]) is $$\label{SY}
\hat{Y}_\omega(x,t)=\hat{Y}^N_\omega(x,t)-\int dx'\,G_\omega(x-x',t-t')\,f(\omega)\,\hat{\varphi}(x',t'),$$ where $G_\omega (x-x',t-t')$ is the Green’s function of Eq.(\[QY\]) satisfying $$\label{EG}
(\partial^2+\omega^2)\,G_\omega(x-x',t-t')=-\delta(x-x')\delta(t-t').$$ To find the explicit form of the Green’s function $G_\omega (x-x',t-t')$, we first take the temporal Laplace transform of both sides of Eq.(\[EG\]), putting the initial conditions equal to zero $$\label{LG}
\frac{d^2}{dx^2}\,\tilde{G}_\omega (x-x',s)-(s^2+\omega^2)\,\tilde{G}_\omega (x-x',s)= \delta(x-x'),$$ the solution of Eq.(\[LG\]) is $$\label{SLG}
\tilde{G}_\omega (x-x',s)=-\frac{1}{2\sqrt{s^2+\omega^2}}\,e^{-\sqrt{s^2+\omega^2}\,|x-x'|}.$$ Now taking the inverse Laplace transform we find [@Gradshteyn] $$\label{Green}
G_\omega (x-x',t-t')=-{{\frac{1}{2}}}\,\theta(t-t'-|x-x'|)\,J_0 (\omega\sqrt{(t-t')^2-|x-x'|^2}\,),$$ where $\theta(x)$, is Heaviside step function and $J_0 (x)$, is Bessel function of the first kind and zero order.
The homogeneous solution $\hat{Y}^N_\omega (x,t)$ satisfies the free space Klein-Gordon equation $$\label{EYN}
(\partial^2+\omega^2)\,\hat{Y}^N_\omega (x,t)=0,$$ and can be interpreted as the noise fields or quantum vacuum fluctuating fields. The noise fields can be expanded in terms of the eigen-modes of the Klein-Gordon equation as $$\begin{aligned}
\label{SYN}
\hat{Y}^N_\omega (x,t) &=& \int \frac{dk}{\sqrt{2\pi (2\omega_k )}}\,\bigg[\hat{a}_k (\omega)\,e^{i (kx-\omega_k t)}
+\hat{a}^{\dag}_k (\omega)\,e^{-i (kx-\omega_k t)} \bigg],\end{aligned}$$ where $\omega_k=k_0=\sqrt{k^2+\omega^2}$, and the creation $(\hat{a}^\dag_k)$ and annihilation $(\hat{a}_k)$ operators satisfy the usual commutation relations induced from the canonical commutation relation Eq.(\[comYi\]) $$[\hat{a}_k (\omega),\,\hat{a}^\dag_{k'} (\omega')]=\delta(\omega-\omega')\delta(k-k').$$ From Eqs.(\[SY\],\[SYN\]), we can rewrite Eq.(\[Qfi\]) as $$\label{Lanfi}
(\partial^2+m^2)\,\hat{\varphi}(x,t)+\int\int dx'\,dt'\,\gamma(x-x',t-t')\,\hat{\varphi}(x',t')=\hat{J}^N(x,t),$$ where the memory function $\gamma(x,t)$ or the susceptibility of the medium is defined by $$\label{mem1}
\gamma(x-x',t-t')=\int_0^\infty d\omega\,f^2(\omega)\,G_\omega(x-x',t-t'),$$ and the noise current $\hat{J}^N (x,t)$ is given by $$\label{noise}
\hat{J}^N(x,t)=\int_0^\infty d\omega\,f(\omega)\,\hat{Y}^N_\omega(x,t).$$ From Eq.(\[noise\]), we find the commutation relation $$\label{fluc}
[\hat{J}^N(x,t),\,\hat{J}^N(x',t')]=i\,[\theta(t-t')\,\gamma(x-x',t-t')-\theta(t'-t)\,\gamma(x'-x,t'-t)].$$ Using Eq.(\[fluc\]), one can easily show that $$\label{fluc-diss}
{{\langle}}0|\hat{J}^N_{+}(x,\omega)\,\hat{J}^N_{-}(x,\omega')|0{{\rangle}}=4\pi\delta(\omega-\omega')\mbox{Im}[\tilde{\gamma}(x-x',\omega)],$$ known as the fluctuation-dissipation theorem. In Eq.(\[fluc-diss\]), $\hat{J}^N_{+}(x,\omega)$ and $\hat{J}^N_{-}(x,\omega)$, are the Fourier transform of the positive and negative frequency parts of the field $\hat{J}^N(x,t)$ defined by Eqs.(\[SYN\],\[noise\]). By inserting Eq.(\[Green\]) for the Green’s function into Eq.(\[mem1\]) we find $$\label{mem2}
\gamma(x-x',t-t')=-{{\frac{1}{2}}}\,\theta(t-t'-|x-x'|)\int_0^\infty d\omega\,f^2(\omega)\,J_0 (\omega\sqrt{(t-t')^2-|x-x'|^2}\,).$$ Eq.(\[mem2\]) is a Hankel transform and can be inverted to find the coupling function in terms of the memory function as $$\label{coupling}
f^2 (\omega)=-\omega\,\int_0^\infty du\,u\,\gamma(u)\,J_0 (\omega u).$$ The left hand side of Eq.(\[coupling\]) is positive and this condition will impose a physical constraint on acceptable memory functions, that is the right hand side of Eq.(\[coupling\]) should be positive. In non-relativistic models the constraint was the positivity of the imaginary part of the memory function or the susceptibility, in order to have dissipation in the system. Negativity of the imaginary part of the susceptibility leads to gain in the system and we do not discuss it here.
Taking the Fourier-Laplace transform of both sides of Eq.(\[Lanfi\]) with respect to spatial-temporal dependence respectively, leads to $$\begin{aligned}
\label{fiks}
\tilde{\hat{\varphi}}(k,s) &=& \frac{s}{s^2+k^2+m^2+\tilde{\gamma}(k,s)}\,\hat{\varphi}(k,0)+\frac{1}{s^2+k^2+m^2+\tilde{\gamma}(k,s)}\,\dot{\hat{\varphi}}(k,0)\nonumber\\
&+& \frac{\tilde{\hat{J}}^N (k,s)}{s^2+k^2+m^2+\tilde{\gamma}(k,s)},\end{aligned}$$ where $$\label{gks}
\tilde{\gamma}(k,s)=-\int_0^\infty d\omega\,\frac{f^2(\omega)}{s^2+\omega^2+k^2}.$$ In deriving Eq.(\[gks\]) we used $$\label{}
\tilde{G}_\omega(k,s)=-\frac{1}{s^2+\omega^2+k^2},$$ which can be easily obtained from Eq.(\[LG\]) using spatial Fourier transform. The Fourier-Laplace transform of the noise field is given by $$\label{}
\tilde{\hat{J}}^N (k,s)=\int_0^\infty d\omega\,f(\omega)\,\frac{2\pi}{\sqrt{4\pi \omega_k}}\,\bigg\{\frac{1}{s+i\omega_k}\,\hat{a}_k (\omega)+
\frac{1}{s-i\omega_k}\,\hat{a}^\dag_k (\omega)\bigg\}.$$ For later convenience, let us define the functions $\beta(k,t)$ and $\alpha(k,t)$ by $$\label{beta}
\beta(k,t)=L^{-1}\bigg[\frac{s}{s^2+k^2+m^2+\tilde{\gamma}(k,s)}\bigg],$$ $$\label{alfa}
\alpha(k,t)=L^{-1}\bigg[\frac{1}{s^2+k^2+m^2+\tilde{\gamma}(k,s)}\bigg]=\int_0^t dt'\,\beta(k,t'),$$ where $L^{-1}$ is the inverse laplace transform operator. From Eq.(\[fiks\]) and definitions Eqs.(\[beta\],\[alfa\]), we have $$\begin{aligned}
\label{fikt}
\tilde{\hat{\varphi}}(k,t) &=& \beta(k,t)\,\tilde{\hat{\varphi}}(k,0)+\alpha(k,t)\,\dot{\tilde{\hat{\varphi}}}(k,0)\nonumber\\
&& + \int_0^t dt'\,\alpha(k,t')\int_0^\infty d\omega\,f(\omega)\sqrt{\frac{\pi}{\omega_k}}\,\bigg(\hat{a}_k(\omega)e^{-i\omega_k (t-t')}+
\hat{a}^\dag_{-k}(\omega)e^{i\omega_k (t-t')}\bigg).\end{aligned}$$ Therefore, the explicit form of the field $\hat{\varphi}(x,t)$ is given by $$\begin{aligned}
\label{fi}
\hat{\varphi}(x,t) &=& \int_{-\infty}^{\infty}\frac{dk}{2\pi}\,e^{ikx}\,\beta(k,t)\,\tilde{\hat{\varphi}}(k,0)+\int_{-\infty}^{\infty}\frac{dk}{2\pi}\,e^{ikx}\,\alpha(k,t)\,
\dot{\tilde{\hat{\varphi}}}(k,0)\nonumber\\
&& + \int_0^t dt'\,\int_0^\infty d\omega\,f(\omega)\,\int_{-\infty}^{\infty} \frac{dk}{\sqrt{2\pi\,2\omega_k}}\,\alpha(k,t')\,\bigg\{\hat{a}_k(\omega)e^{i[kx-\omega_k (t-t')]}+
\hat{a}^\dag_{k}(\omega)e^{-i[kx-\omega_k (t-t')]}\bigg\}.\end{aligned}$$ At the large-time limit $(t\gg1/\omega_k)$, we have $$\begin{aligned}
\label{alfalim}
\int_0^{t\gg1/\omega_k} dt'\,\alpha(k,t')\,e^{i\omega_k t'} &\approx &\tilde{\alpha}(k,s=-i\omega_k),\nonumber\\
&& =\frac{1}{m^2-\omega^2+\tilde{\gamma}(k,-i\omega_k)},\end{aligned}$$ where we made use of Eq.(\[alfa\]). Also from Eq.(\[gks\]), we have $$\label{gg}
\tilde{\gamma}(k,-i\omega_k)=-\int_0^\infty d\omega'\,\frac{f^2(\omega')}{\omega'^2-\omega^2}.$$ Note that due to the dissipation, the dominant term in Eq.(\[fi\]) is the last term, this term could also be obtained directly from Fourier transform instead of taking Laplace transform but in deriving Eq.(\[fi\]) we were interested in the derivation of the complete solution containing the homogeneous part in addition to the particular solution. Now, using Eq.(\[alfalim\]) we find the large-time limit of the field $(\hat{\varphi}_{c}(x,t))$ as $$\begin{aligned}
\hat{\varphi}_{c}(x,t)=
\int_{-\infty}^{\infty} \frac{dk}{\sqrt{2\pi\,2\omega_k}}\, \int_0^\infty d\omega\,\frac{f(\omega)}{m^2-\omega^2+\tilde{\gamma}(k,-i\omega_k)}\,
\bigg\{\hat{a}_k(\omega)e^{i[kx-\omega_k t]}+
\hat{a}^\dag_{k}(\omega)e^{-i[kx-\omega_k t]}\bigg\}.\end{aligned}$$ Knowing the explicit form of the quantum field, one can proceed and construct Fock’s space or find important quantities like the two-point function or Feynman Green function. Including finite temperature corrections is also straightforward since the field is expressed in terms of the creation and annihilation operators of the medium which can be easily thermalized. A more realistic problem appears when one considers an interacting field theory where a path integral approach seems to be more effective, at least from perturbative point of view.
Conclusions
===========
Caldeira-Leggett model of reservoir is generalized to a reservoir modeled by a continuum of real Klein-Gordon fields instead of harmonic oscillators. A quantum Langevin type dissipative equation is obtained for the scalar field. The susceptibility of the medium is defined in terms of the reservoir Green’s function and the coupling function satisfying causality condition. The connection between the coupling function and the susceptibility of the medium is found to be a Hankel transform from which the coupling function can be determined in terms of the susceptibility of the medium. Noise currents and their fluctuation-dissipation relation are obtained. In a homogeneous medium or reservoir, explicit form of the quantum scalar field, and its large-time limit, are found. Knowing the explicit form of the quantum field, one can proceed and construct Fock’s space or find important quantities like the two-point function or Feynman Green’s function. Based on the structure of the article, the generalization to an interacting dissipative field theory is straightforward in the framework of path integrals, at least perturbatively.
[22]{} D.A.R. Dalvit, P.W. Milonni, D.C. Roberts and F.S.S. Rosa eds., Lecture Notes in Physics 834 (Springer-Verlag, 2011) p.345. A. H. Guth, Phys. Rev D**23**, 347 (1981). Brandenberger, Rev. of Mod. Phys. 57, 1 (1985). B. Huttner and S. M. Barnett, Phys.Rev. A**46**, 4306 (1992). F. Kheirandish and M. Soltani, Phys.Rev. A**78**, 012102 (2008); F. Kheirandish and M. Jaffari, Phys.Rev. A**86**, 022503 (2012); F. Kheirandish, M. Soltani, and M. Jafari, Phys.Rev. A**84**, 062120 (2011). M. Amooshahi, Eur. Phys. J. D**54**, 115 (2009). A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981); Ann. Phys. 149, 374 (1983); Physica 121A, 587 (1983). U. Weiss, Quantum Dissipative Systems (1993, World Scientific) and the references therein. R. F. Voss and R. A. Webb, Phys. Rev. Lett. 47, 265 (1981). J. M. Martinis, M. H. Devoret and J. Clarke, Phys. Rev. Lett. 55, 1543 (1985). I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 2007).
|
---
abstract: 'We investigate the method for the reconstruction of $f(R)$ gravity models both from the background evolution observations and from the large scale structure measurements. Due to the lack of the first principles, one needs to rely on the observations to build the $f(R)$ gravity models. We show that the general $f(R)$ models can be specified if the 5% accuracy level large scale structure formation observations will be available in the near future.'
address: 'The Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Republic of Korea'
author:
- Seokcheon Lee
title: 'Reconstruction of f(R) gravity models from observations'
---
modified gravity ,accelerating universe ,large scale structure ,background evolution
Introduction
============
The current accelerating expansion of the Universe can be well explained by the cosmological constant. However the cosmological constant suffers from the so-called the fine tuning and the coincidence problems. Thus, both dark energy (DE) with an exotic equation of state (e.o.s) and a modification of gravity (MG) on large scales are investigated as the alternative solutions to the cosmological constant [@2012PhR...513....1C; @2014arXiv1401.0046M; @2016ARNPS..66...95J]. Two theories are distinguished by the violation of the so-called the strong equivalence principle and one is not able to probe whether this is correct or not. Thus, the origin of the accelerating expansion of the Universe can be probed by observations.
The background evolution of both DE and MG models are degenerated, described by the Hubble parameter $H$ and the integrations of its inverse with respect to the redshift. This degeneracy can be broken by observing the evolution of cosmological perturbations obtained from the large scale structure formation. Thus, one needs to have accurate observations both in the background evolution and in the large scale structure in order to reconstruct the viable MG models. Traditionally, one specifies the specific form of MG and fits the model parameters from observations of the background evolution. From these fixed parameters of the specific models, one predicts the observations of the large scale structure. However, this method is quite ambiguous. Also, if one includes the errors on the measurements then the probing the specific form of model is quite difficult.
The background evolution of the Universe can be specified by the DE e.o.s, $\omega$. To be used for the various models, one usually adopts the parametrization of $\omega$ [@2001IJMPD..10..213C; @2003PhRvL..90i1301L]. Also the evolution of the cosmological perturbation is usually parametrized by using the so-called growth rate index parameter, $\gamma$. We also adopt the specific form of $\gamma$ to generalize the growing of the matter perturbation [@2011JCAP...03..021L].
In this [*Letter*]{}, we focus on the $f(R)$ gravity models among various MG models. In stead of specifying the form of $f(R)$-gravity models, we reconstruct the model from the observed quantities of parametrization of $\omega$ and $\gamma$. In the next section, we briefly review both the background and the perturbation equations in general $f(R)$ gravity models in the metric formalism [@2010RvMP...82..451S]. We show the reconstruction method of $f(R)$-gravity models from the observed quantities and the results in Sec. 3. We derive detail formulae for the model functions as a function of $\omega$ and $\gamma$ in the appendix.
Review on Background and Perturbation Evolutions
================================================
In this section, we review the general feature of f(R) gravity models. Both the background evolution equations and the matter perturbation evolution equations are reviewed.
Model
-----
The action of the general f(R) gravity theories are given by [@1980PhLB...91...99S] S = d\^4 x ( + [L]{}\_[[[m]{}]{}]{} ) d\^4 x ( + [L]{}\_[[[m]{}]{}]{} ) , \[S\] where $f(R)$ is a general function of the Ricci scalar R and ${\cal L}_{{\rm{m}}}$ is the matter Lagrangian. The Ricci scalar is given by R = 6 (H\^2 + ) = 8 G ( \_[[[m]{}]{}]{} + ( 1 - 3 \_[[[eff]{}]{}]{} ) \_[[[eff]{}]{}]{} ) , \[R\] where $\rho_{{\rm{m}}} (\rho_{{\rm{eff}}})$ is the mass density of the matter (the effective dark energy) component and $\omega_{{\rm{eff}}}$ is the e.o.s of the effective DE which is derived from the modification of the Einstein-Hilbert action. It has been considered that the form of $f(R)$ can be constrained in order to satisfy both cosmological and solar-system tests [@2007PhRvD..76f4004H]. It has been believed that following two conditions are required to satisfy observational tests \_[R]{} (R) &=& ,\
\_[R0]{} (R) &=& 0 , \[finfty0\] which lead to several viable $f(R)$ models shown in the table \[tab-1\].
[c ccc]{}\
Model & $f(R)$ & parameters & Ref\
\
Hu-Sawicki & $R- R_{\text{HS}} {\frac{c_1 (R/ R_{\text{HS}})^{p}}{c_2 (R/ R_{\text{HS}})^{p} +1}} $ & $R_{\text{HS}} >0, c_1, c_2, p >0$ & [@2007PhRvD..76f4004H]\
\
Starobinsky & $R - \lambda R_{\text{S}} \left[ 1 - \left(1+ {\frac{R^2}{R_{\text{S}}^2}} \right)^{-n} \right] $ & $\lambda > 0, n > 0, \lambda R_{\text{S}} \sim 2 \Lambda$ & [@2007JETPL..86..157S]\
\
Tsujikawa & $R - \mu R_{\text{T}} \tanh \left[{\frac{R}{R_{\text{T}}}} \right] $ & $\mu >0, R_{\text{T}} > 0$ & [@2008PhRvD..77b3507T]\
\
Exponential Gravity & $R - \beta R_{\text{E}} \left( 1 - e^{-R}{R_{\text{E}}} \right)$ & $\beta >1, R_{\text{E}} > 0$ & [@2008PhRvD..77d6009C]\
There are also constraints on the derivatives of $f(R)$ to satisfy the attractive gravity and the stability of the model. F && f\_[,R]{} = = 1 + \_[,R]{} > 0 , \[F\]\
F\_[,R]{} &=& f\_[,RR]{} = = \_[,RR]{} > 0 R m\^2 . \[FR\] One can refer the recent review on these models [@2017arXiv171005634P].
Background Evolution Equations
------------------------------
We are interested in the evolution of the Universe after the matter dominated epoch. The gravitational field equation is obtained from the variation of action, Eq(\[S\]) with respect to the metric F R\_ - f g\_ - F\_[,; ]{} + F g\_ = 8G T\^[([[m]{}]{})]{}\_ , \[Feq\] where $T^{({\rm{m}})}_{\mu\nu}$ is an energy-momentum tensor of the pressureless matter. In a flat Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) metric with a scale factor a, one obtain the following background equations 3 F\_[0]{} H\^2 &=& ( F R - f ) - 3 H + 3 H\^2 ( F\_[0]{} - F ) + 8 G \_[[[m]{}]{}]{} , \[G00\]\
- 2 F\_[0]{} &=& - H - 2 ( F\_[0]{} - F ) + 8 G ( \_[[[m]{}]{}]{} + P\_[[[m]{}]{}]{} ) ,\[Gii\]\
\_[[[m]{}]{}]{} &=& - 3 H ( \_[[[m]{}]{}]{} + P\_[[[m]{}]{}]{} ) \[BI\] , where $F_0$ denotes the present value of $F$ and the pressure of matter component, $P_{{\rm{m}}}$ will be ignored.
Perturbation Equations
----------------------
The equation for the matter perturbation at the sub-horizon scale limit, ${\frac{k^2}{a^2 H^2}} \gg 1$ is given by [@2002PhRvD..66h4009H] \_[[[m]{}]{}]{} + 2H \_[[[m]{}]{}]{} - \_[[[m]{}]{}]{} ( ) \_[[[m]{}]{}]{} = 0 , M = \[dotdeltam\] . The growth rate of the matter perturbation is well parameterized as d \_[[[m]{}]{}]{} / d a \_[[[m]{}]{}]{}\^ , \[growthrate\] where the growth rate index, $\gamma$ can be parameterized as [@2011JCAP...03..021L] = \_[0]{} + \_[a]{} (1 - e\^[n]{} ) . \[gamma\] The term in the perturbation equation, $M$ can be divided for three limits
(I) $M \ll 1$ :\
These limit corresponds scales where the large scale structure is formed, ($R M \ll {\frac{c^2 k^2}{a^2 R}} R M \ll 1$). The solutions for the matter perturbation are given by $\delta_{{\rm{m}}} = \exp \left[ \int \alpha dn \right]$ and using the approximation $|\alpha'| \ll \alpha^2$, one obtains the growing mode solution [@2007PhRvD..76b3514T] \_[+]{} = 1 + M = 1 + 1 + m , \[alphap\] where $m$ is the dimensionless quantity. If $m$ is constant, then $a^2 R \propto a^{-1}$ during the matter dominated era. Thus, the growing mode of the matter perturbation is given by \_[[[m]{}]{}+]{} a\^[1+ (3/5) \_[+]{}]{} , \_[+]{} . \[deltam\] $F_{,R} = {\frac{F'}{R'}} > 0$ at high $R$ is required for the stability of the solution. This means $F' < 0$ at high $z$ because $R' < 0$.\
(II) $M = 0$ :\
In this case, $F = \text{const.}$ ($F_{,R} = 0$). This case is same as the $\Lambda$CDM model with general relativity.\
(III) $M \gg 1$ :\
For $M \geq 10$, one can approximate \[Mapprox1\] with less than 1% error. In this case, $\omega_{{\rm{eff}}} = {\frac{1}{3}}$ and $\Omega_{{\rm{m}}}=2$ when ${\frac{R F_{,R}}{F}} = -{\frac{1}{4}}$ which corresponds to a $\phi$ matter dominated epoch ($\phi$MDE) [@2007PhRvL..98m1302A]. We will use the large scale structure formation observation to constrain the model and thus we will not consider this limit.
Field Equations
---------------
Impact of $f(R)$ can be investigated as a field equation for $F$. The trace of Eq.(\[Feq\]) is given by F = ( 2f - R F - 8 G \_[[[m]{}]{}]{} ) . \[BoxF\] Thus, the effective potential has an extremum at 2f - RF = R + 2 - R \_[R]{} = 8 G \_[[[m]{}]{}]{} . \[R0\]
Model Reconstruction from Observations
======================================
It is convenient to rewrite the both background evolution equations (\[G00\])-(\[Gii\]) and the matter perturbation equation (\[dotdeltam\]) as a function of the number of e-folding, $n \equiv \ln a$. Then both Friedmann equations and the matter perturbation equations become 3 &=& ( - ) - 3 ( + - 1 - \_[[[m]{}]{}]{} ) , \[G00n2\]\
-2 &=& ( - ) + ( +2 - 2 ) + 3 \_[[[m]{}]{}]{} , \[Giin2\]\
\_[[[m]{}]{}]{}\^[”]{} &=& - ( 2 + ) \_[[[m]{}]{}]{}’ + \_[[[m]{}]{}]{} ( ) \_[[[m]{}]{}]{} , \[deltamn\] as shown in Eqs. (\[G00n2A\]), (\[Giin2A\]), and (\[deltamnA\]).
Now one can rewrite the model functions as functions of observed quantities $\omega$ and $\gamma$. One can refer the appendix for the detail derivations of this section. We adopt the so-called Chevallier-Polarski-Linder (CPL) parameterization of the DE e.o.s $\omega_{{\rm{DE}}} = \omega_{0} + \omega_{a} \left(1 - e^{n}\right)$ [@2001IJMPD..10..213C; @2003PhRvL..90i1301L] to match the background evolution equations (\[G00n2\])-(\[Giin2\]) with (\[H2oH02A\])-(\[HoHpA\]). &=& ( 1 + ) = \_[[[m]{}]{}0]{} e\^[-3n]{} (1+g) \[H2oH02\] ,\
&=& - ( 1 + \_[[[DE]{}]{}]{} ) - (1 + Q)\[HoHp\] . From these equations, one can relate model functions $f$ and $F$ with the DE e.o.s, $\omega$. However, one is not able to constrain the evolutions of model functions with these equations only. However, one also can parametrize the growth rate of the matter perturbation as $d \ln \delta_{{\rm{m}}} / d \ln a \equiv \Omega_{{\rm{m}}}^{\gamma}$ and we adopt the parametrization of the growth rate index parameter as $\gamma = \gamma_{0} + \gamma_{a} \left(1 - e^{n} \right) $ given in [@2011JCAP...03..021L]. One more equation required to fully constrain the model functions is obtained from the matter perturbation equation as shown in Eqs.(\[PA\])-(\[FpoF0A\]) &=& () \[FoF0n\] ,\
[P]{}&& \_[[[m]{}]{}]{}\^ ( \_[[[m]{}]{}]{}\^ + ’ \_[[[m]{}]{}]{} + 3 Q + ) \[P\] ,\
M = &=& A = \[M\] ,\
= M &=& ( ) \[FpoF0\] . All of the quantities in the above equations $F/F_{0}$, ${\cal P}$, $M$, and $A$ are dimensionless.
We list the reconstructed model functions as a function of $\omega_0, \omega_{a}, \gamma_0$, and $\gamma_{a}$ in the table \[tab-4\] in the appendix. Observational measurements can be parameterized by these values. However, the measurement on $\gamma_{0}$ and $\gamma_{a}$ from galaxy redshift surveys are still not accurate enough. Thus, we first assume that one can measure the exact values of ($\gamma_{0}$, $\gamma_{a}$) when ($\omega_{0}$, $\omega_{a}$) measured very accurately. We numerically calculate the ($\gamma_{0}$, $\gamma_{a}$) values from Eq.(\[deltamn\]) for the given values of $\Omega_{{\rm{m}}0}$, $\omega_{0}$, $\omega_{a}$, and $M_{0}$. We assume the $\Lambda$CDM like background evolution ([*i.e.*]{} ($\omega_{0}, \omega_{a}) = (-1, 0)$) for the different values of $\Omega_{{\rm{m}}0}$ to obtain the values of ($\gamma_{0}$, $\gamma_{a}$). One also need to specify the present value of $M_{0}$ to obtain the growth index parameters ($\gamma_{0}$, $\gamma_{a}$). The best fit values for the different models are given in the table \[tab-2\]. For the given value of the matter energy density contrast, $\Omega_{{\rm{m}}0}$ the magnitudes of $\gamma_{0}$ are increased as the magnitudes of $M_{0}$ decrease. Also as the magnitudes of $M_0$ decrease, so do the absolute values of $\gamma_{a}$. This is due to the fact that model gets close to $\Lambda$CDM as $M$ decreases and thus the growth index parameter of $f(R)$ model becomes that of $\Lambda$CDM. For example, ($\gamma_{0}, \gamma_{a}$) = ($0.519, -0.162$) when $\Omega_{{\rm{m}}0} = 0.32$ and $M_0 = 0.1$. But they become ($0.554, -0.016$) for the same value of $\Omega_{{\rm{m}}0}$ when $M_0 = 10^{-4}$ as shown in the table \[tab-2\]. One can also find that the values of $\gamma_{0}$ is increased as those of $\Omega_{{\rm{m}}0}$ decrease for the same value of $M_0$. With the given values of $\omega_0, \omega_a, \gamma_{0}$, and $\gamma_{a}$ one can solve the differential equation (\[FpoF0\]) with the initial condition ${\frac{F}{F_{0}}} = 1$ to reconstruct the $f(R)$ models.
[c cc cc cc]{}\
& & &\
& $\gamma_{0}$ & $\gamma_{a}$ & $\gamma_{0}$ & $\gamma_{a}$ & $\gamma_{0}$ & $\gamma_{a}$\
\
$10^{-1}$ & $0.519$ & $-0.162$ & $0.522$ & $-0.151$ & $0.525$ & $-0.135$\
\
$5 \times 10^{-2}$ & $0.534$ & $-0.099$ & $0.536$ & $-0.093$ & $0.538$ & $-0.085$\
\
$10^{-2}$ & $0.550$ & $-0.035$ & $0.550$ & $-0.034$ & $0.552$ & $-0.033$\
\
$10^{-3}$ & $0.554$ & $-0.017$ & $0.554$ & $-0.018$ & $0.555$ & $-0.019$\
\
$10^{-4}$ & $0.554$ & $-0.016$ & $0.555$ & $-0.016$ & $0.555$ & $-0.017$\
We repeat the same numerical calculation to obtain the best fit values of ($\gamma_{0}, \gamma_{a}$) for the same models. However, we consider the measurement errors on the growth rate, ${\frac{d \ln \delta}{d \ln a}}$ at this moment. We assume the measurement error on the growth rate from -10 % to + 10 % when we obtain the growth index parameters ($\gamma_{0}, \gamma_{a}$). These are shown in the table \[tab-3\]. The values of ($\gamma_{0}, \gamma_{a}$) are increased when the growth index values are decreased by including the decreased values of the growth index due to the measurement errors. For example, the measured values of ($\gamma_0, \gamma_a$) should be (0.600, 0.186) for $\Omega_{{\rm{m}}0} =0.32$ and $M_0 = 0.1$ model when the measured value of ${\frac{d \ln \delta}{d \ln a}}$ is 10 % smaller than true value. When the measurement on the growth index is larger than the true value of it, then the values of ($\gamma_{0}, \gamma_{a}$) are smaller than the true values of the growth index parameters.
[c cc cc cc]{}\
& & &\
in ${\frac{d \ln \delta}{d \ln a}} \, (\%)$ & $\gamma_{0}$ & $\gamma_{a}$ & $\gamma_{0}$ & $\gamma_{a}$ & $\gamma_{0}$ & $\gamma_{a}$\
\
$-10$ & $0.600$ & $0.186$ & $0.598$ & $0.169$ & $0.596$ & $0.146$\
\
$-5$ & $0.559$ & $0.008$ & $0.559$ & $0.005$ & $0.559$ & $0.001$\
\
Exact & $0.519$ & $-0.162$ & $0.522$ & $-0.151$ & $0.525$ & $-0.135$\
\
$+5$ & $0.482$ & $-0.323$ & $0.486$ & $-0.299$ & $0.492$ & $-0.265$\
\
$+10$ & $0.447$ & $-0.476$ & $0.453$ & $-0.440$ & $0.461$ & $-0.390$\
Results
-------
Now, one can reconstruct the $f(R)$ gravity models from the obtained values of ($\omega_{0}, \omega_{a}$) and ($\gamma_{0}, \gamma_{a}$) without and with considering the measurement errors on $\gamma$. One can numerically solve the equation (\[FpoF0\]) by using the initial condition $F/F_{0} = 1$ and values of $\omega_{0}, \omega_{a}, \gamma_{0}$, and $\gamma_{a}$ given in the tables \[tab-2\] and \[tab-3\].
(I) Without measurement error :\
-------------------------------------------- --------------------------------------------
{width="0.5\linewidth"} {width="0.5\linewidth"}
-------------------------------------------- --------------------------------------------
First, we investigate the model reconstruction without considering measurement error. We probe the effective DE e.o.s, $\omega_{{\rm{eff}}}$ given in Eq. (\[omegaDEMG\]). If we assume that all the necessary measurements are measured without errors, then one can reconstruct $\omega_{{\rm{eff}}}$ without any uncertainty. In Fig.\[fig-1\], we show the evolution of the effective DE e.o.s for the different values of $M_{0}$ and $\Omega_{{\rm{m}}0}$ when the background evolution mimics that of $\Lambda$CDM. In the left panel of Fig.\[fig-1\], we adopt ($\omega_{0}, \omega_{a}, M_{0}$) = (-1.0, 0, 0.1). For different values of $\Omega_{{\rm{m}}0} = (0.32, 0.30, 0.27)$, one obtains ($\gamma_{0}, \gamma_{a}$) = (0.519, -0.162), (0.522, -0.151), (0.525, -0.135) as shown in the table \[tab-2\]. Thus, one can solve the differential equation (\[FpoF0\]) to obtain evolution behaviors of $\omega_{{\rm{eff}}}$ given in Eq. (\[omegaDEMG\]). The solid, dashed, and dot-dashed lines depict $\omega_{{\rm{eff}}}$ for $\Omega_{{\rm{m}}0} = 0.32, 0.30$, and 0.27, respectively. All the models consistent with $\omega_{{\rm{eff}}} = -1.0$ within percent level accuracies. Even though the larger the value of $\Omega_{{\rm{m}}0}$, the larger the deviation of $\omega_{{\rm{eff}}}$ from $\omega = -1.0$, all of those deviations are sub-percent level and one will not be able to distinguish $\omega_{{\rm{eff}}}$ from -1. In the right panel of Fig.\[fig-1\], we show the effective DE e.o.s for $M_{0} = 10^{-2}$ with the same notations as the left panel of that figure. One obtains the smaller deviation of $\omega_{{\rm{eff}}}$ from -1 with smaller values of $M_{0}$. This fact is shown in the right panel of Fig.\[fig-1\].
-------------------------------------------- --------------------------------------------
{width="0.5\linewidth"} {width="0.5\linewidth"}
{width="0.5\linewidth"} {width="0.5\linewidth"}
-------------------------------------------- --------------------------------------------
Now, one can reconstruct model functions as a function of the Ricci scalar. In other words, we show the behaviors of $f/(F_0 H_{0}^2)$ and $F/F_{0}$ as a function of $R/H_{0}^2$. For this purpose, we first show the behaviors of $R/H_{0}^2$ as a function of redshift, $z$ for the different values of $\Omega_{{\rm{m}}0}$. This is given by Eq.(\[RoH02\]). We depict the dimensionless normalized Ricci scalar, $R/H_0^2$ as a function of $z$ in the upper left panel of Fig.\[fig-2\]. For $z \leq 0.58$, the smaller the values of $\Omega_{{\rm{m}}0}$, the larger the value of $R/H_0^2$. For larger values of $z$, $R/H_{0}^2$ are larger for the larger values of $\Omega_{{\rm{m}}0}$. The solid, dashed, and dot-dashed lines correspond $\Omega_{{\rm{m}}0} = 0.32, 0.30$, and 0.27, respectively. Thus, for the same redshift period, the larger value of $\Omega_{{\rm{m}}0}$ covers the wider range of $R/H_{0}^2$. $R/H_{0}^2$ varies from 9.12 (9.3, 9.8) to 15.8 (15.6, 15.2) for $\Omega_{{\rm{m}}0} = 0.32$ (0.30, 0.27) during redshift $0 \leq z \leq 1.0$. The evolutions of $F/F_{0}$ for the different values of $\Omega_{{\rm{m}}0}$ as a function of $R/H_{0}^2$ are shown in the upper right panel of Fig.\[fig-2\]. As the initial condition $F/F_{0} |_{n=0} = 1$ is required. We put $M_{0} = 10^{-2}$ in this case. Deviation of $F/F_{0} $ from 1 indicates the modification of the Hilbert-Einstein action. To be consistent with current observation, the deviation of $ \partial f(R)/ \partial R$ from 1 is an order of magnitude $-5$ in these models as shown in the figure. The solid (dashed, dot-dashed) line corresponds to $\Omega_{{\rm{m}}0} = 0.32$ (0.30, 0.27). $F/F_{0}$ varies most rapidly for $\Omega_{{\rm{m}}0} = 0.30$ model. We also shows the behaviors of dimensionless normalized $f/(F_{0} H_{0}^2)$ in the lower left panel of Fig.\[fig-2\]. With $M_{0} = 10^{-2}$, the solid, dashed, and dot-dashed lines correspond $\Omega_{{\rm{m}}0} = 0.32, 0.30$, and 0.27, respectively. The larger the value of $\Omega_{{\rm{m}}0}$, the larger the magnitude of $f/(F_{0}H_{0}^2)$. The deviations of action from the Hilbert-Einstein action for the different models are depicted in the lower right panel of Fig.\[fig-2\]. This is given by - = ( - 1 ) \[tf\] . Thus, the values shown in the figure show the magnitude of the deviation of $f$ from $R$ multiplied by the normalized Ricci scalar. As shown in the figure, the observations show that the deviation should be negative and the variation of it must be very small. The variation is around -4.1 (-4.2, -4.4) for $\Omega_{{\rm{m}}0} = 0.32$ (0.30, 0.27).
(II) With measurement error :\
-------------------------------------------- --------------------------------------------
{width="0.5\linewidth"} {width="0.5\linewidth"}
-------------------------------------------- --------------------------------------------
In the previous consideration, we do not include measurement errors in both $\omega$ and $\gamma$. However, in the real observations, measured values of them include measurement errors. Compared to $\gamma$, one might ignore the errors on $\omega$. In future observations, the error on the growth rate $d \ln \delta / d \ln a$ might be reduced to 5 - 10 % level. Thus, we consider the $f(R)$ model reconstruction including measurement errors on the growth rate. We numerically obtain the growth index parameters with including measurement errors on $d \ln \delta / d \ln a$ in the table \[tab-3\]. We obtain ($\gamma_{0}, \gamma_{a}$) for the different models with the measurement errors on the growth rate from -10 % to + 10 %. For example, when the measurement errors on the growth rate is +5 %, then ($\gamma_{0}, \gamma_{a}$) = (0.482, -0.323) when $\Omega_{m0} = 0.32$. As shown in the table \[tab-3\], if the measurements on the growth rate become smaller than the true values, then both $\gamma_{0}$ and $\gamma_{a}$ become larger than true values for all models. Also if the measurements on the growth rate are larger than the true values, then both $\gamma_{0}$ and $\gamma_{a}$ are smaller than true values of them.
We show the evolutions of $\omega_{{\rm{eff}}}$ and $F/F_{0}$ with including the measurement errors on the growth rate in Fig.\[fig-3\]. We investigate the models with $(\omega_{0}, \Omega_{{\rm{m}}0}, M) = (-1.0, 0.32, 0.1)$. In the left panel of Fig.\[fig-3\], we show the evolutions of the effective DE e.o.s when we include the measurement errors on $d \ln \delta / d \ln a$. The dashed line correspond the case when the measurement of the growth rate is 5 % smaller than the true value. The smaller values of the growth rate than the true one induce the smaller change in the modification terms of the model and thus produce smaller change in $\omega_{{\rm{eff}}}$. The dot-dashed line represents the case when the growth rate measurement is 5% larger than that of the true value. In this case, the larger deviation of the $F/F_{0}$ induces the larger deviation of $\omega_{{\rm{eff}}}$ from -1. The deviation can be as large as 3 % at $z \sim 2$ in this case. Thus, one might have inconsistent values of $\omega_{{\rm{eff}}}$ obtained from the background evolution and the large scale structure. In the right panel of Fig.\[fig-3\], we show the evolutions of $F/F_0$ when we include the measurement errors. The dot-dashed line indicates the evolution of $F/F_0$ when the measurement error on the growth rate is +5 %. This case shows the rapid change of $F/F_0$ compared to the real model. This might give the totally different functional form of the model compared to the original model. Also when the error on the growth rate is - 5%, the $F/F_0$ can be smaller than 1. And this case, one might conclude the gravitational force is smaller than the present one. This gives the opposite result compared to the original model. Thus, the interpretation of $f(R)$ model with measurement errors requires more carel.
Comparison with Viable model
----------------------------
{width="0.7\linewidth"}
We list the viable $f(R)$ gravity models in the table\[tab-1\]. We compare the reconstructed models with the so-called “Hu-Sawicki (HS)” model [@2007PhRvD..76f4004H]. If one converts the notations of HS model to ours, then one obtains R\_[[[HS]{}]{}]{} &=& F\_[0]{} H\_[0]{}\^2 \_[[[m]{}]{}0]{} , \[RHS\]\
&=& 6 , \[c1\]\
c\_[2]{} &=& -p ( -1 )\^[-p-1]{} \[c2\] , where $p$ is the natural number and $\tilde{f}_{R0}$ is the initial input parameter. The figure \[fig-4\] depicts the evolutions of $\tilde{f}_{R} = F/F_0 -1$ for both the reconstructed model and the HS models. We adopt $(\omega_{0}, \omega_{a}, \Omega_{{\rm{m}}0}) = (-1, 0, 0.32)$ in this figure. The solid line indicates the evolution of reconstructed $\tilde{f}_{R}$ when $M_{0} = 10^{-2}$. The dashed (dot-dashed) line shows the evolution of $\tilde{f}_{R}$ of HS models when $p = 1$ (2) with $\tilde{f}_{R0} = 10^{-4}$. All of them show the same behaviors as $F/F_0 -1$ increase as $z$ increases. The main difference between our reconstruction method and HS model is the rate of $\tilde{f}_{R}$ which depends on $\gamma_{0}$ and $\gamma_{a}$ in our method. The reconstruction method can provide general function of $f(R)$ gravity models which consistent with observations.
Conclusions {#conclusions .unnumbered}
===========
We illustrate how to determine the general $f(R)$ gravity models from cosmological observations. Thus, if cosmological observables are accurate enough, then one can constrain the model functions. Especially, if the growth rate index is determined with high accuracy, then one can well specify the model. Even if the future observations are consistent with $\Lambda$CDM in background evolutions, one might have a chance to specify $f(R)$ gravity if the growth rate of the large scale structure is different from that of $\Lambda$CDM. Our method depends on only observations and can be used for various model search after the more accurate large scale structure data are available in the near future.
One can rewrite evolution equations for both the background and the matter perturbation Eqs.(\[G00\])-(\[dotdeltam\]) as a function of $n \equiv \ln a$ in order to relate the$f(R)$ gravity models as a function of observed quantities.
Background Evolution
--------------------
First, we rewrite the background evolution equations Eqs.(\[G00\])-(\[Gii\]) as a function of $n$ 3 F\_[0]{} H\^2 &=& ( FR - f ) - 3H\^2 (F’ + F - F\_[0]{} ) + 8 G \_[[[m]{}]{}]{} , \[G00n\]\
-2 F\_[0]{} H H’ &=& H\^2 ( F” - F’ ) + H H’ (F’ +2F - 2F\_[0]{} ) + 8 G \_[[[m]{}]{}]{} , \[Giin\] where primes mean the derivative with respect to $n$. One can rewrite the above equations 3 &=& ( - ) - 3 ( + - 1 - \_[[[m]{}]{}]{} ) , \[G00n2A\]\
-2 &=& ( - ) + ( +2 - 2 ) + 3 \_[[[m]{}]{}]{} , \[Giin2A\] From the above Eqs.(\[G00n\]) and (\[Giin\]), one can define the energy density, $\rho_{{\rm{eff}}}$ the pressure, $P_{{\rm{eff}}}$, and the equation of state, $\omega_{{\rm{eff}}}$ of the effective dark energy derived from the modification of the theory \_[[[eff]{}]{}]{} &=& ( - 3(F’ + F - F\_[0]{} ) ) , \[rhoeff\]\
P\_[[[eff]{}]{}]{} &=& ( + (F” + 2F’ +3F -3F\_[0]{}) )\
&+& ( (F’+2F -2F\_[0]{}) ) , \[Peff\]\
\_[[[eff]{}]{}]{} &=& -1 +\
&=& -1 + . \[weff\] We adopt the so-called Chevallier-Polarski-Linder (CPL) parameterization of the dark energy e.o.s $\omega_{{\rm{DE}}}$ [@2001IJMPD..10..213C; @2003PhRvL..90i1301L] to match the background evolution equations (\[G00n\])-(\[Giin\]) 3 F\_[0]{} H\^2 &=& 8G ( \_[[[m]{}]{}]{} + \_[[[DE]{}]{}]{} ) 8G \_[[[cr]{}]{}]{} \[G00cpl\] ,\
-2 F\_[0]{} H H’ &=& 8G ( \_[[[m]{}]{}]{} + \_[[[DE]{}]{}]{} + P\_[[[DE]{}]{}]{} ) \[Giicpl\] ,\
\_[[[DE]{}]{}]{} &=& \_[[[DE]{}]{}0]{} e\^[-3(1+\_0+\_a )n + 3\_[a]{}(e\^[n]{}-1)]{} \[rhocpl\] ,\
P\_[[[DE]{}]{}]{} &=& \_[[[DE]{}]{}]{} \_[[[DE]{}]{}]{} \[Pcpl\] , where $\rho_{{\rm{cr}}}$ is the critical density. Thus, one can parameterize model $f(R)$ and its derivatives with respect to $n$ as a function of $\omega_{0}$ and $\omega_{a}$ \_[[[DE]{}]{}]{}&=& ( - ) - ( + - 1 ) \[OmgeaDEMG\] ,\
&& 1 - \_[[[m]{}]{}]{} ,\
g&=& e\^[-3(\_[0]{}+\_[a]{})n +3\_[a]{}(e\^n-1)]{} = , \[gwowa\]\
1 + &=& \[omegaDEMG\] . Thus, one can also rewrite the Friedmann equations &=& ( 1 + ) = \_[[[m]{}]{}0]{} e\^[-3n]{} (1+g) \[H2oH02A\] ,\
&=& - ( 1 + \_[[[DE]{}]{}]{} ) - (1 + Q)\[HoHpA\] ,\
&=& 6 ( 2 + ) = 3 \_[[[m]{}]{}0]{} e\^[-3n]{} (1+g) (1-3Q) \[RoH02\] . Even though one represents $f$, $F =\equiv {\frac{\partial f}{\partial R}}$, and derivatives of $F$ with respect to $n$ as a function of $\omega_{{\rm{DE}}}$, this is not enough to constrain the evolution equations of them. Thus, one needs to use further constraints from the perturbation evolution equations to specify the models.
Perturbation Evolution
----------------------
We also rewrite the matter perturbation equation (\[deltam\]) as a function of $n$ \_[[[m]{}]{}]{}\^[”]{} = - ( 2 + ) \_[[[m]{}]{}]{}’ + \_[[[m]{}]{}]{} ( ) \_[[[m]{}]{}]{} . \[deltamnA\] One can parametrize the growth rate of the matter perturbation, $d \ln \delta_{{\rm{m}}} / d \ln a \equiv \Omega_{{\rm{m}}}^{\gamma}$ and the growth rate index, $\gamma$ can be parameterized as [@2011JCAP...03..021L] = \_[0]{} + \_[a]{} (1 - e\^[n]{} ) . \[gamma\] By using Eqs.(\[H2oH02\])-(\[gamma\]), one rewrite the Eq.(\[deltamn\]) to obtain && \_[[[m]{}]{}]{}\^ ( \_[[[m]{}]{}]{}\^ + ’ \_[[[m]{}]{}]{} + 3 Q + ) \[PA\] ,\
&=& () \[FoF0nA\] ,\
M = &=& A = \[MA\] ,\
= M &=& ( ) \[FpoF0A\] . One can solve non-linear differential equation (\[FpoF0A\]) to solve the evolution of $F/F_{0}$ with the initial condition $F[n=0]/F_{0} =1$. Both $\omega[\omega_0,\omega_a]$ and $\gamma[\gamma_0,\gamma_a]$ can be obtained from cosmological observations and thus one can obtain the k-dependent $F/F_0$ values from the observations as shown in Eq.(\[FoF0n\]).
The below table shows the all the required quantities as functions of $\omega_{0}, \omega_{a}, \gamma_{0}$, and $\gamma_{a}$.
[c cc]{}\
Obtained Quantities & Measured Quantities & Functions\
\
$\Omega_{{\rm{m}}}\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}$ & $\left( 1 + {\frac{1-\Omega_{{\rm{m}}0}}{\Omega_{{\rm{m}}0}}} e^{-3\left(\omega_0+\omega_a \right)n + 3\omega_{a}\left(e^{n}-1\right)} \right)^{-1}$\
\
${\frac{H^2}{H_{0}^2}} = {\frac{H^2\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}{H^2\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},0\right]}}$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}$ & $\Omega_{{\rm{m}}0} e^{-3n} + \left( 1 - \Omega_{{\rm{m}}0}\right) e^{-3\left(1+\omega_0+\omega_a \right)n + 3\omega_{a}\left(e^{n}-1\right)} $\
\
$Q\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}$ & $\omega\left[\omega_{0},\omega_{a},n\right] \left(1-\Omega_{{\rm{m}}}\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, n \right] \right) $\
\
${\frac{H^{'}\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}{H\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}}$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}$ & $-{\frac{3}{2}} \left( 1 + Q\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]\right) $\
\
${\frac{R\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}{H_{0}^2\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}}$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}$ & $6 {\frac{H^2\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}{H^2\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},0\right]}} \left(2 + {\frac{H^{'}\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}{H\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a},n\right]}} \right)$\
\
${\cal P}\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, n \right] $ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}$ & $\Omega_{{\rm{m}}}^{\gamma} \left( \Omega_{{\rm{m}}}^{\gamma} + \gamma' \ln \Omega_{{\rm{m}}} + 3 \gamma Q + {\frac{\left(1-3Q\right)}{2}} \right) $\
\
${\frac{F}{F_{0}}} = {\frac{F\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k, n \right]}{F\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k, 0 \right]}} $ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k$ & ${\frac{3}{2}} {\frac{\Omega_{{\rm{m}}}}{{\cal P}}} \left({\frac{1+4M}{1+3M}}\right) $\
\
$A\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k, n \right]$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k$ & ${\frac{2}{3}}{\frac{F}{F_{0}}} {\frac{{\cal P}}{\Omega_{{\rm{m}}}}} $\
\
$M\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k, n \right]$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k$ & ${\frac{1-A}{3A-4}} $\
\
${\frac{F'}{F_{0}}} = {\frac{F'\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k, n \right]}{F\left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k, 0 \right]}} $ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k$ & ${\frac{e^{2n} H_{0}^2}{k^2}} {\frac{R'}{H_{0}^2}} M {\frac{F}{F_0}}$\
\
$\omega \left[\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k, n \right]$ & $\Omega_{{\rm{m}}0}, \omega_{0}, \omega_{a}, \gamma_{0}, \gamma_{a}, k$ & $-1 + {\frac{\left( {\frac{F''}{F_0}} - {\frac{F'}{F_0}} \right) + {\frac{H'}{H}} \left({\frac{F'}{F_0}} +2{\frac{F}{F_0}} -2\right) }{3 \left(1 - \Omega_{{\rm{m}}} \right)}}$\
Acknowledgments {#acknowledgments .unnumbered}
===============
SL is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (Grant No. NRF-2015R1A2A2A01004532) and (NRF-2017R1A2B4011168.).
References {#references .unnumbered}
==========
[99]{}
T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, [Phys. Rept ]{}[**513**]{}, 1 (2012). M. J. Mortonson, D. H. Weinberg, and M. White, \[ArXiv:1401.0046\]. A. Joyce, L. Lombriser, and F. Schmidt, Ann. Rev. Nuc. Par. Sci, [**66**]{}, 95 (2016). M. Chevallier and D. Polarski, Int. Jour. Mod. Phys. D [**10**]{}, 213 (2001) E.-V. Linder, Phys. Rev. Lett [**90**]{}, 091301 (2003). S. Lee, [J. Cosmol. Astropart. Phys]{}[**3**]{}. 021 (2011).
T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys [**82**]{}, 451 (2010). A. A. Starobinsky, Phys. Lett. B [**91**]{}, 99 (1980) W. Hu and I. Sawick, Phys. Rev. D [**76**]{}, 064004 (2007) A. A. Starobinsky, Jour. Exp. Theor. Phys. Lett [**86**]{}, 157 (2007) S. Tsujikawa, Phys. Rev. D [**77**]{}, 023507 (2008) G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini, Phys. Rev. D [**77**]{}, 046009 (2008) J. P[é]{}rez-Romero and S. Nesseris, \[ArXiv:1710.05634\]. J.-C. Hwang and H. Noh, Phys. Rev. D [**66**]{}, 084009 (2002) S. Tsujikawa, Phys. Rev. D [**76**]{}, 023514 (2007)
L. Amendola, D. Polarski, and S. Tsujikawa, Phys. Rev. Lett, [**98**]{}, 131302 (2007)
|
---
abstract: 'The thermodynamic implications for the out-of-equilibrium dynamics of quantum systems are to date largely unexplored, especially for quantum many-body systems. In this paper we investigate the paradigmatic case of an array of nearest-neighbor coupled quantum harmonic oscillators interacting with a thermal bath and subjected to a quench of the inter-oscillator coupling strength. We study the work done on the system and its irreversible counterpart, and characterize analytically the fluctuation relations of the ensuing out-of-equilibrium dynamics. Finally, we showcase an interesting functional link between the dissipated work produced across a two-element chain and their degree of general quantum correlations. Our results suggest that, for the specific model at hand, the non-classical features of a harmonic system can influence significantly its thermodynamics.'
author:
- 'A. Carlisle'
- 'L. Mazzola'
- 'M. Campisi'
- 'J. Goold'
- 'F. L. Semião'
- 'A. Ferraro'
- 'F. Plastina'
- 'V. Vedral'
- 'G. De Chiara'
- 'M. Paternostro'
title: Out of equilibrium thermodynamics of quantum harmonic chains
---
The out-of-equilibrium dynamics of quantum systems offer a very interesting stage for the study of the thermodynamic properties [@esposito; @campisi; @seifert]. The establishment of quantum fluctuation theorems represents a milestone in the link between arbitrarily fast quantum dynamics and equilibrium figures of merit of thermodynamic relevance, such as feee energy changes, heat, work, and entropy [@Tasaki; @Crooks; @Jarzynski]. The definition of such quantities from a genuine quantum mechanical standpoint, the formulation of their operational interpretations, and the design of experimental techniques for their quantitative assessment are some of the drives of current research on the thermodynamic properties of quantum systems and processes [@lutz; @varie; @Abah; @Dorner; @Joshi; @Huber; @Heyl; @Ngo; @CPmaps; @Oxf; @KavanJohn; @JohnMauroKavan; @Batalhao]. An extensive programme of investigations aimed at understanding and characterising the non-equilibrium thermodynamics of simple, paradigmatic systems is currently underway, including exactly solvable extended spin models [@Joshi; @Dorner; @Smacchia; @Joshi13EPJB86; @Sindona; @dudu; @Fusco], which have offered an interesting platform for the study of the emergence of irreversible thermodynamics from quantum many-body features [@Joshi; @Dorner].
In this context, a rather privileged role is played by the quantum oscillator, which offers the possibility for the (either exact or approximate) analytical assessment of non equilibrium features in an ample range of situations, including external driving and special nonlinear cases [@Galve; @varieoscillator]. However, to the best of our knowledge, little is known on composite systems consisting of more than a single harmonic oscillator. This is an interesting case to study, as it would enable the assessment of the scaling properties of thermodynamically relevant quantities with the size of the system, as well as the study of processes involving either the whole system or only part of it, which in principle would result in different behaviors and manifestations.
This is precisely the context within which the investigation reported in this paper lies. We aim at addressing the effects that a global quench of the inter-particle coupling strength has on the phenomenology of thermodynamic quantities such as (irreversible) work and free energy differences. We study the case of an open-ended array of quadratically coupled quantum harmonic oscillators, in contact with a thermal reservoir. By allowing for a global quantum quench, we address the scaling of both the average work and the free energy differences, providing exact analytic expressions for the dissipated work, which is an important figure of merit to gauge the deviations of the actual state of the array after the quench from its counterpart at thermodynamic equilibrium. It thus gives us information about the effects of non-adiabaticity. However, this study offers even more opportunities for exploration: by calculating explicitly the amount of quantum correlations shared by the elements of a two-oscillator system, we illustrate the existence of a clear functional relation between dissipated work and quantum correlations. For the specific case of the coupling model at hand, this hints at the interdependence of quantum and thermodynamic features in quadratically coupled harmonic chains. This is a tantalising possibility that will deserve future in-depth explorations.
The remainder of this paper is organised as follows: Sec. \[modello\] introduces the harmonic model and illustrates an interferometric approach to the exact determination of the characteristic function of work distribution [@campisi] resulting from a sudden quench of the inter-oscillator coupling strength. This opens the way to the assessment of quantum fluctuation relations [@Tasaki; @Crooks; @Jarzynski] and the fully analytic calculation of the average work, free energy change and other figures of merit for the characterization of irreversibility. This study allows us to identify the degree of squeezing generated by the oscillators’ quadratic coupling as a very important resource for the ability of the process to do work on the system (see Ref. [@Galve] for a different analysis of this point made on a single harmonic oscillator). Our calculations, which are valid for chains of an arbitrary number of elements, allow for the clear identification of “classical" and “quantum" parts of both the change of free energy, which are then related to the degree of quantum correlations across a two-element chain in Sec. \[correlations\]. Finally, in Sec. \[conc\] we draw our conclusions and discuss briefly the questions opened by our study. Two appendices summarize the most technical part of our calculations.
Description of the coupling model and analysis of nonequilibrium thermodynamics {#modello}
===============================================================================
We consider coupled harmonic oscillators in an open linear configuration \[cf. Fig. \[equivalente\] [**(a)**]{}\]. While in this part of our analysis we will mostly concentrate on the case of only two coupled oscillators, the generalization to a multi-element register is addressed later on. We start from a Hooke-like coupling model between two harmonic oscillators in contact with a heat bath at temperature $T$. The model is described by the following Hamiltonian (we assume units such that $\hbar=1$ across the manuscript) $$\label{model}
\hat{\cal H}_1(g_t)=\frac{\Omega}{2}\sum^2_{j=1}(\hat x^2_j+\hat p^2_{j})+g_t(\hat x_1-\hat x_2)^2$$ with $\Omega$ the frequency of the oscillators (assumed for simplicity to be identical and with a unit mass) and $g$ the (possibly) time-dependent interaction strength. Here, $\hat x_j$ and $\hat p_j$ are the position- and momentum-like operators of oscillator $j=1,2$ (satisfying the commutation relations $[\hat x_j,\hat p_j]=i$). Within the context of our analysis we will assume that, after detaching the system from the heat bath, the coupling strength is abruptly turned on to the value $g_0>0$, namely $g_t=g_0\Theta(t)$, where $\Theta(t)$ is the Heaviside step function. This process embodies a sudden quench of the interaction between the harmonic oscillators. A straightforward calculation shows that the post-quench time evolution operator $\hat{\cal U}(t>0)=e^{-i\hat{\cal H}_1t}$ generated by Eq. can be written as $$\label{deco}
\hat{\cal U}(t>0)=\hat{\cal B}^\dag\hat{\cal S}^\dag(r)[\hat{\cal R}_1(\theta_1(t))\otimes\hat{\cal R}_2(\theta_2(t))]\hat{\cal S}(r)\hat{\cal B},$$ where $\hat{\cal B}=\exp[\pi(\hat x_1\hat p_2-\hat x_2\hat p_1)/4]$ is the $50:50$ beam-splitter operator, $\hat{\cal S}(r)=\hat\openone_1\otimes\hat{\cal S}_2(r)$ describes the local squeezing of oscillator $2$ by a degree $r=(1/4)\ln\!\sqrt{1+2g_0/\omega}$ performed by the squeezing operator $\hat{\cal S}_2(r)=\exp[i{\rm Im}(r)(\hat x^2_2-\hat p^2_2)-i{\rm Re}(r)(\hat x_2\hat p_2+\hat p_2\hat x_2)]$, $\omega=\Omega/2$ and $\hat{\cal R}_j(\theta_j)=\exp[-i\theta_j(\hat{x}^2_j+\hat{p}^2_j)]$ accounts for phase-space rotations by the angle $\theta_j$ ($j=1,2$). In the specific case of our problem we have $\theta_1(t)=\omega t$ and $\theta_2(t)=\omega t\sqrt{1+2g_0/\omega}$. In light of such decomposition, which accounts for the free evolution (each occurring at the respective frequency) of the centre-of-mass and relative-motion modes of the system, the time-evolution of the two-oscillator system can be understood as the result of the action of a Mach-Zehnder interferometer endowed with an [*active*]{} element, embodied by the local squeezer, on one of its arms \[cf. Fig. \[equivalente\](b)\]. This establishes quantum correlations between the harmonic oscillators. Our first goal here is to show that such correlations are linked with the work that is irreversibly generated in the process due to the non-adiabatic nature of the quench.
1.3cm[**(a)**]{}7.8cm\
{width="0.5\linewidth"} {width="0.8\columnwidth"}\
In order to accomplish this goal, let us briefly sketch the way to compute the characteristic function of the work probability distribution associated with the process that takes abruptly the Hamiltonian from $\hat {\cal H}_i\equiv\hat{\cal H}_1(0)$ to $\hat{\cal H}_f=\hat{\cal H}_1(g_0)$ at time $t=0$. As we will show, $\chi(u)$ can be understood in terms of the thermal convolution of inner products between displaced squeezed vacuum states. For the sudden switch of the work parameter that we are considering here, the expression for the characteristic function of work distribution takes the form $$\chi(u)={\rm Tr}[e^{iu\hat{\cal H}_f}e^{-iu\hat{\cal H}_i}\rho^{th}_S(0)],$$ where $\rho^{th}_S(0)=e^{-\beta\hat{\cal H}_i}/{\cal Z}_0$ is a pre-quench thermal-equilibrium state of the two harmonic oscillators at inverse temperature $\beta$ and ${\cal Z}_0={\rm Tr}[e^{-\beta\hat{\cal H}_1(0)}]$ is the associated partition function. In light of the structure shown in Eq. , it is convenient to decompose the pre-quench state over the single-oscillator coherent-state basis as $\rho^{th}_S(0)=\int\,d^2\alpha_1\,d^2\alpha_2\prod^2_{j=1}P^{th}_{V}(\alpha_j){\left\vert\alpha_1,\alpha_2\right\rangle}{\left\langle\alpha_1,\alpha_2\right\vert}_{12}$ with $P^{th}_{V}(\alpha_j)=2[\pi(V-1)]^{-1}\exp[{-{2|\alpha^2_j|}/({V-1})}]$ the thermal $P$-function of oscillator $j$, characterised by the variance $V=2{\overline{n}}+1$ with ${\overline{n}}=(e^{\beta\omega}-1)^{-1}$ the thermal mean occupation number. Here, $\vert{\alpha_j}\rangle=\hat{\cal D}_j(\alpha_j){\left\vert0\right\rangle}_j$ is a coherent state generated by the displacement operator $\hat{\cal D}_j(\alpha_j)=\exp[\alpha_j\hat a^\dag_j-\alpha^*_j\hat a_j]$ over the vacuum. With this at hand, we have $$\label{total}
\chi(u)=\int\!d^2\alpha_1\,d^2\alpha_2\prod^2_{j=1}P^{th}_{V}(\alpha_j)\,\chi_{\alpha_1,\alpha_2}(u)$$ with $\chi_{\alpha_1,\alpha_2}(u)=\langle{\alpha_1,\alpha_2}\vert e^{i\hat{\cal H}_f u}e^{-i\hat{\cal H}_i u}{\left\vert\alpha_1,\alpha_2\right\rangle}$ the Loschmidt echo corresponding to the evolution of a pair of initial coherent states under the process addressed here. As the interaction between the harmonic oscillators is quadratic, the Gaussian nature of coherent states is preserved across the process, and the thermal convolution in Eq. consists of a four-fold integration over Gaussian functions. We thus focus on the explicit evaluation of $\chi_{\alpha_1,\alpha_2}(u)$, whose details are given in the Appendix, and results in the elegant expression $\chi_{\alpha_1,\alpha_2}(u)=\!{\left\langle\zeta_1;\xi_1\right\vert}{\zeta_2;\xi_2}\rangle$ with $|{\zeta_j;\xi_j}\rangle=\hat{\cal D}_j(\zeta)\hat{\cal S}_j(\xi){\left\vert0\right\rangle}_j$ a displaced squeezed state ($\zeta,\xi\in\mathbb{C}$) [@Caves; @Moeller], which can be calculated analytically to be
$$\chi_{\alpha_1,\alpha_2}(u)=\frac{\exp\left[{\dfrac{[({\zeta_2}-{\zeta_1})\sinh{r}+({\zeta^*_1}-{\zeta^*_2})\cosh{r}][ ({\zeta_2}-{\zeta_1})\cosh{r}+e^{2 i \theta_2 (u)}({\zeta^*_1}-{\zeta^*_2})\sinh{r}]}{2(\cosh ^2r-\sinh^2r\, e^{2 i \theta_2 (u)})}-\dfrac{\zeta_1\zeta^*_2-\zeta^*_1\zeta_2}{2}}\right]}{\sqrt{\cosh^2r-e^{2 i \theta_2 (u)}\sinh^2{r}}}.$$
The expressions for $\zeta_{1,2}$ and $\xi_{1,2}$ are given in the Appendix. Examples of the behavior of the characteristic function for various quench strengths $g_0$ and temperatures of the initial equilibrium states are shown in Fig. \[charfunc\].
\
![(Color online) Panels [**(a)**]{} and [**(b)**]{}: Characteristic function of the work distribution after a sudden quench of the coupling strength between two harmonic oscillators coupled via a Hooke-like model. We show the behavior of ${\rm Re}[\chi(u)]$ \[panel [**(a)**]{}\] and ${\rm Im}[\chi(u)]$ \[panel [**(b)**]{}\] against $\omega{u}$ for $V=1$ and $g_0/\omega=0.1$ (solid line), $1$ (dashed line), $3$ (dotted line), and $10$ (dot-dashed line). Panels [**(c)**]{} and [**(d)**]{}: Same study as in panels [**(a)**]{} and [**(b)**]{} but for $g_0/\omega=0.75$ and $V=1$ (solid line), $3$ (dashed line), and $10$ (dotted one).[]{data-label="charfunc"}](CharFuncThermalG.pdf "fig:"){width="\linewidth"}\
\
![(Color online) Panels [**(a)**]{} and [**(b)**]{}: Characteristic function of the work distribution after a sudden quench of the coupling strength between two harmonic oscillators coupled via a Hooke-like model. We show the behavior of ${\rm Re}[\chi(u)]$ \[panel [**(a)**]{}\] and ${\rm Im}[\chi(u)]$ \[panel [**(b)**]{}\] against $\omega{u}$ for $V=1$ and $g_0/\omega=0.1$ (solid line), $1$ (dashed line), $3$ (dotted line), and $10$ (dot-dashed line). Panels [**(c)**]{} and [**(d)**]{}: Same study as in panels [**(a)**]{} and [**(b)**]{} but for $g_0/\omega=0.75$ and $V=1$ (solid line), $3$ (dashed line), and $10$ (dotted one).[]{data-label="charfunc"}](CharFuncThermal2D.pdf "fig:"){width="\linewidth"}
Looking at Fig. \[charfunc\] ${\bf (c)}$ and ${\bf (d)}$, we see that as the temperature of the initial thermal states increases ([*i.e.*]{}, as $V$ grows), the absolute value of the derivative of both the real and the imaginary part of $\chi(u)$ at $u=0$ grows. This is an important observation in light of the possibility to evaluate the average work extractable from the system after the process as $\langle W\rangle=-i\partial_u\chi(u)\vert_{u=0}$. Although the full-fledged expression of $\chi(u)$ at arbitrary values of $\beta$ is too involved to be reported here, the average work takes the compact expression $\langle W\rangle=g_0 V/2$, which is thus linear in the strength of the quench and takes the frequency-independent value $g_0/2$ in the low temperature limit $\beta\to\infty$ and grows as $g_0/(\beta\omega)$ in the classical limit for very large temperatures.
As a check that our analytic form for the characteristic function is correct we consider the Jarzynski equality $\chi(i\beta)=e^{-\beta\Delta{F}}$. The net change in free energy of the system can be evaluated using the pre- and post-quench partition functions ${\cal Z}_0$ and ${\cal Z}$, whose evaluation we now sketch. While the calculation of the pre-quenched case trivially leads to ${\cal Z}_0=4/\sinh^2(\beta\omega/2)$, in line with the tensor-product nature of the initial equilibrium state, the post-quenched one requires the evaluation of $$\begin{aligned}
{\cal Z}&={\rm Tr}[e^{-\beta\hat{\cal H}(g_0)}]={\rm Tr}[\hat{\cal B}^\dag\hat{\cal S}^\dag e^{-\sum^2_{j=1}\theta_j(\beta)(\hat x^2_j+\hat p^2_j)}\hat{\cal S}\hat{\cal B}]\\
&={\rm Tr}[e^{-\sum^2_{j=1}\theta_j(\beta)(\hat x^2_j+\hat p^2_j)}]=\frac{4}{\sinh(\beta\omega/2)\sinh(\theta_2(\beta)/2)}
\end{aligned}$$ so that $e^{-\beta\Delta{F}}=\sinh\left(\frac{\beta\omega}2\right){\rm csch}\left(\frac{\beta\omega}{2}\sqrt{1+\frac{2g_0}{\omega}}\right)$. This in turn gives us the free-energy change $$\Delta{F}=-\frac{1}{\beta}\ln\left[\frac{\sinh(\beta\omega/2)}{\sinh\left(\frac{\beta\omega}{2}\sqrt{1+2g_0/\omega}\right)}\right].$$ In the classical limit of very high temperature, this expression becomes $\Delta{F}_c\simeq(1/\beta)\ln[\sqrt{1+2g_0/\omega}]$. In the quantum limit of $\beta\to\infty$, on the other hand, the net change in free energy is bound by the asymptotic value $\Delta{F}_{q}\simeq(\omega/2)(\sqrt{1+2g_0/\omega}-1)$, which only depends on the strength of the quench (in units of $\omega$). Although we have not been able to study analytically the Jarzynski identity due to the cumbersome form of $\chi(u)$, we have numerically checked that it is satisfied.
We now analyze the degree of irreversibility of our quench process. This can be quantified by the quantity $$L = \beta W_\text{diss} = \beta [\langle W\rangle-\Delta{F}] = D[\rho_t||\rho_t^{eq}],$$ which accounts for the “nonequilibrium lag” between the actual system state $\rho_t$ and the reference thermal state $\rho_t^{eq}=e^{-\beta \hat{\cal H}(t)}/{\cal Z}(t)$ as measured by the Kullback-Leibler divergence (or relative entropy) between two arbitrary states $\rho$ and $\sigma$ and defined as $D[\rho ||\sigma]=\text{Tr}(\rho\log\rho-\rho\log\sigma)$ [@Bochkov81aPHYSA106; @Schloegl66ZP191; @Vaikuntanathan09EPL87; @Deffner10PRL105]. We find $$L=\frac{\beta g_0}{2} \coth \left(\frac{\beta \omega}{2}\right)+\ln \left[\sinh \left(\frac{\beta\omega }{2}\right) \text{csch}\left(\frac{\beta\omega}{2} \sqrt{\frac{2
g_0}{\omega}+1}\right)\right].$$ Despite being customarily referred to as “nonequilibrium entropy production”, $L$ is in general not equal to the change in thermodynamic entropy [@Joshi13EPJB86], hence we dub it more appropriately the “nonequilibrium lag”. In Fig. \[studio\] we report the analysis of average work, change in free energy, and nonequilbrium lag against the strength of the quench, as well as the assessment of the dependence of $L$ on the inverse temperature and $g$. A remarkable feature is the quasi-linear growth of the nonequilibrium lag at low temperatures \[cf. Fig. \[studio\] [**(b)**]{}\], which will be useful for the analysis reported in Sec. \[correlations\].
Another closely related quantifier of irreversibility, specifically designed for thermally isolated systems, is provided by $$\Delta{\cal E} = {\rm Tr }\, [ \rho_t \hat{\cal E} (t)- \rho_0 \hat{\cal E}(0)],$$ which is defined using the operator $$\hat{\cal E}(t) = \sum_k \ln k |k,t\rangle \langle k,t|$$ built using the eigenstates $|k,t\rangle$ of the instantaneous Hamiltonian $H(t)$. They are ordered by their increasing energy $E_k(t)> E_m(t)$ for $k>m$. The operator $\hat{\cal E}$, first introduced in Ref. [@Michele], is the quantum version of the Gibbs entropy associated with the microcanonical ensemble [@GibbsBook; @Hertz10AP338a; @Einstein11AP34; @BeckerBook; @Campisi08PRE78b; @MuensterBook; @Campisi05SHPMP36; @Dunkel14NATPHYS10]. Just like thermodynamic entropy, it remains unchanged in a slow (adiabatic) protocol and cannot decrease in a generic fast one, provided the initial density matrix is diagonal in the initial Hamiltonian eigenbasis, its eigenvalues are ordered in a non-increasing fashion, and the spectrum is non-degenerate at all times. The quantitative analysis of the behavior of $\Delta{\cal E}$ in our system, which is made possible by the knowledge of the spectrum of $\hat{\cal H}_1$ as obtained in the Appendix, will be presented elsewhere [@tocome].
{width="0.7\columnwidth"}{width="0.7\columnwidth"}{width="0.7\columnwidth"}
We now turn to the assessment of the role that squeezing has on the ability of the system to produce extractable work. In order to do so, we compare the performance of the coupling scheme addressed so far to the ability of the system to perform work when the two harmonic oscillators are coupled via the model $\hat x_1\hat p_2-\hat p_1\hat x_2$ That is, we consider the Hamiltonian $$\begin{aligned}
\hat{\cal H}_2&=\frac{\Omega}{2}\sum^2_{j=1}(\hat x^2_j+\hat p^2_j)+g_t(\hat x_1\hat p_2-\hat p_1\hat x_2).
\end{aligned}$$ There are two fundamental differences between $\hat{\cal H}_1$ and $\hat{\cal H}_2$: first, $\hat{\cal H}_2$ is energy preserving and the corresponding time propagator would not require the squeezing of any harmonic oscillator [@Helen]. As we will argue soon, this gives rise to key differences with respect to the thermodynamic behavior showcased up to this point. Second, consistently with the fact that $\hat{\cal H}_2$ is the rotating-wave form of Eq. , the strength of the quench cannot be arbitrary, as the spectrum of the Hamiltonian acquires an imaginary eigenvalue for $g_0>\Omega$.
Besides this limitation, the characteristic function associated with the process generated by a quench of $\hat{\cal H}_2$ can be worked out in a way similar to what has been sketched before for the case of Eq. . A second-order Taylor expansion of the characteristic function with respect to variable $u$ leads to the approximate expression $\chi_{\hat{\cal H}_2}(u)\simeq1-\frac{g^2_0}{16}(V^2-1)u^2+{\cal O}(u^3)$ where the subscript indicates that model $\hat{\cal H}_2$ is under scrutiny. The first moment of this distribution evaluated in $u=0$, as requested for the calculation of the average work, gives us $\langle W_{\hat{\cal H}_2}\rangle=0$, at variance with the result for the average work valid for Eq. . The reason behind such dissimilarity should be traced back to the energy-conserving nature of model $\hat{\cal H}_2$, which does not give rise to any squeezing of the oscillators. Let us go back now to the case embodied by Hamiltonian $\hat{\cal H}_1$. The results gathered so far for a two-element system can be generalised to an array of arbitrary length. In particular, the change in free energy for an array of $N$ harmonic oscillators interacting according to the Hooke-like model $$\label{modelloN}
\hat{\cal H}_1=\frac{\Omega}{2}\sum^{N}_{j=1}(\hat x^2_j+\hat p^2_{j})+g_t\sum^{N-1}_{j=1}(\hat x_j-\hat x_{j+1})^2$$ reads $$\Delta F_N=-\frac{1}{\beta}\ln\left[\frac{\sinh^N(\beta\omega/2)}{\Pi^N_{j=1}\sinh(\beta\mu_j/2)}\right]$$ with $\mu_j=\omega\sqrt{\lambda_j/\omega}$, $\omega=\Omega/2$ and $\{\lambda_j\}$ the set of eigenvalues of the adjacency matrix representing the Hamiltonian $\hat{\cal H}_1$ (cf. the Appendix). Using the characteristic function for coherent states $\chi_{\{\alpha\}}(u)$ given in Eq. and its first statistical moment, we can easily calculate the average work, which is found to scale with the number of oscillators as $$\label{workN}
\langle W\rangle_N=g_0 V\frac{N-1}{2}.$$ This formula has a very simple interpretation. Each interaction term (there are in total $N-1$ of them) brings in a contribution $g_0 V/2$ to the total work. The factor $N-1$ can also be understood by noticing the fact that, out of the $N$ modes involved in the evolution of the system resulting from the quench, only $N-1$ of them are squeezed. This is proven rigorously in the Appendix, where the spectrum of Eq. is shown to always contain the bare-oscillator value $\omega$ among $N-1$ squeezing-dependent values \[cf. Eq. \]. Physically, this is due to the fact that the centre-of-mass mode of the system of oscillators is always a normal mode of the system itself.
With the average work and the change in free energy, we can finally consider the nonequilibrium lag for $N$ oscillators $$\begin{aligned}
L&=\frac{\beta g_0 V (N-1)}{2}+\ln\left[\sinh^N\left(\frac{\beta\omega}{2}\right)\right]-\sum^N_{j=1}\ln\left[\sinh\left(\frac{\beta\mu_j}{2}\right)\right]\\
&=(N-1)\left(\frac{\beta g_0V}{2}+\ln\left[\sinh\left(\frac{\beta\omega}{2}\right)\right]\right)-\sum^N_{j=2}\ln\left[\sinh\left(\frac{\beta\mu_j}{2}\right)\right].
\end{aligned}$$ The behavior of $L$ against the length of the chain and for three values of the inverse temperature $\beta$ is reported in Fig. \[entorpycontroN\].
![(Color online) Nonequilibrium lag after a quantum quench in an array of $N$ Hooke-like coupled harmonic oscillators with $g=2\omega$ and for three values of the inverse temperature $\beta$.[]{data-label="entorpycontroN"}](EntropyControN.pdf){width="0.85\linewidth"}
Relation with quantum correlations {#correlations}
==================================
In the following, we study the possibility of establishing a direct quantitative link between the nonequilibrium lag produced by the quantum quench under scrutiny and the general quantum correlations shared by the oscillators. We will mainly restrict our attention to a two-oscillator system, so as to avoid unnecessary computational problems.
Fig. \[studio\] and our related analysis have shown the existence of a one-to-one correspondence between temperature and the nonequilibrium lag ${ L}$, which can be considered as a reliable [*thermometer*]{}, in particular in the interesting quantum region of $\beta\gg1$. In a qualitatively analogous way, it is possible to establish a link between $\beta$ and the amount of non-classical correlations (as measured by Gaussian entanglement and discord) shared by the oscillators of our array after the quench.
[**(a)**]{} {width="0.9\columnwidth"}{width="0.95\columnwidth"}
We start by addressing entanglement, which is quantified here using the logarithmic negativity. For a two-mode Gaussian state, such as the one corresponding to the equilibrium state of Hamiltonian in Eq. at inverse temperature $\beta$ , the latter is defined as $$\text{E}=\max[0,-\ln\nu_-].$$ Here, $\nu_-$ is the smallest eigenvalue of the matrix $|i{\bm \Sigma}{\text P}{\bm \sigma}{\text P}|$, where ${\text P}=\text{diag}[1,1,1,-1]$ performs the inversion of momentum of the second harmonic oscillator, ${\bm \Sigma}=i{\bm\sigma}_y\otimes{\bm \sigma}_y$ is the symplectic matrix (with $\sigma_y$ the y-Pauli matrix) and ${\bm \sigma}$ is the covariance matrix of the two-oscillator system [@ferraro]. The latter can be easily calculated using the formal analogy with an optical interferometer discussed above and used to calculate the characteristic function of the work distribution. The results of our calculations are shown in Fig. \[entanglement\], where the logarithmic negativity is plotted against the inverse temperature at three values of the quench amplitude. Analytically $$\text{E}=\max\left[0,-\ln\frac{\sqrt{\left[1+\text{csch}\left(\frac{\beta \omega }{2}\right)\right]\left[1+\text{csch}\left(\frac{\beta\omega}{2}
\sqrt{1+\frac{2 g_0}{\omega }}\right)\right]}}{\sqrt[4]{\left({1+2 g_0/\omega }\right)}}\right],$$ which reaches the maximum value given by $E=\ln\sqrt[4]{1+2g_0/\omega}$ for $\beta\to\infty$. The two-oscillator entanglement disappears above a threshold temperature whose value depends on the ratio $g_0/\omega$.
![(Color online) Comparison between the full form of the nonequilibrium lag $L$ and its classical counterpart $L_c$ shown against the inverse temperature $\beta$ and the dimensionless interaction strength $g_0/\omega$. At high temperature $L\to L_c$, regardless of the strength of the quench.[]{data-label="nonequilag"}](SirrGandBeta.pdf){width="\linewidth"}
We now aim at comparing the behavior of $\text{E}$ to that of the ‘quantum’ part of the nonequilibrium lag, i.e. the part of ${ L}$ that remains after subtracting the high-temperature value $L_c\equiv\lim_{\beta\to0}{ L}=g_0/\omega-\ln\sqrt[4]{1+2g_0/\omega}$. As seen in Fig. \[nonequilag\], at low temperatures and large coupling strengths, the quantum part of $L$ is crucial in determining quantitatively the non equilibrium lag. In Fig. \[entanglement\] [**(b)**]{} we thus plot the logarithmic negativity against the quantum part $L_q\equiv{ L}-L_c$ of the nonequilibrium lag, by eliminating the inverse temperature, showing that a direct relation exists between such quantities, which appear to be in mutual functional dependence. The (in general) involved non-linear relation of each of them with the inverse temperature prevents us from finding such dependence explicitly. However, some insight can be gathered from the behavior shown in Fig. \[entanglement\] [**(b)**]{}, such as the existence of a (quench-dependent) threshold above which the logarithmic negativity becomes insensitive to the actual value of $L_q$. As the inverse temperature embodies the curvilinear abscissa of each of the curves displayed in Figs. \[entanglement\], we can identify the region of insensitivity to the nonequilibrium lag as the low-temperature part of Fig. \[entanglement\] [**(a)**]{}. However, the large-temperature part of Fig. \[entanglement\] [**(b)**]{} is somehow misleading: at large temperature, entanglement is strictly null while $L_q$ might well achieve, in general, non-zero values. As the existence of such a temperature-dependent threshold for the non-nullity of entanglement is an expected common feature of entanglement measures, this induces us to consider entanglement as a somehow unfit figure of merit for a comparison between the behavior of quantum correlations and the nonequilibrium lag produced across the process. We thus turn our attention to the measure of quantum correlations embodied by the Gaussian discord [@GiordaParis]: for a Gaussian state with covariance matrix ${\bm\sigma}=\begin{pmatrix}{\bm \alpha}_1&{\bm \gamma}\\{\bm\gamma}&{\bm\alpha}_2\end{pmatrix}$, discord is defined as $${\rm D}=f(\sqrt{\det{{\bm\alpha}_2}})-f(\nu_-)-f(\nu_+)+\inf_{{\bm\sigma}_0}f(\sqrt{\det\epsilon}).$$ Here, $f(x)=(x+1)/2\ln[(x+1)/2]-(x-1)/2\ln[(x-1)/2]$, $\nu_\pm$ are the symplectic eigenvalues of ${\bm\sigma}$, ${\bm\epsilon}={\bm\alpha}_1-{\bm\gamma}({\bm\alpha}_2+{\bm\sigma}_0)^{-1}{\gamma}^T$ is the Schur complement of ${\bm\alpha}_1$ and ${\bm\sigma}_0$ is the covariance matrix of a single-mode rotated squeezed state.
The results of the calculations are shown in Fig. \[discord\]. First, panel [**(a)**]{} shows that, at variance with entanglement, Gaussian discord allows for no threshold in temperature and it disappears only for $\beta=0$. Second, albeit panel [**(b)**]{} is qualitatively similar to Fig. \[entanglement\] [**(b)**]{}, the analysis of the former is less ambiguous as both $D$ and $L_q$ vanish at infinite temperatures only. Although valid for the specific case of our system and so far limited to a study of only two-body quantum correlations, our analysis suggests the existence of a clear functional link between the amount of general quantum correlations established between two of the interacting harmonic oscillators studied here and the amount of nonequilibrium lag generated in a quantum-quench. It would be interesting to extend our analysis to multipartite figures of merit for quantum correlations. This is, per se, a rather difficult problem due to the current lack of computable quantifiers of genuinely multipartite quantum correlations.
[**(a)**]{} {width="0.95\columnwidth"}{width="\columnwidth"}
Conclusions {#conc}
===========
We have characterised the dynamics of relevant quantum and thermodynamic properties of an array of coupled harmonic oscillators in thermal equilibrium and experiencing a sudden quench in the inter-particle coupling strength. We have provided useful analytic expressions for the characteristic function of work distribution, the reversible and dissipated work, and the variation of free energy, which have allowed us to study quantum fluctuation identities in relation to the degree of squeezing induced by the dynamics. Our results showcase an interesting functional dependence of the irreversible lag with respect to the degree of quantum correlations across a two-oscillator system, thus suggesting a direct influence of quantum correlations in the settling of thermodynamic features.
APPENDIX {#a1 .unnumbered}
========
We aim at evaluating the function $\chi_{\alpha_1,\alpha_2}(u)={}_{12}\langle{\alpha_1,\alpha_2}\vert e^{iu\hat{\cal H}_f}e^{-iu\hat{\cal H}_i}{\left\vert\alpha_1,\alpha_2\right\rangle}_{12}$. In what follows, we will use the decomposition of the time-evolution operator in Eq. and the fact that $\exp[-i\hat{\cal H}_iu]=\bigotimes^2_{j=1}e^{i\theta_j(t)(\hat x^2_j+\hat p^2_j)}$. We find
$$\label{esplicito}
\begin{aligned}
\chi_{\alpha_1,\alpha_2}(u)&=e^{i\frac{\theta_2(u)-\omega u}{2}}{}_1\!{\left\langle\alpha_-\right\vert}{}_2\!{\left\langle\alpha_+\right\vert}\hat{\cal S}^\dag_2(r)\hat{\cal S}_2(r e^{2i\theta_2(u)}){\left\vert\alpha_-\right\rangle}_1\vert{\alpha_+e^{-i\theta_1(u)+i\theta_2(u)}}\rangle_2\\
&=e^{i\frac{\theta_2(u)-\omega u}{2}}{}_2\!{\left\langle0\right\vert}\hat{\cal D}^\dag_2(\alpha_+)\hat{\cal S}^\dag_2(r)\hat{\cal S}_2(re^{2i\theta_2(u)})\hat{\cal D}(\alpha_+e^{-i\theta_1(u)+i\theta_2(u)}){\left\vert0\right\rangle}_2.
\end{aligned}$$
with $\alpha_\pm=(\alpha_1\pm\alpha_2)/\sqrt 2$. Eq. can be put into the form of an overlap between displaced squeezed states by exploiting the operator identity $$\hat{\cal S}(\xi)\hat{\cal D}(\zeta)\hat{\cal S}^\dag(\xi)=\hat{\cal D}(\zeta\cosh|\xi|+\zeta^*e^{i\arg\xi}\sinh|\xi|),$$ which is valid for any $\zeta,\xi\in\mathbb{C}$. The order of squeezing and displacement operators can thus be swapped to get $\chi_{\alpha_1,\alpha_2}(u)=e^{i\frac{\theta_2(u)-\omega u}{2}}\!{\left\langle\zeta_1;\xi_1\right\vert}\zeta_2;\xi_2\rangle$ with $$\begin{aligned}
\zeta_1&={\alpha_+}\cosh r+{\alpha^*_+}\sinh r,\\
\zeta_2&=[{\alpha_+}e^{-i\omega u}\cosh r+{\alpha^*_+}e^{i\omega u}\sinh r]e^{i\theta_2(u)},\\
\xi_1&=r,~\xi_2=r e^{2i\theta_2(u)}.
\end{aligned}$$
We now sketch the formal procedure for the generalization of the approach discussed above to the case of a harmonic chain of an arbitrary number of oscillators coupled through the Hooke-like model $$\label{modelN}
\hat{\cal H}^N_1=\omega\sum^N_{j=1}(\hat x^2_j+\hat p^2_j)+g_t\sum^{N-1}_{j=1}(x_j-x_{j+1})^2,$$ which generalises Eq. . In the basis of the quadratures $\hat{\bm r}=(\hat x_1,\dots,\hat{x}_N,\hat p_1,\dots,\hat p_N)^T$, the Hamiltonian is represented by the block matrix $\hat{\cal H}^N_1=\hat{\bm r}^{T}{H}^N_1\hat{\bm r}$ reading $$H^N_1=
\begin{pmatrix}
{\mathbb V}&{\mathbb O}\\
{\mathbb O}&{\mathbb K}
\end{pmatrix}$$ with ${\mathbb O}$ the identically null matrix, ${\mathbb K}=\omega\openone_N$ the matrix representing the kinetic-energy term and $$\label{V}
{\mathbb V}=
\begin{pmatrix}
\omega+g_t & -g_t & & 0\\
-g_t & \omega+2g_t & -g_t & \\
& \ddots & \ddots & \\
& -g_t & \omega+2g_t & -g_t \\
0 & & -g_t & \omega+g_t \end{pmatrix}$$ that stands the potential energy of the Hamiltonian. Eq. embodies a symmetric quasi-uniform tridiagonal (QUT) matrix, whose spectrum can be fully characterised analytically. In fact, by shifting and rescaling its entries as $-(1/g_t)[{\mathbb V}-(\omega+2g_t)\openone_N]$, we get a special case of the QUT matrices explicitly addressed in Ref. [@Banchi]. The eigenvalues $\{\lambda_j\}$ of such matrix can be analytically computed and give $$\label{spettro}
\lambda_j=\omega+2g_t\left(1-\cos\left[\frac{\pi(j-1)}{N}\right]\right)~~~~j=1,..,N$$ which shows that there is always one eigenvalue equal to the bare oscillator frequency $\omega$. As we will see, this has quite remarkable consequences and is strongly tied with the results valid for the two-oscillator case addressed in the main text. The diagonalization of ${\mathbb V}$ is achieved through an orthogonal matrix ${\mathbb P}$ (which can be fully determined regardless of $N$ [@Banchi]) that leaves ${\mathbb K}$ unaffected. Following the general protocol put forward in Ref. [@Reck], such matrix can be easily broken down into a cascade of beam-splitters and phase rotators. Therefore $$PH^N_1P^{T}\equiv H_{D^N_1}=\begin{pmatrix}
{\mathbb V_D}&{\mathbb O}\\
{\mathbb O}&{\mathbb K}
\end{pmatrix}$$ with $P={\mathbb P}^{T}\oplus{\mathbb P}$ and ${\mathbb V}_D=\text{diag}[\lambda_1,\dots,\lambda_N]$. Matrix $H_{D^N_1}$ corresponds to a Hamiltonian term of the form $$\label{Nm1sq}
\hat{\cal H}_{{\cal D}^N_1}=\omega(\hat X^2_1+\hat P^2_1)+\sum^N_{j=2}[\lambda_j\hat X^2_j+\omega\hat P^2_j]$$ with $(\hat X_j,\hat P_j)$ the new modes of the system. Eq. has been deliberately written in a way to emphasize that only $N-1$ oscillators are squeezed. Therefore, by applying the squeezing operator $\hat{\cal S}^N=\openone_1\otimes\left[\otimes^{N}_{j=2}\hat{\cal S}_j(r_j)\right]$ we can transform the time-evolution operato generated by the initial model as $$\hat{U}^N(t)=e^{-i\hat{\cal H}^N_1 t}=\hat{\cal P}^\dag\hat{\cal S}^{N\dag}\left[\otimes^N_{j=1}\hat{\cal R}_j(\theta_j(t))\right]\hat{\cal S}^N\hat{\cal P}$$ with $\hat{\cal P}$ the operator corresponding to the transformation matrix $P$ and $\theta_j(t)=\lambda_j t$. This is in formal correspondence with what has been illustrated for the two-oscillator case.
Let us concentrate now on the (so far unspecified) operator $\hat{\cal P}$. As mentioned, this can be decomposed into a suitable sequence of beam-splitting and phase-rotation operations. For the sake of completeness, in Fig. \[completo\] [**(a)**]{} and [**(b)**]{} we provide a pictorial representation of the equivalent interferometer and the sequence of beam-splitting and phase-rotation operations needed for the case of four oscillators. However, although useful in order to identify the correct sequence of operations that would realise $\hat{\cal P}$, we do not actually need to determine the full-fetched decomposition in order to be able to understand the effect that such transformation has overall. Indeed, it is enough to have the entries of $P$ to determine the transformation laws of the oscillators’ quadratures as $\hat r_i\to \sum^N_{j=1}P_{ji}\hat r_j$ $(r=x,p)$. It takes a straightforward calculation to check that, when applied to the tensor product of $N$ coherent states $\otimes^N_{i=1}{\left\vert\alpha_i\right\rangle}_i$, this leads to $$\otimes^N_{i=1}{\left\vert\alpha_i\right\rangle}_i\to e^{i\varphi(P)}\bigotimes^N_{i=1}{\left\vert\sum^N_{j=1}P_{ji}\hat \alpha_i\right\rangle}_i,$$ with $\varphi(P)$ a phase that depends on the set of amplitudes $\alpha_i$ and the entries of $P$. Therefore, the calculation of the characteristic function of the work distribution for an initial thermal equilibrium state of $N$ coupled harmonic oscillators can proceed along the lines of the approach sketched in the main text for two modes only, resulting in $$\chi(u)=\int d^2\alpha_1\cdots\int d^2\alpha_n\Pi^N_{j=1}P^{th}_{V}(\alpha_j)\chi_{\{\alpha\}}(u)$$ with $\chi_{\{\alpha\}}(u)$ the characteristic function of work for a collection of $N$ modes, each initially prepared in a coherent states of amplitude $\alpha_j$ and reading $$\label{tantiN}
\chi_{\{\alpha\}}(u)=e^{\frac{i}{2}\sum^N_{j=1}\theta_j(u)-i\frac{N}{2}\omega u}\Pi^N_{j=2}{}\langle\zeta_{1,j};\xi_{1,j}|\zeta_{2,j};\xi_{2,j}\rangle.$$ Here, $\zeta_{1(2),j}$ and $\xi_{1(2),j}$ are the amplitudes of the displacement and squeezing operations, respectively, of the displaced squeezed states of mode $j=2,..,N$ that enter into the definition of $\chi_{\{\alpha\}}(u)$. Their expressions can be gathered easily in a way analogues to what has been done for just two oscillators.
[**(a)**]{}8.5cm[**(b)**]{}\
{width="0.7\columnwidth"}3.5cm{width="0.7\columnwidth"}
MP thanks Leonardo Banchi for useful discussions on the topic of Ref. [@Banchi]. AC acknowledges the Northern Ireland DEL for support. LM is supported by the EU through a Marie Curie IEF Fellowship. MP acknowledges hospitality by the Centro de Ciências Naturais e Humanas at the Universidade Federal do ABC (UFABC) during the early stages of this work. This work has been supported by the UK EPSRC (EP/G004579/1 and EP/L005026/1), the Alexander von Humboldt Stiftung, the John Templeton Foundation (grant ID 43467), and the EU Collaborative Project TherMiQ (Grant Agreement 618074). MC thanks the Volkswagen Foundation (project No. I/83902). FLS is a member of the Brazilian National Institute of Science and Technology of Quantum Information (INCT-IQ) and acknowledges partial support from CNPq (grant nr. 308948/2011-4). FLS and MP are supported by the CNPq “Ciência sen Fronteiras” programme through the “Pesquisador Visitante Especial” initiative (grant nr. 401265/2012-9). VV acknowledges funding from the National Research Foundation (Singapore), the Ministry of Education (Singapore), the EPSRC (UK), the Templeton Foundation, the Leverhulme Trust, the Oxford Martin School and the Fell Fund (Oxford).
[99]{}
M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod. Phys.Ê [**81**]{}Ê 1665–1702Ê (2009).
M. Campisi, P. Hänggi, and P. Talkner, Rev. Mod. Phys. [**83**]{}, 771 (2011), [*i*bid. ]{} [**83**]{}, 1653 (2011).
U. Seifert, Rep. Prog. Phys. [**75**]{}, 126001 (2012).
H. Tasaki, [arXiv:cond-mat/0009244v2]{}; J. Kurchan, [arXiv:cond-mat/0007360v2]{}; S. Mukamel, Phys. Rev. Lett. [**90**]{}, 170604 (2003).
G. E. Crooks, Phys. Rev. E [**60**]{}, 2721 (1999).
C. Jarzynski, Phys. Rev. Lett. [**78**]{}, 2690 (1997).
P. Talkner, E. Lutz and P. Hänggi, Phys. Rev. E [**75**]{}, 050102R (2007).
J. P. Pekola, P. Solinas, A. Shnirman, and D. V. Averin, arXiv:1212.5808 (2012); V. Vedral, arXiv:1204.6168 (2012); J. Phys. A: Math. Theor. [**45**]{}, 272001 (2012); K. Micadei, R. M. Serra, L. C. Celeri, arXiv:1211.0506 (2012); D. Kafri and S. Deffner, Phys. Rev. A [**86**]{}, 044302 (2012).
O. Abah, J. Rossnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and Eric Lutz, Phys. Rev. Lett [**109**]{}, 203006 (2012).
R. Dorner, J. Goold, C. Cormick, M. Paternostro, and V. Vedral, Phys. Rev. Lett. [**109**]{}, 160601 (2012).
A. Silva, Phys. Rev. Lett. [**101**]{},120603 (2008). G. Huber, F. Schmidt-Kaler, S. Deffner and E. Lutz, Phys. Rev. Lett. [**101**]{}, 070403 (2008).
M. Heyl, and S. Kehrein, Phys Rev Lett [**108**]{}, 190601 (2012).
V. A. Ngo, and S. Haas, Phys. Rev. E [**86**]{}, 031127 (2012); T. Albash, D. A. Lidar, M. Marvian, and P. Zanardi, [*ibid.*]{} [**88**]{}, 032146 (2013).
M. Campisi, P. Talkner, and P. Hänggi, Phys. Rev. Lett. [**102**]{}, 210401 (2009);
R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold and V. Vedral, Phys. Rev. Lett. [**110**]{}, 230601 (2013); L. Mazzola, G. De Chiara, and M. Paternostro, Phys. Rev. Lett. [**110**]{}, 230602 (2013); L. Mazzola, G. De Chiara, and M. Paternostro, arXiv:1401.0566 (2014).
J. Goold, and K. Modi, arXiv:1401.4088 (2014).
J. Goold, M. Paternostro, and K. Modi, arXiv:1402.4499 (2014).
T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, arXiv:1308.3241 (2013).
P. Smacchia, and A. Silva, Phys. Rev. E [**88**]{}, 042109 (2013).
D. G. Joshi, M. Campisi, The European Physical Journal B **86**, 157 (2013).
A. Sindona, N. Lo Gullo, J. Goold, and F. Plastina, arXiv:1309.2669 (2013)
E. Mascarenhas, H. Bragan$\c{c}$a, R. Dorner, M. Fran$\c{c}$a Santos, V. Vedral, K. Modi, and J. Goold, arXiv:1307.5544 (2013).
L. Fusco, [*et al.*]{}, (to appear, 2014).
S. Deffner, and E. Lutz, Phys. Rev. E [**77**]{}, 021128 (2008); M. Campisi, [*ibid.*]{} [**78**]{}, 051123 (2008); P. Talkner, P. Sekhar Burada, and P. Hänggi, [*ibid.*]{} [**78**]{}, 011115 (2008); T. Monnai, Phys. Rev. E [**81**]{}, 011129 (2010); S. Deffner, O. Abah, and E. Lutz, Chem. Phys. [**375**]{}, 200 (2010); J. M. Horowitz, Phys. Rev. E [**85**]{}, 031110 (2012).
F. Galve, and E. Lutz, Phys. Rev. A [**79**]{}, 055804 (2009). C. M. Caves, Phys. Rev. D [**23**]{}, 1693 (1981).
K. B. Møller, T. G. Jørgensen, and J. P. Dahl, Phys. Rev. A [**54**]{}, 5378 (1996).
M. Paternostro, H. McAneney, and M. S. Kim, Phys. Rev. Lett. [**94**]{}, 070501 (2005).
L. Banchi, and R. Vaia, J. Math. Phys. [**54**]{}, 043501 (2013).
M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Phys. Rev. Lett. [**73**]{}, 58 (1994).
A. Ferraro, S. Olivares, and M. G. A. Paris, [*Gaussian states in continuous variable quantum information*]{} (Bibliopolis, Napoli, 2005).
P. Giorda and M. G. A. Paris, Phys. Rev. Lett. [**105**]{}, 020503 (2010).
G.N. [Bochkov]{}, Y.E. [Kuzovlev]{}, Physica A **106**, 443 (1981).
F. Schl[ö]{}gl, Z. Phys. **191**, 81 (1966).
S. Vaikuntanathan, C. Jarzynski, EPL **87**, 60005 (2009).
S. Deffner, E. Lutz, Phys. Rev. Lett. **105**, 170402 (2010).
M. Campisi, Stud. Hist. Phil. Mod. Phys.Ê [**39**]{}, 181 (2008).
J. Gibbs, *Elementary Principles in Statistical Mechanics* (Yale U. P., New Haven, 1902)
P. Hertz, Ann. Phys. (Leipzig) **338**, 225 (1910).
A. Einstein, Annalen der Physik **34**, 175 (1911).
R. Becker, *Theory of Heat* (Springer, New York, 1967)
M. Campisi, Phys. Rev. E **78**, 051123 (2008).
A. Münster, *Statistical thermodynamics*, Vol. 1 (Springer, Berlin, 1969)
M. Campisi, Stud. Hist. Phil. Mod. Phys. **36**, 275 (2005).
J. Dunkel, S. Hilbert, Nat Phys **10**, 67 (2014).
A. Carlisle, [*et al.*]{}, to appear (2014).
|
---
abstract: 'We show that a semibounded Toeplitz quadratic form is closable in the space $\ell^2({\Bbb Z}_{+})$ if and only if its entries are Fourier coefficients of an absolutely continuous measure. We also describe the domain of the corresponding closed form. This allows us to define semibounded Toeplitz operators under minimal assumptions on their matrix elements.'
address: |
IRMAR, Université de Rennes I\
Campus de Beaulieu, 35042 Rennes Cedex, FRANCE
author:
- 'D. R. Yafaev'
date: 'Month dd, yyyy'
title: On semibounded Toeplitz operators
---
[Primary 47A05, 47A07; Secondary 47B25, 47B35]{}
Toeplitz quadratic forms, closed operators and quadratic forms, absolutely continuous measures
INTRODUCTION. MAIN RESULTS
===========================
[**1.1.**]{} Toeplitz operators $T$ can formally be defined in the space $\ell^2({\Bbb Z}_{+})$ of sequences $g=(g_{0}, g_{1}, \ldots)$ by the formula $$(T g)_{n}=\sum_{m=0}^\infty t_{n-m} g_m, {\quad}n=0,1,\ldots.
\label{eq:HD}$$ Thus the matrix elements of a Toeplitz operator depend on the difference of the indices only. So it is natural to expect that properties of Toeplitz operators are close to those of discrete convolution operators acting in the space $\ell^2({\Bbb Z})$.
The precise definition of the operator $T$ requires some accuracy. Let $\cal D \subset \ell^2({\Bbb Z}_{+}) $ be the dense set of sequences $g=\{g_{n}\}_{n\in {\Bbb Z}_{+}} $ with only a finite number of non-zero components. If the sequence $t=\{t_{n}\}_{n\in {\Bbb Z}} \in \ell^2({\Bbb Z}) $, then for $g\in \cal D $ sequence [eq:HD]{} belongs to $ \ell^2({\Bbb Z}_{+}) $. In this case the operator $T$ is defined on $\cal D $, and it is symmetric if $t_{n}= {\overline}{t_{-n}}$. Without any a priori assumptions on $t_{n}$, only the quadratic form $$t[g,g] =\sum_{n,m\geq 0} t_{n-m} g_{m}\bar{g}_{n}
\label{eq:QFq}$$ is well defined for $g\in \cal D$.
The theory of Toeplitz operators is a very well developed subject. We refer to the books [@Bo] and [@GGK] (Chapter XII), [@NK] (Chapters B.4 and B.6), [@Pe] (Chapter 3) for basic information on this theory.
Let us state a necessary and sufficient condition for a Toeplitz operator $T$ to be bounded. Below $d{\bf m}$ is always the normalized Lebesgue measure on the unit circle $\Bbb T$.
\[B-H\] A Toeplitz operator $T$ $($defined, possibly, via its quadratic form [eq:QFq]{}$)$ is bounded if and only if the $t_{n}$ are the Fourier coefficients of some bounded function on $\Bbb T$: $$t_{n} =\int_{\Bbb T} z^{-n} w (z) d{\bf m} (z) , {\quad}n\in{\Bbb Z}, {\quad}w \in L^\infty (\Bbb T; d {\bf m}).
\label{eq:BH}$$
However results on unbounded Toeplitz operators are very scarce. We can mention only the paper [@Hartman] by P. Hartman and the recent survey [@Sarason] by D. Sarason; see also references in these articles.
[**1.2.**]{} In this paper, we consider semibounded Toeplitz operators in the space $\ell^2({\Bbb Z}_{+})$. [*We always suppose that $t_{n}= {\overline}{t_{-n}}$ so that the quadratic form [eq:QFq]{} is real and assume that $$t[g,g] \geq \gamma \| g\|^2 ,{\quad}g\in{\cal D}, {\quad}\| g\| =\| g\|_{\ell^2({\Bbb Z}_{+})},
\label{eq:T1}$$ for some* ]{} $\gamma\in \Bbb R$. In this case, we are tempted to define $T$ as a self-adjoint operator corresponding to the quadratic form $t[g,g]$. Such an operator exists if the form $t[g,g]$ is closable in the space $\ell^2 ({\Bbb Z}_{+})$, but as is well known this is not always true (see Example \[ex\], below). We refer to the book [@BSbook] for basic information concerning these notions; they are also briefly discussed in Subsection 2.1. We recall that, by definition, the operator corresponding to the form $t[g,g] + {\delta}\| g\|^2 $ is given by the equality $T_{{\delta}} = T+{\delta}I$ (observe that the identity operator $I$ is a Toeplitz operator). Also by definition, if a form $t[g,g] $ is closable, then all forms $t[g,g] + {\delta}\| g\|^2$ are closable. Therefore we can suppose that the number $\gamma$ in [eq:T1]{} is positive; for definiteness, we choose $\gamma=1$.
We proceed from the following well known result (see, e.g., §5.1 of the book [@AKH]) that is a consequence of the F. Riesz-Herglotz theorem.
\[R-H\] The condition $$\sum_{n,m\geq 0} t_{n-m} g_{m}\bar{g}_{n}\geq 0, {\quad}\forall g\in \cal D,
\label{eq:QF}$$ is satisfied if and only if there exists a non-negative $($finite$)$ measure $dM(z)$ on the unit circle $\Bbb T$ such that the coefficients $t_{n }$ admit the representations $$t_{n} =\int_{\Bbb T} z^{-n} dM (z) , {\quad}n\in{\Bbb Z}.
\label{eq:WH}$$
Equations [eq:WH]{} for the measure $dM (z)$ are known as the trigonometric moment problem. Of course their solution is unique. Note that for the Lebesgue measure $d {\bf m} (z)$ we have $t_{0 }=1$ and $t_{n }=0$ for $n\neq 0$. Therefore the measure corresponding to the form $t[g,g] + {\delta}\| g\|^2 $ equals $dM (z)+ {\delta}d {\bf m} (z)$. So we have a one-to-one correspondence between Toeplitz quadratic forms satisfying estimate [eq:T1]{} and real measures satisfying the condition $M (X)\geq \gamma {\bf m} (X)$ for all Borelian sets $X\subset{\Bbb T}$.
Our goal is to find necessary and sufficient conditions for the form $t[g,g]$ to be closable. The answer to this question is strikingly simple.
\[T1\] Let the form $t[g,g]$ be given by formula [eq:QFq]{} on elements $g\in {\cal D}$, and let the condition [eq:T1]{} be satisfied. Then the form $t[g,g] $ is closable in the space $\ell^2 ({\Bbb Z}_{+})$ if and only if the measure $dM (z)$ in the equations [eq:WH]{} is absolutely continuous.
Of course Theorem \[T1\] means that $dM (z)= w(z) d{\bf m}(z)$ where the function $w \in L^1 (\Bbb T; d{\bf m})$ and $w (z)\geq \gamma$. Thus Theorem \[T1\] extends Theorem \[B-H\] to semibounded operators. The function $w (z)$ is known as the symbol of the Toeplitz operator $T$. So Theorem \[T1\] shows that for a semibounded Toeplitz operator (even defined via the corresponding quadratic form), the symbol exists and is a semibounded function.
[**1.3.**]{} The proof of Theorem \[T1\] will be given in the next section. The closure of the form $t[g,g] $ is described in Theorem \[Clo\].
The discussion of these results as well as their comparison with similar statements for Hankel operators are postponed until Section 3. We there also explain shortly how our results extend to vectorial Toeplitz operators.
PROOF OF THEOREM \[T1\]
=======================
[**2.1.**]{} Let $t[g,g]$ be a quadratic form defined on a set $\cal D$ dense in a Hilbert space $\cal H$ and satisfying inequality [eq:T1]{} where $\| g\|$ is the norm of $g\in\cal H$. Suppose that $\gamma=1$, consider the norm $\| g\|_{T}= \sqrt{t[g,g]}$ and introduce the closure ${\cal D}[t]$ of $\cal D$ in this norm. If ${\cal D}[t]$ can be realized as a subset of $\cal H$, then one says that $t[g,g]$ is closable in the space $\cal H $; this means that the conditions $$\| g^{(k)}\|\to 0 {\quad}{\rm and} {\quad}\| g^{(k)} - g^{(l)}\|_{T}\to 0$$ as $k,l\to\infty$ imply that $\| g^{(k)}\|_{T}\to 0$. It is easy to see that if $T_{0}$ is a symmetric semibounded operator on $\cal D$, then the form $ t[g,g]= ( g, T_{0} g)$ is closable.
If the form $t[g,g]$ is closable, then ${\cal D}[t]\subset \cal H$ is a closed set with respect to the norm $\| \cdot \|_{T}$. In this case $t[g,g]$ is defined by continuity on all $g\in {\cal D}[t]$, and one says that the form $t[g,g]$ is closed on ${\cal D}[t] $. For a closed form there exists a unique self-adjoint operator $T$ such that $T\geq I$ and $$\begin{aligned}
t[g,h] &= ( g, T h), {\quad}\forall g\in {\cal D}[t], {\quad}\forall h\in {\cal D} (T)\subset {\cal D}[t],
\\
t[g,g]&= \| \sqrt{T} g\|^2, {\quad}\forall g\in {\cal D}(\sqrt{T}) = {\cal D}[t].
\end{aligned}$$ Note that the domain $ {\cal D} (T) $ of $T$ does not admit an efficient description.
We are going to use these general definitions for the space ${\cal H} = \ell^2 ({\Bbb Z}_{+})$ and the Toeplitz quadratic forms [eq:QFq]{}.
Of course quadratic forms, in particular, the Toeplitz forms, are not necessarily closable.
\[ex\] Let $t_{n}=1$ for all $n\in {\Bbb Z}$. Adding the term $\| g\|^2$, we obtain the form $$t[g,g]= \big| \sum_{n\geq 0} g_{n}\big|^2 + \sum_{n\geq 0}| g_{n}|^2$$ satisfying inequality [eq:T1]{} with $\gamma=1$. Define the sequence $g^{(k)}\in {\cal D}$ by the equalities $g_{n}^{(k)}=k^{-1}$ for $0\leq n < k$ and $g_{n}^{(k)}=0$ for $ n\geq k$. Then $\| g^{(k)}\|=k^{-1/2}\to 0$ as $k\to\infty$. Since $\sum_{n \geq 0} g_{n}^{(k)}=1$, we have $ \| g^{(k)} - g^{(l)}\|_{T}= \| g^{(k)} - g^{(l)}\|\to 0$ as $k,l\to\infty$. Nevertheless $ \| g^{(k)} \|_{T} \geq 1$.
Note that the measure $d M(z)$ corresponding to the sequence $t_{n}=1$, $\forall n\in {\Bbb Z}$, is supported by the point $1 \in {\Bbb T}$: $M (\{1\})=1$, $M ({\Bbb T} \setminus \{1\})=0$.
On the other hand, we have the following simple assertion.
\[ex1\] If a sequence $\{t_{n}\}_{n\in{\Bbb Z}}\in\ell^2 ({\Bbb Z})$, then the form [eq:QFq]{} is closable.
Now we have $ t[g,g]= ( g, T_{0} g)$ where the symmetric operator $T_{0}$ is defined by formula [eq:HD]{} on the set $\cal D$.
[**2.2.**]{} As already mentioned, by the proof of Theorem \[T1\] we may suppose that estimate [eq:T1]{} is true for $\gamma=1$. According to Theorem \[R-H\] the equations [eq:WH]{} are satisfied with a measure $dM (z)$ such that $M(X)\geq {\bf m} (X)$ for all Borelian sets $X\subset \Bbb T$; in particular, the measure $dM (z)$ is positive.
Our proof relies on the following auxiliary construction. Let $ L^2 (\Bbb T; dM)$ be the space of functions $u (z)$ on $\Bbb T$ with the norm $$\| u\|_{ L^2 (\Bbb T; dM)}=\sqrt{\int_{\Bbb T} | u(z)|^2 dM(z)}.$$ We put $$({\cal A}g) (z)=\sum_{n=0}^\infty g_{n} z^n
\label{eq:A}$$ and observe that ${\cal A} g\in L^2 (\Bbb T; dM)$ for all $g \in\cal D$. Therefore we can define an operator $A \colon \ell^2 ({\Bbb Z}_{+})\to L^2 (\Bbb T; dM)$ on domain $\cal D (A)=\cal D$ by the formula $Ag={\cal A}g$. In view of equations [eq:WH]{}, the form [eq:QFq]{} can be written as $$\begin{gathered}
t [g,g] =\sum_{n,m\geq 0}\int_{\Bbb T}z^{-n+m} d M(z) g_m {\overline}{g_{n} }
\\
=\int_{\Bbb T} | ({\cal A} g) (z)|^2 dM(z) = \| Ag\|^2_{ L^2 (\Bbb T; dM)} , {\quad}g\in \cal D.
\label{eq:A2}\end{gathered}$$ This yields the following result.
\[de\] The form $ t [g,g]$ defined on $\cal D$ is closable in the space $\ell^2 ({\Bbb Z}_{+})$ if and only if the operator $A \colon \ell^2 ({\Bbb Z}_{+})\to L^2 (\Bbb T; dM)$ defined on the domain $\cal D (A)=\cal D$ is closable.
Our next goal is to construct the adjoint operator $A^*$. Observe that for an arbitrary $u\in L^2 (\Bbb T; dM)$, all the integrals $$\int_{\Bbb T} u(z) z^{-n} dM(z)=: u_{n}, {\quad}n\in {\Bbb Z}_{+},
\label{eq:A1}$$ are absolutely convergent and the sequence $ \{u_{n}\}_{n=0}^\infty$ is bounded. We denote by ${\cal D}_{*}\subset L^2 (\Bbb T; dM)$ the set of all $u \in L^2 (\Bbb T; dM)$ such that $\{u_{n}\}_{n=0}^\infty \in \ell^2 ({\Bbb Z}_{+})$.
\[LTM\] The operator $A ^*$ is given by the equality $$(A^ {*}u)_{n}= \int_{\Bbb T} u(z) z^{-n} dM(z), {\quad}n\in {\Bbb Z}_{+},
\label{eq:a1}$$ on the domain ${\cal D}(A^*) ={\cal D}_{*}$.
Obviously, for all $g\in \cal D$ and all $u\in L^2 (\Bbb T; dM)$, we have the equality $$(Ag,u)_{L^2 (\Bbb T; dM)} = \int_{\Bbb T} \sum_{n=0}^\infty g_{n} z^{n} {\overline}{u(z)} dM(z) = \sum_{n=0}^\infty g_{n} \bar{u}_{n}
\label{eq:Y1}$$ where the sequence $u_{n}$ is defined by relation [eq:A1]{}. If $u\in {\cal D}_{*}$, then the right-hand side here equals $(g, A^* u)$. It follows that $ {\cal D}_{*}\subset {\cal D}(A^*) $.
Conversely, if $u \in {\cal D}(A^*) $, then $$|(Ag, u)_{L^2 (\Bbb T; dM)} | = |(g, A^* u)_{ \ell^2 ({\Bbb Z}_{+})} | \leq \| A^* u\|_{ \ell^2 ({\Bbb Z}_{+})} \, \| g\|_{ \ell^2 ({\Bbb Z}_{+})}$$ for all $ g \in {\cal D}$. Therefore it follows from equality [eq:Y1]{} that $$\big| \sum_{n=0}^\infty g_{n} \bar{u}_{n} \big| \leq \| A^* u\|_{ \ell^2 ({\Bbb Z}_{+})} \| g\|_{ \ell^2 ({\Bbb Z}_{+})}, {\quad}\forall g \in {\cal D}.$$ Since $ {\cal D}$ is dense in $ \ell^2 ({\Bbb Z}_{+})$, we see that $\{ u_{n}\}_{n=0}^\infty \in \ell^2 ({\Bbb Z}_{+})$, and hence $u\in {\cal D}_{*}$. Thus ${\cal D}(A^*) \subset {\cal D}_{*}$.
Recall that an operator $A $ is closable if and only if its adjoint operator $$A^* \colon L^2 (\Bbb T; dM) \to \ell^2 ({\Bbb Z}_{+})$$ is densely defined. We use the notation $ {\operatorname{clos}}{\cal D}_{*}$ for the closure of the set ${\cal D}_{*}$ in the space $L^2 ({\Bbb T}; dM)$. So we have obtained an intermediary result.
\[adj\] The operator $A $ and the form $t[g,g]$ are closable if and only if $${\operatorname{clos}}{\cal D}_{*} =L^2 ({\Bbb T}; dM).
\label{eq:D}$$
[**2.3.**]{} Next, we use the Riesz Brothers theorem. We state it in a slightly more general form than in most textbooks.
\[brothers\] For a complex $($finite$)$ measure $d\mu (z)$ on the unit circle $\Bbb T$, put $${\widehat}{\mu} (n)=\int_{\Bbb T} z^{-n}d\mu(z)$$ and suppose that $${\widehat}{\mu} (n)\in \ell^2 ({\Bbb Z}_{+}).
\label{eq:brbr}$$ Then the measure $d\mu (z)$ is absolutely continuous.
Indeed, in view of [eq:brbr]{} the function $$f(z):= \sum_{n=0}^\infty {\widehat}{\mu} (n) z^n
\label{eq:br1}$$ belongs to $L^2 ({\Bbb T}; d{\bf m})$. Let us consider an auxiliary measure $$d\mu_{0}(z)= d \mu(z) - f(z) d {\bf m} (z),
\label{eq:br2}$$ and let $${\widehat}{\mu}_{0} (n) = {\widehat}{\mu} (n) - \int_{\Bbb T} z^{-n} f(z) d {\bf m} (z),{\quad}n\in {\Bbb Z}_{+} ,$$ be its Fourier coefficients. It follows from [eq:br1]{} that $ {\widehat}{\mu }_{0} (n)= 0$ for all $n\geq 0$. So, by the standard version of the Riesz Brothers theorem (see, e.g., [@Hof], Chapter 4), the measure $ d\mu_{0}(z)$ is absolutely continuous. In view of [eq:br2]{}, the same is true for the measure $ d\mu(z)$.
The following assertion is almost obvious.
\[ac\] Suppose that a set ${\cal D}_{*}$ satisfies condition [eq:D]{}. Let the measures $u(z) dM(z)$ be absolutely continuous for all $u\in{\cal D}_{*}$. Then the measure $ dM(z)$ is also absolutely continuous.
Denote by ${\mathbbm{1}}_{X}$ the characteristic function of a Borelian set $X\subset \Bbb T$. It follows from [eq:D]{} that there exists a sequence $u_{n}\in{\cal D}_{*}$ such that $$\lim_{n\to\infty} \| u_{n} -{\mathbbm{1}}_{X} \|_{L^2 ({\Bbb T}; d M)}=0$$ and hence $$\lim_{n\to\infty} \int_{X} u_{n}(z) dM(z)= \int_{X} dM(z)= M(X).$$ If ${\bf m} (X)=0$ and the measures $u_{n}(z) dM(z)$ are absolutely continuous, then the integrals on the left-hand side are zeros so that $M (X)=0$.
Now it is easy to conclude the “only if " part of Theorem \[T1\]. Suppose that the form $t[g,g] $ is closable. Then by Lemma \[adj\] the condition [eq:D]{} is satisfied. By the definition of the set ${\cal D}_{*}$, the Fourier coefficients of the measures $\mu(z)= u(z) dM(z)$ belong to $\ell^2 ({\Bbb Z}_{+})$. Therefore it follows from Theorem \[brothers\] that these measures are absolutely continuous for all $u\in{\cal D}_{*}$. Hence by Lemma \[ac\] the measure $ dM(z)$ is also absolutely continuous.
[**2.4.**]{} It remains to check the converse statement. Let us consider an auxiliary form $${\bf t}[g,g] =\sum_{n,m\in {\Bbb Z}} t_{n-m} g_{m}\bar{g}_{n}
\label{eq:QFQ}$$ in the space $\ell^2({\Bbb Z} ) $ on the set ${\sf D} $ of sequences $g=\{g_{n}\}_{n\in {\Bbb Z}}$ with only a finite number of non-zero components. It is easy to see that the conditions [eq:T1]{} and $${\bf t}[g,g] \geq \gamma \| g\|^2_{\ell^2({\Bbb Z} )},{\quad}g\in{\sf D},
\label{eq:T1x}$$ are equivalent. We again suppose that $\gamma=1$. Then the coefficients $t_{n}$ are given by formula [eq:WH]{} where $M(X)\geq {\bf m} (X)$ for all $X\subset {\Bbb T}$. Let the operator ${\sf A}\colon \ell^2({\Bbb Z}) \to L^2 ({\Bbb T}; dM)$ be defined (cf. [eq:A]{}) by the formula $$( {\sf A} g) (z)= \sum_{n\in {\Bbb Z} } z^{n} g_n=: f(z), {\quad}{\cal D} ({\sf A} )= {\sf D}.
\label{eq:A4}$$ Quite similarly to [eq:A2]{}, we find that for $g\in \sf D$, $${\bf t} [g,g] = \int_{\Bbb T} | ({\sf A} g) (z)|^2 dM (z) =\| {\sf A} g \|^2_{L^2 ({\Bbb T}; dM)} .
\label{eq:A3}$$
For the completeness of our presentation, let us check that the measure $dM(z)$ is absolutely continuous if the form $ {\bf t} [g,g]$ is closable. This can be done similarly to the proof in the previous subsection of the same fact for the form $ t [g,g]$, but now $n\in{\Bbb Z}$ in all formulas and the Riesz Brothers theorem is not needed. The operator ${\sf A}^* \colon L^2 ({\Bbb T}; dM) \to \ell^2 ({\Bbb Z} ) $ acts again by the formula [eq:a1]{}, and it is defined on the set ${\sf D}_{*} $ of all $u\in L^2 ({\Bbb T}; dM) $ such that ${\sf A}^* u \in \ell^2 ({\Bbb Z} ) $. This means that $u\in {\sf D}_{*} $ if and only if $u(z) dM(z)= \varphi (z) d {\bf m}(z)$ for some $\varphi \in L^2 ({\Bbb T}; d {\bf m})$. The form $ {\bf t} [g,g]$ is closable if and only if $${\operatorname{clos}}{\sf D}_{*} =L^2 ({\Bbb T}; dM).$$ Hence Lemma \[ac\] implies that the measure $dM(z)$ is absolutely continuous.
Next, we show that, for absolutely continuous measures $dM(z) $, the forms ${\bf t} [g,g]$ are closable. Now we have $dM(z)= w(z) d {\bf m} (z)$ where $w \in L^1 ({\Bbb T}; d {\bf m})$. Therefore it follows from [eq:A3]{} that $${\bf t} [g,g ] = {\bf s} [{\sf A}g ,{\sf A} g]$$ where $${\bf s} [f,f] = \int_{\Bbb T} w(z) | f (z)|^2 d {\bf m} (z) .
\label{eq:A5}$$ Since the operator ${\sf A}: \ell^2 ({\Bbb Z})\to L^2 ({\Bbb T}; d {\bf m})$ is unitary, the form ${\bf t} [g,g]$ defined on $\sf D$ is closable in $\ell^2 ({\Bbb Z})$ if and only if the form ${\bf s} [f,f]$ defined on the quasi-polynomials [eq:A4]{} is closable in $L^2 ({\Bbb T}; d {\bf m})$. Clearly, ${\bf s} [f,f]$ is the quadratic form of the operator of multiplication by $ w(z)$. So it is closable because $ w\in L^1 ({\Bbb T} ; d {\bf m})$. Moreover, ${\bf s} [f,f]$ is closed on the set of all $f\in L^2({\Bbb T} ; d {\bf m})$ such the integral [eq:A5]{} is finite.
Let us summarize the results obtained for the form $ {\bf t} [g,g]$.
\[bro\] Let the form ${\bf t}[g,g] $ be defined by the equality [eq:QFQ]{} in the space $\ell^2({\Bbb Z} ) $ on the set ${\sf D} $ of sequences $g= \{g_{n}\}_{n\in {\Bbb Z}}$ with only a finite number of non-zero components. Let inequality [eq:T1x]{} hold, and let the measure $dM(z)$ be defined by the relation [eq:WH]{}. Then the form ${\bf t}[g,g] $ is closable if and only if $dM(z)= w(z) d {\bf m} (z)$ for some $w\in L^1 ({\Bbb T})$, $w(z)\geq \gamma$. In this case the closure of ${\bf t}[g,g] $ is given by relations [eq:A4]{}, [eq:A3]{} on the set of all sequences $ g \in \ell^2({\Bbb Z} )$ such that the integral [eq:A3]{} is finite.
[**2.5.**]{} Let us return to Theorem \[T1\]. Obviously, if the form ${\bf t}[g,g]$ is closable in the space $\ell^2({\Bbb Z} ) $, then the same is true for the form $t [g,g]$ in the space $\ell^2({\Bbb Z}_{+} ) $. So if the measure $dM(z)$ is absolutely continuous, then by Proposition \[bro\] both forms ${\bf t}[g,g]$ and $t[g,g]$ are closable. This concludes the proof of Theorem \[T1\].
\[broy\] $\rm(i)$ For the proof of Theorem \[T1\] we used only the “if " part of Proposition \[bro\]. On the other hand, the “only if " part of Proposition \[bro\] is a consequence of the “only if " part of Theorem \[T1\] proven in Subsection 2.3. $\rm(ii)$ Comparing Theorem \[T1\] and Proposition \[bro\], we see that if the form $t[g,g]$ is closable, then the same is true for the form ${\bf t}[g,g]$. As already noted, the converse assertion is evident.
[**2.6.**]{} We now suppose that $dM(z)= w(z) d{\bf m} (z)$ where $w\in L^1 ({\Bbb T}; d {\bf m})$ and $w(z)\geq 1$ so that the form $t[g,g] $ is closable in the space $\ell^2 ({\Bbb Z}_{+})$. To describe its closure, we need a mild additional assumption on $w(z)$. We suppose that the function $w(z)$ is a Muckenhoupt weight; see, e.g., §B 5.7 of the book [@NK] for various definitions of this notion. One of them is given by the condition $$\sup_{X \subset {\Bbb T}} {\bf m}(X)^{-2}\int_{X} w (z) d{\bf m} (z) \int_{X} w (z)^{-1} d {\bf m} (z)<\infty
\label{eq:Mu}$$ where $X$ runs over all subarcs of $\Bbb T$. Let $P_{+}$ be the orthogonal projection of $L^2 ({\Bbb T}; d {\bf m})$ onto the Hardy class $H^2 ({\Bbb T}; d {\bf m})$ of functions analytic in the unit disc. The operator $P_{+}\colon L^2 (\Bbb T; d M) \to L^2 (\Bbb T; dM)$ is bounded if and only if $w (z)$ is a Muckenhoupt weight. Recall that the operator ${\cal A} $ is defined by formula [eq:A]{}. Obviously, ${\cal A}g\in H^2 ({\Bbb T}; d {\bf m}) $ for all $g\in \ell^2 ({\Bbb Z}_{+})$.
\[Clo\] Let the coefficients $t_{n}$ be given by formula [eq:WH]{} where $dM(z)= w(z) d {\bf m}(z)$, $w\in L^1 ({\Bbb T}; d {\bf m})$ and $w (z)\geq 1$. Suppose that $w(z)$ is a Muckenhoupt weight. Then the closure of the form $t[g,g]$ defined on ${\cal D}$ by [eq:QFq]{} is given by the equality $$t [g,g] =\int_{\Bbb T} | ({\cal A} g) (z)|^2 dM(z)
\label{eq:A2x}$$ on the set ${\cal D}[t]$ of all $g\in \ell^2 ({\Bbb Z}_{+})$ such that the right-hand side of [eq:A2x]{} is finite.
Observe that the operator ${\cal A} : \ell^2 ({\Bbb Z}_{+})\to H^2 (\Bbb T; d {\bf m})$ is unitary and ${\cal A} {\cal D} =: {\cal P}$ consists of all polynomials [eq:A]{}. Let ${\operatorname{clos}}{\cal P}$ be the closure of $ {\cal P}$ in $ L^2 (\Bbb T; d M)$. So the assertion of Theorem \[Clo\] is equivalent to the equality $${\operatorname{clos}}{\cal P}= H^2 (\Bbb T; d {\bf m}) \cap L^2 (\Bbb T; d M).
\label{eq:A2x1}$$ Since the convergence in $ L^2 (\Bbb T; d M)$ is stronger than that in $ L^2 (\Bbb T; d{\bf m})$, we have ${\operatorname{clos}}{\cal P} \subset H^2 (\Bbb T; d {\bf m})$ and therefore the left-hand side of [eq:A2x1]{} is contained in its right-hand side.
It remains to prove the opposite inclusion. Recall that if all Fourier coefficients of some complex measure on $\Bbb T$ are zeros, then this measure is also zero. Suppose that $u\in L^2 (\Bbb T; d M)$ is orthogonal in $ L^2 (\Bbb T; d M)$ to the functions $z^n$ for all $n\in{\Bbb Z}$. Then applying the above fact to the measure $u dM$, we see that $u=0$. Therefore quasi-polynomials $f (z)=\sum_{n\in{\Bbb Z}} a_{n } z^n$ (the sum consists of a finite number of terms) are dense in $ L^2 (\Bbb T; d M)$. So for every $u\in L^2 (\Bbb T; d M)$ there exists a sequence of quasi-polynomials $f_{k} (z)$ such that $$\lim_{k\to\infty} \| u- f_{k} \|_{L^2 (\Bbb T; d M)}=0, {\quad}dM=w d{\bf m}.$$ Since $w(z)$ is a Muckenhoupt weight, this implies that $$\lim_{k\to\infty} \| P_{+}u- P_{+} f_{k} \|_{L^2 (\Bbb T; dM)}=0.$$ So if $u=P_{+}u$ and $\varphi_{k} =P_{+} f_{k} \in {\cal P}$, then $\varphi_{k} \to u$ as $k\to\infty$ in $L^2 (\Bbb T; d M)$ which also implies the convergence in $H^2 (\Bbb T; d{\bf m})$.
\[Clo1\] Recall that the operator $A$ was defined by formula [eq:A]{} on the domain ${\cal D} (A)= {\cal D} $. Of course its closure ${\operatorname{clos}}{A}=A^{**}$. Let $A_{\rm max}$ be given by the formula $A_{\rm max}g= {\cal A}g$ on the domain $\cal D(A_{\rm max})$ that consists of all $g\in \ell^2 ({\Bbb Z}_{+})$ such that ${\cal A}g\in L^2 (\Bbb T; dM)$. Then the assertion of Theorem \[Clo\] is equivalent to the equality $${\operatorname{clos}}{A}= A_{\rm max}.$$
DISCUSSION
==========
[**3.1.**]{} Let us state a consequence of Theorem \[T1\] in terms of the entries $t_{n}$ of the form [eq:QFq]{}.
\[T1m\] Suppose that the condition [eq:T1]{} is satisfied. If the form $t[g,g] $ is closable in the space $\ell^2 ({\Bbb Z}_{+})$, then $t_{n}\to 0$ as $|n|\to\infty$. If $ \{t_n\}_{n\in {\Bbb Z}}\in \ell^2 ({\Bbb Z})$, then the form $t[g,g] $ is closable in the space $\ell^2 ({\Bbb Z}_{+})$.
If the form $t[g,g] $ is closable, then, by Theorem \[T1\], the measure $dM(z)$ in the representation [eq:WH]{} is absolutely continuous. Therefore its Fourier coefficients $t_{n}\to 0$ as $| n| \to\infty$. Conversely, if $ \{t_n\}_{n\in {\Bbb Z}} \in \ell^2 ({\Bbb Z})$, then $d M(z)= w (z) d {\bf m} (z)$ with $w \in L^2 ({\Bbb T}; d {\bf m} )$. Therefore, again by Theorem \[T1\], the form $t[g,g] $ is closable (this result was already stated in Lemma \[ex1\]).
There is a gap between necessary and sufficient conditions on $ t_n$ in Proposition \[T1m\]. Apparently it cannot be significantly reduced. Recall that by the Wiener theorem (see, e.g., Theorem XI.114 in [@RS]), if the Fourier coefficients $ t_n$ of some measure $dM(z)$ tend to zero, then this measure is necessarily continuous, but it may be singular with respect to the Lebesgue measure. Thus the condition $ t_n\to 0$ as $|n |\to \infty$ does not imply that the measure $dM(z)$ defined by [eq:WH]{} is absolutely continuous. So in accordance with Theorem \[T1\] the corresponding Toeplitz quadratic form $t[g,g]$ may be unclosable.
Astonishingly, the sufficient condition $ \{t_n\}_{n\in {\Bbb Z}}\in \ell^2 ({\Bbb Z})$ for the absolute continuity of the measure $dM(z)$ turns out to be very sharp. Indeed, for every $l\in{\Bbb Z}_{+}$, O. S. Ivašëv-Musatov constructed in [@I-M] a singular measure such that its Fourier coefficients satisfy the estimate $$t_{n} =O \big( ( n \ln n \ln\ln n\cdots \ln_{(l)} n)^{-1/2} \big)$$ (here $ \ln_{(l)} n$ means that the logarithm is applied $l$ times to $n$). Examples of singular continuous measures of such type go back to D. E. Menchoff [@Men]. A comprehensive survey of various constructions of singular continuous measures with decaying Fourier coefficients can be found in [@Brow].
[**3.2.**]{} There is a certain parallelism between the theories of Toeplitz and Hankel operators. For example, the criteria of boundedness of Toeplitz and of Hankel operators due to Toeplitz (see Theorem \[B-H\]) and to Nehari [@Nehari], respectively, look formally similar. Toeplitz quadratic forms are linked to the trigonometric moment problem while Hankel quadratic forms are linked to the power moment problem. The following result obtained by Hamburger in [@Hamb] plays the role of Theorem \[R-H\].
\[Hamb\] The condition $$q[g,g]=\sum_{n,m\geq 0} q_{n+m} g_{m}\bar{g}_{n}\geq 0, {\quad}\forall g\in \cal D,
\label{eq:QFh}$$ is satisfied if and only if there exists a non-negative measure $d{\sf M}(x)$ on $\Bbb R$ such that the coefficients $q_{n }$ admit the representations $$q_{n } = \int_{-\infty}^\infty x^n d {\sf M}(x), {\quad}\forall n=0,1,\ldots.
\label{eq:WHh}$$
It is interesting to compare Theorem \[T1\] with the corresponding result for Hankel operators. Let us state necessary and sufficient conditions guaranteeing that a Hankel quadratic form is closable.
\[Hamb1\][@Ya Theorem 1.2] Let assumption [eq:QFh]{} be satisfied. Then the following conditions are equivalent:
1. The form $q[g,g]$ defined on $\cal D$ is closable in the space $\ell^2 ({\Bbb Z}_{+})$.
2. The matrix elements $ q_n\to 0$ as $n\to \infty$.
3. The measure $d {\sf M} (x)$ defined by equations [eq:WHh]{} satisfies the condition $${\sf M} ({\Bbb R}\setminus (-1,1) )=0$$ $($to put it differently, ${\operatorname{supp}}{\sf M}\subset [-1,1]$ and ${\sf M}(\{-1\}) = {\sf M}(\{1\})=0)$.
Thus in contrast to Toeplitz quadratic forms, the condition $ q_n\to 0$ as $n\to \infty$ is necessary and sufficient for a Hankel quadratic form [eq:QFh]{} to be closable.
For Hankel quadratic forms, an analogue of Theorem \[Clo\] (see Theorem 3.4 in [@Ya]) is true without additional assumptions on the measure $d {\sf M} (x)$, but its proof requires substantial work.
[**3.3.**]{} Theorem \[T1\] automatically extends to vectorial Toeplitz operators. In this case $g=\{g_{n}\}_{n\in{\Bbb Z}_{+}}$ where $g_{n}$ are elements of some auxiliary Hilbert space $\mathfrak{N}$, and $t_{n}$ are bounded operators in $\mathfrak{N}$. We now suppose that $t_{n}= t_{-n}^*$ and $$t[g,g] =\sum_{n,m\geq 0} {\langle}t_{n-m} g_{m}, g_{n} {\rangle}_{\mathfrak{N}} \geq \gamma \| g\|^2_{L^2 ({\Bbb Z}_{+} ;\mathfrak{N})},{\quad}\forall g\in \cal D.
\label{eq:QFv}$$ The vectorial version of Theorem \[R-H\] means that inequality [eq:QFv]{} for $\gamma=0$ is equivalent to the representation [eq:WH]{} with a non-negative operator valued measure $dM(z)$. Let us state a generalization of Theorem \[T1\] to the vector case.
\[Tv\] Let the condition [eq:QFv]{} be satisfied. Then the form $t[g,g] $ is closable in the space $\ell^2 ({\Bbb Z}_{+};\mathfrak{N})$ if and only if $$t_{n} =\int_{\Bbb T} z^{-n} w(z) d{\bf m} (z) , {\quad}n\in{\Bbb Z}, {\quad}w(z)\colon \mathfrak{N}\to\mathfrak{N},$$ where $w(z)\geq \gamma I_{\mathfrak{N}}$ and the operator valued function $w(z)$ belongs to $ L^1 ({\Bbb T}, d{\bf m} )$.
Theorem \[Clo\] and its proof also extend to the vectorial case provided the projector $P_{+}$ is a bounded operator in the space $ L^2 ({\Bbb T}, w d{\bf m} ;\mathfrak{N})$. Note that there is a necessary and sufficient condition (see the paper [@T-V]) for the boundedness of this operator generalizing the scalar condition [eq:Mu]{}; the result of [@T-V] requires however that $\dim \mathfrak{N}<\infty$.
The author thanks G. Rozenblum for a useful discussion.
[00]{} N. Akhiezer, *The classical moment problem and some related questions in analysis*, Oliver and Boyd, Edinburgh and London, 1965.
M. Sh. Birman, M. Z. Solomyak, *Spectral theory of selfadjoint operators in Hilbert space*, D. Reidel, Dordrecht, 1987.
A. Böttcher, B. Silbermann, *Analysis of Toeplitz operators*, Springer-Verlag, Berlin, 2006.
G. Brown, E. Hewitt, *Continuous singular measures with small Fourier-Stieltjes transfoms*, Advances in Math. [**37**]{} (1980), 27-60.
I. Gohberg, S. Goldberg, M. Kaashoek, *Classes of linear operators*, vol. 1, Birkhäuser, Basel, 1990.
H. Hamburger, *Über eine Erweiterung des Stieltjesschen Momentenproblems*, Math. Ann. [**81**]{} (1920), 235-319; [**82**]{} (1921), 120-164 and 168-187.
P. Hartman, *On unbounded Toeplitz matrices*, Amer. J. Math. [**85**]{}, N 1 (1963), 59-78.
K. Hoffman, [*Banach spaces of analytic functions*]{}, Prentice-Hall, Inc., Englewood Cliffs, New York, 1962.
O. S. Ivašëv-Musatov, *On the Fourier-Stieltjes coefficients of singular functions*, Dokl. Akad. Nauk SSSR [**82**]{} (1952), 9-11 (Russian).
D. E. Menchoff, *Sur l’unicité du développement trigonométrique*, C. R. Acad. Sci. Paris Sér. A-B [**163**]{} (1916), 433-436.
Z. Nehari, *On bounded bilinear forms*, Ann. Math. [**65**]{} (1957), 153-162.
N. K. Nikolski, [*Operators, functions, and systems: an easy reading*]{}, vol. I: Hardy, Hankel, and Toeplitz, Math. Surveys and Monographs, vol. 92, Amer. Math. Soc., Providence, Rhode Island, 2002.
V. V. Peller, *Hankel operators and their applications*, Springer-Verlag, New York, 2003.
M. Reed, B. Simon, [*Methods of modern mathematical physics*]{}, vol. 2, Academic Press, San Diego, CA, 1975.
D. Sarason, *Unbounded Toeplitz operators*, Integral Eq. Op. Th. [**61**]{} (2008), 281-298.
S. Treil, A. Volberg, *Continuous frame decomposition and a vector Hunt-Muckenhoupt-Wheeden theorem*, Ark. Mat. [**35**]{} (1997), 363-385.
D. R. Yafaev, *Unbounded Hankel operator and moment problems*, arXiv: 1601.08042 (2016), Integral Eq. Oper. Th. DOI: 10.1007/s00020-016-2289-y.
|
---
abstract: |
We clarify the notion of Wilsonian renormalization group (RG) invariance in supersymmetric gauge theories, which states that the low-energy physics can be kept fixed when one changes the ultraviolet cutoff, provided appropriate changes are made to the bare coupling constants in the Lagrangian. We first pose a puzzle on how a quantum modified constraint (such as $\mbox{Pf} (Q^i Q^j) =
\Lambda^{2(N+1)}$ in SP($N$) theories with $N+1$ flavors) can be RG invariant, since the bare fields $Q^i$ receive wave function renormalization when one changes the ultraviolet cutoff, while we naively regard the scale $\Lambda$ as RG invariant. The resolution is that $\Lambda$ is [*not*]{} RG invariant [*if*]{} one sticks to canonical normalization for the bare fields as is conventionally done in field theory. We derive a formula for how $\Lambda$ must be changed when one changes the ultraviolet cutoff. We then compare our formula to known exact results and show that their consistency requires the change in $\Lambda$ we have found. Finally, we apply our result to models of supersymmetry breaking due to quantum modified constraints. The RG invariance helps us to determine the effective potential along the classical flat directions found in these theories. In particular, the inverted hierarchy mechanism does not occur in the original version of these models.
---
\#1\#2
LBNL-40346\
UCB-PTH-97/21\
hep-th/9705189\
.1in
[**Renormalization Group Invariance of\
Exact Results in Supersymmetric Gauge Theories**]{}[^1]
0.1in
Nima Arkani-Hamed and Hitoshi Murayama
0.05in
[*Theoretical Physics Group\
Ernest Orlando Lawrence Berkeley National Laboratory\
University of California, Berkeley, California 94720*]{}
0.05in
and
0.05in
[*Department of Physics\
University of California, Berkeley, California 94720*]{}
Introduction
============
The last two years have seen remarkable progress in understanding the dynamics of supersymmetric gauge theories (for a review, see [@IS]). It is now worthwhile to consider model building implications of strong supersymmetric gauge dynamics, especially in the areas of composite models or dynamical supersymmetry breaking. Quantitative results are often required in many phenomenological applications. For instance, the exact vacuum structure and mass spectrum are needed for realistic models of dynamical supersymmetry breaking. Similarly, the Yukawa couplings must be determined in a realistic composite model. It is, therefore, useful to have a closer look at the quantitative results which follow from exactly solved supersymmetric gauge theories.
Actually, a detailed look at these exact results leads to some possible confusions. For instance, the quantum modified constraint in SP($N$) theories with $N+1$ flavors: $$\mbox{Pf} (Q^{i} Q^{j}) = \Lambda^{2(N+1)}$$ appears inconsistent at the first sight. The left-hand side involves quantum fields which acquire wave function renormalization, while the right-hand side appears renormalization group (RG) invariant.
It is the purpose of this paper to clarify possible confusions associated with the RG invariance of exact results. RG analysis always contains two steps. The first step is naive dimensional analysis which changes all dimensionful parameters by the same factor $e^{t}$; it in particular changes the cutoff scale $M$ where the theory is defined to $e^{t} M$. The second is the readjustment of the bare parameters in the Lagrangian as the cutoff scale is changed from $e^{t} M$ back to $M$, keeping the low energy physics fixed. The first part is of course trivial. The second part requires care. We will show that the scale $\Lambda$ actually changes when one changes the cutoff back to $M$, and hence is [*not*]{} RG invariant. We will discuss in detail why this is true and how an improved understanding helps to avoid possible further confusions. Although most of the essential ingredients in this paper are already contained in the seminal work of Shifman and Vainshtein [@SV], we hope that our paper will help in clarifying this subtle issue and in applying RG invariance to practical problems. See also other analyses in Refs. [@DS; @FP].
The main result of the paper is quite simple. When one changes the ultraviolet cutoff $M$ to $M'<M$ by integrating out a momentum slice, and keeps the same form for the Lagrangian, [*i.e.*]{} canonical kinetic terms for the bare chiral superfields,[^2] one needs to replace the holomorphic gauge coupling constant[^3] as $$\frac{8\pi^{2}}{g_{h}^{2}} \rightarrow
\frac{8\pi^2}{g_h^{\prime 2}} =
\frac{8\pi^{2}}{g_{h}^{2}} + b_{0} \ln \frac{M}{M'}
- \sum_{i} T_{F}^{i} \ln Z_{i} (M', M)
\label{eq:result}$$ where $b_{0} = - 3 C_A + \sum_i T_F^i$ is the one-loop $\beta$-function, $T_{F}^{i}$ is the $\beta$-function coefficient for a chiral multiplet $T_{F}^{i} \delta^{ab}= \mbox{Tr}_{i} T^{a} T^{b}$, and $Z_{i} (M',M)$ is the coefficient of the kinetic term for the chiral multiplet $i$ when the modes between $M'$ and $M$ are integrated out. Employing this formula, it is straightforward to check the consistency of various results.
The above formula implies that the dynamical scale which appears in exact results: $$\label{eq:Lambda}
\Lambda(M, 1/g_h^2)
\equiv M \exp(8\pi^{2}/g_{h}^{2}b_{0})$$ is not RG invariant in theories with matter multiplets. Under the change of the cutoff and bare parameters, it changes to $$\label{eq:result2}
\Lambda(M', 1/g_h^{\prime 2}) = \Lambda(M, 1/g_h^2) \prod_i
Z_i(M,M')^{-T^i_F/b_0}.$$ This observation can solve possible confusions about the RG invariance of exact results and effective potentials. We point out that a naive argument (regarding $\Lambda$ RG invariant) gives a qualitatively incorrect conclusion for the vacuum structure of a theory breaking supersymmetry dynamically. The result is interesting for model building: the naive understanding allows the inverted hierarchy mechanism to be realized in these models, whereas the correct understanding shows that this is impossible.
The paper is organized as follows. In the next section, we formulate the Wilsonian renormalization program in the context of supersymmetric gauge theories. We derive the formula Eq. (\[eq:result\]) in this section. The same result is derived by perturbative calculations in section three. Section four describes various examples where our formula guarantees the consistency of known results. In section five, we apply our improved understanding to a particular model where one might naively expect the inverted hierarchy mechanism to work. A careful application of our formalism demonstrates that this is not the case. We conclude in section six.
Wilsonian Renormalization Group {#sec:Wilsonian}
===============================
In this section, we review the notion of Wilsonian Renormalization Group and apply it to supersymmetric gauge theories. Based on the Shifman–Vainshtein [@SV] result that the renormalization of the gauge kinetic term is exhausted at one-loop, and the anomalous Jacobian of the path integral under the rescaling of the quantum fields [@KS], we determine the correct readjustment of the bare couplings to derive Eq. (\[eq:result\]).
A field theory is normally defined by specifying the bare parameters $\lambda^0_i$ and some cutoff scale $M$. All Green’s functions can then be calculated as functions of $\lambda^0_i$ and $M$. To work out Green’s functions at energy scales much below the cutoff $M$, it is convenient to “integrate out” physics between $\mu$ and $M$ in a path integral and write down a new Lagrangian with a cutoff $\mu$ which is close to the energy scale of the interest. It is a non-trivial fact (related to the renormalizability of the theory) that one does not need to specify infinite number of bare couplings for all possible operators: those for relevant ([*i.e.*]{} dimension $\leq 4$) operators are enough to define the theory. Therefore, one can define the RG flow of the finite number of bare parameters as one changes the cutoff gradually by “integrating out” modes.
Practically, we determine the bare couplings from experiments. By measuring the amplitudes corresponding to the relevant operators at low energies $E$ (which we refer to loosely as $\lambda_i(E)$), we can work backwards and determine what values of $\lambda^0_i$ are needed to reproduce the measured $\lambda_i(E)$. If we work with a different cutoff $M'$, but wish to reproduce the same observed values of the $\lambda_i(E)$, a different set of bare parameters $\lambda^{0 \,
\prime}_i$ must be chosen. However, once this choice is made, the predictions for all other low energy amplitudes are identical [^4] whether we work with the theory based on $(\lambda^0_i,M)$ or $ (\lambda^{0 \, \prime}_i,M')$.
We should emphasize that none of our discussions depend on the precise way in which the theory is cutoff at the scale $M$, nor the precise way of integrating out modes. The point is simply that it is possible to change the bare couplings $\lambda_i^0$ with the cutoff $M$ while keeping the low-energy physics fixed. This is the formal definition of the “integrating out modes” procedure. The way in which the $\lambda^0_i$ must change with $M$, while keeping the low energy physics fixed, is encoded in Renormalization Group Equations (RGE’s) for the $\lambda^0_i$: $$M\frac {d}{d M} \lambda^0_i = \beta^0_i(\lambda^0) .
\label{eq:WRGE}$$
All of the usual results of RG analysis follow from the above considerations. The procedure is always the same: for any quantity of interest, first, one rescales all parameters (including the cutoff) by naive dimensional analysis, then one changes the cutoff back to the original one while simultaneously changing the bare couplings in accordance with Eq. (\[eq:WRGE\]). As an example, consider the 1PI 4-point function $\Gamma^{4}(p_i;\lambda^0,M)$ for a $\lambda^0 \phi^4/4!$ theory with cutoff $M$, and with all the (Euclidean) momenta $|p_i| \sim \mu \ll M$. If we just compute $\Gamma^4$ in perturbation theory, we find $$\Gamma^4(p_i;\lambda^0,M) = \lambda^0 - \frac{3}{16 \pi^2} \lambda^{0 \, 2}
\mbox{ln} \frac{M}{\mu} + \cdots$$ where $\cdots$ stands for higher order terms in perturbation theory and non-logarithmic corrections which depend on $p_i$ and $\mu$. For $\mu
\ll M$, the logarithm in the above becomes large and the 1-loop term becomes comparable to the tree-level piece, making perturbation theory unreliable. Let us now apply the procedure outlined above for applying the Wilsonian RGE. First, we rescale everything by dimensional analysis: $$\Gamma^4(p_i;\lambda^0,M) = \Gamma^4(e^t p_i;\lambda^0,e^t M).$$ Next, we use the Wilsonian RGE to bring the cutoff on the RHS of the above back from $e^t M$ to $M$ while changing $\lambda^0$ appropriately: $$\Gamma^4(e^t p_i; \lambda^0,e^t M) = \Gamma^4(e^t p_i;\lambda(\lambda^0;t),M)$$ where $\lambda(\lambda^0;t)$ is the solution of $(d/dt) \lambda = -
\beta^0(\lambda)$ with $\lambda(\lambda^0; 0)=\lambda^0$. We then have $$\begin{aligned}
\Gamma^4(p_i;\lambda^0,M) &=&\Gamma^4(e^t p_i;\lambda(\lambda^0;t),M)
\nonumber \\
&=&
\lambda(\lambda^0;t) - \frac{3}{16 \pi^2} \lambda(\lambda^0;t)^2
\mbox{ln} \frac{M}{\ e^t \mu} + \cdots\end{aligned}$$ and if we choose $t$ so that $e^t \mu \sim M$, the logarithms on the second line of the above are small and the perturbation expansion is reliable. In particular we have the standard result $$\Gamma^4(p_i;\lambda^0,M) =
\lambda(\lambda^0,t \sim \mbox{ln} M/\mu)
+ \mbox{small calculable corrections}.$$
Let us now consider the Wilsonian RGE for supersymmetric gauge theories with matter. With some cutoff $M$, the theory is specified by the bare Lagrangian $${\cal L}(M) = \frac{1}{4} \int d^2\theta \frac {1}{g_h^2} W^a W^a + h.c.
+ \int d^4\theta \sum_i \phi_i^{\dagger} e^{2 V_i} \phi_i ,$$ where $V_i = V^a T^a_i$, and $T^a_i$ are generators in the representation of the chiral superfield $\phi_i$. We are working with the holomorphic normalization for the gauge coupling $$\frac {1}{g_h^2} = \frac{1}{g^2} + i \frac{\theta}{8 \pi^2}.$$ Actually, there is a hidden parameter in the above Lagrangian: the coefficient of the matter field kinetic term $Z_i(M)$. However, we have chosen to work with canonical normalization for the bare matter field kinetic terms and we have set $Z_i(M)=1$. There are two reasons for taking canonical normalization: (1) this is the conventional choice in field theory, (2) it is easy to compare Lagrangians with different cutoffs with fixed normalization of the bare fields, since the naive dimensional analysis part of the RG analysis preserves the normalization of the kinetic term. Now with this choice of the normalization, when we change the cutoff from $M$ to $M'$, how should the bare parameters be changed to keep the low energy physics fixed? Shifman and Vainshtein argued that, to all orders of perturbation theory, the couplings should be changed so that the Lagrangian with cutoff $M'$ becomes $$\begin{aligned}
{\cal L}(M') &=& \frac{1}{4} \int d^2\theta
\left(\frac{1}{g^2_h} +
\frac{b_0}{8 \pi^2} \mbox{ln} \frac{M}{M'} \right)W^a W^a
+ h.c. \nonumber \\
& &
+ \int d^4\theta \sum_i Z_i(M,M') \phi_i^{\dagger} e^{2 V_i} \phi_i .\end{aligned}$$ That is, the holomorphic coupling receives only 1-loop contributions. However, the matter field kinetic terms do not remain canonical in going from $M$ to $M'$.
One can easily understand that the change of $1/g_h^2$ is exhausted at 1-loop in perturbation theory as long as the change is holomorphic. This is because holomorphy and periodicity in $\theta$ demand that one can expand the dependence in Fourier series of $\exp(- 8\pi^2/g_h^2)$, $$\frac{1}{g^2_h} + \sum_{n \geq 0} a_n\left(\frac{M}{M'}\right)
\mbox{exp}\left(-n \frac{8 \pi^2}{g^2_h(M)}\right).$$ The sum is limited to the positive frequencies $n\geq 0$ to ensure that the theory has a well-defined weak coupling limit $g^2_h
\rightarrow 0$. The terms with $n>0$ can never arise in perturbation theory, and we drop them. The function $a_0(M/M')$ must satisfy the consistency condition $a_0(M/M') + a_0(M'/M'') =
a_0(M/M'')$, and hence it must be a logarithm. This proves the one-loop law of the change in holomorphic gauge coupling constant.
The point is, however, that the change in $1/g^2_h$ is holomorphic only when the normalization for the matter field kinetic terms (which is manifestly non-holomorphic, being only a function of $g$) is allowed to change from 1 to $Z(M,M')$.
In order to go back to canonical normalization for the matter fields, one simply redefines $\phi = Z(M,M')^{-1/2} \phi'$. However, the path integral measure $D\phi$ is not invariant under this change, $D(Z(M,M')^{-1/2} \phi') \neq D \phi'$; there is an anomalous Jacobian [@KS]. In our case, $Z(M,M')$ is positive and real, but it is sensible to look at $D(Z^{-1/2} \phi')$ for a general complex number $Z$ since $\phi'$ is a chiral superfield. When $Z=e^{i \alpha}$ is a pure phase, the field redefinition is a chiral rotation on the fermionic component of $\phi'$ and the Jacobian is the one associated with the chiral anomaly. This Jacobian is exactly known [@KS] and is cutoff independent: $$\begin{aligned}
\lefteqn{
D(e^{-i \alpha/2} \phi') D(e^{+i \alpha/2} \phi'^{\dagger}) }
\nonumber \\
&=& D\phi' D\phi^{\prime\dagger}
\exp\left(\frac{1}{4} \int d^4 y \int d^2 \theta
\frac{T_F(\phi)}{8 \pi^2} \ln(e^{i \alpha}) W^a W^a + h.c. \right) .
\label{eq:purephase}\end{aligned}$$ In the case where $Z$ is a general complex number, the Jacobian will in general have $F$ terms and $D$ terms (such as $Re(\ln Z) W^* W^* W
W$). However, since $R$ symmetry is at least good in perturbation theory, the $F$ terms can only contain $W^a W^a$, and its coefficient is the same as in Eq. (\[eq:purephase\]) with $\ln e^{i \alpha}$ replaced by $\ln Z$. The $D$ terms are all higher dimensional operators suppressed by powers of the cutoff and can be neglected.[^5]
Therefore, if we wish to keep canonical normalization for the matter fields in changing the cutoff from $M$ to $M'$, the Lagrangian at cutoff $M'$ must be given by $${\cal L}'(M') = \frac{1}{4}\int d^2\theta \frac{1}{g'^2_h} W^a W^a + h.c.+ \int
d^4\theta \sum_i \phi_i^{\dagger} e^{2 V_i} \phi_i$$ where $$\frac{1}{g'^2_h} = \frac {1}{g_h^2} +
\frac{b_0}{8 \pi^2} \ln \frac{M}{M'}
- \sum_i \frac{T_F(\phi^i)}{8 \pi^2} \ln Z_i(M,M').$$ We can rephrase the above results in terms of the scale $\Lambda(M,
1/g_h^2)$ (see Eq. (\[eq:Lambda\])). If we change the cutoff from $M$ to $M'$, and always work with canonical normalization for the matter fields, we have $$\Lambda(M, 1/g_h^2) \rightarrow \Lambda(M', 1/g'^2_h)
=\Lambda(M, 1/g_h^2) \prod_i Z_i(M,M')^{-T^i_F/b_0}.$$
So far we have considered the case with zero superpotential, but the extension to the general case is obvious. For instance, suppose we add a superpotential term of the form $\int d^2 \theta W=\int d^2 \theta
\lambda^{i j k} \phi_i \phi_j \phi_k$. Then by the non-renormalization theorem, $\lambda^{i j k}$ stays the same if we allow non-canonical kinetic terms We, however, insist on working with canonical kinetic terms, and we must have $\lambda'^{i j k} =
Z_i(M,M')^{-1/2} Z_j (M,M')^{-1/2} Z_k(M,M') ^{-1/2} \lambda^{i j k}$.
Perturbative Derivation
=======================
In this section, we rederive the result obtained in the previous section by perturbative calculations. We first review how one can relate perturbative results to the exact results, and then discuss how we change the bare parameters as we change the ultraviolet cutoff. The final result is the same as Eq. (\[eq:result\]).[^6]
Comparison of the perturbative results to the exact results is a somewhat confusing issue. The so-called anomaly puzzle is one famous example of such a confusion. In supersymmetric theories, the U(1)$_{R}$ current belongs to the same supermultiplet as the trace of the energy-momentum tensor, and hence the chiral anomaly and the trace anomaly are related. On the other hand, the chiral anomaly is exhausted at one-loop (Adler–Bardeen theorem) while the trace anomaly is not in $N=1$ theories. Shifman and Vainshtein made a breakthrough on this question by discriminating two definitions of coupling constants: “canonical” and “holomorphic”.[^7] The holomorphic gauge coupling $g_{h}$ runs only at one-loop, while the canonical gauge coupling $g_{c}$ has higher order $\beta$-functions. There is a simple relation between them, the Shifman–Vainshtein formula, $$\frac{8\pi^{2}}{g_{h}^{2}} = \frac{8\pi^{2}}{g_{c}^{2}}
+ C_{A} \ln g_{c}^{2},
\label{eq:SV}$$ where $f^{acd} f^{bcd} = C_{A} \delta^{ab}$. The difference $C_{A}
\ln g^{2}$ appears due to an anomalous Jacobian in the path integral when one rescales the vector multiplet $V_{h}$ which appears in the field strength $W_{\alpha} = \bar{D}^{2} e^{-2V_{h}} D_{\alpha}
e^{2V_{h}}$ to the one in canonical normalization $V_{h}=g_{c}V_{c}$. The Lagrangian written in terms of $V_{h}$ does not need the gauge coupling constant in the exponent, and hence does not need to separate the $\theta$ angle from the gauge coupling constant. This normalization of the vector multiplet therefore keeps holomorphicity of the gauge coupling constant manifest (holomorphic normalization) while the canonical one requires an explicit dependence on the gauge coupling constant in the exponent.
There still remains the question how the canonical gauge coupling constant $g_c$ in the Wilsonian action is related to the perturbative definitions of the running coupling constant in popular schemes such as $\overline{\rm DR}$. We are not aware of a complete answer to this question,[^8] even though one can work out the relation between the two coupling constants at each order in perturbation theory [@JJN].
There is a known “exact” $\beta$-function in supersymmetric gauge theories by Novikov–Shifman–Vainshtein–Zakharov (NSVZ) [@NSVZ]. Our understanding is that this exact $\beta$-function applies to the canonical gauge coupling constant in a Wilsonian action and is hence appropriate for our analysis [@AM2]. Therefore we employ the NSVZ $\beta$-function for our perturbative analysis to determine the necessary change of the bare parameters to keep the low-energy physics fixed as we change the ultraviolet cutoff. The exact NSVZ $\beta$-function is given by $$\mu \frac{d g^2}{d\mu} =
\beta = - \frac{g^4}{8\pi^2}
\frac{3 C_{A} - \sum_{i} T_{F}^{i}(1-\gamma_{i})}
{1-C_{A} g^2/8\pi^2} ,
\label{eq:NSVZ}$$ with $\gamma_i = (\mu d/d\mu) \ln Z_i(\mu, M)$. Of course the $\beta$-function for the gauge coupling constant is the same up to two-loop order in any schemes. From the strictly perturbative point of view, one can regard our analysis as a two-loop analysis in, say, $\overline{\rm DR}$. The RGE can then be integrated with the NSVZ $\beta$-function, and one finds that $$\begin{aligned}
\lefteqn{
\left( \frac{8\pi^{2}}{g_c^{2}(\mu)} + C_{A} \ln g_c^{2}(\mu)
+ \sum_{i} T_{F}^{i} \ln Z_{i}(\mu,M) \right) }\nonumber \\
& & = \left( \frac{8\pi^{2}}{g_c^{2}(M)} + C_{A} \ln g_c^{2}(M) \right)
+ b_{0} \ln \frac{M}{\mu} .
\label{eq:running}\end{aligned}$$ The combination in the bracket runs only at one-loop, and the wavefunction renormalization factors are by definition unity at the cutoff scale, $Z_{i}(M,M) = 1$. The $\beta$-function coefficient is given by $b_{0} = - 3 C_{A} + \sum_{i} T_{F}^{i}$.
Now the strategy is to change the bare parameters $M$ and $g_c^{2}(M)$ while keeping the low-energy physics ($g_c^{2}(\mu)$) fixed. Naively, the change required appears to come from $b_{0}
\ln (M/\mu)$ in the right-hand side, and the change $$\left( \frac{8\pi^{2}}{g_c^{2}(M')} + C_{A} \ln g^{2}_c(M') \right)
= \left( \frac{8\pi^{2}}{g^{2}_c(M)} + C_{A} \ln g^{2}_c(M) \right)
+ b_{0} \ln \frac{M}{M'}$$ might appear to be enough. However, this is not correct, because the wave function renormalization factor $Z_{i}(\mu,M)$ in the left-hand side also depends on $M$ implicitly due to the boundary condition $Z_{i}(M,M)=1$.
The trick is that the wave function renormalization is multiplicative: $$Z_{i} (\mu, M) = Z_{i} (\mu, M') Z_{i} (M', M).$$ Then Eq. (\[eq:running\]) can be rewritten as $$\begin{aligned}
\lefteqn{
\left( \frac{8\pi^{2}}{g^{2}_c(\mu)} + C_{A} \ln g^{2}_c(\mu)
+ \sum_{i} T_{F}^{i} \ln Z_{i}(\mu,M') \right) } \nonumber \\
&=& \left( \frac{8\pi^{2}}{g^{2}_c(M)} + C_{A} \ln g^{2}_c(M) \right)
+ b_{0} \left(\ln \frac{M'}{\mu} +\ln \frac{M}{M'} \right)
- \sum_{i} T_{F}^{i} \ln Z_{i}(M',M). \nonumber \\
\label{eq:running2}\end{aligned}$$ It is now clear that the correct change of the bare parameters is $$\begin{aligned}
\lefteqn{
\left( \frac{8\pi^{2}}{g^{2}_c(M')} + C_{A} \ln g^{2}_c(M') \right)
} \nonumber \\
& & = \left( \frac{8\pi^{2}}{g^{2}_c(M)} + C_{A} \ln g^{2}_c(M) \right)
+ b_{0} \ln \frac{M}{M'} - \sum_{i} T_{F}^{i} \ln Z_{i}(M',M) ,\end{aligned}$$ which keeps the low-energy physics ($g_c^2(\mu)$) fixed.
The final step is to rewrite the above relation in terms of the holomorphic gauge coupling $g_{h}$ using the Shifman–Vainshtein formula Eq. (\[eq:SV\]), $$\frac{8\pi^{2}}{g_{h}^{2}}(M') =
\frac{8\pi^{2}}{g_{h}^{2}}(M) + b_{0} \ln \frac{M}{M'}
- \sum_{i} T_{F}^{i} \ln Z_{i} (M', M)$$ This is indeed the same relation as obtained in the previous section.
If a chiral superfield has a coupling $\lambda^{ijk}$ in the superpotential, it is renormalized only due to wave function renormalization because of the non-renormalization theorem. The low-energy coupling is given by $$\lambda^{ijk} (\mu) = \lambda^{ijk} Z_i^{-1/2}(\mu, M)
Z_j^{-1/2}(\mu, M) Z_k^{-1/2}(\mu, M) .$$ By using the multiplicative property of the wave function renormalization again, the change of the bare parameter is $$\lambda'^{ijk} = \lambda^{ijk} Z_i^{-1/2}(M', M)
Z_j^{-1/2}(M', M) Z_k^{-1/2}(M', M)
\label{eq:mchange}$$ to keep $\lambda^{ijk}(\mu)$ fixed when one changes the cutoff.
Examples
========
In this section, we apply our result Eqs. (\[eq:result\],\[eq:result2\]) to many examples. The RG invariance is checked usually with two steps, (1) naive dimensional analysis, and (2) the change of cutoff parameters. For simplicity of the presentation, we do not discuss the first part since it is rather trivial. The non-trivial part of the analysis is the correct application of the change of bare parameters as derived in previous sections.
Quantum Modified Moduli Space (I)
---------------------------------
In SP($N$) theories with $N+1$ flavors, Intriligator and Pouliot found the quantum modified constraint [@SP(N)] $$\mbox{Pf}(Q^{i} Q^{j}) = \Lambda^{2(N+1)} .$$ Dine and Shirman [@DS] correctly emphasized that the fields in the left-hand side are bare fields in a Wilsonian action with an ultraviolet cutoff $M$. The Lagrangian of the model is simply $${\cal L} = \int d^{2} \theta \, \frac{1}{4 g_{h}^{2}} W^a W^a + h.c.
+ \int d^{4} \theta \, Q^{i\dagger} e^{2V} Q^{i}$$ in terms of bare fields.
As explained in Section \[sec:Wilsonian\], a Wilsonian RG allows the change of ultraviolet cutoff while keeping the low-energy physics fixed by appropriately changing the bare coupling constants in the theory. With the same Lagrangian given at a different cutoff $M'$, a coupling constant $g'_{h}$, and bare fields $Q^{\prime i}$, we must find $$\mbox{Pf}(Q^{\prime i} Q^{\prime j}) = \Lambda^{2(N+1)} .
\label{eq:inconsistent}$$ if $\Lambda$ were a RG invariant quantity. Note that we need to keep the form of the Lagrangian the same no matter how we change the cutoff; therefore the fields $Q'$ must have canonical kinetic terms as $Q$ do.
The relation between the bare fields in two different Lagrangians, $Q$ and $Q'$ can be calculated. When one integrates out modes between $M'$ and $M$, the original bare fields $Q$ acquire corrections to the kinetic terms by a factor $Z_{Q} (M', M)$. The bare fields $Q'$ have canonical normalization in the Lagrangian with the cutoff $M'$, and hence they are related by $$Q' = Z_{Q}^{1/2} (M', M) Q .$$ Therefore, the left-hand sides of the constraint equations are related by $$\mbox{Pf} (Q'^{i} Q'^{j}) = Z_{Q}^{N+1}(M', M)
\mbox{Pf} (Q^{i} Q^{j}),$$ and hence the right-handed sides must also differ by $Z_{Q}^{N+1}$. Then Eq. (\[eq:inconsistent\]) is inconsistent.
Our result (\[eq:result2\]) says that the dynamical scale of the theory with cutoff $M'$ is related to the original one by $$\Lambda' = \Lambda \left[ Z_Q^{- T_F/b_0}(M', M)\right]^{2 N_f}
= \Lambda Z_Q^{1/2}(M', M)$$ with $T_{F} = 1/2$ and $b_0 = -2(N+1)$. Now it is easy to see that the quantum modified constraint holds between the primed fields and the primed dynamical scale: $$\mbox{Pf}(Q^{\prime i} Q^{\prime j}) = \Lambda^{\prime 2(N+1)} .$$ This is a consistency check that the quantum modified constraint is RG invariant.
Quantum Modified Moduli Space (II)
----------------------------------
It is amusing to see how the quantum modified constraints are RG invariant in more complicated cases. Let us look at SU($2k+1$) models with one anti-symmetric tensor $A$, three fundamentals $Q^{a}$ ($a=1,2,3$) and $2k$ anti-fundamentals $\tilde{Q}_{i}$ ($i=1, \ldots, 2k$) [@PT]. The moduli space can be described by the gauge invariant polynomials $$\begin{aligned}
M_i^a ~&=&~ \tilde{Q}^\alpha_i~ Q_\alpha^a \nonumber \\
X_{ij} ~&=&~A_{\alpha \beta} ~\tilde{Q}^\alpha_i ~\tilde{Q}^\beta_j \nonumber \\
Y^a ~&=&~ Q_{\alpha_{2 k + 1}}^a ~\epsilon^{\alpha_1 ...\alpha_{2 k + 1}}
{}~A_{\alpha_1 \alpha_2} ... A_{\alpha_{2 k - 1} \alpha_{2 k}} \\
Z ~&=&~ \epsilon^{\alpha_1 ...\alpha_{2 k + 1} }~
A_{\alpha_1 \alpha_2} ... A_{\alpha_{2 k - 3} \alpha_{2 k -2 }}
{}~Q^a_{\alpha_{2 k - 1}}~ Q^b_{\alpha_{2 k}} ~ Q^c_{\alpha_{2 k + 1}}~
\epsilon_{a b c} \nonumber ~,
\label{eq:oddinvts}\end{aligned}$$ and the quantum modified constraint $$\label{eq:oddquantumconstraint}
Y \cdot M^2 \cdot X^{k - 1} ~-~\frac{k}{3}~ Z ~
{\rm Pf} X ~=~\Lambda^{4 k + 2}~.$$ By following Eq. (\[eq:result2\]), we find $$\Lambda^{\prime -b_{0}} = \Lambda^{-b_{0}}
Z_{A}^{k-1/2} Z_{Q}^{3/2} Z_{\tilde{Q}}^{k}$$ with $b_{0} = -(2k+1)$. The quantum modified constraint is indeed RG invariant as we change the cutoff from $M$ to $M'$, replacing all fields by primed fields (with canonical kinetic terms) [*and*]{} the dynamical scale $\Lambda$ by $\Lambda'$.
Matching Equations (I)
----------------------
When there is a massive chiral superfield, the gauge coupling constants in a theory with a massive field (high-energy theory) and the other theory where the massive field is integrated out (low-energy theory) are related by matching equations. For SU($N$) gauge group with a single massive vector-like pair in the fundamental ($Q$) and anti-fundamental ($\tilde{Q}$) representations, the holomorphic coupling constants in high-energy ($g^{2}_{h,HE}$) and low-energy ($g^{2}_{h,LE}$) theories are related by $$\frac{8\pi^{2}}{g^{2}_{h,LE}}
= \frac{8\pi^{2}}{g^{2}_{h,HE}} + \ln \frac{M}{m}$$ where $m$ is the bare mass of the field. This form can be completely fixed (up to a possible constant) by the holomorphy in $8\pi^{2}/g_{h}^{2}$ and $m$, and the anomaly under the chiral U(1) rotation of the matter fields. We drop the possible constant in the following equations, and it can be easily recovered if necessary.
Under the change of the cutoff, we rewrite the left-hand side as $$\frac{8\pi^{2}}{g^{\prime 2}_{h,LE}}
= \frac{8\pi^{2}}{g^{2}_{h,LE}} + b_{0,LE} \ln \frac{M}{M'}
- \sum_{i\neq Q,\tilde{Q}} T_{F}^{i} \ln Z_{i}(M', M)$$ where the sum does not include the massive field $Q$, $\tilde{Q}$ which are integrated out in the low-energy theory. The coupling in the high-energy theory is also rewritten as $$\frac{8\pi^{2}}{g^{\prime 2}_{h,HE}}
= \frac{8\pi^{2}}{g^{2}_{h,HE}} + b_{0,HE} \ln \frac{M}{M'}
- \sum_{i} T_{F}^{i} \ln Z_{i}(M', M) ,$$ but here the sum includes the massive field. The $\beta$-functions are related as $b_{0,HE} = b_{0,LE} + 1$. Now the matching equation reads as $$\begin{aligned}
\frac{8\pi^{2}}{g^{\prime 2}_{h,LE}}
&=& \frac{8\pi^{2}}{g^{\prime 2}_{h,HE}} - \ln \frac{M}{M'}
+ \frac{1}{2} (\ln Z_{Q}(M', M) + \ln Z_{\tilde{Q}}(M', M))
+ \ln \frac{M}{m} \nonumber \\
&=& \frac{8\pi^{2}}{g^{\prime 2}_{h,HE}}
+ \ln \frac{M'}{m'}\end{aligned}$$ with $m' = Z^{-1/2}_{Q}(M', M) Z^{-1/2}_{\tilde{Q}}(M', M) m$. Therefore, the matching equation takes the same form with the new cutoff and bare parameters.
One can also check the consistency with the perturbative calculations on matching of canonical gauge coupling constants. For instance in $\overline{\rm DR}$ scheme, the one-loop matching equation[^9] is simply $g^{2}_{c,LE}(m_{r}) =
g^{2}_{c,HE}(m_{r})$, where $m_{r}$ is the renormalized mass of the chiral multiplet. By using the Shifman–Vainshtein relation (\[eq:SV\]) between the canonical and holomorphic gauge couplings and NSVZ exact $\beta$-function (using the integrated form Eq. (\[eq:running\])), the matching condition between the canonical gauge couplings for high-energy and low-energy theories can be obtained as $$\begin{aligned}
\frac{8\pi^{2}}{g^{2}_{c,LE}(\mu)}
&=& \frac{8\pi^{2}}{g^{2}_{c,HE}(\mu)} + \ln \frac{\mu}{m}
+ \frac{1}{2} (\ln Z_{Q}(\mu, M) + \ln Z_{\tilde{Q}}(\mu, M))
\nonumber \\
&=& \frac{8\pi^{2}}{g^{2}_{c,HE}(\mu)} + \ln \frac{\mu}{m(\mu)}\, .\end{aligned}$$ Therefore the gauge coupling constants can be matched at the renormalized mass of the heavy field $\mu = m(\mu) = m Z_{Q}^{-1/2}(\mu,
M) Z^{-1/2}_{\tilde{Q}}(\mu, M)$ as expected.
Matching Equations (II)
-----------------------
When a chiral superfield acquires an expectation value and the Higgs mechanisms occurs, the gauge coupling constants in a theory with the full gauge group (high-energy theory) and the other theory only with unbroken gauge group (low-energy theory) are related by matching equations. For SU($N$) gauge group with an expectation value of a single vector-like pair in the fundamental and anti-fundamental representations $Q$ and $\tilde{Q}$, they can acquire an expectation value along the $D$-flat direction $Q=\tilde{Q}$ and the gauge group breaks down to SU($N-1$). The holomorphic coupling constants in high-energy ($g^{2}_{h,HE}$) and low-energy ($g^{2}_{h,LE}$) theories are related by $$\frac{8\pi^{2}}{g^{2}_{h,LE}}
= \frac{8\pi^{2}}{g^{2}_{h,HE}} - \ln \frac{M^{2}}{\tilde{Q}Q}$$ where $\tilde{Q}$, $Q$ are the bare fields. This form can be completely fixed (up to a possible constant) by the holomorphy in $8\pi^{2}/g_{h}^{2}$ and $\tilde{Q}$, $Q$, non-anomalous vector U(1) symmetry, and and the anomaly under the chiral U(1) rotation of the matter fields. We drop the possible constant in the following equations, and it can be easily recovered if necessary.
Under the change of the cutoff, we rewrite the left-hand side as $$\frac{8\pi^{2}}{g^{\prime 2}_{h,LE}}
= \frac{8\pi^{2}}{g^{2}_{h,LE}} + b_{0,LE} \ln \frac{M}{M'}
- \sum_{i\neq Q,\tilde{Q}} T_{F}^{i} \ln Z_{i}(M', M)$$ where the sum does not include the massive field $Q$, $\tilde{Q}$ which are integrated out in the low-energy theory. The coupling in the high-energy theory is also rewritten as $$\frac{8\pi^{2}}{g^{\prime 2}_{h,HE}}
= \frac{8\pi^{2}}{g^{2}_{h,HE}} + b_{0,HE} \ln \frac{M}{M'}
- \sum_{i} T_{F}^{i} \ln Z_{i}(M', M) ,$$ but here the sum includes the massive field. The $\beta$-functions are related as $b_{0,HE} = b_{0,LE} -2$. Now the matching equation reads as $$\begin{aligned}
\frac{8\pi^{2}}{g^{\prime 2}_{h,LE}}
&=& \frac{8\pi^{2}}{g^{\prime 2}_{h,HE}} + 2 \ln \frac{M}{M'}
+ \frac{1}{2} (\ln Z_{Q}(M', M) + \ln Z_{\tilde{Q}}(M', M))
- \ln \frac{M^{2}}{\tilde{Q}Q} \nonumber \\
&=& \frac{8\pi^{2}}{g^{\prime 2}_{h,HE}}
- \ln \frac{M'^{2}}{\tilde{Q}'Q'}\end{aligned}$$ where the primed fields are defined by $$\tilde{Q}' = Z_{\tilde{Q}}^{1/2}(M',M) \tilde{Q},
\hspace{1cm}
Q' = Z_{Q}^{1/2}(M',M) Q.$$ Therefore, the matching equation takes the same form with the new cutoff and bare fields.
One can also check the consistency with the perturbative calculations on matching of canonical gauge coupling constants. For instance in $\overline{\rm DR}$ scheme, the one-loop matching equation[^10] is simply $g^{2}_{c,LE}(m_V) = g^{2}_{c,HE}(m_V)$, where $m_V$ is the renormalized mass of the heavy gauge multiplet. By using the Shifman–Vainshtein relation (\[eq:SV\]) between the canonical and holomorphic gauge couplings and NSVZ exact $\beta$-function (using the integrated form Eq. (\[eq:running\])), the matching condition between the canonical gauge couplings for high-energy and low-energy theories can be obtained as $$\begin{aligned}
\lefteqn{
\frac{8\pi^{2}}{g^{2}_{c,LE}(\mu)}
+ (N-1) \ln g^{2}_{c,LE}(\mu)
} \nonumber \\
&&= \frac{8\pi^{2}}{g^{2}_{c,HE}(\mu)}
+ N \ln g^{2}_{c,HE}(\mu)
- \ln \frac{\mu^{2}}{\tilde{Q}Q}
+ \frac{1}{2} (\ln Z_{Q}(\mu, M) + \ln Z_{\tilde{Q}}(\mu, M)),
\nonumber \\\end{aligned}$$ and hence $$\begin{aligned}
\frac{8\pi^{2}}{g^{2}_{c,LE}(\mu)}
= \frac{8\pi^{2}}{g^{2}_{c,HE}(\mu)}
+ \left(N-\frac{1}{2}\right)
\ln \frac{g^{2}_{c,HE}(\mu)}{g^{2}_{c,LE}(\mu)}
+ \ln \frac{m_{V}^{2}(\mu)}{\mu^{2}} .\end{aligned}$$ Here, the renormalized gauge boson mass $m_{V}$ is defined by $$m_{V}^{2}(\mu) \equiv g_{c,LE}(\mu) g_{c,HE}(\mu) Z^{1/2}_{Q}(\mu,
M) Z^{1/2}_{\tilde{Q}}(\mu, M) \tilde{Q}Q .$$ The matching is particularly simple: $g^{2}_{c,LE}(\mu) =
g^{2}_{c,HE}(\mu)$ at the renormalized gauge boson mass $\mu =
m_{V}(m_{V})$ as expected.
Affleck–Dine–Seiberg superpotential
-----------------------------------
In SU($N$) gauge theories with $N_{f} < N$, a non-perturbative superpotential is generated, $$W = \frac{\Lambda^{(3N-N_{f})/(N-N_{f})}}
{(\mbox{det}\tilde{Q}^{i} Q^{j})^{1/(N-N_{f})}} \, .$$ Under the change of the cutoff, we find $$\Lambda^{\prime (3N-N_{f})} = \Lambda^{(3N-N_{f})}
Z_{Q}^{N_{f}/2} Z_{\tilde{Q}}^{N_{f}/2},$$ and the superpotential becomes $$W' = \frac{\Lambda^{\prime (3N-N_{f})/(N-N_{f})}}
{(\mbox{det}\tilde{Q}^{\prime i} Q^{\prime j})^{1/(N-N_{f})}}
= \frac{\Lambda^{(3N-N_{f})/(N-N_{f})}
(Z_{Q}Z_{\tilde{Q}})^{N_{f}/2(N-N_{f})}}
{(\mbox{det}\tilde{Q}^{i} Q^{j})^{1/(N-N_{f})}
(Z_{Q}Z_{\tilde{Q}})^{N_{f}/2(N-N_{f})}}
= W .$$ The Affleck–Dine–Seiberg superpotential is RG invariant.
Gaugino Condensate
------------------
When all chiral superfields are massive, they can be integrated out from the theory and the low-energy pure Yang–Mills theory develops a gaugino condensate. After matching the gauge coupling constant at the threshold, the size of the gaugino condensate is a function of the bare mass of the chiral superfields and the bare gauge coupling constant.
If there are $N_{f}$ chiral superfields with the same mass $m$ coupled to SU($N$) gauge group, the size of the gaugino condensate can be calculated as $$\langle \lambda \lambda \rangle = m^{N_{f}/N} \Lambda^{3-N_{f}/N}$$ using holomorphy and U(1)$_R$ symmetry up to an overall constant. Under the change of the cutoff and bare parameters, the scale $\Lambda$ changes to $$\Lambda^{\prime -b_{0}} = \Lambda^{-b_{0}} Z_{Q}^{N_{f}/2}
Z_{\tilde{Q}}^{N_{f}/2} .$$ Here, $b_{0} = -(3 N - N_{f})$. The corresponding change of the bare mass parameter is $$m' = m Z_{Q}^{-1/2} Z_{\tilde{Q}}^{-1/2} .$$ It is easy to see that the gaugino condensate is invariant under these changes.
$N=2$ theories {#sec:N=2}
--------------
An application of our formalism to $N=2$ theories requires care because of a difference in conventions. Take $N=2$ supersymmetric QCD with $N_f$ hypermultiplets in fundamental representation. In $N=1$ language, the particle content of the theory is the vector multiplet $V$, a chiral multiplet in the adjoint representation $\phi$, $N_f$ chiral multiplets $Q_i$ and $\tilde{Q}_i$ ($i=1, \ldots, N_f$) in fundamental and anti-fundamental representations, respectively. The Lagrangian in the conventional normalization of fields in the $N=2$ context is $$\begin{aligned}
{\cal L} &=& \int d^4 \theta \left( Re\left(\frac{1}{g_2^2}\right)
2\mbox{Tr} \phi_2^\dagger e^{2V} \phi_2 e^{-2V} + Q_i^\dagger e^{2V} Q_i
+ \tilde{Q}_i^\dagger e^{-2 V^T} \tilde{Q}_i \right)
\nonumber \\
& & + \int d^2 \theta \left( \frac{1}{4 g_2^2} W^a W^a +
\sqrt{2} \tilde{Q}_i \phi Q_i \right) + h.c.\end{aligned}$$ Note that the normalization of the $\phi$ kinetic term is not canonical. Here we use the notation $1/g_2^2$ to refer to the gauge coupling constant in this normalization. Correspondingly, we refer to the adjoint field in this normalization as $\phi_2$. In this normalization, singularities ([*e.g.*]{}, massless monopoles/dyons, roots of the baryonic branch) occur on the Coulomb branch of the theory where a symmetric polynomial of the eigenvalues of the adjoint field $\phi$ takes special values proportional to the dynamical scale $\Lambda_2$ in $N=2$ normalization: $$\phi_2^k = c_k \Lambda_2^k \equiv c_k
\left(M e^{-8\pi^2/g_2^2(2N_c-N_f)}\right)^k
\label{eq:singularities}$$ where we used $b_0 = -(2N_c - N_f)$, and $c_k$ are appropriate constants.
One can ask the question whether the locations of such singularities are RG invariant. They are indeed RG invariant in an obvious manner in $N=2$ normalization. First of all, we never need to change the normalization of the adjoint field $\phi$ because the normalization always stays $1/g_2^2$ automatically due to the $N=2$ supersymmetry. Therefore, the left-hand side of Eq. (\[eq:singularities\]) is RG invariant. Moreover, there is no $Z_\phi$ contribution to $1/g_2^2$ when one changes the cutoff from $M$ to $M'$. Second, there is no wavefunction renormalization for the hypermultiplets [@WLVP; @APS]. Therefore, there is no $Z_Q$, $Z_{\tilde{Q}}$ contribution to $1/g_2^2$ either. As a result, $\Lambda$ is RG invariant, the right-hand side of Eq. (\[eq:singularities\]) is also RG invariant, and hence Eq. (\[eq:singularities\]) remains the same under the change of the cutoff and bare parameters trivially.
If one employs $N=1$ language, the analysis is far less obvious. First of all, the holomorphic gauge coupling $g_1^2$ in $N=1$ language differs from $g_2^2$ because one scales the adjoint field to make it canonically normalized $\phi_1 = g_2^{-1} \phi_2$, and the anomalous Jacobian [@KS] gives $$\frac{8\pi^2}{g_1^2} = \frac{8\pi^2}{g_2^2} + N_c \ln g_2^2 .$$ The dynamical scale $\Lambda_1= M e^{-8\pi^2/g_1^2(2N_c-N_f)}$ in $N=1$ normalization is then related to that in $N=2$ normalization by $$\Lambda_1 = \Lambda_2 (g_2^2)^{-N_c/(2N_c-N_f)} .$$ When one changes the cutoff from $M$ to $M'$, the adjoint field $\phi_1$ receives a wave function renormalization $$Z_\phi(M', M) = \frac{g_2^2}{g'^2_2} .
\label{eq:Z_phi}$$ such that the superpotential coupling $\int d^2\theta \sqrt{2} g_2
\tilde{Q} \phi_1 Q$ is always related to the gauge coupling constant as required by $N=2$ supersymmetry.
The locations of singularities are now written as $$\phi_1^k = c_k g_2^{-k} \left(\Lambda_1
(g_2^2)^{N_c/(2N_c-N_f)}\right)^k .
\label{eq:singularities2}$$ Now this form is RG invariant under the same analysis as we did before. Under the change of the cutoff and bare parameters, the left-hand side is replaced by $$\phi_1'^k = Z_\phi(M', M)^{k/2} \phi_1^k ,$$ while the right-hand side by $$\begin{aligned}
\lefteqn{
g'^{-k}_2 \left(\Lambda'_1
(g'^2_2)^{N_c/(2N_c-N_f)}\right)^k } \nonumber \\
& & = g'^{-k}_2 \left(\Lambda_1 Z_\phi^{N_c/(2N_c-N_f)}
Z_Q^{N_f/2(2N_c-N_f)} Z_{\tilde{Q}}^{N_f/2(2N_c-N_f)}
(g'^2_2)^{N_c/(2N_c-N_f)}\right)^k . \nonumber \\\end{aligned}$$ The Eq. (\[eq:singularities2\]) remains invariant with $Z_Q =
Z_{\tilde{Q}} = 1$ and Eq. (\[eq:Z\_phi\]).
Inverted Hierarchy
==================
In this section we apply our understanding of the RG in supersymmetric gauge theories to a model of dynamical supersymmetry breaking, where such an understanding is necessary to resolve puzzles about the correct vacuum structure of the theory. The theories we consider are vector-like SP($N$) models with $N+1$ flavors studied by Izawa, Yanagida [@IY] and by Intriligator, Thomas [@IT]. The question we ask is whether the so-called inverted hierarchy mechanism [@Witten] operates in these models. The inverted hierarchy refers to the situation where dynamics forces the expectation value of a scalar field to be exponentially large compared to the energy scale of the potential. This occurs in O’Rafeartaigh type models of supersymmetry breaking where the scalar field is a classical flat direction. The effective potential is modified by perturbative corrections both from the gauge coupling and Yukawa coupling. The potential minimum arises where the two corrections balance against each other.
The particle content of the SP($N$) models consists of 2(N+1) SP($N$) fundamentals $Q^i$ and singlets $S_{ij}$. The superpotential is given by $$W = \frac{1}{2} \lambda S_{ij} Q^{i} Q^{j}.$$ The equation of motion for $S_{ij}$ demands that $Q^{i} Q^{j}=0$, which is in conflict with the quantum modified constraint $\mbox{Pf}(Q^i Q^j) =
\Lambda^{2(N+1)}$, and supersymmetry is broken. For non-zero $S_{ij}$, the flavors become massive and can be integrated out of theory. The resulting low-energy theory is pure SP($N$) with a dynamical scale depending on $S_{ij}$. This theory exhibits gaugino condensation and generates an effective superpotential for $S_{ij}$ $$W_{\!\it eff}(S_{ij}) = (\mbox{Pf}\lambda S_{ij})^{1/2(N+1)}
\Lambda^2.
\label{eq:Weff}$$ If we expand $S_{ij}$ around $S_{ij} = \sigma J_{ij}/\sqrt{N+1}$ where $J_{ij}$ is the symplectic matrix, all components of $S_{ij}$ other than $\sigma$ become massive. The effective potential for $\sigma$ is then $$W_{\!\it eff}(\sigma) = \lambda \sigma \Lambda^2.$$ The $\sigma$ equation of motion shows that supersymmetry is broken, while the tree-level potential for $\sigma$ is $$V_{\it tree} (\sigma) = |\lambda \Lambda^2|^2,
\label{eq:Vtree}$$ and the vev of $\sigma$ is undetermined at this level. Since supersymmetry is broken, we expect that some nontrivial potential will be generated for $\sigma$ at higher orders in perturbation theory. In [@Yuri], it was argued that the potential $V(\sigma)$ is “RG improved" as $V(\sigma)=|\lambda
\Lambda^2|^2 \rightarrow |\lambda(\sigma) \Lambda^2|^2$, where $\lambda(\sigma)$ is the running value of $\lambda$, which receives contributions from both the $S_{ij}$ and $Q^{i}$ wavefunction renormalizations $\lambda(\sigma) = Z_S^{-1/2}(\sigma)
Z_Q(\sigma)^{-1} \lambda$. If this conclusion is correct, it is possible to realize the inverted hierarchy mechanism in this model: $\sigma$ could develop a stable vev much larger than $\Lambda$, since the $Z_S$ factors depend on $\lambda$ and tend to make the potential rise for large $\sigma$, whereas the $Z_Q$ factors depend on asymptotically free SP($N$) gauge coupling and tend to make the potential rise for small $\sigma$.
On the other hand, a host of arguments indicate that this conclusion can not be correct. For instance, the superpotential for $\sigma$ is exact, and so the potential is only modified by the Kähler potential for $\sigma$, which at one loop only depends on $\lambda$. Alternately, $F_\sigma = \lambda \Lambda^2$ generates a non-supersymmetric spectrum for the $Q^i$ but not for the gauge multiplet. If we simply look at the 1-loop effective potential, the only contribution comes from the non-supersymmetric $Q^i$ spectrum and again depends only on $\lambda$, and the potential is monotonically increasing with $\sigma$.
In order to resolve this puzzle, we must carefully consider how the potential is “RG improved". As we will show explicitly in the remainder of the section, the solution is that not only $\lambda$ but also $\Lambda$ runs; the correct RG improvement of the potential is $V(\sigma) = |\lambda(\sigma) \Lambda(\sigma)^2|^2$, and all the $Z_Q$ dependence cancels in the product $\lambda(\sigma)
\Lambda^2(\sigma)$.
Let us go back to the start and carefully define the problem. With cutoff $M$, the Lagrangian is given by $${\cal L} =
\int d^4\theta \left(Q^{i \dagger} e^{2V} Q^{i} +
\frac{1}{2} \mbox{Tr} S^{\dagger} S\right)
+ \int d^2 \theta \left(\frac{1}{4 g^2_h} W^a W^a +
\frac{1}{2} \lambda S_{ij} Q^i Q^j\right) + h.c.$$ In the functional integral, we would like to integrate out the $Q^i$ and the gauge multiplet and be left with an effective Lagrangian for $S_{ij}$: $$\exp\left(-\int d^4x {\cal L}_{\it eff}(S)\right) \equiv
\int DV DQ \mbox{exp}\left(-\int d^4x {\cal L}\right),$$ with $$\begin{aligned}
{\cal L}_{\it eff}(S) = \int d^4 \theta
\left(\frac{1}{2}\mbox{Tr} S^{\dagger} S
+ \delta K(\lambda S, \lambda^{\dagger} S^{\dagger}) \right)
+ \int d^2 \theta W_{\!\it eff} (\lambda S) + h.c.,
\label{eq:Leff}\end{aligned}$$ where both $\delta K$ and $W_{\!\it eff}$ depend further on the gauge coupling and the cutoff, $(1/g^2_h, M)$, or equivalently on $(\Lambda, M)$.
Of course, when we integrate out the $Q,V$ multiplets in perturbation theory, we never generate any effective superpotential. However, a superpotential is generated non-perturbatively, and its form is completely determined by a non-anomalous $R$ symmetry under which $\lambda S_{ij}$ has charge $+2$, an anomalous U(1) symmetry under which $\lambda S_{ij}$ has charge $-2$ and $\Lambda$ has charge $+1$, and the non-anomalous SU($2N+2$) flavor symmetry. Together, these dictate (up to an overall constant) the exact effective superpotential Eq. (\[eq:Weff\]). One finds an effective potential along the $S_{ij}=\sigma J_{ij}/\sqrt{N+1}$ direction: $$V(\sigma;\lambda,\Lambda;M) =
\frac{|\lambda \Lambda^2|^2}{1 + \left.\left(\partial^2 \delta K/\partial S
\partial S^{\dagger}\right)\right|_{S=\sigma J}} \ .
\label{eq:Veff}$$
The corrections to the Kähler potential $\delta K(\lambda
S,\lambda^{\dagger} S^{\dagger})$ are certainly generated in perturbation theory. For instance, at 1-loop there is a contribution from the loop of $Q$’s, which yields (at the leading-log) $$ \delta K(\lambda S, \lambda^{\dagger} S^{\dagger}) =
- 2 \frac{N}{16 \pi^2} (\lambda \sigma)^{*} (\lambda \sigma)
\mbox{ln} \frac{|\lambda\sigma|^2}{M^2}$$ along the $\sigma$ direction and we find: $$V(\sigma;\lambda,\Lambda;M)= |\lambda \Lambda^2|^2
\left[1 + 2 \frac{N\lambda^2}{16\pi^2}
\left(\ln\frac{|\lambda\sigma|^2}{M^2} +4 \right)
+ {\cal O}(\lambda^4) \right] .$$
For $\sigma\ll M$, the large logs in the above expression make perturbation theory unreliable. However, we can use the same technique as in Sec. 2 to deal with this problem. We are interested in $V(\sigma;\lambda,\Lambda; M)$. First, we rescale everything by naive dimensional analysis $$V(\sigma;\lambda,\Lambda;M)=
e^{-4 t} V(e^t \sigma; \lambda,e^t\Lambda,;e^t M).
\label{eq:scaling}$$ Next, we bring the cutoff back from $e^t M$ to $M$ by appropriately changing the couplings. Since the form of the tree potential is known with canonical normalization for the superfields $Q$, $S$, we would like to keep them canonical. Hence, as we have argued in Sec. 2, not only $\lambda$ but also $\Lambda$ must be changed: $$V(e^t \sigma; \lambda, e^t \Lambda; e^t M) =
V(e^t \sigma Z_S^{1/2}(t); \lambda Z_Q^{-1}(t) Z_S^{-1/2}(t),
e^t \Lambda Z_Q(t)^{1/2}; M),$$ where $Z_Q(t)\equiv Z_Q(M, e^t M)$, $Z_S(t) \equiv Z_S(M, e^t M)$. However, we can choose $e^t |\sigma| \sim M$, then the logarithms in the perturbative expansion of the RHS are small and the tree value of the RHS $V_{\it tree} (\sigma; \lambda, \Lambda; M) = |\lambda
\Lambda^2|^2$ is an excellent approximation. Doing this and combining with Eq. (\[eq:scaling\]), we find $$\begin{aligned}
V(\sigma;\lambda,\Lambda;M) &\simeq&
\left|(\lambda Z_Q^{-1}(t)Z_S^{-1/2}(t))(\Lambda Z_Q^{1/2}(t))^2
\right|^2_{t \sim \ln (\lambda \sigma/M)} \nonumber \\
&=& Z_S^{-1}(\lambda\sigma, M) |\lambda \Lambda^2|^2.\end{aligned}$$ As promised, when the RG improvement is done consistently, the $Z_Q$ dependence drops out and we are left with a monotonically rising potential (from the $Z_S$ factor) which does not realize the inverted hierarchy.
One can also formally check that the effective potential $V(\sigma;
\lambda,\Lambda; M)$, or equivalently the effective Lagrangian Eq. (\[eq:Leff\]), is RG invariant, [*i.e.*]{} independent of the choice of the cutoff $M$ as long as one changes the bare couplings appropriately. First of all, the effective superpotential Eq. (\[eq:Weff\]) is invariant by itself, because $$\begin{aligned}
\lefteqn{
\lambda' (\mbox{Pf} S'_{ij})^{1/2(N+1)} \Lambda'^2 } \nonumber \\
&=& \! \! \! \lambda Z_Q^{-1}(M', M) Z_S^{-1/2}(M', M)
(\mbox{Pf} S_{ij})^{1/2(N+1)} Z_S^{1/2}(M', M)
(\Lambda Z_Q^{1/2}(M',M))^2 \nonumber \\
&=& \! \! \!\lambda (\mbox{Pf} S_{ij})^{1/2(N+1)} \Lambda^2.\end{aligned}$$ The Kähler potential along the $\sigma$ direction $$\int d^4 \theta Z_\sigma (\lambda \sigma, M) \sigma^* \sigma$$ is also RG invariant which can be seen as follows. First, the $Z_\sigma$ factor must depend on the renormalized effective mass of $Q$, $m_Q \equiv \lambda \sigma Z_Q^{-1}(m_Q, M)$ because it is generated from integrating out the massive $Q$ field. This combination can be easily seen to be RG invariant. Second, it is multiplicative, $Z_\sigma(m_{Q}, M) =Z_\sigma(m_{Q},
M') Z_\sigma(M', M)$. With the definition $\sigma' = Z_\sigma^{1/2}
(M', M) \sigma$, the Kähler potential is RG invariant. Since the change of the field variable $\sigma$ to $\sigma'$ is given by a field-independent constant $Z_\sigma(M, M')$, the auxiliary equation for the $F_\sigma$ changes only by an overall factor $Z_\sigma(M, M')$. On the other hand the quadratic term of $F_{\sigma}$ also changes the the same factor and hence we conclude that the effective potential (\[eq:Veff\]) is RG invariant.
We have demonstrated that the inverted hierarchy mechanism does not work in these models, contrary to the naive argument of RG improvement [@Yuri]. In order to achieve the inverted hierarchy mechanism as favored from the model building point of view, one needs to make the flat direction fields $S_{ij}$ gauge non-singlet, as was recently done in [@HM; @DDGR].
Conclusion
==========
In this paper, we studied the renormalization group invariance of the exact results in supersymmetric gauge theories. We first clarified the notion of Wilsonian renormalization group (RG) invariance in supersymmetric gauge theories. It is a non-trivial statement that the low-energy physics can be kept fixed when one changes the ultraviolet cutoff with appropriate changes in the bare coupling constants in the Lagrangian. We derived the formula for the changes of bare couplings using two methods: one using strictly Wilsonian actions and holomorphic gauge coupling, the other using the perturbative NSVZ $\beta$-function. We used canonical normalization for the chiral superfields because it allows the most straightforward application of the renormalization group. We find that the scale $\Lambda$ is [ *not*]{} RG invariant. We then compared our formula to known exact results and showed that they actually require the changes in $\Lambda$ we have derived.
Finally, we applied our result to models of supersymmetry breaking due to quantum modified constraints, namely SP($N$) models with $N+1$ flavors. These models have a classically flat direction, and the crucial question is in what way the flat direction is lifted. The RG invariance allowed us to determine the effective potential along the classical flat direction. A naive application of RG improvement of the potential would tell us that the potential along the flat direction is modified perturbatively both by the SP($N$) gauge interaction and the superpotential interaction, and hence that the flat direction may develop an expectation value exponentially larger than the supersymmetry breaking scale (inverted hierarchy mechanism). However, a careful application of our method demonstrates that the inverted hierarchy mechanism does not occur in these models.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the U.S. Department of Energy under Contract DE-AC03-76SF00098, in part by the National Science Foundation under grant PHY-95-14797. NAH was also supported by NERSC, and HM by the Alfred P. Sloan Foundation.
[99]{}
K. Intriligator and N. Seiberg, lectures presented at TASI 95, Boulder, CO, Jun 4-26, 1995 and Trieste Spring School, Trieste, Italy, Mar 27 - Apr 15, 1995 and Trieste Summer School, Trieste, Italy, Jun 12 - Jul 23, 1995 and Cargeśe Summer School, Cargése, France, Jul 11-24, 1995, hep-th/9509066. [*Nucl. Phys. Proc. Suppl.*]{} [**45BC**]{} 1-28 (1996)
M. A. Shifman and A. I. Vainshtein, [*Nucl. Phys.*]{} [**B277**]{}, 456 (1986).
M. Dine and Y. Shirman, [*Phys. Rev.*]{} [**D50**]{}, 5389 (1994).
D. Finnell and P. Pouliot, [*Nucl. Phys.*]{} [ **B453**]{}, 225 (1995).
K. Konishi and K. Shizuya, [*Nuovo Cim.*]{} [**90A**]{} 111 (1985).
N. Arkani-Hamed and H. Murayama, UCB-PTH/97/22, in preparation.
I. Jack, D. R. T. Jones, C. G. North, [*Nucl. Phys.*]{} [**B486**]{} 479 (1997).
V. Novikov et al, [*Nucl. Phys.*]{} [**B229**]{} 381 (1983); [*Phys. Lett.*]{} [**B166**]{} 329 (1986).
K. Intriligator and P. Pouliot, [*Phys. Lett.*]{} [**B353**]{}, 471 (1995).
E. Poppitz and S. P. Trivedi, [*Phys. Lett.*]{} [ **B365**]{}, 125 (1996).
B. de Wit, P. G. Lauwers, and A. Van Proeyen, [ *Nucl. Phys.*]{} [**B255**]{} 569 (1985).
P. C. Argyres, M. R. Plesser, and N. Seiberg, [ *Nucl. Phys.*]{} [**B471**]{}, 159 (1996).
K.-I. Izawa and T. Yanagida, hep-th/9602180, [*Prog. Theor. Phys.*]{} [**95**]{}, 829 (1996).
K. Intriligator and S. Thomas, hep-th/9603158, [ *Nucl. Phys.*]{} [**B473**]{}, 121 (1996).
E. Witten, [*Phys. Lett.*]{} [**105B**]{}, 267 (1981).
Y. Shirman, hep-th/9608147, [*Phys. Lett.*]{} [**B389**]{}, 287 (1996).
H. Murayama, LBL-40286, UCB-PTH/97/20, hep-ph/9705271.
S. Dimopoulos, G. Dvali, R. Rattazzi, and G.F. Giudice, CERN-TH-97-098, hep-ph/9705307.
[^1]: This work was supported in part by the U.S. Department of Energy under Contract DE-AC03-76SF00098, in part by the National Science Foundation under grant PHY-95-14797. NAH was also supported by NERSC, and HM by Alfred P. Sloan Foundation.
[^2]: We stick to canonical normalization for chiral superfields simply because the first part of the RG analysis (naive scaling) preserves the normalization of the fields and therefore this choice makes the application of the RG analysis simpler. When one would like to keep the holomorphy of the gauge coupling constant manifest, one needs to keep track of the wave function renormalization in a different manner. The same results are obtained for physical quantities either way: see Section \[sec:N=2\] for an example.
[^3]: A holomorphic gauge coupling is defined by the coefficient of the $WW$ operator in the Lagrangian with $W_{\alpha} = \bar{D}^{2} e^{-2V_{h}} D_{\alpha}
e^{2V_{h}}$. We will explain how the holomorphic gauge coupling constant is related to the one in canonical normalization $W_{\alpha} = \bar{D}^{2} e^{-2g_c V_{c}} D_{\alpha} e^{2g_c V_{c}}$ used in the perturbation theory in Section three.
[^4]: Actually, for the amplitudes for two theories with cutoffs $M,M'$ to be [*exactly*]{} the same, an infinite number of higher dimension operators will have to be included in the definition of the theory with cutoff $M'$. However, it is always possible to absorb the effects of higher dimension operators into the relevant operators, up to calculable finite corrections suppressed by powers in $E/M$, $E/M'$.
[^5]: In a general non-supersymmetric theory, it is not possible to simply throw away higher dimension operators suppressed by the cutoff, since loops with these operators may contain power divergences which negate the cutoff suppression; what [*can*]{} be done is to set the operators to zero with an appropriate modification of the renormalizable couplings. However, in supersymmetric theories, the non-renormalization theorem makes it impossible for the higher dimensional $D$ terms to ever contribute to the coefficient of $W^a W^a$ which is an $F$ term, and so the higher dimensional $D$ terms really can be dropped [@AM2]
[^6]: Note that the analysis in this section is not independent from the one in the previous section; it is simply a reanalysis in a different language. The one-loop exhaustion of the renormalization of $W^a W^a$ used in the previous section and NSVZ $\beta$-function used in this section are closely related [@AM2].
[^7]: We find the terminology by Shifman and Vainshtein rather confusing. In our understanding, what they call “1PI” coupling constant is not what appears in 1PI effective actions; they are still coupling constants in Wilsonian effective action. The only difference between them is that one employs canonical normalization for gauge field kinetic term in “1PI” couplings while holomorphic normalization in “Wilsonian” couplings. We will rather refer to them as “canonical” and “holomorphic” gauge coupling constants in this paper. We will discuss more on this issue in our forthcoming paper [@AM2].
[^8]: At least in some models, one can define a regularized Wilsonian action of the theory and compare the canonical gauge coupling constant in the Wilsonian action to the perturbative definition [@AM2].
[^9]: Recall that one-loop matching is required when one employs two-loop RGE. We are not aware of $\overline{\rm DR}$ calculations of two-loop matching which can tell us whether $m_{r}$ must be the on-shell mass or $\overline{\rm DR}$ mass where the latter is more likely.
[^10]: Here again we are not aware of $\overline{\rm DR}$ calculations of two-loop matching which can tell us whether $m_V$ must be the on-shell mass or $\overline{\rm DR}$ mass.
|
---
abstract: |
We investigate the behavior near zero of the integrated density of states $\ell$ for random Schrödinger operators $\Phi(-\Delta) + V^{\omega}$ in $L^2(\mathbb R^d)$, $d {\geqslant}1$, where $\Phi$ is a complete Bernstein function such that for some $\alpha \in (0,2]$, one has $ \Phi(\lambda) \asymp \lambda^{\alpha/2}$, $\lambda \searrow 0$, and $V^\omega (x) = \sum_{\mathbf i\in{\mathds{Z}}^d} q_{\mathbf i}(\omega) W(x-\mathbf i)$ is a random nonnegative alloy-type potential with compactly supported single site potential $W$. We prove that there are constants $C, \widetilde C,D, \widetilde D>0$ such that $$-C {\leqslant}\liminf_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{|\log F_q(D \lambda)|}{\log \ell(\lambda)}
\qquad \text{and} \qquad \limsup_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{|\log F_q(\widetilde D \lambda)|}\log \ell(\lambda) {\leqslant}-\widetilde C,$$ where $F_q$ is the common cumulative distribution function of the lattice random variables $q_{\mathbf i}$. In particular, we identify how the behavior of $\ell$ at zero depends on the lattice configuration. For typical examples of $F_q$ the constants $D$ and $\widetilde D$ can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, both local and non-local kinetic terms such as the Laplace operator, its fractional powers and the quasi-relativistic Hamiltonians.
[**MSC Subject Classification (2010):**]{} Primary 82B44, 60K37, 60G51; Secondary 47D08, 47G30
[**Keywords:**]{} Bernstein functions, Lévy processes, Random local and nonlocal Schrödinger operator, Alloy-type potential, non-local Anderson problem, Integrated density of states, Lifshitz tail, Tauberian theorem
address:
- |
Faculty of Pure and Applied Mathematics\
Wroc[ł]{}aw University of Science and Technology\
ul. Wybrze[ż]{}e Wyspia[ń]{}skiego 27, 50-370 Wroc[ł]{}aw, Poland
- |
Institute of Mathematics\
University of Warsaw\
ul. Banacha 2, 02-097 Warszawa, Poland
author:
- Kamil Kaleta
- 'Katarzyna Pietruska-Pa[ł]{}uba'
title: Lifshitz tail for continuous Anderson models driven by Lévy operators
---
[^1]
0.5 cm
Introduction and statement of results
=====================================
Let $\Phi$ be a *complete Bernstein function* such that $\lim_{\lambda \searrow 0} \Phi(\lambda)=0$ and let $$\label{eq:oper-def}
H^\omega=\Phi(-\Delta) + V^\omega$$ be a random Schrödinger operator in $L^2({\mathds{R}}^d)$ with an *alloy-type* potential $$\label{eq:pot-def}
V^\omega (x) = \sum_{\mathbf i\in{\mathds{Z}}^d} q_{\mathbf i}(\omega) W(x-\mathbf i),\quad x\in\mathbb R^d,$$ where $\{q_{\mathbf i}\}_{\mathbf i\in {\mathds{Z}}^d}$ is a sequence of i.i.d. nonnegative and nondegenerate random variables over the probability space $(\Omega, \mathcal A, \mathbb Q),$ with cumulative distribution function $F_q(t)=\mathbb Q[q{\leqslant}t],$ and $W:{\mathds{R}}^d\to\mathbb [0,\infty)$ is a sufficiently regular nonnegative *single-site potential*. The class of kinetic terms $\Phi(-\Delta)$ we consider contains both local operators such as the classical Laplacian $-\Delta$ (for $\Phi(\lambda)=\lambda$) as well as a wide range of non-local pseudo-differential operators which are of great interest in mathematical physics. The most prominent examples in this class are the *fractional Laplacians* $(-\Delta)^{\alpha/2}$ (for $\Phi(\lambda)=\lambda^{\alpha/2}$, $\alpha \in (0,2)$) and the *quasi-relativistic operators* $(-\Delta+m^{2/\vartheta})^{\vartheta/2}-m$ (for $\Phi(\lambda)=(\lambda+m^{2/\vartheta})^{\vartheta/2}-m$, $\vartheta \in (0,2)$, $m>0$) [@bib:CMS; @bib:J; @bib:SSV].
Under the regularity assumptions [**(B)**]{}, [**(Q)**]{} and [**(W)**]{} (stated below) on the Bernstein function $\Phi$, the distribution function $F_q$, and the single-site potential $W$, respectively, we study the asymptotic behavior of [*the integrated density of states*]{} (IDS) for the operator $H^\omega$ at the bottom of its spectrum. The precise definition of IDS is given in Section \[sec:dirichlet\]. We will use the same letter $\ell$ to denote both the IDS (a measure) and its cumulative distribution function, i.e. $\ell(\lambda):=\ell([0,\lambda]).$
In 1965, I.M. Lifshitz discovered, on physical grounds, that the density of states $\ell(\lambda)$ of certain random Hamiltonian $H^{\omega}= H_0 + V^{\omega}$ displays unusually fast decay near the bottom $\lambda_0$ of its spectrum $\sigma(H^{\omega})$: it behaves roughly as $\exp(-c(\lambda-\lambda_0)^{-d/2})$ [@bib:Lif]. Since then, such a behavior has been called ‘the Lifshitz tail’. For $H_0= -\Delta$ (or $-\Delta$ with periodic potential) and for various classes of random potentials $V^{\omega}$ it has been widely studied and rigourously proven in both continuous (Benderskii and Pastur [@bib:BP], Friedberg and Luttinger [@bib:FL], Luttinger [@bib:Lut], Nakao [@bib:Nak], Pastur [@bib:Pas], Kirsch and Martinelli [@bib:KM1; @bib:KM2], Mezincescu [@bib:Mez], Kirsch and Simon [@bib:KS], Kirsch and Veselić [@bib:KV]) and discrete (Fukushima [@bib:F], Fukushima, Nagai and Nakao [@bib:FNN], Nagai [@bib:Nag], Romerio and Wreszinski [@bib:RW], Simon [@bib:Sim]) settings (both of these lists are far from being complete). Note in passing that these random Hamiltonians typically exhibit the so-called spectral localization (see e.g. Combes and Hislop [@bib:Com-His], Bourgain and Kenig [@bib:Bou-Ken],Germinet, Hislop and Klein [@bib:Ger-His-Kle], and the references in these papers). It is known that the Lifshitz singularity is a strong indication for this property to hold and rigorous proofs of localization often use the approximation of the IDS resulting from the Lifshitz asymptotics (see e.g. the discussion in the papers by Klopp [@bib:Klopp1; @bib:Klopp2] and Kirsch and Veselić [@bib:KV]).
One of the best studied cases in the classical setting are the Poisson-type potentials. These random potentials, defined as $$V^{\omega}(x) = \int_{{\mathds{R}}^d} W(x-y) \mu^{\omega}({\rm d}y),$$ where $W$ is a sufficiently regular non-negative profile function and $\mu^{\omega}$ is the Poisson random measure on ${\mathds{R}}^d$ with intensity $\nu >0$, were first rigorously investigated in the papers of Nakao [@bib:Nak] and Pastur [@bib:Pas]. This special framework allowed to apply the Donsker-Varadhan large deviations technique to show the following strong statement: $$\begin{aligned}
\label{eq:limit-Poiss}
\lim_{\lambda \searrow 0} \lambda^{d/2} \log \ell(\lambda) = - C(d)\,\nu,\end{aligned}$$ where $C(d)$ is an explicit constant. Later, this result was extended by Okura [@bib:Oku] to more general operators $H_0=-L$, where $L$ is a pseudo-differential operator with sufficiently regular Fourier symbol $\Psi(\xi)$ (this class includes $\Delta$ and many other non-local operators $-\Phi(-\Delta)$ studied in the present paper), with Poissonian random potential $V^{\omega}$. It was also the subject of research on less regular spaces such as fractals, see [@bib:KPP1; @bib:KK-KPP2].
Back to the lattice (alloy-type) potentials, let us emphasize that the Lifshitz behavior for $H_0=-\Delta$ and non-negative potentials as in (with $\lambda_0=0$) was typically established in a form somewhat weaker than , namely $$\begin{aligned}
\label{eq:logloglimit}
\lim_{\lambda \searrow 0} \frac{\log |\log \ell(\lambda)|}{\log \lambda}= -\kappa.\end{aligned}$$ Below, this will be referred to as the ‘loglog statement’. Here $\kappa = d/2$ or $\kappa=d/\beta$, where $\beta >0$ is the parameter describing the decay rate of the single site potential $W$ at infinity (see e.g. Kirsch and Martinelli [@bib:KM2 Theorem 7], Kirsch and Simon [@bib:KS Theorem 1]). Kirsch and Martinelli also gave some sufficient conditions for the existence of constants $C, \widetilde C>0$ such that $e^{-C \lambda^{-d/2}} {\leqslant}\ell(\lambda) {\leqslant}e^{-\widetilde C \lambda^{-d/2}}$, for $\lambda$ close to zero.
Recently, Kaleta and Pietruska-Pa[ł]{}uba [@bib:KK-KPP-alloy-stable] considered the case of $H_0 = (-\Delta)^{\alpha/2}$, $\alpha \in (0,2]$, and alloy-type potentials $V^{\omega}$ as in with bounded, compactly supported single-site potentials that are separated from zero in a vicinity of zero, under the assumption that $F_q(\kappa)>0$ for $\kappa>0$. The authors were able to prove that $$\begin{aligned}
\label{eq:alloy_atom}
\lim_{\lambda \searrow 0} \lambda^{d/\alpha}\log \ell(\lambda) = -C(d,\alpha) \, \log\frac{1}{F_q(0)} .\end{aligned}$$ Clearly, this limit is finite if and only if $F_q(0)>0$, i.e. when the distribution of the lattice random variables has an atom at zero. If this holds, the resulting asymptotics is very close to that in , known for the Poissonian model (see the more detailed discussion in the introduction of the cited paper). This result shows also that when $F_q(0)=0$, then the limit is infinite, so $\lambda^{d/\alpha}$ is not a correct normalization term for . The correct rate for $\ell(\lambda)$ should be faster and depend on the behavior of the distribution of the lattice random variables near zero.
This feature motivated our research in the present paper. Our main result, addressing this problem, is given in Theorem \[th:IDS-asymp\] below. Interestingly, quite often, such a delicate influence is lost by taking the double logarithm in the ‘loglog statements’ as in (cf. Example \[ex:distr\] (1)-(2) below). As we will see below, a more detailed analysis is required in this case.
Before we pass to the presentation of our results, we introduce the framework assumptions. They read as follows.
- $\Phi$ is a complete Bernstein function satisfying $$\begin{aligned}
\label{eq:basic_ass_BF}
\lim_{\lambda \to \infty} \frac{\Phi(\lambda)}{\log \lambda} = \infty.
\end{aligned}$$ and such that there there exist $\alpha\in (0,2],$ $C_1,C_2,\lambda_0>0$ for which $$\label{eq:assum-phi-close-to-0}
C_1 \lambda^{ \alpha/2}{\leqslant}\Phi(\lambda){\leqslant}C_2\lambda^{ \alpha/2},\quad \lambda<\lambda_0.$$
- The random variables $ q_{\mathbf i},$ $\mathbf i\in\mathbb Z^d,$ defined on a probability space $(\Omega,\mathcal A, \mathbb Q),$ are independent copies of a non-negative and non-degenerate (i.e. not equal to a constant a.s.) random variable $q$. Moreover, denoting by $F_q$ the cumulative distribution function of $q$, i.e. $F_q(\kappa):= \mathbb Q(q {\leqslant}\kappa)$, we assume that $F_q(\kappa)>0$ for all $\kappa >0$ and that there exists $\kappa_0>0$ such that $F_q\big|_{[0,\kappa_0]}$ is continuous (left discontinuity at 0 is permitted).
- The single-site potential $W:{\mathds{R}}^d\to {\mathds{R}}_+$ has compact support included in $[-M_0,M_0]^d,$ for certain $M_0\in{\mathds{Z}}_+,$ $W\in L^2({\mathds{R}}^d),$ and $\|W\|_2>0.$ Moreover, $W$ belongs to the Kato class of the operator $\Phi(-\Delta).$
For precise definitions of Bernstein functions, Kato class etc. we refer the reader to Section \[sec:prel\] below. Note that our assumption [**(B)**]{} covers a wide range of complete Bernstein functions $\Phi$ leading to various important classes of operators $\Phi(-\Delta)$. Some of them are listed in Example \[ex:bernstein\] in Section \[sec:bernstein\]. For instance, if $\Phi(\lambda)=\lambda^{\alpha/2}$, $\alpha \in (0,2]$ (the Laplace operator and its fractional powers), then clearly holds with the same $\alpha$. For $\Phi(\lambda)=(\lambda+m^{2/\vartheta})^{\vartheta/2}-m$, with $\vartheta \in (0,2)$ and $m>0$ (the quasi-relativistic operators), we have to choose $\alpha=2$ in .
The assumption [**(W)**]{} is also quite general. It automatically holds for nonnegative $W$’s that are bounded and of compact support. However, singular functions $W$ are allowed as well (further details are given in Example \[ex:singular\] in Section \[sec:potentials\]).
Our main result reads as follows.
\[th:IDS-asymp\] Let the assumptions [**(B)**]{}, [**(Q)**]{} and [**(W)**]{} hold and let $\ell$ be the IDS of the random Schrödinger operator defined in –. Then there exist constants $C, \widetilde C,D, \widetilde D>0$ such that $$-C {\leqslant}\liminf_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{g(D/\lambda)}{\log \ell(\lambda)}$$ and $$\limsup_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{g(\widetilde D/\lambda)}\log \ell(\lambda) {\leqslant}-\widetilde C,$$ where $g(x)= \log\frac{1}{F_q(D_0/x)}$ and $D_0$ is another nonnegative constant defined in .
Interestingly, the rate for $\log \ell(\lambda)$ can be fully factorized: it depends on the kinetic term $\Phi(-\Delta)$ through $\lambda^{d/\alpha}$ and, separately, on the potential $V^{\omega}$ – through the function $g(x)$ describing the common distribution of lattice random variables. To the best of our knowledge, such a general description of the behavior of the IDS at the bottom of the spectrum, involving the dependence of the lattice configuration, was not known before. This is illustrated by Example \[ex:distr\] below.
Moreover, for functions $g$ with sufficiently regular growth at infinity (see Example \[ex:distr\] below), the constants $D,\widetilde D$ in Theorem \[th:IDS-asymp\] can be eliminated and we arrive at a single statement: $$-C {\leqslant}\liminf_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{\log\frac{1}{F_q(\lambda)}}\,\log\ell(\lambda) {\leqslant}\limsup_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{\log\frac{1}{F_q(\lambda)}}\,\log\ell(\lambda) {\leqslant}-\widetilde C.$$ In particular, if the distribution of the lattice random variables has an atom at zero, i.e.$F_q(0)>0$, then the statement simplifies to $$-C \log\frac{1}{F_q(0)} {\leqslant}\liminf_{\lambda \searrow 0} {\lambda^{d/\alpha}}{\log \ell(\lambda)}{\leqslant}\limsup_{\lambda \searrow 0} {\lambda^{d/\alpha}}\log \ell(\lambda) {\leqslant}-\widetilde C \log\frac{1}{F_q(0)},$$ i.e. we obtain the rate known from the continuous Poisson model (cf. [@bib:Oku]).
It is useful to give the following interpretation of our main result (cf. [@bib:Szn1 Remark 3.6(1)] and the comments following [@bib:KK-KPP-alloy-stable Theorem 1.1]).
[**(Interpretation)**]{} Suppose $B(r_{\lambda})$ is a ball with radius $r_{\lambda} \asymp \lambda^{-1/\alpha}$. Clearly, $|B(r_{\lambda})| \asymp \lambda^{-d/\alpha}$, and due to the condition and [@bib:CS Theorem 4.4] the principal Dirichlet eigenvalue of the operator $\Phi(-\Delta)$ constrained to $B(r_{\lambda})$ is proportional to $\lambda$ as $\lambda \searrow 0$. Then, assuming that $D = \widetilde D = 1$ (this is often the case – see Example \[ex:distr\] below) our main result from Theorem \[th:IDS-asymp\] can be written as $$F_q(D_0\lambda)^{C |B(r_{\lambda})|} {\leqslant}\ell(\lambda) {\leqslant}F_q(D_0\lambda)^{\widetilde C |B(r_{\lambda})|}, \quad \text{for} \quad \lambda \searrow 0.$$ Since $F_q(D_0\lambda)$ is the probability of the $q_{{\bf i}}$ being not larger than $D_0\lambda$ at any given lattice point, $\ell(\lambda)$ behaves roughly as the probability that in the ball with ground state eigenvalue comparable to $\lambda$, all random variables $q_{{\bf i}}$’s are smaller than $D_0 \lambda$.
A direct consequence of Theorem \[th:IDS-asymp\] is the following ’loglog statement’ which generalizes (see Corollary \[coro:loglog\]): if $\lim_{x \to \infty} \frac{\log g(x)}{\log x}$ exists, then $$\lim_{\lambda \searrow 0}\frac{\log|\log\ell(\lambda)|}{\log \lambda}= -\frac{d}{\alpha}- \lim_{x\to \infty}\frac{\log g(x)}{\log x}.$$
Our main result is illustrated by four different examples of distribution functions $F_q,$ yielding distinct asymptotics of the IDS near zero (more detailed discussion of these examples can be found in Section \[sec:examples\]). Roughly speaking, the faster the decay of $F_q$ at zero, the faster the decay of $\ell(\lambda)$ at zero as well.
\[ex:distr\] Suppose that the assumptions [**(B)**]{} and [**(W)**]{} hold. Then there exist constants $C, \widetilde C >0$ such that:
- **atom at zero:** if there exists $\kappa_0 >0$ such that $F_q$ is continuous on $[0,\kappa_0]$ and $F_q(0) > 0$, then $$-C{\leqslant}\liminf_{\lambda \searrow 0}\lambda^{d/\alpha}\log \ell(\lambda){\leqslant}\limsup_{\lambda \searrow 0}\lambda^{d/\alpha}\log \ell(\lambda){\leqslant}-\widetilde C\quad\mbox{ and }\quad \lim_{\lambda \searrow 0}\frac{\log|\log \ell(\lambda)|}{\log \lambda } = - \frac{d}{\alpha}.$$
- **polynomial decay at zero:** if there exists $\kappa_0 >0$ such that $F_q$ is continuous on $[0,\kappa_0]$ and $c_1\kappa^{\gamma_1}{\leqslant}F_q(\kappa){\leqslant}c_2\kappa^{\gamma_2}$, $\kappa \in [0,\kappa_0]$, for some $\gamma_1, \gamma_2, c_1, c_2>0$, then $$-C{\leqslant}\liminf_{\lambda \searrow 0}\frac{\lambda^{d/\alpha}}{\log \lambda}\log \ell(\lambda){\leqslant}\limsup_{\lambda \searrow 0}\frac{\lambda^{d/\alpha}}{\log \lambda}\log \ell(\lambda){\leqslant}-\widetilde C\quad\mbox{ and }\quad \lim_{\lambda \searrow 0}\frac{\log|\log \ell(\lambda)|}{\log \lambda } = - \frac{d}{\alpha}.$$
- **exponential decay at zero:** if $F_q(\kappa) = {\rm e}^{-\frac{1}{\kappa^\gamma}}$, $\kappa>0,$ for some $\gamma >0$, then $$-C {\leqslant}\liminf_{\lambda \searrow 0} \lambda^{\frac{d}{\alpha}+\gamma}\log \ell(\lambda) {\leqslant}\limsup_{\lambda \searrow 0} \lambda^{\frac{d}{\alpha}+\gamma}\log \ell(\lambda) {\leqslant}-\widetilde C \quad \mbox{and} \quad
\lim_{\lambda \searrow 0} \frac{\log |\log\ell(\lambda)|}{\log \lambda}= -\frac{d}{\alpha}-\gamma.$$
- **double-exponential decay at zero:** if $F_q(\kappa)= {\rm e}^{1-{\rm e}^{
\frac{1}{\kappa}}}$, $\kappa>0$, then there exist constants $D_1,D_2 >0$ such that $$-C{\leqslant}\liminf_{\lambda \searrow 0} \lambda^{d/\alpha}{\rm e}^{ -\frac{D_1}{\lambda}}\log \ell(\lambda) \qquad \mbox{and} \qquad
\limsup_{\lambda \searrow 0} \lambda^{d/\alpha}{\rm e}^{\frac{-D_2}{\lambda}}\log \ell(\lambda){\leqslant}- \widetilde C$$ and $$C{\leqslant}\liminf_{\lambda \searrow 0} \lambda \log|\log \ell(\lambda)| {\leqslant}\limsup_{\lambda \searrow 0} \lambda \log|\log \ell(\lambda)|{\leqslant}\widetilde C.$$ We call this behavior [*the super-Lifchitz tail*]{}.
As Example \[ex:distr\] (3)-(4) above indicates, the contribution coming from the lattice configuration might not be just the correction term (i.e. a lower order term). Actually, its order may be polynomial or even faster, so this term may be the leading one. Such an effect was not observed before. Note also that the distribution functions $F_q$ as in Example \[ex:distr\] (1)-(2) satisfy the assumptions of the paper by Kirsch and Simon [@bib:KS Theorem 1], who established the Lifshitz singularity in the ‘loglog’ form for the random Schrödinger operators based on Laplacian (i.e. for $\Phi(\lambda)=\lambda$). For the case (1), a version of this results was first obtained by Kirsch and Martinelli [@bib:KM2 Theorem 7]. Our present work extends and improves these results to the case of compactly supported single-site potentials (see Section \[ex:log-rate\] for a broader discussion).
Our approach in the present paper is based on a combination of analytic and probabilistic methods. To make the paper easier accessible to the analytic community, in Section \[sec:prel\] we have included a detailed description of subordinate processes and their evolution semigroups, Kato classes, Schrödinger operators, Feynman-Kac formula etc. The reader familiar with those topics may just give a cursory look at this section and start the lecture from Section \[sec:upper\], maybe coming back to the previous section for the notation.
Our argument is constructed as follows. We first find proper estimates for the Laplace transform $L(t)$ of the IDS (Sections \[sec:upper\], \[sec:lower\]), and then we transform them to the bounds on the IDS itself (Section \[sec:tauber\]). The proof of the upper bound for $L(t)$ (Theorem \[th:upper-short\]) is the most demanding part of this work. It consists of several steps which are presented in Sections \[sec:trace\], \[sec:temple\] and \[sec:conclusion\], respectively. In the first step, using the stochastic Feynman-Kac representation and monotonicity, we estimate $L(t)$ by the trace of the evolution semigroup of the Schrödinger operator $\Phi(\Delta_M) + V^{\omega}_M$, where $\Delta_M$ is the Laplace operator on a torus of given size $M {\geqslant}M_0$, and $V^{\omega}_M$ is an alloy-type potential defined for the periodized lattice configuration. With this preparation, in the next step, we are able to apply in our framework a beautiful idea we learned from the papers by Simon [@bib:Sim] and Kirsch and Simon [@bib:KS], which is based on an application of the Temple’s inequality (Proposition \[prop:temple\]). In [@bib:KS Proposition 3] the authors proved that if the ground state eigenvalue of a random Schrödinger operator constrained to a box of size $L$ (with Neumann boundary conditions) is smaller than a given number $\lambda>0$, then the number of those lattice random variables in this box which are less than $4\lambda$ is larger than $L^d/2$. In general, our approach in the present paper is different from that in the cited papers (we estimate the Laplace transform directly and we do not employ the Dirichlet-Neumann bracketing), but we came up with a version of this implication suitable for our setting (Lemma \[lem:lambda-on-a-delta\]). With a given control level $\delta>0$ and fixed $M,$ we divide all lattice configurations into two disjoint subsets: $\mathcal A_{M,\delta}$ and $\mathcal A_{M,\delta}^c$. In the first set, there is a lot of built-in randomness, which permits to find a proper lower-scaling bound for the ground state eigenvalue of the operator $\Phi(\Delta_M) + V^{\omega}_M$ with $\omega \in \mathcal A_{M,\delta}$. The probability of $ \mathcal A_{M,\delta}^c$ is estimated by means of a Bernstein-type estimate for the binomial distribution (Lemma \[lem:bernoulli\]). We then need to balance the two – by optimization we find $M=M(t)$ for which both summands are of the same order. This leads us directly to the identification of the correct rate function for $\log L(t)$.
The proof of the lower bound for the Laplace transform $L(t)$ (Theorem \[th:lower\] of Section \[sec:lower\]) is more direct. We restrict the integration to the set of special lattice configurations for which we can reduce the problem to a careful analysis of the evolution semigroups associated to non-random Schrödinger operators $\Phi(-\Delta) + \frac{C}{M^{\alpha}}\sum_{{\mathbf i\in [-M,2M)^d}} W(x-\mathbf i)$ constrained to the box of size $M$ (with Dirichlet conditions).
As the last step, in Section \[sec:tauber\] we transform the statements concerning the asymptotical behavior of $L(t)$ at infinity into statements for $\ell(\lambda)$ near zero. This is done by an application of a Tauberian-type theorem. Let us emphasize that the the asymptotic rates for $\log L(t)$ identified in our Theorems \[th:upper-short\] and \[th:lower\] are typically more complicated than $t^{\gamma}$, $\gamma \in (0,1)$ (which are the rates e.g. in the Poissonian case), as they may comprise also lower order terms (see the more detailed discussion of the specific cases in Section \[sec:examples\]). This causes additional difficulties as the Tauberian theorems available in the literature did not cover such a case. Therefore, we had to prove a more general Tauberian theorem which is specialized to work in our present framework (Theorem \[eq:taub-lower-assump\]).
Convention concerning constants {#convention-concerning-constants .unnumbered}
-------------------------------
There are four structural constants of this paper - $C_1,$ $C_2$ of Assumption [**(B)**]{}, $M_0$ of assumption [**(W)**]{}, and $D_0$ of formula . Their values are kept fixed throughout the paper. The value of other roman-type constants (both lower- and upper-case) is not relevant and can change at each appearance. When we need to keep track of the dependence between technical constants, we number them inside the proofs consecutively as $c_1,c_2, \ldots.$
Bernstein functions and corresponding Schrödinger operators {#sec:prel}
===========================================================
As indicated in the Introduction, the approach of this paper is based on a combination of probabilistic and analytic methods. The Schrödinger semigroups we consider are represented by the Feynman–Kac formula with respect to Lévy processes that are obtained via subordination (random time change) of the standard Brownian motion in ${\mathds{R}}^d$. We start our preparation by giving the necessary preliminaries on Bernstein functions, related stochastic processes, and unbounded operators, then we discuss the class of random potentials studied in this paper. Finally, we introduce the corresponding Schrödinger operators and discuss their properties.
Bernstein functions and subordinators
-------------------------------------
A function $\Phi:(0,\infty) \to [0,\infty)$ is called *completely monotone* if it is smooth and satisfies $(-1)^n \Phi^{(n)}(x) {\geqslant}0$, for every $x >0$ and $n \in {\mathds{Z}}_+$. We call $\Phi$ a *Bernstein function* if it is a nonnegative and smooth function with completely monotone derivative. Our standard reference to Bernstein functions, corresponding operators, and stochastic processes is the monograph [@bib:SSV].
It is known that every Bernstein function $\Phi$ admits the representation $$\begin{aligned}
\label{eq:def_Phi}
\Phi(\lambda) = a+b \lambda + \int_{(0,\infty)}(1-{\rm e}^{-\lambda u}) \rho({\rm d}u),\end{aligned}$$ where $a, b {\geqslant}0$ and $\rho$ is a Lévy measure, i.e. a nonnegative Radon measure on $(0,\infty)$ such that $\int_{(0,\infty)} (u \wedge 1)\rho({\rm d}u) < \infty$. A Bernstein function is said to be a *complete Bernstein function* if its Lévy measure has a completely monotone density with respect to the Lebesgue measure.
Bernstein functions $\Phi$ with $\lim_{\lambda \searrow 0} \Phi(\lambda) = 0$ (i.e. $a = 0$) are in one-to-one correspondence with subordinators. The stochastic process $S=(S_t)_{t {\geqslant}0}$ on a probability space $(\Omega_0, \mathcal{F}, \mathcal{P})$ is called a *subordinator* if it is a nondecreasing Lévy process in ${\mathds{R}}_+$, i.e. a process with càdlàg paths (right continuous with left limits finite) starting from $0$, with stationary and independent increments. The laws of $S,$ given by $\eta_t({\rm d}u):=\mathcal{P}(S_t \in {\rm d}u),$ $t{\geqslant}0,$ form a convolution semigroup of probability measures on $[0,\infty)$ which is uniquely determined by the Laplace transform $$\begin{aligned}
\label{eq:laplace}
\int_{[0,\infty)} {\rm e}^{-\lambda u} \eta_t({\rm d}u) = {\rm e}^{-t \Phi(\lambda)}, \quad \lambda>0,\end{aligned}$$ where the Laplace exponent $\Phi$ is a Bernstein function such that $\lim_{\lambda \searrow 0} \Phi(\lambda) = 0$. The number $b$ and the measure $\rho$ are called the drift term and the Lévy measure of the subordinator $S,$ respectively.
Under we have $\lim_{\lambda \to \infty} \Phi(\lambda) = \infty$, and therefore either $b>0$ or $\int_{(0,\infty)} \rho({\rm d}u) = \infty$. We also easily see from that in this case $\eta_t(\left\{0\right\})=0$, for every $t>0$. The following lemma will be an important tool below. It is based on standard calculations, but we include here a short proof for the reader’s convenience.
\[lem:basic\_subord\] For every $\gamma>0$ there is a constant $C=C(\gamma)$ such that $$\int_{[0,\infty)} u^{-\gamma} \eta_t({\rm d}u) = C \int_0^{\infty} {\rm e}^{-t \Phi(\lambda^{1/\gamma})} {\rm d} \lambda, \quad t>0.$$ Under the assumption ****, for every $t_0>0$ there exists a constant $\widetilde C=\widetilde C(t_0)$ such that $$\int_{(0,\infty)} u^{-\gamma} \eta_t({\rm d}u) {\leqslant}\widetilde C t^{-2\gamma/\alpha}, \quad t {\geqslant}t_0.$$ In particular, for every $t_0>0$, $$\sup_{t {\geqslant}t_0} \int_{(0,\infty)} u^{-\gamma} \eta_t({\rm d}u) < \infty.$$
First note that by we have ${\rm e}^{-t \Phi(\lambda^{1/\gamma})}=\int_0^{\infty} {\rm e}^{-(\lambda u^{\gamma})^{1/\gamma}} \eta_t({\rm d}u)$, $\lambda, t>0$. Then, by Fubini-Tonelli and the substitution $\vartheta = u^{\gamma}\lambda$, we get $$\int_0^{\infty} {\rm e}^{-t \Phi(\lambda^{1/\gamma})} {\rm d} \lambda =\int_0^{\infty} \int_0^{\infty} {\rm e}^{-(\lambda u^{\gamma})^{1/\gamma}} {\rm d}\lambda \, \eta_t({\rm d}u) = \int_0^{\infty} {\rm e}^{-\vartheta^{1/\gamma}} d\vartheta \, \int_0^{\infty} u^{-\gamma} \eta_t({\rm d}u),$$ which gives the first equality with $C=\big(\int_0^{\infty} {\rm e}^{-\vartheta^{1/\gamma}} {\rm d}\vartheta\big)^{-1}$. Moreover, for every fixed $t_0>0$, $$\sup_{t {\geqslant}t_0} \int_0^{\infty} u^{-\gamma} \eta_t({\rm d}u) = C \sup_{t {\geqslant}t_0} \int_0^{\infty} {\rm e}^{-t \Phi(\lambda^{1/\gamma})} {\rm d} \lambda {\leqslant}C \int_0^{\infty} {\rm e}^{-t_0 \Phi(\lambda^{1/\gamma})} {\rm d} \lambda.$$ Assume now **** and fix $t_0>0$. Observe that by there exists $\widetilde \lambda_0>\lambda_0$ such that $t_0 \Phi(\lambda^{1/\gamma}) {\geqslant}2 \log \lambda$, for $\lambda {\geqslant}\widetilde \lambda_0$. Also, by decreasing the constant $C_1>0$ if needed, we may assume that the lower bound in holds with $\lambda_0$ replaced with $\widetilde \lambda_0$ (this is possible due to monotonicity and strict positivity of $\Phi$ on $(0,\infty)$). With this in mind, $$\begin{aligned}
\int_0^{\infty} u^{-\gamma} \eta_t({\rm d}u) = C \int_0^{\infty} {\rm e}^{-t \Phi(\lambda^{1/\gamma})} {\rm d} \lambda & {\leqslant}C \left(\int_0^{\widetilde \lambda_0} {\rm e}^{-t \Phi(\lambda^{1/\gamma})} {\rm d} \lambda + {\rm e}^{-(t-t_0) \Phi(\widetilde \lambda_0^{1/\gamma})} \int_{\widetilde \lambda_0}^{\infty} {\rm e}^{-t_0 \Phi(\lambda^{1/\gamma})} {\rm d} \lambda \right)\\
& {\leqslant}C \left(\int_0^{\widetilde \lambda_0} {\rm e}^{-t A_1 \lambda^{\alpha/(2\gamma)}} {\rm d} \lambda + {\rm e}^{-(t-t_0) \Phi(\widetilde \lambda_0^{1/\gamma})} \int_{\widetilde \lambda_0}^{\infty} {\rm e}^{-2 \log \lambda } {\rm d} \lambda \right).\end{aligned}$$ Using the substitution $\vartheta = t^{2\gamma/\alpha} \lambda$ for the first integral and the fact that there exists a constant $c = c(t_0) >0$ such that $e^{-(t-t_0) \Phi(\widetilde \lambda_0^{1/\gamma} )} {\leqslant}c t^{-2\gamma/\alpha}$, for $t {\geqslant}t_0$, we finally get $$\begin{aligned}
\int_0^{\infty} u^{-\gamma} \eta_t({\rm d}u) {\leqslant}C \left(\int_0^{\infty} {\rm e}^{-A_1 \vartheta^{\alpha/(2\gamma)}} {\rm d} \vartheta + \frac{c}{\widetilde \lambda_0} \right) t^{-2\gamma/\alpha} = \widetilde C t^{-2\gamma/\alpha}.\end{aligned}$$ This completes the proof.
Operators $\Phi(-\Delta)$ and subordinate Brownian motions {#sec:bernstein}
-----------------------------------------------------------
Denote by $\big\{ G_t: t {\geqslant}0\big\}$ the classical heat semigroup acting on $L^2({\mathds{R}}^d)$, i.e. $$G_t f(x) = \int_{{\mathds{R}}^d} g_t(x-y) f(y) {\rm d}y, \quad f \in L^2({\mathds{R}}^d), \ t>0,$$ where $g_t(x)= (4 \pi t)^{-d/2} e^{-|x|^2/4t}$ is the Gauss-Weierstrass kernel. We have $G_t = e^{t \Delta}$, where $\Delta$ is the classical Laplace operator. Recall that it is an unbounded, non-positive definite, self-adjoint operator in $L^2({\mathds{R}}^d)$. On the probabilistic side, $\big\{ G_t: t {\geqslant}0\big\}$ serves as the transition semigroup of the standard Brownian motion $Z=(Z_t)_{t {\geqslant}0}$ in ${\mathds{R}}^d$, running at twice the usual speed.
Suppose now that $\Phi$ is a Bernstein function such that $\lim_{\lambda \searrow 0} \Phi(\lambda) = 0$ and let $\big\{\eta_t: t {\geqslant}0\big\}$ be the convolution semigroup of measures determined by . With this, we can define $$P_t f(x) := \int_{[0,\infty)} G_u f(x) \eta_t({\rm d}u), \quad f \in L^2({\mathds{R}}^d), \ t {\geqslant}0.$$ One can check that $P_t$ form a strongly continuous semigroup of bounded self-adjoint operators in $L^2({\mathds{R}}^d)$ which is referred to as the *subordinate heat semigroup*; under the assumption (giving $\eta_t(\left\{0\right\})=0$, $t>0$) all the $P_t$’s, $t>0$, are integral operators with kernels $p_t(x,y):=p_t(y-x)$, where $$p_t(x) = \int_0^{\infty} g_u(x) \eta_t({\rm d}u), \quad t>0.$$
Under the full assumption ****, by Lemma \[lem:basic\_subord\], we obtain that for every $t_0>0$ there exists $C=C(t_0)$ such that $$\begin{aligned}
\label{eq:diag_on_pt}
p_t(x) {\leqslant}(4 \pi)^{-d/2} \int_0^{\infty} u^{-d/2} \eta_t({\rm d}u) {\leqslant}C t^{-d/\alpha}, \quad x \in {\mathds{R}}^d, \ \ t {\geqslant}t_0.\end{aligned}$$ It is known that $P_t = {\rm e}^{-t\Phi(-\Delta)}$, where $\Phi(-\Delta)$ is the Fourier multiplier with symbol $\Phi(|\xi|^2)$, i.e. the operator $$\Phi(-\Delta) f = \mathcal{F}^{-1} \big(\Phi(|\cdot|^2) \mathcal{F} f(\cdot) \big), \quad f \in \mathcal D(\Phi(-\Delta))$$ with the domain $$\mathcal D(\Phi(-\Delta)) =\{f\in L^2({\mathds{R}}^d): \Phi(|\xi|^2) \mathcal{F} f(\xi)\in L^2({\mathds{R}}^d)\};$$ $\Phi(-\Delta)$ is an unbounded, non-negative definite, self-adjoint operator on $L^2({\mathds{R}}^d)$ and $\mathcal D(\Phi(-\Delta))$ contains $\mathcal D(-\Delta)$. The quadratic form associated with this operator is given by $$\mathcal E(f,f) = \int_{{\mathds{R}}^d} \Phi(|\xi|^2) (\mathcal{F} f)^2(\xi) {\rm d}\xi, \quad f \in \mathcal D(\mathcal E),$$ where $f \in \mathcal D(\mathcal E)$ if and only both of $f(\xi)$ and $\sqrt{\Phi(|\xi|^2)} \mathcal{F} f(\xi)$ are in $L^2({\mathds{R}}^d)$. Note that the choice $\Phi(\lambda) = b \lambda$, $b>0$, leads to the only pure *local* operator in the class we consider, i.e. the operator $-b \Delta$. Whenever the Lévy measure $\rho$ in is non-zero, the resulting operator $\Phi(-\Delta)$ is a *non-local* integral operator (for $b=0$) or an integro-differential operator (for $b>0$).
The semigroup $\big\{P_t: t {\geqslant}0\big\}$ is the transition semigroup (and $-\Phi(-\Delta)$ is the generator) of a Markov process $X = (X_t)_{t {\geqslant}0}$ which is determined by $$X_t = Z_{S_t}, \quad t {\geqslant}0.$$ Such a process is obtained by a random time change of the Brownian motion $Z$ – this procedure is called the *subordination*. A new, random, clock of the process is given by the subordinator $S$ (we always assume that $Z$ and $S$ are independent). The process $X$ is referred to as the *subordinate Brownian motion* in ${\mathds{R}}^d$. It is an isotropic Lévy process [@bib:Sat] with càdlàg paths whose Lévy-Khintchine exponent is equal to $\Phi(|\xi|^2)$. More precisely, we have $${\mathbf{E}}_0 {\rm e}^{i \xi \cdot X_t} = {\rm e}^{-t \Phi(|\xi|^2)}, \quad \xi \in {\mathds{R}}^d, \ \ t>0.$$ By ${\mathbf{P}}_x$ and ${\mathbf{E}}_x$ we denote the probability measure and the corresponding expected value for the process $X$ starting from $x \in {\mathds{R}}^d$. We have ${\mathbf{P}}^x(X_t \in A) = \int_A p_t(x,y) {\rm d}y$, $A \in \mathcal B({\mathds{R}}^d)$, $x \in {\mathds{R}}^d$, $t>0$, i.e. the kernels $p_t(x,y)$ are transition probability densities of the process $X$. It is important that under we also have $\lim_{|\xi| \to \infty} \Phi(|\xi|^2)/ \log |\xi| = \infty$, and it follows from [@bib:KSch2019 Lemma 2.1] that $(t,x) \mapsto p_t(x)$ is a continuous function on $(0,\infty) \times {\mathds{R}}^d$.
Our Assumption [**(B)**]{} is satisfied by a wide class of complete Bernstein functions (and corresponding subordinators). Below we discuss only several, the most popular examples. For further examples we refer the reader e.g. to the monograph [@bib:SSV].
\[ex:bernstein\]
- *Pure drift.* Let $\Phi(\lambda)=b \lambda$, $ b>0$. As mentioned above, this leads to the only subordinate Brownian motion with continuous paths – the *Brownian motion* with speed $b$.
- *$\alpha/2$-stable subordinators.* Let $\Phi(\lambda)=\lambda^{\alpha/2}$, $\alpha \in (0,2)$. The subordination via this subordinator leads to the pure jump *isotropic $\alpha$-stable process*.
- *Mixture of several purely jump stable subordinators.* In this case, $\Phi(\lambda)=\sum_{i=1}^n \lambda^{\alpha_i/2}$, $\alpha_i \in (0,2)$, $n \in {\mathds{Z}}_+$.
- *$\alpha/2$-stable subordinator with drift.* Let $\Phi(\lambda)=b\lambda + \lambda^{\alpha/2}$, $\alpha \in (0,2)$, $b>0$. Clearly, in this case, $\Phi(\lambda) \approx \lambda$ for $\lambda \to 0^{+}$, and $\Phi(\lambda) \approx \lambda^{\alpha/2}$ for $\lambda \to \infty$.
- *Relativistic $\vartheta/2$-stable subordinator.* Let $\Phi(\lambda)=(\lambda+m^{2/\vartheta})^{\vartheta/2}-m$, $\vartheta \in (0,2)$, $m>0$. The subordination via such a subordinator leads to the so-called *relativistic $\vartheta$-stable process*. Similarly as above, we have $\Phi(\lambda) \approx \lambda$ for $\lambda \to 0^{+}$, and $\Phi(\lambda) \approx \lambda^{\vartheta/2}$ for $\lambda \to \infty$.
- If $S$ is a subordinator with Laplace exponent $\Phi(\lambda)=\lambda^{\alpha/2}[\log(1+\lambda)]^{\beta/2}$, $\alpha \in (0,2)$, $\beta \in (-\alpha, 0)$ or $\beta \in (0,2-\alpha)$, then we see that both the conditions and hold as well.
Next, we introduce the bridge measures of the subordinate process that will be needed in our argument. For fixed $t>0$ and $x,y \in {\mathds{R}}^d,$ the bridge measure $\mathbf P_{x,y}^{t}$ is defined by the following property: for any $0 <s<t$ and $A\in \sigma(X_u: u {\leqslant}s),$ $$\label{eq:bridge}
{\mathbf{P}}_{x,y}^{t}[A] = \frac{1}{p_t(x,y)}\mathbf E_x[\mathbf 1_A p_{t-s}(X_s,y)]$$ which is then extended to $s=t$ by weak continuity. The bridge measures can be understood as the laws of the process that starts from $x$ and is conditioned to have $X_t=y,$ $\mathbf P_x-$almost surely. For more detailed information on Markovian bridges we refer to [@bib:Cha-Uri].
The operators and the corresponding subordinate processes on tori {#sec:operators_on_tori}
-----------------------------------------------------------------
Our argument in the present paper mostly uses subordinate semigroups and the related processes on the torus $\mathcal T_M,$ for any $M\in\mathbb Z_+.$ The torus $\mathcal T_M = \mathbb R^d/(M\mathbb Z_+)$ is understood as the box $[0,M)^d$ with reciprocal sides identified. By $\pi_M$ we denote the canonical projection of ${\mathds{R}}^d$ onto $\mathcal T_M$.
Let $\big\{ G^M_t: t {\geqslant}0\big\}$ be the heat semigroup acting on $L^2(\mathcal T_M)$, i.e. $$G^M_t f(x) = \int_{\mathcal T_M} g^M_t(x,y) f(y) {\rm d}y, \quad f \in L^2(\mathcal T_M), \quad t>0,$$ where $$g^M_t(x,y):= \sum_{y^{\prime} \in \pi^{-1}_M(y)} g_t(x,y^{\prime}) = \sum_{\mathbf i \in M{\mathds{Z}}^d} g_t(x,y+\mathbf i), \ t>0$$ is the transition density of the Brownian motion on the torus $\mathcal T_M$ (in this formula, $g_t(x,y) := g_t(y-x)$ denotes the classical Gauss-Weierstrass kernel). One can check that $G^M_t$ form a strongly continuous semigroup of bounded operators in $L^2(\mathcal T_M)$ (the latter fact is an easy consequence of the symmetry $g^M_t(x,y)=g^M_t(y,x)$). The infinitesimal generator of this semigroup (denoted by $\Delta^M$) is an unbounded, self-adjoint operator on $L^2(\mathcal T_M)$. It is important for our applications that the operators $G^M_t$ have certain *scaling property*: since $g_{a^2t}(ax) = g_t(x)$, $a>0$, we also have $$\begin{aligned}
\label{eq:scaling}
g^{kM}_{k^2t}(kx,ky) = g^{M}_t(x,y), \quad x, y \in \mathcal T_M, \ t>0, \ k \in {\mathds{Z}}_+.\end{aligned}$$ The *subordinate heat semigroup on the torus* is defined in the same way as its free counterpart in ${\mathds{R}}^d$. For a Bernstein function $\Phi$ such that $\lim_{\lambda \searrow 0} \Phi(\lambda) = 0$ and the convolution semigroup of measures $\big\{\eta_t: t {\geqslant}0\big\}$ determined by , we let $$P^M_t f(x) := \int_{[0,\infty)} G^M_u f(x) \eta_t({\rm d}u), \quad f \in L^2(\mathcal T_M), \ t {\geqslant}0;$$ $P_t^M$ form a strongly continuous semigroup of bounded self-adjoint operators in $L^2(\mathcal T_M)$. Under the assumption all the $P_t$’s, $t>0$, are integral operators with kernels given by $$p^M_t(x,y) = \int_0^{\infty} g^M_u(x,y) \eta_t({\rm d}u), \quad t>0.$$ Due to Fubini-Tonelli we have $$\begin{aligned}
\label{eq:pM_def}
p^M_t(x,y) = \sum_{y^{\prime} \in \pi^{-1}_M(y)} \int_0^{\infty} g_u(x,y^{\prime}) \eta_t({\rm d}u) = \sum_{y^{\prime} \in \pi^{-1}_M(y)} p_t(x,y^{\prime}) = \sum_{\mathbf i \in M{\mathds{Z}}^d} p_t(x,y+\mathbf i).\end{aligned}$$ We have $P^M_t = {\rm e}^{-t\Phi(-\Delta^M)}$, where both the operators $\Phi(-\Delta^M)$ and ${\rm e}^{-t\Phi(-\Delta^M)}$ are understood through the spectral representation of unbounded self-adjoint operators.
As shown in Lemma \[lem:regularity\] below, for every fixed $t>0$ both the kernels $g^M_t(x,y)$ and $p^M_t(x,y)$ are bounded functions on $\mathcal T_M \times \mathcal T_M$. Together with the fact that $|\mathcal T_M| = M^d < \infty$, this gives that the operators $G_t^M$ and $P_t^M$ are Hilbert-Schmidt on $L^2(\mathcal T_M)$. In consequence, all the operators considered in this section have purely discrete spectral decompositions. Indeed, for $M=1,2,...,$ the spectrum of the operator $-\Delta^M$ consists of a sequence of eigenvalues $$0 {\leqslant}\mu_1^M < \mu_2^M {\leqslant}\mu_3^M {\leqslant}... \to \infty,$$ each of finite multiplicity, and the corresponding eigenfunctions $\big\{\psi_k^M\big\}_{k=1}^{\infty}$ form a complete orthonormal system in $L^2(\mathcal T_M)$. We have $$-\Delta^M \psi^M_k = \mu_k^M \psi^M_k \qquad \text{and} \qquad G_t^M \psi^M_k = {\rm e}^{-t \mu_k^M} \psi^M_k, \ \ t>0, \ \ k =1,2,\ldots,$$ and due to the conservativeness of the semigroup $\big\{G^M_t: t {\geqslant}0\big\}$, $$\mu_1^M = 0 \qquad \text{and} \qquad \psi_1^M \equiv \frac{1}{\sqrt{|\mathcal T_M|}} = \frac{1}{M^{d/2}}.$$ One can directly check that the eigenvalues of the operators $-\Delta^M$ inherit from the following scaling property: $$\begin{aligned}
\label{eq:scaling_eig}
\mu_k^M = M^{-2} \mu_k^1, \quad k = 1,2,3,\ldots.\end{aligned}$$ Due to the spectral theorem, the spectrum of the operator $\Phi(-\Delta^M)$ consists of eigenvalues $\lambda_1^M < \lambda_2^M {\leqslant}\lambda_3^M {\leqslant}... \to \infty$ satisfying $$\begin{aligned}
\label{eq:eigenvalue_subord}
\lambda_1^M = 0 \qquad \text{and} \qquad \lambda_k^M = \Phi(\mu_k^M), \ \ k = 2,3,\ldots,\end{aligned}$$ and the corresponding eigenfunctions are exactly the same as above. More precisely, we have $$\begin{aligned}
\label{eq:eigenvalue_deltaM}
\Phi(-\Delta^M) \psi^M_k = \lambda_k^M \psi^M_k \qquad \text{and} \qquad P_t^M \psi^M_k = {\rm e}^{-t \lambda_k^M} \psi^M_k, \ \ t>0, \ \ k =1,2,\ldots.\end{aligned}$$
Our present work requires additional regularity properties of the kernels $p^M_t(x,y)$ such as continuity, boundedness, and on-diagonal estimates, gathered in the following Lemma.
\[lem:regularity\] The following hold.
- There exists a constant $C >0$ such that for every $M, n \in {\mathds{Z}}_+$, $x,y \in [0,M)^d$ and $t>0$ we have $$\sum_{\mathbf i \in M{\mathds{Z}}^d \atop \mathbf i \notin [-nM,nM]^d } g_t(x,y+\mathbf i) {\leqslant}\frac{C}{M^{d}} \left(\frac{Mn}{\sqrt{t}} \vee 1\right)^{d-2} {\rm e}^{-\frac{1}{16}\left(\frac{Mn}{\sqrt{t}} \vee 1\right)^2}.$$ In particular, the series defining the kernel $g^M_t(x,y)$ is uniformly convergent in $(t,x,y)$ on every cuboid $[u,v] \times [0,M)^d \times [0,M)^d$, $0 < u < v < \infty$, and there exists a universal constant $\widetilde C >0$ such that for every $M \in {\mathds{Z}}_+$ we have $$g^M_t(x,y) {\leqslant}\widetilde C \left(t^{-d/2} \vee M^{-d}\right), \qquad x,y \in \mathcal T_M, \ t>0.$$
- Under the assumption the function $(t,x,y) \mapsto p_t^M(x,y)$ is continuous on $(0,\infty) \times \mathcal T_M \times \mathcal T_M$. Moreover, there exists a universal constant $C>0$ such that for every $M \in Z_+$ we have $$p^M_t(x,y) {\leqslant}C \left(\int_0^{\infty} {\rm e}^{-t \Phi(\lambda^{2/d})}{\rm d} \lambda \vee M^{-d}\right), \qquad x,y \in \mathcal T_M, \ t>0.$$ In particular, $p_t^M(x,y)$ is bounded on every cuboid $[t_0,\infty) \times \mathcal T_M \times \mathcal T_M$, $t_0>0$.
\(1) For every $x,y \in [0,M)^d$, $t>0$ and $M, n \in {\mathds{Z}}_+$ we have $$\begin{aligned}
\sum_{\mathbf i \in M{\mathds{Z}}^d \atop \mathbf i \notin [-nM,nM]^d } g_t(x,y+\mathbf i) & {\leqslant}& \sum_{k {\geqslant}n} \frac{(2(k+1))^d-(2k)^d}{(4\pi t)^{d/2}} {\rm e}^{-\frac{(kM)^2}{4t}} \\
& {\leqslant}&\frac{c}{\sqrt{t} M^{d-1}}\sum_{k {\geqslant}n} \left(\frac{kM}{\sqrt{t}}\right)^{d-1}{\rm e}^{-\left(\frac{(k+1)M}{4\sqrt{t}}\right)^2} \\
& {\leqslant}&\frac{c}{\sqrt{t} M^{d-1}}\int_n^{\infty}\left(\frac{Mx}{\sqrt{t}}\right)^{d-1}{\rm e}^{-\frac{1}{ 16} \left(\frac{Mx}{\sqrt{t}}\right)^2} {\rm d}x,\end{aligned}$$ with an absolute constant $c>0$. By substitution, the latter expression is equal to $$\frac{c}{M^{d}}\int_{\frac{Mn}{\sqrt{t}}}^{\infty}y^{d-1} {\rm e}^{-\frac{y^2}{16}} {\rm d}y.$$ Using the elementary estimate $$\int_a^{\infty} y^{d-1} {\rm e}^{-\frac{y^2}{ 16}} {\rm d}y {\leqslant}c_1 (a \vee 1)^{d-2} {\rm e}^{-\frac{1}{ 16}(a \vee 1)^2}, \quad a >0,$$ where $c_1>0$ is a uniform constant, we finally get $$\sum_{\mathbf i \in M{\mathds{Z}}^d \atop \mathbf i \notin [-nM,nM]^d } g_t(x,y+\mathbf i) {\leqslant}\frac{c_2}{M^{d}} \left(\frac{Mn}{\sqrt{t}} \vee 1\right)^{d-2} {\rm e}^{-\frac{1}{ 16}\left(\frac{Mn}{\sqrt{t}} \vee 1\right)^2},$$ which is exactly the first assertion of part (1). The uniform convergence follows directly from this uniform bound for the tail of the series. To prove the other assertion of (1), we write $$g_t^M(x,y) {\leqslant}c_3 t^{-d/2} + \sum_{\mathbf i \in M{\mathds{Z}}^d \atop \mathbf i \notin [-M,M]^d } g_t(x,y+\mathbf i).$$ The second term can be easily estimated by using the bound proven above with $n=1$. We have two cases. If $\sqrt{t} {\geqslant}M$, then $g_t^M(x,y) {\leqslant}c_3 t^{-d/2}+c_2M^{-d} {\leqslant}(c_2+c_3) M^{-d}$. If $\sqrt{t} < M$, then $$\left(\frac{M}{\sqrt{t}} \vee 1\right)^{d-2} {\rm e}^{-\frac{1}{ 16}\left(\frac{M}{\sqrt{t}} \vee 1\right)^2} = \left(\frac{M}{\sqrt{t}}\right)^{d-2} {\rm e}^{-\frac{1}{ 16}\left(\frac{M}{\sqrt{t}}\right)^2} {\leqslant}c_4,$$ and, similarly as above, $g_t^M(x,y) {\leqslant}c_3 t^{-d/2}+c_2c_4M^{-d} {\leqslant}(c_2c_4+c_3) t^{-d/2}$. This implies the second estimate in (1).
\(2) We first show the estimate and the boundedness. By the upper estimate for the kernel $g_t^M(x,y)$ proven above, for $M \in {\mathds{Z}}_+$, $x, y \in \mathcal T_M,$ and $t>0$, we have $$p^M_t(x,y) {\leqslant}c_4 \left(\int_0^{M^2} u^{-d/2} \eta_t({\rm d}u) + M^{-d} \eta_t[M^2,\infty)\right) {\leqslant}c_4 \left(\int_0^{\infty} u^{-d/2} \eta_t({\rm d}u) + M^{-d}\right).$$ We then derive from Lemma \[lem:basic\_subord\] that $$p^M_t(x,y) {\leqslant}c_5 \left(\int_0^{\infty} {\rm e}^{-t \Phi(\lambda^{2/d})} {\rm d}\lambda +M^{-d} \right),$$ and, for every $t_0>0$, $$\sup_{(t,x,y) \in [t_0,\infty) \times \mathcal T_M \times \mathcal T_M} p^M_t(x,y) < \infty.$$ We now prove the continuity. Since the function $(t,x,y) \mapsto p_t(x,y)$ is continuous on $(0,\infty) \times {\mathds{R}}^d \times {\mathds{R}}^d$, it is enough to justify that the series $$\sum_{\mathbf i \in M{\mathds{Z}}^d} p_t(x,y+\mathbf i)$$ is uniformly convergent on every cuboid $[t_0, t_1] \times [0,M)^d \times [0,M)^d$, $0<t_0 < t_1 < \infty$. We only need to prove that the tail $$\sum_{\mathbf i \in M{\mathds{Z}}^d \atop \mathbf i \notin [-nM,nM]^d } p_t(x,y+\mathbf i)$$ goes to zero as $n \to \infty$, uniformly in $(t,x,y) \in [t_0, t_1] \times [0,M)^d \times [0,M)^d$. Using the tail estimate from part (1), Fubini-Tonelli an the fact that $c_6:= \sup_{r {\geqslant}1} r^{ d-2} {\rm e}^{- r^2/16} < \infty$, we get $$\begin{aligned}
\sum_{\mathbf i \in M{\mathds{Z}}^d \atop \mathbf i \notin [-nM,nM]^d } & p_t(x,y+\mathbf i) {\leqslant}\frac{c_7}{M^{d}} \int_0^{\infty}\left(\frac{Mn}{\sqrt{u}} \vee 1\right)^{d-2} {\rm e}^{-\frac{1}{16}\left(\frac{Mn}{\sqrt{u}} \vee 1\right)^2} \eta_t({\rm d}u) \\
& {\leqslant}\frac{c_7}{M^{d}} \left(\int_{(0,n]}\left(\frac{Mn}{\sqrt{u}}\right)^{d-2} {\rm e}^{-\frac{1}{16}\left(\frac{Mn}{\sqrt{u}}\right)^2} \eta_t({\rm d}u) + c_6 \eta_t\big(n, \infty\big)\right) \\
& {\leqslant}\frac{c_7}{M^2} n^{ d-1} {\rm e}^{-\frac{n}{16}} \int_{(0,n]} u^{-d/2} \eta_t({\rm d}u) + c_6c_7 \eta_t(n,\infty).\end{aligned}$$ Now, by Lemma \[lem:basic\_subord\], the integral $\int_0^\infty u^{-d/2} \eta_t({\rm d}u)$ is uniformly bounded for $t {\geqslant}t_0$. Moreover, by following the argument in [@bib:KK-KPP1 Lemma 2.2], we obtain $$\int_0^n {\rm e}^{-\lambda u} \eta_t(u,\infty) {\rm d}u {\leqslant}\frac{t \Phi(\lambda)}{\lambda}, \quad \lambda, t>0,$$ which yields
$$\eta_t(n,\infty) (1-{\rm e}^{-\lambda n}) {\leqslant}t \Phi(\lambda), \quad \lambda, t>0.$$ By taking $\lambda = 1/n$, we get $\eta_t(n,\infty) {\leqslant}(1-{\rm e}^{-1})^{-1} t \Phi(1/n) {\leqslant}(1-{\rm e}^{-1})^{-1} t_1 \Phi(1/n)$, whenever $t {\leqslant}t_1$. This implies the claimed uniform convergence, completing the proof of the lemma.
The semigroup $\big\{P^M_t: t {\geqslant}0\big\}$ determines a conservative Markov process $(X^M_t)_{t {\geqslant}0}$ on the torus $\mathcal T_M$. If we denote by $\mathbf P_x^M$ the measure concentrated on trajectories that start from $x \in \mathcal T_M$, then $$\mathbf P_x^M(X^M_t \in A) = P^M_t {\mathds{1}}_A(x) = \int_A p^M_t(x,y) {\rm d}y, \quad A \in \mathcal B(\mathcal T_M), \ x \in \mathcal T_M, \ t >0.$$ It is a symmetric Feller process with continuous and bounded transition probability densities $p^M_t(x,y)$. Due to this process can be identified pathwise as $$X^M_t = \pi_M(X_t), \quad t>0,$$ where $(X_t)_{t {\geqslant}0}$ is the subordinate Brownian motion in ${\mathds{R}}^d$, introduced in the previous section. Throughout the paper we call this process the *subordinate Brownian motion on the torus $\mathcal T_M$*.
For given $t{\geqslant}0$ and $x,y\in\mathcal T_M,$ the bridge measure of the process $X^M,$ conditioned to have $ X_t^M=y,$ $\mathbf P_x^M-$almost surely, is defined by a relation similar to . These measures are denoted by $\mathbf P_{x,y}^{M,t}.$
The bridge measures for the process in ${\mathds{R}}^d$ and on the torus $\mathcal T_M$ are related through the following identity.
\[lm:rotation\] For every $t>0$, $x,y \in {\mathds{R}}^d$, $M=1,2,...$ and any set $A \in \mathcal B(D([0,t],\mathcal T_M)$ we have $$\label{eq:rot1}
p^M_t(\pi_M(x),\pi_M(y))\mathbf P^{M,t}_{\pi_M(x),\pi_M(y)}[A]=
\sum_{y'\in\pi_M^{-1}(\pi_M(y))} p_t(x,y')\mathbf
P^t_{x,y'}[\pi_M^{-1}(A)]$$ ($D([0,t],\mathcal T_M))$ is the Skorohod space).
This statement is readily seen for cylindrical sets and then extended to the desired range of $A$’s by the Monotone Class Theorem. Its fractal counterpart was discussed in [@bib:KK-KPP1 Lemma 2.6].
Random Anderson (alloy-type) potentials {#sec:potentials}
---------------------------------------
Our approach in the present paper allows us to study the alloy-type random fields $$\begin{aligned}
\label{eq:alloy}
V^{\omega}(x) = \sum_{{\bf i} \in {\mathds{Z}}^d} q_{{\bf i}}(\omega) W(x-{\bf i}), \quad x \in {\mathds{R}}^d,\end{aligned}$$ with possibly singular single-site potentials $W$ of bounded support which are in Kato classes corresponding to the operators considered. The main part of our argument is based on an application of certain periodization of such potentials: for given $M{\geqslant}1$ we define $$\begin{aligned}
\label{eq:szn-per-of-V}
V_M^{\omega}(x)&:= &\sum_{{\bf i}\in[0, M)^d} \left(q_{\bf i}(\omega) \sum_{{\bf i'}\in \pi_{M}^{-1}({\bf i})} W(x-{\bf i'})\right) \nonumber\\
&=& \sum_{ {\bf i} \in {\mathds{Z}}^d} q_{\pi_M({\bf i})} W(x-{\bf i}),\quad x\in{\mathds{R}}^d.\end{aligned}$$ This means that we first periodize the lattice random variables $\{q_{\mathbf i}\}_{\mathbf i\in {\mathds{Z}}^d}$ with respect to $\pi_M$, and then, based on that, we construct a new random potential which is also periodic in the usual sense: $V_M^{\omega}(x+{\bf i}) = V_M^{\omega}(x)$, ${\bf i} \in M {\mathds{Z}}^d$. We call it the [*Sznitman-type periodization*]{} of $V^{\omega}$. For simplicity, we will use the same letter for the restriction of this potential to $\mathcal T_M.$
Recall that the Kato class $\mathcal K$ associated with the operator $\Phi(-\Delta)$ consists of those Borel functions $f:{\mathds{R}}^d \to {\mathds{R}}$ for which $$\lim_{t \searrow 0} \sup_{x \in {\mathds{R}}^d} \int_0^t P_s|f|(x)\,{\rm d}s = 0.$$ Similarly, a Borel function $f:\mathcal T_M \to {\mathds{R}}$ belongs to the Kato class $\mathcal K^M$ associated with $\Phi(-\Delta^M)$ if $$\lim_{t \searrow 0} \sup_{x \in \mathcal T_M} \int_0^t P^M_s|f|(x)\,{\rm d}s = 0.$$ Moreover, we say that a Borel function $f$ belongs to the local Kato class $\mathcal K_{{\mathrm{loc}}}$ if its restriction to an arbitrary bounded Borel subset of ${\mathds{R}}^d$ is in $\mathcal K$. Note that the torus $\mathcal T_M$ is a compact space and so the local Kato class for $\mathcal T_M$ would agree with $\mathcal K^M$. Therefore there is no need to define it separately. One can check that $L^{\infty}({\mathds{R}}^d) \subset \mathcal K$ and $L^{\infty}_{{\mathrm{loc}}}({\mathds{R}}^d) \subset \mathcal K_{{\mathrm{loc}}}$. Moreover, $\mathcal K_{{\mathrm{loc}}} \subset L^1_{{\mathrm{loc}}}({\mathds{R}}^d)$ and $\mathcal K^M \subset L^1(\mathcal T_M) = L^1_{{\mathrm{loc}}}(\mathcal T_M)$.
We now show that the alloy-type random potentials $V^{\omega}$ and $V^{\omega}_M$ inherit the Kato-regularity from their profiles $W$.
\[prop:Kato\_regularity\] Let $W \in \mathcal K$, $W {\geqslant}0$, be of bounded support and let the assumption **** hold. Then, for every $M \in {\mathds{Z}}_+$ and $\omega \in \Omega$, we have
- $V^{\omega} \in \mathcal K_{{\mathrm{loc}}}$,
- $V^{\omega}_M \in \mathcal K_{{\mathrm{loc}}}$,
- $V^{\omega}_M \in \mathcal K^M$.
Fix $\omega \in \Omega$ and suppose $\operatorname{supp}W \subset [-M_0,M_0]^d$, for some $M_0 \in {\mathds{Z}}_+$.
\(1) Denote $V^{n,\omega} = {\mathds{1}}_{[-n,n]^d} V^{\omega}$, $n \in {\mathds{Z}}_+$. Observe that $$V^{n,\omega}(x) {\leqslant}\sum_{{\bf i} \in {\mathds{Z}}^d \cap [-M_0-n,M_0+n]^d} q_{{\bf i}}(\omega) W(x-{\bf i}), \quad x \in {\mathds{R}}^d.$$ We have $$\begin{aligned}
\int_0^t P_s V^{n,\omega}(x)\,{\rm d}s & {\leqslant}\sum_{{\bf i} \in {\mathds{Z}}^d \cap [-M_0-n,M_0+n]^d} q_{{\bf i}}(\omega) \int_0^t P_s W (x-{\bf i})\,{\rm d}s \\
& {\leqslant}\sup_{y \in {\mathds{R}}^d} \int_0^t P_s W (y) \left(\sum_{{\bf i} \in {\mathds{Z}}^d \cap [-M_0-n,M_0+n]^d} q_{{\bf i}}(\omega)\right)\,{\rm d}s.\end{aligned}$$ The sum on the right hand side has finitely many terms and $W \in \mathcal K$. Therefore by taking the supremum over $x \in {\mathds{R}}^d$ on the left hand side and then letting $t \searrow 0$, we get that $$\sup_{x \in {\mathds{R}}^d} \int_0^t P_s V^{n,\omega}(x) \to 0,$$ for arbitrary $n \in {\mathds{Z}}_+$. Hence $V^{\omega} \in \mathcal K_{{\mathrm{loc}}}$.
\(2) The proof is a minor modification of that of (1) as we only need to replace $q_{{\bf i}}(\omega)$ with $q_{\pi_M({\bf i})}(\omega)$ in the sum defining the potential.
\(3) Fix $M \in {\mathds{Z}}_+$. By the definition of the operators $P_t^M$ and the potential $V^{\omega}_M$, for every $x \in \mathcal T_M$ and $s>0$ we have $$\begin{aligned}
P_s^M V^{\omega}_M(x) & = \int_{\mathcal T_M} p^M_s(x,y) V^{\omega}_M(y) {\rm d}y \\
& = \int_{[0,M)^d} \left(\sum_{{\bf i} \in M{\mathds{Z}}^d} p_t(x,y+{\bf i}) \right)V^{\omega}_M(y) {\rm d}y \\
& = \int_{[0,M)^d} \left(\sum_{{\bf i} \in M{\mathds{Z}}^d \atop {\bf i} \in [-M,M]^d} p_t(x,y+{\bf i}) \ + \sum_{{\bf i} \in M{\mathds{Z}}^d \atop {\bf i} \notin [-M,M]^d} p_t(x,y+{\bf i}) \right)V^{\omega}_M(y) {\rm d}y \\
& = P_t \big({\mathds{1}}_{[-M,M]^d}V^{\omega}_M\big)(x)
+ \int_{[0,M)^d} \left(\sum_{{\bf i} \in M{\mathds{Z}}^d \atop {\bf i} \notin [-M,M]^d} p_t(x,y+{\bf i}) \right)V^{\omega}_M(y) {\rm d}y.\end{aligned}$$ By Fubini-Tonelli and the tail estimate in Lemma \[lem:regularity\] (1) $$\sum_{{\bf i} \in M{\mathds{Z}}^d \atop {\bf i} \notin [-M,M]^d} p_t(x,y+{\bf i})
= \int_0^{\infty} \left(\sum_{{\bf i} \in M{\mathds{Z}}^d \atop {\bf i} \notin [-M,M]^d} g_t(x,y+{\bf i}) \right) \eta_t({\rm d}u) {\leqslant}cM^{-d},$$ which gives $$P_s^M V^{\omega}_M(x) {\leqslant}P_t \big({\mathds{1}}_{[-M,M]^d}V^{\omega}_M\big)(x) + cM^{-d} \int_{[0,M)^d}V^{\omega}_M(y) {\rm d}y.$$ By part (2) we have $V^{\omega}_M \in \mathcal K_{{\mathrm{loc}}}$. In particular, $V^{\omega}_M \in L^1_{{\mathrm{loc}}}({\mathds{R}}^d)$. Hence $$\sup_x \int_0^t P_s^M V^{\omega}_M(x)\,{\rm d}s {\leqslant}\sup_x \int_0^t P_t \big({\mathds{1}}_{[-M,M]^d}V^{\omega}_M\big)(x)\,{\rm d}s + c t M^{-d} \int_{[0,M)^d}V^{\omega}_M(y)\, {\rm d}y \longrightarrow 0,$$ as $t \searrow 0$. This means that $V^{\omega}_M \in \mathcal K^M$.
As mentioned in the introduction, every bounded function with compact support is automatically in the Kato class ${\mathcal{K}}$. We now provide examples of singular functions from ${\mathcal{K}}$.
\[ex:singular\] Let $\Phi(\lambda) = \lambda^{\alpha/2}$, $\alpha \in (0,2]$ (i.e. we either consider the Laplace operator $-\Delta$ or the fractional Laplace operators $(-\Delta)^{\alpha/2}$, $\alpha \in (0,2)$). For simplicity, assume additionally that $\alpha < d \in {\mathds{Z}}_{+}$. It is known (see e.g. [@bib:BB]) that in this case $$f \in {\mathcal{K}}\qquad \Longleftrightarrow \qquad \lim_{r \searrow 0} \sup_{x \in {\mathds{R}}^d} \int_{|x-y|<r} \frac{f(y)}{|x-y|^{d-\alpha}} {\rm d}y = 0.$$ The same is true for $\Phi(\lambda) = (\lambda+m^{2/\alpha})^{\alpha/2}-m$, $\alpha \in (0,2)$, $m>0$, i.e. for the quasi-relativistic operators. If we now take $W(y):= {\mathds{1}}_{B(0,1)}(y) |y|^{-\beta}$, $\beta >0$, then we see that $W \in {\mathcal{K}}$ if and only if $\beta < \alpha$. In view of the assumption [**(W)**]{} it is also instructive to verify that $W \in L^2({\mathds{R}}^d)$ if and only if $\beta < d/2$. In particular, $W \in {\mathcal{K}}\cap L^2({\mathds{R}}^d)$ if and only if $\beta < \alpha \wedge d/2$.
This example indicates that the intersection ${\mathcal{K}}\cap L^2({\mathds{R}}^d)$ is typically a fairly non-trivial function space, but in general there are no inclusions between ${\mathcal{K}}$ and $L^2({\mathds{R}}^d)$.
Schrödinger operators and the Feynman–Kac formula
-------------------------------------------------
We now introduce the class of random Schrödinger operators based on $\Phi(-\Delta)$ and $-\Phi(-\Delta^M)$, and we discuss their spectral properties. Our standard reference here will be the monograph of Demuth and van Casteren [@bib:DC] which is concerned with the spectral theory of self-adjoint Feller operators.
In the previous sections we have verified that the *subordinate semigroups* $\big\{P_t:t {\geqslant}0\big\}$ and $\big\{P^M_t:t {\geqslant}0\big\}$, determined by the kernels $p_t(x,y)$ and $p^M_t(x,y)$, respectively, satisfy the *basic assumptions of spectral stochastic analysis* (BASSA in short) and in consequence the operators $-\Phi(-\Delta)$ and $-\Phi(-\Delta^M)$ are *(free) Feller generators* [@bib:DC Assumptions A1-A4 and Definition 1.3 in Section 1.B].
Throughout this section we assume that $V^{\omega}$ and $V^{\omega}_M$ are random alloy-type potentials given by and , constructed for a compactly supported and nonnegative single-site potential $W \in \mathcal K$ and lattice random variables $\big\{q_{\bf i}\big\}_{{\bf i} \in {\mathds{Z}}^d}$ satisfying the assumption **(Q)**. Thus, by Proposition \[prop:Kato\_regularity\], we have $V^{\omega} \in \mathcal K_{{\mathrm{loc}}}$ and $V^{\omega}_M \in \mathcal K^M$, for every realization of lattice configuration. This allows us to define the random Schrödinger operators $$H^{\omega} = \Phi(-\Delta)+V^{\omega} \qquad \text{and} \qquad H^{\omega}_M = \Phi(-\Delta^M)+V^{\omega}_M$$ as positive self-adjoint operators on $L^2({\mathds{R}}^d)$ and $L^2(\mathcal T_M)$, respectively [@bib:DC Theorem 2.5]. It is decisive for this work that the evolution semigroups of these operators can be represented probabilistically with respect to subordinate processes $(X_t)_{t {\geqslant}0}$ and $(X^M_t)_{t {\geqslant}0}$. More precisely, the following Feynman–Kac formulas hold: $$P_t^{V^{\omega}} f(x): ={\rm e}^{-tH^{\omega}} f(x) = {\mathbf{E}}_x\left[{\rm e}^{-\int_0^t V^{\omega}(X_s) {\rm d}s} f(X_t)\right], \quad f \in L^2({\mathds{R}}^d), \quad t>0,$$ and $$P_t^{M,V^{\omega}_M} f(x) :={\rm e}^{-tH^{\omega}_M} f(x) = {\mathbf{E}}_x^M\left[{\rm e}^{-\int_0^t V^{\omega}_M(X^M_s) {\rm d}s} f(X^M_t)\right], \quad f \in L^2(\mathcal T_M), \quad t>0.$$ Both $P_t^{V^{\omega}}$ and $P_t^{M,V^{\omega}_M}$, $t>0$, are integral operators with bounded and symmetric kernels $$p^{V^{\omega}}_t(x,y) = p_t(x,y) {\mathbf{E}}_{x,y}^t\left[{\rm e}^{-\int_0^t V^{\omega}(X_s) {\rm d}s}\right],$$ and $$\label{eq:FK-torus}
p^{M,V^{\omega}}_t(x,y) = p^M_t(x,y) {\mathbf{E}}_{x,y}^{M,t}\left[{\rm e}^{-\int_0^t V^{\omega}_M(X^M_s) {\rm d}s}\right],$$ where ${\mathbf{E}}_{x,y}^t$ and ${\mathbf{E}}_{x,y}^{M,t}$ are expected values with respect to bridge measures ${\mathbf{P}}_{x,y}^t$ and ${\mathbf{P}}_{x,y}^{M,t}$ introduced in previous sections.
Observe that $p^{M,V^{\omega}}_t(x,y) {\leqslant}p^M_t(x,y)$ and recall that for every fixed $t>0$ the kernel $p^M_t(x,y)$ is a bounded function on $\mathcal T_M \times \mathcal T_M$. Since $|\mathcal T_M| = M^d < \infty$, this gives that the operators $P_t^{M,V^{\omega}_M}$, $t>0$, are Hilbert-Schmidt in $L^2(\mathcal T_M)$. This implies that the spectrum of the Schrödinger operator $H^{\omega}_M$ is discrete – it consists of a sequence of eigenvalues $$0{\leqslant}\lambda_1^{M, V^{\omega}} < \lambda_2^{M, V^{\omega}} {\leqslant}\lambda_3^{M, V^{\omega}} {\leqslant}... \to \infty$$ with finite multiplicities, and the corresponding eigenfunctions $\big\{\psi_k^{M, V^{\omega}}\big\}_{k=1}^{\infty}$ form a complete orthonormal system in $L^2(\mathcal T_M)$.
Dirichlet Schrödinger operators and the integrated density of states {#sec:dirichlet}
--------------------------------------------------------------------
Denote by $H^{\omega}_\Lambda$ the operator $H^{\omega}$ constrained to a bounded, nonempty region $\Lambda\subset {\mathds{R}}^d$ (we consider Dirichlet conditions on $\Lambda^c$ in the non-local case and on $\partial \Lambda$ in the local case) and let $\big\{{\rm e}^{-tH^{\omega}_{\Lambda}}; t{\geqslant}0\big\}$ be its evolution semigroup on $L^2(\Lambda)$. Then we have the following Feynman–Kac formula: $${\rm e}^{-tH^{\omega}_{\Lambda}} = P_t^{V^{\omega},\Lambda} f(x):= {\mathbf{E}}_x\left[{\rm e}^{-\int_0^t V^{\omega}(X_s) {\rm d}s} f(X_t); t<\tau_{\Lambda}\right], \quad f \in L^2(\Lambda, {\rm d}x), \quad t>0.$$ Here $\tau_{\Lambda}:=\inf\{t{\geqslant}0: \, X_t\notin \Lambda\}$ denotes the first exit time of the process from the domain $\Lambda$. All the $P_t^{V^{\omega},\Lambda}$, $t>0$, are integral operators with bounded and symmetric kernels $$\begin{aligned}
\label{def:sem-dir-kernel-bridge}
p^{V^{\omega}, \Lambda}_t(x,y) = p_t(x,y) \ {\mathbf{E}}^t_{x,y}\left[{\rm e}^{-\int_0^t V^{\omega}(X_s) {\rm d}s} ; t<\tau_{\Lambda}\right].\end{aligned}$$ Again, since $|\Lambda|<\infty$, the operators $P_t^{V^{\omega},\Lambda}$, $t>0$, are Hilbert-Schmidt. In particular, there exists a complete orthonormal system, consisting of eigenfunctions of the operator $H^{\omega}_{\Lambda}$. The corresponding eigenvalues satisfy $0{\leqslant}\lambda_1^{V^{\omega}}(\Lambda) < \lambda_2^{V^{\omega}}(\Lambda){\leqslant}\lambda_3^{V^{\omega}}(\Lambda) {\leqslant}\ldots \to \infty$; each $\lambda_k^{V^{\omega}}(\Lambda)$ is of finite multiplicity and the ground state eigenvalue $\lambda_1^{V^{\omega}}(\Lambda)$ is simple.
We are now in a position to give the formal definition of the IDS. For a given bounded domain $\Lambda\subset {\mathds{R}}^d$, let $$\ell_{\Lambda}^{\omega}(\cdot)=\frac{1}{|\Lambda|}\sum_{k=1}^\infty\delta_{\lambda_k^{V^{\omega}}(\Lambda)}
(\cdot)$$ be the counting measure on the spectrum of $H_{\Lambda}^{\omega},$ normalized by the volume. Under the assumption **(Q)**, the random alloy-type potential $V^{\omega}$ is stationary with respect to ${\mathds{Z}}^d$. Therefore if we restrict our attention to sets $\Lambda$ composed of unit cubes with vertices in ${\mathds{Z}}^d,$ then it follows from the maximal ergodic theorem (see e.g. [@bib:Car-Lac Remark VI.1.2]) that the measures $\ell_{\Lambda}^{\omega}$ converge vaguely, as $\Lambda \nearrow {\mathds{R}}^d,$ to a nonrandom measure $\ell,$ which is called the integrated density of states of $H^\omega.$ The vague convergence of $\ell_{\Lambda}^{\omega}$ when $\Lambda \nearrow {\mathds{R}}^d$ amounts to the convergence of their Laplace transforms $$\begin{aligned}
L_\Lambda^\omega(t)&=&\frac{1}{|\Lambda|} \int_{[0,\infty)}{\rm e}^{-t\lambda}\ell_{\Lambda}^{\omega}({\rm d}\lambda)= \frac{1}{|\Lambda|}{\rm Tr}\, P_t^{V^{\omega},\Lambda}
=
\frac{1}{|\Lambda|}\int_\Lambda p^{V^{\omega},\Lambda}_t(x,x) {\rm d}x
\\
&=&
\frac{p_t(0,0)}{|\Lambda|}\int_\Lambda {\mathbf{E}}_{x,x}^t\left[{\rm e}^{-\int_0^t V^\omega(X_s){\rm d}s}; t < \tau_\Lambda \right] {\rm d}x,
\end{aligned}$$ for any fixed $t>0.$ Denoting by $L$ the Laplace transform of the measure $\ell,$ we have $$\label{eq:el-almost-everywhere}
L(t) = \lim_{\Lambda \nearrow {\mathds{R}}^d} \mathbb E^{\mathbb Q}L_\Lambda^\omega(t), \quad t>0.$$ Let us note that from the ${\mathds{Z}}^d-$stationarity of the potential we have that for any $\Lambda$ as above, $$\label{eq:IDS-formula}
L(t)=\frac{p_t(0,0)}{|\Lambda|}\int_\Lambda \mathbb E^{\mathbb Q}\mathbf E_{x,x}^t\left[{\rm e}^{-\int_0^t V^\omega(X_s){\rm d}s}\right]{\rm d}x.$$ In particular, for any $M\in{\mathds{Z}}_+,$ $$\label{eq:IDS-formula_2}
L(t)=\frac{p_t(0,0)}{M^d}\int_{[0,M)^d} \mathbb E^{\mathbb Q}\mathbf E_{x,x}^t\left[{\rm e}^{-\int_0^t V^\omega(X_s){\rm d}s}\right]{\rm d}x.$$
In the next two sections we will determine the rate of decay, as $t\to\infty,$ of the Laplace transform $L(t)$ of the measure $\ell.$
The upper bound for the Laplace transforms {#sec:upper}
==========================================
We start with the upper bounds, as they will determine the correct rate(s) in the asymptotics. We have more flexibility with lower bounds, thus the crucial step is to get a correct upper bound. To put oneself in a proper perspective, let us recall that for Lévy operators satisfying [**(B)**]{} perturbed by a Poissonian-type potential, the decay of $L(t)$ was of order ${\rm e}^{-Ct^{\frac{d}{d+\alpha}}}.$ We obtained a similar rate for the Anderson model for the fractional Laplacians, provided the distribution of the random variables $q_{\mathbf i}$ had an atom at zero. However, when the atom at zero is not present, then our earlier work [@bib:KK-KPP-alloy-stable] indicates that an extra multiplicative input is needed in the decay rate. As Theorems \[th:upper-short\] and \[th:lower\] show, this is indeed the case.
The rate function and the statement of the upper bound (Theorem \[th:upper-short\]) {#sec:rate_der}
-----------------------------------------------------------------------------------
We start with the definition of the function $h(t)$ which will appear in the rate.
Let $\alpha \in(0,2]$ and $C_1>0$ be the scaling exponent and the constant from the assumption [**(B)**]{}, and let $\kappa_0$ and $M_0$ be the parameters appearing in the assumptions [**(Q)**]{} and [**(W)**]{}, respectively. Moreover, recall that by $\mu_2^1$ we have denoted the second eigenvalue of the operator $-\Delta^1$ (see Section \[sec:operators\_on\_tori\]). Denote $$\label{eq:c-zero}
D_0=\frac{1}{2}\frac{C_1 (\mu_2^1)^{\frac{\alpha}{2}}\|W\|_1}{\|W\|_1^2+(2M_0)^d\|W\|_2^2} > 0.$$ As it will be seen below, in fact $D_0$ can be choosen to be an arbitrary constant for which $$\label{eq:D-def}
\|W\|_1 > \frac{D_0(2M_0)^d\|W\|_2^2}{C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1}>0,$$ but for more clarity we prefer to keep $D_0$ fixed as in .
For $x>0$ let $$\begin{aligned}
\label{eq:Phi}
g(x)= \log\frac{1}{F_q(D_0/x)} = -\log \mathbb Q[q{\leqslant}D_0/x],\quad \mbox{ and } \quad j(x)= x^{d+\alpha}g({x^\alpha}).\end{aligned}$$ Due to the assumption [**(Q)**]{} the function $j(x)$ is increasing, and continuous for $x{\geqslant}x_0:= (D_0/\kappa_0)^{1/\alpha} .$ Therefore $j^{-1}(t)$ is well-defined for $t{\geqslant}t_0:= j(x_0).$ Let $x_t=j^{-1}(t),$ $t{\geqslant}t_0.$ Finally, denote $$\label{eq:def-ht}
h(t) = g(x_t^{\alpha}),\quad t{\geqslant}t_0.$$ For later use, observe that $t$ and $x_t$ are related through the relation $$\label{eq:t-and-xt}
t= x_{t}^{d+\alpha}\log\left(\frac{1}{F_q(D_0/x_t^\alpha)}\right)= x_t^{d+\alpha}h(t), \quad t {\geqslant}t_0.$$ Moreover, the function $t\mapsto x_t$ is increasing and since $
\lim_{x\to\infty}j(x)=\infty,$ we have $\lim_{t\to\infty}x_t=\infty$ as well. This implies that $$\label{eq:lim-h-1}
\lim_{t\to\infty}\frac{h(t)}{t} =0.$$ The limit $\lim_{t \to \infty} h(t)$ always exists and $$\begin{aligned}
\label{eq:lim_h}
\lim_{t \to \infty} h(t) = \begin{cases}
\infty \quad \, \ \ \ \ \ \ \text{if} \quad F_q(0) = 0, \\
\log\frac{1}{F_q(0)} \quad \text{if} \quad F_q(0) \in (0, 1).
\end{cases}\end{aligned}$$ In Section \[sec:examples\] we give examples of such functions $h.$
We are now ready to present the main theorem of this section.
\[th:upper-short\] Assume [**(B)**]{}, [**(Q)**]{}, and [**(W)**]{}. Let $h$ be given by . Then there exists $C>0$ such that $$\label{eq:upper-1}
\limsup_{t\to\infty} \frac{\log L(t)}{t^{\frac{d}{d+\alpha}} (h(t))^{\frac{\alpha}{d+\alpha}}} {\leqslant}-C.$$ In particular, when the distribution of $q$ has an atom at zero, i.e. $F_q(0) >0$, then $$\label{eq:upper-2}
\limsup_{t\to\infty}\frac{\log L(t)}{t^{\frac{d}{d+\alpha}}}{\leqslant}-C \left(\log\frac{1}{F_q(0)}\right)^{\frac{\alpha}{d+\alpha}} .$$
The proof of the theorem is split into three parts which are presented in Sections \[sec:trace\], \[sec:temple\] and \[sec:conclusion\] below.
Preparatory steps in the proof of Theorem \[th:upper-short\] - the trace estimate {#sec:trace}
---------------------------------------------------------------------------------
To begin the proof, we proceed as in [@bib:KK-KPP-alloy-stable Proof of Theorem 4.1, the first page of Section 5.3]. As a corollary of [@bib:KK-KPP-alloy-stable Lemma 5.1] we get that for any given $M\in{\mathds{Z}}_+$ and any $t>0$ we have the following relation between the exponential functionals of the subordinate Brownian motion $X = (X_t)_{t {\geqslant}0}$ in ${\mathds{R}}^d$ with the un-periodized and periodized potentials (cf. -): $$\mathbb E^\mathbb Q\left[{\rm e}^{-\int_0^t V^\omega(X_s(w)){\rm d}s}\right] {\leqslant}\mathbb E^\mathbb Q\left[{\rm e}^{-\int_0^t V_M^\omega(X_s(w)){\rm d}s}\right],$$ and further, invoking Lemma \[lm:rotation\], $$\label{eq:old-argument}
L(t){\leqslant}\frac{1}{M^d}
\mathbb E^{\mathbb Q}\int_{\mathcal T_M} p^M_t(x,x)\mathbf E_{x,x}^{M,t}\left[{\rm e}^{-\int_0^t V_M^{\omega}(X_s^M){\rm d}s}\right]{\rm d}x$$ (recall that $X^M=(X_s^M)_{s{\geqslant}0}$ is the subordinate Brownian motion on the torus $\mathcal T_M,$ $p_t^M(\cdot,\cdot)$ are its transition densities, and $\mathbf E_{x,x}^{M,t}$ - its bridge measures).
The integral at the right-hand side of is the trace of the random operator $P_t^{M,V^{\omega}_M}$ with integral kernel defined in . Consequently, for $t>1$, $$\begin{aligned}
L(t)&{\leqslant}&\frac{1}{M^d} \mathbb E^{\mathbb Q}\mbox{Tr}\,P_t^{M,V^{\omega}_M}
= \frac{1}{M^d} \mathbb E^{\mathbb Q}\sum_{n=1}^\infty {\rm e}^{-t\lambda_n^{{M, V^{\omega}_M}}}
\\
&{\leqslant}& \frac{1}{M^d} \mathbb E^{\mathbb Q}\left[{\rm e}^{-(t-1)\lambda_1^{M, V^{\omega}_M}}\mbox{Tr}\,P_1^{M,V^{\omega}_M}\right]
\\
&{\leqslant}&
\mathbb E^{\mathbb Q}\left[{\rm e}^{-(t-1)\lambda_1^{M, V^{\omega}_M}}\right] \frac{1}{M^d} \int_{\mathcal T_M} p^M_1(x,x){\rm d}x.
\end{aligned}$$ From Lemma \[lem:regularity\](2) there exists a constant $C>0$ independent of $M$ for which $p^M_1(x,x){\leqslant}C,$ $x\in \mathcal T_M,$ so that we are led to the bound $$\label{eq:stop}
L(t){\leqslant}C
\mathbb E^{\mathbb Q}\left[{\rm e}^{-(t-1)\lambda_1^{M, V^{\omega}_M}}\right], \quad t > 1.$$
Temple’s inequality and the lower scaling of the ground state eigenvalue {#sec:temple}
------------------------------------------------------------------------
In this section we find an appropriate lower estimate for the ground state eigenvalue $\lambda_1^{M,V^{\omega}_M}$ of the Schrödinger operator $H^{\omega}_M$. We will use the following inequality.
\[prop:temple\] Suppose $H$ is a self-adjoint operator on a Hilbert space with inner product $\langle \cdot,\cdot \rangle$ such that $\lambda_1:=\inf\sigma(H)$ is an isolated eigenvalue and let $\mu{\leqslant}\inf(\sigma(H)\setminus \{\lambda_1\}).$ Then for any $\psi\in\mathcal D(H)$ which satisfies $$\label{eq:temple-condition}
\langle \psi, H\psi\rangle <\mu\quad\mbox{ and } \quad \|\psi\|=1$$ the following bound holds: $$\label{eq:temple}
\lambda_1{\geqslant}\langle \psi, H\psi\rangle - \frac{\langle H\psi, H\psi\rangle-\langle \psi,H\psi\rangle^2}{\mu-\langle \psi, H\psi\rangle}.$$
For given $M\in\mathbb Z_+, $ consider truncated random variables $$\begin{aligned}
\label{eq:trunc_qs}
\widetilde{q}_{\mathbf i}=q_{\mathbf i}\wedge \frac{D_0}{M^{\alpha}},\quad \mathbf i\in\mathbb Z^d,
\end{aligned}$$ and random Schrödinger operators $$\label{eq:tilde-H}
\widetilde{H}^\omega_M= \Phi(-\Delta^M)+\widetilde V_M^\omega,$$ where $\widetilde {V}_M^\omega$ is the Sznitman-periodization of $$\widetilde{V}^\omega(x)=\sum_{\mathbf i \in {\mathds{Z}}^d}\widetilde{q}_{\mathbf i}(\omega) W(x-\mathbf i),$$ cf. -. We have a lemma.
\[lem:lambda\] Let the assumptions [**(B)**]{} and [**(W)**]{} hold. Then for any $M{\geqslant}M_0$ and any constant $D_0>0$ satisfying we have $$\label{eq:lambda-temple}
\lambda_1^{M, {V}_M^\omega} {\geqslant}\lambda_1^{M, \widetilde {V}_M^\omega} {\geqslant}\frac{1}{M^d}\left[\int_{\mathcal T_M} \widetilde{
V}_M^{\omega}(x){\rm d}x - \frac{\int_{\mathcal T_M}(\widetilde{V}_M^{\omega}(x))^2{\rm d}x}{\big(C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1\big)\cdot M^{-\alpha}}\right].$$
Let $M{\geqslant}M_0$ be fixed. Since $\widetilde{V}^\omega_M{\leqslant}V_M^\omega,$ the leftmost inequality in is clear. We now apply Temple’s inequality to the operator $
\widetilde{H}^\omega_M$ acting in $L^2(\mathcal T_M)$, defined in , and $\mu=\lambda_{2}^M.$ The spectrum of $\widetilde{H}^\omega_M$ is purely discrete and it is clear that $$\mu {\leqslant}\lambda_2^{M, \widetilde {V}_M^\omega}= \inf \left(\sigma(\widetilde {H}^\omega_M) \setminus \left\{\lambda_1^{M, \widetilde {V}_M^\omega}\right\}\right).$$ Let $\psi = \psi_1^M \equiv \frac{1}{M^{d/2}}$. We have $\|\psi\|_2=1$ and, by -, $\Phi(-\Delta^M) \psi=0$. Consequently, $$\langle \psi, \widetilde{H}^\omega_M\psi\rangle = \langle \psi, \Phi(-\Delta^M)\psi\rangle + \langle \psi, \widetilde{V}_M^{\omega}\psi\rangle\\
=\langle \psi, \widetilde{V}_M^{\omega}\psi\rangle= \frac{1}{M^d}\int_{\mathcal T_M} \widetilde{V}_M^{\omega}(x)\,{\rm d}x.$$ By the definition of $\widetilde{V}_M^{\omega}$, we have $$\begin{aligned}
\label{eq:per_int}
\int_{\mathcal T_M} \widetilde{V}_M^{\omega}(x)\,{\rm d}x & = \int_{\mathcal T_M} \sum_{\mathbf i\in [0,M)^d} \widetilde{q}_{\mathbf i}(\omega)\left(\sum_{\mathbf i'\in\pi_M^{-1}(\mathbf i)} W(x-\mathbf i')\right){\rm d}x\nonumber \\
&=\sum_{\mathbf i\in [0,M)^d} \widetilde{q}_{\mathbf i}(\omega)\left(\sum_{\mathbf i'\in\pi_M^{-1}(\mathbf i)} \int_{[0,M)^d -\mathbf i'} W(x)\,{\rm d}x\right) = \|W\|_1 \left(\sum_{\mathbf i\in [0,M)^d} \widetilde{q}_{\mathbf i}(\omega)\right). \end{aligned}$$ Hence, by and , $$\langle \psi, \widetilde{H}^\omega_M\psi\rangle {\leqslant}\frac{1}{M^d}\,\|W\|_1\cdot M^d\cdot \frac{D_0}{M^\alpha}= \frac{D_0\|W\|_1}{M^\alpha}
< \frac{C_1 (\mu_2^1)^{\frac{\alpha}{2}}}{M^\alpha}.$$ On the other hand, from a combination of the lower bound in and - it follows that $$\frac{C_1 (\mu_2^1)^{\frac{\alpha}{2}}}{M^\alpha} {\leqslant}\Phi(M^{-2} \mu_2^1) = \Phi(\mu_2^M) = \lambda_{2}^M,$$ and therefore condition is satisfied. For the ingredients of we have: $$\begin{aligned}
\langle \psi,\widetilde{H}^\omega\psi\rangle &=& \frac{1}{M^d}
\int_{\mathcal T_M}\widetilde {V}_M^\omega(x)\,{\rm d}x,\\
\langle \widetilde{H}^\omega\psi,\widetilde{H}^\omega\psi\rangle &=&
\frac{1}{M^d}
\int_{\mathcal T_M}\left(\widetilde {V}_M^\omega(x)\right)^2{\rm d}x,\\
\mu-\langle \psi,\widetilde{H}^\omega\psi\rangle &=& \lambda_2^M- \langle \psi,\widetilde{H}^\omega\psi\rangle
{\geqslant}\frac{C_1 (\mu_2^1)^{\frac{\alpha}{2}}}{M^\alpha}-\frac{D_0\|W\|_1}{M^\alpha} = \frac{C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1}{M^\alpha}.\end{aligned}$$ Inserting these inside we get $$\begin{aligned}
\lambda_1^{M, {V}_M^\omega}&{\geqslant}& \lambda_1^{M, \widetilde {V}_M^\omega}\\
&{\geqslant}& \frac{1}{M^d} \int_{\mathcal T_M}
\widetilde{V}_M^{\omega}(x)\,{\rm d}x - \frac{\frac{1}{M^d} \int_{\mathcal T_M}
\left(\widetilde{V}_M^{\omega}(x)\right)^2\,{\rm d}x -\left(\frac{1}{M^d} \int_{\mathcal T_M}
\widetilde{V}_M^{\omega}(x)\,{\rm d}x\right)^2}{(C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1) M^{-\alpha}}\\
&{\geqslant}& \frac{1}{M^d}\left[ \int_{\mathcal T_M}
\widetilde{V}_M^{\omega}(x)\,{\rm d}x-\frac{1}{(C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1)M^{-\alpha}} \int_{\mathcal T_M}
\left(\widetilde{V}_M^{\omega}(x)\right)^2\,{\rm d}x\right],\end{aligned}$$ which is the desired statement.
This lemma will be useful when there is a lot of randomness in the picture, namely when the random variables $q_{\mathbf i}$ are bigger than $D_0/ M^{\alpha}$ on a substantial part of sites in $\mathcal T_M.$ To quantify this behavior, fix $\delta\in(0,1)$ (its actual value will be decided later) and consider the set $$\label{eq:a-delta-def}
\mathcal A_{M,\delta}=\left\{\omega:\#\left\{\mathbf i\in [0,M)^d: q_{\mathbf i}(\omega) > \frac{D_0}{M^\alpha}\right\}{\geqslant}\delta M^d \right\}.$$ We have the following estimate.
\[lem:lambda-on-a-delta\] Let the assumptions [**(B)**]{} and [**(W)**]{} hold and let $\delta>0$ be fixed. Suppose $\omega\in\mathcal A_{M,\delta}.$ Then for any $M {\geqslant}M_0$ $$\label{eq:lambda-on-a-delta}
\lambda_1^{M, {V}_M^\omega} {\geqslant}D_0\cdot\delta \left[\|W\|_1-\frac{(2M_0)^dD_0\|W\|_2^2}{C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1}\right]\cdot\frac{1}{M^\alpha}.$$
We have already shown in that $\int_{\mathcal T_M}
\widetilde{V}_M^{\omega}(x)\,{\rm d}x = \|W\|_1 \left(\sum_{\mathbf i\in [0,M)^d} \widetilde{q}_{\mathbf i}(\omega)\right)$. Under present assumptions, we will also find a nice etimate on $\int_{\mathcal T_M}
\left(\widetilde{V}_M^{\omega}(x)\right)^2\,{\rm d}x$ and then we will apply Lemma \[lem:lambda\]. Observe that because of the assumption $\mbox{supp}\,W\subset [-M_0,M_0]^d,$ in the sum defining $\widetilde{V}_M^\omega,$ $$\widetilde{V}_M^{\omega}(x) = \sum_{\mathbf i\in[0,M)^d} \widetilde{q}_{\mathbf i}(\omega)\sum_{\mathbf i'\in\pi_M^{-1}(\mathbf i)}W(x-\mathbf i')$$ there are at most $(2M_0)^d$ nonzero terms and for every fixed $\mathbf i\in[0,M)^d$ the range of the summation $\pi_M^{-1}(\mathbf i)$ in the inner sum contains at most one element. Consequently, $$\left(\widetilde{V}_M^{\omega}(x)\right)^2 {\leqslant}(2M_0)^d \sum_{\mathbf i\in[0,M)^d} \widetilde{q}_{\mathbf i}(\omega)^2\sum_{\mathbf i'\in\pi_M^{-1}(\mathbf i)}W(x-\mathbf i')^2,$$ and further, as in the proof of , $$\int_{\mathcal T_M}
\left(\widetilde{V}_M^{\omega}(x)\right)^2\,{\rm d}x{\leqslant}(2M_0)^d\sum_{\mathbf i\in[0,M)^d} \widetilde{q}_{\mathbf i}^2(\omega) \|W\|_2^2.$$ Inserting these estimates inside we obtain: $$\begin{aligned}
\lambda_1^{M, {V}_M^\omega}(\mathcal T_M) &{\geqslant}& \frac{1}{M^d}\left[
\|W\|_1\sum_{\mathbf i\in [0,M)^d}\widetilde{q}_{\mathbf i}(\omega) -
\frac{(2M_0)^d}{(C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1)M^{-\alpha}}\|W\|_2^2\sum_{\mathbf i\in[0,M)^d} \widetilde{q}_{\mathbf i}^2(\omega)\right]\\
&=& \frac{1}{M^d}\sum_{\mathbf i\in[0,M)^d}\widetilde{q}_{\mathbf i}(\omega)\left[ \|W\|_1-\frac{(2M_0)^d\|W\|_2^2 \sum_{\mathbf i\in[0,M)^d} \widetilde{q}_{\mathbf i}^2(\omega)}{(C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1)M^{-\alpha}} \left(\sum_{\mathbf i\in[0,M)^d} \widetilde{q}_{\mathbf i}(\omega)\right)^{-1}\right].\end{aligned}$$
Now: in the sum $\sum_{\mathbf i\in[0,M)^d}\widetilde{q}_{\mathbf i}(\omega)$ we keep only those $\mathbf i$’s for which $q_{\mathbf i} > \frac{D_0}{M^\alpha}.$ Because of the assumption $\omega\in\mathcal A_{M,\delta},$ this leads to $$\sum_{\mathbf i\in[0,M)^d}\widetilde{q}_{\mathbf i}(\omega) {\geqslant}\frac{D_0}{M^\alpha}\cdot \delta M^d,$$ and for $\sum_{\mathbf i\in[0,M)^d}\widetilde{q}^2_{\mathbf i}(\omega)$ we write $$\sum_{\mathbf i\in[0,M)^d}\widetilde{q}^2_{\mathbf i}(\omega){\leqslant}\frac{D_0}{M^\alpha}\sum_{\mathbf i\in[0,M)^d}\widetilde{q}_{\mathbf i}(\omega).$$ Finally, $$\begin{aligned}
\lambda_1^{M,{V}_M^\omega} &{\geqslant}& \frac{D_0\delta}{M^\alpha}\left[\|W\|_1-\frac{(2M_0)^dD_0\|W\|_2^2}{C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1}\right].
\end{aligned}$$
Conclusion of the proof of Theorem \[th:upper-short\] - derivation of the rate function. {#sec:conclusion}
----------------------------------------------------------------------------------------
To conclude the proof we continue with the estimate of $L(t)$ from , splitting the $\mathbb Q-$integration into two parts: over $\mathcal A_{M,\delta}$ and over its complement.
For the integral over $\mathcal A_{M,\delta}$ we have the estimate from Lemma \[lem:lambda-on-a-delta\], and the integral over $\mathcal A_{M,\delta}^c$ is not bigger than $\mathbb Q[\mathcal A_{M,\delta}^c],$ whose probability can be estimated by the following Bernstein-type inequality on the binomial distribution. Its proof is an exercise from elementary probability, but we give here a short proof for the reader’s convenience.
\[lem:bernoulli\] Let $(\Omega, \mathcal F, \mathbb P)$ be a given probability space and let $S_n:
\Omega\to\mathbb R$ be a random variable with the binomial distribution $B(n,p),$ $n{\geqslant}1, p\in(0,1).$ Then, for any $p,\gamma\in(0,1)$ such that $\gamma>p,$ $$\label{eq:bern-exp}
\mathbb P[S_n {\geqslant}\gamma n]{\leqslant}\left(\left(\frac{1-p}{1-\gamma}\right)^{1-\gamma}\left(\frac{p}{\gamma}
\right)^\gamma\right)^n.$$
For any given $t>0,$ we have the following estimate, deduced from the Markov inequality: $$\begin{aligned}
\mathbb P[S_n {\geqslant}\gamma n]& = & \mathbb P\left[{\rm e}^{tS_n} {\geqslant}{\rm e}^{t\gamma n}\right]
{\leqslant}\frac{\mathbb E{\rm e}^{tS_n}}{{\rm e}^{t\gamma n}}
= \left(\frac{p{\rm e}^t+1-p}{{\rm e}^{t\gamma}}\right)^n.
\end{aligned}$$ The minimal value of the right-hand side is taken at $t_\gamma= \log\left(\frac{\gamma}{p}\,\frac{1-p}{1-\gamma}\right),$ which is positive for $\gamma>p.$ This value is equal to $\left(\left(\frac{1-p}{1-\gamma}\right)^{1-\gamma}\left(\frac{p}{\gamma}
\right)^\gamma\right)^n,$ as claimed.
For an arbitrary $\delta \in (0,1)$ we can write $$\label{eq:stop1}
L(t){\leqslant}C\mathbb E^{\mathbb Q}\left[{\rm e}^{-(t-1)\lambda_1^{M, {V}_M^\omega}}; \mathcal A_{M,\delta}\right]+ C\mathbb Q[\mathcal A _{M,\delta}^c], \quad t > 1, \ \ M {\geqslant}M_0.$$ To make use of Lemma \[lem:bernoulli\] observe $$\begin{aligned}
\mathcal A_{M,\delta}^c&=&\left\{\omega: \#\left\{\mathbf i\in[0,M)^d: q_{\mathbf i}(\omega)>\frac{D_0}{M^\alpha}\right\}< \delta M^d \right\}\\
&=&
\left\{\omega: \#\left\{\mathbf i\in[0,M)^d: q_{\mathbf i}(\omega){\leqslant}\frac{D_0}{M^\alpha}\right\} {\geqslant}(1-\delta) M^d \right\}.\end{aligned}$$ The events $A_{\mathbf i}=\{q_{\mathbf i}{\leqslant}\frac{D_0}{M^\alpha}\}$ are independent and have common probability $ p_M=F_q(\frac{D_0}{M^\alpha}).$ Therefore we use the lemma with $n=M^d,$ $ p = p_M$ as above, and $\gamma=(1-\delta).$ We only need to make sure that $(1-\delta)>p_M,$ i.e. $\delta < 1-p_M.$ As eventually we will let $M\to\infty$ and the distribution of the random variable $q$ is not concentrated at $0,$ this will not be a problem. It then follows that for every $M{\geqslant}M_0$ and $\delta < 1-p_M$ it holds $$\begin{aligned}
\mathbb Q[\mathcal A_{M,\delta}^c]{\leqslant}\left[\left(\frac{1-p_M}{\delta}\right)^{\delta}\left(\frac{p_M}{1-\delta}\right)^{1-\delta}\right]^{M^d}
{\leqslant}\left[\frac{1}{1-\delta}\left(\frac{1}{\delta}\right)^{\frac{\delta}{1-\delta}}\cdot p_M \right]^{(1-\delta)M^d}.\end{aligned}$$ Since $p_M = F_q(\frac{D_0}{M^\alpha}) \to F_q(0) \in [0,1)$ as $M \to \infty$ and $\frac{1}{1-\delta}\left(\frac{1}{\delta}\right)^{\frac{\delta}{1-\delta}} \searrow 1$ as $\delta \searrow 0$, we can easily find $M_1 {\geqslant}M_0$ and $\delta_0 < 1-p_{M_1}$ such that for every $M {\geqslant}M_1$ $$\frac{1}{1-\delta_0}\left(\frac{1}{\delta_0}\right)^{\frac{\delta_0}{1-\delta_0}}\cdot \sqrt{p_M} {\leqslant}1$$ (in particular, $\delta_0 < 1-p_{M}$, for $M {\geqslant}M_1$ as $p_M$ is nonincreasing in $M$). It gives $$\begin{aligned}
\label{eq:q-of-ac}
\mathbb Q[\mathcal A_{M,\delta_0}^c] {\leqslant}{p_M}^{(1-\delta_0)M^d/2} = {\rm e}^{-\frac{1-\delta_0}{2} \, M^d \, \log (1/p_M)}, \quad M {\geqslant}M_1.\end{aligned}$$
Denote $c_1= \frac{D_0\delta_0}{2}\left[\|W\|_1-\frac{(2M_0)^dD_0\|W\|_2^2}{(C_1 (\mu_2^1)^{\frac{\alpha}{2}}-D_0\|W\|_1)}\right]$ and $c_2 = (1-\delta_0)/2$. We now insert the bounds and inside and obtain that there exist $t_0>1$ such that for every $t {\geqslant}t_0$ and $M {\geqslant}M_1$ we have $$\begin{aligned}
L(t) &{\leqslant}& C\left({\rm e}^{-\frac{2c_1(t-1)}{M^\alpha}} +{\rm e}^{-c_2 M^d \log\frac{1}{F_q(D_0/M^\alpha)}}\right) \nonumber\\
&{\leqslant}& C\left({\rm e}^{- \frac{c_1t}{(M-1)^\alpha}} +{\rm e}^{-c_2M^d\log\frac{1}{F_q(D_0/M^\alpha)}}\right) {\leqslant}C\left({\rm e}^{- \frac{c_3t}{(M-1)^\alpha}} +{\rm e}^{-c_3 M^d\log\frac{1}{F_q(D_0/M^\alpha)}}\right) \label{eq:el-t},\end{aligned}$$ with $c_3=\min(c_1,c_2).$
So far, the bound we obtained was valid for any $M>M_1.$ We will now make $M$ depend on $t,$ in such a manner that $M\to\infty$ when $t\to\infty.$ We will use the function $ j(x)=x^{d+\alpha}\log \frac{1}{F_q(D_0/x^\alpha)}$ from . For $t{\geqslant}t_0$ (we may increase $t_0$ if necessary), the function $x_t:= j^{-1}(t)$ is well defined and obeys . Let us take $$\label{eq:M-def}
M=\lfloor x_t\rfloor+1, \quad t{\geqslant}t_0,$$ i.e. $M$ is the unique integer satisfying $x_{t}-1<M-1{\leqslant}x_t.$ Clearly, there is $t_1{\geqslant}t_0$ such that for $t{\geqslant}t_1$ one has $M{\geqslant}M_1.$ Consequently, by , $$(M-1)^\alpha{\leqslant}x_t^\alpha= \left(\frac{t}{\log\frac{1}{F_q(D_0/x_t^\alpha)}}\right)^{\frac{\alpha}{d+\alpha}}$$ so that$$\frac{t}{(M-1)^\alpha} {\geqslant}t^{\frac{d}{d+\alpha}}\left(\log\frac{1}{F_q(D_0/x_t^\alpha)}\right)^{\frac{\alpha}{d+\alpha}},\quad t{\geqslant}t_1.$$ Next, as the function $x\mapsto x^d\log\frac{1}{F_q(D_0/x^\alpha)}$ is increasing and $x_t<M,$ for the other exponent in we have $$M^d\log\frac{1}{F_q(D_0/M^\alpha)}{\geqslant}x_t^d\log\frac{1}{F_q(D_0/x_t^\alpha)}= t^{\frac{d}{d+\alpha}}\left(\log\frac{1}{F_q(D_0/x_t^\alpha)}\right)^{\frac{\alpha}{d+\alpha}}.$$ Consequently, $$L(t){\leqslant}2C {\rm e}^{-D t^{\frac{d}{d+\alpha}}\left(\log\frac{1}{F_q(D_0/x_t^\alpha)}\right)^{\frac{\alpha}{d+\alpha}}}=2C{\rm e}^{-D t^{\frac{d}{d+\alpha}} (h(t))^{\frac{\alpha}{d+\alpha}}},$$ which yields . The second assertion is an easy consequence of and .
The lower bound for the Laplace transforms {#sec:lower}
==========================================
The matching lower bound will be obtained by restricting the integration in the integrals leading to $L(t)$ (see ) to a smaller set, on which we will be able to control the expressions from below, and whose probability will be manageable. On this set we replace our random potential $V^{\omega}$ with deterministic potentials $$V_{\kappa}(x) := \kappa \sum_{\mathbf i\in[-M,2M)^d} W(x- \mathbf i), \quad \text{where \ $\kappa >0$ \ is some specially chosen parameter,}$$ and then we estimate from the above the ground state eigenvalues of the Schrödinger operators $$\Phi(-\Delta) + V_{\kappa}$$ constrained to large boxes in ${\mathds{R}}^d$. Note that in this section we work with the operators $\Phi(-\Delta)$ and the subordinate Brownian motions in ${\mathds{R}}^d$ and its sub-domains only, and we need not consider the operators and the processes on tori. We first recall necessary notation (cf. Section \[sec:dirichlet\]).
If $V \in \mathcal K_{{\mathrm{loc}}}$ is a nonnegative potential and $\Lambda$ is a bounded domain in ${\mathds{R}}^d$, then by $H_{\Lambda}$ we denote the Schrödinger operator $H = \Phi(-\Delta) + V$ constrained to $\Lambda$ (i.e. with Dirichlet conditions on $\Lambda^c$ in the non-local case and on $\partial \Lambda$ in the local case) and by $P_t^{V,\Lambda} = e^{-t H_{\Lambda}}$ the operators of its evolution semigroup. The ground state eigenvalue $\lambda_1^{V}(\Lambda)$ of $H_{\Lambda}$ can be represented through the variational formula $$\begin{aligned}
\label{eq:dirichlet_variat}
\lambda_1^V(\Lambda)=\inf \left\{\mathcal E(\varphi,\varphi) + \int_{\Lambda} V(x) \varphi^2(x) dx: \varphi \in L^2(\Lambda), \|\varphi\|_2=1\right\}.\end{aligned}$$ This infimum is achieved for $\varphi_1^{V,\Lambda}$, the ground state eigenfunction of $H_{\Lambda}$. Below we also consider the case when $V \equiv 0$ for which we use simpler notation: $P_t^{\Lambda}$, $\lambda_1(\Lambda)$ and $\varphi_1^{\Lambda}$.
We start with an auxiliary lemma.
\[lem:lambda-f\] Let $V:\mathbb R^d\to\mathbb R_+$ be a potential that belongs to $\mathcal K_{\rm loc}\cap L^1(\mathbb R^d).$ Then for any domain $\Lambda \subset {\mathds{R}}^d$ we have $$\lambda_1^V(\Lambda){\leqslant}\lambda_1(\Lambda) + {\rm e} \, p_{s}(0) \, \|V\|_1, \quad \text{with} \ \ s:= \frac{1}{\lambda_1(\Lambda)}.$$
Choosing $s=\frac{1}{\lambda_1(\Lambda)}$ in the eigenequation $P_{s/2}^{\Lambda}\varphi_1^{\Lambda}={\rm e}^{-(s/2) \lambda_1(\Lambda)}\varphi_1^{\Lambda},$ we get $$\begin{aligned}
\varphi_1^{\Lambda}(x) &= & \sqrt{{\rm e}} \int_{\Lambda}\varphi_1^{\Lambda}(y) p^{\Lambda}_{s/2}(x,y){\rm d}y
{\leqslant}\sqrt{{\rm e}}\left(\int_{\Lambda}\left(\varphi_1^{\Lambda}(y)\right)^{2}{\rm d}y\right)^{1/2}
\left(\int_{\Lambda} (p^{\Lambda}_{s/2}(x,y))^2{\rm d}y\right)^{1/2}\\
&{\leqslant}& \sqrt{{\rm e}} \, \|\varphi_1^{\Lambda}\|_2 \, (p_s(x,x))^{1/2} = \sqrt{{\rm e} \, p_s(0)} .
\end{aligned}$$ By we then obtain $$\begin{aligned}
\label{eq:lambda-f-1}
\lambda_1^V(\Lambda){\leqslant}\mathcal E(\varphi_1^{\Lambda},\varphi_1^{\Lambda}) +\int_{\Lambda} V(x)\big(\varphi_1^{\Lambda}(x)\big)^2{\rm d}x
{\leqslant}\lambda_1(\Lambda) +{\rm e} \, p_s(0) \, \|V\|_1.
\end{aligned}$$
The following is the main theorem of this section.
\[th:lower\] Assume [**(B)**]{}, [**(Q)**]{}, and [**(W)**]{}. Let $h$ be given by . Then there exists $C>0$ such that $$\label{eq:lower-1}
\liminf_{t\to\infty} \frac{\log L(t)}{t^{\frac{d}{d+\alpha}} (h(t))^{\frac{\alpha}{d+\alpha}}} {\geqslant}-C.$$ In particular, when the distribution of $q$ has an atom at zero, i.e. $F_q(0) >0$, then $$\label{eq:lower-2}
\liminf_{t\to\infty}\frac{\log L(t)}{t^{\frac{d}{d+\alpha}}} {\geqslant}-C \left(\log\frac{1}{F_q(0)}\right)^{\frac{\alpha}{d+\alpha}} .$$
Let $\Lambda_M:= [0,M)^d$, $M \in {\mathds{Z}}_+$. Recall that by , for any $t > 0$, we have $$\begin{aligned}
\label{eq:lt}
L(t)&=& \frac{p_t(0,0)}{M^d}\int_{\Lambda_M}\mathbf E_{x,x}^t\mathbb E^{\mathbb Q}\left[{\rm e}^{-\int_0^t V^\omega(X_s)\,{\rm d}s}\right]{\rm d}x
\nonumber\\
&=& \frac{1}{M^d}\int_{\Lambda_M} p_t(x,x)\mathbf E_{x,x}^t\mathbb E^{\mathbb Q}\left[{\rm e}^{-\int_0^t V^\omega(X_s)\,{\rm d}s}\right]{\rm d}x, \quad M\in{\mathds{Z}}_+.
\end{aligned}$$ For given $M{\geqslant}M_0$ (recall that $M_0$ comes from the Assumption [**(W)**]{}) and $\kappa> 0$ let $$\mathcal A_{\kappa}^M=\{\omega:\,\forall \, \mathbf i\in [-M, 2M)^d \mbox{ we have } q_{\mathbf i}(\omega){\leqslant}\kappa\}$$ and restrict the inner expectations in to the set $\mathcal A_{\kappa}^M\cap\{t< \tau_{ \Lambda_M} \}.$ For later use, observe that $$\label{eq:est-qa}
\mathbb Q[\mathcal A_{\kappa}^M]=(F_q(\kappa))^{ (3M)^d}={\rm e}^{-(3M)^d\log \frac{1}{F_q(\kappa)}}.$$ On the set $\{t< \tau_{ \Lambda_M}\}$ we have $$V^\omega(X_s)= \sum_{\mathbf i\in[-M,2M)^d}q_{\mathbf i}(\omega)W(X_s-\mathbf i),\quad s{\leqslant}t.$$ This is due to the fact that when $x- y \notin [-M,M]^d,$ then $W(x-y)=0.$ Next, for $\omega\in\mathcal A_{\kappa}^M,$ all the $q_{\mathbf i}$’s in the first sum above are not bigger than $\kappa.$ It follows that $$\begin{aligned}
\label{eq:low-2}
\mathbb E^{\mathbb Q} \left[{\rm e}^{-\int_0^tV^{\omega}(X_s)\,{\rm d}s};\mathcal A_{\kappa}^M\right] &{\geqslant}&
\mathbb E^{\mathbb Q} \left[{\rm e}^{-\int_0^t\kappa\sum_{i\in[-M,2M)^d} W(X_s-\mathbf i){\rm d}s};\mathcal A_{\kappa}^M\right]\nonumber\\
&{\geqslant}&{\rm e}^{-\int_0^t\kappa\sum_{i\in[-M,2M)^d} W(X_s-\mathbf i){\rm d}s}{\mathbb Q}[\mathcal A_{\kappa}^M].
\end{aligned}$$
We see that for $M{\geqslant}M_0$ $$\begin{aligned}
L(t)&{\geqslant}& \frac{1}{M^d} \mathbb Q[\mathcal A_{\kappa}^M] \int_{ \Lambda_M} p(t,x,x) \mathbf E_{x,x}^t\left[
{\rm e}^{-\kappa\int_0^t \sum_{\mathbf i\in [-M,2M)^d}W(X_s-\mathbf i){\rm d}s}; t< \tau_{ \Lambda_M}\right]{\rm d}x.
\end{aligned}$$ In the integral over $ \Lambda_M $ we recognize the trace of the operator $P_t^{ V_{\kappa}, \Lambda_M }$ (cf. ) on $L^2( \Lambda_M ),$ corresponding to the potential $$V_{\kappa}(x) = \kappa\sum_{{\mathbf i\in [-M,2M)^d}} W(x-\mathbf i).$$
Therefore this integral is not bigger than the principal eigenvalue of the operator $P_t^{V_{\kappa}, \Lambda_M},$ which in turn can be estimated by Lemma \[lem:lambda-f\]: $$\label{eq:tr}
\mbox{Tr}\, P_t^{V_{\kappa}, \Lambda_M} {\geqslant}{\rm e}^{-t \lambda_1^{ V_{\kappa}}( \Lambda_M)} {\geqslant}{\rm e}^{-t(\lambda_1(\Lambda_M)+ \, {\rm e} \, p_{s}(0)\|V_{\kappa}\|_1)}, \quad \text{with} \ \ s = \frac{1}{\lambda_1(\Lambda_M)}.$$ It remains to estimate the $L^1-$norm of $ V_{\kappa}.$ We have: $$\begin{aligned}
\label{eq:norm-of-f}
\|V_{\kappa}\|_1 & = & \kappa\int_{ {\mathds{R}}^d} \sum_{\mathbf i\in[-M, 2M)^d} W(x-\mathbf i) \,{\rm d}x =\kappa \sum_{\mathbf i\in[-M, 2M)^d} \int_{ {\mathds{R}}^d}W(x){\rm d}x\nonumber\\
& = & \kappa (3M)^d \|W\|_1.
\end{aligned}$$ From the estimates , , and we then obtain that for $M{\geqslant}M_0$ and $t>0$ $$L(t){\geqslant}\frac{1}{M^d}{\rm e}^{-(3M)^d\log \frac{1}{F_q(\kappa)}} {\rm e}^{-t(\lambda_1(\Lambda_M)+ \, {\rm e} \, \kappa \, p_{s}(0) (3M)^d\|W\|_1)}, \quad \text{with} \ \ s = \frac{1}{\lambda_1(\Lambda_M)}.$$ Furthermore, it follows from [@bib:CS Theorem 3.4] that $$\lambda_1(\Lambda_M) {\leqslant}\Phi(\mu_1(\Lambda_M)),$$ where $\mu_1(\Lambda_M)$ is the ground state eigenvalue of the Laplace operator $-\Delta$ on $\Lambda_M$ with Dirichlet boundary conditions. Since $\mu_1(\Lambda_M) = M^{-2} \mu_1(\Lambda_1)$, by the upper bound in the assumption [**(B)**]{} we obtain that there exist $M_1 {\geqslant}M_0$ and a constant $c_1>0$ such that $$\lambda_1(\Lambda_M){\leqslant}\frac{c_1}{M^\alpha},\quad M {\geqslant}M_1.$$ In particular, by , there is a constant $c_2>0$, for which we get $$p_s(0)\big|_{s = \frac{1}{\lambda_1(\Lambda_M)}} {\leqslant}\frac{c_2}{M^d}, \quad \quad M {\geqslant}M_1.$$ Consequently, by choosing $\kappa=\frac{D_0}{M^\alpha},$ where $D_0$ comes from (same as in the proof of the upper bound), we obtain $$L(t){\geqslant}\frac{1}{M^d} {\rm e}^{-c_3\left(\frac{t}{(M+1)^\alpha}+M^d \left(\frac{M}{M+1}\right)^{\alpha} \log \frac{1}{F_q(D_0/M^\alpha)}\right)},\quad M{\geqslant}M_1,$$ for some constant $c_3>0.$ The exponent can be written as $$\label{eq:expo}
-\,\frac{c_3}{(M+1)^\alpha}\left(t+ M^{d+\alpha}\log \frac{1}{F_q(D_0/M^\alpha)}\right).$$ Again, assume $t{\geqslant}t_0,$ let $x_t=j^{-1}(t)$ (for a definition of the function $j$ and $t_0$ see the formula and the two sentences following it), and choose $M=\lfloor x_t\rfloor$. It is the unique integer for which $$M{\leqslant}x_t < M+1,\quad {\rm i.e.} \quad j(M){\leqslant}t<j(M+1).$$ Condition $j(M){\leqslant}t$ reads $M^{d+\alpha}\log\frac{1}{F_q(D_0/M^\alpha)}{\leqslant}t.$ Moreover, by , $$\frac{t}{(M+1)^\alpha}{\leqslant}\frac{t}{x_t^\alpha} = t^{\frac{d}{d+\alpha}}\left(\log\frac{1}{F_q(D_0/x_t^\alpha)}\right)^{\frac{\alpha}{d+\alpha}}=t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}}.$$ Consequently, there is a number $t_1{\geqslant}t_0$ such that for $t{\geqslant}t_1$ $$L(t){\geqslant}\frac{1}{M^d} {\rm e}^{-2c_3 t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}}},$$ with $M$ chosen as above, i.e. $$\begin{aligned}
\label{eq:lower_final}
\frac{\log L(t)}{t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}}} {\geqslant}-2c_3 - d \frac{\log \lfloor x_t\rfloor}{t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}}}.\end{aligned}$$ To conclude, we only need to verify that $$\lim_{t\to\infty} \frac{\log \lfloor x_t\rfloor}{t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}}}=0.$$ This is clear as, by , $t=x_t^{d+\alpha}h(t),$ and further $(d+\alpha)\log x_t +\log h(t)=\log t, $ so that $$\frac{\log x_t}{t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}}}=\frac{1}{(d+\alpha)} \frac{\log t- \log h(t)}{t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}}}\to 0, \quad \text{as} \ \ t \to \infty,$$ because $\lim_{t \to \infty} h(t)$ always exists and is strictly positive (possibly infinite, see ). Denoting $C=2c_3$ and taking $\liminf$ in , we obtain . The second assertion is again a direct consequence of .
Tauberian theorems and the asymptotics of the IDS {#sec:tauber}
=================================================
We will now transform the estimates for the Laplace transform $ L(t) $ of the IDS obtained in Sections \[sec:upper\] and \[sec:lower\] into statements concerning the IDS itself. When $\log L(t) \asymp -t^{\gamma}$ as $t \to \infty$, with $\gamma \in (0,1)$, then one just uses the exponential Tauberian theorem [@bib:F Theorem 2.1], to get $\log \ell(\lambda) \asymp -\lambda^{-\gamma/(1-\gamma)},$ $\lambda\searrow 0,$ as it was done previously in [@bib:Nak; @bib:Oku; @bib:Szn1; @bib:KPP1; @bib:KK-KPP2; @bib:KK-KPP-alloy-stable]. However, the rate we identified in Theorems \[th:upper-short\], \[th:lower\] is more general (a correction term is present) and the Tauberian theorems existing in the literature are not sufficient to deal with it. Therefore we first need to state and prove a version of exponential Tauberian theorem which can be applied in our situation.
Tauberian theorem
-----------------
The setting is as follows. Let $\rho({\rm d}x)$ be a $\sigma-$finite Borel measure on $[0,\infty)$ and let $L(t):= \int_{[0,\infty)} e^{-tx} \rho({\rm d}x)$ be its Laplace transform. We assume that $L(t) < \infty$ for every $t>0$. We will use the same letter $\rho$ for the cumulative distribution function of the measure $\rho,$ i.e. $\rho(x) = \rho([0,x])$, $x {\geqslant}0$. Moreover, let $ g:(0,\infty)\to (0,\infty) $ be a nondecreasing function, continuous on $[x_0, \infty)$, $x_0{\geqslant}0$. Let $\alpha,d>0$ be two given numbers. For $t{\geqslant}t_0:=x_0^{d+\alpha}g(x_0^\alpha)$ there is a unique number $x_t$ such that $t=x_t^{d+\alpha}g(x_t^\alpha).$ Since $x^{d+\alpha}g(x^\alpha) \to \infty$ as $x \to \infty$, we also have $x_t \to \infty$ as $t \to \infty$. Finally, let $ h(t)=g(x_t^{\alpha}),$ $t{\geqslant}t_0.$ Clearly, $\lim_{t \to \infty} h(t)$ exists and $\lim_{t \to \infty} h(t) \in (0,\infty]$. In particular, $t^{\frac{d}{d+\alpha}} (h(t))^{\frac{\alpha}{d+\alpha}} \to \infty$ as $t \to \infty$.
\[th:tauberian\] Using the notation introduced above we have the following.
\(i) If $$\label{eq:taub-lower-assump}
\liminf_{t\to\infty}\frac{\log L(t)}{t^{\frac{d}{d+\alpha}} (h(t))^{\frac{\alpha}{d+\alpha}}} {\geqslant}-A_1,$$ with certain constant $A_1\in(0,\infty),$ then for any $B_1>A_1$ we have $$\label{eq:taub-lower}
\liminf_{x\to 0^+} \frac{x^{d/\alpha}}{g(B_1/x)}\,\log \rho(x){\geqslant}-A_1 B_1^{d/\alpha}.$$
\(ii) If $$\label{eq:taub-upper-assump}
\limsup_{t\to\infty}\frac{\log L(t)}{t^{\frac{d}{d+\alpha}} (h(t))^{\frac{\alpha}{d+\alpha}}} {\leqslant}-A_2,$$ with certain constant $A_2\in(0,\infty),$ then for any $B_2<A_2$ $$\label{eq:taub-upper}
\limsup_{x\to 0^+} \frac{x^{d/\alpha}}{g(B_2/x)}\,\log \rho(x){\leqslant}-(A_2-B_2) B_2^{d/\alpha}.$$
\(i) Assume that holds. To shorten the notation, denote $\gamma=\frac{d}{d+\alpha}.$ Then the rate in the denominator of (and of ) is equal to $t^\gamma (h(t))^{1-\gamma}.$ Let $$\widetilde L(t):=\int _0^\infty {\rm e}^{-tx} \rho(x){\rm d}x =\frac{1}{t}\,L(t)$$ (the last identity is obtained via integration by parts). It follows that is satisfied for $\widetilde L(t) $ as well. Let $\epsilon >0$ be given. Then there is $t_\epsilon>0$ such that for $t>t_\epsilon$ one has $$\label{eq:el-tilde}
\widetilde L(t){\geqslant}{\rm e}^{-(A_1+\epsilon)t^\gamma (h(t))^{1-\gamma}}.$$ Next, take $B_1>A_1$ and write $$\label{eq:taub-1}
\int_0^{B_1t^{\gamma-1}(h(t))^{1-\gamma}} {\rm e}^{-tx}\rho(x)\,{\rm d}x = \widetilde L(t) -
\int_{B_1t^{\gamma-1}(h(t))^{1-\gamma}}^\infty {\rm e}^{-tx}\rho(x)\,{\rm d}x.$$ The left-hand side of is not bigger than $$\rho(B_1t^{\gamma-1}(h(t))^{1-\gamma}) \int_0^{B_1t^{\gamma-1}(h(t))^{1-\gamma}} {\rm e}^{-tx}{\rm d}x =
\rho(B_1t^{\gamma-1}(h(t))^{1-\gamma}) \frac{1- {\rm e}^{-B_1t^\gamma(h(t))^{1-\gamma}}}{t}.$$ Moreover, $$\begin{aligned}
\int_{B_1t^{\gamma-1}(h(t))^{1-\gamma}}^\infty {\rm e}^{-tx}\rho(x)\,{\rm d}x&=&
\int_{B_1t^{\gamma-1}(h(t))^{1-\gamma}}^\infty {\rm e}^{-t\epsilon x}{\rm e}^{-tx(1-\epsilon)}\rho(x)\,{\rm d}x\\
&{\leqslant}& {\rm e}^{-(1-\epsilon)B_1t^\gamma (h(t))^{1-\gamma}}\widetilde L(\epsilon t),\end{aligned}$$ which in the light of yield $$\begin{aligned}
\rho(B_1t^{\gamma-1}(h(t))^{1-\gamma}) \frac{1- {\rm e}^{-B_1t^\gamma(h(t))^{1-\gamma}}}{t}&{\geqslant}& {\rm e}^{-(A_1+\epsilon)t^\gamma (h(t))^{1-\gamma}}-\widetilde L(\epsilon t){\rm e}^{-B_1(1-\epsilon)t^\gamma h(t)^{1-\gamma}}\\
&=& {\rm e}^{-(A_1+\epsilon)t^\gamma (h(t))^{1-\gamma}}\left(1- \widetilde L(\epsilon t){\rm e}^{t^\gamma h(t)^{1-\gamma}[-B_1(1-\epsilon)+(A_1+\epsilon)]}\right)\\
&=& {\rm e}^{-(A_1+\epsilon)t^\gamma (h(t))^{1-\gamma}}(1+o(1)),\quad t\to\infty,
\end{aligned}$$ provided $\epsilon<\frac{B_1-A_1}{B_1+1}.$ It follows $$\begin{aligned}
\log \rho(B_1t^{\gamma-1}(h(t))^{1-\gamma})&{\geqslant}& -(A_1+\epsilon)t^\gamma (h(t))^{1-\gamma}\\
&& +\log(1+o(1)) +\log t -\log(1-{\rm e}^{B_1t^\gamma (h(t))^{1-\gamma}})\end{aligned}$$
As mentioned above, $t^\gamma (h(t))^{1-\gamma}\to\infty$ as $t\to\infty,$ and consequently, for any $B_1>A_1$ and $\epsilon$ sufficiently small, $$\label{eq:taub-lower-2}
\liminf_{t\to\infty}\frac{\log \rho(B_1t^{\gamma-1}(h(t))^{1-\gamma})}{t^\gamma (h(t))^{1-\gamma}}{\geqslant}-(A_1+\epsilon).$$ Now, the number $\epsilon$ on the right-hand side can be sent to zero and eliminated.
To conclude the proof of part (i), substitute $x=B_1t^{\gamma-1}(h(t))^{1-\gamma}$. We need to write $t^\gamma (h(t))^{1-\gamma}$ as a function of $x.$ Recall that $$t=x_{t}^{d+\alpha} g(x_t^\alpha) = x_t^{d+\alpha}h(t),\quad \mbox{i.e.}\quad \frac{h(t)}{t}= x_t^{-(d+\alpha)}.$$ It means $$x=B_1\left(\frac{h(t)}{t}\right)^{1-\gamma}=\frac{B_1}{x_t^\alpha}.$$ Consequently, $$t^\gamma (h(t))^{1-\gamma}= \frac{tx}{B_1}=\frac{t}{x_t^\alpha}=x_t^d g(x_t^\alpha) = \frac {B_1^{d/\alpha}}{x^{d/\alpha}} g(\frac{B_1}{x}).$$ Assertion follows.
\(ii) Assume now that holds. By an argument identical as above, is satisfied for $\widetilde L(t).$ Then for any $\epsilon>0$ there is $t_\epsilon>0$ such that for $t>t_\epsilon$ $$\widetilde L(t){\leqslant}{\rm e}^{-(A_2-\epsilon)t^\gamma (h(t))^{1-\gamma}},$$ and on the other hand, with any $B_2>0,$ $$\widetilde L(t){\geqslant}\int_{B_2t^{\gamma-1}(h(t))^{1-\gamma}}^\infty {\rm e}^{-tx}{\rho(x)\,{\rm d}x {\geqslant}\rho(B_2t^{\gamma-1}(h(t))^{1-\gamma})\cdot \frac{1}{t} {\rm e}^{-B_2t^\gamma (h(t))^{1-\gamma}}}.$$ Using both these inequalities, taking logarithm and rearranging we arrive at $$\frac{\log\rho(B_2t^{\gamma-1}(h(t))^{1-\gamma})}{t^\gamma (h(t))^{1-\gamma}} {\leqslant}\frac{\log t}{t^\gamma (h(t))^{1-\gamma}}-(A_2-\epsilon-B_2),\quad t>t_\epsilon$$ and further $$\label{eq:taub-upper-1}
\limsup_{t\to\infty}\frac{\log\rho(B_2t^{\gamma-1}(h(t))^{1-\gamma})}{t^\gamma (h(t))^{1-\gamma}}{\leqslant}-(A_2-\epsilon-B_2),$$ which yields a viable result for $B_2<A_2$ (again, $\epsilon$ can be eliminated).
To conclude the proof, similarly as in part (i), we substitute $x=B_2t^{\gamma-1}(h(t))^{1-\gamma}$, getting $$t^\gamma (h(t))^{1-\gamma}= \frac{tx}{B_2}=\frac{t}{x_t^\alpha}=x_t^d g(x_t^\alpha) = \frac {B_2^{d/\alpha}}{x^{d/\alpha}} g(\frac{B_2}{x}).$$ This gives and completes the proof.
[In the exponential Tauberian theorems without a correction term (cf. [@bib:F Theorem 2.1]), one was able to handle constants $B_1$ and $B_2.$ Now we do not know, in general, what the function $g$ looks like and how it behaves asymptotically. We will be able to get rid of those constants in particular cases only. ]{}
Asymptotics of the IDS
----------------------
Finally, applying the Tauberian theorem from the previous section, we give the formal proof of Lifshitz tail asymptotics of the IDS in Theorem \[th:IDS-asymp\] stated in the Introduction.
The result follows directly from Theorems \[th:upper-short\], \[th:lower\] and \[th:tauberian\] with $g(x)=\log \frac{1}{F_q(D_0/x)},$ $j(x)=x^{d+\alpha}g(x^{\alpha}),$ $x_t=j^{-1}(t),$ and $h(t)=g(x_t^\alpha)$ as in Section \[sec:rate\_der\].
We complement our presentation with less precise statements (the ‘loglog’ regime), matching the usual statement of the Lifshitz tail sometimes found in the literature.
First we show that the behavior of $\log g(x)$, $\log x_t$ and $\log h(t)$ (see Section \[sec:rate\_der\]) at infinity are closely related.
\[lem:h-and-g\] Let $g(x)=\log \frac{1}{F_q(D_0/x)},$ $j(x)=x^{d+\alpha}g(x^{\alpha}),$ $x_t=j^{-1}(t),$ and $h(t)=g(x_t^\alpha)$. The following three conditions are equivalent:
- $\lim_{x\to\infty}\frac{\log g(x)}{\log x} $ exists and is equal to $a\in[0,\infty];$
- $\lim_{t\to\infty}\frac{\log x_t}{\log t} $ exists and is equal to $b\in[0,1/(d+\alpha)];$
- $\lim_{t\to\infty}\frac{\log h(t)}{\log t} $ exists and is equal to $c\in[0,1].$
Numbers $a$, $b$ and $c$ are related through $$b = \frac{1}{d+(a+1)\alpha} \quad \text{and} \quad c = 1 - (d+\alpha)b = 1- \frac{d+\alpha}{d+(a+1)\alpha}$$ (here we use the standard convention $1/+\infty = 0$ and $1/0^{+} = + \infty$).
Recall that by $$t= x_{t}^{d+\alpha}g(x_t^\alpha)= x_t^{d+\alpha}h(t), \quad t {\geqslant}t_0.$$ In particular, $$\begin{aligned}
\label{eq:first_aux}
\log h(t)=\log t-(d+\alpha)\log x_t, \quad t {\geqslant}t_0,\end{aligned}$$ and $$\begin{aligned}
\label{eq:second_aux}
\frac{\log g(x_t^{\alpha})}{\log x_t^{\alpha}} = \frac{\log h(t)}{\log x_t^{\alpha}} = \frac{1}{\alpha} \frac{\log t}{ \log x_t} - \frac{d+\alpha}{\alpha}, \quad t {\geqslant}t_0.\end{aligned}$$ It follows directly from that (ii) and (iii) are equivalent and $c= 1 - (d+\alpha)b$ (or, equivalently, $b = (1-c)/(d+\alpha)$). Moreover, by and by the fact that $x_t \to \infty$ as $t \to \infty$, we see that (i) implies (ii) and then $b = 1/(d+(a+1)\alpha)$ (In particular, (i) gives (iii) with $c = 1-(d+\alpha)/(d+(a+1)\alpha).$) The converse implication (ii) $\Rightarrow$ (i) also follows from by the fact that $[t_0, \infty) \ni t \mapsto x_t^{\alpha}$ is a continuous and increasing function onto $[x_{t_0}^{\alpha}, \infty)$.
\[coro:loglog\] Suppose that [**(B)**]{}, [**(Q)**]{}, and [**(W)**]{} hold true. If $\lim_{x\to\infty}\frac{\log g(x)}{\log x}$ exists, then $$\lim_{x\to 0^+}\frac{\log|\log\ell(x)|}{\log x}= -\frac{d}{\alpha}- \lim_{x\to \infty}\frac{\log g(x)}{\log x}$$ and $$\lim_{t\to\infty}\frac{\log|\log L(t)|}{\log t} = 1- \frac{\alpha}{d+\left(1+\lim_{x\to \infty}\frac{\log g(x)}{\log x}\right)\alpha}.$$ In particular, when $g(x)$ is of order lower than power-law (i.e. $\lim_{x\to \infty}\frac{\log g(x)}{\log x} = 0$), then $$\lim_{x\to 0^+}\frac{\log|\log\ell(x)|}{\log x}= -\frac{d}{\alpha}$$ and $$\lim_{t\to\infty}\frac{\log|\log L(t)|}{\log t} = \frac{d}{d+\alpha}.$$
The assertion for the IDS follows directly from the estimates in Theorem \[th:IDS-asymp\] and the definition of $\limsup$ and $\liminf$. For a proof of the second assertion, for $L(t)$, observe that by Theorems \[th:upper-short\] and \[th:lower\], the definition of $\limsup$ and $\liminf$, and , we have $$\lim_{t\to\infty}\frac{\log|\log L(t)|}{\log t} = \frac{d}{d+\alpha}+\frac{\alpha}{d+\alpha}\lim_{t\to\infty}\frac{\log h(t)}{\log t} = 1- \alpha \lim_{t \to \infty} \frac{\log x_t}{\log t}.$$ An application of Lemma \[lem:h-and-g\] completes the proof.
Discussion and examples {#sec:examples}
=======================
We now discuss several specific classes of distributions $F_q$ to which our results apply directly. Recall the notation: $g(x)=\log \frac{1}{F_q(D_0/x)},$ $j(x)=x^{d+\alpha} g(x^{\alpha}), $ $x_t=j^{-1}(t),$ $h(t)=g(x_t^\alpha).$ For more clarity, our discussion will be divided into four subsections.
Distribution functions $F_q$ with an atom at zero {#ex:atom}
--------------------------------------------------
Suppose there exists $\kappa_0 >0$ such that $F_q$ is continuous on $[0,\kappa_0]$ and $F_q(0) > 0$. Then there are constants $C, \widetilde C>0$ such that $$\label{eq:stat-atom}
-C{\leqslant}\liminf_{ \lambda \searrow 0}\lambda^{d/\alpha}\log \ell(\lambda){\leqslant}\limsup_{\lambda \searrow 0}\lambda^{d/\alpha}\log \ell(\lambda){\leqslant}-\widetilde C\quad\mbox{ and }\quad \lim_{\lambda \searrow 0}\frac{\log|\log \ell(\lambda)|}{\log \lambda } = - \frac{d}{\alpha}.$$ Note that in this case we simply have $g(x) \asymp 1$ and $j(x) \asymp x^{d+\alpha}$ for large $x$, and therefore $$x_t=j^{-1}(t) \asymp t^{\frac{1}{d+\alpha}} \qquad \text{and} \qquad
h(t) \asymp 1 ,\quad t\to\infty.$$ In [@bib:KK-KPP-alloy-stable] we used Sznitman’s coarse-graining method (the ‘enlargement of obstacles method’) to derive the Lifschitz tail in this case - for alloy-type potentials with random variables $q_{\mathbf i}$ having an atom at 0. The paper was concerned primarily with $\Phi(\lambda)=\lambda^{\alpha/2}$, $\alpha \in (0,2]$ (i.e. with the fractional powers of the Laplace operator and the Laplace operator itself) - in this case we were able to prove the existence of the limit $\lim_{\lambda \searrow 0}\lambda^{d/\alpha}\log \ell(\lambda)$ and to derive its actual value. The value of this limit was coherent with that obtained for Poisson-type potentials in [@bib:Oku; @bib:Szn1]. The method of [@bib:KK-KPP-alloy-stable] is also suitable to cover the case of some other subordinate processes, but with no precise scaling of principal Dirichlet eigenvalues at hand, in general we would be able to obtain only the statements for the $\limsup$ and $\liminf,$ exactly as in (cf. [@bib:KK-KPP2]).
Distribution functions $F_q$ with polynomial decay at zero {#ex:log-rate}
----------------------------------------------------------
This section consists of two parts.
\(1) Suppose that there exist $\gamma_1, \gamma_2 > 0$, $\kappa_0>0$ and constants $B_1, B_2>0$ such that $$B_1\kappa^{\gamma_1}{\leqslant}F_q(\kappa){\leqslant}B_2\kappa^{\gamma_2}, \quad \kappa \in [0,\kappa_0].$$ This example covers all absolutely continuous distributions whose densities near zero behave polynomially or explode at most logarithmically fast (e.g. uniform, exponential, one-side normal, Weibull, arcsin, and many other distributions). In this case, $$g(x)=\log \frac{1}{F_q(D_0/x)} \asymp \log x, \quad \text{for large \ $x.$}$$ We then have $$j(x)= x^{d+\alpha} g(x^{\alpha}) \asymp x^{d+\alpha} \log x,\quad x\to\infty,$$ giving $$x_t= j^{-1}(t) \asymp \left(\frac{t}{\log t}\right)^{\frac{1}{d+\alpha}},\quad t\to\infty$$ and $$h(t)= g(x_t^{\alpha}) \asymp \frac{\alpha}{d+\alpha}\log t ,\quad t\to\infty$$ i.e. for some constants $C, \widetilde C>0$ $$-C {\leqslant}\liminf_{t\to\infty}\frac{\log L(t)}{t^{\frac{d}{d+\alpha}}(\log t)^{\frac{\alpha}{d+\alpha}}}{\leqslant}\limsup_{t\to\infty}\frac{\log L(t)}{t^{\frac{d}{d+\alpha}}(\log t)^{\frac{\alpha}{d+\alpha}}}{\leqslant}-\widetilde C.$$ Finally, for certain constants $C,\widetilde C>0$, $$-C{\leqslant}\liminf_{\lambda \searrow 0}\frac{\lambda^{d/\alpha}}{\log \lambda} \log \ell(\lambda){\leqslant}\limsup_{\lambda \searrow 0}\frac{\lambda^{d/\alpha}}{\log \lambda} \log \ell(\lambda){\leqslant}- \widetilde C\quad\mbox{ and }\quad \lim_{\lambda \searrow 0}\frac{\log|\log \ell(\lambda)|}{\log \lambda } = - \frac{d}{\alpha}.$$
\(2) The asymptotics of the IDS in the loglog regime has been previously established by Kirsch ans Simon in [@bib:KS] for random Schrödinger operators $-\Delta+\sum_{{\bf i} \in {\mathds{Z}}^d} q_{{\bf i}}(\omega) W(x-{\bf i})$ with bounded random variables $q_{{\bf i}}$ satisfying the one-sided bound $B_1\kappa^{\gamma_1}{\leqslant}F_q(\kappa),$ under somewhat different assumptions on the single-site potential $W .$ Observe that such a one-sided bound is not sufficient for determining the term $h(t)$ needed in the ‘log’ regime, even asymptotically: for example, when there is an atom at zero (cf. Section \[ex:atom\] above), then the one-sided bound holds, but $h(t)\asymp 1$, $t \to \infty,$ while still $F_q(\kappa){\geqslant}\mathbb Q[q=0]{\geqslant}B_1\kappa^{\gamma_1},$ $\kappa{\leqslant}\kappa_0$ - which should be contrasted with the results from part (1) above.
Our present approach generalizes the results of Kirch and Simon: we are able to derive the ’loglog statements’ for both the integrated density of states and its Laplace transform from the more delicate statements in the ‘log’ regime. Indeed, as in this case there is a constant $c>0$ such that $$g(x)=\log \frac{1}{F_q(D_0/x)}{\leqslant}c \log x, \quad \mbox{for large $x$},$$ it follows that $$\lim_{x\to \infty}\frac {\log g(x)}{\log x} = 0,$$ and from Corollary \[coro:loglog\] and Lemma \[lem:h-and-g\] we get $$\lim_{\lambda \searrow 0} \frac{\log |\log\ell(\lambda)|}{\log \lambda}= -\frac{d}{\alpha} \qquad \mbox{and} \qquad
\lim_{t\to \infty}\frac{\log|\log L(t)|}{\log t}= \frac{d}{d+\alpha}.$$
Note also that the result for the Laplace transform is new.
Distribution functions $F_q$ with exponential decay at zero
-----------------------------------------------------------
We now give an example what can happen when the decay of $F_q$ near zero is faster than polynomial. For a fixed $\gamma >0$ we let $$F_q(\kappa) = {\rm e}^{-\frac{1}{\kappa^\gamma}},\quad \kappa>0.$$ We verify that in this case $$g(x)= (x/D_0)^{\gamma},$$ and further $$j(x)= D_0^{-\gamma} x^{d+\alpha(1+\gamma)} \;\mbox{ and }\; x_t= (D_0^\gamma t)^{\frac{1}{d+\alpha(1+\gamma)}},$$ which gives $$h(t)= c t^{\frac{\alpha\gamma}{d+\alpha(1+\gamma)}}=ct^{1-\frac{d+\alpha}{d+\alpha(1+\gamma)}}.$$ Consequently, the rate of decay for the Laplace transform of the IDS is $$t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}} = t^{1-\frac{\alpha}{d+\alpha(1+\gamma)}}$$ and there exist two constants $C, \widetilde C>0$ for which $$-C{\leqslant}\liminf_{\lambda \searrow 0} \lambda^{\frac{d}{\alpha}+\gamma}\log \ell(\lambda) {\leqslant}\limsup_{\lambda \searrow 0} \lambda^{\frac{d}{\alpha}+\gamma}\log \ell(\lambda) {\leqslant}-\widetilde C.$$ The loglog limit is $$\lim_{\lambda \searrow 0} \frac{\log |\log\ell(\lambda)|}{\log \lambda}= -\frac{d}{\alpha}-\gamma.$$ It means that we observe an increase in the power of the exponent which is due to the fast decay of the cumulative distribution function of $q.$ It should be noted that when $\gamma\to\infty,$ then the rate of decay of the Laplace transform approaches $t,$ which is the upper bound for the rate possible.
Distribution functions $F_q$ with double-exponential decay near zero
--------------------------------------------------------------------
From we see that $h(t)/t= x_t^{-d-\alpha} \to 0$ as $t\to\infty,$ therefore the rate $$t^{\frac{d}{d+\alpha}}(h(t))^{\frac{\alpha}{d+\alpha}} \qquad \mbox{with} \qquad h(t)\asymp t, \ \ \ \ t \to \infty,$$ is never possible. However, it can happen that $\frac{\log h(t)}{\log t}\to 1$ as $t\to\infty,$ which is illustrated by this example.
For the distribution whose CDF is given by $$F_q[\kappa]= {\rm e}^{1-{\rm e}^{
\frac{1}{\kappa}}},\quad \kappa>0$$ we have $$g(x)= -\log F_q(D_0/x)= {\rm e}^{\frac{x}{D_0}}-1.$$ It then follows from Theorem \[th:IDS-asymp\] that there exist constants $C, \widetilde C, D, \widetilde D >0$ such that $$-C{\leqslant}\liminf_{\lambda \searrow 0} \lambda^{d/\alpha}{\rm e}^{ -{D}/{\lambda}}\log \ell(\lambda) \qquad \mbox{and} \qquad
\limsup_{\lambda \searrow 0} \lambda^{d/\alpha}{\rm e}^{-{\widetilde D}/{\lambda}}\log \ell(\lambda){\leqslant}-\widetilde C.$$ Consequently, we do not have the usual Lifschitz tail; the rate of decay of $\ell(\lambda)$ to zero is double exponential: we see that $$\widetilde D{\leqslant}\liminf_{\lambda \searrow 0} \lambda \log|\log \ell(\lambda)| {\leqslant}\limsup_{\lambda \searrow 0} \lambda \log|\log \ell(\lambda)|{\leqslant}D.$$ This justifies the name: [*super-Lifchitz*]{} tail.
To determine the asymptotical rate for the Laplace transform $L(t)$ first observe that the function $$k(t):=\left(D_0 \log \left(\frac{t}{(D_0 \log t)^{(d+\alpha)/\alpha}} + 1\right) \right)^{1/\alpha}$$ is the asymptotic inverse of the function $j(x) = x^{d+\alpha} g(x^{\alpha}) = x^{d+\alpha}\big({\rm e}^{\frac{x^{\alpha}}{D_0}}-1\big)$ as $x\to+\infty$. Therefore, $$x_t \asymp k(t) \qquad \mbox{and} \qquad h(t)\asymp \frac{t}{(\log t)^{\frac{d}{\alpha}+1}}, \quad t \to \infty,$$ resulting in the asymptotics $$-C {\leqslant}\liminf_{t\to\infty}\frac{\log L(t)}{t/\log t}{\leqslant}\limsup_{t\to\infty}\frac{\log L(t)}{t/\log t}{\leqslant}-\widetilde{C}$$ and $$\lim_{t\to\infty} \frac{\log|\log L(t)|}{\log t} =1.$$ As the last remark observe that the last ’log log assertion’ also follows directly from Corollary \[coro:loglog\], without the prior knowledge of the asymptotic behavior of the functions $x_t$ and $h(t)$.
[99]{}
M. Benderskii, L. Pastur: [*On the spectrum of the one-dimensional Schrödinger equation with random potential*]{}, Mat. Sb. 82 (1970) 245-256.
K. Bogdan, T. Byczkowski: [*Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domains*]{}, Studia Math. 133 (1999) 53-92.
J. Bourgain, C.E. Kenig: [*On localization in the continuous Anderson-Bernoulli model in higher dimension*]{} Invent. Math. 161 (2005), 389-426.
R. Carmona, J. Lacroix: [*Spectral theory of random Schrödinger operators*]{}. Probability and its Applications. Birkhäuser, Boston, Inc., Boston, MA, 1990.
R. Carmona, W.C. Masters, B. Simon: *Relativistic Schrödinger operators: asymptotic behaviour of the eigenfunctions*, J. Funct. Anal. 91, 1990, 117-142.
L. Chaumont, G. Uribe Bravo: *Markovian bridges: weak continuity and pathwise constructions*, Ann. Probab. 39 (2), 2011, 609-647.
Z.-Q. Chen, R. Song: *Two-sided eigenvalue estimates for subordinate processes in domains*, J. Funct. Anal. 226 (2005) 90–113.
J.M. Combes, P.D. Hislop: [*Localization for some continuous, random Hamiltonians in d-dimensions.*]{} J. Funct. Anal. 124, 149-180 (1994).
M. Demuth, J.A. van Casteren: *Stochastic Spectral Theory for Self-adjoint Feller Operators. A Functional Analysis Approach*. Birkhäuser, Basel 2000.
R. Friedberg, J. Luttinger: [*Density of electronic energy levels in disordered systems*]{}, Phys. Rev. B 12 (1975) 4460-4474.
M. Fukushima: *On the spectral distribution of a disordered system and a range of a random walk*, Osaka J. Math. 11, 1974, 73-85.
M. Fukushima, H. Nagai, and S. Nakao: [*On an asymptotic property of spectra of a random difference operator*]{}, Proc. Japan Acad. 51 (1975) 100-102.
F. Germinet, P.D. Hislop, A. Klein: [*Localization for Schrödinger operators with Poisson random potential.*]{} J. Eur. Math. Soc. (JEMS) 9 (2007), no. 3, 577-607.
N. Jacob: *Pseudo-Differential Operators and Markov Processes: Markov Processes and Applications. Vol. I, II, III*. Imperial College Press, London 2001–2005.
K. Kaleta, K. Pietruska-Pałuba: [*Integrated density of states for Poisson–Schrödinger perturbations of subordinate Brownian motions on the Sierpiński gasket,*]{} Stoch. Proc. Appl. 125 (2015), 1244-1281.
K. Kaleta, K. Pietruska-Pałuba: [*Lifschitz singularity for subordinate Brownian motions in presence of the Poissonian potential on the Sierpiński gasket*]{}, Stoch. Proc. Appl. 128 (2018), 3897-3939.
K. Kaleta, K. Pietruska-Pałuba: [*Lifschitz tail for alloy-type models driven by the fractional Laplacian*]{}, preprint 2019, available at arXiv:1906.03419.
K. Kaleta, R.L. Schilling: [*Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators*]{}, preprint 2019, available at arXiv:1903.12004
W. Kirsch, F. Martinelli: [*On the density of states of Schrödinger operators with a random potential*]{}, J. Phys. A 15 (1982) 2139-2156.
W. Kirsch, F. Martinelli: [*Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians*]{}, Commun. Math. Phys. 89 (1983) 27-40.
W. Kirsch, B. Simon: *Lifshitz Tails for Periodic Plus Random Potentials*, J. Stat. Phys. 42:5/6 (1986) 799-808.
W. Kirsch, I. Veselić: [*Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential*]{}, Lett. Math. Phys. 94 (2010) 27-39.
F. Klopp: [*Weak Disorder Localization and Lifshitz Tails*]{}, Commun. Math. Phys. 232 (2002) 125-155.
F. Klopp: [*Weak Disorder Localization and Lifshitz Tails: Continuous Hamiltonians*]{}, Ann. Henri Poincaré 3 (2002) 711-737.
I.M. Lifshitz, [*Energy spectrum structure and quantum states of disordered condensed systems*]{}, Soviet Physics Uspekhi [**7**]{} (1965), 549-573.
J.M. Luttinger: [*New variational method with applications to disordered systems*]{}, Phys. Rev. Lett. 37 (1976) 609-612.
G. Mezincescu: [*Bounds on the integrated density of electronic states for disordered Hamiltonians*]{}, Phys. Rev. B 32 (1985) 6272-6277.
H. Nagai: [*On an exponential character of the spectral distribution function of a random difference operator*]{}, Osaka J. Math. 14 (1977) 111-116.
S. Nakao: [*On the spestral distribution of the Schrödinger operator with random potential*]{}, Japan J. Math. Vol. 3 (1977), 111-139.
L.A. Pastur: [*The behavior of certain Wiener integrals as $t\to\infty$ and the density of states of Schrödinger equations with random potential*]{}, (Russian) Teoret. Mat. Fiz. 32 (1977) 88-95.
K. Pietruska-Pa[ł]{}uba: [*The Lifschitz singularity for the density of states on the Sierpiński gasket*]{}, Probab. Theory Related Fields 89 (1991), no. 1, 1-33.
H. Okura: [*On the spectral distributions of certain integro-differential operators with random potential*]{}, Osaka J. Math. 16 (1979), no. 3, 633-666.
M. Reed, B. Simon: *Methods on Modern Mathematical Physics. Vol. 4: Analysis of Operators*, Academic Press, 1978.
M. Romerio, W. Wreszinski: [*On the Lifshitz singularity and the tailing in the density of states for random lattice systems*]{}, J. Stat. Phys. 21 (1979) 169-179.
R. Schilling, R. Song, Z. Vondraček: *Bernstein Functions*, Walter de Gruyter, 2010.
K. Sato: *Lévy Processes and Infinitely Divisible Distributions*. Cambridge University Press, Cambridge 1999.
B. Simon: *Lifshitz tails for the Anderson model*, J. Stat. Phys. 38 (1985) 65-76.
A.S. Sznitman: *Lifschitz tail and Wiener sausage I*, J. Funct. Anal. [**94**]{} (1990), 223-246.
[^1]: Research supported by the National Science Center, Poland, grant no. 2015/17/B/ST1/01233.
|
---
abstract: 'We present the results of an identification campaign of unassociated sources from the $Fermi$ Large Area Telescope 3FHL catalog. Out of 200 unidentified sources, we selected 110 sources for which archival $Swift$-XRT observations were available, 52 of which were found to have exactly one X-ray counterpart within the 3FHL 95% positional uncertainty. In this work, we report the X-ray, optical, IR, and radio properties of these 52 sources using positional associations with objects in various catalogs. The $Wide$-$field$ $Infrared$ $Survey$ $Explorer$ color-color plot for sources suggests that the most of these belong to the blazar class family. The redshift measurements for these objects range from $z=$0.277 to $z=$2.1. Additionally, under the assumption that the majority of these sources are blazars, three machine-learning algorithms are employed to classify the sample into flat spectrum radio quasars or BL Lacertae objects. These suggest that the majority of the previously unassociated sources are BL Lac objects, in agreement with the fact the BL Lac objects represent by far the most numerous population detected above 10GeV in 3FHL.'
author:
- 'A. Kaur, M. Ajello, S. Marchesi,N. Omodei'
bibliography:
- 'bibliography.bib'
title: |
Identifying the 3FHL catalog: I.\
Archival $Swift$ Observations and Source Classification
---
Introduction
============
The $\gamma$-ray sky provides us with a unique opportunity to reveal the nature of the most extreme environments in the universe. The Energy Gamma Ray Experiment Telescope [EGRET; @Fichtel1993] on the Compton Gamma Ray Observatory (CGRO) performed a survey of the $\gamma$-ray sky above 50MeV, revealing multiple high-energy astrophysical phenomena, such as active galactic nuclei (AGN), Supernova remnants, Gamma-ray bursts, pulsars. The large area telescope [LAT; @Atwood2009] aboard the [*Fermi*]{} satellite was launched in 2008 and revolutionized this area of astrophysics by detecting thousands of sources in the $\gamma$-ray energy band. The latest released, broad-band, all-sky LAT catalog [3FGL; @Acero2015] consists of 3033 sources detected in the energy range 0.1-300 GeV. Other two catalogs [1FHL and 2FHL; @Ackermann2013a; @Ackermann2016b] exploited the high-energy $\gamma$-ray sensitivity of [*Fermi*]{}-LAT, reporting sources detected above 10 GeV (514 objects) and 50 GeV (360). The most recently published 3FHL [*Fermi*]{}-LAT catalog [@Ajello2017] represents a significant upgrade of the 1FHL catalog and lists 1556 sources detected between 10GeV and 2TeV, utilizing 7 years of LAT data.
While the majority ($\sim$78%) of the 3FHL sources has already been associated with a counterpart, 200 objects in this catalog lack any information on association or classification. The knowledge of the properties of these extremely high-energy sources is fundamental for the studies of extragalactic background light [EBL, @Dominguez2015] and to constrain the origin of the extragalactic $\gamma$-ray background [EGB, see e.g. @Ajello2015; @Ackermann2016a]. Furthermore, the 3FHL catalog will likely represent the main reference for future observations with the upcoming Cherenkov Telescope Array [CTA; @Hassan2017].\
The biggest challenge in finding associations to $\gamma$-ray sources are the rather large positional uncertainties ($\sim$ arcminutes). This issue can be minimized by conducting X-ray observations of these $\gamma$-ray source fields, which has been done in the past by various authors [see @Stroh2013; @Parkinson2016; @Paiano2017 for recent studies with this approach]. In the context of this work, we utilize observations performed with the X-ray Telescope (XRT) telescope mounted on the Neil Gehrels [*Swift*]{} Observatory [@Gehrels2004]. This paper is organized as follows: Section 2 describes the source selection criteria and analysis procedure for the $Swift$-XRT data, Section 3 lists the catalogs (radio/IR/optical) and the procedure used to find likely associations these 3FHL objects. Section 4 explains the machine learning methods employed to further classify these sources. Section 5 describes the results of the classification process and Section 6 comprises of the discussion and conclusions based on our analysis.
$Swift$-XRT Data Selection {#sec:obs}
==========================
Firstly, we queried the HEASARC[^1] database for [*Swift*]{} satellite observations of the fields corresponding to the unassociated 200 3FHL sources and we found 110 3FHL source fields observed with $Swift$-XRT. For 48 fields we found multiple observations. All these observations were stacked before proceeding for the $Swift$-XRT analysis. All the XRT data reduction processes, i.e., summing data from different observations, creating images and spectra for all the sources were performed with the online $Swift$-XRT product builder[^2] following the methods described in @Evans2007 [@Evans2008]. This procedure was performed using standard tools in HEASOFT version 6.19. All the generated images were investigated to identify X-ray sources within the 95% confidence interval for the [*Fermi*]{}-LAT positions. In three cases (3FHL J0316.5-2610, 3FHL J1248.8+5128, 3FHL J2042.7+1520) we found exactly one X-ray counterpart outside but close to the 95% uncertainty (within 2 arcmin) and we have included these in our final sample. For another source, 3FHL J1553.8-2425, we found two bright counterparts, one within the 95% region and one known cataclysmic variable about 2.5 arcmin outside the Fermi uncertainty region. We proceeded by including the counterpart inside the 95% region as the associated source in our sample. In 55 fields observed with XRT, no source was detected within the 3FHL uncertainty radius. We point out that the 3FHL sources, where no X-ray counterpart was detected, have on average smaller exposure times (mean 1300 sec, median 1800 sec) than the ones where an X-ray source was detected (mean 5000 sec, median 4000 sec). Furthermore, images with more than one source ( three total) were excluded from the sample and left to a future study. These criteria lead to a total number of 52 X-ray candidate counterparts in our final sample. Each image was then investigated manually to estimate a rough position of the object, which was then provided in the above mentioned online tool to calculate the exact counterpart centroid employing the standard position method. These localized X-ray positions are listed in Table \[tab:swift\]. Spectral fitting was performed using XSPEC version 12.9.1 [@Arnaud1996] utilizing the source and background files generated by the online tool. All the spectra were fitted with a power law in conjunction with the Tuebingen-Boulder ISM absorption model ([tbabs]{}). The Galactic column densities were obtained with the HEASoft online tool[^3] [@Kalberla2005].The resulting parameters from this analysis are presented in Table \[tab:swift\].
Correlation with other databases {#sec:data}
================================
We used the $Swift$-XRT positions to cross-correlate our sources with multiple catalogs, selected in different bands. See Fig. \[fig:corr\] Based on the $Swift$-XRT typical positional accuracy, we allowed a maximum position uncertainty of 5, except for the cross-correlation with the ROSAT catalog [1RXS, @Voges1999 see below]. The resulting associations are presented in Table \[tab:corr\]. The different catalogs used for the association are reported in the following sections.
1RXS
----
The ROSAT All-Sky Survey Source Catalogs for bright and faint sources [1RXS @Voges1999; @Voges2000] contain 18806 and 105924 sources, respectively. The positional uncertainties for these sources are of the order of 30, therefore, we compare the 1RXS positions with the $Swift$-XRT positions allowing a maximum uncertainty of 30to find the possible associations. This lead to 8 positional correlations, as listed in Table \[tab:corr\]. The 1RXS positional error is reported in parenthesis for each association in this table.
BZCat
-----
The 5th Roma-BZCat catalog is the largest known blazar sample [@Massaro2015a; @Massaro2016]. Three out of 52 sources were found spatially coincident with BZCat sources within 5and all of them are classified as BL Lacs: 3FHL J0316.5-2610, 3FHL J1248.8-5128 and 3FHL J1553.8-2425 with redshifts, $z$= 0.443, 0.351 and 0.332, respectively.
AllWISE
-------
We cross-correlated our sample with AllWISE, the complete Wide-field Infrared Survey Explorer (WISE) point source catalog [@Cutri2013], and WIBRaLS, a catalog of radio-loud candidate $\gamma$-ray emitting blazars with WISE colors similar to the colors of confirmed $\gamma$-ray blazars [@DAbrusco2014]. 39 sources were found coincident with AllWISE positions and 3 identified as BL Lacs in the WIBRals catalog, which are the same found in BZCat. (see Table \[tab:corr\]). 31 of these sources were detected in all the W1, W2, W3 and W4 filters (3.4, 4.6, 12 and 22 $\mu$m, respectively). However, only upper limits for W3 and W4 were provided for 8 sources. See Fig. \[fig:wise\]
### WISE color index classification
@Massaro2012 introduced a method to identify blazars of uncertain type using a four-filters WISE color-color diagram [@Wright2010]. These authors identified a particular region in the diagram, which separates blazars from other source classes. They termed this region the WISE blazar strip (WBS) [see Fig 1, 2 in @Massaro2012]. Moreover, they found that BL Lacs and FSRQs follow a bimodal distribution, such that the former occupy the bluer part of the color-color diagram. Utilizing the information obtained from the spatial correlation with the AllWISE catalog in our sample of 39 sources with WISE counterpart, we compare their position on the WISE color-color plot with that of the 915 known blazars from the 3FHL catalog. It is evident from Fig. \[fig:wise\] that $>$80% of our sources lie within the WISE Gamma-ray Strip Projections [@Massaro2012] for BL Lacs and FSRQs; and the majority of these occupy the BL Lac region.
Million Quasar catalog
----------------------
The catalog published by @Flesch2017 presents type I and II QSOs , AGNs and BL Lacs reported in various catalogs from the literature before 21 June, 2016. This list also includes the candidates based on the SDSS photometric quasar catalogs. Ten sources in our sample have an association in the Million Quasar catalog: two were coincident with QSO type I, three with BL Lacs and the other five with possible QSOs with likelihood $>$85%. Finally, the match to the Million Quasar catalog yielded redshifts for two sources (see Table \[tab:corr\]). These redshifts were derived using the photometric method utilizing the SDSS DR12 catalog [@Alam2015], details of which are provided in the Half Million Quasar catalog [@Flesch2015]. Seven of these ten sources are also spatially coincident with a radio counterpart.
6dFGS
-----
The 6dF Galaxy Survey (6dFGS) catalog [@Jones2009] provides the redshift map of the Southern hemisphere for nearby objects ($z$ $\lesssim$ 0.1). None of the sources in this catalog are positionally coincident with our sample sources within a 5$^{\prime\prime}$ uncertainty radius circle.
Radio catalogs
--------------
### NVSS
The NRAO VLA Sky Survey (NVSS) [@Condon1998] constitutes the radio observations of celestial objects with declination greater than -40 $\deg$ at 1.4 GHz. The radio positions of the 22 counterparts of the 3FHL sources is presented in Table \[tab:corr\].
### SUMSS
The Sydney University Molonglo Sky Survey [@Mauch2003 SUMSS;] with radio observations at 843 MHz consists of Southern Hemisphere objects. Two sources in our sample have a radio counterpart in this catalog (see Table \[tab:corr\]).
### 2WHSP
@Chang2016 assembled the largest known catalog of WISE High Synchrotron Peak blazars (2WHSP) which comprises 1691 sources. These authors cross-correlated AllWISE catalog with various other wavelength surveys (radio, IR, X-ray). Utilizing this multiwavelength information, they identified blazars, calculated their peak synchrotron frequencies ($\nu_{syn}^{pk}$) and listed the HSPs ($\nu_{syn}^{pk}$ $>$ 15 Hz) in the 2WHSP catalog. We found 9 sources from our sample to be spatially coincident with HSP blazars in this catalog. The 2WHSP identifications of these sources are provided in Table \[tab:corr\].
Miscellaneous
-------------
Finally, we checked for potential counterparts using the two largest online databases of astronomical objects, i.e., SIMBAD [@Wenger2000] and the NASA/IPAC Extragalactic Database (NED)[^4]. We found the following possible associations, which were not found in any of the above mentioned catalogs:
**3FHL J0438.0-7328, 3FHL J1249.2-2809**: These two sources are positionally coincident with galaxies LEDA 255538 and LEDA 745327, respectively. These associations are derived from HYPERLEDA, a catalog of about one million galaxies brighter than B=18mag [@Paturel2003]. No redshift information was found for these two objects.
**3FHL J1234.8-0435**: This 3FHL source spatially coincides with a galaxy listed in the 2dF Galaxy Redshift Survey @Colless2001. The redshift, $z$=0.277, provided by this catalog was obtained from both absorption and emission features.
Classification with machine learning
====================================
Multiwavelength data analysis is typically required for every [*Fermi*]{} detected source to be correctly classified. This process is highly time consuming and the lack of this information has led to an increasing fraction of unidentified sources in every new [*Fermi*]{} catalog release. However, $\sim$ 80% of the objects in the 3FHL catalog are associated with blazars (FSRQ, BL Lac or BCU), and this fraction increases to $\sim$ 90% if sources along the Galactic plane ($|b|\leq 10^{\circ}$) are not considered. In our sample, 36 out of 52 sources are high-latitude objects, therefore we assume that all these 36 unknown sources in our sample are blazars. This assumption is also justified by the fact that, as seen in Fig. \[fig:wise\], most of the sources are coincident with the WISE-blazar strip, except three outliers: 3FHL J0427.5-6705 (W3 measurement S/N=0.1, no radio), 3FHL J1650.9+0430 (W3 measurement S/N =0.1, no radio) and 3FHL J1958.1+2437.\
Although it is quite evident from Fig. \[fig:wise\] that most of our sources lie within the blazar region, in particular, BL Lac region, these results are based on only two parameters. We thus further analyze our findings by employing various other properties, to refine the way in which we differentiate BL Lacs from FSRQs. In order to accomplish this multi-parameter space classification, we employ three different machine learning algorithms. The parameters employed in these methods were derived from the 3FGL, 1SXPS [@Evans2014] and AllWISE catalogs, and are discussed in detail in Section \[sec:params\]. Various machine learning techniques have been successfully applied to $Fermi$ unidentified sources, e.g., @Ackermann2012 [@Mirabal2012; @Mirabal2016; @Parkinson2016; @Salvetti2017]. From a wide variety of available methods used by these authors, we chose to apply three most commonly employed methods: Decision Tree [@Quinlan1990], Support Vector Machines [@Hearst1998] and Random Forest [@Breiman2001].
Decision Tree
-------------
The decision tree classifier (DT) is an example of supervised machine learning algorithm which separates a dataset into two or more categories based on certain parameters associated with the input data. The data is continuously split into nodes and branches until every data point is assigned to one or the other category. The decision of splitting into separate nodes is based on the Gini index, an impurity measurement. The Gini impurity parameter provides a measurement of the probability of incorrectly labeling a randomly chosen element in the given dataset. The decision tree algorithm works towards minimizing this value and splits the sample into branches until this index reaches zero. Mathematically, it is defined as $$G=1-\Sigma_{i=1}^{J}p_{i}^{2}$$ which calculates the Gini impurity for a dataset with $J$ categories with $p_{i}$ being the fraction of items labeled with category $i$ in the sample. Higher values of G imply higher inequality between two classes for a given parameter. The decision tree is split until the Gini index reaches a minimum value equal to zero, thereby assigning a particular class to the underlying items. This method employs a dataset with known classification as training dataset and trains the classifier. The accuracy obtained for a trained dataset is calculated to evaluate its usage on a sample with no classification.
Support Vector Machines
-----------------------
Support Vector Machine (SVM) is another supervised learning method for separating a dataset into two categories. The underlying principle for this method is that for any data point, $i$ or $j$ (two categories), one or a set of maximum margin hyperplane are found such that the distance between this plan and the nearest point in either category is maximized.
Mathematically, a hyperplane for a set of points with category , $i$ (say $\vec{x}$) is defined as following: $$\vec{w}.\vec{x}-b=0,$$ such that the parameter $\frac{b}{||\vec{w}||}$ defines the distance of this plane from the origin along $\vec{w}$, where $\vec{w}$ is the normal vector to the hyperplane. This is an example of linear kernel classification for SVM. A non-linear SVM employs polynomial or rbf kernels to classify any dataset for higher dimensions. In the context of our work, we employ a polynomial kernel and a non-linear SVM to classify the sample into two kind of blazars, i.e., BL Lacs or FSRQs.
Random Forest
-------------
A random forest is one of the most commonly employed supervised machine learning method used for both classification and regression analysis. A random forest classifier operates as an ensemble algorithm based on the principle of a decision tree classifier. This method constructs various decision tree algorithms and assigns a class to a source for every iteration. An aggregate of these predicted classes is assigned as the final resulting class for that particular source. This method has an advantage over running a single decision tree, since it utilizes the multitude of decision trees, thereby solving the problem of overfitting [@Hastie2009], which is usually observed in the latter case. We employ this method to classify our sample into BL Lacs and/or FSRQs. This yields probabilities for each source to be associated as a BL Lac or an FSRQ.
Sample and Parameter Section {#sec:params}
----------------------------
We employed the DT classifier, SVM classifier and Random Forest implemented in [sklearn0.20.0]{} library [@Pedregosa2011] in [python 2.7]{} on a sample of 152 3FHL blazars (115 BLLacs and 37 FSRQs). This sample was chosen as a subset of all the BL Lacs and FSRQs in the 3FHL catalog for which all the six parameters listed in Table \[tab:pars\] were available. The reason for the selection of these six properties was based on the fact that these have been observed to distinguish BL Lacs from FSRQs. In general, BL Lacs exhibit harder spectrum in Gamma-rays (e.g. @Abdo2010b [@Ackermann2015b]) and softer in X-rays (e.g. @Donato2001) as compared to FSRQs , therefore, we select the spectral indices in Gamma-ray (3FHL and 3FGL) and X-ray. The WISE colors, as already discussed in the text, clearly differentiate the two classes of blazars. FSRQs can be distinguished from BL Lacs on the basis of variability. In the 3FHL catalog [@Ajello2017], a parameter called Variability Bayes Blocks is provided, which lists the number of Bayesian blocks from variability analysis. The values of this parameter range from -1 to 15, where -1 implies no variability and 15 implies high variability. FSRQs exhibit higher values for this parameter, implying higher variability as compared to BL Lacs. This sample was divided into training and test datasets in order to check the accuracy of the method employed. The training and test datasets comprised of 102 blazars (77 BL Lacs and 25 FSRQs) and 50 blazars (38 BL Lacs and 12 FSRQs), respectively. Since the total sample contains $\sim$ 75% BL Lacs and only 25% FSRQs, which being highly imbalanced could yield inaccurate results biased towards the major class, when a machine learning method is applied. We, therefore, employed a technique called SMOTE (Synthetic Minority Over-sampling Technique) [@Chawla2002]. This method creates synthetic minority class using k nearest neighbors algorithm, thereby generating equal number of sources in each class. An an example, in this case, the training dataset has 77 BL Lacs and 25 FSRQs. Implementation of SMOTE method generated 52 synthetic FSRQs, thereby balancing the two classes (77 sources for each class) before application of a classification method.
Results {#sec:results}
=======
The resulting decision tree from the training sample is shown in Fig. \[fig:dttrainer\]. We employed this trained classifier on the test dataset (38 BL Lacs and 12 FSRQs) which yielded an accuracy of 86%. This classifier was then applied to the unknown 3FHL sample, which yielded results suggesting that 31 out of the 36 high-latitude unassociated sources are BL Lacs and the rest are FSRQs. For SVM, a receiver operating characteristic curve (ROC) is used to evaluate the accuracy of this binary classifier and it is shown in Fig. \[fig:svm\_roc\]. The ROC is constructed by plotting the true positive rate (TPR, number of correct positive results) against the false positive rate (FPR, number of incorrect positive results) at various thresholds. The classification accuracy is $\sim$ 90% which was evaluated as the area below the curve for the given sample. The SVM analysis on our sample suggests that all the unknown sources are likely BL Lacs. The Random Forest classifier yielded results consistent with the SVM classifier suggesting that all the unassociated 36 high-latitude sources are BL Lacs. The accuracy obtained on the test sample in this case was 98%. The receiver operating characteristic curves for both the training and the test sample are shown in Fig. \[fig:rf\_roc\]. A comparison of the results yielded by all these methods is displayed in Table \[tab:mlcomp\].
Discussion and Conclusions
==========================
The immediate objective of this work was to identify the nature of unassociated sources reported in the latest $Fermi$ high energy catalog, the 3FHL. In an attempt to find associations to these sources, $Swift$ HEASARC archive was used to derive the accurate positions of the sources for which data were available, which lead to a sample of 52 3FHL unassociated sources with a single bright X-ray counterpart. The X-ray source positions were cross-matched with various catalogs from radio to X-ray wavelengths (see Section \[sec:data\]), leading to the identification of the likely counterpart for 12 out of the 52 objects (6 out of 52 objects are identified as QSOs, 3 as BL Lacs and 3 as galaxies). Six of these 12 sources also have confirmed redshift measurements ranging from $z$=0.277 to $z$=2.1. In addition, the WISE color-color plot, as shown in Fig. \[fig:wise\], suggests that majority of the 3FHL sources are likely blazars. Moreover, 90% of the high-latitude $|b|\geq$ 10$^{\circ}$ objects in the 3FHL sample are associated with the blazar population. 36 objects from our source sample are high-latitude objects and assuming that this subsample comprises only blazars, we employed machine learning techniques (DT, SVM and RF) to classify these objects into two kind of blazars, i.e., BL Lacs and FSRQs. The DT classifier yielded results showing that 31 of the high-latitude sources are BL Lacs. The SVM and the RF classifier predict that all these sources are BL Lacs. For details, please see Table \[tab:mlcomp\]. The inconsistency between the results from DT vs SVM/RF could be attributed to potential overfitting in the former method as discussed earlier.\
In nutshell, this work provides classification for 36/200 sources, which reduces the incompleteness of the 3FHL by 18%. While the redshift info is scarce (12%), our group is working on an optical spectroscopic campaign to observe these unassociated sources with 4m and 8m class telescopes, to obtain redshifts for a significant fraction of them and confirm their nature (see @Marchesi2018, where the first results of this campaign are reported). In addition, our recent successful proposal (Swift Cycle 14, prop ID 1417063 PI: Ajello) in an effort to obtain more XRT sources for the unknown/unassociated 3FHL sources is currently in progress. All these continuing studies will drastically reduce the incompleteness of the 3FHL catalog in a few years timescale.\
[ccccccccccc]{}
J0049.0+4224 & 00:49:05 & +42:24:12 & 00:48:59.14 & +42:23:47.40 & 4106& 1.216& 2.54$\pm$0.24 & 0.12 & 0.68 & 48.31/47$^{*}$\
J0121.9$-$3917 & 01:21:56 & $-$39:17:13 & 01:21:52.51 & $-$39:15:44.64 & 5736 & 0.080 & 1.79$\pm$0.08 & 0.02 & 3.45 & 19.21/19\
J0156.2$-$2419 & 01:56:16 & $-$24:19:30 & 01:56:24.31 & $-$24:20:06.72 & 17280& 0.012 & 2.59$\pm$0.13 & 0.01 & 0.98 & 5.13/9\
J0213.9$-$6950 & 02:13:58 & $-$69:50:12 & 02:13:58.44 & $-$69:51:38.16 & 4098 & 0.100 & 1.87$\pm$0.08 & 0.05 & 4.44 & 17.31/17\
J0251.2$-$1830 & 02:51:13 & $-$18:30:30 & 02:51:11.31 & $-$18:31:15.57 & 7572 & 0.014& 2.03$\pm$0.16 & 0.03 & 0.64 & 68.78/88$^{*}$\
J0316.5$-$2610 & 03:16:32 & $-$26:10:03 & 03:16:14.95 & $-$26:07:57.36 & 4885 & 0.152 & 2.52$\pm$0.06 & 0.01 & 4.82 & 18.90/30\
J0350.4$-$5143 & 03:50:26 & $-$51:43:44 & 03:50:28.37 & $-$51:44:54.96 & 1708 & 0.160 & 2.08$\pm$0.13 & 0.01 & 6.26 & 10.12/10\
J0350.8$-$2814 & 03:50:50 & $-$28:14:02 & 03:50:51.34 & $-$28:16:33.96 & 11010& 0.068 & 1.92$\pm$0.06 & 0.01 & 3.03 & 34.24/32\
J0401.0$-$5355 & 04:01:03 & $-$53:55:11 & 04:01:11.45 & $-$53:54:56.52 & 904 & 0.020 & 1.95$\pm$0.33 & 0.01 & 0.78 & 11.55/15$^{*}$\
J0427.5$-$6705 & 04:27:35 & $-$67:05:49 & 04:27:49.51 & $-$67:04:34.68 & 14500& 0.008 & & 0.04 & & $\dagger$\
J0438.0$-$7328 & 04:38:05 & $-$73:28:26 & 04:38:37.30 & $-$73:29:22.20 & 3868 & 0.007 & 2.10$\pm$0.42 & 0.09 & 0.19 & 13.89/25\
J0506.9+0323 & 05:06:55 & +03:23:42 & 05:06:50.09 & +03:23:59.28 & 17280& 0.012 & 2.47$\pm$0.15 & 0.07 & 0.57 & 3.7/7$^{*}$\
J0541.1$-$4855 & 05:41:10 & $-$48:55:40 & 05:41:07.03 & $-$48:54:10.45 & 1224 & 0.008 & 2.04$\pm$0.76 & 0.03 & 0.47 & 3.8/8$^{*}$\
J0559.6+3045 & 05:59:40 & +30:45:44 & 05:59:40.42 & +30:42:32.75 & 4680 & 0.004& 2.04$\pm$0.76 & 0.36 & 0.48 & 9.6/18$^{*}$\
J0706.1+0247 & 07:06:08 & +02:47:55 & 07:06:10.86 & +02:44:50.82 & 2432 & 0.038 & 1.88$\pm$0.23 & 0.36 & 2.91 & 68.18/74$^{*}$\
J0725.7$-$0548 & 07:25:44 & $-$05:48:53 & 07:25:47.69 & $-$05:48:27.00 & 4872 & 0.036 & 2.37$\pm$0.18 & 0.26 & 2.39 & 10.22/6\
J0739.7$-$6720 & 07:39:45 & $-$67:20:09 & 07:39:27.74 & $-$67:21:37.04 & 3192 & 0.041 & 2.05$\pm$0.16& 0.09 & 2.36 & 82.00/96$^{*}$\
J0747.7$-$4927 & 07:47:44 & $-$49:28:00 & 07:47:25.30 & $-$49:26:32.64 & 4086 & 0.023 & 2.44$\pm$0.18 & 0.09 & 1.18 & 66.75/71$^{*}$\
J0813.7$-$0353 & 08:13:46 & $-$03:53:57 & 08:13:38.02 & $-$03:57:17.64 & 3157 & 0.066 & 1.71$\pm$0.15 & 0.05 & 3.37 & 5.89/7\
J0820.2$-$2803 & 08:20:15 & $-$28:03:04 & 08:20:14.69 & $-$28:03:03.60 & 1763 & 0.022 & 2.49$\pm$0.35 & 0.27 & 1.95 & 26.96/36$^{*}$\
J0928.5$-$5256 & 09:28:33 & $-$52:56:05 & 09:28:18.98 & $-$52:57:05.40 & 3579 & 0.006 & & 0.94 & & $\dagger$\
J0937.8$-$1434 & 09:37:52 & $-$14:34:16 & 09:37:54.86 & $-$14:33:43.56 & 4480 & & & 0.05 & & $\dagger$\
J1016.2$-$4245 & 10:16:14 & $-$42:45:34 & 10:16:21.02 & $-$42:47:24.36 & 3883 & 0.020& 2.39$\pm$0.19 & 0.05 & 0.89 & 54.36/65$^{*}$\
J1024.5$-$4543 & 10:24:33 & $-$45:43:47 & 10:24:32.78 & $-$45:44:27.96 & 4158 & 0.057& 2.48$\pm$0.13 & 0.12 & 2.87 & 81.9/70\
J1033.4$-$5033 & 10:33:26 & $-$50:33:39 & 10:33:32.28 & $-$50:35:28.68 & 3836 & 0.029& 2.25$\pm$0.17 & 0.19 & 1.78 & 83.36/98$^{*}$\
J1034.8$-$4645 & 10:34:53 & $-$46:45:09 & 10:34:38.45 & $-$46:44:01.92 & 1733 & 0.063 & 1.97$\pm$0.19 & 0.14 & 3.40 & 3.36/4\
J1047.9$-$3738 & 10:47:56 & $-$37:38:41 & 10:47:57.02 & $-$37:37:35.76 & 3621 & 0.022 & 2.49$\pm$0.20 & 0.04 & 1.00 & 49.22/67$^{*}$\
J1117.2$-$5338 & 11:17:15 & $-$53:38:03 & 11:17:14.74 & $-$53:38:15.04 & 3361 & 0.012 & 1.97$\pm$0.33 & 0.15 & 0.85 & 40.52/37$^{*}$\
J1125.0$-$5806 & 11:25:05 & $-$58:06:42 & 11:25:04.49 & $-$58:05:40.78 & 3094 & 0.019 & 2.49$\pm$0.24 & 0.37 & 2.15 & 58.74/55$^{*}$\
J1145.9$-$0637 & 11:45:54 & $-$06:37:18 & 11:46:00.65 & $-$06:38:51.36 & 2343 & 0.024 & 1.98$\pm$0.24 & 0.03 & 3.01 & 54.65/48$^{*}$\
J1220.1$-$2459 & 12:20:11 & $-$24:59:07 & 12:20:14.54 & $-$24:59:52.08 & 4233 & 0.061 & 2.00$\pm$0.12 & 0.08 & 3.07 & 9.56/10$^{*}$\
J1220.4$-$3714 & 12:20:26 & $-$37:14:31 & 12:20:19.78 & $-$37:14:10.68 & 1354 & 0.051 & 2.35$\pm$0.20 & 0.08 & 2.43 & 39.28/54$^{*}$\
J1234.8$-$0435 & 12:34:50 & $-$04:35:11 & 12:34:47.96 & $-$04:32:46.26 & 3424 & 0.007 & 2.15$\pm$0.29 & 0.03 & 0.29 & 22.67/23$^{*}$\
J1240.5$-$7148 & 12:40:36 & $-$71:48:40 & 12:40:21.19 & $-$71:48:57.24 & 4695 & 0.272 & 1.84$\pm$0.04 & 0.14 & 14.98 & 78.43/55$^{*}$\
J1248.8+5128 & 12:48:54 & +51:28:08 & 12:48:34.30 & +51:28:06.96 & 6832 & 0.010 & 1.83$\pm$0.20 & 0.01 & 0.47 & 17.25/21$^{*}$\
J1249.2$-$2809 & 12:49:13 & $-$28:09:37 & 12:49:19.42 & $-$28:08:38.04 & 5956 & 0.101 & 2.17$\pm$0.07 & 0.06 & 4.30 & 25.68/27\
J1447.0$-$2657 & 14:47:04 & $-$26:57:50 & 14:46:57.41 & $-$26:57:01.80 & 4106 & 0.239 & 1.89$\pm$0.05 & 0.09 & 13.8 & 21.88/42\
J1517.0+2638 & 15:17:02 & +26:38:31 & 15:16:49.82 & +26:36:42.48 & 3514 & 0.005 & 2.13$\pm$0.46 & 0.04 & 0.23 & 18.49/17$^{*}$\
J1541.7+1413 & 15:41:44 & +14:13:48 & 15:41:49.85 & +14:14:44.52 & 2878 & 0.005 & 2.61$\pm$0.62 & 0.03 & 0.21 & 16.01/14$^{*}$\
J1553.8$-$2425 & 15:53:51 & $-$24:25:14 & 15:53:31.25 & $-$24:22:05.16 & 5384 & 0.013& 1.39$\pm$0.27 & 0.11 & 0.92 & 35.75/60\
J1650.9+0430 & 16:51:00 & +04:30:05 & 16:50:58.87 & +04:27:34.92 & 14020 &0.001 & & 0.06 & & $\dagger$\
J1704.5$-$0527 & 17:04:34 & $-$05:27:25 & 17:04:34.10 & $-$05:28:37.92 & 6309 & 0.063& 1.88$\pm$0.09 & 0.13 & 3.56 & 24.32/18\
J1856.1$-$1221 & 18:56:07 & $-$12:21:16 & 18:56:06.67 & $-$12:21:50.40 & 3286 & 0.023& 1.93$\pm$0.28 & 0.16 & 1.63 & 21.00/21$^{*}$\
J1958.1+2437 & 19:58:07 & +24:37:51 & 19:58:00.38 & +24:38:03.84 & 3529 & 0.019& 2.39$\pm$0.30 & 0.69 & 3.13 & 59.42/55$^{*}$\
J2036.0+4901 & 20:36:03 & +49:01:11 & 20:35:51.24 & +49:01:39.72 & 3521 & 0.004& 2.01$\pm$1.17 &0.66 & 0.13 & 16.01/11$^{*}$\
J2042.7+1520 & 20:42:45 & +15:20:03 & 20:42:59.71 & +15:21:06.48 & 17600 & 0.035& 2.23$\pm$0.08 & 0.07 & 1.89 & 29.92/26\
J2109.7+0440 & 21:09:42 & +04:40:11 & 21:09:40.08 & +04:39:59.40 & 354 & 0.025 & 1.58$\pm$0.73 & 0.07 & 1.69 & 6.19/6$^{*}$\
J2115.2+1218 & 21:15:15 & +12:18:29 & 21:15:22.10 & +12:18:02.88 & 3756 & 0.007& 2.95$\pm$0.36 & 0.05 & 0.37 & 21.28/25$^{*}$\
J2142.3+3659 & 21:42:22 & +36:59:40 & 21:42:26.50 & +36:59:48.12 & 4365 & 0.022& 2.16$\pm$0.14 & 0.17 & 6.73 & 6.69/7\
J2142.7+1959 & 21:42:44 & +19:59:22 & 21:42:47.45 & +19:58:09.84 & 1628 & 0.123& 2.99$\pm$0.22 & 0.07 & 1.88 & 64.00/69$^{*}$\
J2151.5+4155 & 21:51:32 & +41:55:47 & 21:51:22.90 & +41:56:32.28 & 3284 & 0.022& 2.81$\pm$0.32 & 0.23 & 1.53 & 26.37/62\
J2308.8+5424 & 23:08:54 & +54:24:22 & 23:08:48.62 & +54:26:08.88 & 1051 & 0.022& 2.24$\pm$0.53 & 0.28 & 2.11 & 18.41/20$^{*}$\
& & & & & & & &\
[lccccccccr]{} J0049.0+4224 & & N J004859+422350 & J004859.15+422351.1 & & & 004859.0+422351 & &\
J0121.9$-$3917 & & S J012152$-$391542 & J012152.69$-$391544.2 & & & & &\
J0156.2$-$2419 & & & J015624.54$-$242003.7 & & J015624.6$-$242004 & & QSO 93% &\
J0213.9$-$6950 & & & J021358.69$-$695137.0 & & & & &\
J0251.2$-$1830 & & N J025111$-$183112 & J025111.52$-$183112.7 & & & & &\
J0316.5$-$2610 & & N J031615$-$260755 & J031614.94$-$260757.2 & J0316$-$2607 & RXS J03162$-$2607 & 031614.8-260757 & BL LAC & 0.443\
J0350.4$-$5143 & & & J035028.30$-$514454.3& & & & &\
J0350.8$-$2814 & & & J035051.32$-$281632.8 & & & 035051.2-281632 & &\
J0401.0$-$5355 & & & J040111.20$-$535458.5 & & J040111.2$-$535458 & & QSO 86% &\
J0427.5$-$6705 & & & J042749.72$-$670434.7 & & & &\
J0438.0$-$7328 & & S J043836$-$732921 & J043837.07$-$732921.6 & & && LEDA 255538 &\
J0506.9+0323 & & N J050650+032401 & J050650.14+032358.7 & & & &\
J0541.1$-$4855 & & & J054106.92$-$485410.3 & & J054106.9$-$485412 & & QSO 91% &\
J0559.6+3045 & & N J055941+304227 & & & & & &\
J0706.1+0247 & & N J070610+024449 & & & & & &\
J0725.7$-$0548 & & & & & & & &\
J0739.7$-$6720 & J073928.1$-$672147(15) & & & & & & &\
J0747.7$-$4927 & J074725.1$-$492626(12) & & J074724.74-492633.1 & & & 074724.7-492633 & &\
J0813.7$-$0353 & & N J081338$-$035716 & J081338.07$-$035716.7 & & & & &\
J0820.2$-$2803 & & & & & & & &\
J0928.5$-$5256 & & & J092818.63$-$525703.1 & & & & &\
J0937.8$-$1434 & & & & & & & &\
J1016.2$-$4245 & J101620.6$-$424733(14) & & J101620.67$-$424722.6 & & & & &\
J1024.5$-$4543 & & & J102432.37$-$454426.9 & & & & &\
J1033.4$-$5033 & J103332.0$-$503539(14) & & J103332.15$-$503528.8 & & & & &\
J1034.8$-$4645 & & & J103438.49$-$464403.5 & & & & &\
J1047.9$-$3738 & & & J104756.94$-$373730.8 & & & 104756.8-373730 & &\
J1117.2$-$5338 & & & & & & & &\
J1125.0$-$5806 & J112502.3$-$580547(16) & & J112503.99$-$580539.9 & & & & &\
J1145.9$-$0637 && & J114600.85$-$063854.9 & & & & &\
J1220.1$-$2459 && N J122014$-$245949 & J122014.53$-$245948.6 & & J122014.5$-$245948 & & &\
J1220.4$-$3714 & & N J122020$-$371411 & J122019.81$-$371414.2 & & & 122019.8-371413 & &\
J1234.8$-$0435 & & & J123448.05$-$043245.2 & & & & Galaxy &0.277\
J1240.5$-$7148 & J124015.2$-$714859(21) & & J124021.21$-$714857.7 & & & & &\
J1248.8+5128 & J124832.1+512817(12) & N J124834+512807 & J124834.29+512807.8 & J1248+5128 & SDSS J124834.30+512807.8 & & BL LAC & 0.351\
J1249.2$-$2809 & & N J124919$-$280833 & J124919.31$-$280834.4 & & & 124919.3-280834 & LEDA 745327 &\
J1447.0$-$2657 & & & & & & & &\
J1517.0+2638 & & & & & & & &\
J1541.7+1413 & & & & & & & &\
J1553.8$-$2425 & & N J155331.6$-$242206 & & & PKS 1550$-$242 & & BL Lac & 0.332\
J1650.9+0430 & & & J165058.69+042734.9 & & && &\
J1704.5$-$0527 & & & J170433.83$-$052840.7 & & & 170433.7-052840& &\
J1856.1$-$1221 & & N J185606$-$122148 & & & & &&\
J1958.1+2437 & & N J195800+243802 & J195800.50+243800.9 & & && &\
J2036.0+4901 & & N J203551+490143 & & & & & &\
J2042.7+1520 & & N J204259+152107 & J204259.72+152108.1 & & J204259.7+152108 & & QSO 81 &\
J2109.7+0440 & & N J210939+044000 & J210940.12+044000.6 & & SDSS J210940.12+044000.3 & & QSO TYPE 1 &1.4$^{\dagger}$\
J2115.2+1218 & & N J211522+121802 & J211522.00+121802.6 & & SDSS J211522.00+121802.8 & & QSO TYPE 1 &2.1$^{\dagger}$\
J2142.3+3659 & & N J214226+365949 & J214226.49+365949.7 & & &214226.4+365948& &\
J2142.7+1959 & & N J214247+195810 & J214247.62+195810.9 & & & & &\
J2151.5+4155 & & N J215122+415632 & J215123.22+415633.9 & & & & &\
J2308.8+5424 & & N J230848+542612 & J230848.74+542611.2 & & & & &\
Parameter Catalog References
---------------------------- ---------------------------------------- -------------------------
X-ray Spectral Index Table \[tab:swift\] for unknown sample See Table \[tab:swift\]
1SXPS for training set @Evans2014
Variability Bayes Blocks 3FHL @Ajello2017
w1-w2 AllWISE @Cutri2013
w2-w3 AllWISE @Cutri2013
Gamma-ray Spectral Index 1 3FGL @Acero2015
Gamma-ray Spectral Index 2 3FHL @Ajello2017
: Parameters for Blazars for classification \[tab:pars\]
\[tab:mlcomp\]
---------------- ------ ----- ------ ----- ------ --
J0049.0+4224 fsrq bll 1.0 bll 0.84
J0121.9-3917 bll bll 1.0 bll 0.85
J0156.2-2419 bll bll 0.94 bll 0.96
J0213.9-6950 bll bll 0.98 bll 0.69
J0251.2-1830 bll bll 0.95 bll 0.99
J0316.5-2610 bll bll 0.98 bll 0.99
J0350.4-5143 bll bll 0.98 bll 0.92
J0350.8-2814 bll bll 1.0 bll 0.85
J0401.0-5355 fsrq bll 0.99 bll 0.8
J0427.5-6705 fsrq bll 1.0 bll 0.81
J0438.0-7328 bll bll 1.0 bll 0.79
J0506.9+0323 bll bll 0.98 bll 0.95
J0541.1-4855 fsrq bll 1.0 bll 0.77
J0739.7-6720 bll bll 1.0 bll 0.74
J0747.7-4927 bll bll 1.0 bll 0.88
J0813.7-0353 bll bll 1.0 bll 0.85
J0937.8-1434 bll bll 1.0 bll 0.9
J1016.2-4245 bll bll 0.98 bll 1.0
J1047.9-3738 bll bll 0.98 bll 0.91
J1145.9-0637 bll bll 1.0 bll 0.84
J1220.1-2459 bll bll 0.99 bll 0.87
J1220.4-3714 bll bll 0.98 bll 1.0
J1234.8-0435 bll bll 0.57 bll 0.93
J1248.8+5128 bll bll 0.87 bll 0.95
J1249.2-2809 bll bll 1.0 bll 0.86
J1447.0-2657 bll bll 0.99 bll 0.73
J1517.0+2638 bll bll 1.0 bll 0.84
J1541.7+1413 bll bll 1.0 bll 0.86
J1553.8-2425 bll bll 0.98 bll 0.84
J1650.9+0430 fsrq bll 1.0 bll 0.77
J1704.5-0527 bll bll 0.95 bll 0.97
J2042.7+1520 bll bll 0.99 bll 0.84
J2109.7+0440 bll bll 0.99 bll 0.88
J2115.2+1218 bll bll 0.89 bll 0.97
J2142.3+3659 bll bll 1.0 bll 0.84
J2142.7+1959 bll bll 0.99 bll 0.87
: Machine Learning Results
- [Predicted Class by the Decision Tree Algorithm]{}
- [Predicted Class by the SVM Algorithm]{}
- [Probability to be a BLL associated by the SVM Method]{}
- [Predicted Class by the Random Forest (RF) Algorithm]{}
- [Probability to be a BLL associated by the RF Method]{}
This publication made use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This work also utilized the data supplied by UK $Swift$ Data Centre at the University of Leicester as well as of TOPCAT software[@Taylor2005]. This research has also made use of the SIMBAD database, operated at CDS, Strasbourg, France; and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory.\
We are grateful to Eric D. Feigelson for discussions and suggestions regarding the correction for the imbalanced classes part for the machine learning methods.
[^1]: https://heasarc.nasa.gov/
[^2]: http://www.swift.ac.uk/user\_objects/index.php
[^3]: https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl
[^4]: https://ned.ipac.caltech.edu/
|
---
abstract: 'Recently, the frustrated XY model for spins-1/2 on the honeycomb lattice has attracted a lot of attention in relation with the possibility to realize a chiral spin liquid state. This model is relevant to the physics of some quantum magnets. Using the flexibility of ultra-cold atoms setups, we propose an alternative way to realize this model through the Mott regime of the bosonic Kane-Mele-Hubbard model. The phase diagram of this model is derived using the bosonic dynamical mean-field theory. Focussing on the Mott phase, we investigate its magnetic properties as a function of frustration. We do find an emergent chiral spin state in the intermediate frustration regime. Using exact diagonalization we study more closely the physics of the effective frustrated XY model and the properties of the chiral spin state. This gapped phase displays a chiral order, breaking time-reversal and parity symmetry, but is not topologically ordered (the Chern number is zero).'
author:
- Kirill Plekhanov
- Ivana Vasić
- Alexandru Petrescu
- Rajbir Nirwan
- Guillaume Roux
- Walter Hofstetter
- Karyn Le Hur
bibliography:
- 'main.bib'
title: 'Emergent Chiral Spin State in the Mott Phase of a Bosonic Kane-Mele-Hubbard Model'
---
The last few decades have seen a growing interest in the quest for exotic spin states and quantum spin liquids [@Lhuillier2002]. Significant progress has been made both from the theoretical and experimental sides [@Balents2010; @Norman2016; @SavaryBalents2017]. The best candidates for spin liquids are found in two-dimensional systems. Disordered phases are expected to occur in complex geometries, such as the Kagome lattice [@Lecheminant1997; @YanHuseWhite2011; @Depenbrock2012], or in frustrated bipartite lattices, such as the square lattice with second-neighbor couplings [@SchulzZiman1992; @Schulz1996]. Among basic lattices, the honeycomb one hosts free Majorana fermions due to Kitaev anisotropic interactions [@Kitaev2006], and raises questions when starting from the Hubbard model [@Meng2010; @Sorella2012; @Assaad2013]. In such context and motivated by quantum magnets [@Flint2013], frustrated Heisenberg models on the honeycomb lattice have been recently explored [@FouetSindzingreLhuillier2001; @Wang2010; @MudlerEtAl2010; @Clark2011; @Albuquerque2011; @Cabra2011; @Reuther2011; @Mezzacapo2012; @Zhang2013; @Ganesh2013; @GongShengEtAl2013; @Zhu2013; @GonfZhuBalentsSheng2015; @Ferrari2017]. In parallel, the XY version of this model was also tested for the possibility to realize a chiral spin liquid state, but with seemingly contradictory results [@VarneySunGalitskiRigol2011; @VarneySunGalitskiRigol2012; @ZhuHuseWhite2013; @ZhuWhite2014; @BishopLiCampbell2014; @CarrasquillaEtAl2013; @CioloEtAl2014; @NakafujiIchinose2017]. As suggested in Ref. , in the intermediate frustration regime the ground-state physics could be mapped to a fermionic Haldane model [@Haldane1988] with topological Bloch bands at a mean-field level, as a result of Chern-Simons (ChS) gauge fields [@Fradkin1989; @AnbjornSemenoff1989; @LopezRojoFradkin1994; @Misguichjolicoeurgirvin2001; @SunKumarFradkin2015]. However, the topological nature of this spin state is still elusive.
Our objectives are two-fold in this Letter. Motivated by cold atoms experiments [@BlochDalibardNascimbene2012; @GoldmanBudichZoller2016], we first show that the Mott regime of the bosonic Kane-Mele-Hubbard (BKMH) model allows for a tunable realization of the frustrated XY model on the honeycomb lattice. Second, we study its phase diagram and in particular its magnetic properties, using bosonic dynamical mean-field theory (B-DMFT) [@GeorgesKotliarKrauthRozenberg1996; @ByczukVollhardt2008; @HuTong2009; @HubenerSnoekHofstetter2009; @AndersGullPolletTroyerWerner2010; @suppMaterial], exact diagonalization (ED) and theoretical arguments. The Kane-Mele model [@KaneMele2005] is the standard model with spin-orbit coupling that displays ${\mathbb{Z}}_2$ topology. Still, it has not yet been studied for interacting bosons. Importantly, we recall that, for interacting fermions and at the Mott transition, the Kane-Mele model becomes magnetically ordered in the $xy$-plane, with quantum fluctuations stabilizing the Néel ordering [@RachelLehur2010; @WuRachelLiuLehur2012; @Hohenadler2012].
We start our analysis with the bosonic version of the Kane-Mele model [@KaneMele2005] on the honeycomb lattice (Fig. \[fig:0\](a)), which contains two species of bosons labelled by $\sigma = \uparrow, \downarrow$. In the presence of Bose-Hubbard interactions, the Hamiltonian reads: $$\begin{aligned}
\label{eq:bkmhModel}
H =
& - t_1 \!\!\sum\limits_{\sigma, \Braket{ij}} [b^\dag_{\sigma,{\bm{r}}_i} b_{\sigma,{\bm{r}}_j} + {\textrm{h.c.}}]
+ i t_2 \!\!\!\!\sum\limits_{\sigma, \Braket{\Braket{ik}}}
\!\!\!\! \nu^\sigma_{ik} [ b^\dag_{\sigma,{\bm{r}}_i}
b_{\sigma,{\bm{r}}_k} - {\textrm{h.c.}}]
\notag \\
& + \frac{U}{2} \sum\limits_{\sigma,i} n_{\sigma, {\bm{r}}_i}
(n_{\sigma, {\bm{r}}_i} - 1) + U_{\uparrow \downarrow}
\sum\limits_{i} n_{\uparrow, {\bm{r}}_i} n_{\downarrow, {\bm{r}}_i}
\;.\end{aligned}$$ Here, $b^\dag_{\sigma,{\bm{r}}_i} (b_{\sigma,{\bm{r}}_i})$ are creation (annihilation) operators at site $i$ of the honeycomb lattice, and $n_{\sigma,{\bm{r}}_i} = b^\dag_{\sigma,{\bm{r}}_i}
b_{\sigma,{\bm{r}}_i}$ is the density operator. $t_1\ \text{(resp. } t_2$) is the amplitude of hopping to the first (resp. second) neighbors and $\nu^\uparrow_{ik} = -\nu^\downarrow_{ik} = 1\ (\text{resp. } -1)$ for hoppings corresponding to a left-turn (resp. right-turn) on the honeycomb lattice. We assume a filling of one boson per site $\Braket{n_{\uparrow,{\bm{r}}_i} + n_{\downarrow,{\bm{r}}_i}} = 1$. The Haldane model [@Haldane1988] for spinless fermions has been realized through Floquet engineering in cold atoms [@Jotzu2014]. Similarly, spin-orbit models have been proposed in optical lattices setups [@KennedyEtAl2013; @StruckEtAl2013; @Yan2015] and experimentally achieved with photons [@HafeziEtAl2011; @SalaEtAl2015; @LuJoannopoulosSoljacic2014; @LeHurEtAl2016]. All the ingredients required for a successful implementation of are thus available.
.[]{data-label="fig:0"}](0.pdf){width="39.00000%"}
*I. B-DMFT on BKMH model.* The ground-state phase diagram of the BKMH model obtained from B-DMFT [@GeorgesKotliarKrauthRozenberg1996; @ByczukVollhardt2008; @HuTong2009; @HubenerSnoekHofstetter2009; @AndersGullPolletTroyerWerner2010] is shown in Fig. \[fig:0\](b). In order to address unusual states that break translational symmetry, we use real-space B-DMFT [@LiHof; @HeLiHof; @HeJiHof; @suppMaterial]. Local effective problems represented by the Anderson impurity model are solved using exact diagonalization [@suppMaterial]. As found for the bosonic Haldane model with same filling [@VasicEtAl2015], three phases are competing: a uniform superfluid (SF), a chiral superfluid (CSF) and a Mott insulator (MI) (they are sorted out from the behaviors of the order parameter $\braket{b_{\sigma,{\bm{r}}_i}}$ and the local currents $J^\sigma_{ij} = {\mathfrak{Im}} \braket{b^\dag_{\sigma,{\bm{r}}_i}
b_{\sigma,{\bm{r}}_j}}$ [@suppMaterial]).
We now focus on the MI phase. As shown in Fig. \[fig:0\](b), the system enters the Mott phase when intra-species ($U$) and inter-species ($U_{\uparrow \downarrow}$) interactions become strong enough. Applying standard perturbation theory [@KuklovSvistunov2003], one rewrites the Hamiltonian in terms of pseudo spin-$1/2$ operators $S^+_{{\bm{r}}_i} = S^x_{{\bm{r}}_i} + i S^y_{{\bm{r}}_i} =
b^\dag_{\uparrow,{\bm{r}}_i} b_{\downarrow,{\bm{r}}_i}$, $S^-_{{\bm{r}}_i} = S^x_{{\bm{r}}_i} - i S^y_{{\bm{r}}_i} =
b^\dag_{\downarrow,{\bm{r}}_i} b_{\uparrow,{\bm{r}}_i}$ and $S^z_{{\bm{r}}_i} = ( n_{\uparrow,{\bm{r}}_i} - n_{\downarrow,{\bm{r}}_i}
) / 2$ as follows: $$\begin{aligned}
\label{eq:xyModel}
H =
& - \sum\limits_{\Braket{ij}} \left[ J_1 \left( S_{{\bm{r}}_i}^+ S_{{\bm{r}}_j}^-
+ {\textrm{h.c.}}\right) - K_1 S_{{\bm{r}}_i}^z S_{{\bm{r}}_j}^z \right]
\notag \\
& + \sum\limits_{\Braket{\Braket{ik}}} \left[ J_2 \left( S_{{\bm{r}}_i}^+ S_{{\bm{r}}_k}^-
+ {\textrm{h.c.}}\right) + K_2 S_{{\bm{r}}_i}^z S_{{\bm{r}}_k}^z \right]
\;,\end{aligned}$$ where $J_i = t_i^2 / U_{\uparrow \downarrow}$ and $K_i = t_i^2 \left( 1/U_{\uparrow \downarrow} - 2/U \right)$. We observe that the spin-$1/2$ frustrated XY model is realized when $U = 2 U_{\uparrow \downarrow}$ (for which $K_i = 0$). Frustration is associated with the positive sign of the $J_2$-term, which combines the sign of the bosonic exchange and the phase of $\pi$ accumulated in the hoppings between second neighbors. The fermionic Kane-Mele model does not include such frustrating terms [@YoungLeeKallin2008; @RachelLehur2010]. The properties of this effective XY model depend only on the ratio $ J_2 / J_1 = \left( t_2 / t_1 \right)^2 $. In the classical limit, a coplanar ansatz [@RastelliTassiReatto1979; @FouetSindzingreLhuillier2001; @suppMaterial] provides the following phase diagram: the ferromagnetic phase is stable for $J_2 / J_1 \leq 1/6$, above which degenerate incommensurate spiral waves become energetically favoured. Their wave-vectors leave on closed contours in the Brillouin zone. In the case of the Heisenberg model, quantum fluctuations were predicted to lift this degeneracy via an order by disorder mechanism [@MudlerEtAl2010].
![Results of the B-DMFT for different values of $\left( t_2/t_1 \right)^2 = J_2/J_1$ for $h_z/U = 10^{-3}$, $U_{\uparrow \downarrow}/U = 0.5, t_1/U = 0.025$ on a lattice of 24 sites. **(a-d)** Different spin configurations. The color palette gives $\braket{S^z_{{\bm{r}}_i}}$, while arrows depict ordering in the $xy$-plane. **(a)** Uniform state with FM ordering; **(b)** CSS (chiral spin state) with no coplanar order; **(c)** A configuration of spiral states, in which each pseudo spin is aligned with only one of its three first neighbors and anti-aligned with two of its six second neighbors; **(d)** A $120^\circ$ configuration. **(e)** Absolute value of $\left|\braket{S^z_{{\bm{r}}_i}}\right|$. For each ratio $(t_2/t_1)^2$ we plot the result for all 24 sites and compare it to the classical solution. “Pentagons” mark results presented in (a-d). Note that for finite values of $h_z$ the border between the $120^\circ$ Mott state and CSF is slightly shifted in favour of the Mott state.[]{data-label="fig:1"}](1.pdf){width="46.00000%"}
Deviations from this classical picture are already captured by B-DMFT in the BKMH model. In Fig. \[fig:1\](a-d), we study the local coplanar spin ordering (arrows), in the presence of an external staggered magnetic field $h_z$, breaking the parity ${\mathcal{P}}$ symmetry (reflection which maps the sublattice $A$ to the sublattice $B$): $$\label{eq:perturbationHz}
H_z
= h_z \Big( \sum\limits_{i\in{A}} S^z_{{\bm{r}}_i}
-\sum\limits_{j\in{B}} S^z_{{\bm{r}}_j} \Big)
\;.$$ It corresponds to a staggered chemical potential in the boson language and we will understand its role hereafter. We directly infer some of the ordered phases: at low $ J_2 / J_1 $, all spins are aligned in a ferromagnetic (FM) order, while at large $ J_2 / J_1 $, we recover a $120^\circ$ spiral order. For $U_{\uparrow \downarrow}/U = 0.5, t_1/U = 0.025$ in the range $0.36 \lesssim J_2/J_1 \lesssim 1.23$ we observe a different configuration of spiral waves (Fig. \[fig:1\](c)). In addition, we find an exotic intermediate regime when $0.25 \lesssim J_2/J_1 \lesssim 0.36$ (we notice that positions of phase boundaries are affected by $h_z$), characterized by a chiral spin state (CSS) (this definition will be justified later) with no coplanar magnetic order (Fig. \[fig:1\](b)). This is reminiscent of the debated intermediate phase found in numerical studies on the XY spin model [@VarneySunGalitskiRigol2011; @VarneySunGalitskiRigol2012; @CarrasquillaEtAl2013; @CioloEtAl2014; @NakafujiIchinose2017; @ZhuHuseWhite2013; @ZhuWhite2014; @BishopLiCampbell2014]. On one hand, density matrix renormalization group [@ZhuHuseWhite2013; @ZhuWhite2014] and coupled cluster method [@BishopLiCampbell2014] results evidenced an antiferromagnetic Ising ordering along the $z$-axis, breaking ${\mathcal{P}}$ while preserving translational invariance. On the other hand, this observation was not reported in ED [@VarneySunGalitskiRigol2011; @VarneySunGalitskiRigol2012] nor variational Monte-Carlo [@CarrasquillaEtAl2013; @CioloEtAl2014; @NakafujiIchinose2017] analyses, raising questions about the exact nature of this intermediate phase.
Mapping the model onto a fermionic one and performing a mean-field analysis [@SedrakyanGlazmanKamenev2015; @suppMaterial], it was proposed that an intermediate frustration stabilizes a phase with spontaneously broken parity ${\mathcal{P}}$ and time-reversal ${\mathcal{T}}$ symmetries. This phase is characterized by antiferromagnetic correlations and ChS fluxes staggered within the unit cell as in the celebrated Haldane model [@Haldane1988] and the authors suggested that it realizes the chiral spin liquid state of Kalmeyer-Laughlin [@KalmeyerLaughlin1987; @KalmeyerLaughlin1989]. In this context, we plot in Fig. \[fig:1\](e), the response for the magnetization $\braket{S^z_{{\bm{r}}_i}}$ with respect to the field $h_z$. All phases except the CSS are characterized by a trivial response to the perturbation: $\braket{S^z_{{\bm{r}}_i}} \sim h_z$, whereas $\braket{S^z_{{\bm{r}}_i}}$ is strongly fluctuating in the CSS (however we do not observe spontaneous symmetry breaking with B-DMFT). These results cannot be explained in the context of a simple coplanar ansatz, but could be related to a breaking of the degeneracy between two mean-field solutions in the ChS field theory description [@suppMaterial].
*II. ED on frustrated XY model.* We complete the study of the effective frustrated XY model using ED and previously unaddressed probes such as the responses to ${\mathcal{P}}$ and ${\mathcal{T}}$ breaking perturbations and the topological description of the ground-state. We consider lattices of $24-32$ sites, with periodic boundary conditions, and fixed total magnetization $S^z_\textrm{Tot} = 0$ if not stated otherwise. First, we determine the phase boundaries using the fidelity metric [@ZanardiPaunkovic2006; @ShiJian2010; @VarneyEtAl2010; @suppMaterial] $g$. The phase diagram of the XY model deduced from the ED calculations is given in Fig. \[fig:2\](a). In agreement with the B-DMFT analysis and previous numerical studies, we observe three phase transitions at $J_2/J_1 \approx 0.21, 0.36$ and $1.32$. Small deviations from the B-DMFT results could be due to a finite size of ED clusters or non-perturbative interaction effects ($XY$ model does not describe correctly the physics of the Mott phase when $t_i / U$ are not small enough). The nature of the phases detected with the ED is verified by looking at the coplanar static structure factor $$\label{eq:sFactorSpiral}
S_\textrm{Spiral}\left( {\bm{q}} \right) = 2 \sum\limits_{i, j \in A}
e^{i{\bm{q}} \cdot \left( {\bm{r}}_i - {\bm{r}}_j \right)}
\braket{S^x_{{\bm{r}}_i} S^x_{{\bm{r}}_j}} \;.$$ Spiral waves display a maximum of $S_\textrm{Spiral}({\bm{q}})$ at some wave-vector(s) ${\bm{q}}$ in the first Brillouin zone. In the bosonic language, this is interpreted as a macroscopic occupation of the corresponding momentum state(s). We observe [@suppMaterial] that the phase in the region $J_2/J_1 \lesssim 0.21$ corresponds to the FM order since $S_\textrm{Spiral}\left( {\bm{q}} \right)$ has a peak at ${\bm{q}} = {\bm{\Gamma}}$. The phase at $0.36 \lesssim J_2/J_1 \lesssim 1.32$ corresponds to a spiral wave with collinear order (structure factor has maxima at three ${\bm{M}}$ points) as expected from the order by disorder mechanism. At $1.32 \lesssim J_2/J_1$ the ground-state is the $120^\circ$ order spiral wave (structure factor has a peak at two Dirac points ${\bm{K}}$). In the intermediate frustration regime ($0.21 \lesssim J_2/J_1 \lesssim 0.36$) the coplanar static structure factor is flat in the reciprocal space and we expect the ground-state to be disordered in the $xy$-plane. Notice that the ground-state in all phases is located in the same sector of the total momentum at point ${\bm{\Gamma}}$. Based on the ChS field theory predictions, the order by disorder arguments and numerical observations, the CSS – collinear order and collinear order – $120^\circ$ order phase transitions are expected to be of the first order, whereas the FM – CSS phase transition – of the second order.
![**(a)** Phase diagram of the frustrated XY model from ED. **(b-d)** Variation of the observables with the dimensionless parameter $J_2/J_1$ for different values of $h_z$, with $J'_2 = 0.01 J_1$, on a lattice of $6 \times 2$ unit cells. **(b)** Difference of the average Ising magnetization on two sublattices $m$. **(c)** Scalar spin chirality $\chi$. **(d)** Pseudo-spin density wave structure factor $S_\text{PSDW}({\bm{\Gamma}})$. **(e)** Schematic representation of the perturbation term $H_{J'_2}$. []{data-label="fig:2"}](2.pdf){width="40.00000%"}
As for the B-DMFT study, we analyze the linear response to external perturbations breaking ${\mathcal{P}}$ and ${\mathcal{T}}$ symmetries. We are interested in the relative magnetization between the two sublattices $m = \Braket{m_{{\bm{r}}_i}} = \Braket{ S^z_{{\bm{r}}_i} -
S^z_{{\bm{r}}_i + {\bm{u}}_3}}$, as well as the scalar spin chirality $\chi = \Braket{{\bm{S}}_{{\bm{r}}_i} \cdot \left( {\bm{S}}_{{\bm{r}}_i +
{\bm{u}}_1} \times {\bm{S}}_{{\bm{r}}_i + {\bm{u}}_2}
\right)}$. Here we suppose that $i \in A$ and ${\bm{u}}_i$ are vectors between first neighbor sites defined in Fig. \[fig:0\](a). When calculating the chirality $\chi$, we add a perturbation corresponding to the second-neighbor hopping of the Haldane model, of amplitude $J'_2$ and phase $\pi/2$ (as shown in Fig. \[fig:2\](e)): $$\label{eq:perturbationDj2}
H_{J'_2} = J'_2 \sum\limits_{\Braket{\Braket{ik}}} \big( e^{\pm i
\pi/2} S_{{\bm{r}}_i}^+ S_{{\bm{r}}_k}^- + {\textrm{h.c.}}\big)
\;.$$ We are interested in the limit $h_z, J'_2 \ll J_1$. Results of the ED calculations are presented in Figs. \[fig:2\](b-c). The CSS reveals itself by sharp responses to such external fields. Moreover, the renormalized quantities $m / h_z$ and $\chi / (h_z J'_2)$ tend to diverge in weak-coupling limit, giving a strong indication for spontaneous symmetry breaking. This justifies our definition of the CSS, which properties can be observed experimentally by tracking on-site populations of bosons $n_{\sigma, {\bm{r}}_i}$ and currents $J^\sigma_{ij} = {\mathfrak{Im}}\Braket{b^\dag_{\sigma, {\bm{r}}_i} b_{\sigma,
{\bm{r}}_j}}$[@Atala2014]. One can probe the antiferromagnetic ordering without breaking ${\mathcal{P}}$ and ${\mathcal{T}}$ by calculating the pseudo-spin density wave (PSDW) structure factor [@VarneySunGalitskiRigol2011; @VarneySunGalitskiRigol2012]: $$ S_\textrm{PSDW}({\bm{q}}) =\!\! \sum\limits_{i,j} e^{i{\bm{q}} \cdot \left(
{\bm{r}}_i - {\bm{r}}_j \right)}\Braket{m_{{\bm{r_i}}} m_{{\bm{r_j}}}}.$$ We observe in Fig. \[fig:2\](d) that $S_\textrm{PSDW}({\bm{q}})$ has a peak at ${\bm{q}} = \bm{\Gamma}$ in the intermediate frustration regime. These features are hardly affected by moderate Ising interactions $K_i/J_1 \sim 0.1$ in Eq. [@Li2014].
The observed spin configuration of the CSS could describe the chiral spin liquid of Kalmeyer and Laughlin [@KalmeyerLaughlin1987; @KalmeyerLaughlin1989]. Yet, we know that chiral spin liquids are characterized by a topological degeneracy in the thermodynamic limit on a compact space with genus $G$ [@WenWilczekZee1989; @Wen1989; @Wen1995]. This property can be checked using ED in a system with periodic boundaries: as $G = 1$ for a torus, one should have a four-fold degenerate ground-state with two topological degeneracies per chirality sector. Still, because of finite size effects, one only expects an approximate degeneracy in simulations.
In Fig. \[fig:3\](a-b), we show the low-energy spectrum as a function of $J_2/J_1$, resolved in different sectors of total momentum ${\bm{Q}}$. As mentioned previously, the ground-state always belongs to the sector ${\bm{Q}} = \bm{\Gamma}$. In the intermediate frustration regime, we clearly observe the onset of a doubly-degenerate ground-state manifold, well separated from higher energy states. The first excited state has the same momentum ${\bm{Q}} = \bm{\Gamma}$, but lies in the opposite sector of spin-inversion symmetry $S^z_{{\bm{r}}_i} \rightarrow -S^z_{{\bm{r}}_i}$ or reflection symmetry (that coincides with ${\mathcal{P}}$) for some particular lattices. Low-lying excited state also moves away in energy when the perturbations $H_z$ and $H_{J'_2}$ are switched on.
![ED calculations of the low energy spectra as a function of $J_2/J_1$ **(a)** on a lattice of $4 \times 3$ unit cells for various $S^z_\text{Tot}$; **(b)** on a lattice of $4 \times 4$ unit cells in the $S^z_\text{Tot} = 0$ sector only. **(c)** Low energy spectrum as a function of the twist angle $\theta_1$ for $J_2/J_1 = 0.3$ and $\theta_2 = 0$ on a lattice of $4 \times 3$ unit cells. **(d)** Berry curvature calculated using the non-abelian formalism resulting in a vanishing Chern number shown for $J_2/J_1 = 0.3$, $h_z/J_1 = J_2'/J_1 = 0.02$ on a lattice of $4 \times 3$ unit cells.[]{data-label="fig:3"}](3.pdf){width="48.00000%"}
We probe the robustness of the low energy quasi-degenerate state sector by performing the Laughlin’s gedanken experiment and pumping a quantum of magnetic flux through one of the non-trivial loops of the torus [@Laughlin1981; @Laughlin1983; @Thouless1989]. Numerically, this is achieved using twisted boundary conditions in a translational symmetry preserving manner. The results are given in Fig. \[fig:3\](c). We observe that the same states in the sector ${\bm{Q}} = \bm{\Gamma}$ are non-trivially gapped for all twists. For a pumping of a single flux quantum we could not observe a crossing of states in the ground-state manifold, that however does not imply that the manifold is topologically trivial [@WangGuCongSheng2011; @HickeyCincioParamekanti2016; @KumarChanglaniClarkFradkin2016]. The topological nature of the ground-state manifold is unambiguously determined by calculating the Chern number [@NiuThoulessWu1985; @Kohmoto1985; @Hatsugai2004; @Hatsugai2005]: $$\begin{aligned}
\label{eq:chernNumber}
C = \frac{1}{2\pi} \int\limits_{0}^{2\pi} \int\limits_{0}^{2\pi}
B(\theta_1, \theta_2) \textrm{d}\theta_1 \textrm{d}\theta_2 \;.\end{aligned}$$ Here $\theta_1$ and $\theta_2$ are two angles of twisted boundary conditions and $B(\theta_1, \theta_2)$ is the Berry curvature [@Berry1984]. We notice that two phases $\theta_i$ ($i=1,2$) introduced in the spin language would correspond to four phases $\theta^\sigma_i$ in the language of bosons of the BKMH model, for which the spin component $\theta^\uparrow_i - \theta^\downarrow_i = \theta_i$ is fixed and the $U(1)$ component $\theta^\uparrow_i + \theta^\downarrow_i$ is free [@FuKane2006]. Since the two quasi degenerate ground-states lie in the same symmetry sector and cannot be separated unless twists are trivial (reflection and spin-inversion symmetry can not be used with twisted boundary conditions), we evaluate the Berry curvature using the gauge-invariant non-abelian formulation [@YuQiBernevigFangDai2011; @ShapourianClark2016; @KrishnaHiteshClarkFradkin2016]: $B(\theta_1, \theta_2) \delta \theta_1 \delta \theta_2 = {\mathfrak{Im}} \ln
{\mathfrak{Det}} \left( {\mathcal{M}}(\theta_1, \theta_2) \right)$, where elements of the matrix ${\mathcal{M}}$ are obtained as follows: $$\begin{aligned}
\label{eq:berryCurvature}
{\mathcal{M}}_{ij}(\theta_1, \theta_2)
&=
\Braket{\phi_i(\theta_1, \theta_2) | \phi_{\mu_1}(\theta_1 + \delta
\theta_1, \theta_2)}
\notag \\
& \times
\Braket{\phi_{\mu_1}(\theta_1 + \delta \theta_1, \theta_2) | \phi_{\mu_2}(\theta_1 + \delta
\theta_1, \theta_2 + \delta \theta_2)}
\notag \\
& \times
\Braket{\phi_{\mu_2}(\theta_1 + \delta \theta_1, \theta_2 + \delta
\theta_2) | \phi_{\mu_3}(\theta_1, \theta_2 + \delta \theta_2)}
\notag \\
& \times
\Braket{\phi_{\mu_3}(\theta_1, \theta_2 + \delta \theta_2) |
\phi_j(\theta_1, \theta_2)}
\;.\end{aligned}$$ Here $\delta \theta_1$ and $\delta \theta_2$ refer to the numerical mesh along the $\theta_1$ and $\theta_2$. $i, j, \mu_i = 1, 2$ are indices of states $\Ket{\phi_1}$ and $\Ket{\phi_2}$ in the ground-state manifold and the summation over $\mu_i$ is implicit. In Fig. \[fig:3\](d), we show a typical shape of the Berry curvature. We find that the Chern number is zero in the intermediate frustration regime. This result suggests that the intermediate phase in the frustrated XY model is most likely to be a CSS with no topological order, as suggested in Refs. and not the Kalmeyer-Laughlin state, with gauge fluctuations beyond the mean-field solution making the phase topologically trivial as in the fermionic Kane-Mele model case [@RachelLehur2010; @WuRachelLiuLehur2012; @Hohenadler2012].
To conclude, we studied the phase diagram of the bosonic Kane-Mele-Hubbard model on the honeycomb lattice. We have shown that an effective frustrated XY model appears in the Mott insulator phase. This model possesses an intermediate frustration regime with a non-trivial chiral spin state, which breaks both ${\mathcal{P}}$ and ${\mathcal{T}}$ symmetries. It displays a finite scalar spin chirality order and an antiferromagnetic ordering between first-neighbor sites, while remaining translationally invariant. Measuring the Chern number associated with this state reveals its non-topological nature.
We thank Loïc Herviou, Grégoire Misguich, Stephan Rachel, Cécile Repellin, Tigran Sedrakyan for insightful discussions. This work has also benefitted from discussions at CIFAR meetings in Canada and Société Française de Physique.
Support by the Deutsche Forschungsgemeinschaft via DFG FOR 2414, DFG SPP 1929 GiRyd, and the high-performance computing center LOEWE-CSC is gratefully acknowledged. This work was supported in part by DAAD (German Academic and Exchange Service) under project BKMH. I. V. acknowledges support by the Ministry of Education, Science, and Technological Development of the Republic of Serbia under projects ON171017 and BKMH, and by the European Commission under H2020 project VI-SEEM, Grant No. 675121. Numerical simulations were partly run on the PARADOX supercomputing facility at the Scientific Computing Laboratory of the Institute of Physics Belgrade. K. L. H. acknowledges support from Labex PALM.
**Supplemental Material: Emergent Chiral Spin State in the Mott Phase of a Bosonic Kane-Mele-Hubbard Model**
B-DMFT details
==============
For completeness, in this Section we briefly describe the B-DMFT method along the lines of references . In particular, in order to be able to address exotic states that break translational invariance, we implement real-space B-DMFT . The essence of DMFT is mapping of the full lattice model onto a set of local models whose parameters are determined through a self-consistency condition. The self-consistency is imposed on the level of single–particle Green’s functions that can be written in the Nambu notation as $$G_{ij}(\tau, \eta)\equiv G_{ij}(\tau-\eta)= -T_{\tau, \eta}\left\langle\left(\begin{array}{cccc}
b_{\uparrow,{\bm{r}}_i}(\tau) b_{\uparrow,{\bm{r}}_j}^{\dagger} (\eta) & b_{\uparrow,{\bm{r}}_i}(\tau) b_{\uparrow,{\bm{r}}_j}(\eta) & b_{\uparrow,{\bm{r}}_i}(\tau) b_{\downarrow,{\bm{r}}_j}^{\dagger}(\eta) & b_{\uparrow, {\bm{r}}_i}(\tau) b_{\downarrow,{\bm{r}}_j}(\eta)\\
b_{\uparrow,{\bm{r}}_i}^{\dagger}(\tau) b_{\uparrow,{\bm{r}}_j}^{\dagger} (\eta) & b_{\uparrow,{\bm{r}}_i}^{\dagger}(\tau) b_{\uparrow,{\bm{r}}_j}(\eta) & b_{\uparrow,{\bm{r}}_i}^{\dagger}(\tau) b_{\downarrow, {\bm{r}}_j}^{\dagger}(\eta) & b_{\uparrow, {\bm{r}}_i}^{\dagger}(\tau) b_{\downarrow, {\bm{r}}_j}(\eta)\\
b_{\downarrow,{\bm{r}}_i}(\tau) b_{\uparrow, {\bm{r}}_j}^{\dagger}(\eta)& b_{\downarrow, {\bm{r}}_i}(\tau) b_{\uparrow, {\bm{r}}_j}(\eta) & b_{\downarrow, {\bm{r}}_i}(\tau)b_{\downarrow, {\bm{r}}_j}^{\dagger}(\eta) & b_{\downarrow, {\bm{r}}_i}(\tau) b_{\downarrow, {\bm{r}}_j}(\eta)\\
b_{\downarrow,{\bm{r}}_i}^{\dagger}(\tau) b_{\uparrow, {\bm{r}}_j}^{\dagger}(\eta)& b_{\downarrow, {\bm{r}}_i}^{\dagger}(\tau) b_{\uparrow, {\bm{r}}_j}(\eta) & b_{\downarrow, {\bm{r}}_i}^{\dagger}(\tau)b_{\downarrow, {\bm{r}}_j}^{\dagger}(\eta) & b_{\downarrow, {\bm{r}}_i}^{\dagger}(\tau) b_{\downarrow, {\bm{r}}_j}(\eta)
\end{array}\right)\right\rangle \;.$$ In the following we express the Green’s functions in terms of Matsubara frequencies $\omega_n=2\pi n/\beta$, where $\beta $ is the inverse temperature (in the zero temperature limit $\beta\rightarrow\infty$) and $G_{ij}(i\omega_n) = \int\, d{\tau} \exp(i \omega_n \tau)
G_{ij}(\tau)$.
In real-space B-DMFT we decompose the full lattice problem into a set of local single-site effective problems. The approximation is such that local correlations are fully taken into account, while non-local correlations are treated at the mean-field level. At each site $i$, we attach a bath described by orbital degrees of freedom. The effective local Hamiltonian is given by a bosonic Anderson impurity (AI) model $$\begin{aligned}
\mathcal{H}^\mathrm {AI}_i
&=& \sum_{l=0}^L \left[\varepsilon_l
a_l^{\dagger} a_l +
\sum_{\sigma}\left(V_{l, \sigma}
a_l^{\dagger} b_{\sigma, {\bm{r}}_i} +
V_{l, \sigma}^* a_l b_{\sigma,
{\bm{r}}_i}^{\dagger} + W_{l, \sigma} a_l
b_{\sigma, {\bm{r}}_i} + W_{l, \sigma}^*
a_l^{\dagger} b_{\sigma,
{\bm{r}}_i}^{\dagger}\right)\right]
\nonumber\\
&+& \sum_{\sigma}\left(-\psi_{\sigma, {\bm{r}}_i}^{\mathrm {AI}*} b_{\sigma,
{\bm{r}}_i} - \psi_{\sigma, {\bm{r}}_i}^{\mathrm {AI}} b^{\dagger}_{\sigma, {\bm{r}}_i} +
\frac{U}{2} n_{\sigma, {\bm{r}}_i} ( n_{\sigma, {\bm{r}}_i}-1)-\mu_{\sigma}
n_{\sigma, {\bm{r}}_i}\right) + U_{\uparrow \downarrow} n_{\uparrow, {\bm{r}}_i} n_{\downarrow,
{\bm{r}}_i},
\label{eq:Anderson}\end{aligned}$$ where the index $l$ labels the Anderson orbitals with energies $\varepsilon_l$ and we allow for complex values of the Anderson parameters $V_{l, \sigma}$ and $W_{l, \sigma}$ that couple orbital degrees of freedom with impurity atoms. We use $L=4$; we check that results are the same for $L=5$ and $6$. Local interaction terms proportional to $U$ and $U_{\uparrow \downarrow}$ come directly from the initial lattice model and, as we work in the grand canonical ensemble, we introduce chemical potentials $\mu_{\sigma}\equiv\mu$. We define hybridization functions of the Anderson impurity model as $$\begin{aligned}
\Delta_{11}^{\nu \mu}(i \omega_n)
&=& \sum_l\left(\frac{V_{l, \nu
}^*V_{l, \mu
}}{\epsilon_l-i\omega_n}+\frac{W_{l, \nu
}^*W_{l, \mu
}}{\epsilon_l+i\omega_n}\right),\\
\Delta_{22}^{\nu \mu}(i \omega_n)
&=& \sum_l\left(\frac{W_{l, \mu
}^*W_{l, \nu
}}{\epsilon_l-i\omega_n}+\frac{V_{l, \mu
}^*V_{l, \nu
}}{\epsilon_l+i\omega_n}\right),\\
\Delta_{12}^{\nu \mu}(i \omega_n)
&=& \sum_l\left(\frac{V_{l, \nu
}^*W^*_{l,\mu
}}{\epsilon_l-i\omega_n}+\frac{V_{l, \mu
}^*W^*_{l, \nu
}}{\epsilon_l+i\omega_n}\right),\\
\Delta_{21}^{\nu \mu}(i \omega_n)
&=& \sum_l\left(\frac{V_{l, \mu}W_{l, \nu}
}{\epsilon_l-i\omega_n}+\frac{V_{l,\nu
} W_{l, \mu
}}{\epsilon_l+i\omega_n}\right),
\label{eq:hyb}\end{aligned}$$ and introduce a $4 \times 4 $ matrix $\Delta (i \omega_n) $ as $$\Delta (i \omega_n) \equiv \left(
\begin{array}{cccc}
\Delta_{11}^{\uparrow \uparrow} & \Delta_{12}^{\uparrow \uparrow} & \Delta_{11}^{\uparrow \downarrow}& \Delta_{12}^{\uparrow \downarrow}\\
\Delta_{21}^{\uparrow \uparrow} & \Delta_{22}^{\uparrow \uparrow} & \Delta_{21}^{\uparrow \downarrow}& \Delta_{22}^{\uparrow \downarrow}\\
\Delta_{11}^{\downarrow \uparrow}& \Delta_{12}^{\downarrow \uparrow} & \Delta_{11}^{\downarrow \downarrow} & \Delta_{12}^{\downarrow \downarrow}\\
\Delta_{21}^{\downarrow \uparrow}& \Delta_{22}^{\downarrow \uparrow} & \Delta_{21}^{\downarrow \downarrow} & \Delta_{22}^{\downarrow \downarrow}
\end{array}
\right).$$ The following relations for the hybridization functions hold true: $$\begin{aligned}
\Delta_{22}^{\uparrow\uparrow}(i\omega_n) =
\Delta_{11}^{{\uparrow\uparrow}*}(i\omega_n), \quad
\Delta_{21}^{\uparrow\uparrow}(i\omega_n) =
\Delta_{12}^{{\uparrow\uparrow}*}(i\omega_n), \quad
\Delta_{22}^{\downarrow\downarrow}(i\omega_n) =
\Delta_{11}^{{\downarrow\downarrow}*}(i\omega_n), \quad
\Delta_{21}^{\downarrow\downarrow}(i\omega_n) =
\Delta_{12}^{{\downarrow\downarrow}*}(i\omega_n), \nonumber\\
\Delta_{22}^{\uparrow\downarrow}(i\omega_n) =
\Delta_{11}^{{\uparrow\downarrow}*}(i\omega_n), \quad
\Delta_{21}^{\uparrow\downarrow}(i\omega_n) =
\Delta_{12}^{{\uparrow\downarrow}*}(i\omega_n), \quad
\Delta_{22}^{\downarrow\uparrow}(i\omega_n) =
\Delta_{11}^{{\downarrow\uparrow}*}(i\omega_n), \quad
\Delta_{21}^{\downarrow\uparrow}(i\omega_n) =
\Delta_{12}^{{\downarrow\uparrow}*}(i\omega_n).\end{aligned}$$ The terms $\psi_{\sigma, {\bm{r}}_i}^{\mathrm{AI}}$ used in Eq. (\[eq:Anderson\]) incorporate a correction with respect to the mean–field result and they read : $$\begin{aligned}
\psi_{\uparrow,{\bm{r}}_i}^{\mathrm{ AI}}
&=& \sum_j t_{\uparrow, ij}\phi_{\uparrow, {\bm{r}}_j}-\phi_{\uparrow, {\bm{r}}_i}^*\Delta_{21}^{\uparrow\uparrow}(0)-\phi_{\downarrow, {\bm{r}}_i}^*\Delta_{21}^{\downarrow\uparrow}(0)-\phi_{\uparrow, {\bm{r}}_i} \Delta_{11}^{\uparrow\uparrow}(0)-\phi_{\downarrow, {\bm{r}}_i} \Delta_{11}^{\downarrow\uparrow}(0),\\
\psi_{\downarrow,{\bm{r}}_i}^{\mathrm{ AI}}
&=& \sum_j t_{\downarrow, ij}\phi_{\downarrow, {\bm{r}}_j}-\phi_{\uparrow,{\bm{r}}_i}^*\Delta_{21}^{\uparrow\downarrow}(0)-\phi_{\downarrow,{\bm{r}}_i}^*\Delta_{21}^{\downarrow\downarrow}(0)-\phi_{\uparrow, {\bm{r}}_i}
\Delta_{11}^{\uparrow\downarrow}(0)-\phi_{\downarrow, {\bm{r}}_i} \Delta_{11}^{\downarrow\downarrow}(0),\end{aligned}$$ where the condensate order parameters are defined as $$\phi_{\sigma, {\bm{r}}_i} = \langle b_{\sigma, {\bm{r}}_i}\rangle,
\label{eq:con_order}$$ and $t_{\sigma,ij} $ are hopping amplitudes of the two species defined in the initial lattice model.
By exact diagonalization of the local model (\[eq:Anderson\]) we obtain the values of the local Green’s functions. From here, the local self–energy is obtained from the local Dyson equation $$(G^{\mathrm{AI}})^{-1}_{ii} (i \omega_n) =
\left(
\begin{array}{cccc}
i \omega_n +\mu& & &\\
& -i \omega_n + \mu & &\\
& &i \omega_n +\mu & \\
& & & -i \omega_n + \mu
\end{array}
\right)+ \Delta(i\omega_n) - \Sigma_i^{\mathrm{AI}}.
\label{eq:localdysonequation}$$ The approximate real-space Dyson equation takes the following form: $$G^{-1}_{ij, \mathrm{latt}} (i \omega_n) = \left(
\begin{array}{cccc}
\!\!\left(i \omega_n +\mu \right)\delta_{ij}+ t_{\uparrow,ij}& & &\\
\! & \!\!\left(-i \omega_n + \mu \right)\delta_{ij}+t^*_{\uparrow, ij}\!\! &\!&\!\\
\! &\!&\!\!\left(i \omega_n \!+\!\mu \right)\delta_{ij}+ t_{\downarrow, ij}& \!\!\\
\! &\!&\!& \left(-i \omega_n \!+\! \mu \right)\delta_{ij}+t^*_{\downarrow, ij}\\
\end{array}\right)-\delta_{ij} \Sigma_i^{\mathrm{AI}},
\label{eq:realspacedyson}$$ where we approximate the self–energy by a local contribution from Eq. (\[eq:localdysonequation\]). The last step represents the main approximation of DMFT. Finally, we need a criterion to set the values of the parameters $\varepsilon_l$, $V_{l, \sigma}$ and $W_{l, \sigma}$ in Eq. (\[eq:Anderson\]). To this end, a condition is imposed on the hybridization functions (\[eq:hyb\]). These functions should be optimized such that the two Dyson equations, (\[eq:localdysonequation\]) and (\[eq:realspacedyson\]), yield the same values of the local Green’s functions. In practice, we iterate a self–consistency loop to fulfill this condition, starting from arbitrary initial values. At the same time we impose a simple self consistency on the local condensate order parameters $\phi_{\sigma, {\bm{r}}_i}$.
Once that the self-consistency is achieved and values of Anderson parameters $\varepsilon_l, V_{l, \sigma}$ and $W_{l, \sigma}$ are fixed, by solving the local model (\[eq:Anderson\]) we obtain results for local condensate order parameters (\[eq:con\_order\]) and the expectation values of the pseudo spin operators $$\begin{aligned}
\langle S^x_{{\mathbf r}_i} \rangle
&=& \langle b_{\uparrow,{\bm{r}}_i}^{\dagger} b_{\downarrow,{\bm{r}}_i}+b_{\downarrow,{\bm{r}}_i}^{\dagger} b_{\uparrow,{\bm{r}}_i}\rangle/2,
\label{eq:spin_order_x}\\
\langle S^y_{{\mathbf r}_i} \rangle
&=& i \langle b_{\uparrow,{\bm{r}}_i}^{\dagger} b_{\downarrow,{\bm{r}}_i}-b_{\downarrow,{\bm{r}}_i}^{\dagger} b_{\uparrow,{\bm{r}}_i}\rangle/2,
\label{eq:spin_order_y}\\
\langle S^z_{{\mathbf r}_i} \rangle
&=& \langle b_{\uparrow,{\bm{r}}_i}^{\dagger} b_{\uparrow,{\bm{r}}_i}-b_{\downarrow,{\bm{r}}_i}^{\dagger} b_{\downarrow,{\bm{r}}_i}\rangle/2.
\label{eq:spin_order_z}\end{aligned}$$ We work with a finite lattice consisting of 96 sites and periodic boundary conditions that provide a proper sampling of the Brillouin zone that includes its corners [@VarneyEtAl2010]. The values of the chemical potential terms in Eq. (\[eq:Anderson\]) are fixed to $\mu_{\sigma} = U_{\uparrow\downarrow}/2$.
![Color maps: Real-space distribution of condensate order parameters of the two bosonic species in (a) uniform superfluid (SF) ($t_1/U = 0.056, t_2/U = 0.005, U_{\uparrow\downarrow}/U = 0.5,
\mu/U_{\uparrow \downarrow} = 0.5$), and (b) chiral superfluid (CSF) ($t_1/U = 0.018, t_2/U = 0.032, U_{\uparrow\downarrow}/U = 0.5,
\mu/U_{\uparrow \downarrow} = 0.5$). Local condensate order parameters are aligned in the SF. In contrast, they exhibit $2\pi/3$ winding in the CSF. The “winding direction” is opposite for the two species and for the two sublattices, implying that for each sublattice the two species condense in the two different Dirac points. The choice of the Dirac points is opposite for the two sublattices. (c) The condensate order parameters as a function of $t_2/U$ for several values of $t_1/U$.[]{data-label="Fig:Figsf"}](7.pdf){width="\linewidth"}
Finite values of condensate order parameters (\[eq:con\_order\]) mark a superfluid phase, while vanishing values correspond to a Mott insulator state (MI). We further distinguish a uniform superfluid (SF), where the order parameters of the two species on both sublattices are aligned, Fig. \[Fig:Figsf\](a), and a chiral superfluid (CSF) with $2\pi/3$ winding of the order parameters, Fig. \[Fig:Figsf\](b). For the parameters studied in the paper, we find that the absolute values of the order parameters are the same for the two species on all lattice sites, yet for CSF state winding directions are opposite for the two species on the two sublattices, Fig. \[Fig:Figsf\](b). Moreover, in CSF phase condensate order parameters on the two sublattices and for the two species are determined up to a relative phase, Fig. \[Fig:Figsf\](b). We also expect that similarly to the case of the bosonic Haldane model [@VasicEtAl2015] the SF – CSF phase transition is of the first order, whereas the SF (CSF) – MI phase transition is of the second order.
In Fig. \[Fig:Figsf\](c) we plot absolute values of the order parameters (\[eq:con\_order\]) (which are uniform throughout the lattice) as functions of $t_2/U$ for several values of $t_1/U$. For the case of $t_1/U = 0$, we find a transition from the Mott state into the chiral superfluid state at $ t_2/U\approx 0.027$. At $t_1/U = 0.03$, the transition sets in at a slightly higher value $t_2/U \approx 0.0285$. The most interesting behavior is found for $t_1/U = 0.056$, where for small values of $t_2$ we find a uniform superfluid. With an increase in $t_2$, at $t_2/U \approx 0.0265 $ the Mott insulator state is reached due to competing effects of $t_1$ and $t_2$, and finally at $t_2/U \approx 0.0315$ the system becomes a chiral superfluid. These results are summarized in the phase diagram of BKMH model (Fig. 1(b) of the main text).
Different magnetic orderings within the Mott domain, as discussed in Fig. 1, are distinguished based on the order parameters defined in Eqs. (\[eq:spin\_order\_x\]) and (\[eq:spin\_order\_y\]). In Fig. 2 we show the results of a calculation on a 24-site lattice. We monitor magnetic ordering in $z$-direction marked by finite values of order parameter (\[eq:spin\_order\_z\]) that are introduced by a finite value of $h_z$ as defined in equation (\[eq:perturbationHz\]). We have checked that a four-fold increase in lattice size (96 vs. 24 lattice sites) introduces a shift in the position of the “intermediate region” borders of the order of $\Delta t_2/U \sim 2 \times 10^{-4} $ or less than $2\%$ in relative units.
Classical solution
==================
We consider an ansatz for the classical ($S \rightarrow \infty$) solution of the spin problem defined as follows: $$\label{eq:classicalAnsatz}
\bm{S}_{{\bm{r}}_i} = S
\begin{pmatrix}
\sin \left( \theta_\mu \right) \cos \left( \phi_{\mu, i} \right) \\
\sin \left( \theta_\mu \right) \sin \left( \phi_{\mu, i} \right) \\
\cos \left( \theta_{\mu} \right)
\end{pmatrix}
\;.$$ Here $\mu \in [A, B]$ is the sublattice index and a free parameter $\theta_{\mu}$ characterizes the orientation of the spin on the sublattice $\mu$ with respect to the $z$-axis. It verifies $0 \leq \theta_{\mu} \leq \pi$ ($\sin \theta_\mu$ is always positive). Similarly to the Refs. , we define phases $\phi_{A, i} = \bm{q}\cdot\bm{R}_i$ and $\phi_{B, i} = \bm{q}\cdot\bm{R}_i + \eta$, where ${\bm{q}}$ is the spiral wave vector and $\eta$ describes the relative orientation of spins on sublattices $A$ and $B$ at the same unit cell. The (anti-) ferromagnetic ordering between first-neighbor sites in the $XY$-plane is thus described by $\bm{q} = 0$, $\eta = 0(\pi)$ and $\theta_\mu = \pi/2$. The Ising antiferromagnetic ordering is defined by $\theta_A = 0$, $\theta_B = \pi$ and its ${\mathbb{Z}}_2$ symmetric solution $\theta_A = \pi$, $\theta_B = 0$.
Zero external magnetic field $h_z$
----------------------------------
We write the energy per spin in terms of parameters of the Hamiltonian $H$ in Eq. for $K_i = 0$: $$\begin{aligned}
\epsilon =
&
- J_1 S^2 \sin \theta_A \sin \theta_B \left[
\cos \eta + \cos \left( \eta - Q_1 \right) + \cos \left( \eta + Q_2 \right) \right]
\notag \\
&
+ J_2 S^2 \left( \sin^2 \theta_A + \sin^2 \theta_B \right) \left[
\cos Q_1 + \cos Q_2 + \cos \left( Q_1 + Q_2 \right) \right] \;.\end{aligned}$$ Here for simplicity we defined $Q_i = {\bm{q}} \cdot \bm{v}_i$ with ${\bm{v}}_i$ – 3 second-neighbor vectors. By minimizing the energy per spin with respect to all the parameters that we introduced, we obtain that only coplanar solutions with $\theta_\mu = \pi/2$ will survive. In this case we recover [@RastelliTassiReatto1979; @MudlerEtAl2010] $$\begin{aligned}
\label{eq:spiralWaveSolution}
& \cos \eta = \frac{2 J_2}{J_1} \left( 1 + \cos Q_1 + \cos Q_2 \right)
\;, \notag \\
& \sin \eta = \frac{2 J_2}{J_1} \left( \sin Q_1 - \sin Q_2 \right)
\;, \notag \\
& \cos Q_1 + \cos Q_2 + \cos \left( Q_1 + Q_2 \right) =
\frac{1}{2} \left( \frac{J_1^2}{4J_2^2} - 3 \right)
\;.\end{aligned}$$ The uniform solution at $\bm{q} = {\bm{\Gamma}}$ and $\eta = 0$ is valid until $J_2 / J_1 \leq 1/6$. Spiral waves solution is valid in the regime $J_2 / J_1 > 1/6$ for $J_1 \neq 0$. When two sublattices are decoupled ($J_1 = 0$), the solution corresponds to the $120^\circ$ order. The energy per spin of the uniform solution is $\epsilon_\text{cl} = - 3 S^2 \left( J_1 - 2 J_2 \right)$, whereas the energy corresponding to the spiral wave state is $\epsilon_\text{sp} = - S^2 J_1 \left( \frac{J_1}{4J_2} +
\frac{3J_2}{J_1} \right)$.
Effect of the external magnetic field $h_z$
-------------------------------------------
Now we are interested in the effect of the external magnetic field $h_z$ on the stabilization of the out-of plane (PSDW) solution. We calculate the energy per spin when the perturbation term $H_z$ of Eq. is added to the Hamiltonian: $$\begin{aligned}
\epsilon =
& - J_1 S^2 \sin \theta_A \sin \theta_B \left[
\cos \eta + \cos \left( \eta - Q_1 \right) + \cos
\left( \eta + Q_2 \right) \right]
\notag \\
& + J_2 S^2 \left( \sin^2 \theta_A + \sin^2
\theta_B \right) \left[
\cos Q_1 + \cos Q_2 + \cos \left( Q_1 + Q_2 \right) \right]
- \frac{h_z S}{2} \left( \cos \theta_A - \cos \theta_B \right)
\;.\end{aligned}$$ We suppose that the angle $\theta_\mu$ is close to $\pi/2$ (the solution is almost coplanar) for small values of $h_z$ and we perform the expansion in powers of $\tilde{\theta}_\mu = \pi/2 -
\theta_\mu$. At the first order in the expansion we observe that the coplanar degree of freedom and the degree of freedom along the $z$-axis become decoupled. Values of $\eta$, $Q_1$ and $Q_2$ correspond to the spiral wave solution and parameters $\tilde{\theta}_A$ and $\tilde{\theta}_B$ are deduced using the following relation: $$\begin{aligned}
& \tilde{\theta}_A + \tilde{\theta}_B = 0
\;, \notag \\
& \tilde{\theta}_A - \tilde{\theta}_B = \frac{h_z}{J_1 \left[ \cos \eta + \cos
\left( \eta - Q_1 \right) + \cos \left( \eta + Q_2 \right) \right]
- 2 J_2 \left[ \cos Q_1 + \cos Q_2 +
\cos \left( Q_1 + Q_2 \right) \right]}
\;.\end{aligned}$$ In the regime $J_2 / J_1 \leq 1/6$ we obtain $$\tilde{\theta}_A = - \tilde{\theta}_B = \frac{h_z}{6 \left( J_1 - 2 J_2 \right)}
\;,$$ and in the regime $J_2 / J_1 > 1/6$ $$\tilde{\theta}_A = - \tilde{\theta}_B = \frac{2 h_z J_2}{
\left( J_1^2 + 12 J_2^2 \right)}
\;.$$ We see thus that for the classical ansatz the linear response of the spin to the applied magnetic field $h_z$ is supposed to be small and of the order of $h_z$.
Mean-field solution and the ChS field theory description
========================================================
According to the Ref. one can preform a mapping of the spin problem onto the problem of spinless fermions coupled to ChS gauge fields [@Fradkin1989; @AnbjornSemenoff1989; @LopezRojoFradkin1994; @Misguichjolicoeurgirvin2001; @SunKumarFradkin2015]. At the mean-field level, the system is stabilized in the chiral spin state by forming the anti-ferromagnetic order and staggered ChS fluxes within the unit cell identical to the fluxes of the Haldane model [@Haldane1988]. This allowed authors of the Ref. to suggest that the resulting solution (that breaks spontaneously ${\mathcal{P}}$ and ${\mathcal{T}}$ symmetries) could be a chiral spin liquid state of Kalmeyer-Laughlin and deduce the phase boundaries, that were in good agreement with the numerical data [@VarneySunGalitskiRigol2011; @VarneySunGalitskiRigol2012; @CarrasquillaEtAl2013; @CioloEtAl2014; @NakafujiIchinose2017; @ZhuHuseWhite2013; @ZhuWhite2014; @BishopLiCampbell2014]. Below, we represent analytical arguments that lead to this suggestion.
Zero external magnetic field $h_z$
----------------------------------
The problem of the Eq. can be rewritten in the fermionic language using the following transformation: $$S^+_{{\bm{r}}_j} = c^\dag_{{\bm{r}}_j} e^{i\alpha_{{\bm{r}}_j}}, \quad
\alpha_{{\bm{r}}_j} = \sum\limits_{k \neq j} B_{jk} n_{{\bm{r}}_k}, \quad
n_{{\bm{r}}_k} = c^\dag_{{\bm{r}}_k} c_{{\bm{r}}_k} = S^z_{{\bm{r}}_k} + 1/2 \;.$$ Here $c^\dag_{{\bm{r}}_j}$ and $c_{{\bm{r}}_j}$ are fermionic creation and annihilation operators and $$B_{jk} = \text{arg}\left( \tau_k - \tau_j \right) =
{\mathfrak{Im}} \ln \left( \tau_k - \tau_j \right) \;,$$ with the complex number $\tau_j = x_{j} + i y_{j}$ associated to each point on the lattice defined by the vector $\bm{r}_j = x_{j}\bm{e}_x + y_{j}\bm{e}_y$. $B_{jk}$ could be interpreted as the angle that the vector $\bm{r}_k - \bm{r}_j$ forms with the $x$-axis. The Hamiltonian can now be rewritten as $$\begin{aligned}
H =& \left( -
J_1 \sum\limits_{\Braket{ij}} c^\dag_{{\bm{r}}_i}
e^{i\left(\alpha_{{\bm{r}}_i}-\alpha_{{\bm{r}}_j}\right)} c_{{\bm{r}}_j} +
J_2 \sum\limits_{\Braket{\Braket{ik}}} c^\dag_{{\bm{r}}_i}
e^{i\left(\alpha_{{\bm{r}}_i}-\alpha_{{\bm{r}}_k}\right)}
c_{{\bm{r}}_k} + \text{h.c.} \right)
\;.\end{aligned}$$
We introduce a vector field $\bm{A} \left( {\bm{r}}_k \right)$ defined as $$\Braket{\alpha_{{\bm{r}}_j} - \alpha_{{\bm{r}}_i}} =
\int\limits_{\bm{r}_i}^{\bm{r}_j} \textrm{d}\bm{r}_k
\cdot \bm{A} \left( {\bm{r}}_k \right) \;,$$ and a ChS magnetic field $\bm{B} \left( {\bm{r}}_i \right) = B \left( {\bm{r}}_i \right)
{\bm{e}}_z$ such that $$B \left( {\bm{r}}_i \right) = \text{curl} \bm{A} \left( {\bm{r}}_i
\right) = 2 \pi \Braket{n_{{\bm{r}}_i}} = 2 \pi n({\bm{r}}_i) \;.$$ We remove exponential string operators by introducing the $\delta$-function imposing a constraint on the ChS magnetic field through the Lagrange multiplier $A^0(\bm{r}_i)$: $$2\pi \delta\left(B(\bm{r}_i)/(2\pi) - n({{\bm{r}}_i})\right) =
\int { \textrm{d} A^0(\bm{r}_i)
\exp \left\lbrace i A^0(\bm{r}_i) \left[ B(\bm{r}_i)/(2\pi) -
n({{\bm{r}}_i}) \right] \right\rbrace }
\;.$$ We write down the resulting action $$\begin{aligned}
S = \int \textrm{d}t
& \left[
\sum\limits_i \bar{\psi}(\bm{r}_i) \left( i\partial_t -
A^0(\bm{r}_i) \right) \psi(\bm{r}_i) +
\frac{1}{2\pi} \sum\limits_i A^0(\bm{r}_i) B(\bm{r}_i)
\right.
\notag \\
& \left. -
J_1 \sum\limits_{\Braket{ij}} \bar{\psi}(\bm{r}_i) \psi(\bm{r}_j)
e^{i\left\langle\alpha_{{\bm{r}}_i}-\alpha_{{\bm{r}}_j}\right\rangle} +
J_2 \sum\limits_{\Braket{\Braket{ik}}}
\bar{\psi}(\bm{r}_i) \psi(\bm{r}_k)
e^{i\left\langle\alpha_{{\bm{r}}_i}-\alpha_{{\bm{r}}_k}\right\rangle}
+ \text{h.c.}
\right]
\;.\end{aligned}$$ The functional integration with respect to the ChS magnetic field $B(\bm{r}_i)$, the Lagrange multiplier $A^0(\bm{r}_i)$ playing the role of the scalar potential and Grassman variables $\bar{\psi}(\bm{r}_i)$ and $\psi(\bm{r}_i)$ associated to fermionic creation and annihilation operators is considered. One can integrate out Grassmann variables. At the mean-field level we express the fermionic free energy functional $W(\lbrace A^0(\bm{r}_i), B(\bm{r}_i) \rbrace)$ as a sum over eigenvalues of the single-particle problem up to the Fermi energy in such a way that the total filling of fermions equals $1/2$: $$\begin{aligned}
W(\lbrace A^0(\bm{r}_i), B(\bm{r}_i) \rbrace)
=& \sum\limits_k
E_k(\lbrace A^0(\bm{r}_i), B(\bm{r}_i) \rbrace)
\Theta( E_k - E_F)
\;, \notag \\
N_c
=& \sum\limits_k \Theta( E_k - E_F)
\;.\end{aligned}$$ Here $N_c$ is the total number of unit cells in the lattice, $\Theta$ is the Heaviside function and $E_F$ is the Fermi energy, that is calculated self-consistently. We suppose that the solution is translation invariant. In particular, $n({{\bm{r}}_i}) = n_A$ or $n_B$. We allow however the breaking of the symmetry between two sublattices: $n_A \neq n_B$. The condition of being at total filling $1/2$ implies $n_A + n_B = 1$. The first-neighbor hopping terms are sensitive only to the total flux through the unit cell $\Phi_\text{Tot} = 2 \pi$ (each unit cell containing precisely 1 site of the sublattice $A$ and 1 site of the sublattice $B$), that is gauge equivalent to zero. Second-neighbor hoppings exhibit Haldane modulations of the flux through big triangles formed by second-neighbor links. In order to see this more clearly, we separate a symmetric $(+)$ and an antisymmetric $(-)$ components of the scalar potential and the magnetic field: $ B_\pm = B_A \pm B_B, \ A^0_\pm = A^0_A \pm A^0_B $. The flux configuration due to the symmetric component is also gauge equivalent to zero for second-neighbor links, whereas the antisymmetric component leads to $\Phi_A = - \Phi_B = B_{-}$. Here $\Phi_A$ and $\Phi_B$ are fluxes through the smallest triangles formed by second-neighbor sites of the sublattice $A$ or $B$. For consistency with the notation of the Ref. , we also define $\phi = B_- / 3$. The resulting effective Lagrangian for the ChS magnetic field and the scalar potential is $$\label{eq:Leff}
\mathcal{L}_\text{eff}( A^0_-, \phi ) = W( A^0_-, \phi ) +
\frac{3N_c}{2 \pi} A^0_- \phi
\;.$$ The effective mean-field model for free fermions is the Haldane model [@Haldane1988]. We use the saddle-point approximation to find the values of $A^0_-$ and $\phi$: $$\delta_{A^0_-} S_\text{eff} = 0,\quad \delta_{\phi} S_\text{eff} = 0
\;.$$ Solutions of these equations correspond to the extrema of the functional $\mathcal{L}_\text{eff}$, as shown in Fig. \[fig:6\]. By calculating this functional for different values of $J_2/ J_1$, we deduce three different regimes in the phase diagram. In the region $J_2 / J_1 \lesssim 0.21$ the functional $\mathcal{L}_\text{eff}$ has only one point where both equations are verified, that is the saddle point at $A^0_- = 0$, $\phi = 0$. In the region $0.21 \lesssim J_2 / J_1 \lesssim 0.36$ there are three solutions of the equations for the minimization. The solution at $A^0_- = 0$, $\phi = 0$ corresponds to a local maximum of the functional $\mathcal{L}_\text{eff}$, whereas two symmetric solutions not located at zero become new saddle point solutions. These solutions moves continuously with $J_2/J_1$, starting from zero, that corresponds to a second order phase transition. In the region $0.36 \lesssim J_2 / J_1$ again only the local minimum of $\mathcal{L}_\text{eff}$ remains as a solution at $A^0_- = 0$, $\phi = 0$, that corresponds to a first order phase transition.
![**(a-d)** The functional $\mathcal{L}_\text{eff}(\phi, A^0_-)$ of Eq. plotted in units of $J_1$ for different values of $J_2/J_1$, $h_z = 0$. **(e-h)** The effect of the ${\mathcal{P}}$ breaking term $H_z$ on the functional $\mathcal{L}(\phi, A^0_-)$ for a fixed value $J_2 / J_1 = 0.3$. We can see that one of the non-trivial minima shifts in energy with respect to another one, explicitly breaking the symmetry between two degenerate solution from the $h_z = 0$ case.[]{data-label="fig:6"}]({6}.pdf){width="85.00000%"}
Effect of the external magnetic field $h_z$
-------------------------------------------
We consider the effect of adding an external magnetic field $h_z$ to the mean-field solution. In the expression of the fermionic single-particle spectrum this term appears as a Semenoff mass term [@Semenoff1984]. By doing the numerical minimization, we see that the effect of this perturbation consists in breaking the symmetry between two non-trivial solutions in the regime $0.2 \lesssim J_2 / J_1 \lesssim 0.36$. This effect is presented in Fig. \[fig:6\].
Exact diagonalization: Classical phases of the frustrated spin-1/2 $XY$ model
=============================================================================
In order to determine the phase boundaries of the frustrated spin-1/2 $XY$ model, we calculate the fidelity metric $g$ [@ZanardiPaunkovic2006; @ShiJian2010; @VarneyEtAl2010]. The result of this calculation on the lattice of $4 \times 3$ unit cells is shown in Fig. \[fig:4\].
![ED calculation of the fidelity metric $g$ on a lattice of $4 \times 3$ unit cells for hard-core bosons at filling $1/2$ ($S^z_\text{Tot} = 0$).[]{data-label="fig:4"}](4.pdf){width="48.00000%"}
Classical phases are studied by looking at the correlation functions $\Braket{S_{{\bm{r}}_i}^\mu S_{{\bm{r}}_j}^\nu}$ and the related coplanar structure factor $$S_\textrm{Spiral}\left( {\bm{q}} \right) = 2 \sum\limits_{i, j \in A}
e^{i{\bm{q}} \cdot \left( {\bm{r}}_i - {\bm{r}}_j \right)}
\braket{S^x_{{\bm{r}}_i} S^x_{{\bm{r}}_j}} \;.$$ The result of such analysis is presented in Fig. \[fig:5\].
. **(a)** In the FM phase ($J_2 / J_1 = 0.1$ row) the structure factor is piked at ${\bm{q}} = {\bm{\Gamma}}$. **(b)** The systems seems to be disordered in the $xy$ plane in the intermediate frustration regime ($J_2 / J_1 = 0.25$ row). **(c)** We observe a formation of the collinear order for $J_2 / J_1 = 0.6$. We notice however the significant difference of the result on the lattice $4 \times 3$. This is explained by the fact that this lattice does not contain all the ${\bm{M}}$ points in the reciprocal space. **(d)** In the case $J_2 / J_1 = 1.5$ the system forms a $120^\circ$ order. We notice that similarly to the previous case the lattice $4 \times 4$ does not contain Dirac points ${\bm{K}}$ in the reciprocal space, that results in the impossibility to recover correctly the $120^\circ$ phase: two rightmost figures in the bottom line do not differ almost at all.[]{data-label="fig:5"}](5.pdf){width="95.00000%"}
|
---
abstract: 'A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler’s decomposition is non-constructive—a problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions—inner and rational inner functions. In this paper, we show that most of the important structure present in these special cases extends to general bounded analytic functions. We give characterizations of all Agler decompositions, we prove the existence of coisometric transfer function realizations with natural state spaces, and we characterize when Schur functions on the bidisk possess analytic extensions past the boundary in terms of associated Hilbert spaces.'
address:
- 'Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30308'
- 'Bucknell University, Department of Mathematics, Lewisburg, PA 17837'
- 'Washington University in St. Louis, Department of Mathematics, St. Louis, MO 63130'
author:
- Kelly Bickel
- Greg Knese
title: Canonical Agler Decompositions and Transfer Function Realizations
---
[^1]
Introduction
============
Let $E$ and $E_*$ be separable Hilbert spaces and recall that the *Schur class $\mathcal{S}_d(E, E_*)$* is the set of holomorphic functions $\Phi: \D^d \rightarrow
\mathcal{L}(E,E_*)$ such that each $\Phi(z): E \rightarrow E_*$ is a linear contraction. In one variable, the structure of these functions is well-understood and they play key roles in many areas of both pure and applied mathematics. For example, they are objects of interest in $H^{\infty}$ control theory, act as scattering functions of single-evolution Lax-Phillips scattering systems, and serve as the transfer functions of one-dimensional dissipative, linear, discrete-time input/state/output (i/s/o) systems [@bsv05; @he74; @hj99]. Moreover, every $\Phi \in \mathcal{S}_1(E, E_*)$ can actually be realized as both a scattering function of a Lax-Phillips scattering system and a transfer function of a dissipative, linear, discrete-time i/s/o system. For simplicity, we omit the discussion of the connection to the interesting topic of von Neumann inequalities; see [@ampi; @bsv05; @gkvw08].
The situation in several variables is more complicated; although Schur functions are still the scattering functions of $d$-evolution scattering systems and transfer functions of $d$-dimensional dissipative, linear, discrete-time i/s/o systems, the converse is not always true; there are functions in $\mathcal{S}_d(E,E_*)$ that cannot be realized as transfer functions of dissipative i/s/o systems. To make this precise, let $\mathcal{M}= \mathcal{M}_1 \oplus \dots
\oplus \mathcal{M}_d$ be a separable Hilbert space, and for each $z \in \mathbb{D}^d$, define the multiplication operator $\mathcal{E}_z: = z_1P_{\mathcal{M}_1} + \dots + z_d
P_{\mathcal{M}_d},$ where each $P_{\mathcal{M}_r}$ is the projection onto $\mathcal{M}_{r}$.
Let $\Phi \in \mathcal{S}_d(E,E_*).$ A *Transfer Function Realization* (T.F.R.) of $\Phi$ consists of a Hilbert space $\mathcal{M}= \mathcal{M}_1 \oplus \dots
\oplus \mathcal{M}_d$ and a contraction $U: \mathcal{M} \oplus E \rightarrow
\mathcal{M} \oplus E_*$ such that if $U$ is written as $$U =
\left[ \begin{array}{cc}
A & B \\
C & D
\end{array} \right] :
\left[ \begin{array}{c}
\mathcal{M} \\
E
\end{array} \right]
\rightarrow
\left[ \begin{array}{c}
\mathcal{M} \\
E_* \end{array} \right],$$ then $\Phi(z) = D + C\left( I_{\mathcal{M}} - \mathcal{E}_zA \right)^{-1}\mathcal{E}_z B$. The Hilbert space $\mathcal{M}$ is called the *state space* and the contraction $U$ is called the *colligation*. One can associate a d-dimensional dissipative, linear, discrete-time $i/s/o$ system with the pair $(\mathcal{M}, U)$. The transfer function realization is called isometric, coisometric, or unitary whenever $U$ is isometric, coisometric, or unitary.
In [@ag1; @ag90], J. Agler showed that every function in $\mathcal{S}_2(E,E_*)$ has a T.F.R. and used the realizations to generalize the Pick interpolation theorem to two variables. Since Agler’s seminal results, these formulas have been used frequently to both generalize one-variable results and address strictly multivariate questions on the polydisc as in [@agmc_isb; @agmc_dv; @mcc10a; @amy10a; @baltre98; @kn07b; @kn08ua; @mcc12]. There is also a simple relationship between transfer function realizations and positive kernels:
\[thm1\] *(Agler [@ag90]).* Let $\Phi \in \mathcal{S}_d(E,E_*)$. Then, $\Phi$ has a transfer function realization if and only if there are positive holomorphic kernels $K_1, \dots, K_d: \mathbb{D}^d \times \mathbb{D}^d \rightarrow \mathcal{L}(E_*)$ such that for all $z,w \in \D^d$ $$I_{E_*} - \Phi(z) \Phi(w)^* = (1-z_1 \bar{w}_1) K_1(z,w) + \dots + (1-z_d
\bar{w}_d) K_d(z,w).$$
This decomposition using positive kernels is called an *Agler decomposition* of $\Phi$. In two variables, it is convenient to reverse the ordering, and throughout this paper, positive kernels $(K_1, K_2)$ are called *Agler kernels of* $\Phi \in \mathcal{S}_2(E,E_*)$ if for all $z,w \in \D^2$ $$\label{eqn:agdecomp} I_{E_*} - \Phi(z) \Phi(w)^* = (1 -z_1 \bar{w}_1) K_2(z,w) +
(1-z_2\bar{w}_2) K_1(z,w).$$
Agler proved the existence of a pair of Agler kernels for each function in $\SEE$ and then showed this gives a transfer function realization via Theorem \[thm1\]. It is often easier to go from kernels to realizations because positive kernels immediately bring operator theory and reproducing kernel Hilbert space methods into the picture. We review some of these concepts related to positive kernels below.
Recall that $K : \Omega \times \Omega \rightarrow \mathcal{L}(E)$ is a *positive kernel on $\Omega$* if for each $N \in \NN$ $$\sum_{i,j =1}^N \LL K(x_i,x_j) \eta_j, \eta_i \RR_{E} \ge 0$$ for all $x_1, \dots, x_N \in \Omega$ and $\eta_1, \dots, \eta_N \in
E.$ Similarly, $\mathcal{H}$ is a *reproducing kernel Hilbert space on $\Omega$* if $\mathcal{H}$ is a Hilbert space of functions on defined $\Omega$ such that evaluation at $x$ is a bounded linear operator for each $x \in \Omega.$ Then there is a unique positive kernel $K: \Omega \times \Omega \rightarrow \mathcal{L}(E)$ with $$\LL f, K(\cdot, y) \eta \RR_{\mathcal{H}} = \LL f(y), \eta
\RR_{E} \qquad \forall \ f \in \mathcal{H}, y \in \Omega, \text{ and }
\eta \in E.$$ Conversely, given any positive kernel $K$ on $\Omega$, there is a reproducing kernel Hilbert space, denoted $\mathcal{H}(K)$, on $\Omega$ with $K$ as its reproducing kernel. For details, see [@bv03b].
The kernels $K_1,K_2$ are written in reverse order in because upon dividing the equation through by $(1-z_1\bar{w}_1)(1-z_1\bar{w}_2)$, an Agler decomposition can be given a much more natural interpretation in terms of de Branges-Rovnyak spaces.
Assume $(K_1, K_2)$ are Agler kernels of $\Phi$ and rewrite (\[eqn:agdecomp\]) as follows: $$\label{eqn:agdecomp2} \frac{I -\Phi(z) \Phi(w)^*}{(1-z_1\bar{w}_1) (1-z_2\bar{w}_2)}
= \frac{K_1(z,w)}{1-z_1\bar{w}_1} + \frac{K_2(z,w)}{1-z_2\bar{w}_2}.$$ Each term in (\[eqn:agdecomp2\]) is a positive kernel and so, we can define the following reproducing kernel Hilbert spaces: $$\mathcal{H}_{\Phi}:= \mathcal{H} \left( \frac{I -\Phi(z) \Phi(w)^*}{(1-z_1\bar{w}_1)
(1-z_2\bar{w}_2)} \right) \ \ \text{ and } \ \ H_j : = \mathcal{H} \left( \frac{K_j(z,w)}{1-z_j\bar{w}_j}
\right),$$ for $j=1,2.$ The Hilbert space $\mathcal{H}_{\Phi}$ is *the two-variable de Branges-Rovnyak space associated* to $\Phi$. For $j=1,2,$ define the function $Z_j$ by $Z_j(z):= z_j$. Then the $H_j$ Hilbert spaces have the following properties:
- $Z_j H_j \subseteq H_j$ and multiplication by $Z_j$ on $H_j$ is a contraction.
- The reproducing kernels of the $H_j$ sum to the kernel of $\mathcal{H}_{\phi}$.
Basic facts about reproducing kernels imply that if Hilbert spaces $H_1$ and $H_2$ satisfy $(1)$ and $(2)$, then the numerators of their reproducing kernels are Agler kernels of $\Phi.$
Agler used non-constructive methods to obtain Agler kernels, and a major stride was made in this theory when Ball-Sadosky-Vinnikov proved the existence of Agler kernels through constructive Hilbert space geometric methods. Indeed, our analysis is motivated by their work on two-evolution scattering systems and scattering subspaces associated to $\Phi \in \SEE.$ In [@bsv05], they showed that such scattering subspaces have canonical decompositions into subspaces $S_1$ and $S_2$, each invariant under multiplication by $Z_1$ or $Z_2.$ This work was continued in [@gkvw08] where a specific scattering subspace associated to $\Phi$, denoted $\Kphi$, was used to show that canonical decompositions of $\Kphi$ yield Agler kernels $(K_1,K_2)$ of $\Phi$. The analysis from [@bsv05] was also extended in [@bkvsv]; here, many results from [@bsv05] are illuminated or extended via the theory of formal reproducing kernel Hilbert spaces.
While more explicit, the approaches so far do not shed much light on the actual structure of the Hilbert spaces $\mathcal{H}(K_j)$ and the functions contained therein for general Schur functions. The spaces $\mathcal{H}(K_j)$ have been shown to possess a very rich structure when $\Phi$ is an *inner* function or a *rational inner* function [@bic12; @bk12; @colwer99; @knapde]. This has led to applications in the study of two variable matrix monotone functions in [@mcc10] and in the study of *three* variable rational inner functions in [@bk12]. This structure is also important in the Geronimo-Woerdeman characterizations of bivariate Fejér-Riesz factorizations as well as the related bivariate auto-regressive filter problem [@gw04]. The theory is much simpler in these cases because Agler kernels can be constructed directly from orthogonal decompositions of $\Hphi$. Therefore, the major goal of this paper is to show directly that the rich Agler kernel structure present when $\Phi$ is inner is still present when $\Phi$ is not an inner function. A direct application of this will be to prove that every function in $\SEE$ possesses a *coisometric* transfer function realization with state space ${\mathcal{H}}(K_1)\oplus {\mathcal{H}}(K_2)$ for some pair of Agler kernels $(K_1,K_2)$; this construction answers a question posed by Ball and Bolotnikov in [@bb11]. We also generalize classical work of Nagy-Foias connecting regularity of $\Phi \in \mathcal{S}_1(E,E_{*})$ on the boundary to the regularity of functions in its associated de Branges-Rovnyak space. See [@sar94] for a discussion.
We now outline the rest of the paper. The structure of $\Hphi$ is revealed by embedding an isometric copy into the larger scattering subspace $\Kphi$ alluded to above. The reader need not know anything about scattering theory—the basic facts we need are built from scratch in Section \[sect:scattering\]. In Section \[sect:construction\], canonical orthogonal decompositions of $\Kphi$ are projected down to canonical decompositions of $\Hphi$ and these yield certain pairs of extremal Agler kernels of $\Phi$ denoted $$(K^{max}_1, K^{min}_2) \ \ \text{ and } \ \ (K^{min}_1,
K^{max}_2).$$ These pairs are related by a positive kernel $G:\D^2\times \D^2 \to \mathcal{L}(E_*)$ $$G(z,w) := \frac{K_1^{max}(z,w) - K_1^{min}(z,w)}{1-z_1\bar{w}_1} =
\frac{K_2^{max}(z,w) - K_2^{min}(z,w)}{1-z_2\bar{w}_2}.$$ In section 4, we show that all Agler kernels of $\Phi$ can be characterized in terms of the special kernels $K_1^{min}, K_2^{min}, G$:
Let $\Phi \in \mathcal{S}_2(E, E_*)$ and let $K_1, K_2: \D^2 \times \D^2 \rightarrow \mathcal{L}(E_*)$. Then $(K_1, K_2)$ are Agler kernels of $\Phi$ if and only if there are positive kernels $G_1,G_2: \D^2 \times \D^2 \rightarrow \mathcal{L}(E_*)$ such that $$\begin{aligned}
K_1(z,w) =& K_1^{min}(z,w) + (1-z_1 \bar{w}_1) G_1(z,w) \\
K_2(z,w) =& K_2^{min}(z,w) + (1-z_2 \bar{w}_2) G_2(z,w)
\end{aligned}$$ and $G = G_1 + G_2.$
While Ball-Sadosky-Vinnikov [@bsv05] proved the existence of analogous maximal and minimal decompositions in the scattering subspace $\Kphi$, our contribution here is to show that many of these extremality properties also hold in the space of interest $\Hphi$. On the path to our regularity result, we obtain explicit characterizations of the spaces $\mathcal{H}(K^{max}_j)$ and $\mathcal{H}(K^{min}_j)$ and use those to show that all $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ are contained inside “small”, easily-studied subspaces of $\Hphi$. Section \[sect:functions\] has the details.
In Section 5, we consider applications of this Agler kernel analysis. When $\Phi$ is square matrix valued, the containments allow us to characterize when $\Phi$ and the elements of $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ extend analytically past portions of $\partial \D^2$, thus generalizing the regularity result of Nagy-Foias mentioned above. A key point is that $\Hphi$ is too big of a space for these characterizations, and it really is necessary to study Agler kernels to investigate the regularity of $\Phi$.
We now state the main regularity theorem found in Section \[sect:extensions\]. Let $X \subseteq \mathbb{T}^2$ be an open set and define the sets $$\begin{aligned}
X_1 & := \left \{ x_1 \in \mathbb{T} : \text{ such that } \exists \ x_2 \text{ with }
(x_1, x_2) \in X \right \} \\
X_2 & := \left \{ x_2 \in \mathbb{T} : \text{ such that } \exists \ x_1 \text{ with } (x_1, x_2) \in X \right \}
\end{aligned}$$ using $X$ and the sets $\mathbb{E} := \mathbb{C} \setminus \overline{\D}$ and $S := \left \{ 1 / \bar{z}: \det \Phi(z) = 0 \right\}.$ Then, we obtain the following result:
Let $\Phi \in \mathcal{S}_2(E, E_*)$ be square matrix valued. Then the following are equivalent:
- $\Phi$ extends continuously to $X$ and $\Phi$ is unitary valued on $X$.
- There is some pair $(K_1,K_2)$ of Agler kernels of $\Phi$ such that the elements of $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ extend continuously to $X.$
- There exists a domain $\Omega$ containing \^2 X (X\_1 ) (X\_2) (\^2 S ) such that $\Phi$ and the elements of $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ extend analytically to $\Omega$ for every pair $(K_1, K_2)$ of Agler kernels of $\Phi.$ Moreover the points in the set $\Omega$ are points of bounded evaluation of every $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2).$
In Section \[sect:tfr\], we return to the setting of transfer function realizations. We use the canonical Agler kernels $(K^{max}_1, K^{min}_2)$ to construct a T.F.R. of $\Phi$ with refined properties. Specifically we prove:
Let $\Phi \in
\mathcal{S}_2(E,E_*)$ and consider its Agler kernels $(K^{max}_1,
K^{min}_2).$ Then, $\Phi$ has a coisometric transfer function realization with state space $\mathcal{H}(K^{min}_2) \oplus
\mathcal{H}(K^{max}_1).$
This construction answers a question posed by Ball and Bolotnikov in [@bb11]. We also obtain additional information about the block operators $A, B, C,$ and $D$ of the associated coisometric colligation $U$. In Section \[sect:opkernels\], we provide an appendix outlining results concerning operator valued reproducing kernels used in the paper. We supply the commonly used symbols and table of contents below for convenience.
Decompositions of Scattering Subspaces {#sect:scattering}
======================================
For brevity, this paper only outlines the structure of particular scattering systems defined for $\Phi \in \SEE$. Many details of these scattering systems also appear in [@bsv05] and [@bkvsv]. For a review of the general theory of one- and multi-evolution scattering systems, see [@bsv05].
Notation and Operator Ranges
----------------------------
Before proceeding to scattering systems, we require some notation. Let $E$ be a Hilbert space. Then $L^2(E):= L^2(\mathbb{T}^2) \otimes
E$, i.e. the space of $E$ valued functions on ${\mathbb{T}}^2$ with square summable Fourier coefficients. Similarly, $H^2(E) := H^2(\D^2) \otimes
E$ denotes the space of $E$ valued holomorphic functions on $\D^2$ whose Taylor coefficients around zero are square summable. Recall that $Z_1,
Z_2$ denote the coordinate functions $Z_j(z_1,z_2) = z_j$. We will define some standard subspaces of $L^2(E)$ according to their Fourier series support. Let $\ZZ_+ = \{0,1,2,\dots\}$ and $\ZZ_{-} =
\{-1,-2,-3,\dots\}$. If $N\subset \ZZ^2$ and $f \in L^2(E)$, the statement ${\text{supp}}(\hat{f}) \subset N$ means $\hat{f}(n_1,n_2) = 0$ for $(n_1,n_2) \not \in N$. Now define $$\begin{aligned}
L^2_{++}(E) &:= \{f \in L^2(E): {\text{supp}}(\hat{f}) \subset \ZZ_+ \times \ZZ_+ \}\\
L^2_{+\bullet}(E) &:= \{f \in L^2(E): {\text{supp}}(\hat{f}) \subset \ZZ_+ \times \ZZ \} \\
L^2_{-\bullet}(E) &:= \{f \in L^2(E): {\text{supp}}(\hat{f}) \subset \ZZ_{-} \times \ZZ\} \\
L^2_{+-}(E) &:= \{f \in L^2(E): {\text{supp}}(\hat{f}) \subset \ZZ_{+} \times \ZZ_{-}\} \\
L^2_{--}(E) &:= \{f \in L^2(E): {\text{supp}}(\hat{f}) \subset \ZZ_{-} \times \ZZ_{-}\},
\end{aligned}$$ and similarly one can define $L^2_{\bullet +}(E),$ $L^2_{\bullet
-}(E)$, and $L^2_{-+} (E).$ It is well-known that associating an $H^2(E)$ function $f$ with the $L^2$ function whose Fourier coefficients agree with the Taylor coefficients of $f$ maps $f$ unitarily to its radial boundary value function in $L^2_{++}(E).$ We will denote both the function in $H^2$ and the associated function in $L^2_{++}$ by $f$.
We also require the following definition and simple lemma about operator ranges; for more details, see the first chapter of [@sar94].
Let $\mathcal{K}$ be a Hilbert space and let $T: \mathcal{K} \rightarrow \mathcal{K}$ be a bounded linear operator on $\mathcal{K}$. Then the *operator range* of T, denoted $\mathcal{M}(T)$, is the Hilbert space consisting of elements in the image of $T$ endowed with the inner product $$\LL T x, Ty \RR_{\mathcal{M}(T)} := \LL P_{(\ker T)^{\perp} } x, y \RR_{\mathcal{K}}
\qquad \forall \ x,y \in \mathcal{K}.$$
\[lem:oprange\] Let $\mathcal{K}$ be a Hilbert space and let $T: \mathcal{K}
\rightarrow \mathcal{K}$ be a bounded linear self-adjoint operator on $\mathcal{K}$. Then the operator range $\mathcal{M}(T)$ is the closure of the image of $T^2$ in the $\mathcal{M}(T)$ norm and $\LL T x, T^2 y \RR_{\mathcal{M}(T)} = \LL T x, y
\RR_{\mathcal{K}},$ for all $x,y \in \mathcal{K}.$
We show that if $\eta \in \mathcal{M}(T)$ and $\eta \perp T^2 \mathcal{K}$, then $\eta \equiv 0.$ Fix such an $\eta$ and choose $x \in ( \ker T )^{\perp}$ such that $Tx
=\eta.$ Then, for each $y \in \mathcal{K}$, $$0 = \LL \eta, T^2 y \RR_{\mathcal{M}(T)} = \LL x, T y \RR_{\mathcal{K}} =
\LL Tx, y \RR_{\mathcal{K}} =\LL \eta, y \RR_{\mathcal{K}} ,$$ which implies $\eta \equiv 0.$ Moreover, for any $x,y \in \mathcal{K}$, $$\LL T x, T^2 y \RR_{\mathcal{M}(T)} = \LL P_{(ker T)^{\perp}} x, Ty \RR_{\mathcal{K}}
= \LL T P_{(ker T)^{\perp}} x, y \RR_{\mathcal{K}} = \LL T x, y \RR_{\mathcal{K}},$$ as desired.
\[ex:hphi\] Let $\Phi \in \SEE$. The two-variable de Branges-Rovnyak space $\Hphi$ is also the operator range of the bounded linear self adjoint operator $$\Dphi : =( I - \Phi P_{H^2(E)} \Phi^*)^{1/2}: H^2(E_*) \rightarrow
H^2(E_*). \nomenclature{$\Dphi$}{The operator $(I - \Phi P_{H^2(E)} \Phi^*)^{1/2}$}$$ To see this notice first that by Lemma \[lem:oprange\], $\Dphi ^2 H^2(E_*)$ is dense in $\mathcal{M}(\Dphi)$ and $$\LL \Dphi f, \Dphi^2 g
\RR_{\mathcal{M}(\Dphi)} = \LL \Dphi f, g \RR_{H^2(E_*)}$$ for all $f,g \in
H^2(E_*).$ Let $k_z$ be the Szegő kernel on the bidisk. Then, the reproducing kernel of $H^2(E_{*})$ is $k_z\otimes I_{E_*}$. Given $f
\in \mathcal{M}(\Dphi)$, $z \in \D^2, v \in E_{*}$, we see that $$\LL f, \Dphi^2 k_z v \RR_{\mathcal{M}(\Dphi)} = \LL f, k_z v \RR_{H^2(E_{*})} = \LL
f(z), v \RR_{E_{*}}$$ and therefore the operator range of $\Dphi$ is a reproducing kernel Hilbert space on $\D^2$ with reproducing kernel $$\frac{I-\Phi(z)\Phi(w)^*}{(1-z_1\bar{w}_1)(1-z_2\bar{w}_2)}.$$ Specifically, $\mathcal{M}(\Dphi)$ is equal to the de Branges-Rovnyak space associated to $\Phi$, which is $\Hphi.$ This follows from the standard identity for reproducing kernels $P_{H^2} \Phi^* k_z v = \Phi(z)^* k_z v$ and the computation $\Dphi^2 k_z v = (I-\Phi P_{H^2} \Phi^*) k_z v = k_z v -\Phi
\Phi(z)^*k_z v$.
The following consequence of Douglas’s lemma [@do66] is found on page 3 of [@sar94].
\[lem:douglas\] Let ${\mathcal{K}}$ be a Hilbert space and let $A:{\mathcal{K}}\to {\mathcal{K}}, B:{\mathcal{K}}\to
{\mathcal{K}}$ be bounded linear operators. Then, $\mathcal{M}(A) =
\mathcal{M}(B)$ if and only if $AA^* = BB^*$.
The de Branges-Rovynak Models
-----------------------------
Now we proceed to scattering systems:
A *two-evolution scattering system* $\mathcal{S} = (\scrH, \mathcal{U}_1,
\mathcal{U}_2, \mathcal{F}, \mathcal{F}_*)$ consists of a Hilbert space $
\scrH$, two unitary operators $\mathcal{U}_1$, $\mathcal{U}_2: \scrH \rightarrow
\scrH,$ and two wandering subspaces $\mathcal{F}, \mathcal{F}_* \subseteq
\scrH$ of $\mathcal{U}_1$ and $\mathcal{U}_2$, i.e. $$\mathcal{F} \perp \U_1^{n_1} \U_2^{n_2} \F \ \ \text{ and }
\ \ \ \mathcal{F}_* \perp \U_1^{n_1} \U_2^{n_2} \F_* \ \qquad \forall \ (n_1, n_2) \in \mathbb{Z}^2
\setminus(0,0).$$
Given any $\Phi \in \SEE$, one can define the de Branges-Rovnyak model for $\Phi$. This is a concrete transcription of the (almost) unique minimal scattering system whose scattering function coincides with $\Phi.$ See [@bsv05] for the proof and additional theory.
The *de Branges-Rovnyak model for $\Phi \in \SEE$* consists of the operator range, denoted $\scrH$, of the following bounded linear self-adjoint operator: $$\left[ \begin{array}{cc} I & \Phi \\
\Phi^*& I \end{array} \right]^{1/2} :
\left[ \begin{array}{c}
L^2(E_*) \\
L^2(E) \end{array} \right] \rightarrow \left[ \begin{array}{c}
L^2(E_*) \\
L^2(E) \end{array} \right].$$ Then $\scrH$ has inner product given by $$\left \langle \left[ \begin{array}{cc} I & \Phi \\
\Phi^*& I \end{array} \right]^{1/2}
\left[ \begin{array}{c}
f \\
g \end{array} \right] ,
\left[ \begin{array}{cc} I & \Phi \\
\Phi^*& I \end{array} \right]^{1/2}
\left[ \begin{array}{c}
f' \\
g' \end{array} \right] \right \rangle_{\scrH}
:= \left \langle P_{Q^{\perp}} \left[ \begin{array}{c}
f \\
g \end{array} \right], \left[ \begin{array}{c}
f' \\
g' \end{array} \right] \right \rangle_{L^2(E_*) \oplus L^2(E)},$$ where $Q = \ker \left[ \begin{array}{cc} I & \Phi \\
\Phi^*& I \end{array} \right]^{1/2}.$ Lemma \[lem:oprange\] implies the image of the operator ${\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}$ is dense in $\scrH$ and that $$\left \langle {\begin{bmatrix} f \\ g \end{bmatrix}}, {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}{ \left[ \begin{array}{c}
f' \\ g' \end{array} \right]}\right \rangle_{\scrH}
= \left \langle {\begin{bmatrix} f \\ g \end{bmatrix}}, { \left[ \begin{array}{c}
f' \\ g' \end{array} \right]}\right \rangle_{L^2(E_*) \oplus L^2(E)},\qquad \forall
\ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \scrH.$$ The de Branges-Rovnyak model also contains the following two subspaces of $\scrH$: $$\F: = \left[ \begin{array} {c}
\Phi \\
I \end{array} \right] E = \left[ \begin{array}{cc} I & \Phi \\
\Phi^*& I \end{array} \right]
\left[ \begin{array}{c} 0 \\ E \end{array} \right]
\ \ \text{ and } \ \
\F_*: = \left[ \begin{array} {c}
I \\
\Phi^* \end{array} \right] E_* = \left[ \begin{array}{cc} I & \Phi \\
\Phi^*& I \end{array} \right]
\left[ \begin{array}{c} E_* \\ 0 \end{array} \right]$$ and the two operators $\U_1, \U_2: \scrH \rightarrow \scrH$ defined by $$\U_j := \left[ \begin{array}{cc}
Z_j I_{E_*} & 0 \\
0 & Z_j I_E \end{array} \right] \qquad \text{ for } j=1,2.$$ Each $\U_j$ is onto since $$\U_j {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}^{1/2} = {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}^{1/2} \U_j \ \ \text{ and }
\ \ \U_j \Big( L^2(E_*) \oplus L^2(E) \Big) = L^2(E_*) \oplus L^2(E).$$ To see that $\U_j$ is isometric, observe that $\U_j$ preserves the $\scrH$ norm on the image of ${\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}$ since: $$\begin{aligned}
\Bigg \| \U_j {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}{\begin{bmatrix} f \\ g \end{bmatrix}}\Bigg \|^2_{\scrH} &
= \LL \U_j {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}{\begin{bmatrix} f \\ g \end{bmatrix}}, \U_j {\begin{bmatrix} f \\ g \end{bmatrix}}\RR_{L^2(E_*) \oplus L^2(E)} \\
&= \LL {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\begin{bmatrix}
Z_j f \\
Z_j g \end{bmatrix} ,
\begin{bmatrix}
Z_j f \\
Z_j g \end{bmatrix} \RR_{L^2(E_*) \oplus L^2(E)} \\
& = \Bigg \| {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}{\begin{bmatrix} f \\ g \end{bmatrix}}\Bigg \|^2_{\scrH}. \end{aligned}$$ Since said image is dense in $\scrH$, each $\U_j$ is unitary. Observe that $\F$ is *wandering* for $\U_1$ and $\U_2$ since if $\eta, \nu \in E$ and $(n_1,n_2) \ne (0,0)$, then $$\begin{aligned} \LL { \left[ \begin{array}{c}
\Phi \\ I \end{array} \right]}\eta, \ \U_1^{n_1} \U_2^{n_2} { \left[ \begin{array}{c}
\Phi \\ I \end{array} \right]}\nu \RR_{\scrH} &= \LL {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\left[ \begin{array}{c} 0 \\ \eta \end{array} \right], {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\left[ \begin{array}{c}
0 \\ Z_1^{n_1}Z_2^{n_2} \nu \end{array} \right] \RR_{\scrH} \\
& \\
&= \LL {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\left[ \begin{array}{c} 0 \\ \eta \end{array} \right],
\left[ \begin{array}{c} 0 \\ Z_1^{n_1}Z_2^{n_2} \nu \end{array} \right] \RR_{L^2(E_*) \oplus L^2(E)} \\
&\\
& = \LL \eta, \ Z_1^{n_1}Z_2^{n_2} \nu \RR_{L^2(E)},
\end{aligned}$$ which is zero. Analogous arguments show $\F_*$ is wandering. We will usually just write $\U_j = Z_j$, unless we wish to emphasize the connection to scattering systems.
The following remarks detail additional facts about $\scrH$.
\[rem:hspace\] **Alternate Characterization of $\scrH$.** Define the bounded linear self-adjoint operators $$\begin{aligned} \Delta: &= (I - \Phi^*\Phi)^{1/2}: L^2(E) \rightarrow L^2(E) \\
\Delta_*: &= (I - \Phi \Phi^*)^{1/2}: L^2(E_*) \rightarrow L^2(E_*). \end{aligned}
\nomenclature{$\Delta, \Delta_{*}$}{ $(I-\Phi^*\Phi)^{1/2}, (I-\Phi\Phi^*)^{1/2}$}$$ By Lemma \[lem:douglas\], the factorizations $$\begin{bmatrix} I & \Phi \\ \Phi^* &
I \end{bmatrix}^{1/2} \begin{bmatrix} I & \Phi \\ \Phi^* &
I \end{bmatrix}^{1/2} = \begin{bmatrix} I & 0 \\ \Phi^* &
\Delta \end{bmatrix} \begin{bmatrix} I & \Phi \\ 0 &
\Delta \end{bmatrix}
= \begin{bmatrix} \Delta_{*} & \Phi \\ 0 &
I \end{bmatrix} \begin{bmatrix} \Delta_{*} & 0 \\ \Phi^{*} &
I \end{bmatrix}$$ show that $$\begin{aligned}
\scrH &= \mathcal{M} \left( \begin{bmatrix} I & 0 \\ \Phi^* &
\Delta \end{bmatrix} \right) = \left\{\begin{bmatrix} f \\ g \end{bmatrix}:
f\in L^2(E_*), g \in L^2(E), g- \Phi^* f \in \Delta L^2(E)
\right\} \label{eqn:hphichar} \\
&= \mathcal{M} \left( \begin{bmatrix} \Delta_{*} & \Phi \\ 0 &
I \end{bmatrix} \right) = \left\{\begin{bmatrix} f \\ g \end{bmatrix}:
f\in L^2(E_*), g \in L^2(E), f- \Phi g \in \Delta_{*} L^2(E_{*})
\right\}.\end{aligned}$$ where the equality is on the level of Hilbert spaces, not just as sets.These characterizations of $\scrH$ can be used to show that the linear maps
f\
g
f
f\
g
g are contractive operators from $\scrH$ onto $L^2(E_*)$ and $L^2(E)$ respectively. To see this, note that for each element in $\scrH$, there is an $h \in L^2(E)$ such that $${\begin{bmatrix} f \\ g \end{bmatrix}}=
\begin{bmatrix} I & 0 \\ \Phi^* &
\Delta \end{bmatrix}
\begin{bmatrix}
f \\
h \end{bmatrix}, \text{ where }
\begin{bmatrix}
f \\
h \end{bmatrix} \perp \ker
\begin{bmatrix} I & 0 \\ \Phi^* &
\Delta \end{bmatrix}.$$ Since $\scrH$ and the operator range of $ \begin{bmatrix}
I & 0 \\ \Phi^* & \Delta \end{bmatrix}$ coincide as Hilbert spaces, $$\label{eqn:fnorm}
\left \| {\begin{bmatrix} f \\ g \end{bmatrix}}\right \|^2_{\scrH} = \| f \|^2_{L^2(E_*)} + \|h \|^2_{L^2(E)}
\ge \| f \|^2_{L^2(E_*)}.$$ Similarly, the equality between $\scrH$ and the operator range of $\begin{bmatrix} \Delta_{*} & \Phi \\ 0 &
I \end{bmatrix}$ shows that for each element of $\mathscr{H}$, $$\label{eqn:gnorm} \| g \|_{L^2(E)} \le \left \|
{\begin{bmatrix} f \\ g \end{bmatrix}}\right \|_{\scrH}.$$
The following remark discusses additional subspaces of $\scrH$ that are important for the structure of the scattering system:
\[rem:kphi\] **The Scattering Subspace $\Kphi.$** The incoming subspace $\W_*$ and outgoing subspace $\W$ of the de Branges-Rovnyak model are defined as follows: $$\begin{aligned}
\W_*&:= \bigoplus_{n \in \ZZ^2 \setminus \mathbb{Z}^2_+}
\U_1^{n_1} \U_2^{n_2} \F_* = { \left[ \begin{array}{c}
I \\ \Phi^* \end{array} \right]}L^2 \ominus H^2(E_*) \\
\W &: = \bigoplus_{n \in \mathbb{Z}^2_+} \U_1^{n_1} \U_2^{n_2} \ \F =
{ \left[ \begin{array}{c}
\Phi \\ I \end{array} \right]}H^2(E). \end{aligned}$$ An easy calculation shows $\W \perp \W_*$ in $\scrH$. This means $\scrH$ decomposes as $$\scrH = \W_* \oplus \Kphi \oplus \W,$$ where $\Kphi := \scrH \ominus (\W \oplus \W_*)$ is called the *scattering subspace*. A simple computation shows that $${\begin{bmatrix} f \\ g \end{bmatrix}}\perp \W_* \text{ iff } f \in H^2(E_*) \text{ and } {\begin{bmatrix} f \\ g \end{bmatrix}}\perp \W
\text{ iff } g \in L^2\ominus H^2(E).$$ This means that the scattering subspace $$\begin{aligned} \Kphi &: = \scrH \ominus ( \W \oplus \W_*) \\
& = \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \scrH : f \in H^2(E_*), \ g \in L^2\ominus H^2(E) \right \}.
\end{aligned}$$ Using the alternate characterizations of $\scrH$ from Remark \[rem:hspace\], it follows that $$\begin{aligned} \Kphi & = \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}: f \in H^2(E_*), \ g \in L^2\ominus H^2(E)
, \ g-\Phi^*f \in \Delta L^2(E) \right \} \\
& = \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}: f \in H^2(E_*), \ g \in L^2\ominus H^2(E), \ f- \Phi g \in
\Delta_* L^2(E_*) \right \}.
\end{aligned}$$ The following operator gives the orthogonal projection onto $\Kphi:$ $$P_{\Phi} : =
\left[ \begin{array}{cc}
P_+ & -\Phi P_+ \\
-\Phi^*P_{-} & P_{-} \end{array} \right],$$ where $P_+ = P_{H^2}$, and $P_{-} =P_{L^2 \ominus H^2}$, for either $L^2 \ominus H^2(E)$ or $L^2 \ominus H^2(E_*).$ It is easy to check that $P_{\Phi}^2 =
P_{\Phi}$, $P_{\Phi}|_{\Kphi} \equiv I$ and $ P_{\Phi}|_{\W \oplus
\W_*} \equiv 0.$
When $\Phi$ is an inner function, namely when $\Phi^*\Phi = I,
\Phi\Phi^* = I$ a.e. on ${\mathbb{T}}^2$, the above machinery simplifies significantly and scattering systems are not really necessary. In this case, $\Delta = 0, \Delta_{*} = 0$, so that $$\Kphi =
\left\{ \begin{bmatrix} f \\ \Phi^* f \end{bmatrix} : f \in
H^2(E_*), \Phi^*f \in L^2\ominus H^2(E) \right\}.$$ Evidently, the first component in this space is $f \in H^2(E_{*})$ such that $\Phi^*f \in L^2\ominus H^2(E)$. This is equivalent to saying $f
\in H^2(E_{*}) \ominus \Phi H^2(E)$. This space is the usual model space associated to the inner function $\Phi$; it is studied in [@bsv05] and is studied in great depth in [@bk12]. Although in this paper we recover many results from [@bk12], there are many results related to rational inner functions in [@bk12] that we do not mention here. In general, the paper [@bk12] is a more accessible introduction to the present material.
Decompositions of $\Kphi$
-------------------------
In [@bsv05 Theorem 5.5], Ball-Sadosky-Vinnikov prove the following canonical decomposition of $\Kphi.$ For completeness, we include a simple proof here as well.
\[thm:kdecomp\] Define these subspaces of the scattering subspace $\Kphi$: $$\begin{aligned}
S^{max}_1 &= \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \Kphi: Z_1^k {\begin{bmatrix} f \\ g \end{bmatrix}}\in \Kphi \ \forall \
k \in \NN \right \} \
S^{min}_1 = \text{closure }P_{\Phi} { \left[ \begin{array}{c}
\Phi \\ I \end{array} \right]}L^2_{+-}(E) \\
S^{max}_2 &= \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \Kphi: Z_2^k {\begin{bmatrix} f \\ g \end{bmatrix}}\in \Kphi \ \forall \
k \in \NN \right \} \
S^{min}_2 = \text{closure }P_{\Phi} { \left[ \begin{array}{c}
\Phi \\ I \end{array} \right]}L^2_{-+}(E),
\end{aligned}
\nomenclature{$S_j^{max}, S_j^{min}$}{Subspaces of the scattering subspace}$$ where each closure is taken in $\Kphi.$ Then, each $S^{max}_j$ and $S^{min}_j$ is invariant under multiplication by $Z_j$ and $$\begin{aligned}
\label{eqn:kdecomp} \Kphi = S^{max}_1 \oplus
S^{min}_2 = S^{min}_1 \oplus S^{max}_2. \end{aligned}$$
Our first observation is that $S_1^{max}$ is equal to $$\begin{gathered}
\left(\begin{bmatrix} I \\ \Phi^* \end{bmatrix} L^2\ominus
H^2(E_*)\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{\bullet+}(E)\right)^{\perp} \\
=
\left\{ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \scrH: f \in H^2(E_*), g \in
L^2_{\bullet-}(E)\right\}\end{gathered}$$ since $Z_1^k {\begin{bmatrix} f \\ g \end{bmatrix}}\perp \begin{bmatrix} \Phi \\ I \end{bmatrix}
H^2(E)$ for all $k\geq 0$ if and only if ${\begin{bmatrix} f \\ g \end{bmatrix}}\perp \begin{bmatrix}
\Phi \\ I \end{bmatrix} L^2_{\bullet+}(E)$, which is equivalent to saying $g \in L^2_{\bullet-}(E)$. Therefore, $S_1^{max}$ is equal to $$\begin{gathered}
\left(\begin{bmatrix} I \\ \Phi^* \end{bmatrix} L^2\ominus
H^2(E_*)\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} H^2(E)\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{-+}(E)\right)^{\perp} \\
= \Kphi \ominus
P_{\Phi} \left(\begin{bmatrix} \Phi \\ I \end{bmatrix}
L^2_{-+}(E)\right). \end{gathered}$$ Hence, $$\Kphi \ominus S_{1}^{max} = \text{closure } P_{\Phi}
\left(\begin{bmatrix} \Phi \\ I \end{bmatrix} L^2_{-+}(E)\right)
=S_2^{min},$$ which shows $\Kphi = S_1^{max} \oplus S_2^{min}$ and similarly $\Kphi
= S_1^{min} \oplus S_2^{max}$. It is also clear that $S_j^{max}$ is invariant under $Z_j$ for $j=1,2$. Showing the same is true for $S_j^{min}$ requires more work. Define the following subspace of $\scrH$ $${\mathcal{Q}}= \left(\begin{bmatrix} I \\ \Phi^* \end{bmatrix} L^2_{\bullet -}(E_*)
\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{\bullet+}(E)\right)^{\perp}$$ and notice that ${\mathcal{Q}}$ is invariant under both $Z_1$ and $\bar{Z}_1$. Projection onto ${\mathcal{Q}}$ is given by $$P_{{\mathcal{Q}}} = \begin{bmatrix} P_{\bullet+} & -\Phi P_{\bullet+} \\
-\Phi^* P_{\bullet -} & P_{\bullet-} \end{bmatrix}$$ where $P_{\bullet\pm}$ is projection onto the appropriate $L^2_{\bullet\pm}$ space; the proof of this fact is similar to the proof of the formula for $P_{\Phi}$. Now it can be directly checked that $$P_{\Phi} \left(\begin{bmatrix} \Phi \\ I \end{bmatrix}
L^2_{+-}(E)\right) = P_{{\mathcal{Q}}} \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{+-}(E)\right).$$ The key things to notice are that since $\Phi L^2_{+-}(E) \subset
L^2_{+\bullet}(E_*)$, it follows that $P_{\bullet+} \Phi
L^2_{+-}(E) = P_{+} \Phi L^2_{+-}(E)$, $P_{\bullet+}L^2_{+-} =0 =
P_{+}L^2_{+-}$, $P_{\bullet-} \Phi L^2_{+-}(E) = P_{-} \Phi
L^2_{+-}(E)$, and $P_{\bullet-} L^2_{+-}(E)= P_{-} L^2_{+-}$. However, since ${\mathcal{Q}}$ is invariant under $Z_1$ and $\bar{Z}_1$, it follows that $P_{{\mathcal{Q}}}$ commutes with $Z_1$. Since $\begin{bmatrix}
\Phi \\ I \end{bmatrix} L^2_{+-}(E)$ is invariant under $Z_1$, we see that $$P_{{\mathcal{Q}}} \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{+-}(E)\right)$$ is invariant under $Z_1$, and hence so is its closure. This shows $S_1^{min}$ is invariant under $Z_1$ and the proof that $S_2^{min}$ is invariant under $Z_2$ is similar.
\[defn:residspace\] **The Residual Subspace ${\mathcal{R}}$.** It is also useful to consider the residual subspace ${\mathcal{R}}$ of $\Kphi$ defined initially as $ {\mathcal{R}}:=
S^{max}_1 \ominus S^{min}_1.$ Using the decomposition in (\[eqn:kdecomp\]), it is basically immediate that $${\mathcal{R}}= S^{max}_2 \ominus S^{min}_2 = S^{max}_1
\cap S^{max}_2.$$
Constructing Agler Decompositions {#sect:construction}
=================================
Connections between $\Kphi$ and $\Hphi$
---------------------------------------
The decompositions of $\Kphi$ into $S^{max}_j$ and $S^{min}_j$ can be used to construct similar decompositions of $\Hphi.$ The following results link $\Kphi$ and $\Hphi$.
\[lem:isom\] There is an isometry $V: \Hphi \rightarrow \Kphi$ such that $$\begin{aligned} Vf = {\begin{bmatrix} f \\ g \end{bmatrix}}\text{ for some } g \in L^2 \ominus H^2(E) \text{ and }
V^* {\begin{bmatrix} f \\ g \end{bmatrix}}= f \ \ \forall g \text{ with } {\begin{bmatrix} f \\ g \end{bmatrix}}\in \Kphi.
\end{aligned}$$
As was mentioned in Example \[ex:hphi\], the set $\Dphi^2 H^2(E_*)$ is dense in $\Hphi$. Define the operator $V$ on $\Dphi^2 H^2(E_*)$ by $$V \Dphi^2 h = P_{\Phi} { \left[ \begin{array}{c}
I \\ \Phi^* \end{array} \right]}h \qquad \forall \ h \in H^2(E_*).$$ Notice that this equals $$\begin{bmatrix} P_{+} & -\Phi P_{+} \\ -\Phi^* P_{-} &
P_{-} \end{bmatrix} \begin{bmatrix} I \\ \Phi^* \end{bmatrix} h =
\left[ \begin{array}{c}
\Dphi^2 h \\
P_- \Phi^* h \end{array}
\right]
= {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\begin{bmatrix} h \\
-P_{+} \Phi^* h \end{bmatrix}.$$ The computation $$\begin{aligned}
\left \|{\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\begin{bmatrix} h \\
-P_{+} \Phi^* h \end{bmatrix} \right\|^2_{\scrH} &= {\left\langle {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\begin{bmatrix} h \\
-P_{+} \Phi^* h \end{bmatrix}, \begin{bmatrix} h \\
-P_{+} \Phi^* h \end{bmatrix} \right\rangle}_{L^2(E_*) \oplus L^2(E)} \\
&=
{\left\langle \begin{bmatrix} \Dphi^2 h \\ P_{-} \Phi^* h\end{bmatrix}, \begin{bmatrix} h \\
-P_{+} \Phi^* h \end{bmatrix} \right\rangle}_{L^2(E_*) \oplus L^2(E)} \\
&= {\left\langle \Dphi^2 h, h \right\rangle}_{L^2(E_{*}) }\\
&= \|\Dphi^2 h\|^2_{\Hphi}
\end{aligned}$$ at once shows that $V$ is well-defined ($\Dphi^2 h=0$ implies $V\Dphi^2 h = 0$) and isometric, and therefore extends to an isometry from $\Hphi$ to $\Kphi$. To see that the first component of $Vf$ is always $f$, it suffices to notice that since the projection $\pi:{\begin{bmatrix} f \\ g \end{bmatrix}}\mapsto f$ is bounded from $\scrH$ to $L^2(E_*)$ and since we have $\pi V f = f$ for the dense set of $f
\in \Dphi^2 H^2(E_*)$, the identity $\pi V f = f$ must hold for all $f\in \Hphi$ by boundedness of $\pi V$.
Now, $V^*$ is a partial isometry from $\K_{\phi}$ onto $\Hphi$, and $$\text{ker } V^* =(\text{range } V)^{\perp} = \left\{ \begin{bmatrix} 0
\\ g \end{bmatrix} : g \in L^2\ominus H^2(E) \cap \Delta L^2(E)\right\}.$$ The latter equality can be seen from the following computation. If ${\begin{bmatrix} f \\ g \end{bmatrix}}\in \Kphi$ is orthogonal to the range of $V$, then for any $h \in
H^2(E_*)$ $$\begin{aligned}
0={\left\langle {\begin{bmatrix} I & \Phi \\ \Phi^* & I \end{bmatrix}}\begin{bmatrix} h \\ -P_{+} \Phi^* h \end{bmatrix}, \begin{bmatrix} f\\ g \end{bmatrix} \right\rangle}_{\Kphi} &=
{\left\langle \begin{bmatrix} h \\ -P_{+} \Phi^* h \end{bmatrix}, \begin{bmatrix} f\\ g \end{bmatrix} \right\rangle}_{L^2(E_*)\oplus L^2(E)}\\
& = {\left\langle h, f \right\rangle}_{L^2(E_*)},
\end{aligned}$$ since $f \in H^2(E_*)$ and $g \in L^2\ominus H^2(E).$ Upon setting $h=f$, this yields $f=0$. On the other hand, the above computation shows that if $\begin{bmatrix} 0 \\ g \end{bmatrix} \in
\Kphi,$ then this element is orthogonal to the range of $V$. So, the action of $V^*$ on $\Kphi$ can be directly computed as follows. Any ${\begin{bmatrix} f \\ g \end{bmatrix}}\in \Kphi$ can be written as $Vf + \begin{bmatrix} 0
\\ h \end{bmatrix}$ for some $h \in L^2\ominus H^2(E) \cap \Delta L^2(E)$. Then, $V^* {\begin{bmatrix} f \\ g \end{bmatrix}}= f$.
An immediate corollary of the above theorem is:
As sets, $\Hphi = \left \{ f \in H^2(E_*): \text{ there is a } g \text{ with } {\begin{bmatrix} f \\ g \end{bmatrix}}\in
\Kphi \right \}.$
Hilbert Spaces in $\Hphi$
-------------------------
Using the partial isometry $V^*$ and the decompositions of $\Kphi$ given in Theorem \[thm:kdecomp\], we can construct Hilbert spaces yielding Agler decompositions. First, we make some general observations. Let $K$ be a closed subspace of $\Kphi$, and denote the operator range of $V^*|_K$ by $H_K$. Then, $f \in H_K$ if and only if there exists $g$ such that ${\begin{bmatrix} f \\ g \end{bmatrix}}\in K$. Essentially by the definition of operator range, $V^*\mid_{K}$ is a unitary from $K\ominus (K\cap
\ker V^*)$ onto $H_K$, and the inverse of this unitary will be of the form $f \mapsto {\begin{bmatrix} f \\ A_K f \end{bmatrix}}$ where $A_K:H_K \to L^2(E)$ is some linear operator. By , $A_K$ is contractive, i.e. : $$\label{eqn:genhk}
\|A_K f\|_{L^2(E)} \leq \left\| {\begin{bmatrix} f \\ A_K f \end{bmatrix}}\right\|_{\scrH} =
\|f\|_{H_{K}}$$ and it is worth pointing out the following representation of the norm $$\|f\|_{H_K} = \min \left\{ \left\|
{\begin{bmatrix} f \\ g \end{bmatrix}}\right\|_{\scrH}: g \text{ satisfies } {\begin{bmatrix} f \\ g \end{bmatrix}}\in K\right\}.$$ Let $$k_w (z) = \frac{I}{(1-z_1\bar{w}_1)(1-z_2\bar{w}_2)}$$ be the Szegő kernel on $H^2(E_{*})$.
\[lem:opkern\] The reproducing kernel for $H_K$ is given by $$V^* P_{K} V \Dphi^2 k_w(z).$$ Moreover, if $K$ is an orthogonal direct sum, $K =
\bigoplus_{j=1}^{\infty} K_j$, then the reproducing kernel for $H_K$ is the sum of the reproducing kernels for $H_{K_j}$.
Take any $f\in H_K$; this means $f = V^* {\begin{bmatrix} f \\ g \end{bmatrix}}$, for some ${\begin{bmatrix} f \\ g \end{bmatrix}}\in K\ominus \left [ K\cap \ker V^*
\right]$. Then, for $w\in \D^2$ and $v \in E_*$ $$\begin{aligned}
{\left\langle f, V^* P_{K} V \Dphi^2 k_w v \right\rangle}_{H_K} &= {\left\langle {\begin{bmatrix} f \\ g \end{bmatrix}}, V \Dphi^2 k_w
v \right\rangle}_{\Kphi} \\
&= {\left\langle V^* {\begin{bmatrix} f \\ g \end{bmatrix}}, \Dphi^2 k_ w v \right\rangle}_{\Hphi} \\
&= {\left\langle f, k_w v \right\rangle}_{H^2(E_*)} =
{\left\langle f(w), v \right\rangle}_{E_*}.
\end{aligned}$$ The assertion about direct sums follows from noticing $P_K =
\sum_{j=1}^{\infty} P_{K_j}$ in the strong operator topology.
The Hilbert spaces of primary interest are defined as follows:
Define the Hilbert spaces $H^{max}_j$ and $H^{min}_j$ to be the operator ranges of $V^*|_{S^{max}_j}$ and $V^*|_{S^{min}_j}$. Then $$f \in H^{max}_j \text{ if and only if } \exists \ g \text{ with }
{\begin{bmatrix} f \\ g \end{bmatrix}}\in S^{max}_j,$$ and the $H^{max}_j$ norm is given by $$\| f \|_{H^{max}_j} := \left \| P_{S^{max}_j \ominus \left
[S^{max}_j \cap \ker V^* \right ]} {\begin{bmatrix} f \\ g \end{bmatrix}}\right \|_{S^{max}_j}
= \min \left \{ \left \| \begin{bmatrix}
f \\
\tilde{g}
\end{bmatrix} \right \|_{S^{max}_j} : \begin{bmatrix}
f \\
\tilde{g}
\end{bmatrix} \in S^{max}_j \right \}.$$
$$S^{max}_j = \bigoplus_{n \in \NN} Z_j^n \big( S^{max}_j \ominus Z_j S^{max}_j \big )
\ \oplus M_j^{max}$$ $$S^{min}_j = \bigoplus_{n \in \NN} Z_j^n \big( S^{min}_j \ominus Z_j S^{min}_j \big )
\ \oplus M^{min}_j$$ where $M_j^{max}, M_j^{min} \subset \ker V^*$.
Since multiplication by $Z_j$ is an isometry on $S^{max/min}_j$, the classical Wold decomposition says that $S_j^{max}, S_j^{min}$ can be decomposed as above where $$M_j^{max} = \bigcap_{n\geq 0} Z_j^n S_j^{max} \text{ and } M_j^{min} =
\bigcap_{n\geq 0} Z_j^n S_j^{min}$$ so the only thing to show is $M_j^{max} \subset \ker V^*$, since $M_j^{min} \subset M_j^{max}$. So, if ${\begin{bmatrix} f \\ g \end{bmatrix}}\in \bigcap_{n\geq 0}
Z_1^n S_1^{max}$, then $\bar{Z}_1^n f \in H^2(E_*)$ for all $n\geq 0$, which can only happen if $f=0$. This shows ${\begin{bmatrix} f \\ g \end{bmatrix}}\in \ker V^*$.
\[lem:hkernels\] Let $K_j^{max}, K_j^{min}$ be the reproducing kernels for the operator ranges of $V^*\mid_{S_j^{max} \ominus Z_j S_j^{max}}, V^*\mid_{S_j^{min} \ominus Z_j S_j^{min}}$. Then, the reproducing kernels for $H_j^{max}$ and $H_j^{min}$ are given by $$\frac{K_j^{max}(z,w)}{1-z_j \bar{w}_j} \text{ and }
\frac{K_j^{min}(z,w)}{1-z_j \bar{w}_j}.$$ In addition, if $G$ is the reproducing kernel for the operator range of $V^*|_{{\mathcal{R}}}$, then $$\label{eqn:KKG}
\frac{K_j^{max}(z,w)}{1-z_j \bar{w}_j} = \frac{K_j^{min}(z,w)}{1-z_j
\bar{w}_j} + G(z,w).$$
We can focus on $H_1^{max}$ which has reproducing kernel $V^*
P_{S_1^{max}} V \Dphi^2 k_w$ by previous remarks. Let $P_1$ denote orthogonal projection onto $S_1^{max} \ominus Z_1 S_1^{max}$. Then, orthogonal projection onto $Z_1^n(S_1^{max} \ominus Z_1 S_1^{max})$ is given by $Z_1^nP_1 \bar{Z}_1^n$. We now claim that the reproducing kernel for the operator range of $V^*$ restricted to $Z_1^n(S_1^{max}
\ominus Z_1 S_1^{max})$ satisfies $$V^*Z_1^n P_1 \bar{Z}_1^n V \Dphi^2 k_{w}v = \bar{w}_1^n Z_1^n V^* P_1
V \Dphi^2 k_{w}v.$$ Now for ${\begin{bmatrix} f \\ g \end{bmatrix}}\in S_1^{max},$ we have $Z_1^n V^* {\begin{bmatrix} f \\ g \end{bmatrix}}= V^* Z_1^n
{\begin{bmatrix} f \\ g \end{bmatrix}}$. This means $V^* Z_1^n P_1 = Z_1^n V^* P_1$ and so, for any $f
\in \Hphi$, $v \in E_*$, $$\begin{aligned}
{\left\langle f, V^* Z_1^n P_1 \bar{Z}_1^n V \Dphi^2 k_{w}v \right\rangle}_{\Hphi} &=
{\left\langle V^* Z_1^n P_1 \bar{Z}_1^n V f, \Dphi^2 k_{w}v \right\rangle}_{\Hphi} \\
&= {\left\langle Z_1^n V^* P_1 \bar{Z}_1^n V f, \Dphi^2 k_{w}v \right\rangle}_{\Hphi}
\\
&= w_1^n {\left\langle f, V^* Z_1^n P_1 V \Dphi^2 k_{w}v \right\rangle}_{\Hphi} \\
&= {\left\langle f, \bar{w}_1^n Z_1^n V^* P_1 V \Dphi^2
k_{w}v \right\rangle}_{\Hphi},
\end{aligned}$$ so that $V^*Z_1^n P_1 \bar{Z}_1^n V \Dphi^2 k_{w}v = \bar{w}_1^n Z_1^n
V^* P_1 V \Dphi^2 k_{w}v$. If we break up $S_1^{max}$ according to its Wold decomposition, then since $V^*$ annihilates $M_1^{max}$, then Lemma \[lem:opkern\] implies that the reproducing kernel of $H_1^{max}$ is given by $$\sum_{n \geq 0} \bar{w}_1^n z_1^n V^* P_1 V \Dphi^2 k_{w}(z) =
\frac{V^* P_1 V \Dphi^2 k_{w}(z)}{1-z_1 \bar{w}_1} =
\frac{K_1^{max}(z,w)}{1-z_1\bar{w}_1}.$$ The formulas for $H_2^{max}$ as well as the $H_j^{min}$ follow similarly. The formula follows from the orthogonal decomposition $S_j^{max} = S_j^{min} \oplus {\mathcal{R}}$ and Lemma \[lem:opkern\].
Construction of Agler Kernels
-----------------------------
As above, let $K_j^{max}, K_j^{min}$ be the reproducing kernels for the operator ranges of $V^*|_{S_j^{max} \ominus Z_j S_j^{max}}$ and $V^*|_{S_j^{min} \ominus Z_j S_j^{min}}$ respectively.
\[thm:agdecomp\]
The pairs $(K^{max}_1, K^{min}_2)$ and $(K^{min}_1, K^{max}_2)$ are Agler kernels of $\Phi$, i.e. for all $z,w \in \D^2,$ $$\label{eqn:hdecomp}
\frac{I_{E_*} - \Phi(z) \Phi(w)^*}{(1-z_1\bar{w}_1)(1-z_2\bar{w}_2)}
\ = \ \frac{K^{max}_1(z,w)}{1-z_1\bar{w}_1} + \frac{K^{min}_2(z,w)}{1-
z_2\bar{w}_2} \ = \ \frac{K^{min}_1(z,w)}{1-z_1\bar{w}_1} +
\frac{K^{max}_2(z,w)} {1-z_2\bar{w}_2}.$$
The reproducing kernel of $\Hphi$, namely $$\Dphi^2 k_w(z) =
\frac{I_{E_*}-\Phi(z)\Phi(w)^*}{(1-z_1\bar{w}_1)(1-z_2\bar{w}_2)}$$ is the sum of the kernels for $H_1^{max}$ and $H_2^{min}$ by Lemma \[lem:opkern\], and these kernels are given by $$V^* P_{S_1^{max}} V \Dphi^2 k_w(z) \text{ and } V^* P_{S_2^{min}} V
\Dphi^2 k_w(z).$$ By Lemma \[lem:hkernels\], these kernels can be computed directly in terms of the reproducing kernels of $K_1^{max}$ and $K_2^{min}$ to give us the formula .
We remark that by we get the formula $$\frac{I_{E_*}-\Phi(z)\Phi(w)^*}{(1-z_1\bar{w}_1)(1-z_2\bar{w}_2)}
= \frac{K_1^{min}(z,w)}{1-z_1\bar{w}_1} + \frac{K_2^{min}(z,w)}{1-z_2 \bar{w}_2}
+ G(z,w)
$$ where $G(z,w) = V^* P_{{\mathcal{R}}} V\Dphi^2 k_{w}(z)$ is the reproducing kernel of $H_{{\mathcal{R}}}$, the operator range of $V^*|_{{\mathcal{R}}}.$
General Agler Kernels
=====================
Characterizations of General Agler Kernels {#sect:characterization}
------------------------------------------
Assume $(K_1, K_2)$ are Agler kernels of $\Phi \in
\mathcal{S}_2(E, E_*)$ and define the Hilbert spaces $$\label{eqn:hspaces} H_1 : = \mathcal{H}
\left( \frac{K_1(z,w) }{1-z_1\bar{w}_1} \right) \text{ and } H_2
: = \mathcal{H} \left( \frac{K_2(z,w) }{1-z_2\bar{w}_2}
\right).$$ Our goal is to use these auxiliary Hilbert spaces $H_1$ and $H_2$ to characterize $(K_1,K_2)$ in terms of the extremal kernels $K^{max/min}_1$ and $K^{max/min}_2.$ The first main result is the following theorem:
\[thm:mincontain\] Let $\Phi \in
\mathcal{S}_2(E,E_*)$ and let $(K_1,K_2)$ be Agler kernels of $\Phi$. Define $H_1, H_2$ as in . Then $$H_1 \subseteq H_1^{max} \ \text{ and } \ H_2 \subseteq
H_2^{max}$$ and these containments are contractive, i.e. for $j=1,2$ $$\|f \|_{H_j^{max}} \le \| f \|_{H_j} \ \qquad \forall \ f \in
H_j.$$
Let $f \in H_1$ and assume $\|f\|_{H_1} = 1$. Then for all $n\geq 0$, $Z_1^n f \in H_1 \subset \Hphi$ and $\|Z_1^n f\|_{\Hphi} \leq \|Z_1^n
f\|_{H_1} \leq 1$, since multiplication by $Z_1$ is a contraction in $H_1$. For each $n$ we can choose $g_n \in L^2\ominus H^2(E)$ such that ${\begin{bmatrix} Z_1^n f \\ g_n \end{bmatrix}} \in \Kphi \ominus \ker V^*$ and $$\left\|{\begin{bmatrix} f \\ \bar{Z}_1^n g_n \end{bmatrix}}\right\|_{\scrH} = \left\|{\begin{bmatrix} Z_1^n f \\ g_n \end{bmatrix}}\right\|_{\scrH} =
\|Z_1^n f\|_{\Hphi}
\leq 1.$$ Notice $F_n:= {\begin{bmatrix} f \\ \bar{Z}_1^n g_n \end{bmatrix}} \in \Kphi \ominus \bar{Z}_1^n
\ker V^*,$ since $\bar{Z}_1^n g_n \in \bar{Z}_1^n (L^2\ominus H^2(E))$. The sequence $\{F_n\}\subset \Kphi$ is bounded in norm and therefore has a subsequence $\{F_{n_j}\}$ that converges weakly to some $F:=
{ \left[ \begin{array}{c}
f' \\ g' \end{array} \right]}$. We claim that $f=f'$ and $g' \in L^2_{\bullet-}(E)$. Since $${\left\langle F_{n_j}, { \left[ \begin{array}{c}
I \\ \Phi^* \end{array} \right]}h \right\rangle}_{\scrH} = {\left\langle f, h \right\rangle}_{L^2(E_*)} \to
{\left\langle F, { \left[ \begin{array}{c}
I \\ \Phi^* \end{array} \right]}h \right\rangle}_{\scrH} = {\left\langle f', h \right\rangle}_{L^2(E_*)} \text{ as } j \to
\infty$$ for all $h \in L^2(E_*)$, we see that $f=f'$. Next, for any $v \in E$ and $n\in \ZZ, m\geq 0$ $${\left\langle F_{n_j}, { \left[ \begin{array}{c}
\Phi \\ I \end{array} \right]}Z_1^n Z_2^mv \right\rangle}_{\scrH} = {\left\langle \bar{Z}_1^{n_j}
g_{n_j}, Z_1^n Z_2^mv \right\rangle}_{L^2(E)} = 0$$ for $j$ large enough that $n_j + n \geq 0$ since $g_{n_j} \perp
H^2(E)$. By weak convergence, the above expression converges to $${\left\langle F, { \left[ \begin{array}{c}
\Phi \\ I \end{array} \right]}Z_1^n Z_2^mv \right\rangle}_{\scrH} = {\left\langle g', Z_1^nZ_2^m v \right\rangle}_{L^2(E)}
= {\left\langle \widehat{g'}(n,m), v \right\rangle}_{E} = 0$$ so we see that $g' \perp L^2_{\bullet+}(E)$ and therefore $g' \in
L^2_{\bullet-}(E)$. Hence we conclude that $$F = {\begin{bmatrix} f \\ g' \end{bmatrix}} \in S_1^{max}$$ and so $f = V^{*}F$ must be in $H_1^{max}$. To show $\|f\|_{H_1^{max}} \leq 1$, observe that $$|{\left\langle F_{n_j}, F \right\rangle}_{\scrH}| \to \|F\|^2_{\scrH}$$ and $$|{\left\langle F_{n_j}, F \right\rangle}_{\scrH}| \leq \|F_{n_j}\|_{\scrH} \|F \|_{\scrH} \leq
\|F\|_{\scrH}$$ so that $\| F\|_{\scrH} \leq 1$. Finally, $\|f\|_{H_1^{max}} \leq \|
F\|_{\scrH} \leq 1$ as desired. Thus, $H_1$ is contractively contained in $H_1^{max}$.
Using the previous result, it is possible to characterize all Agler kernels in terms of the canonical kernels $K^{min}_1$, $K^{min}_2$ and $G$ as follows:
\[thm:maxmin\] Let $\Phi \in \mathcal{S}_2(E, E_*)$ and let $K_1, K_2: \D^2 \times \D^2 \rightarrow \mathcal{L}(E_*)$. Then $(K_1, K_2)$ are Agler kernels of $\Phi$ if and only if there are positive kernels $G_1,G_2: \D^2 \times \D^2 \rightarrow \mathcal{L}(E_*)$ such that $$\begin{aligned}
K_1(z,w) =& K_1^{min}(z,w) + (1-z_1 \bar{w}_1) G_1(z,w) \\
K_2(z,w) =& K_2^{min}(z,w) + (1-z_2 \bar{w}_2) G_2(z,w)
\end{aligned}$$ and $G = G_1 + G_2.$
($\Rightarrow$) Assume $(K_1,K_2)$ are Agler kernels of $\Phi$. By Theorem \[thm:mincontain\] and Theorem \[thm:kerdiff\], there are positive kernels $G_1, G_2: \D^2 \times \D^2 \rightarrow
\mathcal{L}(E_*)$ such that $$\begin{aligned}
G_1(z,w) &= \frac{K_1^{max}(z,w)}{1-z_1\bar{w}_1} -
\frac{K_1(z,w)}{1-z_1\bar{w}_1} =\frac{K_1(z,w)}{1-z_1\bar{w}_1} -
\frac{K^{min}_1(z,w)}{1-z_1\bar{w}_1} \\
G_2(z,w) &= \frac{K_2^{max}(z,w)}{1-z_2\bar{w}_2} -
\frac{K_2(z,w)}{1-z_2\bar{w}_2}= \frac{K_2(z,w)}{1-z_2\bar{w}_2} -
\frac{K^{min}_2(z,w)}{1-z_2\bar{w}_2}.\end{aligned}$$ To show $G_1 + G_2 = G$, recall that since $(K_1,K_2)$ are Agler kernels of $\Phi$, $$\begin{aligned}
\frac{K_1^{min}(z,w)}{1-z_1 \bar{w}_1 }+ G_1(z,w) +
\frac{K_2^{min}(z,w)}{1-z_2 \bar{w}_2} &+ G_2(z,w)
= \frac{K_1(z,w)}{1-z_1 \bar{w}_1} + \frac{K_2(z,w)}
{1-z_2 \bar{w}_2} \\
&\\
& = \frac{ I_{E_*} - \Phi(z) \Phi(w)^*}{(1-z_1 \bar{w}_1)
(1-z_2 \bar{w}_2)} \\
&\\
& = \frac{K_1^{min}(z,w)}{1-z_1 \bar{w}_1 } +\frac{K_2^{min}
(z,w)}{1-z_2 \bar{w}_2}+ G(z,w),
\end{aligned}$$ which implies $G = G_1 +G_2.$\
($\Leftarrow$) Now assume $(K_1,K_2)$ are positive kernels with positive kernels $G_1,G_2: \D^2 \times
\D^2 \rightarrow \mathcal{L}(E_*)$ satisfying $$\begin{aligned}
K_j(z,w) =& K_j^{min}(z,w) + (1-z_j
\bar{w}_j) G_j(z,w)
\end{aligned}$$ for $j=1,2$ and $G= G_1 +G_2.$ Then $$\begin{aligned} \frac{K_1(z,w)}{1-z_1 \bar{w}_1} +
\frac{K_2(z,w)}{1-z_2 \bar{w}_2}
& = \frac{K_1^{min}(z,w)}{1-z_1 \bar{w}_1 } +
\frac{K_2^{min}(z,w)}{1-z_2 \bar{w}_2}+ G(z,w) \\
&\\
& =\frac{ I_{E_*} - \Phi(z) \Phi(w)^*}{(1-z_1
\bar{w}_1)(1-z_2 \bar{w}_2)},
\end{aligned}$$ which implies $(K_1,K_2)$ are Agler kernels of $\Phi.$
Containment Properties of $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ {#sect:functions}
-------------------------------------------------------------------
In this section, we consider the set of functions that can be contained in $\mathcal{H}(K_1)$ or $\mathcal{H}(K_2).$ This result generalizes a result about inner functions from [@bk12]. We require two additional subspaces ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ of $\scrH$, defined as follows: $${\mathcal{R}}_j = \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}: f \in H^2(E_*), \ g \in Z_j L^2_{-
-}(E), f-\Phi g \in \Delta_{*} L^2(E_*) \right \}$$ for $j=1,2.$ These are slight enlargements of the residual subspace ${\mathcal{R}}.$ We can now state the result:
\[thm:containment\] Let $\Phi \in \mathcal{S}_2(E,E_*)$. Then for $j=1,2$ $$\begin{aligned}
{ \mathcal{H}(K^{max}_j)}&= \left \{ f : \text{ there exists } g \text{ with }
{\begin{bmatrix} f \\ g \end{bmatrix}}\in {\mathcal{R}}_j \ominus Z_j {\mathcal{R}}\right \} \\ {\mathcal{H}(K^{min}_j)}&=
\left \{ f : \text{ there exists } g \text{ with } {\begin{bmatrix} f \\ g \end{bmatrix}}\in
{\mathcal{R}}_j \ominus {\mathcal{R}}\right \}. \end{aligned}$$ If $(K_1,K_2)$ are general Agler kernels of $\Phi,$ then for $j=1,2$ $$\begin{aligned}
\mathcal{H}(K_j) &\subseteq \left \{ f : \text{ there exists } g \text{ with }
{\begin{bmatrix} f \\ g \end{bmatrix}}\in {\mathcal{R}}_j \right \} \\
&= \left \{ f \in H^2(E_*): f\in \big ( \Phi
Z_j L^2_{--}(E) + \Delta_* L^2(E_*) \big ) \right \}. \end{aligned}$$
The proof of this result requires several auxiliary results about the functions in $S^{max}_j \ominus Z_j S^{max}_j$ and $S^{min}_j \ominus
Z_j S^{min}_j.$
\[prop:Smin\] For $j=1,2$, the following equality holds: $$S^{min}_j \ominus Z_j S^{min}_j = {\mathcal{R}}_j \ominus {\mathcal{R}}.$$
We prove the result for $S^{min}_1.$ We shall make use of the proof of Theorem \[thm:kdecomp\]. Recall the space ${\mathcal{Q}}$ defined there: $${\mathcal{Q}}= \left(\begin{bmatrix} I \\ \Phi^* \end{bmatrix} L^2_{\bullet -}(E_*)
\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{\bullet+}(E)\right)^{\perp}.$$ We define and manipulate a related space $$\begin{aligned}
{\mathcal{M}}&= \left(\begin{bmatrix} I \\ \Phi^* \end{bmatrix} L^2_{\bullet -}(E_*)
\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2\ominus L^2_{--}(E)\right)^{\perp} \\
&=\left(\begin{bmatrix} I \\ \Phi^* \end{bmatrix} L^2_{\bullet -}(E_*)
\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{\bullet+}(E)\right)^{\perp} \cap \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{+-}(E)\right)^{\perp}\\
&= {\mathcal{Q}}\ominus P_{{\mathcal{Q}}} \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{+-}(E)\right).
\end{aligned}$$ Also, note ${\mathcal{M}}= \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \scrH : f \in L^2_{\bullet +}(E_*), g \in L^2_{- -}(E)\right \}$. Then, $$\begin{aligned}
{\mathcal{Q}}\ominus {\mathcal{M}}&= \text{closure}_{\scrH} P_{{\mathcal{Q}}} \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{+-}(E)\right) \\
&= \text{closure}_{\scrH} P_{\Phi} \left(\begin{bmatrix} \Phi \\
I \end{bmatrix} L^2_{+-}(E)\right) = S_1^{min},
\end{aligned}$$ using the proof of Theorem \[thm:kdecomp\]. Observe that $\mathcal{M} \subseteq Z_1 \mathcal{M} \subseteq
\mathcal{Q}$ and $Z_1 \mathcal{Q} = \mathcal{Q}$. Since multiplication by $Z_1$ is an isometry on $\scrH$, we can calculate $$\begin{aligned} S^{min}_1 \ominus Z_1 S^{min}_1
&= \left( \mathcal{Q} \ominus \mathcal{M} \right) \ominus Z_1
\left( \mathcal{Q} \ominus \mathcal{M} \right)\\
& = \left( \mathcal{Q} \ominus \mathcal{M} \right) \ominus
\left( Z_1 \mathcal{Q} \ominus Z_1 \mathcal{M} \right)\\
& = \left( \mathcal{Q} \ominus \mathcal{M} \right) \ominus
\left( \mathcal{Q} \ominus Z_1 \mathcal{M} \right)\\
& = Z_1 \mathcal{M} \ominus \mathcal{M}.
\end{aligned}$$ As $S^{min}_1 \ominus Z_1 S^{min}_1 \subseteq S^{max}_1$, we can conclude $$\begin{aligned}
S^{min}_1 \ominus Z_1 S^{min}_1 =& \big( Z_1 \mathcal{M}
\cap S^{max}_1 \big) \ominus \big( \mathcal{M} \cap S^{max}_1 \big) \\
=& \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \scrH : f \in H^2(E_*), \ g \in Z_1 L^2_{--}(E) \right \} \\
&\ominus \left \{ {\begin{bmatrix} f \\ g \end{bmatrix}}\in \scrH : f \in H^2(E_*), \ g \in
L^2_{--}(E) \right \} \\
=& {\mathcal{R}}_1 \ominus {\mathcal{R}},
\end{aligned}$$ as desired. The proof follows similarly for $S^{min}_2.$
We also obtain similar characterizations of $S^{max}_j \ominus Z_j S^{max}_j$.
\[prop:smax\] For $j=1,2$ the following equalities hold: $$S^{max}_j \ominus Z_j S^{max}_j = {\mathcal{R}}_j \ominus Z_j {\mathcal{R}}.$$
Recall that $S^{max}_j = {\mathcal{R}}\oplus S_j^{min}$ and ${\mathcal{R}}, Z_j {\mathcal{R}}\subseteq {\mathcal{R}}_j.$ Now $$\begin{aligned}
S_j^{max} &= (S_j^{max} \ominus Z_j S_j^{max}) \oplus Z_j S_j^{max} \\
&= (S_j^{max} \ominus Z_j S_j^{max}) \oplus Z_j {\mathcal{R}}\oplus Z_j
S_j^{min}
\end{aligned}$$ while $S_j^{max}$ can also be decomposed as $$\begin{aligned}
& {\mathcal{R}}\oplus (S_j^{min} \ominus Z_j S_j^{min}) \oplus Z_j S_j^{min}
\\
=& {\mathcal{R}}\oplus ({\mathcal{R}}_j \ominus {\mathcal{R}}) \oplus Z_j S_j^{min} \\
=& {\mathcal{R}}_j \oplus Z_j S_j^{min}.
\end{aligned}$$ Together these show $S_j^{max} \ominus Z_j
S_j^{max} = {\mathcal{R}}_j \ominus Z_j{\mathcal{R}}$.
Now we can prove Theorem \[thm:containment\].
The definitions of ${ \mathcal{H}(K^{max}_j)}$ and ${\mathcal{H}(K^{min}_j)}$ combined with Propositions \[prop:Smin\] and \[prop:smax\] imply that
$$\begin{aligned}
{ \mathcal{H}(K^{max}_j)}&= \left \{ f : \text{ there exists } g \text{ with }
{\begin{bmatrix} f \\ g \end{bmatrix}}\in {\mathcal{R}}_j \ominus Z_j {\mathcal{R}}\right \} \\
{\mathcal{H}(K^{min}_j)}&= \left \{ f : \text{ there exists } g \text{ with }
{\begin{bmatrix} f \\ g \end{bmatrix}}\in {\mathcal{R}}_j \ominus {\mathcal{R}}\right \}, \end{aligned}$$
and then the definition of ${\mathcal{R}}_j$ implies: $$\begin{aligned}
\mathcal{H}(K^{max/min}_j ) &\subseteq \left \{ f : \text{ there exists } g \text{ with }
{\begin{bmatrix} f \\ g \end{bmatrix}}\in {\mathcal{R}}_j \right \} \\
& = \left \{ f \in H^2(E_*): f\in \big ( \Phi Z_j L^2_{--}(E)
+ \Delta_* L^2(E_*) \big ) \right \}.
\end{aligned}$$ Now let $(K_1,K_2)$ be any pair of Agler kernels of $\Phi$. By Theorem \[thm:maxmin\], there are positive kernels $G_1, G_2$ such that each $$K_j(z,w) = K^{min}_j(z,w) + (1-z_j\bar{w}_j)G_j(z,w)$$ and $G= G_1 + G_2.$ This means $$\Big( K^{min}_1 (z,w) + G(z,w) \Big) - K_1(z,w)
= G_2(z,w) + z_1 \bar{w}_1 G_1(z,w)$$ is a positive kernel. Similar results hold for $K_2$, so that Theorem \[thm:kerdiff\] implies $
\mathcal{H}(K_j)$ is contained contractively in $\mathcal{H}(K^{min}_j + G).$ But then, Theorem \[thm:kersum\] implies that each $f \in \mathcal{H}(K_j)$ can be written as $f = f_1 + f_2$, for $f_1 \in {\mathcal{H}(K^{min}_j)}$ and $f_2 \in \mathcal{H}(G).$ Our above arguments give the desired result for $f_1$ and the definition of $\mathcal{H}(G)$ gives the desired result for $f_2.$ This means $$\begin{aligned}
\mathcal{H}(K_j) &\subseteq \left \{ f : \text{ there exists } g \text{ with }
{\begin{bmatrix} f \\ g \end{bmatrix}}\in {\mathcal{R}}_j \right \} \\
& = \left \{ f \in H^2(E_*): f\in \big ( \Phi Z_j L^2_{--}(E)
+ \Delta_* L^2(E_*) \big ) \right \}.
\end{aligned}$$ as desired.
Applications
============
Analytic Extension Theorem {#sect:extensions}
--------------------------
In this section, we restrict to the situation where $E$ and $E_*$ are finite dimensional with equal dimensions, so after fixing orthonormal bases of $E$ and $E_*$, we can assume $\Phi$ is a square matrix of scalar valued $H^\infty (\D^2)$ functions. The containment results in Theorem \[thm:containment\] allow us to give conditions for when such $\Phi$ and the elements of any $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ associated to Agler kernels of $\Phi$ extend analytically past portions of $\partial \D^2$. We first make some preliminary comments about defining functions in the canonical spaces outside of the bidisk.
Any Hilbert space contractively contained in $H^2(E_*)$ clearly has bounded point evaluations at points of $\D^2$. On the other hand, for the spaces ${\mathcal{R}},{\mathcal{R}}_1,{\mathcal{R}}_2$ we can construct points of bounded evaluation at certain points of ${\mathbb{E}}^2$, where ${\mathbb{E}}= {\mathbb{C}}\setminus
\overline{\D}$. Using the notation of , there is a unitary map from $H_{{\mathcal{R}}}$ onto ${\mathcal{R}}\ominus ({\mathcal{R}}\cap \ker V^*)$ of the form $$f \mapsto {\begin{bmatrix} f \\ A_{{\mathcal{R}}} f \end{bmatrix}}$$ where $A_{{\mathcal{R}}}$ is a contractive linear map from $H_{{\mathcal{R}}}$ to $L^2_{--}(E)$. If $f\in H_{\mathcal{R}}$, then ${\begin{bmatrix} f \\ A_{{\mathcal{R}}} f \end{bmatrix}} \in \scrH$ and so $$f = \Phi A_{{\mathcal{R}}} f + (I-\Phi \Phi^*)^{1/2} h \text { by } \eqref{eqn:hphichar}$$ for some $h \in L^2(E_*)$. Let $$S = \{z \in {\mathbb{E}}^2: \Phi(1/\bar{z}) \text{ is not invertible} \}.$$ Since $A_{{\mathcal{R}}} f \in L^2_{--}(E)$, we can write $A_{{\mathcal{R}}} f = \overline{Z_1 Z_2 g}$ for $g \in H^2(E)$ and then evaluation at $z \in {\mathbb{E}}^2 \setminus S$ is defined by $$\label{extdef}
f(z) := (\Phi(1/\bar{z})^*)^{-1} \frac{1}{z_1
z_2}\overline{g(1/\bar{z})}.$$ Since $\D^2$ and ${\mathbb{E}}^2$ are disjoint, for the moment this is just a formal definition. However, with additional assumptions on $\Phi$, it is this definition of $f$ in ${\mathbb{E}}^2$ that provides a holomorphic extension of $f$. This evaluation is bounded since $|g(1/\bar{z})|
\leq C \|g\|_{H^2(E)} = C \|A_{{\mathcal{R}}} f \|_{L^2(E)}$ for some $C>0$ and then $$|f(z)| \leq C \frac{1}{|z_1z_2|}\|(\Phi(1/\bar{z})^*)^{-1}\|
\|A_{{\mathcal{R}}} f\|_{L^2(E)} \leq C
\frac{1}{|z_1z_2|}\|(\Phi(1/\bar{z})^*)^{-1}\| \| f\|_{H_{{\mathcal{R}}}}.$$ This shows evaluation at $z \in {\mathbb{E}}^2\setminus S$ is a bounded linear functional of $H_{{\mathcal{R}}} = \mathcal{H}(G)$.
Analogous analysis can be applied to ${\mathcal{R}}_1,{\mathcal{R}}_2$ so that $H_{{\mathcal{R}}_1}, H_{{\mathcal{R}}_2}$ possess bounded point evaluations at points of ${\mathbb{E}}^2 \setminus S$. In the case of $f \in H_{{\mathcal{R}}_1}$, since $A_{{\mathcal{R}}_1}f \in Z_1 L^2_{--}$, we can write $f = Z_1 \overline{Z_1Z_2
g} = \bar{Z}_2 \bar{g}$ for some $g \in H^2(E_*)$ and then we replace with $$f(z) := (\Phi(1/\bar{z})^*)^{-1} \frac{1}{z_2}\overline{g(1/\bar{z})}$$ for $z \in {\mathbb{E}}^2 \setminus S$. For $H_{{\mathcal{R}}_2}$ we simply switch the roles of $z_1,z_2$. Since $\mathcal{H}(K_j^{max/min})$ is contractively contained in $H_{{\mathcal{R}}_j}$, we can define point evaluations at points of ${\mathbb{E}}^2\setminus S$ for the canonical Agler kernel spaces as well.
We proceed to study analytic extensions of $\Phi$ past the boundary. Let $X \subseteq \mathbb{T}^2$ be an open set and define the related sets $$\begin{aligned} X_1 & := \left \{ x_1 \in \mathbb{T} : \text{ such that }
\exists \ x_2 \text{ with } (x_1, x_2) \in X \right \} \\
X_2 & := \left \{ x_2 \in \mathbb{T} : \text{ such that } \exists \ x_1
\text{ with } (x_1, x_2) \in X \right \} .
\end{aligned}$$ Then we have the following result:
\[thm:extension\] Let $\Phi \in \mathcal{S}_2(E, E_*)$ be square matrix valued. Then the following are equivalent:
- $\Phi$ extends continuously to $X$ and $\Phi$ is unitary valued on $X$.
- There is some pair $(K_1,K_2)$ of Agler kernels of $\Phi$ such that the elements of $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ extend continuously to $X.$
- There exists a domain $\Omega$ containing \^2 X (X\_1 ) (X\_2) (\^2 S ) such that $\Phi$ and the elements of $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ extend analytically to $\Omega$ for every pair $(K_1, K_2)$ of Agler kernels of $\Phi.$ Moreover the points in the set $\Omega$ are points of bounded evaluation of every $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2).$
We prove $(i) \Rightarrow (iii) \Rightarrow (ii) \Rightarrow (i).$ A similar result for inner functions appears as Theorem 1.5 in [@bk12]. Many of the arguments in this situation are similar. Thus, we outline the proof and provide more details on the points where the two proofs diverge.
Since most of the work occurs in $(i) \Rightarrow (iii),$ let us consider this implication first. The proof involves $3$ claims.\
**Claim 1: $\Phi$ extends analytically to $\Omega.$**\
Since $\Phi$ extends continuously to $X$ and is unitary valued there, there is a neighborhood $W^+ \subseteq \D^2$ such that $\Phi$ is invertible on $W^+$ and $X \subseteq \overline{ W^+}.$ Then $$\label{eqn:phiextend}
\Phi(z): = \left[ \Phi \left ( 1 / \bar{z} \right)^* \right]^{-1}$$ defines an analytic function on $\mathbb{E}^2 \setminus S$ that is meromorphic on $\mathbb{E}^2$. Define $W^-
= \left \{ 1 / \bar{z} : z \in W^+ \right \}.$ Then $\Phi$ is analytic on $W^+ \cup W^-$ and continuous on $W^+ \cup X \cup W^-.$ By Rudin’s continuous edge-of-the-wedge theorem, which appears as Theorem A in [@rudeow], there is a domain $\Omega_0$ containing $W^{+}\cup X \cup W^{-},$ where $\Phi$ extends analytically. This domain only depends on $X, W^{\pm}.$ Also $\Phi$ is already holomorphic on $\D^2$, meromorphic on $\mathbb{E}^2$, and holomorphic on $\mathbb{E}^2 \setminus S$ using definition (\[eqn:phiextend\]).
We can extend this domain further using Rudin’s Theorem 4.9.1 in [@rud69]. It roughly says that if a holomorphic function $f$ on $\D^2$ extends analytically to a neighborhood $N_x$ of some $x=(x_1,x_2) \in \mathbb{T}^2,$ then $f$ extends analytically to an open set containing $\{x_1\} \times \D$ and $\D \times \{x_2\}.$ As the edge-of-the-wedge theorem guarantees $\Phi$ extends to a neighborhood $N_x$ of each $x \in X$, Rudin’s Theorem 4.9.1 implies $\Phi$ extends analytically to an open set $\Omega_1$ containing $(X_1 \times \D )
\cup (\D \times X_2).$ The *proof* of Theorem 4.9.1 implies that $\Omega_1$ only depends on the $\{N_x\}_{x \in X}$. Thus, $\Phi$ extends analytically to $$\Omega := \D^2 \cup \Omega_1 \cup \Omega_0 \cup
\left( \mathbb{E}^2 \setminus S \right).$$
**Claim 2: Elements of $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$ extend analytically to $\Omega.$**\
Let $(K_1,K_2)$ be Agler kernels of $\Phi$ and let $f \in \mathcal{H}(K_1).$ By the containment result in Theorem \[thm:containment\], $$f =\Phi A_{{\mathcal{R}}_1}f + (I - \Phi \Phi^*)^{1/2} h,$$ for some $h \in L^2(E_*)$ and $A_{{\mathcal{R}}_1}f \in Z_1 L^2_{--}(E).$ Then $g:=
\overline{Z_2 A_{{\mathcal{R}}_1} f}
\in H^2(E),$ and we can define $f$ analytically on $\mathbb{E}^2 \setminus S$ as before: $$f(z) = \Phi(z) \frac{1}{z_2} \overline{g(1/\bar{z})}.$$ Then $f$ is analytic on $W^+ \cup W^-$ and $f=\Phi A_{{\mathcal{R}}_1}f $ on $X$. As in the proof of Theorem 1.5 in [@bk12], we can use the distributional edge-of-the-wedge theorem, which appears as Theorem B in [@rudeow], to extend $f$ to $\Omega_0.$ As before, by an application of Rudin’s Theorem 4.9.1 in [@rud69], we can analytically extend $f$ to $\Omega_1$, the set containing $X_1 \times \D$ and $\D \times X_2$ mentioned earlier. As $f$ is already holomorphic in $\D^2 \cup (\mathbb{E}^2 \setminus S),$ we can conclude that every $f \in \mathcal{H}(K_1)$ is holomorphic in $\Omega$.\
**Claim 3: Points in $\Omega$ are points of bounded evaluation in $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2).$**\
The proof for inner functions given in [@bk12] essentially goes through to give bounded point evaluations in $\Omega$. Recall from the previous section that points of $\D^2$ and ${\mathbb{E}}^2\setminus S$ are points of bounded evaluation for $\mathcal{H}(K_1)$ and $\mathcal{H}(K_2)$. The next step is to show that the set of points of bounded evaluation is relatively closed in $\Omega$. This follows using the uniform boundedness principle as in [@bk12]. To show evaluation at points of $\Omega_0$ are bounded, we merely note as we did in [@bk12] that the proof of the edge-of-the-wedge theorem in [@rudeow] produces the extended values via an integral over a compact subset $K$ of $W^{+}\cup X \cup W^{-}$. Since evaluation at any point of $K$ is bounded in $\mathcal{H}(K_j)$ and since elements of $\mathcal{H}(K_j)$ are analytic in a neighborhood of $K$, $$\sup\{ \|f(z)\|_{E_*}: z \in K\} < \infty$$ for each $f \in \mathcal{H}(K_j)$ and therefore by the uniform boundedness principle there exists $M$ such that $$\|f(z)\|_{E_*} \leq M \|f\|_{\mathcal{H}(K_j)} \qquad \forall \ f \in \mathcal{H}(K_j)$$ and $z \in K$. So, since values of $f$ in $\Omega_0 $ are given by an integral of $f$ over $K$, it follows that evaluation at points in $\Omega_0$ are bounded in $\mathcal{H}(K_j)$. Now consider the points in $\Omega_1.$ As Rudin’s Theorem 4.9.1 in [@rud69] also constructs the extension of $f$ using values of $f$ at points in compact sets $K \subset \Omega_0$, the uniform boundedness principle implies that the points in $\Omega_1$ are also points of bounded evaluation.\
$(iii) \Rightarrow (ii)$ is immediate.\
Now consider $(ii) \Rightarrow (i)$.\
First, we will show that there is a point $w\in \D^2$ where $\Phi(w)$ is invertible. To do this, take any sequence $\{z^n\} \subset \D^2$ converging to a point $x \in X \subset {\mathbb{T}}^2$. Since elements of ${\mathcal{H}}(K_j)$ extend continuously to $X$, for each fixed $f \in
{\mathcal{H}}(K_j)$ the set $$\{\|f(z^n)\|_{E*}: n =1,2,\dots\}$$ is bounded. Therefore by the uniform boundedness principle for each $j=1,2$ the set $$\{ \|f(z^n)\|_{E_*}: f \in {\mathcal{H}}(K_j), \|f\|_{{\mathcal{H}}(K_j)}\leq 1, n=1,2,\dots\}$$ is bounded by say $M>0$, and this is enough to show evaluation at $x\in X$ is bounded in ${\mathcal{H}}(K_j)$ and $$\| K_j(z^n,z^n) \|_{E_* \rightarrow E_*} \leq M^2 \text{ for each } n \text{ and } \| K_j(x,x)\|_{E_* \rightarrow E_*}
\leq M^2$$ for $j=1,2.$ It follows immediately that $$\label{limsup}
\limsup_{n\to \infty} (1-|z_1^n|^2) K_2(z^n,z^n) = 0 \ \ \text{ and } \ \ \limsup_{n\to \infty} (1-|z_2^n|^2) K_1(z^n,z^n) = 0.$$ This shows that $$\lim_{n\to \infty} I-\Phi(z^n) \Phi(z^n)^* = \lim_{n\to \infty}
(1-|z_1^n|^2) K_2(z^n,z^n)+ (1-|z_2^n|^2) K_1(z^n,z^n) = 0$$ and therefore for some $N \in \mathbb{N}$, $I-\Phi(z^N) \Phi(z^N)^* \leq \frac{1}{2}
I$, which implies $\Phi(z^N)$ is invertible. Set $w=z^N$. Since $\Phi$ satisfies $$I-\Phi(z) \Phi(w)^* = (1-z_1\bar{w}_1) K_{2,w}(z) + (1-z_2 \bar{w}_2)
K_{1,w} (z)$$ we can extend $\Phi$ continuously to $X$ via the formula $$\Phi(z) = (I-(1-z_1\bar{w}_1) K_{2,w}(z) - (1-z_2 \bar{w}_2)
K_{1,w} (z))(\Phi(w)^*)^{-1}$$ since the right hand side is assumed to be continuous.
Finally, $\Phi$ is unitary on $X$ since for any $x\in X$, if we take a sequence $\{z^n\}$ in $\D^2$ converging to $x$ as above, then we will again get the result in . However, now that we know $\Phi$ is continuous at $x$, $$0=\lim_{n\to \infty} I-\Phi(z^n) \Phi(z^n)^* = I-\Phi(x)\Phi(x)^*,$$ which completes the proof.
Canonical Realizations {#sect:tfr}
----------------------
Unlike the previous section, we no longer assume $E,E_{*}$ are finite dimensional. Let $\Phi \in \mathcal{S}_1(E,E_*)$ and define its de Branges-Rovnyak space $\mathcal{H}_{\Phi}$ to be the Hilbert space with reproducing kernel $$K_{\Phi}(z,w) := \frac{ I -\Phi(z) \Phi(w)^*} {1-z\bar{w}}.$$ Then, $\Phi$ has an (almost) unique coisometric transfer function realization with state space equal to $\mathcal{H}_{\Phi}$ and colligation defined by $$U :=
\left[ \begin{array}{cc}
A & B \\
C & D
\end{array} \right] :
\left[ \begin{array}{c}
\mathcal{H}_{\Phi} \\
E
\end{array} \right]
\rightarrow
\left[ \begin{array}{c}
\mathcal{H}_{\Phi} \\
E_*
\end{array} \right]$$ with block operators given by $$\begin{aligned}
A&: f(z) \mapsto \frac{f(z) - f(0)}{z}\ \ && B: e \mapsto \frac{\Phi(z) - \Phi(0)}{z} e \\
C &: f(z) \mapsto f(0) \ \ &&D: e \mapsto \Phi(0)e.
\end{aligned}$$ Then, $ \Phi(z) = D + Cz \left( I - Az \right)^{-1}B$, and this representation is unique up to a minimality condition and unitary equivalence [@bb11].
In two variables, transfer function realizations are more complicated and rarely unique. Traditionally, T.F.R.’s associated to $\Phi \in \mathcal{S}_2(E,E_*)$ are constructed using Agler kernels $(K_1,K_2)$ of $\Phi$. In [@bb11], Ball-Bolotnikov studied T.F.R.’s defined using pairs of Agler kernels and obtained partial characterizations of the associated block operators $A$, $B$, $C$, and $D.$ Refined results about unitary T.F.R.’s for a subclass of $\mathcal{S}_d(\mathbb{D}^d)$ appear in [@bkvsv]; these are constructed in the related, but different setting of minimal augmented Agler decompositions.
Nevertheless, open questions about the structure of Agler kernels often go hand in hand with open questions about the structure of T.F.R.’s. In this section, we use our previous analysis to clear up one such question. Specifically, we use the concrete Agler kernels $(K^{max}_1,
K^{min}_2)$ to construct a coisometric T.F.R. with an explicit state space $\mathcal{M}$ and colligation $U.$ The construction answers a question posed by Ball and Bolotnikov in [@bb11].
\[rem:contfr\]**Constructing Transfer Function Realizations.** There is a canonical way to obtain transfer function realizations from Agler kernels. To illustrate this method, let $(K_1,K_2)$ be Agler kernels of $\Phi$. Then, they satisfy $$\label{eqn:agform2}
I_{E_*} - \Phi(z) \Phi(w)^*
= (1 -z_1 \bar{w}_1) K_2(z,w) + (1-z_2\bar{w}_2) K_1(z,w).$$ Define the kernel functions $K_{j,w}\nu (z):= K_j(z,w) \nu$ and define the operator $V$ by $$V: \begin{bmatrix} \bar{w}_1 K_{2,w} \nu \\
\bar{w}_2 K_{1,w} \nu \\
\nu \end{bmatrix} \mapsto
\begin{bmatrix} K_{2,w} \nu \\
K_{1,w} \nu \\
\Phi(w)^* \nu \end{bmatrix}
\quad \forall \ w \in \mathbb{D}^2, \ \nu \in E_*.$$ Then (\[eqn:agform2\]) guarantees that V can be extended to an isometry mapping the space \_V := \_[w \^2, E\_\*]{}
|[w]{}\_1 K\_[2,w]{}\
|[w]{}\_2 K\_[1,w]{}\
(K\_2) (K\_1) E\_\* onto the space $${\mathcal{R}}_V := \bigvee_{w \in \D^2, \nu \in E_*}
\begin{bmatrix} K_{2,w} \nu \\
K_{1,w} \nu \\
\Phi(w)^* \nu \end{bmatrix} \subseteq \mathcal{H}(K_2) \oplus
\mathcal{H}(K_1) \oplus E.$$ Transfer function realizations with state space $\mathcal{H}(K_2) \oplus \mathcal{H}(K_1)$ are obtained by extending $V$ to a contraction from $$\mathcal{H}(K_2) \oplus \mathcal{H}(K_1) \oplus E \rightarrow
\mathcal{H}(K_2) \oplus \mathcal{H}(K_1) \oplus E_*$$ and setting $U=V^*.$ In Ball-Bolotnikov [@bb11], such a $U$ is called a *canonical functional model (c.f.m.) colligation* of $\Phi$ associated to $(K_1, K_2).$ Similarly, coisometric transfer function realizations are obtained by extending $V$ to an isometry mapping $$\mathcal{H}(K_2) \oplus \mathcal{H}(K_1) \oplus \mathcal{H} \oplus E \rightarrow
\mathcal{H}(K_2) \oplus \mathcal{H}(K_1) \oplus \mathcal{H} \oplus E_*,$$ where $\mathcal{H}$ is an arbitrary infinite dimensional Hilbert space only added in when required, and $U$ is defined to be $V^*.$
Let $\Phi \in \mathcal{S}(E, E_*)$. Currently, it is an open question as to whether there always exists a coisometric transfer function realization of $\Phi$ with state space $\mathcal{H}(K_2) \oplus \mathcal{H}(K_1)$ for every pair of Agler kernels $(K_1,K_2)$. In Section 3.2 of [@bb11], Ball-Bolotnikov posed the following related question, which was originally stated in the d-variable setting:
Let $\Phi \in \mathcal{S}_2(E, E_*)$. Is there *any* pair of Agler kernels $(K_1,K_2)$ of $\Phi$ such that $\Phi$ has a *coisometric* c.f.m. colligation associated to $(K_1, K_2)$?
This is equivalent to asking if the construction in Remark \[rem:contfr\] gives a coisometric transfer function realization of $\Phi$ with state space $\mathcal{H}(K_2) \oplus \mathcal{H}(K_1).$
The following theorem answers that question in the affirmative.
\[thm:canonicalcmf\] Let $\Phi \in \mathcal{S}_2(E,E_*)$ and consider its Agler kernels $(K^{max}_1, K^{min}_2).$ The construction in Remark \[rem:contfr\] gives a unique, coisometric transfer function realization of $\Phi$ with state space $\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1).$
Consider the construction in Remark \[rem:contfr\] using Agler kernels $(K^{max}_1, K^{min}_2)$. The operator $V$ is initially defined by V:
|[w]{}\_1 K\^[min]{}\_[2,w]{}\
|[w]{}\_2 K\^[max]{}\_[1,w]{}\
K\^[min]{}\_[2,w]{}\
K\^[max]{}\_[1,w]{}\
(w)\^\*
w \^2, E\_\* and extended to an isometry on the space \_V := \_[w \^2, E\_\*]{}
|[w]{}\_1 K\^[min]{}\_[2,w]{}\
|[w]{}\_2 K\^[max]{}\_[1,w]{}\
(K\^[min]{}\_2) (K\^[max]{}\_1) E\_\*. Then, transfer function realizations with state space $\mathcal{H}(K_2) \oplus
\mathcal{H}(K_1)$ are obtained by extending $V$ to a contraction on $\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1)
\oplus E_*$. We will show $\mathcal{D}_V =
\mathcal{H}(K^{min}_2)
\oplus \mathcal{H}(K^{max}_1) \oplus E_*.$ Then, the result will follow because $V$ will already be an isometry on $\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1)
\oplus E_*$ and so we can immediately set $U=V^*$. Define $$\mathcal{D} := \bigvee_{w \in \D^2, \nu \in E_*}
\begin{bmatrix} \bar{w}_1 K^{min}_{2,w} \nu \\
\bar{w}_2 K^{max}_{1,w}\nu \end{bmatrix}
\subseteq \mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1).$$ Examining the case $w=0$ shows that $\mathcal{D}_V$ coincides with $\mathcal{D} \oplus E_*$, so it suffices to show $\mathcal{D}=
\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1).$ Assume
f\_2\
f\_1
. Then for each $w \in \mathbb{D}^2$ and $\nu \in E_*$, $$\begin{aligned}
0 &= \LL \begin{bmatrix} f_2 \\ f_1
\end{bmatrix}, \begin{bmatrix} \bar{w}_1 K^{min}_{2,w} \nu \\
\bar{w}_2 K^{max}_{1,w}\nu \end{bmatrix} \RR_{\mathcal{H}(K^{min}_2)
\oplus \mathcal{H}(K^{max}_1)} \\
&& \\
&= w_1 \LL f_2, K^{min}_{2,w}\nu \RR_{\mathcal{H}(K^{min}_2)}+
w_2 \LL f_1, K^{max}_{1,w}\nu \RR_{\mathcal{H}(K^{max}_1)} \\
&\\
& = \LL w_1f_2(w) + w_2f_1(w), \nu \RR_{E_*},\end{aligned}$$ which implies $Z_1f_2 + Z_2f_1 = 0.$ Thus, there is some $F \in H^2(E_*)$ such that $f_1 =Z_1 F.$ Now, since $f_1 \in \mathcal{H}(K^{max}_1)$, there is a $g_1 \in Z_1 L^2_{--}(E)$ such that $$\label{eqn:maxcon} \begin{bmatrix}
f_1 \\
g_1
\end{bmatrix} \in {\mathcal{R}}_1
\ominus Z_1 {\mathcal{R}}.$$ This also gives $g_1 - \Phi^* f_1 \in \Delta L^2(E)$ and a $G \in L^2_{--}(E)$ with $g = Z_1 G.$ Since $\Delta L^2(E)$ is invariant under $Z^*_1$, is is clear that $G - \Phi^* F\in \Delta L^2(E)$ as well. Then
f\_1\
g\_1
= Z\_1
F\
G
F\
G
. Given this, (\[eqn:maxcon\]) forces $f_1 \equiv 0$, so $f_2 \equiv 0$ and $\mathcal{D}
= \mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1).$
**The Canonical Block Operators.** Let $U$ be the operator associated to the transfer function realization given in Theorem \[thm:canonicalcmf\]. Much can be said about its block operators $A,B,C,D$. In the setting of general $(K_1,K_2)$, much of this analysis already appears in [@bb10] and [@bb11]. We will first give the formulas for $A,B,C,D$ and then discuss the derivations. Specifically, for every $ f := \begin{bmatrix}
f_1 \\
f_2
\end{bmatrix} \in \mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1)$ and $\eta \in E$, $$C:
\begin{bmatrix}
f_1 \\
f_2
\end{bmatrix}
\mapsto
f_1(0) + f_2(0)
\ \text{ and } \
D: \eta \mapsto \Phi(0) \eta.$$ For $A$ and $B$, let us first simplify notation by setting $$\begin{bmatrix}
(Af)_1 \\
(Af)_2
\end{bmatrix}
:=A
\begin{bmatrix}
f_1 \\
f_2
\end{bmatrix}
\ \text{ and }
\begin{bmatrix}
(B \eta)_1 \\
(B \eta)_2
\end{bmatrix}
:= B \eta.$$ Then $(Af)_2$ and $(B \eta)_2$ are the unique functions in $\mathcal{H}(K^{max}_1)$ satisfying $$\begin{aligned}
\left(Af \right)_2 (0, w_2) &= \frac{ f_1(0, w_2) - f_1(0) + f_2(0,w_2)-f_2(0)}{w_2} \\
\left( B \eta \right)_2(0,w_2) & = \frac{\Phi(0,w_2)- \Phi(0)}{w_2} \eta,\end{aligned}$$ for all $w_2 \in \mathbb{D} \setminus \{0\},$ and $(Af)_1$ and $(B \eta)_1$ are the unique functions in $\mathcal{H}(K^{min}_2)$ satisfying $$\begin{aligned}
\left( Af \right)_1(w) &= \frac{f_1(w) - f_1(0) + f_2(w)-f_2(0) -w_2 \left(Af \right)_2 (w)}{w_1} \\
\left( B \eta \right)_1(w) & = \frac{\left( \Phi(w)- \Phi(0) \right)\eta - w_2 \left( B \eta \right)_2(w)}{w_1},\end{aligned}$$ for all $w \in \mathbb{D}^2$ with $w_1 \ne 0.$ The results for $C$ and $D$ follow because, by definition $$U^* =
\begin{bmatrix}
A^* & C^* \\
B^* & D^*
\end{bmatrix}
: \begin{bmatrix} \bar{w}_1 K_{2,w}^{min} \nu \\
\bar{w}_2 K_{1,w}^{max} \nu \\
\nu \end{bmatrix} \mapsto
\begin{bmatrix} K_{2,w}^{min} \nu \\
K_{1,w}^{max} \nu \\
\Phi(w)^* \nu \end{bmatrix}
\quad \forall \ w \in \mathbb{D}^2, \ \nu \in E_*.$$ Setting $w=0$ immediately implies that $$C^*: \nu \mapsto
\begin{bmatrix}
K^{min}_{2,0}\nu \\
K^{max}_{1,0} \nu
\end{bmatrix}
\text{ and }
D^*: \nu \mapsto \Phi(0)^*\nu$$ for all $\nu \in E_*$. Then the calculations $$\LL C
\begin{bmatrix} f_1 \\
f_2
\end{bmatrix},
\nu
\RR_{E_*}
=\LL
\begin{bmatrix} f_1 \\
f_2
\end{bmatrix},
\begin{bmatrix}
K^{min}_{2,0} \nu \\
K^{max}_{1,0}\nu
\end{bmatrix} \RR_{\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1)}
=
\LL f_1(0) + f_2(0), \nu \RR_{E_*}$$ and $$\LL D \eta, \nu \RR_{E_*}
= \LL \eta, D^* \nu \RR_{E}
=\LL \eta, \Phi(0)^* \nu \RR_{E}
=\LL \Phi(0) \eta, \nu \RR_{E_*}$$ give the formulas for $C$ and $D$. Moreover, The results about $C^*$ and $D^*$ imply that $$A^*:
\begin{bmatrix}
\bar{w}_1 K_{2,w}^{min} \nu \\
\bar{w}_2 K_{1,w}^{max} \nu
\end{bmatrix}
\mapsto
\begin{bmatrix}
\left( K_{2,w}^{min} - K^{min}_{2,0} \right) \nu \\
\left( K_{1,w}^{max} - K^{max}_{1,0} \right) \nu
\end{bmatrix}$$ and $$B^*:
\begin{bmatrix}
\bar{w}_1 K_{2,w}^{min} \nu \\
\bar{w}_2 K_{1,w}^{max} \nu
\end{bmatrix}
\mapsto
\left( \Phi(w)^* - \Phi(0)^* \right) \nu.$$ Then $$\begin{aligned}
\LL w_1(Af)_1 (w) + w_2(Af)_2 (w), \nu \RR_{E_*} &= \LL Af, \begin{bmatrix}
\bar{w}_1 K_{2,w}^{min} \nu \\
\bar{w}_2 K_{1,w}^{max} \nu
\end{bmatrix}\RR_{\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1)}\\
&\\
& = \LL \begin{bmatrix}
f_1 \\
f_2
\end{bmatrix},
\begin{bmatrix}
\left( K_{2,w}^{min} - K^{min}_{2,0} \right) \nu \\
\left( K_{1,w}^{max} - K^{max}_{1,0} \right) \nu
\end{bmatrix} \RR_{\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1)}
\\
&\\
& = \LL f_1(w) - f_1(0) + f_2(w) - f_2(0), \nu \RR_{E_*},
\end{aligned}$$ and similarly, $$\LL w_1(B \eta)_1 (w) + w_2(B \eta)_2 (w), \nu \RR_{E_*}
= \LL \left( \Phi(w) - \Phi(0) \right)\eta, \nu \RR_{E_*}.$$ Therefore, we have $$\begin{aligned}
\label{eqn:gleason1} w_1 \left( Af \right)_1(w) + w_2 \left(Af \right)_2 (w) &= f_1(w) - f_1(0) + f_2(w)-f_2(0) \\
\label{eqn:gleason2} w_1 \left( B \eta \right)_1(w) + w_2 \left( B \eta \right)_2(w) & = \left( \Phi(w)- \Phi(0) \right)\eta.\end{aligned}$$ Operators that solve $(\ref{eqn:gleason1})$ or $(\ref{eqn:gleason2})$ are said to solve the structured Gleason problem for $\mathcal{H}(K^{min}_2) \oplus \mathcal{H}(K^{max}_1)$ or for $\Phi$, respectively. In general, such operators are not unique. However, in this situation, $A$ and $B$ are uniquely determined. The proof of this rests on two observations. First, when $w_1=0$ and $w_2 \ne 0,$ $(\ref{eqn:gleason1})$ and $(\ref{eqn:gleason2})$ become $$\begin{aligned}
\label{eqn:gleason3} \left(Af \right)_2 (0, w_2) &= \frac{ f_1(0, w_2) - f_1(0) + f_2(0,w_2)-f_2(0)}{w_2} \\
\label{eqn:gleason4} \left( B\eta \right)_2(0,w_2) & = \frac{ \Phi(0,w_2)- \Phi(0)}{w_2}\eta.\end{aligned}$$ It is also true that the set $\{ (0,w_2) : w_2 \in \mathbb{D} \setminus \{0\}\}$ is a set of uniqueness for $\mathcal{H}(K^{max}_1).$ Indeed, suppose two functions $g_1, g_2 \in \mathcal{H}(K^{max}_1)$ satisfy $g_1(0,w_2) = g_2(0,w_2)$ for all $w_2 \ne 0$. This immediately implies $g_1(0,0)=g_2(0,0)$ and $$g_1- g_2 =Z_1 h$$ for some $h \in H^2(E_*).$ Arguments identical to those in the proof of Theorem \[thm:canonicalcmf\] show that $h$ must be zero, so $g_1=g_2.$ As $\left(Af \right)_2$ and $\left( B \eta \right)_2$ are in $\mathcal{H}(K^{max}_1)$, they must be the unique such functions satisfying $(\ref{eqn:gleason3})$ and $(\ref{eqn:gleason4})$ respectively. Then, the other components $\left(Af \right)_1$ and $\left( B \eta \right)_1$ are uniquely determined by $(\ref{eqn:gleason1})$ and $(\ref{eqn:gleason2}).$ In one-variable, $Af$ and $B \eta$ can be explicitly written in terms of $f$ and $\eta.$ Given that, our characterizations of $A$ and $B$ seem slightly unsatisfying. This motivates the question
Assume $g \in \mathcal{H}(K^{max}_1)$. Is there an explicit way to construct $g$ using only the function $g(0,w_2)?$
A clean answer would also provide nice formulas for the operators $A$ and $B$. It seems possible that the refined results in [@bkvsv] about unitary T.F.R.’s associated to minimal augmented Agler decompositions might suggest methods of answering this question.
Appendix: Vector valued RKHS’s {#sect:opkernels}
==============================
In this section, we record several facts about vector valued reproducing kernel Hilbert spaces that were used in earlier sections. The results are well-known in the scalar valued case. See, for example [@aro50], [@bv03b], Chapter 2 in [@alp01], and Chapter 2 in [@ampi]. We outline how the needed vector valued results follow from the known scalar valued results. Let $\Omega$ be a set and $E$ be a separable Hilbert space. We will frequently use the following observation:
For each function $f: \Omega \rightarrow E$ there is an associated scalar valued function $\tilde{f}: \Omega \times E
\rightarrow \mathbb{C}$ defined as follows: $$\tilde{f}(z, \eta) := \LL f(z), \eta \RR_{E}.$$ If functions $f,g: \Omega \rightarrow E$ and $\tilde{f} \equiv \tilde{g}$, then $f \equiv g.$
\[defn:scalarhs\] Let $\mathcal{H}(K)$ be a reproducing kernel Hilbert space of $E$ valued functions on $\Omega$. For $w\in \Omega$ and $\nu \in E$, define the function $K_w \nu:= K( \cdot, w)\nu.$ An associated reproducing kernel Hilbert space of scalar valued functions on $\Omega \times E$ can be defined as follows: Define the set of functions $$\mathcal{H} := \left \{ \tilde{f}: f \in \mathcal{H}(K) \right \}$$ and equip $\mathcal{H}$ with the inner product $$\LL \tilde{f}, \tilde{g} \RR_{\mathcal{H}}
= \LL f ,g \RR_{\mathcal{H}(K)}.$$ It is routine to show that $\mathcal{H}$ is a Hilbert space with this inner product and since $$\tilde{f}(w,\nu) = \LL f(w), \nu \RR_{E}
= \LL f, K_w \nu \RR_{\mathcal{H}(K)}
= \LL \tilde{f}, \widetilde{ K_w \nu} \RR_{\mathcal{H}},$$ $\mathcal{H}$ is a reproducing kernel Hilbert space with reproducing kernel $$L \big( (z,\eta), (w,\nu) \big) := \widetilde{K_w\nu} ( z, \eta)
= \LL K(z,w) \nu, \eta \RR_{E} = \eta^* K(z,w) \nu.$$ Then $f \in \mathcal{H}(K)$ if and only if $\tilde{f} \in \mathcal{H}(L)$. It is also clear that $\|f \|_{\mathcal{H}(K)} = \| \tilde{f}\|_{\mathcal{H}(L)}.$
The following results are well-known for scalar valued reproducing kernel Hilbert spaces and follow easily for vector valued reproducing kernel Hilbert spaces.
\[thm:kerdiff\] Let $\mathcal{H}(K)$ and $\mathcal{H}(K_1)$ be reproducing kernel Hilbert spaces of $E$ valued functions on $\Omega$. Then $\mathcal{H}(K_1) \subseteq \mathcal{H}(K)$ contractively if and only if $$K(z,w) - K_1(z,w) \text{ is a positive kernel.}$$
As in Definition \[defn:scalarhs\], consider the Hilbert spaces $\mathcal{H}(L)$ and $\mathcal{H}(L_1)$ of scalar valued functions on $\Omega \times E$ with reproducing kernels given by $$L \big( (z,\eta), (w,\nu) \big) :=\eta^* K(z,w) \nu \ \ \text{ and }
L_1 \big( (z,\eta), (w,\nu) \big) :=\eta^* K_1(z,w) \nu.$$ It is routine to show that $\mathcal{H}(K_1) \subseteq \mathcal{H}(K)$ contractively if and only if $\mathcal{H}(L_1) \subseteq
\mathcal{H}(L)$ contractively. It follows from well-known scalar results, which appear on page 354 of [@aro50], that $\mathcal{H}(L_1) \subseteq \mathcal{H}(L)$ contractively if and only if $$L(z,w) - L_1(z,w) \text{ is a positive kernel.}$$ The result follow from the fact that $L(z,w) - L_1(z,w)$ is a positive kernel if and only if $K(z,w)-K_1(z,w)$ is a positive kernel.
Similarly, the following two results can be deduced from the scalar-valued case:
\[thm:kermult\] Let $\mathcal{H}(K)$ be a reproducing kernel Hilbert space of $E$ valued functions on $\Omega$ and let $\psi: \Omega \rightarrow \mathbb{C}$. Then $\psi$ is a multiplier of $\mathcal{H}(K)$ with multiplier norm bounded by one if and only if $$\big( 1 - \psi(z) \overline{ \psi(w)} \big) K(z,w) \text{ is a positive kernel}.$$
When we say “$\psi$ is a multiplier of $\mathcal{H}(K)$," we mean that $\psi \otimes I_{\mathcal{H}(K)}$ maps $\mathcal{H}(K)$ into $\mathcal{H}(K).$
Now, using the definition of $\mathcal{H}(L)$, it is easy to show that $\psi$ is a multiplier of $\mathcal{H}(K)$ with multiplier norm bounded by one if and only if $\psi$ is a multiplier of $\mathcal{H}(L)$ with multiplier norm bounded by one. By the analogous scalar valued result, which appears as Corollary 2.3.7 in [@ampi], it follows that $\psi$ is a multiplier of $ \mathcal{H}(L)$ with multiplier norm bounded by one if and only if $$\big( 1 - \psi(z) \overline{ \psi(w)} \big) L \big( (z, \eta),
(w, \nu) \big) \text{ is a positive kernel}.$$ The result then follows by using the definition of a positive kernel to show that $\big( 1 - \psi(z) \overline{ \psi(w)} \big) L \big( (z, \eta),
(w, \nu) \big)$ is a positive kernel if and only if $\big( 1 - \psi(z) \overline{ \psi(w)} \big) K(z,w) $ is a positive kernel.
\[thm:kersum\] Let $\mathcal{H}(K_1),
\mathcal{H}(K_2)$ be reproducing kernel Hilbert spaces of $E$ valued functions on $\Omega$. Then $\mathcal{H}(K_1 +
K_2)$ is precisely the Hilbert space composed of the set of functions $$\mathcal{H}(K_1) + \mathcal{H}(K_2) := \left \{ f_1 +f_2 :
f_j \in \mathcal{H}(K_j) \right \}.$$ equipped with the norm $$\| f \|^2_{\mathcal{H}(K_1+K_2)} = \min_{\substack{ f = f_1 + f_2 \\ f_j
\in \mathcal{H}(K_j)}} \|f_1\|^2_{\mathcal{H}(K_1)}
+ \|f_2 \|^2_{\mathcal{H}(K_2)}.$$
As before consider the related scalar valued reproducing kernel Hilbert spaces $\mathcal{H}(L_1)$ and $\mathcal{H}(L_2)$, where $$L_1 \big( (z,\eta), (w,\nu) \big) :=\eta^* K_1(z,w) \nu \ \ \text{ and }
L_2 \big( (z,\eta), (w,\nu) \big) :=\eta^* K_2(z,w) \nu.$$ The analogous scalar valued result, which appears on page 353 in [@aro50], states $\mathcal{H}(L_1 + L_2)$ is precisely the Hilbert space composed of the set of functions $$\mathcal{H}(L_1) + \mathcal{H}(L_2) := \left \{ f_1 +f_2 : f_j \in
\mathcal{H}(L_j) \right \}.$$ equipped with the norm $$\| f \|^2_{\mathcal{H}(L_1+L_2)} = \min_{\substack{ f = f_1 + f_2
\\ f_j \in \mathcal{H}(L_j)}} \|f_1\|^2_{\mathcal{H}(L_1)} +
\|f_2 \|^2_{\mathcal{H}(L_2)},$$ Using this and the connections between $\mathcal{H}(L_j)$ and $\mathcal{H}(K_j)$, it is easy to deduce the desired result. The details are left as an exercise.
[10]{} J. Agler. Some interpolation theorems of [Nevanlinna-Pick]{} type. Preprint, 1988.
J. Agler. On the representation of certain holomorphic functions defined on a polydisc. In [*Topics in operator theory: [E]{}rnst [D.]{} [H]{}ellinger memorial volume*]{}, volume 48 of [*Oper. Theory Adv. Appl.*]{}, pages 47–66. [Birkhäuser Verlag]{}, Basel, 1990.
J. Agler and J.E. M.45exCarthy. Interpolating sequences on the bidisk. , 12(9):1103–1114, 2001.
J. Agler and J.E. M.45exCarthy. . American Mathematical Society, Providence, RI, 2002.
J. Agler and J.E. M.45exCarthy. Distinguished varieties. , 194:133–153, 2005.
J. Agler and J.E. M.45exCarthy. What can [H]{}ilbert spaces tell us about bounded functions in the bidisk? In [*A glimpse at Hilbert space operators*]{}, volume 207 of [ *Oper. Theory Adv. Appl.*]{}, pages 81–97. [Birkhäuser Verlag]{}, Basel, 2010.
J. Agler, J.E. M.45exCarthy, and N. J. Young. Operator monotone functions and [L]{}oewner functions of several variables. , 176(3):1783–1826, 2012.
J. Agler, J.E. M.45exCarthy, and N.J. Young. A [Carathéodory]{} theorem for the bidisk via [Hilbert]{} space methods. , 352(3):581–624, 2012.
D. Alpay. . American Mathematical Society, Providence, RI, 2001.
N. Aronszajn. Theory of reproducing kernels. , 68:337–404, 1950.
J.A. Ball and V. Bolotnikov. , Providence, RI, 2010.
J.A. Ball and V. Bolotnikov. In [*A panorama of modern operator theory and related topics: the Israel Gohberg memorial volume*]{}, volume 218 of [*Oper. Theory Adv. Appl.,*]{} pages 75-212. , Basel, 2012.
J.A. Ball, D.S. Kaliuzhnyi-Verbovetskyi, C. Sadosky, and V. Vinnikov. Scattering systems with several evolutions and formal reproducing kernel [H]{}ilbert spaces. Preprint, 2013.
J.A. Ball, C. Sadosky, and V. Vinnikov. Scattering systems with several evolutions and multidimensional input/state/output systems. , 52:323–393, 2005.
J.A. Ball and T.T. Trent. Unitary colligations, reproducing kernel [Hilbert]{} spaces, and [Nevanlinna-Pick]{} interpolation in several variables. , 197:1–61, 1998.
J.A. Ball and V. Vinnikov. Formal reproducing kernel [H]{}ilbert spaces: the commutative and noncommutative settings. In [*Reproducing kernel spaces and applications*]{}, volume 143 of [*Oper. Theory Adv. App,*]{} pages 77–134. [Birkhäuser]{}, Basel, 2003.
K. Bickel. Fundamental [A]{}gler decompositions. , 74(2):233–257, 2012.
K. Bickel and G. Knese. Inner functions on the bidisk and associated [H]{}ilbert spaces. , 265, no. 11, 2753–2790, 2013.
B.J. Cole and J. Wermer. theorem and sums of squares. , 48:767–791, 1999.
R.G. Douglas. On majorization, factorization, and range inclusion of operators on [H]{}ilbert space. , 17: 413-415, 1966.
J.S. Geronimo and H.J. Woerdeman. Positive extensions, [F]{}ejér-[R]{}iesz factorization and autoregressive filters in two variables. (2) 160, no.3, 839–906, 2004.
J.W. Helton. , 16:15–38, 1974.
J.W. Helton and M.R. James. in series [*Advances in Design and Control*]{}, SIAM, Philadelphia, 1999.
A. Grinshpan, D. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, and H. Woerdeman. Classes of tuples of commuting contractions satisfying the multivariable [von Neumann]{} inequality. , 256(9):3035–3054, 2009.
G. Knese. A [S]{}chwarz lemma on the polydisk. , 135:2759–2768, 2007.
G. Knese. Bernstein-[S]{}zeg[ő]{} measures on the two dimensional torus. , 57(3):1353–1376, 2008.
G. Knese. Polynomials defining distinguished varieties. , 362(11):5635–5655, 2010.
G. Knese. A refined [A]{}gler decomposition and geometric applications. , 60:1831–1842, 2011.
G. Knese. Polynomials with no zeros on the bidisk. , 3, no. 2, 109–149, 2010.
J. E. McCarthy. Shining a [H]{}ilbertian lamp on the bidisk. In [*Topics in Complex Analysis and Operator Theory*]{}, volume 561 of [*Contemp. Math.*]{}, pages 49–65. Amer. Math. Soc., Providence, RI, 2012.
W. Rudin. . Benjamin, New York, 1969.
W. Rudin. . American Mathematical Society, Providence, RI, 1971.
D. Sarason. . University of Arkansas Lecture Notes, Wiley, New York, 1994.
[^1]: GK supported by NSF grant DMS-1048775
|
---
abstract: 'The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an “algebraic” refinement of the small object argument, cast in terms of Grandis and Tholen’s natural weak factorisation systems, which rectifies each of these three deficiencies.'
author:
- |
Richard Garner[^1]\
Department of Mathematics, Uppsala University,\
Box 480, S-751 06 Uppsala, Sweden
bibliography:
- 'biblio.bib'
title: Understanding the small object argument
---
Ł
Introduction
============
The concept of *factorisation system* provides us with a way of viewing a category ${{\mathcal C}}$ as a compositional product of two subcategories ${{\mathcal L}}$ and ${{\mathcal R}}$. The two key ingredients are an axiom of *factorisation*, which affirms that any map of ${{\mathcal C}}$ may be written as a map of ${{\mathcal L}}$ followed by a map of ${{\mathcal R}}$, and an axiom of *orthogonality*, which assures us that this decomposition is unique up to unique isomorphism. From these two basic axioms a very rich theory can be developed, and a very useful one, since most categories arising in mathematical practice will admit at least a few different factorisation systems.
However, in those mathematical areas where the primary objects of study are themselves higher-dimensional entities – most notably, topology and higher dimensional category theory – the notion of factorisation system is frequently too strong, since we would like factorisations be unique, not up to isomorphism, but up to something weaker. Thus in a 2-category, we might want uniqueness up-to-equivalence; or in a category of topological spaces, uniqueness up-to-homotopy.
The usual way of achieving this is to pass from factorisation systems to *weak factorisation systems*. The modifier “weak” has the familiar effect of turning an assertion of unique existence into an assertion of mere existence, here in respect to the diagonal fill-ins which are guaranteed to us by the axiom of orthogonality.
From this, we would not necessarily expect the factorisations in a weak factorisation system (henceforth w.f.s.) to be unique up to anything at all: but remarkably, each weak factorisation system generates its own notion of “equivalence” which respect to which its factorisations *are* unique. The framework within which this is most readily expressed is that of Quillen’s *model categories* [@Quillen:homotopical], which consist in a clever interaction of two w.f.s.’s on a category: but we can make do with a single w.f.s., and for the purposes of this paper, we will.
Whilst in many respects, the theory of w.f.s.’s is similar to the theory of factorisation systems (which we will henceforth call *strong* factorisation systems to avoid ambiguity), there are some puzzling aspects to it: and notable amongst these is the manner in which one typically constructs a w.f.s.
In the case of strong factorisation systems, there is a very elegant theory which, given a sufficiently well-behaved category ${{\mathcal C}}$, can generate a strong factorisation system from any set of maps $J \subset {{\mathcal C}}^\mathbf 2$. The ${{\mathcal R}}$-maps will be the maps which are *right orthogonal* to each of the maps in $J$ (in a sense which we recall more precisely in Section \[Sec:nwfs\]); and the ${{\mathcal L}}$-maps, those which are left orthogonal to each of the maps in ${{\mathcal R}}$. The key difficulty is how we should build the factorisations, and for this we are able bring to bear a well-established body of knowledge concerning transfinite constructions in categories, on which the definitive word is [@Ke80].
There is a corresponding theory for weak factorisation systems. Again, we suppose ourselves given a well-behaved ${{\mathcal C}}$ and a set of maps $J$, but this time we take for ${{\mathcal R}}$ the class of maps *weakly* right orthogonal to $J$, and for ${{\mathcal L}}$, the class of maps *weakly* left orthogonal to ${{\mathcal R}}$. To obtain a weak factorisation system, we must also have factorisation of maps: and for this, we apply a construction known as the *small object argument*, introduced by Quillen [@Quillen:homotopical], and first given in its full generality by Bousfield [@Bous].
The problem lies in divining the precise nature of the small object argument. It is certainly some kind of transfinite construction: but it is a transfinite construction which does not converge, has no universal property, and does not seem to be an instance of any other known transfinite construction.
In this paper, we present a modification of the small object argument which rectifies each of these deficiencies: it is guaranteed to converge; the factorisations it provides are freely generated by the set $J$, in a suitable sense; and it may be construed as an instance of a familiar free monoid construction.
To make this possible, we must adopt a rather different perspective on weak factorisation systems. The definition of a w.f.s. specifies classes of maps ${{\mathcal L}}$ and ${{\mathcal R}}$ together with axioms which affirm properties: that *there exist* factorisations, or that *there exist* certain diagonal fill-ins. But a key tenet of category theory is that anything we specify in terms of properties should have an equally valid expression in terms of structure: and in the case of w.f.s.’s, a suitable “algebraic” reformulation is given by Tholen and Grandis’ notion of *natural weak factorisation system* [@nwfs].
The extra algebraicity provided by natural w.f.s.’s allows us a clearer view of what is actually going on in the small object argument. We now have a *functor* from the category of natural w.f.s.’s on ${{\mathcal C}}$ into ${\mathbf{CAT}}$ which sends each natural w.f.s. to its category of ${{\mathcal L}}$-maps; and we can factor this functor through ${\mathbf{CAT}} / {{\mathcal C}}^\mathbf 2$. We may view the resultant functor ${\mathbf{NWFS}}({{\mathcal C}}) \to {\mathbf{CAT}} / {{\mathcal C}}^\mathbf 2$ as being the “semantics” side of a syntax/semantics adjunction: for which the syntax side is precisely our refinement of the small object argument.
Although all our arguments will be cast in terms of natural w.f.s.’s, we will see that there are ramifications for plain [w.f.s.]{}’s as well, since our refined version of the small object argument can equally well be applied there, giving rise to factorisations which are less redundant than the original argument, and in many cases can be easily calculated by hand.
**Acknowledgements**. My foremost thanks go to the organisers of CT ’07 for providing such a pleasant and stimulating environment within which to present this material. Further thanks go to Clemens Berger, Eugenia Cheng, Jeff Egger, André Hirschowitz, Martin Hyland, Joachim Kock, Mike Shulman, Carlos Simpson, Walter Tholen, and members of the Stockholm-Uppsala Logic Seminar for useful discussions and comments.
Notions of factorisation system {#Sec:nwfs}
===============================
In this section, we describe in detail the various sorts of factorisation system mentioned in the Introduction.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Most familiar is the notion of *strong factorisation system* $({{\mathcal L}},
{{\mathcal R}})$ on a category ${{\mathcal C}}$, introduced by Freyd and Kelly in [@FreydKelly:continuous]. This is given by two classes of maps ${{\mathcal L}}$ and ${{\mathcal R}}$ in ${{\mathcal C}}$ which are each closed under composition with isomorphisms, and which satisfy the axioms of
(factorisation)
: Every map $e \colon X \to Y$ in ${{\mathcal C}}$ can be written as $e = gf$, where $f \in {{\mathcal L}}$ and $g \in {{\mathcal R}}$; and
(orthogonality)
: $f \mathbin\bot g$ for all $f \in {{\mathcal L}}$ and $g \in
{{\mathcal R}}$, where $f \mathbin \bot g$ means that for every commutative square $$\label{fillinsquare}
{\vcenter{\hbox{\xymatrix{
A
\ar[r]^-h
\ar[d]_f &
C
\ar[d]^g \\
B
\ar[r]_-k &
D
}}}}$$ in ${{\mathcal C}}$, there is a unique map $j \colon B \to C$ such that $gj = k$ and $jf = h$.
Instead of writing $f \mathbin \bot g$, we may also say that $f$ is *left orthogonal* to $g$ or that $g$ is *right orthogonal* to $f$; moreover, given a class ${{\mathcal A}}$ of maps in ${{\mathcal C}}$, we write $${}^\bot {{\mathcal A}}= {\left\{\,g \in {{\mathcal C}}^\mathbf 2 \mid f \mathbin \bot g \text{ for all $g \in {{\mathcal A}}$}\,\right\}}
\quad\text{and} \quad
{{\mathcal A}}^\bot = {\left\{\,f \in {{\mathcal C}}^\mathbf 2 \mid f \mathbin \bot g \text{ for all $f \in {{\mathcal A}}$}\,\right\}}\text;$$ and this sets up a Galois connection on the collection of all classes of maps in ${{\mathcal C}}$. In a strong factorisation system, we have ${{\mathcal R}}= {{\mathcal L}}^\bot$ and ${{\mathcal L}}=
{}^\bot {{\mathcal R}}$, so that the classes ${{\mathcal L}}$ and ${{\mathcal R}}$ determine each other.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}\[defwfs\] We arrive at the notion of a *weak factorisation system* [@Bous] by making two alterations to the above definition. One is minor: we require that ${{\mathcal L}}$ and ${{\mathcal R}}$ are closed under retracts in the arrow category ${{\mathcal C}}^\mathbf 2$, rather than merely closed under isomorphism. The other is more far-reaching: we replace the orthogonality condition with
(weak orthogonality)
: $f \mathbin\pitchfork g$ for all $f \in {{\mathcal L}}$ and $g \in {{\mathcal R}}$, where $f \mathbin\pitchfork g$ means that for every commutative square as in , there exists a (not necessarily unique) fill-in $j \colon B \to C$ such that $gj = k$ and $jf = h$.
We now have a Galois connection ${}^{\pitchfork}(\ ) \mathbin \dashv
(\ )^\pitchfork$; and again, the classes ${{\mathcal L}}$ and ${{\mathcal R}}$ of a w.f.s. determine each other by the equations ${{\mathcal L}}= {}^{\pitchfork}{{\mathcal R}}$ and ${{\mathcal R}}=
{{\mathcal L}}^\pitchfork$. However, the classes ${{\mathcal L}}$ and ${{\mathcal R}}$ need not determine the factorisation of a map, even up to isomorphism, as the following examples show:
\[exs1\]
(i) (Epi, Mono) is a strong factorisation system on ${\mathbf{Set}}$; but (Mono, Epi) is a weak factorisation system. For the latter, there are two natural choices of factorisation for a map $f \colon X \to Y$: the *graph* factorisation which goes via $X \times Y$; and the *cograph* factorisation, which goes through $X + Y$.
(ii) There is a weak factorisation system on ${\mathbf{Cat}}$ given by (injective equivalences, isofibrations). A *injective equivalence* is a functor which is both injective on objects and an equivalence of categories; whilst an *isofibration* is a functor along which all isomorphisms have liftings.
(iii) There is a weak factorisation system (anodyne extensions, Kan fibrations) on ${\mathbf{SSet}} = [\Delta^{\mathrm{op}}, {\mathbf{Set}}]$, the category of simplicial sets. The Kan fibrations are easy to describe: they are precisely the maps which are weakly right orthogonal to the set of *horn inclusions* $\Lambda^k[n] \to \Delta[n]$. The anodyne extensions are the class of maps weakly left orthogonal to all Kan fibrations; more explicitly, they are obtained by closing the set of horn inclusions under countable composition, cobase change, coproduct and retract.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}As we mentioned in the Introduction, Grandis and Tholen’s *natural weak factorisation systems* [@nwfs] provide an algebraisation of the notion of weak factorisation system. In order to motivate the definition, we first give a similar algebraisation of the notion of strong factorisation system.
So suppose that we are given a strong factorisation system $({{\mathcal L}}, {{\mathcal R}})$ on a category ${{\mathcal C}}$, together with for each map of ${{\mathcal C}}$ a choice of factorisation: $$X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y \quad \mapsto \quad X {\ext@arrow 01{20}0\rightarrowfill@{}{\lambda_f}} Kf {\ext@arrow 01{20}0\rightarrowfill@{}{\rho_f}} Y\text,$$ where $\lambda_f \in {{\mathcal L}}$ and $\rho_f \in {{\mathcal R}}$. It follows from the orthogonality property that this assignation may be extended in a unique way to a *functorial factorisation*: which is to say, a functor $F \colon
{{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 3$ (where $\mathbf 2$ and $\mathbf 3$ are the ordinals $(0 \leqslant 1)$ and $(0 \leqslant 1 \leqslant 2)$ respectively) which splits the “face map” $d_1 \colon {{\mathcal C}}^\mathbf 3 \to {{\mathcal C}}^\mathbf 2$ given by $$d_1(X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y {\ext@arrow 01{20}0\rightarrowfill@{}{g}} Z) = (X {\ext@arrow 01{20}0\rightarrowfill@{}{gf}} Z)\text.$$ [[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}This face map is induced by the functor $\delta_1 \colon \mathbf 2 \to
\mathbf 3$ picking out the unique arrow $0 \to 2$: we have $d_1 =
{{\mathcal C}}^{\delta_1}$. There are two other functors $\delta_0, \delta_2 \colon \mathbf
2 \to \mathbf 3$, and homming these into ${{\mathcal C}}$ induces two further face maps $d_0, d_2 \colon {{\mathcal C}}^\mathbf 3 \to {{\mathcal C}}^\mathbf 2$, with $$d_0(X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y {\ext@arrow 01{20}0\rightarrowfill@{}{g}} Z) = (Y {\ext@arrow 01{20}0\rightarrowfill@{}{g}} Z)\quad \text{and}\quad
d_2(X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y {\ext@arrow 01{20}0\rightarrowfill@{}{g}} Z) = (X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y)\text.$$ Postcomposing our functorial factorisation $F \colon {{\mathcal C}}^\mathbf 2 \to
{{\mathcal C}}^\mathbf 3$ with these induces functors $L, R \colon {{\mathcal C}}^\mathbf 2 \to
{{\mathcal C}}^\mathbf 2$, which send an object $f$ of ${{\mathcal C}}^\mathbf 2$ to $\lambda_f$ and $\rho_f$ respectively.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}\[furtherstructure\] There is further structure in ${\mathbf{Cat}}(\mathbf
2, \mathbf 3)$ which we can make use of: we have natural transformations $\gamma_{2, 1} \colon \delta_2 \Rightarrow \delta_1$ and $\gamma_{1, 0} \colon
\delta_1 \Rightarrow \delta_0$, and by homming these into ${{\mathcal C}}$, we induce natural transformations $c_{2, 1} \colon d_2 \Rightarrow d_1$ and $c_{1, 0}
\colon d_1 \Rightarrow d_0$. Postcomposing $F \colon {{\mathcal C}}^\mathbf 2 \to
{{\mathcal C}}^\mathbf 3$ with these now gives us natural transformations $\Phi \colon L
\Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}$ and $\Lambda \colon {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}
\Rightarrow R$ with components $$\Phi_f =
{\vcenter{\hbox{\xymatrix@1{
X \ar[d]_{\lambda_f} \ar[r]^{{\mathrm{id}}_X} &
X \ar[d]^{f} \\
Kf \ar[r]_{\rho_f} &
Y
}}}}
\qquad \text{and} \qquad \Lambda_f =
{\vcenter{\hbox{\xymatrix@1{
X \ar[d]_{f} \ar[r]^{\lambda_f} &
Kf \ar[d]^{\rho_f} \\
Y \ar[r]_{{\mathrm{id}}_Y} &
Y\text.
}}}}$$ [[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Now, because $F \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 3$ arose from a strong factorisation system, the corresponding $\Lambda : {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}
\Rightarrow R$ will provide the unit for a reflection of ${{\mathcal C}}^\mathbf 2$ into the full subcategory of ${{\mathcal C}}^\mathbf 2$ spanned by the ${{\mathcal R}}$-maps. To see this, consider a morphism $${\vcenter{\hbox{\xymatrix@1{
X \ar[d]_{f} \ar[r]^{h} &
W \ar[d]^{g} \\
Y \ar[r]_{k} &
Z
}}}}$$ from $f$ to $g$ in ${{\mathcal C}}^\mathbf 2$, with $g$ an ${{\mathcal R}}$-map. Then applying orthogonality to the square $${\vcenter{\hbox{\xymatrix{
X \ar[d]_{\lambda_f} \ar[r]^{h} &
W \ar[d]^{g} \\
Kf \ar[r]_{k . \rho_f} &
Z
}}}}$$ we obtain a map $j \colon Kf \to W$ making both triangles commute; and so the map $(h, k) \colon f \to g$ factors uniquely through $\Lambda_f$ as $$f {\ext@arrow 01{20}0\rightarrowfill@{}{\Lambda_f}} \rho_f {\ext@arrow 01{20}0\rightarrowfill@{}{(j, k)}} g\text.$$
Thus the subcategory spanned by the ${{\mathcal R}}$-maps is a full, replete, reflective subcategory of ${{\mathcal C}}^\mathbf 2$, via the reflector $\Lambda \colon
{\mathrm{id}}_{{{\mathcal C}}^\mathbf 2} \Rightarrow R$, and so $(R, \Lambda)$ extends uniquely to an idempotent monad ${\mathsf R}= (R, \Lambda, \Pi)$ whose category of ${\mathsf R}$-algebras may be identified with this subcategory. Dually, the pair $(L, \Phi)$ may be extended uniquely to an idempotent comonad ${\mathsf L}= (L, \Phi, \Sigma)$ whose category of coalgebras is isomorphic to the full subcategory of ${{\mathcal C}}^\mathbf 2$ spanned by the ${{\mathcal L}}$-maps. Thus we have proved:
\[swfs\] There is a bijective correspondence between strong factorisation systems $({{\mathcal L}}, {{\mathcal R}})$ on a category ${{\mathcal C}}$ for which a choice of factorisation for every map has been made, and functorial factorisations $F \colon {{\mathcal C}}^\mathbf 2 \to
{{\mathcal C}}^\mathbf 3$ for which the corresponding pointed endofunctor $(R, \Lambda)$ underlies an idempotent monad and the corresponding copointed endofunctor $(L,
\Phi)$ underlies an idempotent comonad.
The notion of natural weak factorisation system now arises by generalising the situation of this Proposition in a very obvious way: by dropping the requirement of idempotency.
[@nwfs] A *natural weak factorisation system* on a category ${{\mathcal C}}$ is given by a functorial factorisation $F \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 3$, together with an extension of the corresponding pointed endofunctor $(R, \Lambda)$ to a monad ${\mathsf R}= (R, \Lambda, \Pi)$; and an extension of the corresponding copointed endofunctor $(L, \Phi)$ to a comonad ${\mathsf L}= (L, \Phi, \Sigma)$.
Observe that we can reconstruct $F$ from ${\mathsf L}$ and ${\mathsf R}$, and thus we may speak simply of a natural weak factorisation system $({\mathsf L}, {\mathsf R})$.
\[exs2\]
(i) There is a natural w.f.s. on ${\mathbf{Set}}$ whose underlying functorial factorisation is the graph factorisation of Examples \[exs1\](i): $$X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y \quad \mapsto \quad X {\ext@arrow 01{20}0\rightarrowfill@{}{{{\left<{id, f}\right>}}}} X \times Y {\ext@arrow 01{20}0\rightarrowfill@{}{\pi_2}} Y\text.$$ Dually, there is a natural w.f.s. on ${\mathbf{Set}}$ which factors $f$ through $X+Y$. These examples generalise to any category with products or coproducts, as the case may be.
(ii) There is a natural w.f.s. on ${\mathbf{Cat}}$ whose underlying functorial factorisation is given by $${{\mathcal C}}{\ext@arrow 01{20}0\rightarrowfill@{}{F}} {{\mathcal D}}\quad \mapsto \quad {{\mathcal C}}{\ext@arrow 01{20}0\rightarrowfill@{}{\lambda_F}} {{\mathcal D}}\downarrow F {\ext@arrow 01{20}0\rightarrowfill@{}{\rho_F}} {{\mathcal D}}\text,$$ where ${{\mathcal D}}\downarrow F$ is the *comma category* whose objects are triples $(c, d, f \colon d \to Fc)$; $\lambda_F$ is the functor sending $c$ in ${{\mathcal C}}$ to $({\mathrm{id}}\colon Fc \to Fc)$ in ${{\mathcal D}}\downarrow F$; and $\rho_F$ is the functor sending $(f \colon d \to Fc)$ in ${{\mathcal D}}\downarrow
F$ to $d$. There are variations on this theme: we can replace ${{\mathcal D}}\downarrow F$ with the dual comma category $F \downarrow {{\mathcal D}}$; or with the *iso-comma category* ${{\mathcal D}}\downarrow_{\cong} F$, which is the full subcategory of ${{\mathcal D}}\downarrow F$ whose objects are the invertible arrows. These examples generalise to any 2-category with comma objects.
(iii) By Proposition \[swfs\], any strong factorisation system on ${{\mathcal C}}$ gives rise to a natural weak factorisation system on ${{\mathcal C}}$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}It is not immediately clear that a natural w.f.s. deserves the name of weak factorisation system. To show that this is so, we must exhibit suitable analogues of the axioms of factorisation and weak orthogonality; for which we must first identify what the ${{\mathcal L}}$-maps and ${{\mathcal R}}$-maps are. Now, for a strong factorisation system, we can reconstruct the ${{\mathcal L}}$- and ${{\mathcal R}}$-maps from the associated comonad ${\mathsf L}$ and monad ${\mathsf R}$ as their respective coalgebras and algebras; and thus it is natural to define:
Let $({\mathsf L}, {\mathsf R})$ be a natural w.f.s. on ${{\mathcal C}}$. We write ${\mathsf L\text-{\mathbf{Map}}}$ for the category of ${\mathsf L}$-coalgebras, and call its objects *${\mathsf L}$-maps*; and write ${\mathsf R\text-{\mathbf{Map}}}$ for the category of ${\mathsf R}$-algebras and call its objects *${\mathsf R}$-maps*.
Note that being an ${\mathsf L}$- or ${\mathsf R}$-map is structure on, and not a property of, a map of ${{\mathcal C}}$.
\[exs3\]
(i) For the natural w.f.s. on ${\mathbf{Set}}$ which factors $f \colon X \to
Y$ through $X + Y$, an ${\mathsf R}$-map structure on $g \colon C \to D$ is a splitting for $g$: that is, a morphism $g^\ast \colon Y \to X$ with $gg^\ast = {\mathrm{id}}_Y$. An ${\mathsf L}$-map structure on $f \colon A \to B$ exists just when $f$ is a monomorphism, and in this case is uniquely determined: thus the comonad ${\mathsf L}$ is “property-like”, though not idempotent.
(ii) For the natural w.f.s. on ${\mathbf{Cat}}$ which factors $F \colon {{\mathcal C}}\to
{{\mathcal D}}$ through ${{\mathcal D}}\downarrow F$, an ${\mathsf R}$-map is a *split fibration*: that is, a Grothendieck fibration with chosen liftings that compose up strictly. An ${\mathsf L}$-map is, roughly speaking, an inclusion of a reflective subcategory: more precisely, an ${\mathsf L}$-map structure on a functor $F \colon {{\mathcal C}}\to {{\mathcal D}}$ is given by specifying a functor $F^\ast
\colon {{\mathcal D}}\to {{\mathcal C}}$ and a natural transformation $\eta \colon 1_{{\mathcal D}}\Rightarrow FF^\ast$ satisfying $F^\ast F = 1_{{\mathcal D}}$, $F^\ast \eta =
{\mathrm{id}}_{F^\ast}$ and $\eta F = {\mathrm{id}}_F$. For the n.w.f.s. which factors through $F \downarrow {{\mathcal D}}$ instead, the ${\mathsf R}$-algebras are split opfibrations and the ${\mathsf L}$-coalgebras, inclusions of coreflective subcategories; whilst if we factor through ${{\mathcal D}}\downarrow_{\cong} F$, then ${\mathsf R}$-algebras are split isofibrations, and ${\mathsf L}$-coalgebras are retract equivalences.
(iii) If we view a strong factorisation system $({{\mathcal L}}, {{\mathcal R}})$ on ${{\mathcal C}}$ as a natural w.f.s., then the ${\mathsf L}$-maps and ${\mathsf R}$-maps reduce to ${{\mathcal L}}$-maps and ${{\mathcal R}}$-maps. In this particular case, being an ${\mathsf L}$- or ${\mathsf R}$-map returns to being a mere property; and this is because the comonad ${\mathsf L}$ and monad ${\mathsf R}$ are idempotent.
Further details on these examples may be found in [@nwfs].
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}\[factdefn\] With this definition of ${\mathsf L}$-map and ${\mathsf R}$-map, it is now clear that any natural w.f.s. $({\mathsf L}, {\mathsf R})$ admits an axiom of factorisation: given a map $f \colon C \to D$, we obtain an ${\mathsf L}$-map structure on $\lambda_f
\colon C \to Kf$ by applying the cofree functor ${{\mathcal C}}^\mathbf 2 \to {\mathsf L\text-{\mathbf{Map}}}$, and an ${\mathsf R}$-map structure on $\rho_f \colon Kf \to D$ by applying the free functor ${{\mathcal C}}^\mathbf 2 \to {\mathsf R\text-{\mathbf{Map}}}$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}\[worthdefn\] More interestingly, we also have an axiom of weak orthogonality. To see this, suppose that we are given a square like together with an ${\mathsf L}$-coalgebra structure on $f$ and an ${\mathsf R}$-algebra structure on $g$. Thus we have a coaction morphism $e \colon f
\to Lf$ and an action morphism $m \colon Rg \to g$, which the (co)algebra axioms force to be of the following forms: $$e = {\vcenter{\hbox{\xymatrix{
A \ar[d]_f \ar[r]^{{\mathrm{id}}_A} & A \ar[d]^{\lambda_f} \\
B \ar[r]_{s} & Kf
}}}} \qquad \text{and} \qquad
m = {\vcenter{\hbox{\xymatrix{
Kf \ar[d]_{\rho_g} \ar[r]^{p} & C \ar[d]^{g} \\
D \ar[r]_{{\mathrm{id}}_D} & D\text.
}}}}$$ Furthermore, we may view the square as a map $(h, k)
\colon f \to g$ in ${{\mathcal C}}^\mathbf 2$; and so applying the functorial factorisation of $({\mathsf L}, {\mathsf R})$ yields an arrow $K(h, k) \colon Kf \to Kg$ in ${{\mathcal C}}$. We now obtain a diagonal fill-in for as the composite $$B {\ext@arrow 01{20}0\rightarrowfill@{}{s}} Kf {\ext@arrow 01{20}0\rightarrowfill@{}{K(h, f)}} Kg {\ext@arrow 01{20}0\rightarrowfill@{}{p}} C\text.
\label{fillineq}$$ Note that this fill-in is canonically determined by the ${\mathsf L}$-map structure on $f$ and the ${\mathsf R}$-map structure on $g$. Indeed, it is reasonable to view an ${\mathsf L}$-map structure as encoding a coherent choice of lifting opposite every ${\mathsf R}$-map, and vice versa.
Let us see how we obtain diagonal fill-ins for the natural w.f.s. on ${\mathbf{Cat}}$ which factors $F \colon {{\mathcal C}}\to {{\mathcal D}}$ through ${{\mathcal D}}\downarrow F$. We suppose ourselves given a square of functors $${\vcenter{\hbox{\xymatrix{
{{\mathcal A}}\ar[r]^-H
\ar[d]_F &
{{\mathcal C}}\ar[d]^G \\
{{\mathcal B}}\ar[r]_-K &
{{\mathcal D}}\text,
}}}}$$ with $F$ an ${\mathsf L}$-coalgebra and $G$ an ${\mathsf R}$-algebra. The ${\mathsf L}$-coalgebra structure on $F$ provides us with a functor $F^\ast \colon {{\mathcal B}}\to {{\mathcal A}}$ and a natural transformation $\eta \colon 1 \Rightarrow FF^\ast$. Thus we can define a functor $HF^\ast \colon {{\mathcal B}}\to {{\mathcal C}}$ and a natural transformation $${\vcenter{\hbox{\xymatrix{
{{\mathcal B}}\ar[rr]^{HF^\ast} \ar[dr]_K & {}{\ar@{}[d] \ar@{=>}?(0.5)+/l 0.2cm/;?(0.5)+/r 0.2cm/^{\alpha}} & {{\mathcal C}}\ar[dl]^G \\ & {{\mathcal D}}\text;
}}}}$$ indeed, we have $GHF^\ast = KFF^\ast$, and so can take $\alpha = K \eta \colon
K \Rightarrow KFF^\ast$. Now using the ${\mathsf R}$-algebra structure on $G$, we may factorise this 2-cell as: $${\vcenter{\hbox{\xymatrix@R+1em{
{{\mathcal B}}\ar@/^0.8em/[rr]^{HF^\ast} \ar@/_0.8em/[rr]_J \ar[dr]_K & {}{\ar@{}[d] \ar@{=>}?(0)+/d 0.2cm/;?(0)+/u 0.2cm/_{\overline \alpha}} {\ar@{}[d] \save ?(0.5)*{=}\restore} & {{\mathcal C}}\ar[dl]^G \\ & {{\mathcal D}}\text,
}}}}$$ where $J$ is given by reindexing $HF^\ast$ along $\alpha$. It is not hard to see that this functor $J \colon {{\mathcal C}}\to {{\mathcal D}}$ is precisely the fill-in specified by equation above.
\[underlyingplain\] It follows from the observations of §\[factdefn\] and §\[worthdefn\] that any natural w.f.s. $({\mathsf L}, {\mathsf R})$ on a category ${{\mathcal C}}$ has an underlying plain w.f.s. For if we define ${{\mathcal L}}$ to be the class of arrows in ${{\mathcal C}}$ which admit some ${\mathsf L}$-coalgebra structure and ${{\mathcal R}}$ to be the class of arrows admitting some ${\mathsf R}$-algebra structure, then $({{\mathcal L}}, {{\mathcal R}})$ will satisfy all the axioms required of a w.f.s., expect possibly for closure under retracts. So we take $\bar {{\mathcal L}}$ and $\bar {{\mathcal R}}$ to be the respective retract-closures of ${{\mathcal L}}$ and ${{\mathcal R}}$; and now the pair $(\bar {{\mathcal L}}, \bar {{\mathcal R}})$ gives a w.f.s. on ${{\mathcal C}}$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}\[strengthenslightly\] It turns to be very useful to strengthen the notion of natural w.f.s. slightly. For this, we consider the natural transformations $\Pi \colon RR \Rightarrow R$ and $\Sigma \colon L \Rightarrow
LL$ associated to a natural w.f.s. $({\mathsf L}, {\mathsf R})$. We may denote their respective components at $f \in {{\mathcal C}}^\mathbf 2$ by $$\Pi_f = {\vcenter{\hbox{\xymatrix{
K\rho_f \ar[d]_{\rho_{\rho_f}} \ar[r]^{\pi_f} & Kf \ar[d]^{\rho_f} \\
B \ar[r]_{{\mathrm{id}}_B} & B
}}}} \qquad \text{and} \qquad
\Sigma_f = {\vcenter{\hbox{\xymatrix{
A \ar[d]_{\lambda_f} \ar[r]^{{\mathrm{id}}_A} & A \ar[d]^{\lambda_{\lambda_f}} \\
Kf \ar[r]_{\sigma_f} & K\lambda_f\text;
}}}}$$ again, the arrows written as identities are forced to be so by the (co)monad axioms. Now, these maps $\sigma_f$ and $\pi_f$ provide us with the components of a natural transformation $\Delta \colon LR \Rightarrow RL$ whose component at $f$ is given by: $$\Delta_f = {\vcenter{\hbox{\xymatrix{
Kf \ar[d]_{\lambda_{\rho_f}} \ar[r]^{\sigma_f} & K\lambda_f \ar[d]^{\rho_{\lambda_f}} \\
K\rho_f \ar[r]_{\pi_f} & Kf\text.
}}}}$$ (That this square commutes is a consequence of the (co)monad axioms). We will say that a natural w.f.s. *satisfies the distributivity axiom* if this natural transformation $\Delta \colon LR \Rightarrow RL$ defines a distributive law of ${\mathsf L}$ over ${\mathsf R}$ in the sense of [@Beck]. Note that this is a property of a natural w.f.s., rather than extra structure on it.
We may check that each of the natural w.f.s.’s given so far satisfies the distributivity axiom.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}There are important results about n.w.f.s.’s that are true only if we restrict to those for which the distributivity axiom holds. Two such results are Theorem \[diamondtwofold\] and Theorem \[freealgfree\] below; and there is another which allows us to characterise ${\mathsf R}$-maps purely in terms of lifting properties against the ${\mathsf L}$-maps, and vice versa. In order that these results should be valid, we henceforth modify the definition of natural w.f.s.to include the requirement that the distributivity axiom should hold.
Free and algebraically-free natural w.f.s.’s
============================================
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Our goal is to use the theory of natural w.f.s.’s to give a categorically coherent reformulation of the small object argument. As we stated in the Introduction, this argument provides the means by which, starting from a set of maps $J$, one may produce a w.f.s. *cofibrantly generated* by $J$: that is, a w.f.s. $({{\mathcal L}}, {{\mathcal R}})$ for which ${{\mathcal R}}= J^\pitchfork$.
\[exs4\] All the weak factorisation systems of Examples \[exs1\] are cofibrantly generated:
- For the w.f.s. (Mono, Epi) on ${\mathbf{Set}}$, a suitable $J$ is given by the set containing the single map $! \colon 0 \to 1$.
- For the (injective equivalences, isofibrations) w.f.s. on ${\mathbf{Cat}}$, a suitable $J$ is given by the single map ${\left\llcorner{b}\right\lrcorner} \colon
1 \to {\mathbf{Iso}}$, where ${\mathbf{Iso}}$ is the indiscrete category on the set $\{a, b\}$.
- For the w.f.s. (anodyne extensions, Kan fibrations) on ${\mathbf{SSet}}$, a suitable $J$ is given by the set of horn inclusions $\{\Lambda_n^k
\to \Delta_n\}$.
To give our reformulation of the small object argument, we will need to provide a notion of “cofibrantly generated” natural w.f.s. However, a careful analysis reveals two candidates for this notion. In this section, we study these candidates and their relationship to each other.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}We begin by forming the entities that we have met so far into categories. Suppose we are given functorial factorisations $F$ and $F' \colon {{\mathcal C}}^\mathbf 2
\to {{\mathcal C}}^\mathbf 3$ on ${{\mathcal C}}$. We define a *morphism of functorial factorisations* $\alpha \colon F \to F'$ to be a natural transformation $\alpha
\colon F \Rightarrow F'$ which upon whiskering with $d_1 \colon {{\mathcal C}}^\mathbf 3
\to {{\mathcal C}}^\mathbf 2$ becomes the identity transformation ${\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}
\Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}$. To give such a morphism is to give a family of maps $\alpha_f \colon Kf \to K'f$, natural in $f$, and making diagrams of the following form commute: $${\vcenter{\hbox{\xymatrix{
& A \ar[dl]_{\lambda_f} \ar[dr]^{\lambda'_f} \\
Kf \ar[dr]_{\rho_f} \ar[rr]_{\alpha_f} & & K'f \ar[dl]^{\rho'_f} \\ & B\text.}}}}$$
Suppose now that $F$ and $F'$ underlie natural w.f.s.’s $({\mathsf L}, {\mathsf R})$ and $({\mathsf L}', {\mathsf R}')$ on ${{\mathcal C}}$, and consider a morphism of functorial factorisations $\alpha \colon F \to F'$. By whiskering the natural transformation $\alpha
\colon F \Rightarrow F'$ with the other two face maps $d_0, d_2 \colon
{{\mathcal C}}^\mathbf 3 \to {{\mathcal C}}^\mathbf 2$, we induce natural transformations $\alpha_l
\colon L \Rightarrow L'$ and $\alpha_r \colon R \Rightarrow R'$; and we will say that $\alpha \colon F \to F'$ is a *morphism of natural w.f.s.’s* just when $\alpha_l$ is a comonad morphism and $\alpha_r$ a monad morphism.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Let us write ${\mathbf{NWFS}}({{\mathcal C}})$ for the category of n.w.f.s.’s on ${{\mathcal C}}$. We may define a “semantics” functor ${{\mathcal G}}\colon {\mathbf{NWFS}}({{\mathcal C}}) \to
{\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$, which sends a n.w.f.s. $({\mathsf L}, {\mathsf R})$ to its category of ${\mathsf L}$-coalgebras ${\mathsf L\text-{\mathbf{Map}}}$, equipped with the forgetful functor into ${{\mathcal C}}^\mathbf 2$; and sends a morphism $\alpha \colon ({\mathsf L}, {\mathsf R}) \to ({\mathsf L}',
{\mathsf R}')$ of n.w.f.s.’s to the morphism $${\vcenter{\hbox{\xymatrix{
{\mathsf L\text-{\mathbf{Map}}}\ar[rr]^{(\alpha_l)_\ast} \ar[dr]_{U_{{\mathsf L}}} & &
{\mathsf L}'\text-{\mathbf{Map}} \ar[dl]^{U_{{\mathsf L}'}} \\ & {{\mathcal C}}^\mathbf 2
}}}}$$ of ${\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$. Here $(\alpha_l)_\ast$ is the functor which sends an ${\mathsf L}$-coalgebra $x \colon X \to LX$ to the ${\mathsf L}'$-coalgebra $$X {\ext@arrow 01{20}0\rightarrowfill@{}{x}} LX {\ext@arrow 01{20}0\rightarrowfill@{}{(\alpha_l)_X}} L'X\text.$$ Our first candidate for the notion of “cofibrantly generated” n.w.f.s. is now:
\[cand1\] Let $I \colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ be an object of ${\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$, with ${{\mathcal J}}$ small; and let $({\mathsf L}, {\mathsf R})$ be a n.w.f.s. on ${{\mathcal C}}$. We will say that $({\mathsf L}, {\mathsf R})$ is *free on ${{\mathcal J}}$*[^2] if we can provide a morphism $${\vcenter{\hbox{\xymatrix@!C@C-1em{
{{\mathcal J}}\ar[rr]^{\eta} \ar[dr]_{I} & &
{\mathsf L}\text-{\mathbf{Map}} \ar[dl]^{U_{{\mathsf L}}} \\ & {{\mathcal C}}^\mathbf 2
}}}}$$ of ${\mathbf{CAT}} / {{\mathcal C}}^\mathbf 2$ which exhibits $({\mathsf L}, {\mathsf R})$ as a reflection of $I$ along ${{\mathcal G}}$: which is to say that, for any n.w.f.s. $({\mathsf L}', {\mathsf R}')$ on ${{\mathcal C}}$ and functor $F \colon {{\mathcal J}}\to {\mathsf L}'\text-{\mathbf{Map}}$ over ${{\mathcal C}}^\mathbf 2$, there is a unique morphism of n.w.f.s.’s $\alpha \colon ({\mathsf L}, {\mathsf R}) \to ({\mathsf L}', {\mathsf R}')$ for which $F = (\alpha_l)_\ast \circ \eta$.
There is a dual semantics functor ${{\mathcal H}}\colon {\mathbf{NWFS}}({{\mathcal C}}) \to
({\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2)^{\mathrm{op}}$, which sends a n.w.f.s. to its category of ${\mathsf R}$-algebras: and a corresponding notion of an n.w.f.s. being *cofree* on ${{\mathcal J}}$. However, being cofree is significantly less common than being free, primarily because the conditions under which we will construct free n.w.f.s.’s – typically, local presentability or local boundedness – are much more prevalent than their duals.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Whilst Definition \[cand1\] is natural from a categorical perspective, it has an obvious drawback: it provides no analogue of the equation ${{\mathcal R}}=
J^\pitchfork$ which a cofibrantly generated w.f.s. satisfies. Definition \[cand2\], our second candidate for the notion of “cofibrantly generated” n.w.f.s., will rectify this. Before we can give it, we will need a preliminary result.
Let ${{\mathcal C}}$ be a category. Then the Galois connection ${}^{\pitchfork}(\ )
\mathbin \dashv (\ )^\pitchfork$ induced by the notion of weak orthogonality may be lifted to an adjunction $${\vcenter{\hbox{\xymatrix{{\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2 \ar@<4pt>[r]^-{{}^{\pitchfork}({{\mathord{\text{--}}}})} \ar@{}[r]|-{\bot} & ({\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2)^{\mathrm{op}}\ar@<4pt>[l]^-{({{\mathord{\text{--}}}})^\pitchfork}}}}}\text.$$
First we give the functor $({{\mathord{\text{--}}}})^\pitchfork \colon ({\mathbf{CAT}}/{{\mathcal C}}^\mathbf
2)^{\mathrm{op}}\to {\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$. On objects, this sends a category $U \colon
{{\mathcal A}}\to {{\mathcal C}}^\mathbf 2$ over ${{\mathcal C}}^\mathbf 2$ to the following category ${{\mathcal A}}^\pitchfork$ over ${{\mathcal C}}^\mathbf 2$. Its objects are pairs $(g, \phi)$, where $g$ is a morphism of ${{\mathcal C}}$ and $\phi$ is a coherent choice of lifting against every element of ${{\mathcal A}}$: which is to say, a mapping which to each object $a \in
{{\mathcal A}}$ and square $$\label{fillin2}
{\vcenter{\hbox{\xymatrix{
A
\ar[r]^-h
\ar[d]_{Ua} &
C
\ar[d]^g \\
B
\ar[r]_-k &
D
}}}}$$ in ${{\mathcal C}}$, assigns a morphism $\phi(a, h, k) \colon B \to C$ making both triangles commute, and subject to the following naturality condition: if we are given a morphism $\sigma \colon a \to a'$ of ${{\mathcal A}}$ whose image under $U$ is the morphism $${\vcenter{\hbox{\xymatrix{
A
\ar[r]^-s
\ar[d]_{Ua} &
A'
\ar[d]^{Ua'} \\
B
\ar[r]_-t &
B'\text,
}}}}$$ of ${{\mathcal C}}^\mathbf 2$, then we have $\phi(a, hs, kt) = \phi(a', h, k) \circ t$. A morphism of ${{\mathcal A}}^\pitchfork$ from $(g, \phi)$ to $(g', \phi')$ is a morphism $(u, v) \colon g \to g'$ of ${{\mathcal C}}^\mathbf 2$ which respects the choice of liftings in $\phi$ and $\phi'$, in the sense that the equation $u \circ \phi(a,
h, k) = \phi'(a, uh, vk)$ holds for all suitable $a$, $h$ and $k$. The functor exhibiting ${{\mathcal A}}^\pitchfork$ as a category over ${{\mathcal C}}^\mathbf 2$ is the evident forgetful functor.
This defines $({{\mathord{\text{--}}}})^\pitchfork$ on objects of ${\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$; and to extend this definition to morphisms, we consider a further category ${{\mathcal B}}$ over ${{\mathcal C}}^\mathbf 2$ and a functor $F \colon {{\mathcal A}}\to {{\mathcal B}}$ over ${{\mathcal C}}^\mathbf 2$: from which we obtain a map $F^\pitchfork \colon {{\mathcal B}}^\pitchfork \to {{\mathcal A}}^\pitchfork$ over ${{\mathcal C}}^\mathbf 2$ by sending the object $(c, \phi({{\mathord{\text{--}}}}, \mathord \ast,
\mathord ?))$ of ${{\mathcal B}}^\pitchfork$ to the object $(c, \phi(F({{\mathord{\text{--}}}}), \mathord
\ast, \mathord ?))$ of ${{\mathcal A}}^\pitchfork$.
We define the functor ${}^\pitchfork({{\mathord{\text{--}}}})$ in the same way as $({{\mathord{\text{--}}}})^\pitchfork$, but with $Ua$ and $g$ swapped around in equation . It remains only to exhibit the adjointness ${}^\pitchfork({{\mathord{\text{--}}}}) \dashv ({{\mathord{\text{--}}}})^\pitchfork$: for which it is easy to see that, given categories $U \colon {{\mathcal A}}\to {{\mathcal C}}^\mathbf 2$ and $V \colon {{\mathcal B}}\to
{{\mathcal C}}^\mathbf 2$ over ${{\mathcal C}}^\mathbf 2$, we may identify both $$\text{functors } {{\mathcal A}}\to {}^\pitchfork {{\mathcal B}}\qquad \text{and} \qquad \text{functors } {{\mathcal B}}\to {{\mathcal A}}^\pitchfork$$ over ${{\mathcal C}}^\mathbf 2$ with “$({{\mathcal A}}, {{\mathcal B}})$-lifting operations”: that is, functions $\psi$ which given an object $a \in {{\mathcal A}}$, an object $b \in {{\mathcal B}}$ and a commuting square $${\vcenter{\hbox{\xymatrix{
A
\ar[r]^-h
\ar[d]_{Ua} &
C
\ar[d]^{Vb} \\
B
\ar[r]_-k &
D\text,
}}}}$$ provide a morphism $\psi(a, b, h, k) \colon B \to C$ making both triangles commute; and subject to the obvious naturality condition with respect to morphisms of both ${{\mathcal A}}$ and ${{\mathcal B}}$.
In particular, we see from §\[worthdefn\] that any any n.w.f.s. $({\mathsf L},
{\mathsf R})$ comes equipped with a privileged $({\mathsf L\text-{\mathbf{Map}}}$, $ {\mathsf R\text-{\mathbf{Map}}})$-lifting operation: which by the above proof, we may view as a privileged morphism $\textsf{lift}
\colon {\mathsf R\text-{\mathbf{Map}}}\to {\mathsf L\text-{\mathbf{Map}}}^\pitchfork$ over ${{\mathcal C}}^\mathbf 2$.
\[cand2\] Let $I \colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ be a category over ${{\mathcal C}}^\mathbf 2$, and $({\mathsf L}, {\mathsf R})$ a n.w.f.s. on ${{\mathcal C}}$. We say that $({\mathsf L}, {\mathsf R})$ is *algebraically-free* on ${{\mathcal J}}$ if we can provide a morphism $\eta \colon {{\mathcal J}}\to {\mathsf L\text-{\mathbf{Map}}}$ over ${{\mathcal C}}$ for which the functor $$\label{algfreefunctor}
{\mathsf R\text-{\mathbf{Map}}}{\ext@arrow 01{20}0\rightarrowfill@{}{\textsf{lift}}} {\mathsf L\text-{\mathbf{Map}}}^\pitchfork {\ext@arrow 01{20}0\rightarrowfill@{}{\eta^\pitchfork}}
{{\mathcal J}}^\pitchfork$$ is an isomorphism of categories.
The terminology we have chosen deliberately recalls the distinction which is made in [@Ke80] between the *free* and the *algebraically-free* monad generated by a pointed endofunctor. We will partially justify this in Section \[Sec:cfaf\], by showing that algebraic-freeness in our sense can be seen as a special case of algebraic-freeness in the sense of [@Ke80]; and in the Appendix, where we prove the implication “algebraically-free $\Rightarrow$ free” for n.w.f.s.’s.
However, there are some results of [@Ke80] which the author has been unable to find an analogue of: in particular, he has been unable to produce either positive or negative results about the implication “free $\Rightarrow$ algebraically-free”. The corresponding implication does not hold in the theory of monads; and whilst it seems unlikely that it should hold here either, a proof of this fact has been elusive. Despite this, we will be able to show in Section \[Sec:cfaf\] that any free n.w.f.s. which we come across *in mathematical practice* will be algebraically-free.
\[exs5\] The natural w.f.s. on ${\mathbf{Set}}$ which factors $f \colon X \to Y$ through $X
+ Y$ is algebraically-free: we let ${{\mathcal J}}$ be the terminal category and let $I
\colon {{\mathcal J}}\to {\mathbf{Set}}^\mathbf 2$ pick out the object $! \colon 0 \to 1$. It is now easy to see that the category ${{\mathcal J}}^\pitchfork$ consists precisely of the ${\mathsf R}$-algebras: morphisms $g \colon C \to D$ equipped with a splitting $g^\ast
\colon D \to C$.
However, none of the other natural w.f.s.’s described in Examples \[exs2\] are free or algebraically-free: and this despite being close relatives of plain w.f.s.’s which are cofibrantly generated. The problem for these examples is that, although an ${\mathsf R}$-map structure affirms the existence of certain liftings, it also asserts certain coherence conditions between those liftings, which cannot be expressed in the language of orthogonality.
A fair intuition is that the (algebraically)-free natural w.f.s.’s are the natural w.f.s.’s which may be specified by a “signature” ${{\mathcal J}}$ of lifting properties; but subject to no “equations” between these liftings.
We may relate the notion of algebraically-free n.w.f.s. quite directly to that of cofibrantly generated w.f.s., if we assume the axiom of choice in our metatheory:
\[relateback\] Let ${{\mathcal C}}$ be a category and $J$ a set of maps in ${{\mathcal C}}$; and let ${{\mathcal J}}$ denote the set $J$, viewed as a discrete subcategory of ${{\mathcal C}}^\mathbf 2$. If the algebraically-free n.w.f.s. $({\mathsf L}, {\mathsf R})$ on ${{\mathcal J}}\hookrightarrow {{\mathcal C}}^\mathbf 2$ exists, then its underlying plain w.f.s. $(\bar {{\mathcal L}}, \bar {{\mathcal R}})$ is the w.f.s.cofibrantly generated by $J$.
Recall from §\[underlyingplain\] that the class of maps ${{\mathcal R}}$ consists of those maps in ${{\mathcal C}}$ admitting some ${\mathsf R}$-algebra structure; and that $\bar {{\mathcal R}}$ consists of all retracts of maps in ${{\mathcal R}}$. We are required to show that $\bar {{\mathcal R}}= J^\pitchfork$; and since $J^\pitchfork$ is easily seen to be closed under retracts, it will suffice to show that ${{\mathcal R}}= J^\pitchfork$.
Now, since $({\mathsf L}, {\mathsf R})$ is algebraically-free on ${{\mathcal J}}$, we have ${\mathsf R\text-{\mathbf{Map}}}\cong
{{\mathcal J}}^\pitchfork$ over ${{\mathcal C}}^\mathbf 2$; and so a morphism $f \in {{\mathcal C}}^\mathbf 2$ will admit an ${\mathsf R}$-algebra structure, and thus lie in ${{\mathcal R}}$, just when it can be lifted through the forgetful functor ${{\mathcal J}}^\pitchfork \to {{\mathcal C}}^\mathbf 2$. But an object of ${{\mathcal J}}^\pitchfork$ consists of a map of ${{\mathcal C}}$ equipped with a choice of lifting against every map in the set $J$, subject to no further coherence conditions; and so, if we allow ourselves the axiom of choice, $f$ will admit a lifting through ${{\mathcal J}}^\pitchfork$ just when $f \in J^\pitchfork$. Thus we have that ${{\mathcal R}}= J^\pitchfork$ as desired.
Constructing free natural w.f.s.’s {#construct}
==================================
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}We now ready to give our analogue of the small object argument, which will be a general apparatus by means of which we can construct free, and even algebraically-free, n.w.f.s.’s on a category ${{\mathcal C}}$.
For our argument to work, we will at least require ${{\mathcal C}}$ to be cocomplete: but in order to guarantee the convergence of certain transfinite sequences we construct, we must impose some further “smallness” property on ${{\mathcal C}}$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Given a regular cardinal $\alpha$, we say that $X \in {{\mathcal C}}$ is $\alpha$-*presentable* if the representable functor ${{\mathcal C}}(X, {{\mathord{\text{--}}}}) \colon {{\mathcal C}}\to {\mathbf{Set}}$ preserves $\alpha$-filtered colimits. The first smallness property we may consider on ${{\mathcal C}}$ is that:
> (\*) For every $X \in {{\mathcal C}}$, there is an $\alpha_X$ for which $X$ is $\alpha_X$-presentable.
This is certainly the case for any category ${{\mathcal C}}$ which is *locally presentable* in the sense of [@lpk]. However, it does not obtain in categories such as the category of topological spaces, the category of Hausdorff topological spaces, or the category of topological groups: and since we would like our argument to be valid in such contexts, we will require a more general notion of smallness.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Recall that a strong factorisation system $({{\mathcal E}}, {{\mathcal M}})$ on ${{\mathcal C}}$ is said to be *proper* if every ${{\mathcal E}}$-map is an epimorphism and every ${{\mathcal M}}$-map a monomorphism; and is said to be *well-copowered* if every object of ${{\mathcal C}}$ possesses, up-to-isomorphism, a mere set of ${{\mathcal E}}$-quotients. We say that an object $X$ is $\alpha$-*bounded* with respect to a proper $({{\mathcal E}}, {{\mathcal M}})$ if ${{\mathcal C}}(X, {{\mathord{\text{--}}}})$ preserves $\alpha$-filtered unions of ${{\mathcal M}}$-subobjects (in the sense of sending them to $\alpha$-filtered unions of sets). The second smallness property we consider on ${{\mathcal C}}$ supposes some proper, well-copowered $({{\mathcal E}}, {{\mathcal M}})$, and says that:
> () For every $X \in {{\mathcal C}}$, there is an $\alpha_X$ for which $X$ is $\alpha_X$-bounded with respect to $({{\mathcal E}}, {{\mathcal M}})$.
${\mathbf{Top}}$, ${\mathbf{Haus}}$ and ${\mathbf{TopGrp}}$ all satisfy (), with ${{\mathcal M}}$ = the subspace inclusions in the first two cases, and ${{\mathcal M}}= $ the inclusion of subgroups which are also subspaces in the third.
We may now state the main result of the paper.
\[mainthm\] Let ${{\mathcal C}}$ be a cocomplete category satisfying either (\*) or (), and let $I
\colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ be a category over ${{\mathcal C}}^\mathbf 2$ with ${{\mathcal J}}$ small. Then the free n.w.f.s. on ${{\mathcal J}}$ exists, and is algebraically-free on ${{\mathcal J}}$.
In this section, we will prove freeness: in the next, algebraic-freeness.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}We begin by factorising the semantics functor ${{\mathcal G}}$ through a pair of intermediate categories. The first is the category ${\mathbf{LNWFS}}({{\mathcal C}})$ of “left halves of n.w.f.s.’s”. Its objects $(F, {\mathsf L})$ are functorial factorisations $F$ on ${{\mathcal C}}$ together with an extension of the corresponding $(L, \Phi)$ to a comonad ${\mathsf L}$; and its morphisms are maps of functorial factorisations which respect the comonad structure. There is an obvious functor ${{\mathcal G}}_1 \colon
{\mathbf{NWFS}}({{\mathcal C}}) \to {\mathbf{LNWFS}}({{\mathcal C}})$ sending $({\mathsf L}, {\mathsf R})$ to $(F, {\mathsf L})$.
The second category we consider is ${\mathbf{Cmd}}({{\mathcal C}}^\mathbf 2)$, the category of comonads on ${{\mathcal C}}^\mathbf 2$. We have a functor ${{\mathcal G}}_2 \colon {\mathbf{LNWFS}}({{\mathcal C}}) \to
{\mathbf{Cmd}}({{\mathcal C}}^\mathbf 2)$, which sends $(F, {\mathsf L})$ to ${\mathsf L}$; and we have the semantics functor ${{\mathcal G}}_3 \colon {\mathbf{Cmd}}({{\mathcal C}}^\mathbf 2) \to {\mathbf{CAT}} /
{{\mathcal C}}^\mathbf 2$ which sends a comonad to its category of coalgebras, and a comonad morphism $\gamma \colon \mathsf C \to \mathsf C'$ to $\gamma_\ast
\colon \mathsf C\text-{\mathbf{Coalg}} \to \mathsf C'\text-{\mathbf{Coalg}}$. We now have that $${{\mathcal G}}\quad = \quad {\mathbf{NWFS}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{{{\mathcal G}}_1}} {\mathbf{LNWFS}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{{{\mathcal G}}_2}}
{\mathbf{Cmd}}({{\mathcal C}}^\mathbf 2) {\ext@arrow 01{20}0\rightarrowfill@{}{{{\mathcal G}}_3}} {\mathbf{CAT}} / {{\mathcal C}}^\mathbf 2\text,$$ so that we may give a reflection along ${{\mathcal G}}$ by giving a reflection along each functor ${{\mathcal G}}_1$, ${{\mathcal G}}_2$ and ${{\mathcal G}}_3$ in turn. For ${{\mathcal G}}_3$, we have the following well-known result, which was first stated at this level of generality by Dubuc [@Dbc]; but see also [@triples].
\[Kanext\] Let ${{\mathcal C}}$ be cocomplete, and let $U \colon {{\mathcal A}}\to {{\mathcal C}}^\mathbf 2$ be a small category over ${{\mathcal C}}^\mathbf 2$. Then ${{\mathcal A}}$ admits a reflection along ${{\mathcal G}}_3 \colon
{\mathbf{Cmd}}({{\mathcal C}}^\mathbf 2) \to {\mathbf{CAT}} / {{\mathcal C}}^\mathbf 2$.
Because ${{\mathcal A}}$ is small and ${{\mathcal C}}^\mathbf 2$ cocomplete (since ${{\mathcal C}}$ is), we can form the left Kan extension of $U$ along itself: $${\vcenter{\hbox{\xymatrix{
& {{\mathcal A}}\ar[dr]^U \ar[dl]_U {\ar@{}[d] \ar@{=>}?(0.5)+/r 0.2cm/;?(0.5)+/l 0.2cm/^{\theta}} \\
{{\mathcal C}}^\mathbf 2 \ar[rr]_{\operatorname{Lan}_U(U)} & & {{\mathcal C}}^\mathbf 2\text,
}}}}$$ whose defining property is that $\theta$ should provide the unit for a representation $$[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2](\operatorname{Lan}_U(U), {{\mathord{\text{--}}}}) \cong [{{\mathcal A}}, {{\mathcal C}}^\mathbf 2](U, ({{\mathord{\text{--}}}}) \circ U)\text.$$ In particular, corresponding to the identity transformation ${\mathrm{id}}_U \colon U
\Rightarrow U$, we have a natural transformation $\epsilon \colon \operatorname{Lan}_U(U)
\Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}$; whilst corresponding to the composite natural transformation $$U {\ext@arrow 01{20}0\rightarrowfill@{}{\theta}} \operatorname{Lan}_U(U) \circ U {\ext@arrow 01{20}0\rightarrowfill@{}{\operatorname{Lan}_U(U) \circ \theta}} \operatorname{Lan}_U(U) \circ
\operatorname{Lan}_U(U) \circ U$$ we have a natural transformation $\Delta \colon \operatorname{Lan}_U(U) \Rightarrow \operatorname{Lan}_U(U)
\circ \operatorname{Lan}_U(U)$. It is now easy to check that $\epsilon$ and $\Delta$ make $\operatorname{Lan}_U(U)$ into a comonad on ${{\mathcal C}}^\mathbf 2$, the so-called *density comonad* of $U$. This has the property that comonad morphisms $(\operatorname{Lan}_U(U),
\epsilon, \Delta) \to \mathsf T$ are in bijection with left coactions of $\mathsf T$ on $U$, which in turn are in bijection with liftings of $U \colon
{{\mathcal A}}\to {{\mathcal C}}^\mathbf 2$ through the category of $\mathsf T$-coalgebras: and this is precisely the universal property for $\operatorname{Lan}_U(U)$ to be a reflection of $U$ along ${{\mathcal G}}_3$.
Next, we consider reflections along ${{\mathcal G}}_2 \colon {\mathbf{LNWFS}}({{\mathcal C}}) \to
{\mathbf{Cmd}}({{\mathcal C}}^\mathbf 2)$. These exist under very mild hypotheses indeed:
\[reflection\] If ${{\mathcal C}}$ has pushouts, then ${{\mathcal G}}_2 \colon {\mathbf{LNWFS}}({{\mathcal C}}) \to
{\mathbf{Cmd}}({{\mathcal C}}^\mathbf 2)$ has a left adjoint.
Let us say that an endofunctor $F \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 2$ *preserves domains* if ${\mathrm{dom}}\circ F = {\mathrm{dom}}$. Given two such endofunctors $F$ and $F'$, we will say that a natural transformation $\alpha$ between them *preserves domains* if ${\mathrm{dom}}\circ \alpha = {\mathrm{id}}_{\mathrm{dom}}$. Finally, we will say that a comonad $(T, \epsilon, \Delta)$ on ${{\mathcal C}}^\mathbf 2$ *preserves domains* if $T$, $\epsilon$ and $\Delta$ all preserve domains.
It is now a simple but instructive exercise to show that ${\mathbf{LNWFS}}({{\mathcal C}})$ is isomorphic to the full subcategory of ${\mathbf{Cmd}}({{\mathcal C}}^\mathbf2)$ whose objects are the domain-preserving comonads. Thus the Proposition will follow if we can show this subcategory to be reflective.
To do this, we first observe that there is a strong factorisation system on ${{\mathcal C}}^\mathbf 2$ whose left class ${{\mathcal P}}$ consists of the pushout squares, and whose right class consists ${{\mathcal D}}$ of the squares whose domain component is an isomorphism. In fact, if we make a choice of pushouts in ${{\mathcal C}}$, then we obtain a functorial factorisation of every map into a pushout square followed by a square whose domain component is an *identity*.
We can lift the factorisation system $({{\mathcal P}}, {{\mathcal D}})$ to one of the same name on $[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$; and the accompanying functorial factorisation lifts too, allowing us to factor every map of $[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$ as a map whose components are pushouts, followed by one whose domain components are identities.
Suppose now that we are given a comonad $\mathsf S = (S, \epsilon, \Delta)$ on ${{\mathcal C}}^\mathbf 2$: we construct its reflection into domain-preserving comonads as follows. We start by factorising the counit of $\mathsf S$ as $$\epsilon = S {\ext@arrow 04{22}0{\Rightarrowfill@}{}{\phi}} \hat S {\ext@arrow 04{22}0{\Rightarrowfill@}{}{\hat \epsilon}} {\mathrm{id}}_{{{\mathcal C}}^\mathbf
2}\text,$$ where the components of $\phi$ are pushout squares, and the domain components of $\hat \epsilon$ are identities. From this latter fact, we deduce that both $\hat S$ and $\hat \epsilon$ preserve domains. We now consider the following diagram: $${\vcenter{\hbox{\xymatrix{
S \ar@2[r]^{\Delta} \ar@2[d]_{\phi} &
SS \ar@2[r]^{\phi \phi} &
\hat S \hat S \ar@2[d]^{\hat \epsilon \hat S} \\
\hat S \ar@2[rr]_{{\mathrm{id}}_{\hat{S}}} & & \hat S\text.
}}}}$$ Since $\phi$ is in ${{\mathcal P}}$, and $\hat \epsilon \hat S$ in ${{\mathcal D}}$, we obtain by orthogonality a unique diagonal fill-in $\hat \Delta \colon \hat S \Rightarrow
\hat S \hat S$. Since both $\hat \epsilon \hat S$ and ${\mathrm{id}}_{\hat S}$ are domain-preserving, we deduce that $\hat \Delta$ is too.
A little calculus with the unique diagonal fill-in property and the comonad axioms for $(S, \epsilon, \Delta)$ now yields the comonad axioms for $\hat{\mathsf S} = (\hat S, \hat \epsilon, \hat \Delta)$; and it is immediate that $\phi \colon S \Rightarrow \hat S$ then satisfies the necessary axioms for it to lift to a comonad morphism $\phi \colon \mathsf S \to \hat{\mathsf S}$.
We claim that this $\phi$ provides the desired reflection of $\mathsf S$ into domain-preserving comonads. Indeed, suppose we are given another domain-preserving comonad $\mathsf T = (T, e, D)$, and a morphism of comonads $\psi \colon \mathsf S \to \mathsf T$. Then we have the following commutative square: $${\vcenter{\hbox{\xymatrix{
S \ar@2[r]^{\psi} \ar@2[d]_{\phi} &
T \ar@2[d]^{e} \\ \hat S \ar@2[r]_{\hat \epsilon} & {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}\text.
}}}}$$ The map $\phi$ is in ${{\mathcal P}}$, and $e$ is in ${{\mathcal D}}$: so by orthogonality, we induce a unique natural transformation $\hat \psi \colon \hat S \Rightarrow T$. The comonad morphism axioms for $\hat \psi$ now follow from the axioms for $\psi$ and uniqueness of diagonal fill-ins.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}We have thus reduced the problem of constructing free n.w.f.s.’s to the problem of constructing reflections along ${{\mathcal G}}_1 \colon {\mathbf{NWFS}}({{\mathcal C}}) \to
{\mathbf{LNWFS}}({{\mathcal C}})$. The key to constructing these will be to exhibit a monoidal structure on ${\mathbf{LNWFS}}({{\mathcal C}})$ whose corresponding category of monoids is isomorphic to ${\mathbf{NWFS}}({{\mathcal C}})$.
We will deduce the existence of this monoidal structure from a more general result characterising natural w.f.s.’s on ${{\mathcal C}}$ as *bialgebra* objects in the category of functorial factorisations on ${{\mathcal C}}$. Now, usually when one considers bialgebra objects in a category, it is with reference to a symmetric or braided monoidal structure on that category: but here we will need something slightly more general.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}By a *two-fold monoidal category* [@iter1], we mean a category ${{\mathcal V}}$ equipped with two monoidal structures $(\otimes, I, \alpha, \lambda,
\rho)$ and $(\odot, \bot, \alpha', \lambda', \rho')$ in such a way that the functors $\odot \colon {{\mathcal V}}\times {{\mathcal V}}\to {{\mathcal V}}$ and $\bot \colon 1 \to {{\mathcal V}}$, together with the natural transformations $\alpha'$, $\lambda'$ and $\rho'$, are lax monoidal with respect to the $(\otimes, I)$ monoidal structure.
Of course, being lax monoidal is not merely a property of a functor, but extra structure on it: and in this case, the extra structure amounts to giving maps $$m \colon \bot \otimes \bot \to \bot\text, \quad c \colon I \to I \odot I \quad \text{and} \quad j \colon I \to \bot$$ making $(\bot, j, m)$ into a $\otimes$-monoid and $(I, j, c)$ into a $\odot$-comonoid; together with a natural family of maps $$z_{A,B,C,D} \colon (A \odot B) \otimes (C \odot D) \to (A \otimes C) \odot (B \otimes D)$$ obeying six coherence laws. It follows that $\otimes$ and $I$ are oplax monoidal with respect to the $(\odot, \bot)$ monoidal structure; and in fact, we may take this as an alternative definition of two-fold monoidal category.
Any braided or symmetric monoidal category is two-fold monoidal, with the two monoidal structures coinciding; the maps $z_{A, B, C, D}$ are built from braidings/symmetries and associativity isomorphisms: c.f. [@Braided].
If ${{\mathcal V}}$ is a cocomplete symmetric monoidal category, then the functor category $[X \times X, {{\mathcal V}}]$ has a two-fold monoidal structure. The first monoidal structure $(\otimes, I)$ is given by matrix multiplication, whilst the second structure $(\odot, \bot)$ is given pointwise.
Similarly, if ${{\mathcal V}}$ is a cocomplete symmetric monoidal category, then the functor category $[\mathbb N, {{\mathcal V}}]$ has a two-fold monoidal structure on it. The first monoidal structure $(\otimes, I)$ is the *substitution tensor product*, with unit given by $I(n) = 0$ for $n \neq 1$ and $I(1) = I$; and binary tensor given by $$(F \otimes G)(n) = \sum_{\substack{m, k_1, \dots, k_m \\ k_1 + \dots + k_m = n}} F(m) \otimes G(k_1) \otimes \dots \otimes G(k_m)\text.$$ The second monoidal structure $(\odot, \bot)$ is again given pointwise.
Further examples and applications to topology may be found in [@iter1; @iter2].
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}A two-fold monoidal category $({{\mathcal V}}, \otimes, I, \odot, \bot)$ provides a suitable environment to define a notion of bialgebra. Indeed, because the $\odot$-monoidal structure is lax monoidal with respect to the $\otimes$-structure, it lifts to the category ${\mathbf{Mon}}_\otimes({{\mathcal V}})$ of $\otimes$-monoids in ${{\mathcal V}}$. Thus we define the category of *$(\otimes,
\odot)$-bialgebras* to be $${\mathbf{Bialg}}_{\otimes, \odot}({{\mathcal V}}) := {\mathbf{Comon}}_\odot({\mathbf{Mon}}_\otimes({{\mathcal V}}))\text.$$
Now, because the $\otimes$-monoidal structure is also oplax monoidal with respect to the $\odot$-monoidal structure, it lifts to the category of $\odot$-comonoids in ${{\mathcal V}}$; and thus we obtain an alternative definition of bialgebra by setting $${\mathbf{Bialg}}'_{\otimes, \odot}({{\mathcal V}}) := {\mathbf{Mon}}_\otimes({\mathbf{Comon}}_\odot({{\mathcal V}}))\text.$$
However, it is not hard to see that these two constructions yield isomorphic results. Indeed, to give an object of either ${\mathbf{Bialg}}({{\mathcal V}})$ or ${\mathbf{Bialg}}'({{\mathcal V}})$ is to give an object $A$ of ${{\mathcal V}}$; maps $\eta \colon I \to A$ and $\mu \colon A \otimes A \to A$ making it into a $\otimes$-monoid; and maps $\epsilon \colon A \to \bot$ and $\delta \colon A \to A \odot A$ making it into a $\odot$-comonoid; all subject to the commutativity of the following four diagrams: $$\label{bialg}
\begin{gathered}
{\vcenter{\hbox{\xymatrix{
I \ar[r]^-\eta \ar[d]_{c} &
A \ar[d]^\Delta \\
I \odot I \ar[r]_-{\eta \odot \eta} &
A \odot A\text,
}}}} \qquad {\vcenter{\hbox{\xymatrix{
A \otimes A \ar[r]^-\mu \ar[d]_{\epsilon \otimes \epsilon} &
A \ar[d]^\epsilon \\
\bot \otimes \bot \ar[r]_-m & \bot\text,
}}}} \qquad {\vcenter{\hbox{\xymatrix{
& A \ar[dr]^\epsilon \\
I \ar[ur]^\eta \ar[rr]_j & & \bot\text,
}}}}
\\
{\vcenter{\hbox{\xymatrix@C+1.5em{
A \otimes A
\ar[rr]^\mu \ar[d]_{\Delta \otimes \Delta} & &
A \ar[d]^\Delta \\
(A \odot A) \otimes (A \odot A)
\ar[r]_-{z_{A, A, A, A}} & (A \otimes A) \odot (A \otimes A)
\ar[r]_-{\mu \odot \mu} &
A \odot A\text.
}}}}
\end{gathered}$$ Likewise, to give a morphism of either ${\mathbf{Bialg}}({{\mathcal V}})$ or ${\mathbf{Bialg}}'({{\mathcal V}})$ is to give a map $f \colon A \to B$ of ${{\mathcal V}}$ which is both a monoid morphism and a comonoid morphism. We may summarise this by saying that, in the following diamond of forgetful functors $$\label{diamond}
{\vcenter{\hbox{\xymatrix{
& {\mathbf{Bialg}}_{\otimes, \odot}({{\mathcal V}}) \ar[dl] \ar[dr] \\
{\mathbf{Comon}}_\odot({{\mathcal V}}) \ar[dr] & &
{\mathbf{Mon}}_\otimes({{\mathcal V}})\text, \ar[dl] \\ &
{{\mathcal V}}}}}}$$ each west-pointing arrow forgets monoid structure, and each east-pointing arrow forgets comonoid structure.
If view a braided or symmetric monoidal category as a two-fold monoidal category, then a bialgebra in our sense is precisely a bialgebra in the usual sense.
In the two-fold monoidal category $[X \times X, {{\mathcal V}}]$, a $\otimes$-monoid is a ${{\mathcal V}}$-category with object set $X$; an $\odot$-comonoid is an $X \times
X$-indexed family of comonoids in ${{\mathcal V}}$; and a $(\otimes, \odot)$-bialgebra is a *comonoidal* ${{\mathcal V}}$-category with object set $X$: which we may view either as a comonoid in ${{\mathcal V}}\text-{\mathbf{Cat}}$, or as a ${{\mathcal V}}$-category whose homsets are comonoids and whose unit and composition maps are comonoid morphisms.
A bialgebra in the two-fold monoidal category $[\mathbb N, {{\mathcal V}}]$ is what is sometimes called a *Hopf operad*: namely, an operad in ${{\mathcal V}}$ whose objects of $n$-ary operations are comonoids; and whose substitution maps are morphisms of comonoids.
Bialgebras in two-fold monoidal categories play a central role in recent work [@lamarche] of François Lamarche.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Let us write ${\mathbf{FF}}({{\mathcal C}})$ for the category of functorial factorisations on ${{\mathcal C}}$, and let us write ${\mathbf{RNWFS}}({{\mathcal C}})$ for the category dual to ${\mathbf{LNWFS}}({{\mathcal C}})$: so its objects are pairs $(F, {\mathsf R})$ of a functorial factorisation $F$ on ${{\mathcal C}}$ together with an extension of the corresponding $(R,
\Lambda)$ to a monad.
\[diamondtwofold\] There is a two-fold monoidal structure on ${\mathbf{FF}}({{\mathcal C}})$ such that the diamond of forgetful functors is, up-to-isomorphism, the diamond of forgetful functors $${\vcenter{\hbox{\xymatrix{
& {\mathbf{NWFS}}({{\mathcal C}})\ar[dl] \ar[dr] \\
{\mathbf{LNWFS}}({{\mathcal C}}) \ar[dr] & &
{\mathbf{RNWFS}}({{\mathcal C}}) \ar[dl] \\ &
{\mathbf{FF}}({{\mathcal C}})\text{\rlap.}
}}}}$$
We begin by exhibiting two strict monoidal structures on ${\mathbf{FF}}({{\mathcal C}})$. We do this by describing two different categories which are both isomorphic to ${\mathbf{FF}}({{\mathcal C}})$, and which both admit obvious strict monoidal structures: then by transport of structure, we induce the required monoidal structures on ${\mathbf{FF}}({{\mathcal C}})$.
The first category we consider is the category of domain-preserving copointed endofunctors and copointed endofunctor maps on ${{\mathcal C}}^\mathbf 2$. It is easy to see that this category is isomorphic to ${\mathbf{FF}}({{\mathcal C}})$; and that it has a strict monoidal structure $(\odot, \bot)$ on it, with unit $$\bot = ({\mathrm{id}}_{{\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}} \colon {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2} \Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2})$$ and tensor product $$(\Phi \colon L \Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}) \odot (\Phi' \colon L' \Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}) = (\Phi \Phi' \colon LL' \Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2})\text.$$
When we transport this along the isomorphism with ${\mathbf{FF}}({{\mathcal C}})$, we obtain the following monoidal structure. The unit $\bot$ is the functorial factorisation $$X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y \quad \mapsto \quad X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y {\ext@arrow 01{20}0\rightarrowfill@{}{{\mathrm{id}}_Y}} Y$$ and the tensor product $F' \odot F$ of two functorial factorisations $F, F'
\colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 3$ is given by $$X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y \quad \mapsto \quad X {\ext@arrow 01{20}0\rightarrowfill@{}{\lambda'_{\lambda_f}}} K' \lambda_f {\ext@arrow 01{20}0\rightarrowfill@{}{\rho_f \circ \rho'_{\lambda_f}}} Y\text.$$
Furthermore, to give a $\odot$-comonoid structure on some $F \in {\mathbf{FF}}({{\mathcal C}})$ is to give a comonoid structure on the corresponding copointed $(L, \Phi)$; but this is precisely to extend it to a comonad on ${{\mathcal C}}^\mathbf 2$. Thus we may identify ${\mathbf{Comon}}_\odot({\mathbf{FF}}({{\mathcal C}}))$ with ${\mathbf{LNWFS}}({{\mathcal C}})$.
The second category we consider is the category of codomain-preserving pointed endofunctors on ${{\mathcal C}}^\mathbf 2$. Again, this is isomorphic to ${\mathbf{FF}}({{\mathcal C}})$, and again, it has a strict monoidal structure given by composition. When we transport this back to ${\mathbf{FF}}({{\mathcal C}})$, we obtain the monoidal structure whose unit $I$ is the functorial factorisation $$X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y \quad \mapsto \quad X {\ext@arrow 01{20}0\rightarrowfill@{}{{\mathrm{id}}_X}} X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y\text;$$ and whose tensor product $F' \otimes F$ is the functorial factorisation $$X {\ext@arrow 01{20}0\rightarrowfill@{}{f}} Y \quad \mapsto \quad X {\ext@arrow 01{20}0\rightarrowfill@{}{\lambda'_{\rho_f} \circ \lambda_f}} K' \rho_f {\ext@arrow 01{20}0\rightarrowfill@{}{\rho'_{\rho_f}}} Y\text.$$
To make $F \in {\mathbf{FF}}({{\mathcal C}})$ into a monoid with respect to this monoidal structure is now to give an extension of the corresponding $(R, \Lambda)$ to a monad; and so we have ${\mathbf{Mon}}_\otimes({\mathbf{FF}}({{\mathcal C}})) \cong {\mathbf{RNWFS}}({{\mathcal C}})$ as required.
We next show that these two monoidal structures on ${\mathbf{FF}}({{\mathcal C}})$ can be made into a two-fold monoidal structure. Since $I$ is initial and $\bot$ terminal in ${\mathbf{FF}}({{\mathcal C}})$, for this we need only give the family of interchange maps $z_{A, B, C, D} \colon (A \odot B) \otimes (C \odot D) \to (A \otimes C) \odot
(B \otimes D)$: and this we do explicitly. The factorisation $(A \odot B)
\otimes (C \odot D)$ sends a map $f \colon X \to Y$ to $$X {\ext@arrow 01{20}0\rightarrowfill@{}{\textstyle\lambda^A(\lambda^B(\rho^{C \odot D}_f)) \circ
\lambda^C(\lambda^D_f)}} {K}^A(\lambda^B(\rho^{C \odot D}_f))
{\ext@arrow 01{20}0\rightarrowfill@{}{\textstyle\rho^B(\rho^{C \odot D}_f) \circ \rho^A(\lambda^B(\rho^{C \odot D}_f))}} Y\text,$$ where $\rho^{C \odot D}_f$ abbreviates the map $\rho^D_f \circ
\rho^C(\lambda^D_f)$; whilst $(A \otimes C) \odot (B \otimes D)$ sends $f$ to $$X {\ext@arrow 01{20}0\rightarrowfill@{}{\textstyle\lambda^A(\rho^C(\lambda^{B \otimes D}_f)) \circ
\lambda^C(\lambda^{B \otimes D}_f)}} K^A(\rho^C(\lambda^{B \otimes D}_f))
{\ext@arrow 01{20}0\rightarrowfill@{}{\textstyle\rho^B(\rho^D_f) \circ \rho^A(\rho^C(\lambda^{B \otimes D}_f))}} Y\text,$$ where $\lambda^{B \otimes D}_f$ abbreviates the map $\lambda^B(\rho^D_f) \circ
\lambda^D_f$. To give $z_{A, B, C, D}$ we must therefore give suitable maps ${K}^A(\lambda^B(\rho^{C \odot D}_f)) \to K^A(\rho^C(\lambda^{B \otimes
D}_f))$. For this, we consider the following square: $${\vcenter{\hbox{\xymatrix@+1.5em@C+2em{
K^C(\lambda^D_f)
\ar[d]_{\lambda^B(\rho^{C \odot D}_f)}
\ar[r]^-{K^C({\mathrm{id}}_X, \lambda^B(\rho^D_f))} &
K^C(\lambda^{B \otimes D}_f)
\ar[d]^{\rho^C(\lambda^{B \otimes D}_f)} \\
K^B(\rho^{C \odot D}_f)
\ar[r]_-{K^B(\rho^C(\lambda^D_f), {\mathrm{id}}_Y)} &
K^B(\rho^D_f)\text.
}}}}$$ This square commutes, with both sides equal to $$K^C(\lambda^D_f) {\ext@arrow 01{20}0\rightarrowfill@{}{\rho^C(\lambda^D_f)}} K^D f {\ext@arrow 01{20}0\rightarrowfill@{}{\lambda^B(\rho^D_f)}} K^B(\rho^D_f)\text,$$ and so we may view it as a morphism $\lambda^B(\rho^{C \odot D}_f) \to
\rho^C(\lambda^{B \otimes D}_f)$ in ${{\mathcal C}}^\mathbf 2$: applying $K^A$ to which yields the required map ${K}^A(\lambda^B(\rho^{C \odot D}_f)) \to
K^A(\rho^C(\lambda^{B \otimes D}_f))$ in ${{\mathcal C}}$. The (extensive) remaining details are left to the reader.
Thus we have a two-fold monoidal structure $(\otimes, \odot)$ on ${\mathbf{FF}}({{\mathcal C}})$: and to complete the proof, we must show that the corresponding bialgebras are precisely n.w.f.s.’s on ${{\mathcal C}}$. But to equip a functorial factorisation with both a $\otimes$-monoid and an $\odot$-comonoid structure is to give extensions of the corresponding $(R, \Lambda)$ to a monad ${\mathsf R}$, and the corresponding $(L, \Phi)$ to a comonad ${\mathsf L}$; and it is now a short calculation to show that the bialgebra axioms will hold just when the distributivity axiom holds for $({\mathsf L}, {\mathsf R})$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}This Theorem implies that an object $X \in {\mathbf{LNWFS}}({{\mathcal C}})$ will admit a reflection along the functor ${{\mathcal G}}_1 \colon {\mathbf{NWFS}}({{\mathcal C}}) \to {\mathbf{LNWFS}}({{\mathcal C}})$ just when the free $\otimes$-monoid on $X$ exists. But since the unit $I$ of the monoidal structure on ${\mathbf{LWNFS}}({{\mathcal C}})$ is also an initial object, to construct the free monoid on $X$ is equally well to construct the free monoid on the pointed object $! \colon I \to X$. In order to do this, we may employ a standard transfinite construction: which we now describe.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}If ${\mathbf{On}}$ denotes the category of all small ordinals, then a *transfinite sequence* in a category ${{\mathcal V}}$ is a functor $X \colon {\mathbf{On}}
\to {{\mathcal V}}$, whose value at an ordinal $\alpha$ we denote by $X_\alpha$, and whose value at the unique morphism $\alpha \to \beta$ (for $\alpha \leqslant \beta$) we denote by $X_{\alpha, \beta} \colon X_\alpha \to X_\beta$. We say that a transfinite sequence $X \colon {\mathbf{On}} \to {{\mathcal V}}$ *converges* at an ordinal $\gamma$ if the maps $X_{\alpha, \beta}$ are isomorphisms for all $\beta
\geqslant \alpha \geqslant \gamma$.
Let ${{\mathcal V}}$ now be a cocomplete monoidal category. Given a pointed object $t
\colon I \to T$ in ${{\mathcal V}}$, we may form a transfinite sequence $X \colon {\mathbf{On}}
\to {{\mathcal V}}$ which we call the *free monoid sequence* for $(T, t)$. We build this sequence, together with a family of maps $\sigma_\alpha \colon T \otimes
X_\alpha \to X_{\alpha^+}$, by the following transfinite induction:
- $X_0 = I$, $X_1 = T$, $X_{0, 1} = t$, and $\sigma_0 = \rho_T \colon T
\otimes I \to T$;
- For a successor ordinal $\beta = \alpha^+$, we give $X_{\beta}$ and $\sigma_{\beta} \colon T \otimes X_\beta \to X_{\beta^+}$ by the following coequaliser diagram: $${\vcenter{\hbox{\[email protected]{ &
X_{\beta}
\ar@/^0.5em/[dr]^-{t \otimes X_{\beta}}
\\
T \otimes X_\alpha
\ar@/^0.5em/[ur]^-{\sigma_\alpha}
\ar@/_0.5em/[dr]_-{T \otimes t \otimes X_\alpha} & &
T \otimes X_{\beta}
\ar[r]^-{\sigma_{\beta}} &
X_{\beta^{+}}\text,
\\ &
T \otimes T \otimes X_{\alpha}
\ar@/_0.5em/[ur]_-{T \otimes \sigma_{\alpha}} }}}}$$ and give $X_{\beta, \beta^+}$ by the composite $\sigma_\beta \circ (t \otimes X_\beta)$;
- For a non-zero limit ordinal $\gamma$, we give $X_\gamma$ by $\operatorname{colim}_{\alpha < \gamma} X_\alpha$, with connecting maps $X_{\alpha,
\gamma}$ given by the injections into the colimit. We give $X_{\gamma^+}$ and $\sigma_\gamma$ by the following coequaliser diagram: $${\vcenter{\hbox{\xymatrix@C-2em{
& \operatorname{colim}X_{\alpha^+} = X_\gamma
\ar[dr]^{t \otimes X_\gamma} \\
\operatorname{colim}(T \otimes X_{\alpha})
\ar[ur]^{\operatorname{colim}\sigma_\alpha\ }
\ar[rr]_{\textsf{can}} & &
T \otimes \operatorname{colim}X_{\alpha} = T \otimes X_{\gamma}
\ar[rrr]_-{\sigma_{\gamma}} & & &
X_{\gamma^+}
}}}}$$ where “” is the map induced by the cocone $T \otimes X_\alpha
\to T \otimes \operatorname{colim}X_\alpha$. We give $X_{\gamma, \gamma^+}$ by the composite $\sigma_\gamma \circ (t \otimes X_\gamma)$.
The following is now Theorem 23.3 of [@Ke80].
Let ${{\mathcal V}}$ be a cocomplete monoidal category in which each functor $({{\mathord{\text{--}}}})
\otimes X \colon {{\mathcal V}}\to {{\mathcal V}}$ preserves connected colimits; and let $t \colon I
\to T$ be a pointed object of ${{\mathcal V}}$. If the free monoid sequence for $(T, t)$ converges at stage $\gamma$, then $X_\gamma$ is the free monoid on $(T, t)$, with the universal map given by $X_{1, \gamma} \colon T \to X_\gamma$.
In fact, this result is a mild generalisation of [@Ke80], since we require $({{\mathord{\text{--}}}}) \otimes X$ to preserve only connected colimits, rather than all colimits; but it is trivial to check that this does not affect the argument in any way.
In order to apply this result, we observe that:
If ${{\mathcal C}}$ is a cocomplete category, then ${\mathbf{LNWFS}}({{\mathcal C}})$ is also cocomplete; and moreover, each functor $({{\mathord{\text{--}}}}) \otimes X \colon {\mathbf{LNWFS}}({{\mathcal C}}) \to
{\mathbf{LNWFS}}({{\mathcal C}})$ preserves connected colimits.
We first note that the category ${\mathbf{FF}}({{\mathcal C}})$ may be obtained by taking the category $[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}]$, slicing this over the object ${\mathrm{cod}}\colon
{{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}$; and then coslicing this under the object $\upsilon \colon
{\mathrm{dom}}\Rightarrow {\mathrm{cod}}$ given by $\upsilon_f = f$ for all $f \in {{\mathcal C}}^\mathbf 2$. Consequently, ${\mathbf{FF}}({{\mathcal C}})$ will be cocomplete whenever ${{\mathcal C}}$ is. But by Theorem \[diamondtwofold\], the functor $U \colon {\mathbf{LNWFS}}({{\mathcal C}}) \to
{\mathbf{FF}}({{\mathcal C}})$ is a forgetful functor from a category of comonoids, and as such creates colimits, so that ${\mathbf{LNWFS}}({{\mathcal C}})$ is also cocomplete.
In order to see that each functor $({{\mathord{\text{--}}}}) \otimes X \colon {\mathbf{LNWFS}}({{\mathcal C}}) \to
{\mathbf{LNWFS}}({{\mathcal C}})$ preserves connected colimits, we consider the composite $$V := {\mathbf{LNWFS}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{U}} {\mathbf{FF}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{d_0 \circ ({{\mathord{\text{--}}}})}} [{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]\text,$$ where we recall that postcomposing with $d_0$ sends a functorial factorisation $F \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 3$ to the corresponding endofunctor $R
\colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 2$. It is easy to see that $d_0 \circ
({{\mathord{\text{--}}}})$ creates connected colimits; and since $U$ creates all colimits, we conclude that $V$ creates connected colimits.
Now observe that $V$ sends the monoidal structure on ${\mathbf{LNWFS}}({{\mathcal C}})$ to the compositional monoidal structure on $[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$, so that we have the following commutative diagram: $${\vcenter{\hbox{\xymatrix{
{\mathbf{LNWFS}}({{\mathcal C}}) \ar[r]^{({{\mathord{\text{--}}}}) \otimes X} \ar[d]_{V} & {\mathbf{LNWFS}}({{\mathcal C}}) \ar[d]^{V} \\
[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2] \ar[r]_{({{\mathord{\text{--}}}}) \circ VX} & [{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]\text.
}}}}$$ We wish to show that $({{\mathord{\text{--}}}}) \otimes X$ preserves connected colimits: but because $V$ creates them, it suffices to show that the composite around the top preserves connected colimits; and this follows from the fact that both functors $V$ and $({{\mathord{\text{--}}}}) \circ VX$ around the bottom preserve connected colimits.
Thus the free monoid on $X \in {\mathbf{LNWFS}}({{\mathcal C}})$ will exist whenever the free monoid sequence for $! \colon I \to X$ converges. Sufficient conditions for convergence are given by Theorem 15.6 of [@Ke80], which when adapted to the present situation becomes:
\[convergnece\] Let ${{\mathcal V}}$ be a cocomplete monoidal category, and let $t \colon I \to T$ be a pointed object of ${{\mathcal V}}$. If the functor $T \otimes ({{\mathord{\text{--}}}}) \colon {{\mathcal V}}\to {{\mathcal V}}$ preserves either $\lambda$-filtered colimits; or $\lambda$-indexed unions of ${{\mathcal M}}$-subobjects for some proper, well-copowered $({{\mathcal E}}, {{\mathcal M}})$ on ${{\mathcal V}}$, then the free monoid sequence for $(T, t)$ converges.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}There is a problem if we apply this result with ${{\mathcal V}}= {\mathbf{LNWFS}}({{\mathcal C}})$, since the second of the two smallness criteria requires a proper, well-copowered $({{\mathcal E}}, {{\mathcal M}})$ on ${{\mathcal V}}$; and even if we have such an $({{\mathcal E}}, {{\mathcal M}})$ on the category ${{\mathcal C}}$, we will not, in general, be able to lift it to ${\mathbf{LNWFS}}({{\mathcal C}})$. In order to resolve this problem, we consider again the composite $$V := {\mathbf{LNWFS}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{U}} {\mathbf{FF}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{d_0 \circ ({{\mathord{\text{--}}}})}} [{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]\text.$$
We saw above that this preserves both connected colimits and monoidal structure; and so takes the free monoid sequence on $! \colon I \to X$ in ${\mathbf{LNWFS}}({{\mathcal C}})$ to the *free monad sequence* on the underlying pointed endofunctor $\Lambda \colon {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2} \Rightarrow R$ of $X$. Moreover, $V$ reflects isomorphisms: hence the convergence of the latter sequence guarantees the convergence of the former.
Thus, it will suffice to apply Proposition \[convergnece\] for ${{\mathcal V}}=
[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$, which avoids the problem described above, since any proper, well-copowered $({{\mathcal E}}, {{\mathcal M}})$ on ${{\mathcal C}}$ can be lifted without trouble to $[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$. In fact, it will suffice to lift to ${{\mathcal C}}^\mathbf
2$, since when we instantiate Proposition \[convergnece\] at ${{\mathcal V}}=
[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$, the requirement that $T \otimes ({{\mathord{\text{--}}}}) \colon
[{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2] \to [{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$ should preserve $\lambda$-filtered colimits or unions may be safely reduced to the requirement that $T \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 2$ should preserve the same.
We may summarise this argument as follows:
\[reflection2\] Let there be given a cocomplete category ${{\mathcal C}}$; and let $(F, {\mathsf L}) \in
{\mathbf{LNWFS}}({{\mathcal C}})$. If the functor $R = d_0 \circ F \colon {{\mathcal C}}^\mathbf 2 \to
{{\mathcal C}}^\mathbf 2$ preserves either $\lambda$-filtered colimits; or $\lambda$-indexed unions of ${{\mathcal M}}$-subobjects for some proper, well-copowered $({{\mathcal E}}, {{\mathcal M}})$ on ${{\mathcal C}}^\mathbf 2$, then the free monoid sequence for $! \colon I \to
(F, {\mathsf L})$ converges: and in particular, the reflection of $(F, {\mathsf L})$ along ${{\mathcal G}}_1 \colon {\mathbf{NWFS}}({{\mathcal C}}) \to {\mathbf{LNWFS}}({{\mathcal C}})$ exists.
We are now ready to prove the first part of our main theorem:
Let ${{\mathcal C}}$ be a cocomplete category satisfying one of the smallness conditions (\*) or (), and let $I \colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ be a category over ${{\mathcal C}}^\mathbf 2$ with ${{\mathcal J}}$ small. Then the free n.w.f.s. on ${{\mathcal J}}$ exists.
By Proposition \[Kanext\] and Proposition \[reflection\], we may find an object $(F, {\mathsf L}) \in {\mathbf{LNWFS}}({{\mathcal C}})$ which is a reflection of ${{\mathcal J}}$ along ${{\mathcal G}}_3
{{\mathcal G}}_2 \colon {\mathbf{LNWFS}}({{\mathcal C}}) \to {\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$. We now wish to apply Proposition \[reflection2\] to $(F, {\mathsf L})$: so for a ${{\mathcal C}}$ satisfying (\*), we will show that $R = d_0 \circ F \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 2$ preserves $\lambda$-filtered colimits for some $\lambda$; whilst for a ${{\mathcal C}}$ satisfying (), we will show that $R$ preserves $\lambda$-indexed unions of ${{\mathcal M}}$-subobjects for the induced factorisation system $({{\mathcal E}}, {{\mathcal M}})$ on ${{\mathcal C}}^\mathbf
2$. Since the proof is the same in both cases, we restrict our attention to the former.
We begin by considering the following diagram: $${\vcenter{\hbox{\xymatrix@R-2em{ & & {{\mathcal C}}^\mathbf 2 \\
{{\mathcal C}}^\mathbf 2 \ar[r]^{F} & {{\mathcal C}}^\mathbf 3 \ar[ur]^{d_0} \ar[dr]_{d_2} \\ & & {{\mathcal C}}^\mathbf 2
}}}}$$ The upper composite is $R \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 2$, which we are to show preserves $\lambda$-filtered colimits; but since $d_0$ and $d_2$ preserve and reflect connected colimits, we may equally well show that the lower composite $L = d_2 \circ F$ preserves $\lambda$-filtered colimits.
Now, from Proposition \[Kanext\] and Proposition \[reflection\], the functor $L$ has the following explicit description. First we form the left Kan extension of $I \colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ along itself to obtain a functor $M
\colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 2$. We may describe this by the usual coend formula $$M(f) = \int^{j \in {{\mathcal J}}} {{\mathcal C}}^\mathbf 2(I(j), f) \cdot I(j)\text.$$
We now consider the counit transformation $\epsilon \colon M \Rightarrow
{\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}$, whose component at $f$ is the map $$\epsilon_f \colon \int^{j \in {{\mathcal J}}} {{\mathcal C}}^\mathbf 2(I(j), f) \cdot I(j) \to f$$ corresponding to the identity transformation ${{\mathcal C}}^\mathbf 2(I({{\mathord{\text{--}}}}), f)
\Rightarrow {{\mathcal C}}^\mathbf 2(I({{\mathord{\text{--}}}}), f)$; and we factor this transformation $\epsilon$ as $$M {\ext@arrow 04{22}0{\Rightarrowfill@}{}{\xi}} L {\ext@arrow 04{22}0{\Rightarrowfill@}{}{\Phi}} {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2}\text,$$ where each component of $\xi$ is a pushout; and each component of $\Phi$ is the identity in its domain.
Let us first show that $L$ preserves any colimit which $M$ does. Suppose that $A \colon {{\mathcal I}}\to {{\mathcal C}}^\mathbf 2$ is a small diagram whose colimit is preserved by $K$, and consider the following diagram: $$\label{rendering}
{\vcenter{\hbox{\xymatrix@C+2em{
\operatorname{colim}_i MA_i \ar[r]^{\operatorname{colim}_i \xi_{A_i}} \ar[d]_{\textsf{can}_M} &
\operatorname{colim}_i LA_i \ar[r]^{\operatorname{colim}_i \Phi_{A_i}} &
\operatorname{colim}_i A_i \ar[d]^{=} \\
M \operatorname{colim}_i A_i \ar[r]_{\xi_{\operatorname{colim}_i A_i}}&
L \operatorname{colim}_i A_i \ar[r]_{\Phi_{\operatorname{colim}_i A_i}} &
\operatorname{colim}_i A_i\text.
}}}}$$ The class ${{\mathcal P}}$ of morphisms in ${{\mathcal C}}^\mathbf 2$ which are pushout squares is the left class of a strong factorisation system, and hence stable under colimit: and thus not only $\xi_{\operatorname{colim}_i A_i}$, but also $\operatorname{colim}_i \xi_{A_i}$, is in ${{\mathcal P}}$. Likewise, the class ${{\mathcal D}}$ of morphisms in ${{\mathcal C}}^\mathbf 2$ which are domain-isomorphisms is also the left class of a strong factorisation system on ${{\mathcal C}}^\mathbf 2$, whose corresponding right class is the class of codomain-isomorphisms. Hence ${{\mathcal D}}$ is also stable under colimit; and so both $\Phi_{\operatorname{colim}_i A_i}$ and $\operatorname{colim}_i \Phi_{A_i}$ are in ${{\mathcal D}}$.
The orthogonality property for $({{\mathcal P}}, {{\mathcal D}})$ now implies that there is a unique map $\phi \colon \operatorname{colim}_i LA_i \to L \operatorname{colim}A_i$ rendering commutative; and moreover, that $\phi$ is invertible, since $\textsf{can}_M$ is. But the canonical morphism $\textsf{can}_L \colon \operatorname{colim}_i LA_i \to L
\operatorname{colim}A_i$ makes commute; and so we deduce that $\textsf{can}_L = \phi$ is invertible as required.
Thus $L$ preserves any colimit which $M$ does: so we will be done if we can find some $\lambda$ for which $M$ preserves $\lambda$-filtered colimits. Now, for each $j \in {{\mathcal J}}$, we have the morphism $I(j) \colon X \to Y$ of ${{\mathcal C}}^\mathbf
2$: and by condition (\*), we can find a $\lambda_j$ for which both $X$ and $Y$ are $\lambda_j$-presentable; from which it follows that $I(j)$ is $\lambda_j$-presentable in ${{\mathcal C}}^\mathbf 2$. Thus, if we take $\lambda$ to be a regular cardinal larger than each $\lambda_j$, then each $I(j)$ is $\lambda$-presentable in ${{\mathcal C}}^\mathbf 2$.
We now show that $K$ preserves $\lambda$-filtered colimits. Indeed, suppose that $A \colon {{\mathcal I}}\to {{\mathcal C}}^\mathbf 2$ is a $\lambda$-filtered diagram in ${{\mathcal C}}^\mathbf 2$; then we have that $$\begin{aligned}
M(\operatorname{colim}_i A_i) & = \textstyle\int^{j \in {{\mathcal J}}} {{\mathcal C}}^\mathbf 2(I(j), \operatorname{colim}_i A_i) \cdot I(j) \\
& \cong \textstyle\int^{j \in {{\mathcal J}}} (\operatorname{colim}_i {{\mathcal C}}^\mathbf 2(I(j), A_i)) \cdot I(j) \text{\ \ \ (as $I(j)$ is $\lambda$-presentable)}\\
& \cong \operatorname{colim}_i \textstyle\int^{j \in {{\mathcal J}}} {{\mathcal C}}^\mathbf 2(I(j), A_i) \cdot I(j) \text{\ \ \ (as colimits commute with colimits)}\\
& = \operatorname{colim}_i M(A_i)\text,\end{aligned}$$ as desired.
Constructively-free implies algebraically-free {#Sec:cfaf}
==============================================
In this Section, we prove that all free n.w.f.s.’s obtained by the procedure of the previous Section are algebraically-free. In order to do this, we will need to establish a link between our notion of algebraically-free n.w.f.s., and [@Ke80]’s notion of algebraically-free monad. We begin, therefore, by recalling the latter.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Let $\sigma \colon {\mathrm{id}}\Rightarrow S$ be a pointed endofunctor on some category ${{\mathcal V}}$. An *$S$-algebra* is an object $X \in {{\mathcal V}}$ together with a morphism $x \colon SX \to X$ satisfying $x . \sigma = {\mathrm{id}}_X$; and an *$S$-algebra morphism* $(X, x) \to (Y, y)$ is a morphism $f \colon X \to
Y$ of ${{\mathcal V}}$ for which $f . x = y . Sf$. We write $S$-${\mathbf{Alg}}$ for the category of $S$-algebras and $S$-algebra morphisms. A *morphism of pointed endofunctors* $(S, \sigma) \Rightarrow (T, \tau)$ is a natural transformation $\alpha \colon S \Rightarrow T$ satisfying $\tau = \alpha . \sigma$; and any such morphism induces a functor $\alpha^\ast \colon T\text-{\mathbf{Alg}} \to
S\text-{\mathbf{Alg}}$ sending $(X, x)$ to $(X, x . \alpha_X)$.
If we are given a monad $\mathsf T = (T, \eta, \mu)$ on ${{\mathcal V}}$, we can consider its category $\mathsf T$-${\mathbf{Alg}}$ of algebras *qua* monad; or we can consider its category $T$-${\mathbf{Alg}}$ of algebras *qua* pointed endofunctor. Evidently, every $\mathsf T$-algebra is a $T$-algebra, and so we have an inclusion functor $\textsf{inc} \colon \mathsf T\text-{\mathbf{Alg}} \to
T\text-{\mathbf{Alg}}$.
Now let $(S, \sigma)$ be a pointed endofunctor on ${{\mathcal V}}$. We say that a monad $\mathsf T = (T, \eta, \mu)$ is *algebraically-free* on $(S, \sigma)$ if we can provide a morphism of pointed endofunctors $\alpha \colon (S, \sigma)
\Rightarrow (T, \eta)$ such that the composite $$\mathsf T\text-{\mathbf{Alg}} {\ext@arrow 01{20}0\rightarrowfill@{}{\textsf{inc}}} T\text-{\mathbf{Alg}} {\ext@arrow 01{20}0\rightarrowfill@{}{\alpha^\ast}} S\text-{\mathbf{Alg}}$$ is an isomorphism of categories.
The main result we will need about algebraically-free monads is the following, which is Theorem 22.3 of [@Ke80]:
\[kellyconstr\] Let ${{\mathcal V}}$ be a cocomplete category, and let $(S, \sigma)$ be a pointed endofunctor of ${{\mathcal V}}$. If the free monad sequence $X \colon {\mathbf{On}} \to [{{\mathcal V}},
{{\mathcal V}}]$ for $(S, \sigma)$ converges at stage $\gamma$, then the morphism $X_{1,
\gamma} \colon S \Rightarrow X_\gamma$ exhibiting $X_\gamma$ as the free monad on $S$ also exhibits it as the algebraically-free monad on $S$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}We are now ready to prove the second part of our main Theorem. We suppose given a cocomplete ${{\mathcal C}}$, so that any small $I \colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ over ${{\mathcal C}}^\mathbf 2$ has a reflection $(F', {\mathsf L}')$ along ${{\mathcal G}}_3 {{\mathcal G}}_2$; and we now say that the free n.w.f.s. on such a ${{\mathcal J}}$ *exists constructively* just when the free monoid sequence for $(F', {\mathsf L}')$ converges.
\[constralg\] Let ${{\mathcal C}}$ be a cocomplete category, and let $I \colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ be a small category over ${{\mathcal C}}^\mathbf 2$. If the free n.w.f.s. on ${{\mathcal J}}$ exists constructively, then it is algebraically-free on ${{\mathcal J}}$.
Let us write $({\mathsf L}, {\mathsf R})$ for the free n.w.f.s. on ${{\mathcal J}}$, and $(F', {\mathsf L}')$ for the reflection of ${{\mathcal J}}$ along ${{\mathcal G}}_3 {{\mathcal G}}_2$. By constructive existence, we obtain $({\mathsf L}, {\mathsf R})$ as the convergent value $X_\gamma$ of the free monoid sequence on $(F', {\mathsf L}')$; and so if $\eta \colon {{\mathcal J}}\to {\mathsf L\text-{\mathbf{Map}}}$ exhibits $({\mathsf L}, {\mathsf R})$ as the free n.w.f.s. on ${{\mathcal J}}$, then the corresponding morphism $\alpha \colon (F',
{\mathsf L}') \to (F, {\mathsf L})$ of ${\mathbf{LNWFS}}({{\mathcal C}})$ is the map $X_{1, \gamma}$ of this free monoid sequence. Now, applying the functor $$V:= {\mathbf{LNWFS}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{U}} {\mathbf{FF}}({{\mathcal C}}) {\ext@arrow 01{20}0\rightarrowfill@{}{d_0 \circ ({{\mathord{\text{--}}}})}} [{{\mathcal C}}^\mathbf 2, {{\mathcal C}}^\mathbf 2]$$ to this free monoid sequence yields the free monad sequence for the pointed endofunctor $\Lambda' \colon {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2} \Rightarrow R'$: and the convergence of the former guarantees the convergence of the latter. Thus by Proposition \[kellyconstr\], we deduce that the map of pointed endofunctors $\alpha_r \colon (R', \Lambda') \to (R, \Lambda)\text,$ obtained by applying $V$ to $\alpha$, exhibits ${\mathsf R}$ as the algebraically-free monad on $(R', \Lambda')$.
We now consider the following diagram: $$\label{twosquares}
{\vcenter{\hbox{\xymatrix{
\mathsf R\text-{\mathbf{Map}} \ar[r]^{\textsf{\upshape lift}} \ar[d]_{{\mathrm{id}}} & {\mathsf L}\text-{\mathbf{Map}}^{\pitchfork} \ar[r]^-{\eta^\pitchfork} \ar@{.>}[d]^G & {{\mathcal J}}^\pitchfork \ar@{.>}[d]^H \\
\mathsf R\text-{\mathbf{Alg}} \ar[r]_{\textsf{\upshape inc}} & R\text-{\mathbf{Alg}} \ar[r]_{(\alpha_r)^\ast} & R'\text-{\mathbf{Alg}}\text.
}}}}$$ By algebraic-freeness of ${\mathsf R}$, the composite along the bottom is an isomorphism; and we would like to deduce that the composite along the top is an isomorphism. To do this, it suffices to find isomorphisms $G$ and $H$ as indicated which make both squares commute.
We begin by constructing $G$. Recall that an object of ${\mathsf L}\text-{\mathbf{Map}}^{\pitchfork}$ is a pair $(g, \phi)$ consisting of a morphism $g \colon C \to D$ and a mapping $\phi$ which to each object $a \in
{\mathsf L\text-{\mathbf{Map}}}$ and square $${\vcenter{\hbox{\xymatrix{
A
\ar[r]^-h
\ar[d]_{U_{\mathsf L}(a)} &
C
\ar[d]^g \\
B
\ar[r]_-k &
D
}}}}$$ in ${{\mathcal C}}$, assigns a fill-in $\phi(a, h, k) \colon B \to C$ which is natural with respect to morphisms of ${\mathsf L\text-{\mathbf{Map}}}$. Now, to give such a $\phi$ is equally well to give a natural transformation $$\phi \colon {{\mathcal C}}^\mathbf 2(U_{\mathsf L}({{\mathord{\text{--}}}}), g) \Rightarrow {{\mathcal C}}^\mathbf 2(U_{\mathsf L}({{\mathord{\text{--}}}}), {\mathrm{id}}_C) \colon ({\mathsf L\text-{\mathbf{Map}}})^{\mathrm{op}}\to {\mathbf{Set}}$$ which is a section of the natural transformation ${{\mathcal C}}^\mathbf 2(U_{\mathsf L}({{\mathord{\text{--}}}}), {\mathrm{id}}_C) \Rightarrow {{\mathcal C}}^\mathbf 2(U_{\mathsf L}({{\mathord{\text{--}}}}), g)$ induced by postcomposition with $({\mathrm{id}}_C, g) \colon {\mathrm{id}}_C \to g$. But $U_{\mathsf L}\colon {\mathsf L\text-{\mathbf{Map}}}\to {{\mathcal C}}^\mathbf 2$ has a right adjoint given by the cofree functor $C_{\mathsf L}\colon {{\mathcal C}}^\mathbf 2 \to {\mathsf L\text-{\mathbf{Map}}}$; and thus we have an isomorphism $${{\mathcal C}}^\mathbf 2(U_{\mathsf L}({{\mathord{\text{--}}}}), g) \cong {\mathsf L\text-{\mathbf{Map}}}({{\mathord{\text{--}}}}, C_{\mathsf L}(g))\text.$$ So ${{\mathcal C}}^\mathbf 2(U_{\mathsf L}({{\mathord{\text{--}}}}), g)$ is represented by $C_{\mathsf L}(g)$; and thus by the Yoneda Lemma, $\phi$ is uniquely determined by where it sends the counit map $U_{\mathsf L}C_{\mathsf L}g \to g$; which is to say, by the fill-in it provides for the square $${\vcenter{\hbox{\xymatrix{
C
\ar[r]^-{{\mathrm{id}}}
\ar[d]_{\lambda_g} &
C
\ar[d]^g \\
Kf
\ar[r]_-{\rho_g} &
D
}}}}$$ in ${{\mathcal C}}$. But to provide a fill-in for this square is precisely to make $g$ into an algebra for the pointed endofunctor $(R, \Lambda)$. Thus we have an isomorphism between objects of ${\mathsf L\text-{\mathbf{Map}}}^\pitchfork$ and objects of $R\text-{\mathbf{Alg}}$; and it is now straightforward to extend this to the required isomorphism of categories $G \colon {\mathsf L\text-{\mathbf{Map}}}^\pitchfork \to
R\text-{\mathbf{Alg}}$, and to verify that this $G$ makes the left-hand square of commute.
We now complete the proof by constructing the isomorphism $H \colon
{{\mathcal J}}^\pitchfork \to R'\text-{\mathbf{Alg}}$. Proceeding as above, we see that to give an object of ${{\mathcal J}}^\pitchfork$ is to give a morphism $g \colon C \to D$ of ${{\mathcal C}}$ together with a natural transformation $\phi \colon {{\mathcal C}}^\mathbf 2(I({{\mathord{\text{--}}}}), g) \Rightarrow {{\mathcal C}}^\mathbf 2(I({{\mathord{\text{--}}}}), {\mathrm{id}}_C)$ which is a section of the natural transformation ${{\mathcal C}}^\mathbf 2(I({{\mathord{\text{--}}}}), {\mathrm{id}}_C)
\Rightarrow {{\mathcal C}}^\mathbf 2(I({{\mathord{\text{--}}}}), g)$ induced by postcomposing with $({\mathrm{id}}_C, g)
\colon {\mathrm{id}}_C \to g$. Now, if we write $$Mg := \int^{j \in {{\mathcal J}}} {{\mathcal C}}^\mathbf 2(I(j), g) \cdot I(j)$$ and $\epsilon_g$ for the counit map $Mg \to g$ as before, then to give $\phi$ is equivalently to give a morphism $k \colon Mg \to {\mathrm{id}}_C$ satisfying $\epsilon_g = ({\mathrm{id}}_C, g) \circ k$. Furthermore, we obtain $L'g$ from $Mg$ by factorising $\epsilon_g$ as $$\epsilon_g = Mg {\ext@arrow 01{20}0\rightarrowfill@{}{\xi_g}} L'g {\ext@arrow 01{20}0\rightarrowfill@{}{\Phi'_g}} g\text,$$ where $\xi_g$ is a pushout square, and $\Phi'_g$ is the identity in its domain; and so given such a map $k \colon Mg \to {\mathrm{id}}_C$, applying unique diagonalisation to the diagram $${\vcenter{\hbox{\xymatrix{
Mg \ar[r]^{k} \ar[d]_{\xi_g} & {\mathrm{id}}_C \ar[d]^{({\mathrm{id}}_C, g)} \\
L'g \ar[r]_{\Phi'_g} & g
}}}}$$ shows that $k$ is induced by a unique morphism $m \colon L'g \to {\mathrm{id}}_C$. But to give such a morphism is to give a diagonal fill-in for the square $${\vcenter{\hbox{\xymatrix{
C
\ar[r]^-{{\mathrm{id}}}
\ar[d]_{\lambda'_g} &
C
\ar[d]^g \\
K'f
\ar[r]_-{\rho'_g} &
D
}}}}$$ in ${{\mathcal C}}$; which in turn is to make $g$ into an algebra for the pointed endofunctor $(R', \Lambda')$. The remaining details are again straightforward.
Comparison with the small object argument
=========================================
Since we have advertised the argument of Theorem \[mainthm\] as an adaptation of the small object argument, it behooves us to investigate the relationship between the two. To do this, we combine our main Theorem with Proposition \[relateback\] to deduce:
\[compare\] Let ${{\mathcal C}}$ be a cocomplete category satisfying either of the smallness conditions (\*) or (); and let $J$ be a set of maps in ${{\mathcal C}}$. Then the w.f.s. $({{\mathcal L}},
{{\mathcal R}})$ cofibrantly generated by $J$ exists.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Since the two classes of maps ${{\mathcal L}}$ and ${{\mathcal R}}$ of this w.f.s. are entirely determined by the equations ${{\mathcal L}}= {}^{\pitchfork}{{\mathcal R}}$ and ${{\mathcal R}}=
J^\pitchfork$, the content of this Proposition is that we may find an $({{\mathcal L}},
{{\mathcal R}})$-factorisation for every map of ${{\mathcal C}}$. This is also the content of the small object argument, and so we may compare the two by comparing the choices of factorisation which they provide. For a detailed account of the small object argument, we refer the reader to [@Bous] or [@Hovey].
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Suppose we are given a category ${{\mathcal C}}$ and a set of maps $J$ as in the Proposition; and let $g \colon C \to D$ be a morphism of ${{\mathcal C}}$ that we wish to factorise. The first step in both the small object argument and our argument turns out to be the same. In the small object argument, we form the set $S$ whose elements are squares $${\vcenter{\hbox{\xymatrix{
A \ar[r]^{h} \ar[d]_f & C \ar[d]^g \\
B \ar[r]_k & D
}}}}$$ such that $f \in J$. We then form the coproduct $$\label{obvioussquare}
{\vcenter{\hbox{\xymatrix@C+1.5em{
\sum_{x \in S} A_x \ar[r]^-{[h_x]_{x \in S}} \ar[d]_{\sum_{x \in S} f_x} & C \ar[d]^g \\
\sum_{x \in S} B_x \ar[r]_-{[k_x]_{x \in S}} & D
}}}}$$ and define an object $K'g$ and morphisms $\lambda'_g$ and $\rho'_g$ by factorising this square as $$\label{pushoutsquare}
{\vcenter{\hbox{\xymatrix@C+1.5em{
\sum_{x \in S} A_x \ar[r]^-{[h_x]_{x \in S}} \ar[d]_{\sum_{x \in S} f_x} & C \ar[d]^{\lambda'_g} \ar[r]^{{\mathrm{id}}_C} & C \ar[d]^g \\
\sum_{x \in S} B_x \ar[r]_-{\xi_g} & K'g \ar[r]_{\rho'_g} & D\text,
}}}}$$ where the left-hand square is a pushout.
On the other hand, suppose we view $J$ as a discrete subcategory ${{\mathcal J}}$ of ${{\mathcal C}}^\mathbf 2$; and write $I \colon {{\mathcal J}}\hookrightarrow {{\mathcal C}}^\mathbf 2$ for the inclusion functor. Then we may view as the morphism $$\epsilon_g \colon \int^{f \in {{\mathcal J}}} {{\mathcal C}}^\mathbf 2(If, g) \cdot If \to g\text,$$ of ${{\mathcal C}}^\mathbf 2$; which is to say, the component at $g$ of the counit transformation $$\epsilon \colon \operatorname{Lan}_I(I) \Rightarrow {\mathrm{id}}_{{{\mathcal C}}^\mathbf 2} \colon {{\mathcal C}}^\mathbf 2 \to {{\mathcal C}}^\mathbf 2\text.$$ We may then view as the component at $g$ of the factorisation of $\epsilon$ into a map which is componentwise a pushout, followed by a map whose domain components are identities. Thus the assignation $g \mapsto (\lambda'_g, \rho'_g)$ obtained from the small object argument is just the underlying factorisation of the reflection of $I \colon {{\mathcal J}}\hookrightarrow {{\mathcal C}}^\mathbf 2$ along ${{\mathcal G}}_3 {{\mathcal G}}_2$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}At this point, the two arguments under consideration diverge from each other. The small object argument is the more naive of the two: it simply iterates the above procedure, each time replacing the map $g$ with the map $\rho'_g$. This gives rise to the countable sequence $${\vcenter{\hbox{\xymatrix{
C \ar[d]_g \ar[r]^{\lambda'_g} & K'g \ar[d]_{\rho'_g} \ar[r]^{\lambda'_{\rho'_g}} & K'\rho'_g \ar[d]_{\rho'_{\rho'_g}} \ar[r]^{\lambda'_{\rho'_{\rho'_g}}} & \dots \\
D \ar[r]_{{\mathrm{id}}_D} & D \ar[r]_{{\mathrm{id}}_D} & D \ar[r]_{{\mathrm{id}}_D} & \dots\text,
}}}}$$ which we extend transfinitely by taking colimits at limit ordinals. However, as pointed out in [@injective], this sequence *almost never converges*. Instead, the small object argument requires one to choose an arbitrary ordinal at which to stop: or rather, an ordinal which is large enough to ensure that the right part of the corresponding factorisation lies in $J^\pitchfork$.
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}Our argument produces a different transfinite sequence, whose first few terms are: $${\vcenter{\hbox{\xymatrix{
C \ar[d]_g \ar[r]^{\lambda'_g} & K'g \ar[d]_{\rho'_g} \ar[r]^{\lambda''_g} & K''g \ar[d]_{\rho''_g} \ar[r]^{\lambda'''_g} & \dots \\
D \ar[r]_{{\mathrm{id}}_D} & D \ar[r]_{{\mathrm{id}}_D} & D \ar[r]_{{\mathrm{id}}_D} & \dots\text;
}}}}$$ here, $K''g$ is the coequaliser $${\vcenter{\hbox{\xymatrix@C+3em{
K'g \ar@<4pt>[r]^{\lambda'_{\rho'_g}} \ar@<-4pt>[r]_{K'(\lambda'_g, {\mathrm{id}}_D)} & K'\rho'_g \ar[r] & K''g\text,
}}}}$$ and in general, the term at stage $\alpha$ in this sequence will be a quotient of the corresponding term at stage $\alpha$ in the small object argument.
We may understand this quotienting process as follows. The small object argument provides a way of taking a map $g \colon C \to D$, and recursively adding elements to its domain which witness the required lifting properties against the set $J$. This process must be recursive, since the process of adding witnesses can create new instances of the lifting properties: which in turn will require new witnesses to be added, and so on.
However, the small object argument is badly behaved: at each stage it adds new witnesses for *all* instances of the required lifting properties – including those instances for which witnesses were added at a previous stage of the induction. The effect of the quotienting process which our argument carries out is to collapse these superfluous new witnesses back onto their predecessors.
Applications
============
We end the paper with two simple applications of Theorem \[mainthm\].
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}In Examples \[exs1\], we saw that the set $J$ of horn inclusions generates a plain w.f.s. (anodyne extensions, Kan fibrations) on ${\mathbf{SSet}}$. If we view the set $J$ as a discrete subcategory ${{\mathcal J}}\hookrightarrow
{\mathbf{SSet}}^\mathbf 2$, then it also generates a natural w.f.s. $({\mathsf L}, {\mathsf R})$.
By restricting the monad ${\mathsf R}\colon {\mathbf{SSet}}^\mathbf 2 \to
{\mathbf{SSet}}^\mathbf 2$ of this natural w.f.s. to the slice over the terminal object, we obtain a monad $T \colon {\mathbf{SSet}} \to {\mathbf{SSet}}$, whose category of algebras is the category ${\mathbf{AlgKan}}$ of “algebraic Kan complexes”: simplicial sets equipped with a chosen filler for every horn, subject to no further coherence conditions. Since ${\mathbf{AlgKan}}$ is finitarily monadic over ${\mathbf{SSet}}$, it is locally finitely presentable, and hence provides a rich categorical base for further constructions.
Observe that the morphisms of ${\mathbf{AlgKan}}$ are maps of simplicial sets which *strictly* preserve the chosen fillers. Whilst these maps are of some theoretical importance, we are more likely to be interested in the category ${\mathbf{AlgKan}}_\psi$ whose objects are the same, but whose morphisms are arbitrary maps of simplicial sets. We may obtain this category by considering the adjunction $${\vcenter{\hbox{\xymatrix{{\mathbf{AlgKan}} \ar@<4pt>[r]^-{U} \ar@{}[r]|-{\top} & {\mathbf{SSet}} \ar@<4pt>[l]^-{F}\text.}}}}$$ This generates a comonad $FU$ on ${\mathbf{AlgKan}}$; and the corresponding co-Kleisli category is precisely ${\mathbf{AlgKan}}_\psi$. In particular, we deduce that the inclusion functor ${\mathbf{AlgKan}} \hookrightarrow {\mathbf{AlgKan}}_\psi$ has a left adjoint. It is a corresponding result which forms the cornerstone of two-dimensional monad theory [@2dmonad Theorem 3.13].
[[startsection [paragraph]{}[3]{}[0mm]{}[-]{}[-0.4em plus 0.2em minus 0.2em]{}[****]{}]{}]{}For an example even more in the spirit of [@2dmonad], we consider the category ${{\mathcal C}}= {\mathbf{2}}\text-{\mathbf{Cat}}$ and the set of maps $J$ given as follows: $${\vcenter{\hbox{\xymatrix@R+2em{\emptyset \ar@{.>}[d] \\ \bullet}}}}\text; \qquad {\vcenter{\hbox{\xymatrix@R+2em{ \bullet \ar@{}[r]_{}="a" & \bullet \\
\bullet \ar[r]^{}="b" \ar@{.>}"a"; "b"& \bullet}}}}\text; \qquad {\vcenter{\hbox{\xymatrix@R+2em{ \bullet \ar@/^1em/[r] \ar@/_1em/[r]_{}="a" & \bullet \\
\bullet \ar@/^1em/[r]^{}="b" \ar@/_1em/[r] \ar@{}[r] \ar@{=>}?(0.5)+/u 0.15cm/;?(0.5)+/d 0.15cm/ \ar@{.>}"a"; "b"& \bullet}}}}\text; \qquad
{\vcenter{\hbox{\xymatrix@R+2em{ \bullet \ar@/^1em/[r] \ar@/_1em/[r]_{}="a" \ar@{}[r] \ar@{=>}?(0.65)+/u 0.15cm/;?(0.65)+/d 0.15cm/ \ar@{}[r] \ar@{=>}?(0.35)+/u 0.15cm/;?(0.35)+/d 0.15cm/ & \bullet \\
\bullet \ar@/^1em/[r]^{}="b" \ar@/_1em/[r] \ar@{}[r] \ar@{=>}?(0.5)+/u 0.15cm/;?(0.5)+/d 0.15cm/ \ar@{.>}"a"; "b"& \bullet\text.}}}}$$
These maps generate a plain w.f.s. which is one half of the model structure on ${\mathbf{2}}\text-{\mathbf{Cat}}$ described by Lack in [@2catmodel]. Our purpose here will be to consider the corresponding natural w.f.s. $({\mathsf L}, {\mathsf R})$ generated by these maps, where as usual we view $J$ as a discrete subcategory ${{\mathcal J}}\hookrightarrow {{\mathcal C}}^\mathbf 2$.
In particular, if we take the comonad ${\mathsf L}$ for this natural w.f.s. and restrict it to the coslice under the initial object, we obtain a comonad $Q
\colon {\mathbf{2}}\text-{\mathbf{Cat}} \to {\mathbf{2}}\text-{\mathbf{Cat}}$. We can describe $Q$ quite explicitly. Given a 2-category ${{\mathcal K}}$, we first form the free 2-category $FU{{\mathcal K}}$ on the underlying 1-graph of ${{\mathcal K}}$. Then we take the counit 2-functor $\epsilon_{{\mathcal K}}\colon FU{{\mathcal K}}\to {{\mathcal K}}$ and factorise it as $$\epsilon_{{\mathcal K}}= FU{{\mathcal K}}{\ext@arrow 01{20}0\rightarrowfill@{}{\xi_{{\mathcal K}}}} Q{{\mathcal K}}{\ext@arrow 01{20}0\rightarrowfill@{}{\phi_{{\mathcal K}}}} {{\mathcal K}}$$ where $\xi_{{\mathcal K}}$ is bijective on objects and 1-cells, and $\phi_{{\mathcal K}}$ is locally fully faithful. The resultant $Q{{\mathcal K}}$ is precisely the “homomorphism classifier” of ${{\mathcal K}}$: it is characterised by an isomorphism, natural in ${{\mathcal L}}$, between $$\text{2-functors } Q{{\mathcal K}}\to {{\mathcal L}}\qquad \text{and} \qquad \text{pseudofunctors } {{\mathcal K}}\to {{\mathcal L}}\text.$$ It follows from this characterisation that the co-Kleisli category of $Q$ is the category ${\mathbf{2}}\text-{\mathbf{Cat}}_\psi$ of 2-categories and pseudofunctors between them.
Observe that in this example, we at no point had to define what a “pseudofunctor” was: it emerged simply from applying our apparatus for a well-chosen set of maps $J$. Of course, since we already knew what pseudofunctors were, we did not gain much from this; however, it suggests that for a more complex ${{\mathcal C}}$, we may be able to define a suitable notion of “pseudomorphism” simply by applying the above argument for a suitable set of maps $J$.
As an example of this, let us consider the category ${\mathbf{Tricat}}$ of tricategories and (strict) structure-preserving maps between them, and see how this argument allows us to derive the notion of trihomomorphism. By “tricategory”, we will mean [@nicktricats]’s algebraic definition of tricategory, so that ${\mathbf{Tricat}}$ is finitarily monadic over the category ${\mathbf{GSet}}_3$ of 3-dimensional globular sets; and in particular is locally finitely presentable. Let us write $${\vcenter{\hbox{\xymatrix{{\mathbf{Tricat}} \ar@<4pt>[r]^-{U} \ar@{}[r]|-{\top} & {\mathbf{GSet}}_3 \ar@<4pt>[l]^-{F}}}}}$$ for the free/forgetful adjunction. We define a set $J$ of morphisms in ${\mathbf{Tricat}}$ by taking the following set of maps in ${\mathbf{GSet}}_3$: $${\vcenter{\hbox{\xymatrix@R+2em@C+1em{\emptyset \ar@{.>}[d] \\ \bullet}}}}\text; \qquad {\vcenter{\hbox{\xymatrix@R+2em@C+1em{ \bullet \ar@{}[r]_{}="a" & \bullet \\
\bullet \ar[r]^{}="b" \ar@{.>}"a"; "b"& \bullet}}}}\text; \qquad {\vcenter{\hbox{\xymatrix@R+2em@C+1em{ \bullet \ar@/^1em/[r] \ar@/_1em/[r]_{}="a" & \bullet \\
\bullet \ar@/^1em/[r]^{}="b" \ar@/_1em/[r] \ar@{}[r] \ar@{=>}?(0.5)+/u 0.15cm/;?(0.5)+/d 0.15cm/ \ar@{.>}"a"; "b"& \bullet}}}}\text; \qquad
{\vcenter{\hbox{\xymatrix@R+2em@C+1em{ \bullet \ar@/^1em/[r] \ar@/_1em/[r]_{}="a" \ar@{}[r] \ar@{=>}?(0.65)+/u 0.15cm/;?(0.65)+/d 0.15cm/ \ar@{}[r] \ar@{=>}?(0.35)+/u 0.15cm/;?(0.35)+/d 0.15cm/ & \bullet \\
\bullet \ar@{}[r] \ar@3?(0.5)+/l 0.13cm/;?(0.5)+/r 0.13cm/ \ar@/^1em/[r]^{}="b" \ar@/_1em/[r] \ar@{}[r] \ar@{=>}?(0.7)+/u 0.15cm/;?(0.7)+/d 0.15cm/ \ar@{}[r] \ar@{=>}?(0.3)+/u 0.15cm/;?(0.3)+/d 0.15cm/ & \bullet \ar@{.>}"a"; "b"}}}}\text; \qquad
{\vcenter{\hbox{\xymatrix@R+2em@C+1em{ \bullet \ar@{}[r] \ar@3?(0.5)+/l 0.13cm/+/u 0.13cm/;?(0.5)+/r 0.13cm/+/u 0.13cm/ \ar@3?(0.5)+/l 0.13cm/+/d 0.26cm/;?(0.5)+/r 0.13cm/+/d 0.26cm/ \ar@/^1em/[r] \ar@/_1em/[r]_{}="a" \ar@{}[r] \ar@{=>}?(0.7)+/u 0.15cm/;?(0.7)+/d 0.15cm/ \ar@{}[r] \ar@{=>}?(0.3)+/u 0.15cm/;?(0.3)+/d 0.15cm/ & \bullet\\
\bullet \ar@{}[r] \ar@3?(0.5)+/l 0.13cm/;?(0.5)+/r 0.13cm/ \ar@/^1em/[r]^{}="b" \ar@/_1em/[r] \ar@{}[r] \ar@{=>}?(0.7)+/u 0.15cm/;?(0.7)+/d 0.15cm/ \ar@{}[r] \ar@{=>}?(0.3)+/u 0.15cm/;?(0.3)+/d 0.15cm/ & \bullet\text, \ar@{.>}"a"; "b"}}}}$$ and applying the free functor $F$ to each of them. We now proceed as before: we consider this set $J$ as a discrete subcategory ${{\mathcal J}}\hookrightarrow
{\mathbf{Tricat}}^\mathbf 2$ and let $({\mathsf L}, {\mathsf R})$ be the n.w.f.s. generated by ${{\mathcal J}}$; and then let $Q$ be the comonad on ${\mathbf{Tricat}}$ given by the restriction of ${\mathsf L}$ to the coslice under the initial object.
We now *define* a trihomomorphism $\mathcal S \to {{\mathcal T}}$ to be a strict morphism $Q\mathcal S \to {{\mathcal T}}$, and define the category ${\mathbf{Tricat}}_\psi$ of tricategories and trihomomorphisms to be the co-Kleisli category of $Q$. The notion of trihomomorphism we obtain in this way cannot be the one we are used to from [@GPS], since the latter does not admit a strictly associative composition: see [@gg]. Nonetheless, we can show that our new notion of trihomomorphism is equivalent to the old one, in that we can exhibit a biequivalence between a suitably defined 2-category of these new trihomomorphisms and a corresponding bicategory of the usual ones.
The full details of this will be worked out in a forthcoming paper; but for now, let us merely say that this method should immediately extend to (sufficiently algebraic) weak $n$-categories and even weak $\omega$-categories, thereby allowing us to give a notion of “weak morphism of $\omega$-categories” which admits a strictly associative composition.
Algebraically-free implies free
===============================
The purpose of this Appendix is to sketch a proof of the following result:
\[freealgfree\] Let $({\mathsf L}, {\mathsf R})$ be a n.w.f.s. on ${{\mathcal C}}$ which is algebraically-free on $I
\colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$. Then $({\mathsf L}, {\mathsf R})$ is free on ${{\mathcal J}}$.
We first define a monoidal structure on the category ${\mathbf{CAT}} / {{\mathcal C}}^\mathbf
2$. Given $U \colon {{\mathcal A}}\to {{\mathcal C}}^\mathbf 2$ and $V \colon {{\mathcal B}}\to {{\mathcal C}}^\mathbf 2$, their tensor product $W \colon {{\mathcal A}}\otimes {{\mathcal B}}\to {{\mathcal C}}^\mathbf 2$ is obtained by first taking the pullback $${\vcenter{\hbox{\xymatrix@-1em{
{{\mathcal A}}\otimes {{\mathcal B}}\ar[rr]
\ar[dd] {\save*!/dr-1.2pc/dr:(-1,1)@^{|-}\restore}& &
{{\mathcal A}}\ar[d]^{U} \\ & &
{{\mathcal C}}^\mathbf 2 \ar[d]^{{\mathrm{dom}}}\\
{{\mathcal B}}\ar[r]_{V} &
{{\mathcal C}}^\mathbf 2
\ar[r]_{{\mathrm{cod}}} &
{{\mathcal C}}\text,
}}}}$$ and then defining the projection $W$ by $W(a, b) = Ua \circ Vb$. The unit for this tensor product is the object $(s_0 \colon {{\mathcal C}}\to {{\mathcal C}}^\mathbf 2)$, where $s_0$ is the functor induced by homming the unique map $\sigma_0 \colon \mathbf
2 \to \mathbf 1$ into ${{\mathcal C}}$; thus $s_0(c) = {\mathrm{id}}_c \colon c \to c$.
Next, we show that, for any n.w.f.s. $({\mathsf L}, {\mathsf R})$ on ${{\mathcal C}}$, the object $(U_{\mathsf L}\colon {\mathsf L\text-{\mathbf{Map}}}\to {{\mathcal C}}^\mathbf 2)$ is a monoid with respect to this monoidal structure: the key point being that, given ${\mathsf L}$-map structures on $f \colon X
\to Y$ and $g \colon Y \to Z$, we may define an ${\mathsf L}$-map structure on $gf
\colon X \to Z$. Indeed, if these two ${\mathsf L}$-map structures are provided by morphisms $s \colon Y \to Kf$ and $t \colon Z \to Kg$ (as in §\[worthdefn\]), then the ${\mathsf L}$-map structure on the composite $gf$ is given by: $$Z {\ext@arrow 01{20}0\rightarrowfill@{}{t}} Kg {\ext@arrow 01{20}0\rightarrowfill@{}{K(s, {\mathrm{id}}_Z)}} K(g \circ \rho_f) {\ext@arrow 01{20}0\rightarrowfill@{}{K(K(1, g), 1)}} K\rho_{gf}
{\ext@arrow 01{20}0\rightarrowfill@{}{\pi_{gf}}} K(gf)\text.$$ The remaining details are routine; and by dualising, we see that $U_{\mathsf R}\colon
{\mathsf R\text-{\mathbf{Map}}}\to {{\mathcal C}}^\mathbf 2$ is also a monoid in ${\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$.
We may now show that, if $\alpha \colon ({\mathsf L}, {\mathsf R}) \to ({\mathsf L}', {\mathsf R}')$ is a map of n.w.f.s.’s, then the induced functors $(\alpha_l)_\ast \colon {\mathsf L\text-{\mathbf{Map}}}\to
{\mathsf L}'\text-{\mathbf{Map}}$ and $(\alpha_r)^\ast \colon {\mathsf R}'\text-{\mathbf{Map}} \to {\mathsf R\text-{\mathbf{Map}}}$ are maps of monoids; so that the semantics functors ${{\mathcal G}}$ and ${{\mathcal H}}$ may be lifted to functors $$\begin{aligned}
\hat {{\mathcal G}}\colon {\mathbf{NWFS}}({{\mathcal C}}) & \to {\mathbf{Mon}}({\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2) \\
\text{and} \qquad \hat {{\mathcal H}}\colon {\mathbf{NWFS}}({{\mathcal C}}) & \to \big({\mathbf{Mon}}({\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2)\big)^{\mathrm{op}}\text.\end{aligned}$$
We now arrive at a crucial juncture in the proof: we show that $\hat {{\mathcal G}}$ and $\hat {{\mathcal H}}$ are fully faithful. In the case of $\hat {{\mathcal G}}$, for example, we consider n.w.f.s.’s $({\mathsf L}, {\mathsf R})$ and $({\mathsf L}', {\mathsf R}')$ on ${{\mathcal C}}$, and a map of monoids $F \colon {\mathsf L\text-{\mathbf{Map}}}\to {\mathsf L}'\text-{\mathbf{Map}}$ over ${{\mathcal C}}^\mathbf 2$; and must show that there is a unique morphism $\alpha \colon ({\mathsf L}, {\mathsf R}) \to ({\mathsf L}',
{\mathsf R}')$ for which $F = (\alpha_l)_\ast$. To do this, we consider squares of the following form: $${\vcenter{\hbox{\xymatrix{
A \ar[d]_{\lambda_f} \ar[r]^{\lambda'_f} &
K'f \ar[d]^{\rho'_f} \\
Kf \ar[r]_{\rho_f} &
B\text.
}}}}$$ We can make $\rho'_f$ into an ${\mathsf R}'$-map, since it is the free ${\mathsf R}'$-map on $f$. Similarly, we can make $\lambda_f$ into an ${\mathsf L}$-map; and by applying the functor $F \colon {\mathsf L\text-{\mathbf{Map}}}\to {\mathsf L}'\text-{\mathbf{Map}}$, we may make it into an ${\mathsf L}'$-map. Now we apply the lifting operation associated with $({\mathsf L}', {\mathsf R}')$ to obtain a morphism $\alpha_f \colon Kf \to K'f$. These maps $\alpha_f$ provide the components of a morphism between the underlying functorial factorisations of $({\mathsf L}, {\mathsf R})$ and $({\mathsf L}', {\mathsf R}')$: it remains only to check that the comonad and monad structures are preserved. This is just a matter of checking details, but makes essential use of two facts: that $F$ is a map of monoids; and that the distributivity axiom holds in $({\mathsf L}, {\mathsf R})$ and $({\mathsf L}',
{\mathsf R}')$.
Next, we prove that for any category $U \colon {{\mathcal A}}\to {{\mathcal C}}^\mathbf 2$ over ${{\mathcal C}}^\mathbf 2$, the category ${{\mathcal A}}^\pitchfork \to {{\mathcal C}}^\mathbf 2$ is a monoid in ${\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$. The key point is to show that, whenever we equip morphisms $f \colon C \to D$ and $g \colon D \to E$ of ${{\mathcal C}}$ with coherent choices of liftings against the elements of ${{\mathcal A}}$, we induce a corresponding equipment on the composite $gf$. Indeed, given $a \in {{\mathcal A}}$ and a square $${\vcenter{\hbox{\xymatrix{
A
\ar[r]^-h
\ar[dd]_{Ua} &
C
\ar[d]^f \\ &
D
\ar[d]^g \\
B
\ar[r]_-k &
E\text,
}}}}$$ we define $\phi_{gf}(a, h, k) \colon B \to C$ as follows. First we form $j :=
\phi_g(a, fh, k) \colon B \to D$; and now we take $\phi_{gf}(a, h, k) :=
\phi_f(a, h, j) \colon B \to C$.
We may now check that if $F \colon {{\mathcal A}}\to {{\mathcal B}}$ is a morphism of ${\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$, then the morphism $F^\pitchfork \colon {{\mathcal B}}^\pitchfork
\to {{\mathcal A}}^\pitchfork$ respects the monoid structures on ${{\mathcal A}}^\pitchfork$ and ${{\mathcal B}}^\pitchfork$, so that the functors $({{\mathord{\text{--}}}})^\pitchfork$, and dually ${}^\pitchfork({{\mathord{\text{--}}}})$, lift to functors $$\begin{aligned}
({{\mathord{\text{--}}}})^\pitchfork & \colon ({\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2)^{\mathrm{op}}\to {\mathbf{Mon}}({\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2) \\
\text{and} \qquad {}^\pitchfork({{\mathord{\text{--}}}}) & \colon {\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2 \to \big({\mathbf{Mon}}({\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2)\big)^{\mathrm{op}}\text.\end{aligned}$$
Finally, we may show that for any n.w.f.s. $({\mathsf L}, {\mathsf R})$ on ${{\mathcal C}}$, the canonical operation of lifting $\textsf{lift} \colon {\mathsf R\text-{\mathbf{Map}}}\to {\mathsf L\text-{\mathbf{Map}}}^\pitchfork$ is a monoid morphism in ${\mathbf{CAT}}/{{\mathcal C}}^\mathbf 2$. Again, this is simply a matter of checking details.
We now have all the material we need to prove the Theorem. We suppose ourselves given a n.w.f.s. $({\mathsf L}, {\mathsf R})$ which is algebraically-free on $I \colon {{\mathcal J}}\to
{{\mathcal C}}^\mathbf 2$ via the morphism $\eta \colon {{\mathcal J}}\to {\mathsf L\text-{\mathbf{Map}}}$: and are required to show that $({\mathsf L}, {\mathsf R})$ is free on ${{\mathcal J}}$. So consider a further n.w.f.s. $({\mathsf L}',
{\mathsf R}')$ on ${{\mathcal C}}$, and a morphism $F \colon {{\mathcal J}}\to {\mathsf L}'\text-{\mathbf{Map}}$ over ${{\mathcal C}}^\mathbf 2$. We can form the following diagram of functors over ${{\mathcal C}}^\mathbf
2$: $${\vcenter{\hbox{\xymatrix{
{\mathsf R\text-{\mathbf{Map}}}\ar[r]^{\textsf{lift}} & {\mathsf L\text-{\mathbf{Map}}}^\pitchfork \ar[r]^-{\eta^\pitchfork} & {{\mathcal J}}^\pitchfork\text.\\
{\mathsf R}'\text-{\mathbf{Map}} \ar[r]_{\textsf{lift}} & {\mathsf L}'\text-{\mathbf{Map}}^\pitchfork \ar[ur]_{F^\pitchfork}
}}}}$$
By algebraic-freeness, the composite along the top is invertible, and so we obtain from this diagram a functor ${\mathsf R}'\text-{\mathbf{Map}} \to
{\mathsf R}\text-{\mathbf{Map}}$. But every map in the diagram is a map of monoids, and hence the induced functor ${\mathsf R}'\text-{\mathbf{Map}} \to {\mathsf R}\text-{\mathbf{Map}}$ is too; and so is induced by a unique morphism of n.w.f.s.’s $\alpha \colon ({\mathsf L}, {\mathsf R})
\to ({\mathsf L}', {\mathsf R}')$.
It requires a little more work to show $(\alpha_l)_\ast \circ \eta = F$, and that $\alpha$ is the unique morphism of n.w.f.s.’s with this property. The two essential facts that we need are that, for any n.w.f.s. $({\mathsf L}, {\mathsf R})$, the canonical morphism ${\mathsf L\text-{\mathbf{Map}}}\to {}^\pitchfork({\mathsf R\text-{\mathbf{Map}}})$ is a monomorphism; and that, for any morphism of n.w.f.s.’s $\alpha \colon ({\mathsf L}, {\mathsf R}) \to ({\mathsf L}',
{\mathsf R}')$, the following diagram commutes: $${\vcenter{\hbox{\xymatrix{
{\mathsf R\text-{\mathbf{Map}}}\ar[r]^{\textsf{lift}} \ar[d]_{(\alpha_r)^\ast} & {\mathsf L\text-{\mathbf{Map}}}^\pitchfork \ar[d]^{((\alpha_l)_\ast)^\pitchfork} \\
{\mathsf R}'\text-{\mathbf{Map}} \ar[r]_{\textsf{lift}} & {\mathsf L}'\text-{\mathbf{Map}}^\pitchfork\text.
}}}}$$ We leave these details to the reader.
[^1]: Supported by a Research Fellowship of St John’s College, Cambridge and a Marie Curie Intra-European Fellowship, Project No. 040802.
[^2]: Here we commit the usual abuse of notation in denoting a category $I \colon {{\mathcal J}}\to {{\mathcal C}}^\mathbf 2$ over ${{\mathcal C}}^\mathbf
2$ merely by its domain category ${{\mathcal J}}$.
|
---
abstract: 'As a crucial component in task-oriented dialog systems, the Natural Language Generation (NLG) module converts a dialog act represented in a semantic form into a response in natural language. The success of traditional template-based or statistical models typically relies on heavily annotated data, which is infeasible for new domains. Therefore, it is pivotal for an NLG system to generalize well with limited labelled data in real applications. To this end, we present [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}, the first NLG benchmark to simulate the few-shot learning setting in task-oriented dialog systems. Further, we develop the SC-GPT[^1] model. It is pre-trained on a large set of annotated NLG corpus to acquire the controllable generation ability, and fine-tuned with only a few domain-specific labels to adapt to new domains. Experiments on [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} and the large Multi-Domain-WOZ datasets show that the proposed SC-GPT significantly outperforms existing methods, measured by various automatic metrics and human evaluations.'
author:
- |
Baolin Peng, Chenguang Zhu, Chunyuan Li\
**Xiujun Li, Jinchao Li, Michael Zeng, Jianfeng Gao**\
Microsoft Research, Redmond\
`{bapeng,chezhu,chunyl,xiul,jincli,nzeng,jfgao}@microsoft.com`
bibliography:
- 'acl2020.bib'
title: 'Few-shot Natural Language Generation for Task-Oriented Dialog'
---
Introduction
============
Task-oriented dialog systems are becoming increasingly popular, as they can assist users in various daily activities such as ticket booking and restaurant reservations. In a typical task-oriented dialog system, the [*Natural Language Generation*]{} (NLG) module plays a crucial role: it converts a system action (often specified in a semantic form selected by a dialog policy) into a final response in natural language. Hence, the response should be [*adequate*]{} to represent semantic dialog actions, and [*fluent*]{} to engage users’ attention. As the ultimate interface to interacts with users, NLG plays a significant impact on the users’ experience.
Existing methods for NLG can be broadly summarized into two major categories. $(\RN{1})$ [*Template-based methods*]{} require domain experts to handcraft templates for each domain, and the system fills in slot-values afterward [@rule; @halogen]. Thus, the produced responses are often adequate to contain the required semantic information, but not always fluent and nature, hurting users’ experiences. $(\RN{2})$ [*Statistical language models*]{} such as neural networks [@gao2019neural] learn to generate fluent responses via training from labelled corpus. One canonical model is [*semantically conditioned LSTM*]{} (SC-LSTM) [@wen-etal-2015-semantically], which encodes dialog acts with one-hot representations and uses it as an extra feature to inform the sentence generation process. Despite its good performance on simple domains, it requires large amounts of domain-specific annotated data which is not available for many domains in real-world applications. Even worse, this renders severe scalability issues when the number of possible combinations of dialog acts grows exponentially with the number of slots in more complex domains.
We revisit the current research benchmarks for NLG, and notice that each dialog domain is extensively labelled to favor model training. However, this is in contrast to the real-world application scenarios, where only very limited amounts of labelled data are available for new domains. To simulate such a few-shot learning setting, we have developed a new benchmark dataset, called [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}, based on the MultiWOZ [@budzianowski2018multiwoz] and Cambridge NLG datasets [@rnnlg]. [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} consists of dialog utterances from 7 domains. For each domain, we provide less than 50 labeled utterances for fine-tuning. We believe that [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} can better inspire research to address the challenge of learning data-hungry statistical models with very limited amounts of labelled data in real-world scenarios.
To deal with the challenge of few-shot learning, we develop the SC-GPT model. SC-GPT is a multi-layer Transformer neural language model, trained in three steps: $(\RN{1})$ Pre-trained on plain text, similar to GPT-2 [@gpt2]; $(\RN{2})$ Continuously pre-trained on large amounts of dialog-act labeled utterances corpora to acquire the ability of controllable generation; $(\RN{3})$ Fine-tuned for a target domain using very limited amounts of domain labels. Unlike GPT-2, SC-GPT generates semantically controlled responses that are conditioned on the given semantic form, similar to SC-LSTM but requiring much less domain labels to generalize to new domains.
------------------------------------------------------------- --------------------------------------------------
{height="3.0cm"} {height="3.0cm"}
\(a) The overall framework of a task-oriented dialog system \(b) Dialog act & Response
------------------------------------------------------------- --------------------------------------------------
In summary, our key contributions are three-fold:
- A new benchmark [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} is introduced to simulate the few-shot adaptation setting where only a handful of training data from each domain is available.
- We propose a new model SC-GPT. To our best knowledge, this work is the first study of exploiting state-of-the-art pre-trained language models for NLG in task-oriented dialog systems.
- On the MultiWOZ dataset, SC-GPT creates a new SOTA, outperforming previous models by 4 points in BLEU. On [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}, SC-GPT outperforms several strong baselines such as SC-LSTM and HDSA [@chen-etal-2019-semantically], showing that SC-GPT adapts to new domain much more effectively, requiring much smaller amounts of in-domain labels. We release our code[^2] and dataset[^3] for reproducible research.
Background
==========
A typical task-oriented spoken dialog system uses a pipeline architecture, as shown in Figure \[fig:dailog\_system\] (a), where each dialog turn is processed using a four-step procedure. $(\RN{1})$ Transcriptions of user’s input are first passed to the natural language understanding (NLU) module, where the user’s intention and other key information are extracted. $(\RN{2})$ This information is then formatted as the input to dialog state tracking (DST), which maintains the current state of the dialog. $(\RN{3})$ Outputs of DST are passed to the dialog policy module, which produces a dialog act based on the facts or entities retrieved from external resources (such as a database or a knowledge base). $(\RN{4})$ The dialog act emitted by the dialog policy module serves as the input to the NLG, through which a system response in natural language is generated. In this paper, we focus on the NLG component of task-oriented dialog systems, how to produce natural language responses conditioned on dialog acts.
Specifically, [*dialog act*]{} $\Acal$ is defined as the combination of intent $\Imat$ and slot-value pairs $\{(s_i, v_i)\}^P_{i=1}$: $$\Acal = [ \underbrace{~~\Imat_{~_{~}}}_{\text{Intent}}, \underbrace{(s_1, v_1), \cdots, (s_P, v_P)}_{\text{Slot-value pairs} } ]$$ where $P$ is the number of pairs[^4], which varies in different dialog acts.
- [*Intents*]{} are usually used to distinguish different types of system actions. Typical examples include [*inform*]{}, [*request*]{}, [*confirm*]{}, [*select*]{}
- [*Slot-value pairs*]{} indicate the category and content of the information to express in the utterance, respectively.
The goal of NLG is to translate $\Acal$ into a natural language response $\xv = [x_1, \cdots, x_T]$, where $T$ is the sequence length. In Figure \[fig:dailog\_system\] (b), we show an example of the dialog act: $\textit{\texttt{confirm}~(name=Hilton, area=center)}$, and the corresponding natural language response is “[*Let me confirm that you are searching for Hilton in the center area*]{}”.
{width="2\columnwidth"}
Semantically Conditioned GPT
============================
We tackle this generation problem using conditional neural language models. Given training data of $N$ samples $\Dcal=\{(\Acal_n, \xv_n)\}_{n=1}^{N}$, our goal is to build a statistical model parameterized by $\thetav$ to characterize $p_{\thetav}(\xv | \Acal)$. To leverage the sequential structure of response, one may further decompose the joint probability of $\xv$ using the chain rule, casting an auto-regressive generation process as follows: $$p_{\thetav}(\xv|\Acal) = \prod_{t=1}^{T} p_{\thetav}(x_t | x_{<t}, \Acal)
\label{eq:conditional}$$ where $x_{<t}$ indicates all tokens before $t$.
Learning $\thetav$ is performed via maximizing the log-likelihood (MLE) of the conditional probabilities in over the entire training dataset: $$\Lcal_{\thetav}(\Dcal) = \sum_{n=1}^{|\Dcal|} \sum_{t=1}^{T_n} \log p_{\thetav}(x_{t,n} | x_{<t,n}, \Acal_n)$$ In this paper, we employ the Transformers [@transformer] to parameterize the conditionals in . To enable strong generalization and controllable ability for the learned model, we propose the following three-stage procedure as the training recipe.
#### Massive Plain Language Pre-training.
Large models trained on massive training corpus usually generalize better to new domains. Inspired by this, we inherit the GPT-2 architecture [@gpt2] as the backbone language model. GPT-2 is an auto-regressive language model that leverages 12-24 layers of masked, multi-head self-attention Transformers. GPT-2 is pre-trained on extremely massive text data OpenWebText [@gpt2]. It has demonstrated superior performance on characterizing human language data distribution and knowledge transfer. Given text prompts, GPT-2 can often generate realistic sentences.
#### Dialog-Act Controlled Pre-training.
To enable the guidance of dialog act in response generation, we propose to continuously pre-train the GPT-2 model on large amounts of annotated (dialog act, response) pairs. The pre-training dataset[^5] includes annotated training pairs from Schema-Guided Dialog corpus, MultiWOZ corpus, Frame corpus, and Facebook Multilingual Dialog Corpus. The total size of the pre-training corpus is around 400k examples.
We firstly pre-process dialog act $\Acal$ into a sequence of control codes using the following format: $$\Acal^{\prime} = \left[ ~ \Imat~~ (~~ s_1~~ = ~~ v_1~~,~\cdots~~ s_P~ = ~~ v_P~~ )~ \right]
\label{eq:pre_process}$$ Meanwhile, the output sequence $\xv^{\prime}$ is pre-processed via appending $\xv$ with a special start token `[BOS]` and an end token `[EOS]`. Finally, the sequentialized dialog act $\Acal^{\prime}$ is concatenated with its augmented response $\xv^{\prime}$, and then fed into GPT-2. During training, the prediction loss is only computed for $\xv^{\prime}$, and $\Acal^{\prime}$ provides the attended conditions. Since the dialog act represents the semantics of the generated sentences, we follow the naming convention of SC-LSTM, and term our model as [*Semantically Conditioned Generative Pre-training*]{} (SC-GPT). The overall architecture of SC-GPT is illustrated in Figure \[fig:scGPT\].
#### Fine-tuning.
For a new domain, a dialog act usually contains novel intents or slot-value pairs, and annotated training samples are often limited. We fine-tune SC-GPT on limited amounts of domain-specific labels for adaptation. The fine-tuning follows the same procedure of dialog-act controlled pre-training, as described above, but uses only a few dozens of domain labels.
It is worth noticing that the above recipe has several favorable properties:
- [*Flexibility.*]{} SC-GPT operates on a sequence of tokens without delexicalization, which means that SC-GPT does not assume a fixed one-hot or tree-structured dialog act representation vectors. Hence, it has great flexibility in extending to novel dialog acts.
- [*Controllability.*]{} In contrast to GPT-2 that generates natural sentences without high-level semantic guidance, SC-GPT can generate sentences with adequate intent and slot-value information and maintain its fluency.
- [*Generalizability.*]{} SC-GPT is able to generalize significantly better than SC-LSTM, due to the pre-training on massive plain text corpora and annotated dialog datasets.
Dataset: [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}
=====================================================================
#### Revisiting NLG Benchmarks.
The three commonly used NLG datasets in developing and evaluating task-oriented dialog systems are E2E NLG [@e2enlg] BAGEL [@bagel] and RNNLG [@rnnlg], as summarized in Table \[tab:dataset\_compare\]. We observe two issues from their shared statistics: $(\RN{1})$ All the datasets contain a large number of labelled training samples for each domain, ranging from hundreds to tens of thousands. However, the cost of labeling is high in practice, labeling 50 utterances is 5 hours per domain. Creating such an extensively annotated dataset for each new domain is prohibitively expensive. $(\RN{2})$ The percentage of distinct delexicalised dialog acts between training and testing data is quite small. For example, the delexicalised dialog acts in testing is 100% covered by the training set for the E2E NLG dataset. It renders difficulties in evaluating the model’s generalization ability for new domains.
#### [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}.
To build a setting for more pragmatic NLG scenarios, we introduce a new dataset [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} to better reflect real application complexity, and encourage the community to develop algorithms that are capable of generalizing with only a few domain-specific labels for each (new) domain. The dataset statistics are shown in the last column of Table \[tab:dataset\_compare\]. We see that [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} is different from the other datasets in three aspects: $(\RN{1})$ [*More domains*]{}. [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} contains seven domains in total, which is larger than any existing NLG datasets. $(\RN{2})$ [*Less training instances*]{}. Importantly, [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} has a much smaller number of training instances per domain, aiming to evaluate the few-shot learning ability. $(\RN{3})$ [*Lower training/testing overlap*]{}. [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} has only 8.82% overlap, significantly smaller than the other datasets, which amount to more than 90% overlap. The average number of intents per instance in $\mathtt{Attraction}$/ $\mathtt{Taxi}$/ $\mathtt{Train}$ domain is 2, 1.33, and 2.05, respectively. In contrast, there is only one intent for each example in the other datasets. The NLG task defined on [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} requires the models to learn to generalize over new compositions of intents. The details of [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} is shown in Table \[tab:fewshotwoz\].
#### Collection Protocols.
We construct [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} via re-organizing data samples from RNNLG and MultiWOZ datasets [@budzianowski2018multiwoz]. For each domain in RNNLG, we first group utterances according to their delexicalised dialog acts, and keep only one utterance as the target sentence. To ensure diversity, we consider three domains from MultiWOZ: $\mathtt{Attraction}$, $\mathtt{Taxi}$, and $\mathtt{Train}$. Since MultiWOZ is a cross-domain dataset, the dialog act of an utterance may exist in multiple domains. We choose to keep utterances whose dialog act appears only in one domain. Similar delexicalising processing is applied to ensure that each dialog act has only one target utterance. Finally, to simulate the few-shot learning in practice, we randomly sample 50 training examples for each domain, except the $\mathtt{Taxi}$ domain, which has 40 examples.
Related Work
============
#### Pre-trained Models.
Pre-trained language models (PLMs) have substantially advanced the state-of-the-art across a variety of natural language processing (NLP) tasks [@peters2018deep; @devlin2019bert; @yang2019xlnet; @liu2019roberta; @keskar2019ctrl; @raffel2019exploring]. PLMs are often trained to predict words based on their context on massive text data, and the learned models can be fine-tuned to adapt to various downstream tasks. The closest line of research to ours are GPT-2 [@gpt2], CTRL [@keskar2019ctrl] and Grover [@zellers2019defending]. GPT-2 first investigated missive Transformer-based auto-regressive language models with large-scale text data for pre-training. After fine-tuning, GPT-2 achieves drastic improvements on several generation tasks. One drawback of GPT-2 is the lack of high-level semantic controlling ability in language generation. To alleviate this issue, CTRL [@keskar2019ctrl] was introduced to train the model based on pre-defined codes such as text style, content description, and task-specific behavior, meanwhile Grover [@zellers2019defending] was proposed to generate news articles conditioned on authors, dates Although conceptually similar to our SC-GPT, CTRL and Grover cannot be readily applied to NLG in task-oriented dialog systems, as the conditioning codes are quite different. Another controllable generation work for GPT-2 is PPLM [@dathathri2019plug], which provides a decoding scheme to guide the generation process using key-words or classifiers, without re-training the model. In this paper, we focus on pre-training an NLG model conditioned on finer-grained semantic dialog acts, which are more desirable for dialog systems.
#### Dialog.
Various dialog systems have been developed [@gao2019neural], including task-oriented dialog systems such as Rasa[^6], Microsoft Bot Framework[^7], and Conversational Learner[^8], and chit-chat systems such as XiaoIce [@zhou2018design], DialoGPT [@zhang2019dialogpt], Meena [@adiwardana2020towards]. In this paper, we focus on task-oriented systems, particularly the NLG module. With the blooming of deep learning, neural sequential models have shown powerful capability and flexibility in NLG. Extensive efforts have been made, including new architecture choices such as RNNs [@wen-etal-2015-stochastic], attention RNNs [@dusek-jurcicek-2016-sequence], SC-LSTM [@wen-etal-2015-semantically] and its variants [@tran-etal-2017-neural; @tran-nguyen-2017-natural], as well as learning objectives [@zhu-etal-2019-multi]. However, they all require large amounts of annotated data to reach satisfactory performance. A more realistic scenario is to require much less labeling and improve the sample efficiency of models, This is especially important when deploying the models to new domains, where dialog acts need to be labelled from scratch. Our paper aims to formally set up such a research scenario by proposing a new dataset [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}, and a new model SC-GPT.
{height="2.5cm"}
Experiments
===========
In this section, we evaluate the proposed SC-GPT on the [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} and MultiWOZ datasets to answer two research questions: $(\RN{1})$ Is SC-GPT an effective model for strong generalization and controllability in dialog response generation? $(\RN{2})$ Does [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} meet the goal of effectively evaluating the generalization of NLG models in the few-shot learning setting?
Experimental Setup
------------------
#### Implementation details.
The model was built upon Huggingface Pytorch Transformer [@Wolf2019HuggingFacesTS]. We use GPT2-Medium with 345M parameters[^9] as the initial checkpoint, and byte pair encodings [@bpe] for the tokenization. Linear rate scheduler with start rate as 5e-5 was used for both pre-training and fine-tuning. Adam [@kingma2014adam] with weight decay was used to optimize the parameters. For pre-training, the model was trained with a mini-batch of 8 on an 8 Nvidia V100 machine until observing no significant progress on validation loss or up to 20 epochs, whichever is earlier. For fine-tuning on [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}, models were trained on each domain separately with five epochs.
#### Automatic metrics.
Following @wen-etal-2015-semantically, BLEU scores and the slot error rate (ERR) are reported. BLEU score evaluates how natural the generated utterance is compared with human readers. ERR measures the exact matching of the slot tokens in the candidate utterances. $\text{ERR}=(p+q)/M$, where $M$ is the total number of slots in the dialog act, and $p$, $q$ is the number of missing and redundant slots in the given realisation. For each dialog act, we generate five utterances and select the top one with the lowest ERR as the final output.
#### Human evaluation.
We conducted the human evaluation using Amazon Mechanical Turk to assess subjective quality. We recruit master level workers (who have good prior approval rates) to perform a human comparison between generated responses from two systems (which are randomly sampled from comparison systems). The workers are required to judge each utterance from 1 (bad) to 3 (good) in terms of informativeness and naturalness. *Informativeness* indicates the extent to which generated utterance contains all the information specified in the dialog act. *Naturalness* denotes whether the utterance is as natural as a human does. To reduce judgement bias, we distribute each question to three different workers. Finally, we collected in total of 5800 judges.
#### Baselines.
We compare with three baseline methods. $(\RN{1})$ **SC-LSTM** [@wen-etal-2015-semantically] is a canonical model and a strong baseline that uses an additional dialog act vector and a reading gate to guide the utterance generation. $(\RN{2})$ **GPT-2** [@gpt2] is used to directly fine-tune on the domain-specific labels, without pre-training on the large-scale corpus of (dialog act, response) pairs. $(\RN{3})$ **HDSA** [@chen-etal-2019-semantically] is a state-of-the-art model on MultiWOZ. It leverages dialog act structures to enable transfer in the multi-domain setting, showing superior performance than SC-LSTM.
[<span style="font-variant:small-caps;">FewShotWOZ</span>]{}
------------------------------------------------------------
Table \[tab:res\_fewshotwoz\] reports the automatic evaluation performance of different methods on [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}. SC-LSTM fails to learn the generation effectively in this few-shot learning setting. The generated utterances are poor in quality and suffer from inaccurate slot rendering. In addition, GPT-2 performs consistently better than SC-LSTM in all the domains. It reveals the feasibility of using a pre-trained language model for NLG, though only limited annotations are available for fine-tuning. Importantly, SC-GPT performs significantly better than GPT and SC-LSTM in terms of both BLEU and ERR. In all the domains, SC-GPT reduces the ERR to a significantly lower level, revealing its strong controllability power. This verifies the importance of pre-training on large annotated dialog data, as SC-GPT learns how to generate utterances specified by the dialog acts accurately.
Table \[tab:human\_eval\_few\] shows the human assessment on [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}. The results exhibit the same trend with automatic evaluation. SC-GPT outperforms GPT-2 and SC-LSTM significantly in both metrics, SC-GPT can better control the generation to convey information in the dialog act while maintaining good fluency. Note that the gap between SC-GPT and human annotation is still large, indicating that the proposed [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} exhibits an under-explored research area, and provides a large space to encourage future research for improvement.
{height="2.6cm"}
\# **Model** **Generated Responses from Different Models**
------------- ----------------- ---------------------------------------------------------------------------------------------------------------------------------------------
1 ***Input DA*** Laptop{inform(name=satellite proteus 84; type=laptop; memory=8 gb; drive=1 tb; weight=2.3 kg)}
\[2.5pt\] ***Reference*** the satellite proteus 84 is a laptop with a 1 tb drive , 8 gb memory and weighs in at 2.3 kg
\[2.5pt\] SC-LSTM the satellite proteus 84 is a laptop with 8 gb , with a 1 tb drive , and is , and is % [ ** ]{}
\[2.5pt\] GPT-2 the satellite proteus 84 laptop has a 1 tb drive , weighs 2.3 kg and is % [ ** ]{}
\[2.5pt\] SC-GPT the satellite proteus 84 is a laptop with 8 gb memory , 1 tb drive , and a weight of 2.3 kg
\[2.5pt\] 2 ***Input DA*** Restaurant{inform\_only\_match(name=marnee thai and thanh long; pricerange=moderate; near=outer sunset; goodformeal=dinner)}
\[2.5pt\] ***Reference*** marnee thai and thanh long are the only 2 moderately priced restaurants near outer sunset that are good for dinner
\[2.5pt\] SC-LSTM marnee thai and thanh long is is moderate restaurant restaurant and good good good dinner % [ ** ]{}
\[2.5pt\] GPT-2 there is a moderately priced restaurant called marnee thai and thanh long that is near the outer sunset area % [ ** ]{}
\[2.5pt\] SC-GPT marnee thai and thanh long is the only restaurant that serves moderately priced food near outer sunset and good for dinner
\[2.5pt\] 3 ***Input DA*** Train{inform(time=50 minutes)}, hotel{request(stars=?; area=?), inform(choice=5)}
\[2.5pt\] ***Reference*** travel time is 50 minutes . i found 5 hotels you may like . do you have a star rating request or an area you prefer
\[2.5pt\] SC-LSTM there are that are 50 minutes . do you have a preferred ?
% [ ** ]{}
HDSA there are that meet your criteria . do you have a preference on the area or star rating ? % [ ** ]{}
\[2.5pt\] SC-GPT there are 5 hotels that meet your criteria . the trip will last 50 minutes . do you have an area preference or star rating you would like ?
\[2.5pt\]
MultiWOZ
--------
The results on MultiWOZ are shown in Table \[tab:res\_multiwoz\]. Following @chen-etal-2019-semantically, Entity F1 [@wen2016network] is used to evaluate the entity coverage accuracy (including all slot values, days, numbers, and reference, ). Again, SC-GPT achieves the best performance on BLEU score. Note that GPT-2 performs similarly with SC-GPT on the full MultiWOZ dataset, this is because MultiWOZ contains 57k utterances, which is large enough for GPT-2 to achieve good performance. The results also confirm that with enough annotated data, conditional language model formulation performs significantly better than HDSA, a strong competitor that leverages graph/tree-structure information to encode dialog acts.
To study how SC-GPT performs with different training data sizes. We further conduct experiments with varying percentages of training data on MultiWOZ, ranging from 0.1% (50 examples) to 50%. As shown in Table \[tab:res\_multiwoz\_pct\], the observations are consistent with [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}. SC-GPT performs consistently better than GPT-2, HDSA, and SC-LSTM for a wide range of dataset sizes, and the improvement is more substantial when the fewer numbers of in-domain labels are used for fine-tuning. Table \[tab:human\_eval\_multiwoz\] shows the human assessment results on MultiWOZ. The results are consistent with the automatic evaluation. It is interesting to see that $(\RN{1})$ the gap between the new state-of-the-art method (SC-GPT ) and human performance on [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} (as shown in Table \[tab:human\_eval\_few\]) is much larger than that on MultiWOZ; $(\RN{2})$ the human rating on the naturalness of SC-GPT is even higher than humans on MultiWOZ, while there is a visible gap on [<span style="font-variant:small-caps;">FewShotWOZ</span>]{}. These results demonstrate that [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} presents a challenging few-shot learning setting, SG-GPT serves as a simple and strong baseline in this setting, and the combined provides a platform for researchers to develop NLG models that are able to generalize to new domains and generate semantically conditioned and fluent responses.
Analysis
--------
We perform detailed analysis to investigate SG-GPT’s *flexibility*, *controllability* and *generalizability*. The test set is split into two subsets - *seen* and *unseen*. If a dialog act of an example appears in the training set, the example is marked as *seen*; otherwise, it is marked as *unseen*. Table \[tab:res\_seen\_unseen\] compares different models on the seen and unseen subsets in the $\mathtt{restaurant}$ domain. SC-GPT yields higher BLEU and lower ERR, and the improvement is more significant on the unseen set. For example, SC-GPT reduces ERR to 4.96, an order of magnitude lower than SC-LSTM and only 1/3 of GPT-2. This demonstrates that SC-GPT generalizes well to novel dialog acts, and is able to precisely ground in them to compose fluent responses. This is further confirmed by the quantitative comparison in Table \[tab:examples\], where we compare the generated utterance examples of different models. While the baseline methods prone to over-generate or miss important slots, SC-GPT can successfully generate fluent natural language utterances that share precise semantic conditions with the ground-truth references.
\# **Model** **Generated Responses from Different Models**
-------------- ------------------- -----------------------------------------------------------------------------------------------------------------
***Original DA*** ack\_makereservation ( price = moderate ; restaurantcusine = chinese ; location = center ; peoplecount = 2 )
\[2.5pt\] ***Reference*** ok making a reservation for restaurant moderate chinese in center for two people?
\[2.5pt\] 1 ***Input DA*** ack\_makereservation ( price = moderate ; restaurantcusine = ; location = ; peoplecount = 2 )
\[2.5pt\] GPT-2 ok making reservation for moderate japanese restaurant in bellevue square for 2 people?
\[2.5pt\] SC-GPT ok making a reservation for a moderate japanese cuisine restaurant in bellevue square for 2 people?
\[2.5pt\] 2 ***Input DA*** ack\_makereservation ( ; price = moderate ; restaurantcusine = ; location = ; peoplecount = 2 )
\[2.5pt\] GPT-2 ok making a reservation for 2 people sakura moderate japanese restaurant located bellevue sqare for 10 minutes?
\[2.5pt\] SC-GPT ok making a reservation for sakura, moderate japanese cuisine in bellevue square for 2 people?
\[2.5pt\] 3 ***Input DA*** ack\_makereservation ( price = ; restaurantcusine = japanese ; ; )
\[2.5pt\] GPT-2 ok making reservation for japanese cuisine expensive?
\[2.5pt\] SC-GPT ok making reservation for expensive japanese cuisine?
\[2.5pt\] \# **Model** **Generated Responses from Different Models**
***Original DA*** ack\_makereservation ( childrenfriendly = true )
\[2.5pt\] ***Reference*** make reservation for a children friendly restaurant ?
\[2.5pt\] 1 ***Input DA*** ack\_makereservation ( ; childrenfriendly = true ; )
\[2.5pt\] GPT-2 make reservation at sakura child friendly restaurant and make table moderate price?
\[2.5pt\] SC-GPT make reservation for restaurant sakura moderate price and children friendly restaurant?
\[2.5pt\]
We further simulate the process when deploying SC-GPT for a new domain, using the examples provided in the RASA dialog toolkit [^10]. We first fine-tune SC-GPT using a few training examples (only 16 instances in this new domain), and then generate utterances based on novel dialog acts that are unseen in training data, shown in Table \[tab:new\_domain\_examples\]. In practice, it is desirable for an NLG system to deal with an extending domain whose dialog acts change dynamically. We simulate the setting by editing the original input dialog acts, such as inserting or deleting a slot, or substituting a slot value.
Since SC-LSTM is infeasible in the setting of an extending domain, we compare SC-GPT with GPT-2. Results show that SC-GPT produces better utterances than GPT-2. SC-GPT can generate reasonably good natural language responses with different combinations of editing operations, showing its high flexibility to generalize to new dialog acts with very limited training data, and produce controllable responses.
Conclusion and Future Work
==========================
In this paper, we have made two major contributions towards developing a more pragmatic NLG module for task-oriented dialog systems: $(\RN{1})$ A new benchmark [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} is introduced to simulate the few-shot learning scenarios with scarce labelled data in real-world applications. $(\RN{2})$ A new model SC-GPT is proposed to endow the NLG module with strong semantically controlling and generalization ability. Empirical results on both [<span style="font-variant:small-caps;">FewShotWOZ</span>]{} and MultiWOZ show that SC-GPT achieves the best overall performance in both automatic and human evaluations. There are two interesting directions for future work. The first is to design mechanisms to generate more interpersonal responses which are proven to help improve user experiences [@li2016diversity; @zhou2018design]. The other is to generalize the generative pre-training idea to all four modules in the dialog system pipeline for end-to-end training. Since these four modules process information in order, one may organize their input/output as segments, and pre-train a segment-level auto-regressive model.
[^1]: [**S**]{}emantically-[**C**]{}onditioned [**G**]{}enerative [**P**]{}re-[**T**]{}raining
[^2]: <https://github.com/pengbaolin/SC-GPT>
[^3]: Project website: <https://aka.ms/scgpt>
[^4]: In some literature, dialog act denotes only the type of system actions, slot-value pairs are defined as meaning representations. Throughout this paper, we follow the usage in @budzianowski2018multiwoz and use dialog acts to indicate system action and associated slot-value pairs.
[^5]: The domains appearing in fine-tuning are excluded.
[^6]: https://rasa.com/
[^7]: https://dev.botframework.com/
[^8]: https://www.microsoft.com/en-us/research/project/conversation-learner/
[^9]: We also experimented using GPT2 with 117M parameters but observed significant poor performance.
[^10]: https://github.com/RasaHQ/rasa/tree/master\
/examples/restaurantbot
|
---
abstract: 'The collective decay of excitons from initial Bose-Einstein condensate state is investigated theoretically. As practically more interesting case we consider excitons of the yellow series in the semiconductor cuprous oxide where we have collective photon and phonon assisted decay of excitons. It is shown that because of intrinsic instability of recoilless two-boson decay of Bose-Einstein condensate, the spontaneously emitted bosonic pairs are amplified leading to an exponential buildup of a macroscopic population into the certain modes. The collective decay rate has a nonlinear dependence on the excitonic density being comparable or larger than Auger recombination loss rate up to the high densities, which makes obtainable its observation. The considering phenomenon can also be applied for the realization of phonon laser.'
author:
- 'H.K. Avetissian'
- 'A.K. Avetissian'
- 'G.F. Mkrtchian'
- 'B.R. Avchyan'
title: 'Collective two-boson decay of excitons in Bose-Einstein condensate and generation of coherent photon-phonon radiation'
---
Introduction
============
Over the past half-century, excitons were considered as notably interesting candidates for Bose–Einstein condensation (BEC), in which collective coherence may lead to intriguing macroscopic quantum phenomena (see, Ref. \[\] and references therein). Exciton being a bound state of an electron and a hole in a semiconductor is a unique physical system with a rather small mass comparable to the free electron mass. This is a crucial advantage from the experimental point of view since the BEC critical temperature of an excitonic gas is much higher than that of an atom gas with the same number density. [@Pethick] However, the BEC was first successfully realized for trapped alkali atoms, [@BEC] which are several thousand times heavier than excitons. The latter provided additional stimulus for realization of BEC for various condensed matter physics of quasiparticles. In this context, it is worthy to mention realization of BEC of quasiparticles, known as exciton–polaritons, [@Kasp] existing even at room-temperature. [@BECS]
Among the variety of bosonic quasiparticles, the excitons of the yellow series in the semiconductor cuprous oxide ($\mathrm{Cu}_{2}\mathrm{O}$) are still considered as the most promising candidates for pure excitonic BEC. [@Rev1; @Rev2; @Rev3] Experiments in this direction have been done since 1986, [@Exp1; @Exp2; @Exp3; @Exp4; @Exp5] and continued up to present [R1,R2,R3,R4,R5,R6,R7,R8]{} due to several favorable features of excitons in $%
\mathrm{Cu}_{2}\mathrm{O}$. First, the large binding energy of $0.15\
\mathrm{eV}$ which increases the Mott density up to $10^{19}\mathrm{cm}^{-3}$. Second, the ground state of this series splits into the threefold degenerate orthoexciton and the non-degenerate paraexciton. The latter is the lowest energetic state lying below the orthoexciton states. Due to the selection rules one photon decay of paraexciton is forbidden. Its decay is only possible via optical phonon and photon resulting in a long lifetime. [@Mys; @Shi] The latter is in the microsecond range during which BEC may be reached. To achieve excitonic BEC one should create a dense gas of excitons either in a bulk crystal or in a potential trap. However experiments [@Exp1; @Exp2; @Exp3; @Exp4; @Exp5; @R3; @R4; @R5; @R6] did not demonstrate conclusively excitonic BEC. The main reason for this failure is connected with the fact that the lifetime of excitons in $\mathrm{Cu}_{2}\mathrm{O}$ decreases significantly at high gas densities. This effect has been attributed to an Auger recombination process between two excitons resulting in a loss rate $\Gamma _{\mathrm{A}}=\alpha n$, where $\alpha $ is the Auger constant and $n$ is the exciton gas density. However, there is no general consent on the value of Auger constant. The reported values for $\alpha $ range are from $10^{-20}\ \mathrm{cm}^{3}\mathrm{ns}^{-1}$ to $1.8\times
10^{-16}\ \mathrm{cm}^{3}\mathrm{ns}^{-1}$ and differ for orto- and para-excitons. [@D1; @D2; @D3; @D4; @D5; @D6]
As was mentioned above, the isolated exciton in $\mathrm{Cu}_{2}\mathrm{O}$ is unstable and decays into photon and phonon. Due to BEC coherence, one can expect collective radiative effects at the decay of a large number of excitons. The latter may be a tool that evidences the state of the BEC, as well as, it may significantly reduce the lifetime of the BEC state. Such an effect has been revealed for the positronium atoms, [@Mer1; @Mer2] which in some sense resembles excitons. It has been shown that at the coupling of two coherent ensembles of bosons – the BEC of positronium atoms and photons there is an instability at which, starting from the vacuum state of the photonic field, the expectation value of the photon’s mode occupation grows exponentially for a narrow interval of frequencies. For the excitons in $%
\mathrm{Cu}_{2}\mathrm{O}$ one will have coupling between three bosonic fields and it is of interest to investigate how excitonic BEC burst into photons/phonons.
In this paper collective decay of excitons from initial Bose-Einstein condensate state is investigated arising from the second quantized formalism. It is shown that because of intrinsic instability of recoilless two-boson decay of Bose-Einstein condensate, the spontaneously emitted bosonic pairs are amplified, leading to an exponential buildup of a macroscopic population into certain modes. The exponential growth rate has a nonlinear dependence on the BEC density and it is quite large for the experimentally achievable densities. For the elongated condensate, one can reach self-amplification of the end-fire-modes. With the initial monochromatic photonic beam, one can generate the monochromatic phononic beam. Hence, the considered phenomenon may also be applied for realization of phonon laser.
The paper is organized as follows. In Sec. II the main Hamiltonian is introduced. In Sec. III spontaneous decay of exciton is analyzed. In Sec. IV we consider intrinsic instability of recoilless collective two-boson decay of excitonic BEC. Finally, conclusions are given in Sec. V.
Basic Hamiltonian
=================
We start our study with the construction of the Hamiltonian which governs the quantum dynamics of considered process. The total Hamiltonian consists of four parts: $$\widehat{H}=\widehat{H}_{\mathrm{exc}}+\widehat{H}_{\mathrm{phot}}+\widehat{H%
}_{\mathrm{phon}}+\widehat{H}_{\mathrm{d}}. \label{fH}$$Here the first part is the Hamiltonian of free excitons:$$\widehat{H}_{\mathrm{exc}}=\int d\Phi _{\mathbf{p}}\mathcal{E}_{\text{%
\textsc{e}}}\left( \mathbf{p}\right) \widehat{\text{\textsc{e}}}_{\mathbf{p}%
}^{+}\widehat{\text{\textsc{e}}}_{\mathbf{p}}, \label{H_exc}$$where $\widehat{\text{\textsc{e}}}_{\mathbf{p}}^{+}$ ($\widehat{\text{%
\textsc{e}}}_{\mathbf{p}}$) is the creation (annihilation) operator for an exciton. These operators satisfy the Bosonic commutation rules for a relatively small number density $n$ of excitons, that is at $n<n_{M}$, where $n_{M}$ is the Mott density. [@Moskal] For the integration in phase-space we have introduced the notation $d\Phi _{\mathbf{p}}=\mathcal{V}%
d^{3}\mathbf{p}/\left( 2\pi \right) ^{3}$ ($\mathcal{V}$ is the quantization volume). Then, $\mathcal{E}_{\text{\textsc{e}}}\left( \mathbf{p}\right)
=\hbar ^{2}\mathbf{p}^{2}/2m_{\ast }+\hbar \omega _{\mathrm{exc}}$ is the total energy of exciton with the momentum $\hbar \mathbf{p}$ of the center-of-mass motion, $m_{\ast }$ is an exciton mass, $\mathcal{E%
}_{in}$ is the exciton internal energy ($\hbar \omega _{\mathrm{exc}}=%
\mathcal{E}_{G}-\mathcal{E}_{b}$, in terms of the band-gap difference $%
\mathcal{E}_{G}$ and the binding energy $\mathcal{E}_{b}$).
The second term in Eq. (\[fH\]) is the Hamiltonian of the free photons$$\widehat{H}_{\mathrm{phot}}=\int d\Phi _{\mathbf{k}}\hbar \omega \left(
\mathbf{k}\right) \widehat{c}_{\mathbf{k}}^{+}\widehat{c}_{\mathbf{k}},
\label{H_ph}$$where $\widehat{c}_{\mathbf{k}}$ ($\widehat{c}_{\mathbf{k}}^{+}$) is the annihilation (creation) operator of the photon with the momentum $\mathbf{k}$ and dispersion relation $\omega =\omega \left( \mathbf{k}\right) $**.** The third term in Eq. (\[fH\]) is the Hamiltonian of the free phonons with annihilation (creation) operator $\widehat{b}_{\mathbf{q}}$ ($\widehat{b}_{%
\mathbf{q}}^{+}$): $$\widehat{H}_{\mathrm{phon}}=\int d\Phi _{\mathbf{q}}\hbar \omega _{\mathrm{ph%
}}\left( \mathbf{q}\right) \widehat{b}_{\mathbf{q}}^{+}\widehat{b}_{\mathbf{q%
}}. \label{H_op}$$The last term in Eq. (\[fH\])$$\widehat{H}_{\mathrm{d}}=\int d\Phi _{\mathbf{q}}\int d\Phi _{\mathbf{p}}%
\left[ \frac{\hbar \mathcal{M}\left( \mathbf{q},\mathbf{p}\right) }{\mathcal{%
V}^{1/2}}\widehat{b}_{\mathbf{q}}^{+}\widehat{c}_{\mathbf{p-q}}^{+}\widehat{%
\text{\textsc{e}}}_{\mathbf{p}}\right]$$$$+\left. \frac{\hbar \mathcal{M}^{\ast }\left( \mathbf{q},\mathbf{p}\right) }{%
\mathcal{V}^{1/2}}\widehat{\text{\textsc{e}}}_{\mathbf{p}}^{+}\widehat{c}_{%
\mathbf{p-q}}\widehat{b}_{\mathbf{q}}\right] \label{H_2j}$$is the Hamiltonian of the two-boson decay of an exciton. [@Shi] Here we assume that the direct recombination of electrons and holes is very weak and the main decay process is a phonon-assisted recombination process in which an exciton decays emitting an optical phonon, as well as, a photon. The amplitude $\mathcal{M}\left( \mathbf{q},\mathbf{p}\right) $ for an exciton decay can be calculated by the Feynman diagrams.
Spontaneous decay of an exciton
===============================
Before considering collective decay of excitons it will be useful to consider spontaneous decay of a single exciton from the quantum dynamic point of view. For the spontaneous decay we consider initial condition in which the photonic and phononic fields begin in the vacuum state, while excitonic field is prepared in a Fock state with the one exciton in the rest ($\mathbf{p=0}$). Such state can be represented as $|\Psi \left( 0\right)
\rangle =|0\rangle _{\mathrm{phon}}\otimes |0\rangle _{\mathrm{phot}}\otimes
\widehat{\text{\textsc{e}}}_{0}^{+}|0\rangle _{\mathrm{exc}}$. Then the state vector for times $t>0$ is just given by the expansion$$|\Psi \rangle =C_{0}e^{-\frac{i}{\hbar }\mathcal{E}_{\text{\textsc{e}}%
}\left( 0\right) t}|0\rangle _{\mathrm{phon}}\otimes |0\rangle _{\mathrm{phot%
}}\otimes \widehat{\text{\textsc{e}}}_{0}^{+}|0\rangle _{\mathrm{exc}}+\int
d\Phi _{\mathbf{k}}d\Phi _{\mathbf{k}^{\prime }}$$$$\times C_{\mathbf{k};\mathbf{k}^{\prime }}\left( t\right) e^{-i\left( \omega
_{\mathrm{ph}}\left( \mathbf{k}^{\prime }\right) +\omega \left( \mathbf{k}%
\right) \right) t}\widehat{b}_{\mathbf{k}^{\prime }}^{+}|0\rangle _{\mathrm{%
phon}}\otimes \widehat{c}_{\mathbf{k}}^{+}|0\rangle _{\mathrm{phot}}\otimes
|0\rangle _{\mathrm{exc}}, \label{Psit}$$where $C_{\mathbf{k};\mathbf{k}^{\prime }}\left( t\right) $ is the probability amplitude for the photonic and phononic fields to be in the single-particle state, while excitonic field in the vacuum state. From the Schrödinger equation one can obtain evolution equations:$$i\frac{\partial C_{\mathbf{k};\mathbf{k}^{\prime }}\left( t\right) }{%
\partial t}=\frac{\mathcal{M}\left( \mathbf{k}^{\prime },\mathbf{0}\right) }{%
\mathcal{V}^{1/2}}C_{0}\left( t\right) \frac{\left( 2\pi \right) ^{3}}{%
\mathcal{V}}\delta \left( \mathbf{k}+\mathbf{k}^{\prime }\right)$$$$\times e^{i\left( \omega _{\mathrm{ph}}\left( \mathbf{k}^{\prime }\right)
+\omega \left( \mathbf{k}\right) -\omega _{\mathrm{exc}}\right) t}.
\label{ev1}$$Then, according to perturbation theory we take $C_{0}\left( t\right) \simeq 1
$, and for the amplitude $C_{\mathbf{k};\mathbf{k}^{\prime }}\left(
t\rightarrow \infty \right) $ from Eq. (\[ev1\]) we obtain$$C_{\mathbf{k};\mathbf{k}^{\prime }}=\frac{\mathcal{M}\left( \mathbf{k}%
^{\prime },\mathbf{0}\right) }{i\mathcal{V}^{1/2}}\frac{\left( 2\pi \right)
^{4}}{\mathcal{V}}$$$$\times \delta \left( \omega _{\mathrm{ph}}\left( \mathbf{k}^{\prime }\right)
+\omega \left( \mathbf{k}\right) -\omega _{\mathrm{exc}}\right) \delta
\left( \mathbf{k}+\mathbf{k}^{\prime }\right) . \label{pert3}$$For the decay of an exciton the modes laying in the narrow interval of wavenumbers are responsible. Hence, for the dispersion relations we assume $%
\omega _{\mathrm{ph}}\left( \mathbf{k}\right) =\mathrm{const}\equiv \omega _{%
\mathrm{ph}}$ and $\omega \left( \mathbf{k}\right) =kc_{l}$, where $c_{l}$-is the light speed in a semiconductor. Then returning to expansion ([Psit]{}), one can write$$|\Psi \rangle \simeq C_{0}e^{-i\omega _{\mathrm{exc}}t}|0\rangle _{\mathrm{%
phon}}\otimes |0\rangle _{\mathrm{phot}}\otimes |1_{\mathbf{0}}\rangle _{%
\mathrm{exc}}$$$$+\frac{\mathcal{V}^{1/2}\mathcal{M}\left( k_{0},0\right) k_{0}^{2}}{i\left(
2\pi \right) ^{2}c_{l}}e^{-i\omega _{\mathrm{exc}}t}$$$$\times |0\rangle _{\mathrm{exc}}\otimes \int d\widehat{\mathbf{k}}|1_{%
\mathbf{k}}\rangle _{\mathrm{phon}}\otimes |1_{-\mathbf{k}}\rangle _{\mathrm{%
phot}}, \label{fs}$$ where $\widehat{\mathbf{k}}=\mathbf{k/}\left\vert \mathbf{k}\right\vert $, and $$k_{0}=\frac{\omega _{\mathrm{exc}}-\omega _{\mathrm{ph}}}{c_{l}}. \label{k0}$$Hear we have taken into account that the decay amplitude does not depend on the direction of $\mathbf{k}$ and**,** as a result, the final state (\[fs\]) resulting from an exciton decay is a superposition of the states of oppositely propagating photon and phonon with the given momentum $k_{0}$. That is, we have recoilless two-boson decay of exciton, which is crucial for the development of instability in BEC where the lowest energy single particle state is occupied. For the decay rate one can write $$\Gamma =\int d\Phi _{\mathbf{k}}d\Phi _{\mathbf{k}^{\prime }}\frac{%
\left\vert C_{\mathbf{k};\mathbf{k}^{\prime }}\right\vert ^{2}}{t_{\mathrm{%
int}}},$$where $t_{\mathrm{int}}$ is the interaction time. With the help of Eq. ([pert3]{}) we obtain the well known result: $$\Gamma =\frac{\mathcal{M}^{2}}{\pi c_{l}}k_{0}^{2}. \label{456}$$The radiative lifetime of an isolated exciton is $\Gamma ^{-1}$.
Collective decay
================
For analysis of the collective photon-phonon decay of excitons we will use Heisenberg representation, where the evolution operators are given by the following equation $$i\frac{\partial \widehat{L}}{\partial t}=\left[ \widehat{L},\widehat{H}%
\right] , \label{Heis}$$and the expectation values are determined by the initial wave function $\Psi
_{0}$:$$\left\langle \widehat{L}\right\rangle =\langle \Psi _{0}|\widehat{L}|\Psi
_{0}\rangle .$$We will assume that the excitonic field starts up in the Bose-Einstein condensate state, while for photonic and phononic fields we will consider both vacuum state and states with nonzero mean number of particles. Taking into account Hamiltonian (\[fH\]) from Eq. (\[Heis\]) we obtain a set of equations:$$i\frac{\partial \widehat{c}_{\mathbf{k}}}{\partial t}=\omega \left( \mathbf{k%
}\right) \widehat{c}_{\mathbf{k}}+\int d\Phi _{\mathbf{p}}\frac{\mathcal{M}%
\left( \mathbf{p-k},\mathbf{p}\right) }{\mathcal{V}^{1/2}}\widehat{b}_{%
\mathbf{p-k}}^{+}\widehat{\text{\textsc{e}}}_{\mathbf{p}}, \label{M1}$$$$i\frac{\partial \widehat{b}_{\mathbf{k}}}{\partial t}=\omega _{\mathrm{ph}%
}\left( \mathbf{k}\right) \widehat{b}_{\mathbf{k}}+\int d\Phi _{\mathbf{p}}%
\frac{\mathcal{M}\left( \mathbf{k},\mathbf{p}\right) }{\mathcal{V}^{1/2}}%
\widehat{c}_{\mathbf{p-k}}^{+}\widehat{\text{\textsc{e}}}_{\mathbf{p}},
\label{M2}$$$$i\frac{\partial \widehat{\text{\textsc{e}}}_{\mathbf{p}}}{\partial t}=\hbar
^{-1}\mathcal{E}_{\text{\textsc{e}}}\left( \mathbf{p}\right) \widehat{\text{%
\textsc{e}}}_{\mathbf{p}}+\int d\Phi _{\mathbf{q}}\frac{\mathcal{M}^{\ast
}\left( \mathbf{q},\mathbf{p}\right) }{\mathcal{V}^{1/2}}\widehat{c}_{%
\mathbf{p-q}}\widehat{b}_{\mathbf{q}}. \label{M3}$$
These equations are a nonlinear set of equations with the photonic, phononic and excitonic fields’ operators defined self-consistently. As we are interested in the quantum dynamics of considered system in the presence of instabilities we can decouple the excitonic field treating the dynamics of photonic and phononic fields. For this propose we just use the Bogolubov approximation. If the lowest energy single particle state has a macroscopic occupation, we can separate the field operators $\widehat{\text{\textsc{e}}}%
_{\mathbf{p}}$ into the condensate term and the non-condensate components, i.e. the operator $\widehat{\text{\textsc{e}}}_{\mathbf{p}}$ in Eqs. ([M1]{}) and (\[M2\]) is replaced by the c-number as follow$$\widehat{\text{\textsc{e}}}_{\mathbf{p}}=\sqrt{n_{0}}\frac{\left( 2\pi
\right) ^{3}}{\mathcal{V}^{1/2}}\delta \left( \mathbf{p}\right) e^{-i\omega
_{\mathrm{exc}}t}, \label{Bogol}$$where $n_{0}$ is the number density of excitons in the condensate. Making Bogoloubov approximation we arrive at a linear set of the Heisenberg equations$$i\frac{\partial \widehat{c}_{-\mathbf{k}}}{\partial t}=\omega \left(
k\right) \widehat{c}_{-\mathbf{k}}+\chi \left( k\right) \widehat{b}_{\mathbf{%
k}}^{+}e^{-i\omega _{\mathrm{exc}}t}, \label{L1}$$$$i\frac{\partial \widehat{b}_{\mathbf{k}}}{\partial t}=\omega _{\mathrm{ph}}%
\widehat{b}_{\mathbf{k}}+\chi \left( k\right) \widehat{c}_{\mathbf{-k}%
}^{+}e^{-i\omega _{\mathrm{exc}}t}, \label{L2}$$which couples photon modes with momentum $\mathbf{k}$ to the phonons with momentum $-\mathbf{k}$. The coupling constant is $$\chi \left( k\right) =\sqrt{n_{0}}\mathcal{M}\left( k,0\right) . \label{CC}$$
Equations (\[L1\]) and (\[L2\]) compose a set of linearly coupled operator equations that can be solved by the method of characteristics whose eigenfrequencies define the temporal dynamics of the bosonic fields. The existence of an eigenfrequency with an imaginary part would indicate the onset of instability at which the initial spontaneously emitted bosonic pairs are amplified leading to an exponential buildup of a macroscopic mode population. Solving Eqs. (\[L1\]) and (\[L2\]), we obtain$$\widehat{b}_{\mathbf{k}}^{+}=e^{i\left( \omega _{\mathrm{ph}}-\frac{\delta
\left( k\right) }{2}\right) t}\left\{ \widehat{b}_{\mathbf{k}}^{+}\left(
0\right) \cosh \left( \sigma \left( k\right) t\right) +\frac{i}{2\sigma
\left( k\right) }\right.$$$$\left. \times \left( \delta \left( k\right) \widehat{b}_{\mathbf{k}%
}^{+}\left( 0\right) +2\chi ^{\ast }\left( k\right) \widehat{c}_{-\mathbf{k}%
}\left( 0\right) \right) \sinh \left( \sigma \left( k\right) t\right)
\right\} , \label{S1}$$$$\widehat{c}_{-\mathbf{k}}=e^{i\left( \frac{\delta \left( k\right) }{2}%
-\omega \left( k\right) \right) t}\left\{ \widehat{c}_{-\mathbf{k}}\left(
0\right) \cosh \left( \sigma \left( k\right) t\right) -\frac{i}{2\sigma
\left( k\right) }\right.$$$$\left. \times \left( 2\chi \left( k\right) \widehat{b}_{\mathbf{k}%
}^{+}\left( 0\right) +\delta \left( k\right) \widehat{c}_{-\mathbf{k}}\left(
0\right) \right) \sinh \left( \sigma \left( k\right) t\right) \right\} ,
\label{S2}$$where$$\delta \left( k\right) =\omega \left( k\right) -\omega _{\mathrm{exc}%
}+\omega _{\mathrm{ph}} \label{det}$$is the resonance detuning, and $$\sigma \left( k\right) =\sqrt{\left\vert \chi \left( k\right) \right\vert
^{2}-\frac{\delta ^{2}\left( k\right) }{4}}. \label{CCC}$$As is seen from Eqs. (\[S1\])-(\[CCC\]), the condition for the dynamic instability is: $$\left\vert \chi \left( k\right) \right\vert >\frac{\left\vert \delta \left(
k\right) \right\vert }{2}$$leading to the exponential growth of the modes in the narrow interval of wavenumbers$$\omega _{\mathrm{exc}}-\omega _{\mathrm{ph}}-2\left\vert \chi \left(
k_{0}\right) \right\vert <kc_{l}<\omega _{\mathrm{exc}}-\omega _{\mathrm{ph}%
}+2\left\vert \chi \left( k_{0}\right) \right\vert . \label{inter}$$For the interval (\[inter\]) we find that the expectation value of the photonic and phononic modes occupations grow exponentially:$$N_{\mathrm{phot}}\left( \mathbf{k,}t\right) =\langle \Psi _{0}|\widehat{c}_{%
\mathbf{k}}^{+}\widehat{c}_{\mathbf{k}}|\Psi _{0}\rangle$$$$=N_{\mathrm{phot}}\left( \mathbf{k,}0\right) \left( \cosh ^{2}\left( \sigma
\left( k\right) t\right) +\frac{\delta ^{2}\left( k\right) }{4\sigma
^{2}\left( k\right) }\sinh ^{2}\left( \sigma \left( k\right) t\right) \right)$$$$+\frac{\left\vert \chi \left( k\right) \right\vert ^{2}}{\sigma ^{2}\left(
k\right) }\left( 1+N_{\mathrm{phon}}\left( -\mathbf{k,}0\right) \right)
\sinh ^{2}\left( \sigma \left( k\right) t\right) , \label{Exp1}$$$$N_{\mathrm{phon}}\left( \mathbf{k,}t\right) =\langle \Psi _{0}|\widehat{b}_{%
\mathbf{k}}^{+}\widehat{b}_{\mathbf{k}}|\Psi _{0}\rangle$$$$=N_{\mathrm{phon}}\left( \mathbf{k,}0\right) \left( \cosh ^{2}\left( \sigma
\left( k\right) t\right) +\frac{\delta ^{2}\left( k\right) }{4\sigma
^{2}\left( k\right) }\sinh ^{2}\left( \sigma \left( k\right) t\right) \right)$$$$+\frac{\left\vert \chi \left( k\right) \right\vert ^{2}}{\sigma ^{2}\left(
k\right) }\left( 1+N_{\mathrm{phot}}\left( -\mathbf{k,}0\right) \right)
\sinh ^{2}\left( \sigma \left( k\right) t\right) . \label{Exp2}$$For the central wavenumber ($\delta \left( k_{0}\right) =0$) the exponential growth rate is $$G=2\chi \left( k_{0}\right) =2\sqrt{n_{0}}\mathcal{M}\left( k_{0},0\right) .
\label{gain1}$$Taking into account Eq. (\[CC\]) and derived expression (\[456\]) for the decay rate, we obtain compact expression for the exponential growth rate:$$G=\sqrt{\frac{4\pi n_{0}c_{l}\Gamma }{k_{0}^{2}}}. \label{main}$$
As is seen from Eqs. (\[Exp1\]) and (\[Exp2\]), we have an exponential buildup of a macroscopic mode population even for the initial vacuum state $%
N_{\mathrm{phot}}\left( \mathbf{k,}0\right) =N_{\mathrm{phon}}\left( \mathbf{%
k,}0\right) =0$. In this case from Eqs. (\[Exp1\]) and (\[Exp2\]) we have$$N_{\mathrm{phot}}\left( \mathbf{k,}t\right) =N_{\mathrm{phon}}\left( \mathbf{%
k,}t\right) =\frac{4\left\vert \chi \left( k\right) \right\vert ^{2}}{%
4\left\vert \chi \left( k\right) \right\vert ^{2}-\delta ^{2}\left( k\right)
}$$$$\times \left( e^{\sqrt{4\left\vert \chi \left( k\right) \right\vert
^{2}-\delta ^{2}\left( k\right) }t}+e^{-\sqrt{4\left\vert \chi \left(
k\right) \right\vert ^{2}-\delta ^{2}\left( k\right) }t}-2\right) .
\label{izo}$$
We have solved the issue considering uniform BEC without boundary conditions and, as a consequence, according to Eq. (\[izo\]) we have an isotropic exponential gain. Due to the BEC coherence, here we have an absolute instability, i.e., the number of photons/phonons grows at every point within a BEC and the gain is scaled as $\sqrt{n_{0}}$. Here the excitonic BEC burst into photons and phonons. Note, that our approximation is valid for the interaction times $t_{\mathrm{int}}$ at which the total number of photons and phonons are much smaller than the number of excitons in BEC: $N_{\mathrm{%
phot}},N_{\mathrm{phon}}<<N_{\mathrm{exc}}$.
For laserlike action, i.e., for directional radiation, one should take an elongated shape of the BEC. In this case, boundary conditions define interaction time. This can be incorporated into the derived equation ([L1]{}) and (\[L2\]) by introducing mode damping. The latter is simply due to the propagation of the bosonic fields, which escapes from the active medium and is inversely proportional to the transit time of a photon in the active medium. This transit time strictly depends on the propagation direction. The latter is equivalent to the finite interaction time strictly depending on the shape of the BEC.
For the directional radiation decay, one can also consider initial photonic or phononic beam. For the initial monochromatic photonic beam, in the result of the collective decay, one will have backscattered monochromatic phononic beam. Thus, one can realize a coherent source of phonons applying resonant laser beam.
Let us make explicit calculations for the initial photonic beam with the distribution: $$N_{\mathrm{phot}}\left( \mathbf{k,}0\right) =N_{0}\exp \left( -\frac{%
k_{x}^{2}+k_{y}^{2}}{2\delta ^{2}}\right) \exp \left( -\frac{\left(
k_{z}-k_{0}\right) ^{2}}{2\delta ^{2}}\right) ,$$where $N_{0}>>1$ and $\delta $ is the width of distribution in the momentum space $\delta <<k_{0}$. In this case, for the angular distribution of the phonon number density (we assume $N_{\mathrm{phon}}\left( \mathbf{k,}%
0\right) <<1$) we have$$\frac{dn_{\mathrm{phon}}}{d\vartheta }\simeq \frac{N_{0}}{\left( 2\pi
\right) ^{2}}\int_{k_{0}-\frac{G}{c_{l}}}^{k_{0}+\frac{G}{c_{l}}}dk\frac{%
k^{2}\left\vert \chi \left( k\right) \right\vert ^{2}}{\sigma ^{2}\left(
k\right) }\sin \vartheta$$$$\times \exp \left( -\frac{k^{2}\sin ^{2}\vartheta }{2\delta ^{2}}-\frac{%
\left( k\cos \vartheta +k_{0}\right) ^{2}}{2\delta ^{2}}\right) \sinh
^{2}\left( \sigma \left( k\right) t_{\mathrm{int}}\right) , \label{nphon}$$where $t_{\mathrm{int}}$ is the interaction time of the photonic beam with excitonic BEC. As is seen from Eq. (\[nphon\]), phonons are radiated in the opposite to the photonic beam direction and have peak near $\vartheta
\simeq \pi $. For the phonon number density one should integrate Eq. ([nphon]{}) over $\vartheta $. Taking into account that $G<<k_{0}c_{l}$, we obtain: $$n_{\mathrm{phon}}\simeq \frac{\delta ^{2}GN_{0}}{2\pi ^{2}c_{l}}F\left( Gt_{%
\mathrm{int}}\right) , \label{total}$$where $$F\left( Gt_{\mathrm{int}}\right) =\int_{0}^{1}dx\frac{\sinh ^{2}\left( \frac{%
Gt_{\mathrm{int}}}{2}\sqrt{1-x^{2}}\right) }{1-x^{2}} \label{fanct}$$is the amplification factor. The latter is a rapidly increasing function, displayed in Fig. 1.
{width=".50\textwidth"}
Let us consider the parameters required for observation of the considered effect for the excitons of the yellow series in the semiconductor cuprous oxide. In Cu$_{2}$O, the radiative lifetime of an isolated exciton is $%
\Gamma ^{-1}\approx 10^{-5}\mathrm{s}$, the refractive index is approximately $3$ ($c_{l}\approx 10^{10}\mathrm{cm/s}$), the energy of the optical phonon is $10^{-2}$ $\mathrm{eV}$, the energy gap $\mathcal{E}%
_{G}\approx 2$ $\mathrm{eV}$ and the binding energy $\mathcal{E}_{b}\approx
0.15$ $\mathrm{eV}$. Thus,for the exponential growth rate we have:$$G\simeq \left( \frac{n_{0}}{10^{18}\mathrm{cm}^{-3}}\right) ^{1/2}\times
4\times 10^{11}\mathrm{s}^{-1}. \label{Cu2O}$$As is seen from Eq. (\[Cu2O\]), the growth rate is quite large $G\simeq
4\times 10^{11}\mathrm{s}^{-1}$for the experimentally achievable densities $%
n_{0}=10^{18}\mathrm{cm}^{-3}$. Note that collective growth rate is larger than Auger recombination loss rate $\Gamma _{\mathrm{A}}=\alpha n_{0}$ up to high densities $n_{0}<4\times 10^{18}\mathrm{cm}^{-3}$. Let us also estimate possible parameters of coherent phononic beam generated by the photon beam. Taking density $n_{0}=10^{18}\mathrm{cm}^{-3}$ and interaction time $t_{%
\mathrm{int}}\approx 50\ \mathrm{ps\ }$from Fig.1 one can define $F\left(
Gt_{\mathrm{int}}\simeq 20\right) \simeq 3.5\times 10^{7}$. From Eq. ([total]{}) for the phonon number density we have $n_{\mathrm{phon}}\simeq
N_{0}\times 5.6\times 10^{12}\mathrm{cm}^{-3}$. Thus, considered phenomenon may be applied for the realization of a phonon laser.
Conclusion
==========
In conclusion, we have studied the collective two-boson decay of excitons, arising from the second quantized formalism. It was shown that BEC state is unstable because of recoilless two-boson decay. The spontaneously emitted bosonic pairs are amplified leading to an exponential buildup of a macroscopic population into resonant modes. As a practically more interesting case, we have considered the decay of excitons of the yellow series in the semiconductor cuprous oxide, where BEC burst into the photons and phonons with the collective growth rate proportional to the square root of the BEC density. Calculations show that the collective decay rate is comparable or larger than Auger recombination loss rate up to the high densities. Hence, it can be used as a tool that evidences the formation of BEC state in Cu$_{2}$O. We have also studied another application of considered effect – a possible source for generation of coherent phonon beam. For the latter propose one can take an elongated condensate where self-amplification of the end-fire-modes takes place. Otherwise, applying a resonant photonic beam one can generate backscattered intense coherent phonon beam.
This work was supported by the State Committee of Science MES RA, in the frame of the research project SCS 15T-1C013.
[99]{} S. A. Moskalenko and D. W. Snoke, Bose–Einstein Condensation of Excitons and Biexcitons and Coherent Nonlinear Optics with Excitons (Cambridge Univ. Press, New York, 2000).
C. Pethick and H. Smith, *Bose-Einstein Condensation in Dilute Gases*, 2nd ed., Vol. 2 (Cambridge University Press, Cambridge, 2001).
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science **269**, 198 (1995); K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. **75**, 3969 (1995).
J. Kasprzak et all, Nature **443**, 409-414 (2006).
S. O. Demokritov et al., Nature **443**, 430 (2006); J. D. Plumhof et al., Nature Materials **13**, 248 (2014).
D. W. Snoke, Science **298**, 1368 (2002).
D. W. Snoke, Adv. Condens. Matter Phys. **2011**, 938609 (2011).
D. Snoke and G. M. Kavoulakis, Reports on Progress in Physics **77**, 116501 (2014).
D. P. Trauernicht, J. P.Wolfe, and A.Mysyrowicz, Phys. Rev. B **34**, 2561 (1986).
D. W. Snoke, J. P. Wolfe, and A. Mysyrowicz, Phys. Rev. Lett. 64, 2543 (1990); Phys. Rev. B **41**, 11171 (1990).
D. W. Snoke, J. L. Lin, and J. P. Wolfe, Phys. Rev. B **43**, 1226 (1991).
J. L. Lin and J. P. Wolfe, Phys. Rev. Lett. **71**, 1222 (1993).
E. Fortin, S. Fafard, and A. Mysyrowicz, Phys. Rev. Lett. **70**, 3951 (1993).
K. Yoshioka, T. Ideguchi, A. Mysyrowicz, and M. Kuwata-Gonokami, Phys. Rev. B **82**, 041201(R) (2010).
C. Sandfort, J. Brandt, C. Finke, D. Frohlich, M. Bayer, H. Stolz, and N. Naka, Phys. Rev. B **84**, 165215 (2011).
K. Yoshioka, E. Chae, and M. Kuwata-Gonokami, Nat. Commun. **2**, 328 (2011).
R. Schwartz, N. Naka, F. Kieseling, and H. Stolz, New J. Phys. **14**, 023054 (2012).
K. Yoshioka and M. Kuwata-Gonokami, New J. Phys. **14**, 055024 (2012).
K. Yoshioka, Y. Morita, K. Fukuoka, M. Kuwata-Gonokami, Phys. Rev. B **88**, 041201(R) (2013).
J. P. Wolfe, J. I. Jang, New J. Phys. **16**, 123048 (2014).
K. Yoshioka and M. Kuwata-Gonokami, Phys. Rev. B **91**, 195207 (2015).
A. Mysyrowicz, D. Hulin, and A. Antonetti, Phys. Rev. Lett. **43**, 1123 (1979).
H. Shi, G. Verechaka, and A. Griffin, Phys. Rev. B **50**, 1119 (1994).
D. Hulin, A. Mysyrowicz, and C. Benoit a la Guillaume, Phys. Rev. Lett. **45**, 1970 (1980).
K. E. O’Hara, J. R. Gullingsrud, and J. P. Wolfe, Phys. Rev. B **60,** 10872 (1999).
J. T. Warren, K. E. O’Hara, and J. P. Wolfe, Phys. Rev. B **61**, 8215 (2000).
J. P. Wolfe and J. I. Jang, Solid State Commun. **134**, 143 (2005).
K. Yoshioka, T. Ideguchi, A. Mysyrowicz, and M. Kuwata-Gonokami, Phys. Rev. B **82**, 041201(R) (2010).
G. M. Kavoulakis and G. Baym, Phys. Rev. B **54**, 16625 (1996).
H. K. Avetissian, A. K. Avetissian, and G. F. Mkrtchian, Phys. Rev. Lett. **113**, 023904 (2014).
H. K. Avetissian, A. K. Avetissian, and G. F. Mkrtchian, Phys. Rev. A. **92**, 023820 (2015).
|
---
abstract: 'We consider a composite convex minimization problem associated with regularized empirical risk minimization, which often arises in machine learning. We propose two new stochastic gradient methods that are based on stochastic dual averaging method with variance reduction. Our methods generate a sparser solution than the existing methods because we do not need to take the average of the history of the solutions. This is favorable in terms of both interpretability and generalization. Moreover, our methods have theoretical support for both a strongly and a non-strongly convex regularizer and achieve the best known convergence rates among existing nonaccelerated stochastic gradient methods.'
author:
- 'Tomoya Murata[^1]'
- 'Taiji Suzuki[^2]'
bibliography:
- 'bibsvrda.bib'
title:
- Stochastic dual averaging methods using variance reduction techniques for regularized empirical risk minimization problems
- Appendix
---
Introduction {#intro}
============
We consider the following composite convex minimization problem: $$\label{problem}
\underset{x \in \mathbb{R}^d}{\mathrm{min}}\ \ \{P(x)\overset{\mathrm{def}}{=}F(x)+R(x)\},$$ where $F(x)=\frac{1}{n}\sum_{i=1}^n f_{i}(x)$. Here each $f_{i}:\mathbb{R}^d \to \mathbb{R}$ is an $L_i$-smooth convex function and $R:\mathbb{R}^d \to \mathbb{R}$ is a relatively simple and (possibly) nondifferentiable convex function. Problems of this form often arise in machine learning and are known as regularized empirical risk minimization.
A traditional method for solving (\[problem\]) is the (proximal) gradient descent (GD) method. The GD algorithm is very simple and intuitive and achieves a linear convergence rate for a strongly convex regularizer. However, in typical machine learning tasks, the number $n$ can be very large, and then the iteration cost of GD can be quite expensive.
A popular alternative for solving (\[problem\]) is the stochastic gradient descent (SGD) method [@singer2009efficient; @hazan2007logarithmic; @shalev2007logarithmic]. Since the iteration cost of SGD is very cheap, SGD is suitable to many machine learning tasks. However, SGD only achieves a sublinear convergence rate and is ultimately slower than GD.
Recently, a number of (first-order) stochastic gradient methods using variance reduction techniques, which utilize the finite sum structure of problem (\[problem\]), have been proposed [@roux2012stochastic; @schmidt2013minimizing; @johnson2013accelerating; @xiao2014proximal; @nitanda2014stochastic; @defazio2014saga; @allen2015univr]. The iteration costs of these methods are the same as that of SGD, and, moreover, they achieve a linear convergence rate for a strongly convex objective.
The stochastic average gradient (SAG) method [@roux2012stochastic; @schmidt2013minimizing] can be used to treat the special case of problem (\[problem\]) with $R=0$. To the best of our knowledge, SAG is the first variance reduction algorithm that achieves a linear convergence rate for a strongly convex objective. SAGA [@defazio2014saga] is a modified SAG algorithm that not only achieves a linear convergence rate for a strongly convex objective but also can handle a nondifferentiable and non-strongly convex regularizer. However, for a non-strongly convex regularizer, SAGA needs to output the average of the whole history of the solutions for a convergence guarantee whereas SAG and SAGA do not for a strongly convex objective.
In contrast, the stochastic variance reduced gradient (SVRG) method [@johnson2013accelerating; @xiao2014proximal] adopts a different variance reduction scheme from SAG and SAGA, and in Acc-SVRG [@nitanda2014stochastic] a momentum scheme is applied to SVRG. These methods do not have theoretical support for a non-strongly convex regularizer but they achieve a linear convergence rate for a strongly convex objective. (SVRG needs to output the average of the generated solutions in the last stage for a convergence guarantee whereas Acc-SVRG does not.) UniVR [@allen2015univr] is an extension of SVRG and can handle a non-strongly convex regularizer and achieves an $O\left(n\mathrm{log}\frac{1}{\varepsilon}+\frac{\bar L}{\varepsilon}\right)$ rate (where the $O$ notation means the order of the necessary number of the gradient evaluations), which is faster than the $O\left(\frac{n+L_{\mathrm{max}}}{\varepsilon}\right)$ rate of SAGA for $\mathbb{E}[P(x)-P(x_*)] \leq \varepsilon$, $\bar L = (1/n) \sum_{i=1}^{n}L_i$, and $L_{\mathrm{max}} = {\mathrm{max}}\{L_1, \ldots , L_n\}$. However, UniVR also needs to output the average of the generated solutions in the last stage for convergence guarantees for both strongly and non-strongly convex regularizers.
In summary, the algorithms used in these methods often need to output the average of the history of the solutions as a final solution for convergence guarantees (and, especially, for a non-strongly convex regularizer, all of these methods need to take the average). This requirement is unsatisfactory for a sparsity-inducing regularizer because the average of the previous solutions could be nonsparse.
In this paper, we propose two new stochastic gradient methods using variance reduction techniques: the stochastic variance reduced dual averaging (SVRDA) method and the stochastic average dual averaging (SADA) method. Compared to previous stochastic optimization methods, the main advantages of our algorithms are as follows:
- Nice sparsity recovery performance: Our algorithms do not need to take the average of the history of the solutions whereas the existing ones do. This property often leads to sparser solutions than the existing methods for sparsity-inducing regularizers.
- Fast convergence: Our algorithms achieve the best known convergence rates among the existing nonaccelerated stochastic gradient methods for both strongly and non-strongly convex regularizers. Experimentally, our algorithms show comparable or superior convergence speed to that of the existing methods.
Assumptions and notation {#assump}
========================
We make the following assumptions for our theory:
\[assump1\] Each $f_i$ is convex and differentiable, and its gradient is $L_i$-Lipschitz continuous, i.e., $$\begin{aligned}
||\nabla f_i(x)-\nabla f_i(y)||_2 \leq L_i||x-y||_2\hspace{0.5cm}(\forall x, y \in \mathbb{R}^d). \label{L-smooth}\end{aligned}$$
Condition (\[L-smooth\]) is equivalent to the following conditions (see [@nesterov2013introductory]): $$f_i (y) \leq f_i(x) + \langle y-x, \nabla f_i(x)\rangle + \frac{L_i}{2}||x-y||_2^2\hspace{0.5cm}(\forall x, y \in \mathbb{R}^d)$$ and $$f_i(x) + \langle y-x, \nabla f_i(x)\rangle + \frac{1}{2L_i}||\nabla f_i(x) - \nabla f_i(y)||_2^2 \leq f_i(y)\hspace{0.5cm}(\forall x, y \in \mathbb{R}^d).$$
\[assump2\] The regularization function $R$ is $\mu$-strongly convex (and it is possible that $\mu =0$), i.e., $$R(y) \geq R(x) + \xi^{T}(y-x) + \frac{\mu}{2}||y-x||_2^2 \hspace{0.5cm}(\forall x, y \in \mathbb{R}^d, \forall \xi \in \partial R(x)),$$ where $\partial R(x)$ denotes the set of the subgradients of $R$ at $x$.
Observe that, if the regularization function $R$ is $\mu$-strongly convex, then the objective function $P$ is also $\mu$-strongly convex. It is well known that a strongly convex function with $\mu > 0$ has a unique minimizer.
\[assump3\] The regularization function $R$ is relatively simple, which means that the proximal mapping of $R,$ $$\mathrm{prox}_{R}(y)=\underset{x \in \mathbb{R}^d}{\mathrm{argmin}\ } \left\{ \frac{1}{2}||x-y||_{2}^{2} + R(x) \right\},$$ can be efficiently computed.
Since the function $(1/2)||x-y||_{2}^{2} + R(x)$ is $1+\mu$-strongly convex, the function $\mathrm{prox}_{R}$ is well defined regardless of the strong convexity of $R$. Note that $R$ is not necessarily differentiable.
\[assump4\] There exists a minimizer $x_*$ of problem (\[problem\]).
In addition, we define $\bar L = (1/n) \sum_{i=1}^n L_i $. Moreover, we define the probability distribution $Q$ on the set $\{1, 2, \ldots , n\}$ by $Q = \{q_i\}_{i \in \{1, 2, \ldots , n\}} = \left\{\frac{L_i}{n\bar L}\right\}_{i \in \{1, 2, \ldots , n\}}$. This probability distribution is used to randomly pick up a data point in each iteration. By employing nonuniform distribution, we can improve the convergence as in [@xiao2014proximal].
Many regularized empirical risk minimization problems in machine learning satisfy these assumptions. For example, given a set of training examples $(a_1, b_1), (a_2, b_2), \ldots , (a_n, b_n)$, where $a_i \in \mathbb{R}^d$ and $b_i \in \mathbb{R}$, if we set $f_i(x) = (1/2)(a_{i}^{\top}x-b_i)^2$ and $R(x)=\lambda||x||_{1}$, we get Lasso regression. Then the above assumptions are satisfied with $L_i = ||a_i||_2$, $\mu = 0$, and $\mathrm{prox}_{R}(y)=(\mathrm{sign}(y_j)\mathrm{max} \{|y_j|-\lambda, 0\})_{j=1}^d$. If we set $f_i(x)=\mathrm{log}(1+\mathrm{exp}(-b_{i}a_{i}^{\top} x))$ and $R(x)=\lambda_1||x||_1+(\lambda_2/2)||x||_2^2$, we get logistic elastic net regression. Then the above assumptions are satisfied with $L_i = ||a_i||_2^2/4$, $\mu = \lambda_2$, and $\mathrm{prox}_{R}(y)=(1/(1+\lambda_2))(\mathrm{sign}(y_j)\mathrm{max} \{|y_j|-\lambda_1, 0\})_{j=1}^d$.
Related work and our contribution
=================================
In this section, we comment on the relationships between our methods and several closely related methods.
Standard methods for solving problem (\[problem\]) are the GD method and the dual averaging (DA) method [@nesterov2009primal]. These methods take the following update rules: $$\begin{aligned}
x_t=& \mathrm{prox}_{\frac{1}{\eta} R}\left(x_{t-1}-\frac{1}{\eta} \nabla F(x_{t-1})\right )\text{\hspace{1cm}(GD), } \notag \\
x_t =& \mathrm{prox}_{\frac{1}{\eta} tR}\left(x_0 - \frac{1}{\eta} \sum_{\tau =1}^{t} \nabla F(x_{\tau-1})\right) \text{\hspace{0.4cm}(DA), } \notag\end{aligned}$$ where $x_0$ is an initial vector and $1/\eta$ is a constant step size. GD and DA achieve linear convergence rates for a strongly convex regularizer (where, for DA, we need to borrow a multistage scheme as in [@chen2012optimal]). However, when the number of data $n$ is very large, these methods can be quite expensive because they require $O(nd)$ computation for each update.
Effective alternatives are the SGD method [@singer2009efficient; @hazan2007logarithmic; @shalev2007logarithmic] and the regularized dual averaging (RDA) method [@xiao2009dual]. These methods randomly draw $i$ in $\{1,2,\ldots , n\}$ and use $\nabla f_i$ as an estimator of the full gradient $\nabla F$ in each iteration: $$\begin{aligned}
x_t=& \mathrm{prox}_{\frac{1}{\eta_t} R}\left(x_{t-1}-\frac{1}{\eta_t} \nabla f_{i_t}(x_{t-1}) \right)\text{\hspace{1cm}(SGD), } \notag \\
x_t =& \mathrm{prox}_{\frac{1}{\eta_t} tR}\left(x_0 - \frac{1}{\eta_t} \sum_{\tau =1}^{t} \nabla f_{i_{\tau}}(x_{\tau-1})\right) \text{\hspace{0.32cm}(RDA), } \notag\end{aligned}$$ where $1/\eta_t$ is a decreasing step size. These methods only require $O(d)$ computation for each iteration and are suitable for large-scale problems in machine learning. However, though $\nabla f_i$ is an unbiased estimator of $\nabla F$, it generally has a large variance, which causes slow convergence. As a result, these methods only achieve sublinear convergence rates even when the regularizer is strongly convex. One of simple solutions of this problem is to use a mini-batch strategy [@cotter2011better; @dekel2012optimal]. However, a mini-batch strategy still gives sublinear convergence.
In recent years, a number of (first-order) stochastic gradient methods using variance reduction techniques, which utilize the finite sum structure of problem (\[problem\]), have been proposed [@roux2012stochastic; @schmidt2013minimizing; @johnson2013accelerating; @xiao2014proximal; @nitanda2014stochastic; @defazio2014saga; @allen2015univr]. These methods apply a variance reduction technique to SGD. For example, SVRG [@johnson2013accelerating; @xiao2014proximal] takes the following update rules: $$\begin{aligned}
&\widetilde{x} = \widetilde{x}_{s-1} \notag \\
&\text{for } t=1 \text{ to } m \notag \\
&\hspace{0.5cm} \text{Draw } i_t \text{ randomly from } \{1,2,\ldots ,n\} \notag \\
&\hspace{0.5cm} v_t = \nabla f_{i_t}(x_{t-1}) - \nabla f_{i_t}(\widetilde{x}) + \nabla F( \widetilde{x}) \notag \\
&\hspace{0.5cm} x_t = \mathrm{prox}_{\frac{1}{\eta} R}\left(x_{t-1}-\frac{1}{\eta} v_t \right) \notag \\
&\widetilde{x}_{s} = \frac{1}{m}\sum_{t=1}^m x_t. \notag\end{aligned}$$ $v_t$ is an unbiased estimator of $\nabla F(x_{t-1})$ and one can show that its variance is “reduced”: $$\mathrm{E}||v_t - \nabla F(x_{t-1})||^2 \leq 4\bar L [P(x_{t-1}) - P(x_*) + P(\widetilde{x}) - P(x_*)].$$ This means that the variance of the estimator $v_t$ converges to zero as $x_t$ and $\widetilde{x}$ to $x_*$. In this sense, $v_t$ is a better estimator of $\nabla F(x_{t-1})$ than the simple estimator $\nabla f_{i_t} (x_{t-1})$. Indeed, these methods achieve linear convergence rates for a strongly convex regularizer.
However, these methods often need to take the average of the previous solutions for convergence guarantee. For example, SVRG and UniVR [@allen2015univr] require taking the average of the history of the solutions in the last stage. SAGA [@defazio2014saga] also requires taking the average of all previous solutions for a non-strongly convex regularizer, though it does not for a strongly convex regularizer. For a sparsity-inducing regularizer, this requirement is unsatisfactory because taking the average could cause a nonsparse solution even though the optimal solution is sparse.
In contrast, our proposed methods have theoretical convergence guarantees without taking the average of the previous solutions for both strongly and non-strongly convex regularizers. The basic idea of our methods is simple: We apply a variance reduction technique to RDA rather than to SGD. For example, using an analogy to SVRG, we naturally get the following algorithm: $$\begin{aligned}
&\widetilde{x} = \widetilde{x}_{s-1} \notag \\
&\text{for } t=1 \text{ to } m \notag \\
&\hspace{0.5cm} \text{Draw } i_t \text{ randomly from } \{1,2,\ldots ,n\} \notag \\
&\hspace{0.5cm} v_t = \nabla f_{i_t}(x_{t-1}) - \nabla f_{i_t}(\widetilde{x}) + \nabla F( \widetilde{x}) \notag \\
&\hspace{0.5cm} x_t = \mathrm{prox}_{\frac{1}{\eta} tR}\left(x_0 - \frac{1}{\eta} \sum_{\tau =1}^{t} v_{\tau}\right) \notag \\
&\widetilde{x}_{s} = \frac{1}{m}\sum_{t=1}^m x_t. \notag\end{aligned}$$ However, this algorithm is not sufficient because the final solution has to be the average of the previous solutions for convergence guarantees (a situation that is similar to RDA). Hence we borrow a momentum scheme and an additional SGD step. (For more detail, see Section \[algorithm\].) Then the algorithm does not need to take the average of the previous solutions for convergence guarantees even when the regularizer is non-strongly convex. We call this algorithm SVRDA. Similarly, we can apply the dual averaging scheme to SAGA and we call this algorithm SADA.
Comparisons of the properties of these methods are summarized in Table \[table\]. “Gradient complexity” indicates the order of the number of the necessary gradient evaluations for $\mathbb{E}[P(x)-P(x_*)] \leq \varepsilon$ (or $\mathbb{E}||x-x_*||_2^2 \leq \varepsilon$). “Final output” indicates whether the (theoretically guaranteed) final solution is generated from the (weighted) average of previous iterates (Avg) or from the proximal mapping (Prox). For sparsity-inducing regularizers, the solution generated from the proximal mapping is often sparser than the averaged solution. As we can see from Table \[table\], the proposed SVRDA and SADA both possess good properties in comparison with state-of-the-art stochastic gradient methods.
Algorithm description {#algorithm}
=====================
In this section, we illustrate the proposed methods.
The SVRDA method
----------------
We provide details of the SVRDA method in Algorithm \[svrda\]. The SVRDA method adopts a multistage scheme. Step (\[svrg step\]) generates a variance reduced estimator of the full gradient with nonuniform sampling and is the same as SVRG [@johnson2013accelerating; @xiao2014proximal]. Update rules (\[dual averaging step\]) and (\[gradient descent step\]) are the dual averaging update and the gradient descent update, respectively. The SVRDA method combines these two update rules. This idea is similar to the ORDA method [@chen2012optimal]. As in (\[itr\_num\]), for a non-strongly convex regularizer, we have to exponentially increase the iteration number in each inner loop whereas we can use a common fixed iteration number in each inner loop for a strongly convex regularizer. Note that the computational cost of each iteration in the inner loop of the SVRDA method is $O(d)$ rather than $O(nd)$. Also note that SVRDA outputs the solution generated from the proximal mapping rather than the average of previous iterates. For a strongly convex regularizer, SVRDA can output both $\widetilde{x}_S$ (the gradient descent step’s output) and $\widetilde{v}_S$ (the dual averaging step’s output) as a final solution. This is because the convergence of $\mathbb{E}||\widetilde{v}_s-x_*||_2^2$ is guaranteed with a linear convergence rate whereas the theoretical convergence of $\mathbb{E}[P(\widetilde{v}_s)-P(x_*)]$ is not guaranteed. Outputting $\widetilde{v}_S$ experimentally leads to better sparsity recovery performance than outputting $\widetilde{x}_S$ (see Section \[experiments\]).
$\widetilde{x}_{0} \in \mathbb{R}^d$, $\eta > 0$, $m_1 \in \mathbb{N}$, $S \in \mathbb{N}$. $\widetilde{v}_{0} = \widetilde{x}_{0}$ $\alpha = \begin{cases} \frac{1}{4} & (\mu >0) \\ 0 & (\mu=0)\end{cases}$ $x_0=\widetilde{x}_{s-1}$, $v_0=(1-\alpha)\widetilde{v}_{s-1}+\alpha\widetilde{x}_{s-1}, u_0 = v_0$, $\bar{g}_0 = 0$ $$m_s =\begin{cases} m_1 &(\mu>0)\\ 2^{s-1} m_1 &(\mu=0) \end{cases} \label{itr_num}$$ pick $i_t \in \{1,2,\ldots ,n\}$ randomly according to $Q$ $$\begin{aligned}
g_{t}&=(\nabla f_{i_t}(u_{t-1})-\nabla f_{i_t}(x_0))/nq_{{i}_t}+\nabla F(x_0) \hspace{2cm} \label{svrg step}\\
\bar{g}_t &= \left(1-\frac{1}{t}\right)\bar{g}_{t-1} + \frac{1}{t}g_t \notag \\
v_{t}&=\underset{x \in \mathbb{R}^d}{\mathrm{argmin}\ } \left\{ \langle \bar g_t, x \rangle + R(x) + \frac{\eta}{2t}||x-v_0||_2^2 \right\} \notag \\
&=\mathrm{prox}_{\frac{1}{\eta}tR}\left(v_0-\frac{1}{\eta}t\bar{g}_t\right)\label{dual averaging step}\\
x_{t}&= \underset{x \in \mathbb{R}^d}{\mathrm{argmin}\ } \left\{ \langle g_t, x \rangle + R(x) + \frac{\eta t}{2}||x-u_{t-1}||_2^2 \right\} \notag \\
&=\mathrm{prox}_{\frac{1}{\eta t }R}\left(u_{t-1}-\frac{1}{\eta t}g_t\right) \label{gradient descent step}\\
u_t &= \left(1-\frac{1}{t+1}\right)x_t + \frac{1}{t+1}v_t \notag\end{aligned}$$ $\widetilde{x}_s = x_{m_s}$, $\widetilde{v}_s = v_{m_s}$ $\widetilde{x}_S$ or $\widetilde{v}_S$ ($\mu>0$), $\widetilde{x}_S$ ($\mu=0$).
The SADA method
---------------
We provide details of the SADA method in Algorithm \[sada\]. The algorithm is similar to SVRDA (Algorithm \[svrda\]). The main difference from the SVRDA method is the update rule (\[saga step\]). This step reduces the variance of the approximation of the full gradient using a SAGA [@defazio2014saga] type variance reduction technique rather than SVRG. Note that SADA is a multistage algorithm like SVRG and SVRDA whereas SAGA is a single-stage algorithm. To the best of our knowledge, there exists no single-stage dual averaging algorithm that achieves a linear convergence rate for a strongly convex regularizer. This is probably because of the limitations of the single-stage dual averaging algorithms. Also note that we adopt uniform sampling for SADA. Schmidt et al. [@schmidt2015non] have considered a nonuniform sampling scheme for SAGA on the special setting $R=0$ in (\[problem\]), but their methods require two gradient evaluations in one iteration and it is not satisfactory. For this reason, we do not adopt nonuniform sampling schemes for SADA in this paper. SADA has theoretically similar properties to SVRDA except for the difference of the sampling scheme, and experimentally SADA sometimes outperforms SVRDA (see Section \[experiments\]).
$\widetilde{x}_{0} \in \mathbb{R}^d$, $\eta>0$, $m_1 \in \mathbb{N}$, $S \in \mathbb{N}$ $\widetilde{v}_{0} = \widetilde{x}_{0} $ $\alpha = \begin{cases} \frac{1}{4} & (\mu >0) \\ 0 & (\mu=0)\end{cases}$ $x_0=\widetilde{x}_{s-1}$, $v_0 = (1-\alpha)\widetilde{v}_{s-1}+\alpha \widetilde{x}_{s-1}$, $u_0 = v_0$, $\bar{g}_0 = 0$, $\phi_i^0 = x_0 \ (i=1,2, \ldots , n)$ $$m_s =\begin{cases} m_1 &(\mu>0)\\ 2^{s-1} m_1 &(\mu=0)\end{cases}$$ pick $i_t \in \{1,2,\ldots ,n\}$ uniformly at random $$\begin{aligned}
\phi_{i_t}^t &= u_{t-1}, \phi_i^t = \phi_i^{t-1} (i \neq i_t) \notag \\
g_{t}&=\nabla f_{i_t}(\phi_{i_t}^t)-\nabla f_{i_t}(\phi_{i_t}^{t-1})+\frac{1}{n}\sum_{i=1}^{n}\nabla f_i(\phi_{i}^{t-1}) \hspace{2cm}
\label{saga step}\\
\bar{g}_t &= \left(1-\frac{1}{t}\right)\bar{g}_{t-1} + \frac{1}{t}g_t \notag \\
v_{t}&=\underset{x \in \mathbb{R}^d}{\mathrm{argmin}\ } \left\{ \langle \bar g_t, x \rangle + R(x) + \frac{\eta}{2t}||x-v_0||_2^2 \right\} \notag \\
&=\mathrm{prox}_{\frac{t}{\eta}R}\left(v_0-\frac{t}{\eta}\bar{g}_t\right) \notag \\
x_{t}&= \underset{x \in \mathbb{R}^d}{\mathrm{argmin}\ } \left\{ \langle g_t, x \rangle + R(x) + \frac{\eta t}{2}||x-u_{t-1}||_2^2 \right\} \notag \\
&=\mathrm{prox}_{\frac{1}{\eta t}R}\left(u_{t-1}-\frac{1}{\eta t}g_t\right)\notag \\
u_t &= \left(1-\frac{1}{t+1}\right)x_t + \frac{1}{t+1}v_t\notag \end{aligned}$$ $\widetilde{x}_s = x_{m_s}$, $\widetilde{v}_s = v_{m_s}$ $\widetilde{x}_S$ or $\widetilde{v}_S$ ($\mu>0$), $\widetilde{x}_S$ ($\mu=0$).
Convergence analysis
====================
Now we give a convergence analysis of our algorithms. In this section, all norms $||\cdot||$ mean the $L_2$-norm $||\cdot||_2$.
Convergence analysis of SVRDA
-----------------------------
In this subsection, we give the convergence analysis of SVRDA.
\[main thm1\] Suppose that Assumptions \[assump1\], \[assump2\], \[assump3\], and \[assump4\] hold (and it is possible that $\mu = 0$). Let $x_0 \in \mathbb{R}^d$, $\eta = 4\bar L$, $m_1 \in \mathbb{N,}$ and $0 \leq \alpha \leq 1$. Then the SVRDA algorithm satisfies $$\begin{aligned}
&\mathbb{E} \left[P(\widetilde{x}_{s})-P(x_{*})\right] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||\widetilde{v}_{s} - x_*||^2\notag \\
\leq& \frac{1}{2} \mathbb{E} [P(\widetilde{x}_{s-1})-P(x_{*})] + \left(\frac{\alpha\eta }{2m_s}-\frac{\mu}{4}\right)\mathbb{E} ||\widetilde{x}_{s-1} - x_*||^2+\frac{(1-\alpha)\eta }{2m_s}\mathbb{E} ||\widetilde{v}_{s-1} - x_*||^2.\notag \end{aligned}$$
On inequality (\[roughbound\]) in Appendix \[appendix\], we can apply a tighter bound and $\eta$ can be smaller than $4\bar L$ for satisfying Theorem \[main thm1\]. This means that we can get a larger step size $\frac{1}{\eta}$ than $\frac{1}{4\bar L}$ and have a theoretically tighter bound. However, practically, if we tune $\eta$, it makes little difference and thus we omit it in this paper.
The proof of Theorem \[main thm1\] is given in Appendix \[appendix\]. Using this theorem, we derive recursive inequalities relative to $\mathbb{E} [P(\widetilde{x}_{s})-P(x_{*})]$ and $\mathbb{E}||\widetilde{v}_{s} - x_*||^2$. Based on Theorem \[main thm1\], we obtain the linear convergence of SVRDA for $\mu > 0$.
\[corsc\] Suppose that Assumptions \[assump1\], \[assump2\], \[assump3\], and \[assump4\] hold. Moreover, assume that $\mu > 0$. Let $x_0\in \mathbb{R}^d$, $\eta = 4\bar L$, $m_1 = \frac{\eta}{2\mu}$, $S\in \mathbb{N,}$ and $\alpha = \frac{1}{4}$. Then the SVRDA algorithm satisfies $$\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})] + \frac{3\mu}{2} \mathbb{E}||\widetilde{v}_{S} - x_*||^2 \leq \frac{1}{2^S}\left[P(\widetilde{x}_{0})-P(x_{*}) + \frac{3\mu}{2} ||\widetilde{x}_{0} - x_*||^2\right].$$ In addition, the SVRDA algorithm has a gradient complexity of $$O\left(\left(n+\frac{\bar L}{\mu}\right)\mathrm{log } \frac{1}{\varepsilon} \right)$$ for $\mathbb{E}[P(\widetilde{x}_S)-P(x_{*})] \leq \varepsilon$ and $$O\left(\left(n+\frac{\bar L}{\mu}\right)\mathrm{log } \frac{1}{\mu \varepsilon} \right)$$ for $\mathbb{E}||\widetilde{v}_S-x_*||^2 \leq \varepsilon$.
These gradient complexities are essentially the same as the ones obtained by [@roux2012stochastic; @schmidt2013minimizing; @johnson2013accelerating; @xiao2014proximal; @nitanda2014stochastic; @defazio2014saga; @allen2015univr] and are the best known ones among the existing nonaccelerated stochastic gradient methods. Note that the gradient complexity of GD is $O\left(n\frac{\bar L}{\mu}\mathrm{log}\frac{1}{\varepsilon}\right)$ and that of SGD is $O\left(\frac{1}{\mu \varepsilon}\right)$. In a typical empirical risk minimization task, we require that $\varepsilon$ be $O\left(\frac{1}{n}\right)$. Then the gradient complexities of SVRDA, GD, and SGD are $O\left(\left(n+\frac{\bar L}{\mu}\right)\mathrm{log } n \right)$, $O\left(n\frac{\bar L}{\mu}\mathrm{log}n \right)$, and $O\left(\frac{n}{\mu}\right)$, respectively. Hence, SVRDA significantly improves upon the gradient complexities of GD and SGD for $\mu>0$.
By Theorem \[main thm1\] and the definitions of $\eta$, $m_s,$ and $\alpha$, we obtain $$\begin{aligned}
&\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})] + \frac{3\mu}{2} \mathbb{E}||\widetilde{v}_{S} - x_*||^2\notag \\
\leq& \frac{1}{2} \left[\mathbb{E}[P(\widetilde{x}_{S-1})-P(x_{*})] + \frac{3\mu}{2} \mathbb{E}||\widetilde{v}_{S-1} - x_*||^2\right] \notag \\
\leq& \cdots \notag \\
\leq& \frac{1}{2^S} \left[P(\widetilde{x}_{0})-P(x_{*}) + \frac{3\mu}{2} ||\widetilde{v}_{0} - x_*||^2\right] \notag \\
=&\frac{1}{2^S}\left[P(\widetilde{x}_{0})-P(x_{*}) + \frac{3\mu}{2} ||\widetilde{x}_{0} - x_*||^2\right]. \notag \end{aligned}$$ By this inequality, we can see that the order of the necessary number of outer iterations for $\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})] \leq \varepsilon$ is $O\left(\mathrm{log}\frac{1}{\varepsilon}\right)$ and the order of the necessary number of outer iterations for $\mathbb{E} ||\widetilde{v}_{S}-x_{*}||^2 \leq \varepsilon$ is $O\left(\mathrm{log}\frac{1}{\mu \varepsilon}\right)$. Finally, since SVRDA computes $S$ times the full gradient $\nabla F$ and $2m_1=O\left(\frac{\bar L}{\mu}\right)$ times the gradient $\nabla f_i$, the total gradient complexity is $$O\left(\left(n+\frac{\bar L}{\mu}\right)\mathrm{log } \frac{1}{\varepsilon} \right)$$ for $\mathbb{E}[P(\widetilde{x}_S)-P(x_{*})] \leq \varepsilon$ and $$O\left(\left(n+\frac{\bar L}{\mu}\right)\mathrm{log } \frac{1}{\mu \varepsilon} \right)$$ for $\mathbb{E}||\widetilde{v}_S-x_*||^2 \leq \varepsilon$.
Next, we derive the convergence rate for $\mu=0$ from Theorem \[main thm1\] as follows.
\[corsvrganonstrong\] Suppose that Assumptions \[assump1\], \[assump2\], \[assump3\], and \[assump4\] hold (and it is possible that $\mu = 0$). Let $x_0\in \mathbb{R}^d$, $\eta = 4\bar L$, $m_1$, $S\in \mathbb{N},$ and $\alpha = 0$. Then the SVRDA algorithm satisfies $$\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})]\leq \frac{1}{2^S}\left[P(\widetilde{x}_{0})-P(x_{*}) + \frac{4\bar L }{m_{1}}||\widetilde{x}_{0} - x_*||^2\right].$$ In addition, if $m_1=O(\bar L)$, then the SVRDA algorithm has a gradient complexity of $$O\left(n\mathrm{log}\frac{1}{\varepsilon}+ \frac{\bar L }{\varepsilon}\right)$$ for $\mathbb{E}[P(\widetilde{x}_S)-P(x_{*})] \leq \varepsilon$.
The gradient complexity of SVRDA for a non-strongly convex regularizer is the same as that of UniVR [@allen2015univr] and is the best known among the existing stochastic gradient methods. Note that the gradient complexities of GD, SGD, and SAGA [@defazio2014saga] are $O\left(\frac{\bar Ln}{\varepsilon}\right)$, $O\left(\frac{1}{\varepsilon^2}\right)$, and $O\left(\frac{n+L_{\mathrm{max}}}{\varepsilon}\right)$, respectively. In a typical empirical risk minimization task, we require that $\varepsilon$ be $O\left(\frac{1}{n}\right)$. Then the gradient complexities of SVRDA, GD, SGD, and SAGA are $O\left(n\mathrm{log}n+ \bar L n \right)$, $O\left(\bar L n^2\right)$, $O\left(n^2 \right)$, and $O\left(n^2+L_{\mathrm{max}}n \right)$, respectively. Hence, SVRDA significantly improves upon the gradient complexities of GD, SGD, and SAGA for $\mu=0$.
By Theorem \[main thm1\] and the definitions of $\eta$, $m_s$, and $\alpha$, we obtain $$\begin{aligned}
&\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})] + \frac{\eta}{m_{S+1}} \mathbb{E}||\widetilde{v}_{S} - x_*||^2 \notag \\
=&\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})] + \frac{\eta}{2m_{S}} \mathbb{E}||\widetilde{v}_{S} - x_*||^2 \notag \\
\leq& \frac{1}{2} \left[\mathbb{E}[P(\widetilde{x}_{S-1})-P(x_{*})] + \frac{\eta}{m_{S}}\mathbb{E}||\widetilde{v}_{S-1} - x_*||^2\right] \notag \\
\leq& \cdots \notag \\
\leq& \frac{1}{2^S} \left[P(\widetilde{x}_{0})-P(x_{*}) + \frac{\eta}{m_{1}}||\widetilde{v}_{0} - x_*||^2\right] \notag \\
=&\frac{1}{2^S}\left[P(\widetilde{x}_{0})-P(x_{*}) + \frac{4\bar L}{ m_{1}}||\widetilde{x}_{0} - x_*||^2\right], \notag \end{aligned}$$ and therefore $$\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})]\leq\frac{1}{2^S} \left[P(\widetilde{x}_{0})-P(x_{*}) + \frac{4\bar L}{ m_{1}}||\widetilde{x}_{0} - x_*||^2\right].$$ Thus the order of the necessary number of outer iterations for $\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})] \leq \varepsilon$ is $O\left(\mathrm{log} \frac{1}{\varepsilon} \right)$. Finally, since SVRDA computes $S$ times the full gradient $\nabla F$ and $O\left(\sum_{s=1}^S 2m_s\right) =
O\left(2^Sm_1\right)$ times the gradient $\nabla f_i$, the total gradient complexity is $$O\left(n S +\sum_{s=1}^S 2m_s \right) = O\left(n S + 2^Sm_1 \right) = O\left(n\mathrm{log}\frac{1}{\varepsilon} + \frac{\bar L}{\varepsilon} \right).$$
Convergence analysis of SADA
----------------------------
In this subsection, we give the convergence analysis of SADA.
\[main thm2\] Suppose that Assumptions \[assump1\], \[assump2\], \[assump3\], and \[assump4\] hold (and it is possible that $\mu = 0$). Let $x_0\in \mathbb{R}^d$, $\eta = 5L_{\mathrm{max}}$, $m_1 \in \mathbb{N,}$ and $0\leq \alpha \leq 1$. Then the SADA algorithm satisfies $$\begin{aligned}
&\mathbb{E} [P(\widetilde{x}_{s})-P(x_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||\widetilde{v}_{s} - x_*||^2 \notag \\
\leq&\frac{1}{2}\mathbb{E} \left[P(\widetilde{x}_{s-1})-P(x_{*})\right] + \left(\frac{\alpha\eta }{2m_s}-\frac{\mu}{4}\right)\mathbb{E}||\widetilde{x}_{s-1} - x_*||^2 + \frac{(1-\alpha)\eta}{2m_s}\mathbb{E}||\widetilde{v}_{s-1} - x_*||^2. \notag\end{aligned}$$
The proof of Theorem \[main thm2\] is given in Appendix \[appendix2\]. Using this theorem, we derive recursive inequalities relative to $\mathbb{E} [P(\widetilde{x}_{s})-P(x_{*})]$ and $\mathbb{E}||\widetilde{v}_{s} - x_*||^2$. Based on Theorem \[main thm2\], we obtain the linear convergence of SADA for $\mu > 0$.
\[corsc2\] Suppose that Assumptions \[assump1\], \[assump2\], \[assump3\], and \[assump4\] hold. Moreover, assume that $\mu > 0$. Let $x_0\in \mathbb{R}^d$, $\eta = 5L_{{\mathrm{max}}}$, $m_1 = \frac{\eta}{2\mu}$, $S \in \mathbb{N,}$ and $\alpha = \frac{1}{4}$. Then the SADA algorithm satisfies $$\mathbb{E} [P(\widetilde{x}_{S})-P(x_{*})] + \frac{3\mu}{2} \mathbb{E}||\widetilde{v}_{S} - x_*||^2 \leq \frac{1}{2^S}\left[P(\widetilde{x}_{0})-P(x_{*} )+ \frac{3\mu}{2}||\widetilde{x}_{0} - x_*||^2\right].$$ In addition, the SADA algorithm has a gradient complexity of $$O\left(\left(n+\frac{L_{\mathrm{max}}}{\mu}\right)\mathrm{log } \frac{1}{\varepsilon} \right)$$ for $\mathbb{E}[P(\widetilde{x}_S)-P(x_{*})] \leq \varepsilon$ and $$O\left(\left(n+\frac{L_{\mathrm{max}}}{\mu}\right)\mathrm{log } \frac{1}{\mu \varepsilon} \right)$$ for $\mathbb{E}||\widetilde{v}_S-x_*||^2 \leq \varepsilon$.
These gradient complexities are essentially same as the ones obtained by [@roux2012stochastic; @schmidt2013minimizing; @johnson2013accelerating; @xiao2014proximal; @nitanda2014stochastic; @defazio2014saga; @allen2015univr] and the ones of SVRDA and are the best known among the existing nonaccelerated stochastic gradient methods. Note that the gradient complexity of GD is $O\left(n\frac{\bar L}{\mu}\mathrm{log}\frac{1}{\varepsilon}\right)$ and that of SGD is $O\left(\frac{1}{\mu \varepsilon}\right)$. In a typical empirical risk minimization task, we require that $\varepsilon$ be $O\left(\frac{1}{n}\right)$. Then the gradient complexities of SADA, GD, and SGD are $O\left(\left(n+\frac{L_{\mathrm{max}}}{\mu}\right)\mathrm{log } n \right)$, $O\left(n\frac{\bar L}{\mu}\mathrm{log}n \right)$, and $O\left(\frac{n}{\mu}\right)$, respectively. Hence, SADA significantly improves upon the gradient complexities of GD and SGD for $\mu>0$.
The proof of Corollary \[corsc\] is identical to that of Corollary \[corsc\] and we omit it.
\[cornsc2\] Suppose that Assumptions \[assump1\], \[assump2\], \[assump3\], and \[assump4\] hold (and it is possible that $\mu = 0$). Let $x_0\in \mathbb{R}^d$, $\eta = 5 L_{\mathrm{max}}$, $m_1,S \in \mathbb{N}$, and $\alpha=0$. Then the SADA algorithm satisfies $$\mathbb{E}[P(\widetilde{x}_S)-P(x_{*})] \leq \frac{1}{2^{S}} \Bigl[ P(\widetilde{x}_0) - P(x_{*}) + \frac{5L_{\mathrm{max}}}{m_1}||\widetilde{x}_0-x_*||^2 \Bigr].$$ In addition, if $m_1=O(L_{\mathrm{max}})$, then the SADA algorithm has a gradient complexity of $$O\Bigl(n\mathrm{log}\frac{1}{\varepsilon}+ \frac{L_{\mathrm{max}}}{\varepsilon}\Bigr)$$ for $\mathbb{E}[P(\widetilde{x}_S)-P(x_{*})] \leq \varepsilon$.
The gradient complexity of SADA for a non-strongly convex regularizer is the same as that of UniVR [@allen2015univr] and SVRDA and is the best known among the existing stochastic gradient methods. Note that the gradient complexities of GD, SGD, and SAGA [@defazio2014saga] are $O\left(\frac{\bar Ln}{\varepsilon}\right)$, $O\left(\frac{1}{\varepsilon^2}\right)$, and $O\left(\frac{n+L_{\mathrm{max}}}{\varepsilon}\right)$, respectively. In a typical empirical risk minimization task, we require that $\varepsilon$ be $O\left(\frac{1}{n}\right)$. Then the gradient complexities of SVRDA, GD, SGD, and SAGA are $O\left(n\mathrm{log}n+ L_{\mathrm{max}} n \right)$, $O\left(\bar L n^2\right)$, $O\left(n^2 \right)$, and $O\left(n^2+L_{\mathrm{max}}n \right),$ respectively. Hence, SADA significantly improves upon the gradient complexities of GD, SGD, and SAGA for $\mu=0$.
The proof is the same as that of Corollary \[corsvrganonstrong\] and we omit it.
Numerical experiments {#experiments}
=====================
In this section, we provide numerical experiments to demonstrate the performances of SVRDA and SADA. We compare our methods with several state-of-the-art stochastic gradient methods: SVRG [@johnson2013accelerating; @xiao2014proximal], SAGA [@defazio2014saga], and UniVR [@allen2015univr]. For a fair comparison, we compare all different methods using solutions that are theoretically guaranteed. We used nonuniform sampling for SVRG [@johnson2013accelerating; @xiao2014proximal], UniVR [@allen2015univr], and SVRDA. (Zhu et al. [@allen2015univr] have not considered a nonuniform sampling scheme for UniVR, but because there is theoretical justification of nonuniform sampling for UniVR, we adopted nonuniform sampling for UniVR.) However, we used uniform sampling for SAGA [@defazio2014saga] and SADA. (Schmidt et al. [@schmidt2015non] considered nonuniform sampling for SAGA on the special setting $R=0$ in (\[problem\]), but their algorithm require two gradient evaluations in one iteration for a gradient complexity of $O\left(\left(n+\frac{\bar L}{\mu}\right)\mathrm{log } \frac{1}{\varepsilon} \right)$, and thus in our experiment we adopted uniform sampling for SAGA and SADA.)
In this experiments, we focus on the regularized logistic regression problem for binary classification: Given a set of training examples $(a_1, b_1), (a_2, b_2), \ldots , (a_n, b_n)$, where $a_i \in \mathbb{R}^d$ and $b_i \in \left\{+1,-1\right\}$, we find the optimal classifier $x \in \mathbb{R}^d$ by solving$$\underset{x \in \mathbb{R}^d}{\mathrm{min}}\ \ \frac{1}{n} \sum_{i=1}^n \mathrm{log}(1+\mathrm{exp}(-b_{i}a_{i}^{\top}x)) +\lambda_1||x||_1+\frac{\lambda_2}{2}||x||_2^2,$$ where $\lambda_1$ and $\lambda_2$ are regularization parameters.
We used three publicly available data sets in the experiments. Their sizes $n$ and dimensions $d$ are listed in Table \[datatable\]. Each continuous feature vector in these data sets has been normalized to zero mean and unit variance.
Data sets $n$ $d$
--------------- ----------- ---------
covertype $581,012$ $54$
Reuters-21578 5,964 18,933
sido0 $12,678$ $4,932$
: Summary of the data sets used in our numerical experiments[]{data-label="datatable"}
We performed our experiments on a desktop computer (a Windows 7 64-bit machine with an Intel i7-4790 CPU operating at 3.60 GHz and 8 GB of RAM) and implemented all algorithms in MATLAB 2015a.
\
Figures \[covertype\_sc\] and \[covertype\_nsc\] show the comparison of SVRDA and SADA with the different methods described above on the covertype data set for different setups of $\lambda_1$ and $\lambda_2$ (the strongly convex case $\lambda_2=10^{-6} >0$ (Figure \[covertype\_sc\]) and the non-strongly convex case $\lambda_2=0$ (Figure \[covertype\_nsc\])). Objective Gap (left) means $P(x)-P(x_*)$ for the output solution $x$ and NNZs (right) means the number of nonzeros in the output solution. SVRDA-x and SADA-x output the solution generated by the gradient descent update $\widetilde{x}_s$, and SVRDA-v and SADA-v output the one generated by the dual averaging update $\widetilde{v}_s$. We do not report SVRDA-v and SADA-v for a non-strongly convex regularizer, because it has no theoretical convergence guarantee. For a strongly convex regularizer (top), UniVR, SVRDA-x, and SVRDA-v outperform other methods, as indicated by the theories (see Table \[table\]). Observe that the objective gaps of SVRDA-x and SVRDA-v (respectively SADA-x and SADA-v) are very close though SVRDA-v (respectively SADA-v) has no theoretical guarantee for convergence of the objective gap. Note that SVRDA-v (respectively SADA-v) gives sparser solutions than SVRDA-x (respectively SADA-x) and the other methods. For a non-strongly convex regularizer (bottom), SVRDA-x and SADA-x converge more quickly than both UniVR and SAGA. The sparsity pattern of the output solutions of UniVR is unstable and that of SAGA is very poor, because UniVR and SAGA need to average the history of the solutions and the averaged solutions could be nonsparse. In contrast, SVRDA-x and SADA-x show a nice sparsity recovery performance.
\
Figures \[Reuters\_sc\] and \[Reuters\_nsc\] show the comparison of different methods on the Reuters-21578 data set for different setups of $\lambda_1$ and $\lambda_2$ (the strongly convex case $\lambda_2=10^{-4} >0$ (Figure \[Reuters\_sc\]) and the non-strongly convex case $\lambda_2=0$ (Figure \[Reuters\_nsc\])). For a strongly convex regularizer, SVRG type algorithms (SVRG, UniVR, SVRDA-x, and SVRDA-v) show nice convergence behavior whereas SAGA type algorithms (SAGA, SADA-x, and SADA-v) show a slightly unstable behavior. Note that the sparsity pattern of the output solution of SVRG is poor. For a non-strongly convex regularizer, SVRDA-x and SADA-x converge more quickly but a bit more unstably than the other methods. Observe that, when a new stage starts, SVRDA-x and SADA-x lead to a sharp increase in the objective gap followed by a quick drop. This behavior can also be seen in the Multi-stage ORDA [@chen2012optimal]. We can see that the sparsity recovery performances of SVRDA-x and SADA-x are very nice whereas that of UniVR is unstable and poor and that of SAGA is quite poor.
\
Figures \[sido0\_sc\] and \[sido0\_nsc\] show the comparison of different methods on the sido0 data set for different setups of $\lambda_1$ and $\lambda_2$ (the strongly convex case $\lambda_2=10^{-4} >0$ (Figure \[sido0\_sc\]) and the non-strongly convex case $\lambda_2=0$ (Figure \[sido0\_nsc\])). For a strongly convex regularizer, the performances of SAGA, SADA-x, and SADA-v are among the best. Especially, SADA-v shows the best sparsity recovery performance. Note that the sparsity recovery performance of SVRG is very poor. We can see that the convergence of the NNZs of SVRDA-v (respectively SADA-v) is superior to SVRDA-x (respectively SADA-x). For a non-strongly convex regularizer, SVRDA-x and SADA-x outperform both UniVR and SAGA. Especially, SVRDA-x and SADA-x show nice sparsity recovery performances though the solutions of UniVR and SAGA are not sparse at all.
Conclusion and future work
==========================
In this paper, we proposed two stochastic gradient methods for regularized empirical risk minimization problems: SVRDA and SADA. We have shown that SVRDA and SADA achieve $O\left(\left(n+\frac{\bar L}{\mu}\right)\mathrm{log}\frac{1}{\varepsilon}\right)$ and $O\left(\left(n+\frac{L_{\mathrm{max}}}{\mu}\right)\mathrm{log}\frac{1}{\varepsilon}\right)$complexity, respectively, for a strongly convex regularizer and $O\left(n\mathrm{log}\frac{1}{\varepsilon}+\frac{\bar L}{\varepsilon}\right)$ and $O\left(n\mathrm{log}\frac{1}{\varepsilon}+\frac{L_{\mathrm{max}}}{\varepsilon}\right)$ complexity, respectively, for a non-strongly convex regularizer.
In numerical experiments, our methods led to better sparsity recovery than the existing methods for sparsity-inducing regularizers and showed nice convergence behaviors, especially for non-strongly convex regularizers.
An interesting future work is to extend our methods to the alternating directional multiplier method (ADMM) framework. In this paper, we assumed that the proximal mapping of $R$ can be efficiently computed. However, for structured regularization problems (for example, overlapped group lasso, graph lasso, etc.), this assumption is generally not satisfied and our methods cannot be directly applied. In contrast, ADMM can be applied to these problems without this assumption. Suzuki [@suzuki2013dual] has proposed regularized dual averaging-ADMM (RDA-ADMM), which is RDA [@xiao2009dual] for ADMM in an online setting. Furthermore, Suzuki [@suzuki2014stochastic] has proposed stochastic dual coordinate ascent-ADMM (SDCA-ADMM), which is SDCA [@shalev2013stochastic; @shalev2013accelerated] for ADMM in regularized an empirical risk minimization setting, and has shown that it converges exponentially for a strongly convex regularizer. Applying SVRDA to the ADMM framework and showing linear convergence for a strongly convex regularizer would be promising future work.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was partially supported by MEXT Kakenhi (25730013, 25120012, and 26280009), JST-PRESTO and JST-CREST.
Proof of Theorem \[main thm1\] {#appendix}
==============================
In this section, we give the proof of Theorem \[main thm1\]. First we prove the following two easy lemmas.
\[avg lemma\]$$\bar g_t = \frac{1}{t}\sum_{\tau = 1}^{t} g_\tau \ (t \geq 1).$$
For $t=1$, $\bar g_1 = g_1 = \frac{1}{1}\sum_{\tau = 1}^1 g_{\tau}$.\
Assume that the claim holds for some $t \geq 1$. Then $$\begin{aligned}
\bar g_{t+1} &= \left(1-\frac{1}{t+1}\right)\bar g_t + \frac{1}{t+1}g_{t+1} \ \text{(by the definition)} \notag \\
&= \left(1-\frac{1}{t+1}\right)\frac{1}{t}\sum_{\tau = 1}^{t} g_\tau + \frac{1}{t+1}g_{t+1} \ \text{(by the assumption of the induction)} \notag \\
&= \frac{1}{t+1}\sum_{\tau = 1}^{t+1} g_\tau. \notag\end{aligned}$$ This finishes the proof for Lemma \[avg lemma\].
\[smooth lemma\]For every $x$, $u \in \mathbb{R}^d$, $$F(u) + \langle \nabla F(u), x-u \rangle + R(x) \leq P(x) - \frac{1}{2\bar L}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x) - \nabla f_i(u)||^2.$$
Since $f_i$ is $L_i$-smooth, we have (see [@nesterov2013introductory]) $$f_i(u) + \langle \nabla f_i(u), x-u \rangle \leq f_i (x) - \frac{1}{2L_i}||\nabla f_i(x) - \nabla f_i(u)||^2.$$ Summing this inequality from $i=1$ to $n$ and dividing it by $n$ results in $$F(u) + \langle \nabla F(u), x-u \rangle \leq F(x) - \frac{1}{2\bar L}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x) - \nabla f_i(u)||^2.$$ Adding $R(x)$ gives the desired result.
Next we prove the following main lemma.
\[main lemma1\]For the $s$th stage of SVRDA, $$\begin{aligned}
&\mathbb{E} [P({x}_{m_s})-P(x
_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||{v}_{m_s} - x_*||^2 \notag \\
\leq &\frac{1}{2 m_s} \sum_{t=1}^{m_s} \left[ \frac{t}{\eta t - \bar L}\mathbb{E} ||g_t- \nabla F(u_{t-1})||^2 -\frac{1}{\bar L}\mathbb{E} \left[\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(u_{t-1}) - \nabla f_i(x_*)||^2\right]\right] \notag \\
&+\frac{\eta}{2m_s}||{v}_{0} - x_*||^2, \notag\end{aligned}$$ where the expectations are conditioned on all previous stages.
First note that $u_t = \left(1-\frac{1}{t+1}\right)x_t + \frac{1}{t+1}v_t$ for $t \geq 0$ by the definition of $u_0$. We define $$\begin{aligned}
\ell_t (x) &= F(u_{t-1}) + \langle \nabla F(u_{t-1}), x-u_{t-1} \rangle + R(x), \notag \\
\hat \ell_t (x) &= F(u_{t-1}) + \langle g_t, x-u_{t-1} \rangle + R(x). \notag\end{aligned}$$ Observe that $\ell_t \leq P$. For $t \geq 1$, by Lemma \[avg lemma\], we have $$\begin{aligned}
\sum_{\tau=1}^{t} \hat \ell_{\tau} (x) =& \sum_{\tau = 1}^{t} F(u_{t-1}) + \sum_{\tau = 1}^{t}\langle g_{\tau}, x-u_{\tau-1} \rangle + \sum_{\tau = 1}^{t} R(x) \notag \\
=& \langle t \bar g_t, x \rangle + t R(x) + \sum_{\tau = 1}^{t}F(u_{t-1}) - \sum_{\tau = 1}^{t}\langle g_{\tau}, u_{\tau-1} \rangle \notag\end{aligned}$$ and thus we have $v_t = \underset{x \in \mathbb{R}^d} { \mathrm{argmin}\ }\left\{\sum_{\tau=1}^{t} \hat \ell_{\tau} (x) + \frac{\eta}{2}||x-v_0||^2 \right\} $.\
Also note that $x_t = \underset{x \in \mathbb{R}^d} { \mathrm{argmin}\ }\left\{ \hat \ell_{\tau} (x) + \frac{\eta t}{2}||x-u_{t-1}||^2 \right\} $. Since $F$ is $\bar L$-smooth, we have (see [@nesterov2013introductory]) $$F(x_t) \leq F(u_{t-1}) + \langle \nabla F(u_{t-1}), x_t - u_{t-1} \rangle + \frac{\bar L}{2} ||x_t-u_{t-1}||^2,$$and thus $$\begin{aligned}
P(x_t) \leq& \ell_t(x_t) + \frac{\bar L}{2}||x_t-u_{t-1}||^2 \notag \\
=& \hat \ell_t(x_t) + \frac{\eta t}{2}||x_t-u_{t-1}||^2 - \frac{\eta t -\bar L}{2}||x_t-u_{t-1}||^2 -\langle g_t-\nabla F(u_{t-1}), x_t- u_{t-1} \rangle. \notag\end{aligned}$$ Since $x_t$ is the minimizer of $\hat \ell_t (x) + \frac{\eta t}{2}||x-u_{t-1}||^2$, we have $$\begin{aligned}
\hat \ell_t (x_t) +& \frac{\eta t}{2}||x_t-u_{t-1}||^2 \notag \\ \leq&\hat \ell_t \left(\left(1-\frac{1}{t}\right)x_{t-1} + \frac{1}{t}v_t\right) + \frac{\eta t}{2}\left|\left|\left(1-\frac{1}{t}\right)x_{t-1} + \frac{1}{t}v_t-u_{t-1}\right|\right|^2 \notag\end{aligned}$$ and hence $$\begin{aligned}
P(x_t) \leq&
\hat \ell_t \left(\left(1-\frac{1}{t}\right)x_{t-1} + \frac{1}{t}v_t\right) + \frac{\eta t}{2}\left|\left|\left(1-\frac{1}{t}\right)x_{t-1} + \frac{1}{t}v_t-u_{t-1}\right|\right|^2 \notag \\
&- \frac{\eta t - \bar L}{2}||x_t-u_{t-1}||^2- \langle g_t-\nabla F(u_{t-1}), x_t- u_{t-1} \rangle. \notag\end{aligned}$$ Using the convexity of $\hat \ell_t$ and the facts that $\left(1-\frac{1}{t}\right)x_{t-1} + \frac{1}{t}v_t-u_{t-1} = \frac{1}{t}(v_t-v_{t-1})$ and $$\begin{aligned}
- \frac{\eta t - \bar L}{2}||x_t-u_{t-1}||^2-& \langle g_t-\nabla F(u_{t-1}), x_t- u_{t-1} \rangle \leq \frac{1}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2,\notag \end{aligned}$$ we get $$\begin{aligned}
P(x_t) \leq& \left(1-\frac{1}{t}\right)\hat \ell_t (x_{t-1}) + \frac{1}{t}\hat \ell_t (v_t) +\frac{\eta}{2t}||v_t -v_{t-1}||^2
+ \frac{1}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2 \notag \\
=& \left(1-\frac{1}{t}\right) \ell_t (x_{t-1}) + \frac{1}{t}\hat \ell_t (v_t) +\frac{\eta}{2t}||v_t -v_{t-1}||^2 \notag \\
&+ \frac{1}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2
+ \left(1-\frac{1}{t}\right)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle \notag \\
\leq& \left(1-\frac{1}{t}\right)P(x_{t-1}) + \frac{1}{t}\hat \ell_t (v_t) +\frac{\eta}{2t}||v_t -v_{t-1}||^2 \notag \\ &+ \frac{1}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2
+ \left(1-\frac{1}{t}\right)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle. \notag\end{aligned}$$ Multiplying both sides of the above inequality by $t$, we have $$\begin{aligned}
t P(x_t) \leq& (t-1)P(x_{t-1}) + \hat \ell_t (v_t) + \frac{\eta}{2}||v_t -v_{t-1}||^2 \notag \\ &+ \frac{t}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2
+ (t-1)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle. \notag\end{aligned}$$ By the fact that $\sum_{\tau=1}^{t-1} \hat \ell_{\tau} (x) + \frac{\eta}{2}||x-v_0||^2$ is $\eta$-strongly convex and $v_{t-1}$ is the minimizer of $\sum_{\tau=1}^{t-1} \hat \ell_{\tau} (x) +\frac{\eta}{2}||x-v_0||^2$ for $t \geq 2$, we have $$\sum_{\tau=1}^{t-1} \hat \ell_{\tau} (v_{t-1}) +\frac{\eta}{2}||v_{t-1}-v_0||^2 + \frac{\eta}{2}||v_t -v_{t-1}||^2 \leq\sum_{\tau=1}^{t-1} \hat \ell_{\tau} (v_t) + \frac{\eta}{2}||v_t-v_0||^2$$for $t \geq 1$ (and, for $t=1$, we define $\sum_{\tau=1}^0 = 0$). Using this inequality, we obtain $$\begin{aligned}
&t P(x_t) - \sum_{\tau=1}^{t} \hat \ell_{\tau} (v_t) - \frac{\eta}{2}||v_t-v_0||^2 \notag \\\leq& (t-1)P(x_{t-1}) -\sum_{\tau=1}^{t-1} \hat \ell_{\tau} (v_{t-1}) - \frac{\eta}{2}||v_{t-1}-v_0||^2
+ \frac{t}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2 \notag \\ &+ (t-1)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle. \notag\end{aligned}$$ Summing the above inequality from $t=1$ to $m_s$ results in $$\begin{aligned}
m_s P(x_{m_s}) - &\sum_{t=1}^{m_s} \hat \ell_t (v_{m_s}) - \frac{\eta}{2}||v_{m_s}-v_0||^2 \notag \\
\leq& \sum_{t=1}^{m_s}\frac{t}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2 \notag \\
&+ \sum_{t=1}^{m_s}(t-1)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle. \notag\end{aligned}$$ Using $\eta+m_s\mu$-strongly convexity of the function $\sum_{t=1}^{m_s} \hat \ell_t (x) + \frac{\eta}{2}||x-v_0||^2$ and the optimality of $v_{m_s}$, we have $$\sum_{t=1}^{m_s} \hat \ell_t (v_{m_s}) + \frac{\eta}{2}||v_{m_s}-v_0||^2 \leq \sum_{t=1}^{m_s} \hat \ell_t (x_*) + \frac{\eta}{2}||v_0-x_*||^2 - \frac{\eta +m_s \mu}{2}||v_{m_s}-x_*||^2$$ and hence $$\begin{aligned}
&m_s P(x_{m_s}) \notag \\ \leq& \sum_{t=1}^{m_s} \hat \ell_t (x_*) + \frac{\eta}{2}||v_0-x_*||^2 - \frac{\eta +m_s \mu}{2}||v_{m_s}-x_*||^2 \notag \\
&+\sum_{t=1}^{m_s}\frac{t}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2 + \sum_{t=1}^{m_s}(t-1)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle \notag \\
=&\sum_{t=1}^{m_s} \ell_t (x_*) + \frac{\eta}{2}||v_0-x_*||^2 - \frac{\eta +m_s \mu}{2}||v_{m_s}-x_*||^2 \notag \\
&+\sum_{t=1}^{m_s}\frac{t}{2(\eta t - \bar L)}||g_t- \nabla F(u_{t-1})||^2+ \sum_{t=1}^{m_s}(t-1)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle \notag \\
&+\sum_{t=1}^{m_s}\langle g_t-\nabla F(u_{t-1}), x_*- u_{t-1} \rangle. \notag \end{aligned}$$ By Lemma \[smooth lemma\] with $x = x_*$ and $u = u_{t-1}$, we have $$\ell_t(x_*)\leq P(x_*) - \frac{1}{2\bar L}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x_*) - \nabla f_i(u_{t-1})||^2.$$ Applying this inequality to the above inequality yields $$\begin{aligned}
&m_s P(x_{m_s})\notag \\ \leq&m_s P(x_*) +\frac{\eta}{2}||v_0-x_*||^2 - \frac{\eta +m_s \mu}{2}||v_{m_s}-x_*||^2 \notag \\
&+ \frac{1}{2}\sum_{t=1}^{m_s} \left[ \frac{t}{\eta t - \bar L}||g_t- \nabla F(u_{t-1})||^2 -\frac{1}{\bar L}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x_*) - \nabla f_i(u_{t-1})||^2\right] \notag \\
&+ \sum_{t=1}^{m_s}(t-1)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle
+\sum_{t=1}^{m_s}\langle g_t-\nabla F(u_{t-1}), x_*- u_{t-1} \rangle. \notag \end{aligned}$$ Dividing this inequality by $m_s$ results in $$\begin{aligned}
&P(x_{m_s})\notag \\ \leq& P(x_*)+ \frac{\eta}{2m_s}||v_0-x_*||^2 - \frac{\eta +m_s \mu}{2m_s }||v_{m_s}-x_*||^2 \notag \\
&+\frac{1}{2 m_s} \sum_{t=1}^{m_s} \left[ \frac{t}{\eta t - \bar L}||g_t- \nabla F(u_{t-1})||^2 -\frac{1}{\bar L}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x_*) - \nabla f_i(u_{t-1})||^2\right] \notag \\
&+ \frac{1}{m_s}\sum_{t=1}^{m_s}(t-1)\langle g_t-\nabla F(u_{t-1}), x_{t-1}- u_{t-1} \rangle
+\frac{1}{m_s}\sum_{t=1}^{m_s}\langle g_t-\nabla F(u_{t-1}), x_*- u_{t-1} \rangle. \notag \end{aligned}$$ Taking the expectation on both sides yields $$\begin{aligned}
&\mathbb{E} [P({x}_{m_s})-P(x
_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||{v}_{m_s} - x_*||^2 \notag \\
\leq &\frac{1}{2 m_s} \sum_{t=1}^{m_s} \left[ \frac{t}{\eta t - \bar L}\mathbb{E} ||g_t- \nabla F(u_{t-1})||^2 -\frac{1}{\bar L}\mathbb{E} \left[\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x_*) - \nabla f_i(u_{t-1})||^2\right]\right] \notag \\
&+\frac{\eta}{2m_s}||{v}_{0} - x_*||^2. \notag\end{aligned}$$ Here we used the fact that $\mathbb{E}[ g_t-\nabla F(u_{t-1})] = 0$ for $t=1, \ldots , m_s$. This finishes the proof of Lemma \[main lemma1\].
Now we need the following lemma.
\[object bound lemma\]For every $x \in \mathbb{R}^d$, $$\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x) - \nabla f_i(x_*)||^2 \leq 2 \bar L (P(x) - P(x_*)-\frac{\mu}{2}||x-x_*||^2).$$
From the argument of the proof of Lemma \[smooth lemma\], we have $$\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x) - \nabla f_i(x_*)||^2 \leq 2\bar L (F(x)-\langle \nabla F(x_*), x-x_* \rangle - F(x_*)).$$ By the optimality of $x_*$, there exists $\xi_* \in \partial R(x_*)$ such that $\nabla F(x_*) + \xi_*$. Then using $\mu$-strong convexity of $R$, we get $$-\langle \nabla F(x_*), x-x_* \rangle = \langle \xi_*, x-x_* \rangle \leq R(x)-R(x_*)-\frac{\mu}{2}||x-x_*||^2$$ and hence $$\frac{1}{n}\sum_{i=1}^{n}\frac{1}{nq_i}||\nabla f_i(x) - \nabla f_i(x_*)||^2 \leq 2 \bar L (P(x) - P(x_*)-\frac{\mu}{2}||x-x_*||^2).$$
We bound the term $\mathbb{E}||g_t - \nabla F(u_{t-1})||^2$: $$\begin{aligned}
&\mathbb{E}||g_t - \nabla F(u_{t-1})||^2 \notag \\
=& \mathbb{E}\left[ \mathbb{E}_{i_t}\left[||(\nabla f_{i_t}(u_{t-1})-\nabla f_{i_t} (x_0))/n q_{i_t}+\nabla F(x_0)-\nabla F(u_{t-1})||^2|i_1, \ldots , i_{t-1} \right]\right] \notag \\
\leq&\mathbb{E}\left[ \mathbb{E}_{i_t}\left[||(\nabla f_{i_t}(u_{t-1})-\nabla f_{i_t} (x_0))/n q_{i_t}||^2|i_1, \ldots , i_{t-1} \right]\right] \notag \\
=& \mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n \frac{1}{n q_{i}}||\nabla f_{i}(u_{t-1})-\nabla f_{i} (x_0)||^2\right] \notag \\
\leq& 3\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n \frac{1}{n q_{i}}||\nabla f_{i}(u_{t-1})-\nabla f_{i} (x_*)||^2\right] + \frac{3}{2}\left[\frac{1}{n}\sum_{i=1}^n \frac{1}{n q_{i}}||\nabla f_{i}(x_0)-\nabla f_{i} (x_*)||^2\right]. \notag \end{aligned}$$ Combining this inequality with Lemme \[object bound lemma\], we get $$\begin{aligned}
\mathbb{E}||g_t - \nabla F(u_{t-1})||^2 \leq& 3\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n \frac{1}{n q_{i}}||\nabla f_{i}(u_{t-1})-\nabla f_{i} (x_*)||^2\right] \notag \\
&+ 3\bar L (P(x_0) - P(x_*)-\frac{\mu}{2}||x_0-x_*||^2). \notag\end{aligned}$$ Since $\eta = \frac{1}{4 \bar L}$, using the inequality $$\frac{t}{\eta t - \bar L} \leq \frac{1}{3\bar L} \ \ \ \ (\forall t \geq 1), \label{roughbound}$$ by Lemma \[main lemma1\] we obtain $$\begin{aligned}
&\mathbb{E} [P({x}_{m_s})-P(x_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||{v}_{m_s} - x_*||^2\notag \\ \leq& \frac{1}{2}(P(x_0) - P(x_*)-\frac{\mu}{2}||x_0-x_*||^2)
+\frac{\eta}{2m_s}||{v}_{0} - x_*||^2. \notag \end{aligned}$$ Since $x_{m_s}=\widetilde{x}_s$, $v_{m_s}=\widetilde{v}_s$, $x_0=\widetilde{x}_{s-1}$, and $v_0=(1-\alpha)\widetilde{v}_{s-1}+\alpha\widetilde{x}_{s-1}$, we have $$\begin{aligned}
&\mathbb{E} [P(\widetilde{x}_{s})-P(x_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||\widetilde{v}_{s} - x_*||^2\notag \\
\leq& \frac{1}{2} (P(\widetilde{x}_{s-1})-P(x_{*})) + \left(\frac{\alpha\eta }{2m_s}-\frac{\mu}{4}\right)||\widetilde{x}_{s-1} - x_*||^2+\frac{(1-\alpha)\eta }{2m_s}||\widetilde{v}_{s-1} - x_*||^2.\notag \end{aligned}$$ Finally, taking expectations with respect to all previous stages gives the desired result.
Proof of Theorem \[main thm2\] {#appendix2}
==============================
In this section, we give the proof of Theorem \[main thm2\].
\[main lemma2\] For the $s$th stage of SADA, $$\begin{aligned}
&\mathbb{E} [P({x}_{m_s})-P(x_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||{v}_{m_s} - x_*||^2 \notag \\
\leq& \frac{1}{2 m_s} \sum_{t=1}^{m_s} \left[ \frac{t}{\eta t-L_{\mathrm{max}}}\mathbb{E}||g_t - \nabla F(u_{t-1})||^2 - \frac{1}{L_{\mathrm{max}}} \mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}|| \nabla f_i(u_{t-1})-\nabla f_i(x_*) ||^2\right] \right] \notag \\ &+ \frac{\eta}{2m_s}||{v}_{0} - x_*||^2. \notag\end{aligned}$$
The proof of Lemma \[main lemma2\] is identical to the proof of Lemma \[main lemma1\] and we omit it.
First we bound the term $\mathbb{E}||g_t - \nabla f(u_{t-1})||^2$: $$\begin{aligned}
\mathbb{E}||g_t - \nabla F(u_{t-1})||^2 =& \mathbb{E}\left[ \mathbb{E}_{i_t}\left[||\nabla f_{i_t}(u_{t-1})-\nabla f_{i_t} (\phi_{i_t}^{t-1})+\frac{1}{n}\sum_{i=1}^n \nabla f_i(\phi_i^{t-1})-\nabla F(u_{t-1})||^2|i_1, \ldots , i_{t-1} \right]\right] \notag \\
\leq&\mathbb{E}\left[ \mathbb{E}_{i_t}\left[||\nabla f_{i_t}(u_{t-1})-\nabla f_{i_t} (\phi_{i_t}^{t-1})||^2|i_1, \ldots , i_{t-1} \right]\right] \notag \\
=& \mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n ||(\nabla f_{i}(u_{t-1})-\nabla f_{i} (\phi_{i}^{t-1}))||^2\right] \notag \\
\leq& 2 \mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n ||(\nabla f_{i}(u_{t-1})-\nabla f_{i} (x_*))||^2\right] + 2\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n ||(\nabla f_{i}(\phi_{i}^{t-1})-\nabla f_{i} (x_*))||^2\right]. \notag \end{aligned}$$ Next we bound the term $\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n ||(\nabla f_{i}(\phi_{i}^{t-1})-\nabla f_{i} (x_*))||^2\right]$ for $t\geq 1$: $$\begin{aligned}
&\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n ||(\nabla f_{i}(\phi_{i}^{t-1})-\nabla f_{i} (x_*))||^2\right] \notag \\
=& \mathbb{E}\left[ \mathbb{E}_{i_{t-1}}\left[\frac{1}{n}\sum_{i=1}^n ||(\nabla f_{i}(\phi_{i}^{t-1})-\nabla f_{i} (x_*))||^2|i_1, \ldots , i_{t-2}\right]\right]\notag \\
=& \mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n\mathbb{E}_{i_{t-1}}\left[ ||(\nabla f_{i}(\phi_{i}^{t-1})-\nabla f_{i} (x_*))||^2|i_1, \ldots , i_{t-2}\right]\right] \notag \\
=&\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n\left[ \frac{1}{n} ||(\nabla f_{i}(u_{t-2})-\nabla f_{i} (x_*))||^2 + \left(1-\frac{1}{n}\right) ||(\nabla f_{i}(\phi_{i}^{t-2})-\nabla f_{i} (x_*))||^2\right] \right] \notag \\
=& \frac{1}{n}\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(u_{t-2})-\nabla f_{i} (x_*))||^2\right] + \left(1-\frac{1}{n}\right)\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(\phi_{i}^{t-2})-\nabla f_{i} (x_*))||^2\right] \notag \\
=& \cdots \notag \\
=& \sum_{j=1}^{t-1}\frac{1}{n}\left(1-\frac{1}{n}\right)^{t-1-j}\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(u_{j-1})-\nabla f_{i} (x_*))||^2\right] \notag \\
&+ \left(1-\frac{1}{n}\right)^{t-1}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(\phi_{i}^0)-\nabla f_{i} (x_*))||^2\right]. \notag\end{aligned}$$ Here we defined $\sum_{j=1}^0 = 0$ for $t=1$.
Using these inequalities and the definition of $\eta$, we have $$\begin{aligned}
&\sum_{t=1}^{m_s} \left[ \frac{t}{\eta t-L_{\mathrm{max}}}\mathbb{E}||g_t - \nabla F(u_{t-1})||^2 - \frac{1}{L_{\mathrm{max}}}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}||\nabla f_i(u_{t-1})-\nabla f_i(x_*)||^2\right] \right] \notag \\
=&\sum_{t=1}^{m_s} \left[ \frac{1}{2L_{\mathrm{max}}}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n ||(\nabla f_{i}(\phi_{i}^{t-1})-\nabla f_{i} (x_*))||^2\right]- \frac{1}{2L_{\mathrm{max}}}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n ||\nabla f_{i}(u_{t-1})-\nabla f_{i} (x_*)||^2\right] \right] \notag \\
=&\frac{1}{2L_{\mathrm{max}}}\sum_{t=1}^{m_s} \left[ \sum_{j=1}^{t-1}\frac{1}{n}\left(1-\frac{1}{n}\right)^{t-1-j}\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(u_{j-1})-\nabla f_{i} (x_*))||^2\right] \right] \notag \\
&+ \frac{1}{2L_{\mathrm{max}}}\sum_{t=1}^{m_s}\left(1-\frac{1}{n}\right)^{t-1}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(\phi_{i}^0)-\nabla f_{i} (x_*))||^2\right] \notag \\ &- \frac{1}{2L_{\mathrm{max}}}\sum_{t=1}^{m_s}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}||\nabla f_i(u_{t-1})-\nabla f_i(x_*)||^2\right] . \notag \end{aligned}$$ Observe that $$\begin{aligned}
&\sum_{t=1}^{m_s} \left[ \sum_{j=1}^{t-1}\frac{1}{n}\left(1-\frac{1}{n}\right)^{t-1-j}\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(u_{j-1})-\nabla f_{i} (x_*))||^2\right] \right] \notag \\
=&\sum_{t=2}^{m_s}\frac{1}{n}\left(1-\frac{1}{n}\right)^{t-2}\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(u_{0})-\nabla f_{i} (x_*))||^2\right] \notag \\
&+ \sum_{t=3}^{m_s}\frac{1}{n}\left(1-\frac{1}{n}\right)^{t-3}\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(u_{1})-\nabla f_{i} (x_*))||^2\right] \notag \\
&+\cdots \notag \\
&+ \sum_{t=m_{s}}^{m_s}\frac{1}{n}\left(1-\frac{1}{n}\right)^{t-m_s}\mathbb{E}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(u_{m_s-2})-\nabla f_{i} (x_*))||^2\right] \notag \\
\leq&\sum_{t=1}^{m_s}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}||\nabla f_i(u_{t-1})-\nabla f_i(x_*)||^2\right]. \notag \end{aligned}$$ By Lemma \[object bound lemma\] and the definition of $\phi_i^0$, we get $$\begin{aligned}
&\sum_{t=1}^{m_s}\left(1-\frac{1}{n}\right)^{t-1}\left[\frac{1}{n} \sum_{i=1}^n||(\nabla f_{i}(\phi_{i}^0)-\nabla f_{i} (x_*))||^2\right] \notag \\
=&2m_s L_{\mathrm{max}}(P(x_0)-P(x_*)-\frac{\mu}{2}||x_0-x_*||^2).\notag \end{aligned}$$ Hence we get $$\begin{aligned}
&\sum_{t=1}^{m_s} \left[\frac{t}{\eta t-L_{\mathrm{max}}}\mathbb{E}||g_t - \nabla F(u_{t-1})||^2 - \frac{1}{L_{\mathrm{max}}}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}||\nabla f_i(u_{t-1})-\nabla f_i(x_*)||^2\right] \right] \notag \\
\leq&m_s \left(P(x_0)-P(x_*)-\frac{\mu}{2}||x_0-x_*||^2\right).\notag\end{aligned}$$ Combining Lemma \[main lemma2\] with this result yields $$\begin{aligned}
&\mathbb{E} [P({x}_{m_s})-P(x_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||{v}_{m_s} - x_*||^2 \notag \\
\leq& \frac{1}{2}\left(P(x_0)-P(x_*)-\frac{\mu}{2}||x_0-x_*||^2\right)+ \frac{\eta}{2m_s}||{v}_{0} - x_*||^2. \notag\end{aligned}$$ Since $x_{m_s}=\widetilde{x}_s$, $v_{m_s}=\widetilde{v}_s$, $x_0=\widetilde{x}_{s-1}$, and $v_0=(1-\alpha)\widetilde{v}_{s-1}+\alpha\widetilde{x}_{s-1}$, we obtain $$\begin{aligned}
&\mathbb{E} [P(\widetilde{x}_{s})-P(x_{*})] + \frac{\eta+m_s\mu}{2m_s} \mathbb{E}||\widetilde{v}_{s} - x_*||^2 \notag \\
\leq& \frac{1}{2}(P(\widetilde{x}_{s-1})-P(x_*)) + \left(\frac{\alpha \eta}{2m_s}-\frac{\mu}{4}\right)||\widetilde{x}_{s-1} - x_*||^2 \notag \\ &+\frac{(1-\alpha)\eta}{2m_s}||\widetilde{v}_{s-1} - x_*||^2. \notag\end{aligned}$$ Finally, taking expectations with respect to all previous stages gives the desired result.
[^1]: Email: [email protected]
[^2]: Email: [email protected]
|
---
abstract: 'The NA60 experiment has measured low-mass muon pair production in In-In collisions at 158 A GeV with unprecedented precision. We show that this data is reproduced very well by a dynamical model with parameters scaled from fits to measurements of hadronic transverse mass spectra and Hanbury-Brown and Twiss correlations in Pb-Pb and Pb-Au collisions at the same energy. The data is consistent with in-medium properties of $\rho$ and $\omega$-mesons at finite temperature and density as deduced from empirical forward-scattering amplitudes. Inclusion of the vacuum decay of the $\rho$-meson after freeze-out is necessary for an understanding of the mass and transverse momentum spectrum of dimuons with $M \apprle 0.9~{\rm GeV}/c^2$.'
author:
- Jörg Ruppert
- Charles Gale
- Thorsten Renk
- Peter Lichard
- 'Joseph I. Kapusta'
title: Low Mass Dimuons Produced in Relativistic Nuclear Collisions
---
The main goal of the relativistic nuclear collision program is to produce and study strongly interacting matter at high temperature and density. It is hoped that exotic many-body effects may be uncovered, one of them being a quark-gluon plasma (QGP), a state where hadronic matter exhibits partonic behavior. However, other interesting phases may also manifest themselves. In this context, electromagnetic observables - real and virtual photons - constitute a privileged class of probes because of the near absence of final state effects. The radiation will travel essentially unscathed from its production point to the detectors. As the system expands and cools, a quantitative understanding of the net electromagnetic spectrum requires a detailed understanding of the local emissivity as well as knowledge of the space-time evolution of the radiating matter.
The recent NA60 experiment at the CERN Super-Proton Synchrotron (SPS) has measured the production of low-mass muon pairs in In-In collisions at 158 A GeV. In this experiment the spectra of invariant mass ($M$) in the region $M \apprle 1.5$ GeV/$c^2$, and of transverse momentum ($p_T$), were obtained with unprecedented precision [@NA60data]. While the invariant mass spectrum is essential to characterize in-medium changes to the electromagnetic current-current correlation function, the transverse momentum spectrum of dileptons is especially sensitive to the interplay between production processes and collective transverse flow. A simultaneous description of $p_T$ and $M$ spectra therefore constitutes a stringent test of the dynamical evolution of the produced matter and of our understanding of in-medium modifications of vector mesons as revealed through thermal dilepton production. In this paper we pursue a theoretical interpretation of the recent NA60 data that involve folding microscopic dilepton emission rates with a dynamical evolution model. The local pair production rate may be written as [@KGbook] $$\label{eqn1}
\frac{dN}{d^4x d^4q}=\frac{\alpha^2}{12 \pi^4} P(M) R(M,\vec{q}) f_{\rm B}(q_0,T)$$ where $\alpha$ is the electromagnetic fine-structure constant, $T$ is temperature, and $M$, $q_0$ and $\vec{q}$ are the dimuon mass, energy and momentum, respectively. The function $P(M)$ accounts for the phase-space reduction due to the finite rest mass of the muon and does not depend on the medium’s properties. The function $f_{\rm B}$ is the Bose-Einstein distribution. The virtual photon spectral function (averaged over polarizations), $R(M,\vec{q})$, is directly related to the retarded in-medium electromagnetic current-current correlator $\Pi^{\mu \nu}_{\rm em}$ via $R=-(4\pi/M^2) {\rm Im} \Pi^\mu_{{\rm em}, \mu}$.
In the low invariant mass region, $M \apprle 1~{\rm GeV}/c^2$, the in-medium modifications of this correlator in the hadronic phase are directly related to spectral properties of light vector mesons ($\rho$, $\omega$ and $\phi$) through vector meson dominance (VMD) [@sakurai]. Of these, the contribution from the $\rho$-meson is the largest. Our focus here is on contributions from the $\rho$ and $\omega$ mesons; the modifications of the $\phi$-meson will be studied elsewhere. The in-medium vector meson spectral densities are evaluated in an approach where it is assumed that the vector-isovector and vector-isoscalar fields are modified mainly through scattering from nucleons and pions in the heat bath [@Kapusta]. One can infer the finite temperature and density dependence of the spectral functions in the single scattering limit by extracting the leading term in the self-energy expansion. More specifically, the self-energy is related to forward scattering amplitudes which are evaluated in a two-component model that involves the excitation of s-channel resonances upon a background - dual to the Pomeron - which becomes important especially at high energies [@Kapusta; @Martell]. Alternatively, effective hadronic Lagrangian [@Rapp:1999ej; @RG] or chiral reduction techniques [@DZ] may be used.
![The imaginary part of the $\rho$-meson propagator for temperature $T$ = 150 MeV and baryonic density 0.5 and 1 (in units of nuclear matter density). The imaginary part of the vacuum propagator is also shown.[]{data-label="spectral"}](compareKapustaaugust.eps){width="45.00000%"}
The results from both methods yield a negative contribution to the pole mass from interactions with pions and a positive contribution from interactions with nucleons; the net deviation from the vacuum mass being small (at most a few tens of MeV) at temperatures and densities such as those probed in 158 A GeV In-In collisions. The spectral width, however, is increased considerably owing to interactions with the medium [@Kapusta]; these features are shown in Figure \[spectral\]. Collisions with nucleons are the dominant effect but collisions with pions also contribute [@Kapusta; @Martell].
Two additional sources are emission from a thermalized partonic phase and from four-pion annihilation processes. A deconfined phase existing at early times and high temperatures is modeled by means of a quasiparticle model [@FireballMunich]. Previous studies have made clear that quantifying dimuon emission in the invariant mass region $M \apprge 1~{\rm GeV}/c^2$ requires the inclusion of four-pion states [@ligale]. In the present work the required matrix elements were evaluated in the Lagrangian approach of Ref. [@Lichard] where the inverse process, $e^+e^-$ annihilation in the vacuum into four charged pions, was studied. The model was shown to provide reasonable agreement with the measured cross sections. In addition, we included the annihilation of two neutral and two charged pions within the same framework supplemented by intermediate states containing the $\omega$ and $h_1$ mesons.
Two contributions other than radiation from the thermalized expanding medium are essential to the understanding of the experimentally observed spectra in the mass region $M \apprle 1.5~{\rm GeV}/c^2$: the dileptons emitted from hadronic decays after the system has blown apart, and those from correlated charm decays ($D$, $\bar{D}$). The late hadronic decay contribution (for example, the spectral profile contributions from vacuum $\phi$ and $\omega$ decay) has been subtracted in the $M$ and $p_T$ distribution by NA60, but not that of the freeze-out $\rho$’s. The spectra analyzed by NA60 therefore reflect not only the [*excess*]{} dimuon spectrum, but also include a substantial contribution from the decay of vacuum $\rho$-mesons outside of the medium after hadronic freeze-out. Since those $\rho$-mesons are strongly influenced by transverse flow, it is essential to go beyond the statistical model formulation and calculate their momentum spectrum after decoupling using the Cooper-Frye formula [@OurModel]. This again stresses the need for a realistic description of the spatial and temporal evolution of the system.
A detailed description of the fireball evolution model for nucleus-nucleus collisions at the CERN SPS can be found in [@RenkModel; @OurModel]. The main assumption is that an equilibrated system is formed a short time $\tau_0$ after the nuclear impact. The fireball subsequently expands isentropically until mean free paths exceed the system size and particles free-stream to the detectors. This is assumed to occur at a freeze-out time (or temperature) that is the same for all hadronic species. The entropy density $s(\tau,\eta_s,r)$ is described by the product of two Woods-Saxon distributions $s=N R(r,\tau) H(\eta_s,\tau)$ that depend on the spacetime rapidity $\eta_s=\frac{1}{2} {\rm ln} \left(\frac{t+z}{t-z}\right)$ and the transverse-plane radius $r$. The $N$ is a normalization constant. The Woods-Saxon profiles $R(r,\tau)=\left(1+{\rm exp}\left[\left(r-R_c(\tau)\right)/d_{\rm ws}\right]\right)^{-1}$ and $H(\eta_s,\tau)=\left(1+{\rm exp}\left[\left(|\eta_s|-H_c(\tau)\right)/\eta_{\rm ws}\right]\right)^{-1}$ are characterized by the thickness parameters $d_{\rm ws}$ and $\eta_{\rm ws}$ and by the size of the emitting zone as a function of proper time via $R_c(\tau)$ and $H_c(\tau)$. The latter two functions are calculated under the assumption of constant radial and longitudinal acceleration, respectively. This translates into the rapidity of the fireball front being $\eta_s(\tau)=\eta_0+a_\eta \tau$. The parameter $2\eta_0$ is the initial size occupied by the fireball and $a_\eta$ is a longitudinal expansion parameter. The initial radial extension $R_c(0)$ is determined in a calculation of the initial density profile using the Glauber model. Transverse flow is described best if transverse rapidity $\rho_T$ scales like $\sqrt{r}$ [@PTpaper1]. The accelerated longitudinal expansion driven by strong initial compression implies that in general space-time and momentum rapidity are not the same; their mismatch $\Delta \eta$ can be seen as a characterization of how much the solution departs from the ideal Bjorken [@BJ] scenario. The parameters $\tau_0$, $a_{\perp}$, $a_\eta$, $\eta_0$, together with the decoupling temperature $T_f$, set the scale of the spacetime evolution, and $d_{\rm ws}$ and $\eta_{\rm ws}$ specify the details of the entropy density.
The parameters listed above should in principle be fit to hadronic data and then used to predict the dimuon spectra. However, the necessary hadronic data for In-In collisions is not yet available; fortunately, the data for Pb-Pb and Pb-Au collisions at the same beam energy are [@NA49; @CERES]. The same theoretical framework as described here has proven successful in the description of photon and dilepton emission and charmonium supression in those collisions [@RenkModel]. For the collisions of In on In at the SPS, the total entropy in semi-central In-In collisions at $\eta=3.8$ measured by NA60 [@NA60data] is obtained from that in peripheral (30%) Pb-Au collisions with $2.1<\eta<2.55$ measured by CERES [@CERES2] by multiplying by the ratio of charged particle rapidity densities $dN_{\rm ch}/d\eta$ measured in both experiments. The number of participant baryons and initial spatial extent are obtained via geometrical nuclear overlap calculations, while $\eta_0$ is determined under the assumption that stopping power scales approximately with the number of binary collisions per participant. The electromagnetic emission near midrapidity turns out to be quite insensitive to changes in the values of $\eta_{\rm ws}$ and $d_{\rm ws}$: these are therefore not modified. The parameters of the accelerated expansion $a_{\perp}$ and $a_\eta$ as well as the equilibration time are assumed to be primarily determined by the incident system energy so they are kept as in Pb-Pb collisions. The largest uncertainty is the choice of the decoupling temperature. Because the In-In system is smaller than Pb-Pb, a higher decoupling temperature is expected. However, to give an unambiguous answer would only be possible with simultaneous measurements of HBT correlations and transverse mass spectra. Here we choose $T_f=130$ MeV. As alluded to earlier, kinetic equilibrium of all processes is assumed until this universal decoupling temperature is reached. The fact that the four-pion processes contribute all the way down to $T_f$ translates into an upper limit to their contribution, as microscopic descriptions of the dynamics (see, for example, [@Nonaka]) suggest a sequential decoupling of the different channels. The equation of state in the hadronic phase and the chemical potentials $\mu_{\pi}, \mu_{K}$ are inferred from statistical model calculations as described in [@RenkModel; @Renk:2002sz]. The resulting fireball has a peak temperature of about 250 MeV and a lifetime of about 7.5 fm/$c$.
The thermal contributions have to be calculated by folding the space-time evolution of the expanding matter with the thermal rates. For the mass spectra this amounts to $$\nonumber
\frac{dN}{M dM d\eta}= \int d^4x \int d\psi \int dp_T p_T {\cal A}(M,p_T,\eta) \frac{dN}{d^4x d^4q}$$ where ${\cal A}$ represents the detector acceptance of NA60. The space-time evolution enters via the dependence of the thermal rates on temperature, baryon and pion chemical potential, and energy and momentum of the decaying virtual photon in the local rest frame. The equilibrium thermal rates have been augmented by appropiate fugacity factors ${\rm exp}(n \mu_{\pi}/T)$ with $n=2,3,4$ for the thermal $\rho$, $\omega$, and four-pion annihilation contributions.
A comparison of a theoretical calculation of the invariant mass spectra, with its different components, with NA60 measurements in semi-central In-In collisions at the SPS is presented in Fig. \[mspectra\].
![Mass spectrum in semi-central In-In collisions at SPS [@NA60data] compared to theory. The lower panel shows an integration over all $p_T$, the upper panel one over high transverse momenta $1~{\rm GeV}/c<p_T<2~{\rm GeV}/c$. Partial contributions arise from $\rho$ decays in vacuum after freeze-out, thermal in-medium $\rho$ and $\omega$ decays, radiation from a thermalized QGP and from thermal four-pion annihiliation, and from correlated open charm decay.[]{data-label="mspectra"}](twoinone-final-cooper-correctedPRL4-october.eps){width="50.00000%"}
Consider the all $p_T$-data in the lower panel first. The region below $1~{\rm GeV}$ and at masses smaller than the vacuum mass of the $\rho$ meson clearly demonstrates a considerable in-medium broadening of the $\rho$ and $\omega$ mesons. The decays of the vacuum $\rho$ after freeze-out are important, and in certain mass ranges are of the same order as the in-medium $\rho$ and $\omega$ meson decay contributions. The contribution from the QGP phase becomes more important and eventually dominant with higher $M$. We find that four-pion annihilation processes are subdominant even for masses above $M \apprge 1.25~{\rm GeV}/c^2$. This differs from the findings in [@vanHeesRapp] where the four-pion annihilation was derived in the soft pion limit, assuming chiral mixing.
Even though the spectrum is integrated over all $p_T$, and mass is Lorentz invariant, a successful theoretical understanding of these spectra still requires a detailed understanding of the $p_T$ spectra since the acceptance ${\cal A}$ is transverse momentum-dependent. Additional information is obtained if one compares the theoretical prediction for momentum cuts of the mass spectrum with the experimental data. Since the acceptance restricts considerably the information that can be obtained from the mass spectrum at low transverse momenta, this contribution is not shown even though our model does provide a good description of the spectrum in this window of $0<p_T<0.5~{\rm GeV}/c$. At higher transverse momenta, $1~{\rm GeV}/c<p_T< 2~{\rm GeV}/c$, the relative contribution of dimuons from vacuum $\rho$-decays is relatively enhanced as the emission then occurs at a later stage of the evolution where considerable transverse flow has already built up.
![Transverse momentum spectra as obtained from theory for semi-central In-In collision. The acceptance-corrected data are from NA60 [@NA60data].[]{data-label="pspectra"}](NA60p_TspecPRL4_october.eps){width="50.00000%"}
The robustness of our theoretical understanding of the collision can be further assessed from a comparison with the acceptance corrected $p_T$ spectra for different mass windows: see also [@PTpaper1]. Figure \[pspectra\] shows the $p_T$ spectra in three different mass windows for semi-central collisions. (The data are averaged over different centrality classes excluding peripheral collisions [@NA60data]. We also performed this averaging in [@PTpaper1; @PTpaper2] and found that differences between the averaged data and the semi-central collision data are small.)
The mass region $0.4~{\rm GeV}/c^2<M<0.6~{\rm GeV}/c^2$ and the $\rho$-like mass region $0.6~{\rm GeV}/c^2<M<0.9~{\rm GeV}/c^2$ receive most of their contribution from the late hadronic stages, namely, from decays of in-medium vector mesons and the vacuum $\rho$ mesons after freeze-out. In those stages considerable flow has already been built up which implies that the blueshift of the spectra by flow is large. The difference between the $\rho$-like and the lower mass region in the spectra is caused by the different contributions of the in-medium vector mesons and the vacuum $\rho$. The latter receives the maximum flow and predominantly contributes to the $\rho$-like region for momenta above $\sim 1~{\rm GeV}/c$. The slope of the transverse momentum spectrum in the mass region $1.0~{\rm GeV}/c^2<M<1.4~{\rm GeV}/c^2$ is dominated by contributions from the early QGP phase where flow has not yet built up while a contribution from four-pion annihilation processes is subdominant. If four-pion annihilation processes were more substantial this would result in considerable hardening of the $p_T$-spectrum which is not observed [@NA60data; @PTpaper1; @PTpaper2].
In conclusion, we have shown that low mass dimuon production as measured in 158 A GeV In-In collisions at the CERN SPS reflects substantial in-medium broadening of the $\rho$ meson spectral function in the hot and dense nuclear medium. Furthermore, we found that at higher invariant masses thermal radiation with $T>170~{\rm MeV}$ dominates over four pion annihiliation processes. This is especially relevant to a theoretical understanding of intermediate mass dimuon production.
*Acknowledgments.* We are grateful to S. Damjanovic for help concerning the implementation of the NA60 acceptance and to her and H. Specht for discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada, by the Academy of Finland, by Czech Ministry of Education grant LC07050, and by DOE grant DE-FG02-87ER40328.
[10]{} url \#1[\#1]{}urlprefix\[2\]\[\][[\#2](#2)]{}
R. Arnaldi, [*et al.*]{} \[NA60 Collaboration\], Phys. Rev. Lett. [**96**]{}, 162302 (2006); S. Damjanovic, [*et al.*]{} \[NA60 Collaboration\], Nucl. Phys. A [**783**]{}, 327 (2007). J. I. Kapusta and C. Gale, [*Finite-Temperature Field Theory: Principles and Applications*]{}, Cambridge University Press (Cambridge, 2006), and references therein.
J. J. Sakurai, [*Currents and Mesons*]{}, University of Chicago Press (Chicago, 1969).
V. L. Eletsky, M. Belkacem, P. J. Ellis and J. I. Kapusta, Phys. Rev. D [**66**]{}, 010001 (2002). A. T. Martell and P. J. Ellis, Phys. Rev. C [**69**]{}, 065206 (2004).
R. Rapp and J. Wambach, Adv. Nucl. Phys. [**25**]{}, 1 (2000). R. Rapp and C. Gale, Phys. Rev. C [**60**]{}, 024903 (1999). K. Dusling, D. Teaney and I. Zahed, Phys. Rev. C [**75**]{}, 024908 (2007).
T. Renk, R. A. Schneider and W. Weise, Phys. Rev. D [**66**]{}, 010001 (2002). G. Q. Li and C. Gale, Phys. Rev. Lett. [**81**]{}, 1572 (1998); Phys. Rev. C [**58**]{}, 2914 (1998).
P. Lichard and J. Juran, hep-ph/0601234. T. Renk and J. Ruppert, hep-ph/0605130.
T. Renk, J. Phys. G [**30**]{}, 1495 (2004).
T. Renk and J. Ruppert, hep-ph/0612113.
J. D. Bjorken, Phys. Rev. D [**27**]{}, 140 (1983).
S. V. Afanasiev, [*et al.*]{} \[NA49 Collaboration\], Phys. Rev. C [**66**]{}, 054902 (2002).
D. Adamova [*et al.*]{} \[CERES collaboration\], Nucl. Phys. A [**714**]{}, 124 (2003); H. Tilsner and H. Appelshauser \[CERES Collaboration\], Nucl. Phys. A [**715**]{}, 607 (2003).
G. Agakishiev, [*et al.*]{} \[CERES Collaboration\], Phys. Rev. Lett. [**75**]{}, 1272 (1995); G. Agakishiev, [*et al.*]{} \[CERES Collaboration\], Nucl. Phys. A [**638**]{}, 467 (1998); B. Lenkeit, [*et al.*]{} \[CERES-Collaboration\], Nucl. Phys. A [**661**]{}, 23 (1999). C. Nonaka and S. A. Bass, Phys. Rev. C [**75**]{}, 014902 (2007).
T. Renk, Phys. Rev. C [**68**]{}, 064901 (2003).
H. van Hees and R. Rapp, Phys. Rev. Lett. [**97**]{}, 102301 (2006). J. Ruppert and T. Renk, J. Phys. G [**34**]{} S1047 (2007).
|
---
abstract: 'Superconducting qubits often show signatures of coherent coupling to microscopic two-level fluctuators (TLFs), which manifest themselves as avoided level crossings in spectroscopic data. In this work we study a phase qubit, in which we induce Rabi oscillations by resonant microwave driving. When the qubit is tuned close to the resonance with an individual TLF and the Rabi driving is strong enough (Rabi frequency of order of the qubit-TLF coupling), interesting 4-level dynamics are observed. The experimental data shows a clear asymmetry between biasing the qubit above or below the fluctuator’s level-splitting. Theoretical analysis indicates that this asymmetry is due to an effective coupling of the TLF to the external microwave field induced by the higher qubit levels.'
author:
- Jürgen Lisenfeld
- Clemens Müller
- 'Jared H. Cole'
- Pavel Bushev
- Alexander Lukashenko
- Alexander Shnirman
- 'Alexey V. Ustinov'
bibliography:
- 'RabiSpectroscopy.bib'
title: 'Rabi spectroscopy of a qubit-fluctuator system'
---
Spectroscopic analysis of superconducting qubits often shows clear signatures of avoided level crossings, indicating the presence of microscopic two-level fluctuators (TLFs) that can be in resonance with the qubit. Evidence for the existence of TLFs have been found in nearly all known types of superconducting qubits, including phase- [@Simmonds04; @camelback], flux- [@Plourde05; @lupascu08], charge- [@kim08], and transmon qubits [@Schreier08]. Since TLFs are considered to be a source of decoherence [@Simmonds04; @Martinis:2005p1096; @Muller:2009], experiments are usually conducted by biasing the qubit in a frequency range where none of these strongly coupled natural two-level systems are present. Alternatively, one can take advantage of the longer coherence times of TLFs as compared to the qubits for using them as a quantum memory [@Martinis:2008NatPhys]. Here we focus on the dynamics of the qubit-fluctuator system on or near resonance.
There are at least two possible mechanisms explaining the interaction of the TLFs with the qubit: (i) the TLF is an electric dipole which couples to the electric field in the qubit’s Josephson junction [@Martin05; @Martinis:2005p1096]. Nanoscale dipoles could emerge from metastable lattice configurations in the amorphous dielectric of the junction’s tunnel barrier [@Esquinazi]; (ii) the state of the TLF affects the critical current of the qubit’s Josephson junction [@Simmonds04; @deSousa:2009]. In this case the TLF could be related, e.g., to the formation of Andreev bound states at the interface between the superconductor and the insulator [@Faoro05; @deSousa:2009].
In this Letter, we explore the complexity of the dynamical behavior of a driven phase qubit operated in the vicinity of a resonance with a two-level fluctuator. Due to the strong coupling between the qubit and the TLF and equally strong Rabi driving, we observe the dynamics of the resulting 4-level hybrid system consisting of the microscopic defect state and the macroscopic phase qubit. Strong microwave driving of the coupled system leads to coherent oscillations, revealing a characteristic beating pattern which we analyze quantitatively. Our experimental data displays a distinct asymmetry of the system response with respect to the detuning between the qubit and the TLF. We argue that this asymmetry is due to Raman-like virtual processes involving higher quantum levels of the qubit, giving rise to an effective driving of the TLF.
The sample investigated in this study is a phase qubit [@Simmonds04], consisting of a capacitively shunted Josephson junction embedded in a superconducting loop. Its potential energy has the form of a double well for suitable combinations of the junction’s critical current (here, $I_c = 1.1 \mu$A) and loop inductance (here, $L = 720$ pH). For the qubit states, one uses the two Josephson phase eigenstates of lowest energy which are localized in the shallower of the two potential wells, whose depth is controlled by the external magnetic flux through the qubit loop. The qubit state is controlled by an externally applied microwave pulse, which in our sample is coupled capacitively to the Josephson junction. A schematic of the complete qubit circuit is depicted in Fig. \[fig:Splitting\](a). Details of the experimental setup can be found in Ref. . During all measurements presented in this paper, the sample was cooled to a temperature of 35 mK in a dilution refrigerator.
Spectroscopic data taken in the whole accessible frequency range between 5.8 GHz and 8.1 GHz showed only 4 avoided level crossings due to TLFs having a coupling strength larger than 10 MHz, which is about the spectroscopic resolution given by the linewidth of the qubit transition. In this work, we studied the qubit interacting with a fluctuator whose energy splitting was $\epsilon_f / h = 7.805$ GHz. From its spectroscopic signature shown in Fig. \[fig:Splitting\](b), we extract a coupling strength $v_{\perp} / h \approx 25$ MHz. The coherence times of this TLF were measured by directly driving it at its resonance frequency while the qubit was kept detuned[@Lisenfeld2010]. A $\pi$ pulse was applied to measure the energy relaxation time $T_{1, f} \approx 850$ ns, while two delayed $\pi/2$ pulses were used to measure the dephasing time $T_{2, f}^{*} \approx 110$ ns in a Ramsey experiment. To read out the resulting TLF state, the qubit was tuned into resonance with the TLF to realize an iSWAP gate, followed by a measurement of the qubit’s excited state.
![(color online) **(a)** Schematic of the phase qubit circuit. **(b)** Probability to measure the excited qubit state (color-coded) vs. bias flux and microwave frequency, revealing the coupling to a two-level defect state having a resonance frequency of 7.805 GHz (indicated by a dashed line). []{data-label="fig:Splitting"}](Fig1){width="\columnwidth"}
Experimentally, we observe the probability $P({\ensuremath{\left|e\right\rangle}})$ of the qubit being in its excited state after driving it resonantly with a short microwave pulse. Varying the duration $\tau$ of the microwave pulse allows us to observe the evolution of $P({\ensuremath{\left|e\right\rangle}})$ in the time domain. If the energy splitting of the qubit is tuned far away from that of the fluctuator, the qubit remains decoupled from the TLF and $P({\ensuremath{\left|e\right\rangle}})$ displays the usual Rabi oscillations in the form of an exponentially decaying sinusoid having only a single frequency component. For our qubit sample, which has coherence times of $T_{1,q}\approx 120$ ns and $T_{2,q} \approx 90$ ns, these oscillations have the characteristic decay time of about $115$ ns. If, in contrast, the qubit is tuned close to the resonance frequency of a TLF, the probability to measure the excited qubit state shows a complicated time dependence, which is very sensitive to the chosen qubit bias.
![(color online) **(a)** Experimentally observed time evolution of the probability to measure the qubit in the excited state, $P({\ensuremath{\left|e\right\rangle}})(t)$, vs. driving frequency; **(b)** Fourier-transform of the experimentally observed $P({\ensuremath{\left|e\right\rangle}})(t)$. The resonance frequency of the TLF is indicated by vertical lines. **(c)** Time evolution of $P({\ensuremath{\left|e\right\rangle}})$ and **(d)** its Fourier-transform obtained by the numerical solution of Eq. (\[eq:master\_eq\]) as described in the text, taking into account also the next higher level in the qubit. (As the anharmonicity $\Delta/h \sim 100$ MHz in our circuit is relatively small, this required going beyond the second order perturbation theory and analyze the 6-level dynamics explicitly). The qubit’s Rabi frequency $\Omega_{q}/h$ is set to 48 MHz. []{data-label="fig:DataRabi"}](Fig2){width="\columnwidth"}
Figure \[fig:DataRabi\](a) shows a set of time traces of $P({\ensuremath{\left|e\right\rangle}})$ taken for different microwave drive frequencies. Each trace was recorded after adjusting the qubit bias to result in an energy splitting resonant to the chosen microwave frequency. The Fourier transform of this data, shown in Fig. \[fig:DataRabi\](b), allows us to distinguish several frequency components. Frequency and visibility of each component depend on the detuning between qubit and TLF. We note a striking asymmetry between the Fourier components appearing for positive and negative detuning of the qubit relative to the TLF’s resonance frequency, which is indicated in Figs. \[fig:DataRabi\](a,b) by the vertical lines at 7.805 GHz. We argue below that this asymmetry is due to virtual Raman-transitions involving higher levels in the qubit.
To describe the system theoretically, we write down the Hamiltonian, consisting of two parts: $ \hat{H} = \hat{H}_{S} + \hat{H}_{I} $, with $\hat{H}_{S}$ being the system Hamiltonian, representing qubit, TLF and their coupling and $\hat{H}_{I}$ describing the interaction between system and microwave driving. The Hamiltonian of the qubit circuit reads $$H_{S}^{q} = E_{C} \left(n - n_{G} \right)^{2} - E_{J} \cos{\phi} + E_{L} \left( \phi - \phi_{ext} \right)^{2} \,,$$ where $E_{C/J/L}$ are charging/Josephson/inductive energies of the circuit, $\phi$ is the phase difference across the Josephson junction, and $n$ is the dimensionless charge conjugate to $\phi$, i.e., $[\phi,n]=i$. The circuit can be manipulated by applying an [*ac*]{} driving to gate charge $n_{G}$ or the external flux $\phi_{ext}$. The TLF is described as a two level system $H_{S}^{f} = 1/2 \epsilon_{f} \tau_{z}$ which couples either to the electric field across the junction $\propto (n-n_{G})$ or, alternatively, to the Josephson energy $\propto \cos{\phi}$. The coupling can be either transverse, $\propto \tau_{\pm}$, or longitudinal, $\propto \tau_{z}$.
For maximum generality, we first define a *minimal* model needed to describe the splitting of Fig. \[fig:Splitting\]. To this end, we restrict ourselves to the lowest two states of the phase qubit circuit (the qubit subspace) and disregard the longitudinal coupling $\propto \tau_{z}$. Within the rotating wave approximation (RWA) the Hamiltonian reads $$\hat{H}_{S}^{min} = \frac{1}{2} \epsilon_{q} \sigma_{z} + \frac{1}{2} \epsilon_{f} \tau_{z}
+ \frac{1}{2} v_{\perp} \left( \sigma_{-} \tau_{+} + \sigma_{+} \tau_{-} \right) \,,
\label{eq:HsMin}$$ with the Pauli-matrices for the qubit $\sigma$ and for the fluctuator $\tau$. The minimal interaction Hamiltonian couples only the qubit to the driving field via the coupling constant $\Omega_{q}$: $\hat{H}_{I}^{min} = \Omega_{q} \cos{(\omega_{d} t)} \: \sigma_{x}$. The RWA is justified since $\Omega_q \sim v_\perp \ll \epsilon_q \sim \epsilon_f$. Rabi oscillations in this minimal system have been considered earlier [@Ashhab:2006p37; @Galperin:2007p34].
Going to the rotating frame for both qubit and TLF and taking the frequency of the driving to be resonant with the qubit splitting, $\omega_{d} = \epsilon_{q}$, we arrive at the effective 4-level Hamiltonian $$H^{min} =- \frac{1}{2} \delta\omega \tau_{z}
+ \frac{1}{2} v_{\perp} \left( \sigma_{-} \tau_{+} + \sigma_{+} \tau_{-} \right)+
\frac{1}{2} \Omega_{q} \sigma_{x}\ ,
\label{eq:H4Levels}$$ where $\delta\omega = \left( \epsilon_{q} - \epsilon_{f} \right)$. The level structure and the spectrum of possible transitions in the Hamiltonian (\[eq:H4Levels\]) is illustrated in Fig. \[fig:Transitions\]a. The transition frequencies in the rotating frame correspond to the frequencies of the Rabi oscillations observed experimentally.
![(color online) **(a)** Analytically obtained transition spectrum of the Hamiltonian (\[eq:H4Levels\]) in the minimal model for $\Omega_q/h = 40$ MHz and $v_\perp/h = 25$ MHz. Dashed-dotted lines show the transition frequencies while the gray-scale intensity of the thicker lines indicates the weight of the respective Fourier-components in the probability $P({\ensuremath{\left|e\right\rangle}})$. The system shows a symmetric response as a function of the detuning $\delta\omega$. Two of the four lines are double degenerate. **(b)** The same as (a) but including the second order Raman process with $\Omega^{v}_{f} = v_{\perp} \Omega_{q} / \Delta$. The two degenerate transitions in (a) split and the symmetry of the response is broken. **Inset:** Schematic representation of the structure of the Hamiltonian (\[eq:H4Levels\]). We denote the ground and excited states of the qubit as ${\ensuremath{\left|g\right\rangle}}$ and ${\ensuremath{\left|e\right\rangle}}$ and those of the TLF as ${\ensuremath{\left|\downarrow\right\rangle}}$ and ${\ensuremath{\left|\uparrow\right\rangle}}$. Arrows indicate the couplings between qubit and fluctuator $v_{\perp}$ and to the microwave field $\Omega_{q}$ and $\Omega^{v}_{f} $. []{data-label="fig:Transitions"}](Fig3){width="\columnwidth"}
To describe the time evolution of our system we consider the state ${\ensuremath{\left|\Psi (t)\right\rangle}} = \sum_{k} c_{k} e^{-i E_{k} t} {\ensuremath{\left|k\right\rangle}}$, where $E_{k}$ are the eigenvalues and ${\ensuremath{\left|k\right\rangle}}$ the eigenstates of the Hamiltonian (\[eq:H4Levels\]). The coefficients $c_{k}$ are determined by the initial conditions. The eigenvectors ${\ensuremath{\left|k\right\rangle}}$ can be expressed as linear combinations of the measurement basis states ${\ensuremath{\left|g\downarrow\right\rangle}}$, ${\ensuremath{\left|g\uparrow\right\rangle}}$, ${\ensuremath{\left|e\downarrow\right\rangle}}$, ${\ensuremath{\left|e\uparrow\right\rangle}}$, i.e., the mutual eigenstates of $\sigma_z$ and $\tau_z$, which we denote by $\{ {\ensuremath{\left|l\right\rangle}} \}$ with $l=0,1,2,3$. For the expectation value of the operator $\sigma_z$ we get $$\begin{aligned}
{\ensuremath{\left\langle \Psi\right|}} \sigma_z {\ensuremath{\left|\Psi\right\rangle}} &=& \sum_{k,l,m} a^*_{k,l} a_{m,l} e^{-i (E_{m} - E_{k}) t} {\ensuremath{\left\langle l\right|}} \sigma_z{\ensuremath{\left|l\right\rangle}}\ ,
\label{eq:Operator}\end{aligned}$$ where $a_{k,l} = c_{k} {\ensuremath{\left\langle l\vphantom{k}\right.\left|\vphantom{l}k\right\rangle}}$ and we used the fact that $\sigma_z$ is diagonal in basis $\{ {\ensuremath{\left|l\right\rangle}} \}$. From Eq. (\[eq:Operator\]) we can extract the Fourier components of the experimentally measured excited state population $P({\ensuremath{\left|e\right\rangle}})=(1+{\ensuremath{\left\langle \sigma_z\right\rangle}})/2$. There are six components with, in general different, transition frequencies $E_{m} - E_{k}$. These are shown in Fig. \[fig:Transitions\]a for the minimal model. Only four lines are seen due to two double degeneracies. The intensity of the thick lines overlaying the dashed-dotted transition lines corresponds to the amplitude of these Fourier components. The situation depicted in Fig. \[fig:Transitions\] and realized in our experiment corresponds to the qubit and the fluctuator initially in their ground states. It is important to note that the pattern of Fig. \[fig:Transitions\] is characteristic for the regime $\Omega_q \sim v_\perp$.
As seen in Fig. \[fig:Transitions\]a the observed asymmetry in the response can not be explained by the minimal model. We identify three possible mechanisms which could break the symmetry: (i) Longitudinal coupling between qubit and TLF $H_{S}^{long} \sim v_{\parallel} \sigma_{z} \tau_{z}$. We note that the longitudinal coupling is excluded for the electric dipole coupling mechanism in phase and flux qubits, since this term would necessitate an average electric field (voltage) across the junction. The longitudinal coupling might be present if the TLF couples via a change in the critical current [@Simmonds04; @deSousa:2009]. In this case the state of the TLF directly affects the shape of the Josephson potential, therefore modulating the level-splitting of the qubit. For realistic parameters, this might lead to a strong longitudinal coupling $v_{\parallel}$. Such a coupling was, however, ruled out spectroscopically in Ref. as well as by our preliminary spectroscopic data [@Bushev2010]. (ii) Direct coupling of the TLF to the external field $H_{I}^{d} = \Omega_{f}^{d} \cos{(\omega_{d} t)} \: \tau_{x}$. Due to the presumably small size of the TLF this coupling should be negligible. (iii) Effective coupling of the TLF to the external driving field due to a second order Raman-like process in which the next higher level of the qubit ${\ensuremath{\left|e_2\right\rangle}}$ is virtually excited followed by a mutual flip of the TLF and the qubit (back to state ${\ensuremath{\left|e\right\rangle}}$). The energy difference between the states ${\ensuremath{\left|e_2\right\rangle}}$ and ${\ensuremath{\left|e\right\rangle}}$ is given by $\epsilon_q - \Delta$, where $\Delta$ characterizes the anharmonicity of the qubit. This gives an effective coupling $H_{I}^{v} = \Omega_{f}^{v} \cos{(\omega_{d} t)}\: \tau_{x} {\ensuremath{\left|e\right\rangle}}{\ensuremath{\left\langle e\right|}}$, i.e., the coupling is present only when the qubit is excited. For $\delta\omega < \Delta$, we find $\Omega_{f}^{v} \approx v_{\perp}\Omega_{q} / \Delta$. In Fig \[fig:Transitions\]b we show the spectrum of transitions with only the term $H_{I}^{v}$ added to the minimal model, Eq. (\[eq:H4Levels\]), not including longitudinal coupling or direct coupling of the TLF to the driving field.
To fully describe the experiment, we include decoherence in our calculations by solving the time evolution of the system’s density matrix $\rho$ using a standard Lindblad-approach [@Gardiner]. The dynamic equations are given by $$\dot{\rho} = i \left[ \rho, \hat{H} \right] + \sum_{j} \Gamma_{j} \left( L_{j} \rho L_{j}^\dagger -\frac{1}{2} \left\{ L_{j}L_{j}^\dagger, \rho \right\} \right)\,,
\label{eq:master_eq}$$ where the sum is over all possible channels of decoherence with the respective rates $\Gamma_{j}$. The $L_{j}$ are the operators corresponding to each decoherence channel, e.g., pure dephasing of the qubit is described by the operator $\sigma_{z}$. The theoretical spectral response of the system obtained by numerically solving the dynamical equations is shown in Fig. \[fig:DataRabi\](c, d). Relaxation and pure dephasing rates for qubit and TLF have been taken to be equal to the values mentioned earlier. The plot of Fig. \[fig:DataRabi\](c,d) takes into account the third level in the qubit. As the anharmonicity $\Delta/h \sim 100$ MHz is known from other measurements [@Bushev2010], we have no additional fit parameters and quantitatively reproduce the experimental data. Note that we are able to explain the experimental data by assuming $v_{\parallel}=0$, which provides further evidence in favor of the dipole coupling mechanism.
In conclusion, we studied the dynamics of a driven system consisting of a phase qubit strongly coupled to a TLF. The Fourier-analysis of the Rabi oscillation data reveals the characteristic pattern of transition frequencies in the coupled system. This asymmetric pattern is reproduced quantitatively by the presented theoretical model including virtual transitions to the qubit’s higher levels. The apparent absence of the longitudinal coupling between the qubit and the TLF gives a hint about the microscopic nature of the TLFs.
We would like to thank M. Ansmann and J. M. Martinis (UCSB) for providing us with the sample that we measured in this work. This work was supported by the CFN of DFG, the EU projects EuroSQIP and MIDAS, and the U.S. ARO under Contract No. W911NF-09-1-0336.
|
=6.5in =0.03in
=0.7cm =9.0in
[**Convergence Rates of Finite Difference Stochastic Approximation Algorithms [^1]**]{}
Liyi Dai\
Army Research Office\
Research Triangle Park, NC 27703\
[email protected]\
**Abstract**
stochastic approximation, Kiefer-Wolfowitz algorithm, mirror descent algorithm, finite-difference approximation, Monte Carlo methods
=-0.5in
[**1. Introduction.**]{} Let $R$ denote the set of real numbers. Consider a real-valued function $J(\theta)$ of the form $J(\theta)=E_{X}[L(X(\theta))]$ where $\theta$ is a parameter, or a vector of parameters, $L(X)$ is a real-valued function, and $X(\theta)$ is a random variable that depends on $\theta$. For simplicity, throughout this paper we assume that $\theta$ is a scalar and $\theta \in \Theta \subset R$, and $X(\theta)$ is of the form $X(\theta)= X(\theta,\xi)$, where $\xi$ is a random variable independent of $\theta$. In such a formulation, $X(\theta)$ is parameterized on an underlying probability space that is independent of $\theta$. For any two random variables $\eta$ and $\xi$, there exists a Borel function $\phi$ such that $\eta=\phi(\xi)$ \[Shiryayev (1984), p.172\]. Such a representation for $X(\theta,\xi)$ is always possible. Therefore, $J(\theta)$ can be written as $J(\theta) =
E_{\xi}[L(X(\theta,\xi))]$. We are particularly interested in finding an optimal parameter $\theta^{*}\in \Theta$ to optimize, say minimize, $J(\theta)$. This is a challenging problem since the analytical form of $J(\theta)$ is usually unavailable for most problems of interest. What is obtainable are the sample measurements of the random value of $L(X(\theta,\xi))$. We have to use the information on $L(X(\theta,\xi))$ to find $\theta^{*}$. Such stochastic optimization problems can be found in many applications. The main approach to finding the optimal solution is to successively approximate $\theta^{*}$ via algorithms of [*stochastic approximation*]{}. This is a classical and standard approach that has been adopted in practice for decades. The [*Robbins-Monro*]{} (RM) algorithm, the [*Kiefer-Wolfowitz*]{} (KW) algorithm, and the relatively recent mirror descent (MD) algorithm are the most popular algorithms of this class.
The RM algorithm, introduced by Robbins and Monro (1951), finds $\theta^{*}$ in the following way. Let $\theta_{0}$ be selected and $\{a_{n}\}$ a sequence of positive numbers. For each integer $n\geq 0$, let $$\theta_{n+1}=\theta_{n}-a_{n}g_{n}
\label{rm}$$ where $g_{n}$ is an unbiased estimate of the derivative $J'(\theta)$ of $J(\theta)$ with respect to $\theta$. Assume that $J'(\theta)$ exists on $\Theta$ and the variance of $g_{n}$ is uniformly bounded for all $n$. Assume $(\theta-\theta^{*})J'(\theta)>0$ for all $\theta\not= \theta^{*}$, $$\sum_{n}a_{n}=\infty, \;\; \sum_{n}a_{n}^{2}<\infty,$$ and that several other technical conditions are satisfied. Then $\{\theta_{n}\}$ converges to $\theta^{*}$ with probability one. The convergence rate (in terms of root mean square error) is $n^{-1/2}$. Note that this is the best possible rate of convergence for algorithms of the form (1) for stochastic optimization \[see, e.g., Fabian (1971)\].
The KW algorithm, introduced by Kiefer and Wolfowitz (1952), is a modification of the RM algorithm by approximating the gradient using a finite difference and finds $\theta^{*}$ recursively by $$\theta_{n+1} = \theta_{n}-a_{n}h_{n},
\label{kw}$$ where $$h_{n}
=\frac{L(X(\theta_{n}+\delta_{n},\xi_{1,n}))
-L(X(\theta_{n}-\delta_{n},\xi_{2,n}))}{2\delta_{n}},
\label{kw_h}$$ $\{\delta_{n}\}$ is a sequence of positive numbers, $L(X(\theta_{n}+\delta_{n},\xi_{1,n}))$ and $L(X(\theta_{n}-\delta_{n},
\xi_{2,n}))$ are two measurements of $L(X(\theta,\xi))$ at $\theta_{n}
+\delta_{n}$ and $\theta_{n}-\delta_{n}$, and $\xi_{1,n}, \xi_{2,n}$ are corresponding samples of $\xi$. Kiefer-Wolfowitz (1952) proved that if $J(\theta)$ is decreasing for $\theta<\theta^{*}$ and increasing for $\theta>\theta^{*}$, and if $$\delta_{n}\rightarrow 0,\;\; \sum_{n}a_{n}=\infty, \;\;
\sum_{n}a_{n}\delta_{n}<\infty, \;\;
\sum_{n}a_{n}^{2}/\delta_{n}^{2}<\infty,$$ the sequence $\{\theta_{n}\}$ converges to $\theta^{*}$ with probability one under some additional minor conditions. If all entries in $\{\xi_{i,n}\}$ are mutually independent, the best possible convergence rate for the KW algorithm (2) is $n^{-1/3}$ which is achieved by choosing $a_{n}=an^{-1}, \delta_{n}=d n^{-1/6}$ with $a, d > 0$ constants \[e.g. Burkholder (1956); Fabian (1971); Sacks (1958)\]. The rate $n^{-1/3}$ is regarded not satisfactory compared to the best possible rate $n^{-1/2}$ for the RM algorithm.
The MD algorithm, introduced by Nemirovski and Yudin (1983), improves the robustness of gradient based optimization algorithms. At iteration $n\geq 0$, $\theta_{n+1}$ is updated via solving $$\theta_{n+1} = \textrm{argmin}_{\theta\in\Theta}\left\{ <h_n,\theta>+\frac{1}{a_n}D_{\psi}(\theta,\theta_n)\right\},
\label{md}$$ where $h_n$ is an estimate of the derivative $J'(\theta)$, $D: \Theta\times\Theta\rightarrow R^+$ is a Bregman distance defined as $$D(\theta,\tau) := \psi(\theta)-\psi(\tau)-<\psi'(\tau),\theta-\tau> \geq \kappa ||\theta-\tau||^2,
\label{md_bd}$$ where $\psi(.)$ is a distance generating function and $\kappa>0$ is a constant. In (\[md\_bd\]), $||.||$ is a general norm on $R^m$ (and on $R$ in this paper). It has been established by Nemirovski et al. (2009) and Duchi et al. (2012,2013) that if $J(\theta)$ is convex, Lipschitz continuous and $$a_n \rightarrow 0, \;\; \sum_{n}a_{n}=\infty, \;\;$$ $$\hat{\theta}_n = \frac{1}{n}\sum_{i=1}^n \theta_i \textrm{ or } \hat{\theta}_n
=\sum_{i=1}^n \nu_i\theta_i,\;\; \nu_i = \frac{a_i}{\sum_{j=1}^i a_j}, i =1 , 2, ...$$ then $J(\hat{\theta}_n)$ converges to the minimum of $J(\theta)$ and the rate of convergence is $n^{-1/2}$ under mild technical conditions that will be specified in Section 3.
The convergence of these algorithm is fairly understood \[Burkholder (1956); Fabian (1971); Kushner and Clark (1978); Chung (1954); Dupa$\check{c}$ (1957); Dvoretzky (1956); Sacks (1958); Wasan (1969), Nemirovski et al. (2009), Duchi et al. (2012, 2013)\]. The conditions for the convergence of these algorithms can be made substantially weaker than those we have previously mentioned \[e.g. Kushner and Clark (1978); Wasan (1969)\]. The convergence rate for the RM algorithm is much faster than that for the KW algorithm. This is not surprising if we note that the KW algorithm uses the finite difference $h_{n}$ as an approximation to the derivative $J'(\theta)$, while the RM algorithm uses an unbiased estimate of $J'(\theta)$. Therefore, the faster rate is achieved at the cost of obtaining an unbiased estimate of the derivative $J'(\theta)$ that is often challenging in practice. On the other hand, although its convergence is slower, the KW algorithm requires no detailed information on the function $J(\theta)$. It is simple to use and applicable to a wide range of problems. Kesten (1958) suggests that the stepsize $a_{n}$ be chosen according to the fluctuation in the signs of $g_{n}$ and $h_{n}$. A few of other techniques for the acceleration of stochastic approximation algorithms can be found in Wasan (1986). None of these accelerating techniques can improve the rate of convergence of the algorithms under study.
In this paper, we are interested in the acceleration of the KW algorithm and the MD algorithm through controlling the estimation of the derivative using finite differences. Furthermore, we consider the employment of the scheme of common random numbers (CRN) for improving the convergence of the algorithms — that is, the random factors $\xi_{1,n}$ and $\xi_{2,n}$ are chosen in such a manner that $\xi_{1,n}=
\xi_{2,n}=\xi_{n}$. Implementation of CRN in Monte Carlo optimization is rather straightforward. The term “Monte Carlo optimization” is used here to refer to the procedure of finding the optimal solutions through computer simulation where the random factors, represented through a sequence of psuedo-random numbers, can be controlled \[see Bratley et al. (1983)\]. Computer simulation is often necessary when the form of $L(X(\theta,\xi))$ is too complicated. This is the case when $L(X(\theta,\xi))$ represents a performance measure of a stochastic system such as queueing systems, manufacturing systems, transportation systems, and communications networks \[see, e.g. Ho and Cao (1991); Bratley et al. (1983); Law and Kelton (1982)\]. In Section 6, we will give an example where the scheme of common random numbers is feasible.
We use the term CRN in a much narrow sense. The term CRN has more general and sometimes ill-posed meaning than we intend in this paper \[see Glasserman and Yao (1992)\]. In this paper, CRN simply refers to that simulation experiments be performed with the same stream of random numbers. As far as the KW algorithm (\[kw\]) or the MD algorithm (\[md\]) is concerned, CRN requires that estimates of $J(\theta+\delta)$ and $J(\theta-\delta)$ be obtained from simulation experiments using the same stream of random numbers $\{\xi_{n}\}$. Let $F(\theta,x)$ denote the distribution function of $X(\theta,\xi)$. Then any experiments with $h_{n}$ constructed in the following form conform the CRN requirement: $$\frac{L(Y_{1}(\theta_{n},\delta_{n},\xi_{n}))
-L(Y_{2}(\theta_{n},\delta_{n},\xi_{n}))}{2\delta_{n}}
\label{fd_g}$$ where the marginal distributions of $Y_{1}(\theta_{n},\delta_{n},\xi_{n})$ and $Y_{2}(\theta_{n},\delta_{n},\xi_{n})$ are $F(\theta_{n}+\delta_{n},x)$ and $F(\theta_{n}-\delta_{n},x)$, respectively. Note that the joint distribution of $Y_{1}(\theta_{n},\delta_{n},\xi_{n})$ and $Y_{2}(\theta_{n},\delta_{n},\xi_{n})$ is left open, which may be used to improve the estimation variance. For a distribution function $F(\theta,x)$, its inverse function is defined as $F^{-1}(\theta,x) \stackrel{\rm def}{=}
\inf\{u\;|\;F(\theta,u)> x\}$. Cambanis and Simons (1976) and Whitt (1976) proved that the variance of (\[fd\_g\]) is minimized when $Y_{1}(\theta_{n},\delta_{n},\xi_{n})=F^{-1}(\theta_{n}+
\delta_{n},\xi_{n})$ and $Y_{2}(\theta_{n},\delta_{n},\xi_{n})=
F^{-1}(\theta_{n}-\delta_{n},\xi_{n})$. In this paper we assume that the form of $L(X(\theta,\xi))$ is given and fixed. The term CRN merely refers to the special choice of $\xi_{1,n}=\xi_{2,n}$. We will show that the use of CRN can significantly increase the rate of convergence for the KW algorithm (\[kw\]) or the MD algorithm (\[md\]) from $n^{-1/3}$ to at least $n^{-2/5}$. For a large class of functions, the rate can be increased to $n^{-1/2}$, the best possible rate for stochastic approximation algorithms. CRN increases the rate of convergence by reducing the variance of $h_{n}$. Let $Var[X]$ denote the mathematical variance of a random variable $X$. Assume that $Var[L(X(\theta,\xi))]$ is continuous in $\theta$, is bounded from below by a positive constant and from above by a constant. Then if $\xi_{1,n}$ and $\xi_{2,n}$ are independent, the variance of $h_{n}$ is $$(Var[L(X(\theta_{n}+\delta_{n},\xi_{1,n}))]+
Var[L(X(\theta_{n}-\delta_{n},\xi_{2,n}))])/(2\delta_{n})^{2}
=O(1/\delta_{n}^{2})$$ which grows quadratically as $\delta_{n}$ goes to zero. We say a variable $f(s)=O(s)$ if $|f(s)/s|\leq C, C>0$ is a constant independent of $s$ ($f(s)=o(s)$ if $\lim|f(s)/s| = 0$ when $s$ goes to zero or infinity depending on the context). It is such a large variance of $h_{n}$ that slows down the convergence rate since, when $\delta_{n}$ is suitably chosen, the rate would be $n^{-1/2}$ if the variance of $h_{n}$ is bounded. As we will show later, the convergence rate for (\[kw\]) depends on how fast the variance of $h_{n}$ goes to infinity. The slower the variance goes to infinity, the faster the convergence rate for (\[kw\]) is. CRN has been observed effective for variance reduction for decades. It is perhaps the most popular method for variance reduction \[Bratley et al. (1983); Conway (1963); Fishman (1974); Hammersley and Handscomb (1964); Heikes et al. (1976); Kleijnen (1974); Law and Kelton (1982)\]. The rest of the paper is arranged as follows: In Section 2 we examine the rates of convergence for the KW algorithm under a very general setting that covers many interesting situations. the analysis is extended to the MD algorithms in Section 4. In Section 4 we show that the use of CRN can reduce the variance of $h_{n}$ by orders of magnitude, which in turn accelerates the convergence of the KW algorithm. In Section 5, we examine the rate of convergence for the MD algorithm under CRN. In Section 6, we extend the results to multivariates. A practical example is given to illustrate the feasibility of applying CRN in practice. Finally, a summary is provided in Section 7.
[**2. Rates of convergence for the KW algorithm.**]{} In this section, we examine the rates of convergence for the KW algorithm (\[kw\]) under general assumptions on $h_{n}$. We do not assume that $h_{n}$ is of the form (\[kw\_h\]). We will see later that such a treatment covers several important cases.
Assume that $\delta_{n}>0$ goes to zero as $n\rightarrow\infty$ and, for $n\geq n_{0}>1$, $h_{n}$ satisfies the following assumptions: $$E[h_{n}|\theta_{n}]=J'(\theta_{n})+\Delta_{n}, \;\;\;
|\Delta_{n}|\leq b\delta_{n}^{\beta},
\label{h1}$$ and $$Var[h_{n}|\theta_{n}]\leq c\delta_{n}^{\gamma},
\label{h2}$$ where $b, c, \beta$ are real nonnegative numbers, $\gamma\in R$. The form of (\[h1\]) assures that $h_{n}$ is an asymptotically unbiased estimate of $J'(\theta)$ when $\beta>0$. When $\gamma>0$, the variance of the estimate goes to zero as $n\rightarrow\infty$. This is generally impossible in practice. When $\gamma =0$ such as in the RM algorithm, the variance is bounded. In the case that $\gamma<0$, e.g. $\gamma=-2$ if $h_{n}$ is defined by (\[kw\_h\]) and if $\xi_{1,n}$ and $\xi_{2,n}$ are independent, the variance of $h_{n}$ goes to infinity. Next we examine the convergence and the rate of convergence for the KW algorithm (\[kw\]). The commonly used criterion for measuring the convergence of a stochastic sequence $\{\theta_{n}\}$ is the [*root mean square error*]{} (RMSE) defined as $$RMSE_{\theta_{n}} = (E[(\theta_{n}-\theta^{*})^{2}])^{1/2}.$$ If $RMSE_{\theta_{n}} = O(n^{-s}), s>0$, we say that $\{\theta_{n}\}$ converges at the rate of $n^{-s}$ or the convergence rate for $\{\theta_{n}\}$ is $n^{-s}$.
We need the next lemma that was due to Chung (1954) and was formulated in the present form by Fabian (1971).
[Lemma 1]{}. [*Let $s, t, B, A_{n}, b_{n}$ be real numbers, $0<s\leq 1$, $t\geq 0$, $B>0$. Define $b_{+}=0$ if $s<1$ and $b_{+}=t$ if $s=1$ and assume that $c=\lim_{n\rightarrow\infty} A_{n}-b_{+}$ exists and is finite. If for $n\geq n_{0}$, $$b_{n+1}\leq b_{n}(1-\frac{A_{n}}{n^{s}})
+\frac{B}{n^{s+t}}$$ and if $c>0$, then $$\lim_{n\rightarrow \infty}\sup n^{t}b_{n} \leq B/c.$$ The statement remains valid if all the inequalities are reversed and $\lim\sup$ is replaced by $\lim\inf$*]{}.
The following Theorems 1 and 2 give the convergence rate for the KW algorithm (\[kw\]) with $h_{n}$ satisfying (\[h1\])-(\[h2\]):
[Theorem 1]{}.
*Assume that $\{\theta_{n}\}$ is determined by (\[kw\]) and*
- $a_{n}=a n^{-\alpha}, \delta_{n}=d n^{-\eta}$, $0<\alpha
\leq 1$, $\eta>0$, $a, d > 0$;
- $J(\theta)$ is increasing for $\theta<
\theta^{*}$ and decreasing for $\theta>\theta^{*}$, and there exist two constants $K_{1}, K_{2}$, $0<K_{1}\leq K_{2}
<\infty$, such that for all $\theta\in \Theta$, $$K_{1}|\theta-\theta^{*}|\leq |J'(\theta)|\leq
K_{2}|\theta-\theta^{*}|;$$
- conditioned on $\theta_{n}$, $h_{n}$ at the $n$th iteration is independent of those at the other iterations.
Then, if $\sigma=(1/2)\min\{\alpha+\gamma\eta,2\beta\eta\}$ and $0<
\sigma < aK_{1}$, we have $$\lim_{n\rightarrow\infty}\sup
n^{2\sigma}E[(\theta_{n}-\theta^{*})^{2}]\leq C
\label{kw_rate1}$$ where $C > 0$ is a constant. The convergence rate for $RMSE_{\theta_{n}}$ is at least $n^{-\sigma}$
.
[Proof]{}. Without loss of generality, we assume that $\theta^{*}=0$. Then $$\begin{aligned}
E[\theta_{n+1}^{2}] & = &E[\theta_{n}^{2}]-2a_{n}E[\theta_{n}h_{n}]
+a_{n}^{2}E[h_{n}^{2}] \nonumber \\
& = & E[\theta_{n}^{2}]-2a_{n}E[\theta_{n}(J'(\theta_{n})
+\Delta_{n})]+a_{n}^{2}((E[h_{n}])^{2}+Var[h_{n}]).\end{aligned}$$ According to (\[h1\])-(\[h2\]), we have $$E[\theta_{n+1}^{2}] \leq
E[\theta_{n}^{2}]-2a_{n}E[\theta_{n}J'(\theta_{n})]
+2b a_{n}\delta_{n}^{\beta}E[|\theta_{n}|]
+2a_{n}^{2}(E[J'(\theta_{n})]^{2}
+b^{2}\delta_{n}^{2\beta})+c a_{n}^{2}\delta_{n}^{\gamma}.
\label{bound1}$$ By Assumption (A2), $\theta_{n}J'(\theta_{n}) \geq 0$ and $$\theta_{n}J'(\theta_{n})\geq K_{1}\theta_{n}^{2}, \;\;
(J'(\theta_{n}))^{2}\leq K_{2}^{2}\theta_{n}^{2}.
\label{bound_J}$$ Furthermore, for any $\epsilon_{n} > 0$, $|\theta_{n}|\leq \epsilon_{n}+
\theta_{n}^{2}/\epsilon_{n}$ and consequently $$E[|\theta_{n}|] \leq
\epsilon_{n}+\frac{1}{\epsilon_{n}}E[\theta_{n}^{2}].$$ By setting $0< \epsilon< 1$ and $$\epsilon_{n} = \frac{2b\delta_{n}^{\beta}}{K_{1}\epsilon},$$ we have $$E[|\theta_{n}|] \leq
\frac{2b\delta_{n}^{\beta}}{K_{1}\epsilon}
+\frac{K_{1}\epsilon}{2b\delta_{n}^{\beta}}E[\theta_{n}^{2}].
\label{bound_th}$$ Substituting (\[bound\_J\]) and (\[bound\_th\]) into (\[bound1\]), we obtain $$E[\theta_{n+1}^{2}]\leq E[\theta_{n}^{2}][1-(2-\epsilon)K_{1}a_{n}
+2K_{2}^{2}a_{n}^{2}]+2b^{2}a_{n}^{2}\delta_{n}^{2\beta}
+c a_{n}^{2}\delta_{n}^{\gamma}
+\frac{4b^{2}}{K_{1}\epsilon}a_{n}\delta_{n}^{2\beta}.
\label{bound_th2}$$ According to (\[bound\_th2\]), also noting Assumption (A1), we can choose an $n_{1}\geq n_{0}>1$ such that for all $n\geq n_{1}$ $$E[\theta_{n+1}^{2}]\leq E[\theta_{n}^{2}](1-\frac{A_{n}}{n^{\alpha}})
+\frac{B}{n^{\alpha+2\sigma}}$$ where $$A_{n}= (2-\epsilon)a K_{1}
-\frac{2K_{2}^{2}a^{2}}{n^{\alpha}},\;\;\;
B=2 a^{2}b^{2}d^{2\beta}+c a^{2}d^{\gamma}
+\frac{4a b^{2}d^{2\beta}}{K_{1}\epsilon}.$$ If $aK_{1}>\sigma$, we can always choose $\epsilon>0$ so small that $(2-\epsilon)aK_{1}>2\sigma$. Applying Lemma 1, we obtain (\[kw\_rate1\]) with $C=
B/((2-\epsilon)aK_{1})$ if $\alpha<1$ and $C=B/((2-\epsilon)aK_{1}-2\sigma)$ if $\alpha=1$.
------------------------------------------------------------------------
It follows directly from Theorem 1 that $\{\theta_{n}\}$ converges to $\theta^{*}$ as long as $\sigma > 0$, or equivalently, as long as $\alpha
+\gamma\eta > 0$. When $\alpha+\gamma\eta \leq 0$ which is possible only when $\gamma < 0$, the variance of $h_{n}$ grows to infinity at the rate of $n^{t}$ with $t=-\gamma\eta \geq \alpha$. It is obvious from (\[kw\]) that $\{\theta_{n}\}$ does not converge. Another extreme case is that $\gamma > 0$. In this case, $\sigma$ can be made arbitrarily large by choosing appropriate $\eta$. The convergence rate for $\{\theta_{n}\}$ can be made arbitrarily large if $\eta$ can take any value. In fact, by setting $\eta\rightarrow \infty$ in (\[bound\_th2\]) (or equivalently, $\delta_n\rightarrow 0$) and $a_{n}=a$ such that $ 0 < q=1-(2-\epsilon)K_{2}a
+2K_{2}^{2}a^{2} < 1$ for sufficiently large $n$, we have $$E[\theta_{n+1}^{2}] \leq q E[\theta_{n}^{2}].$$ The convergence rate for the sequence $\{\theta_{n}\}$ is that of a geometric progression. Unfortunately, this is a very special case. One should not expect $\gamma > 0$ in practice. Both of the situations $\gamma > 0$ and $\alpha+\gamma\eta\leq 0$ are too special to deserve further study. The most interesting case is when $\gamma$ satisfies $-\alpha/\eta < \gamma \leq 0$.
Theorem 1 shows that, when $h_{n}$ satisfies (\[h1\])-(\[h2\]), $\{\theta_{n}\}$ converges with probability one to the optimal parameter $\theta^{*}$ at a rate of at least $n^{-\sigma}$. We can further prove that $\{\theta_{n}\}$ converges exactly at this rate as interpreted in the following Theorem 2.
[Theorem 2]{}.
*Assume that Assumptions (A1)-(A3) are satisfied.*
1. If $\alpha+\gamma\eta < 2\beta\eta$, $aK_{2}>\sigma$, $E[h_{n}|\theta_{n}]=J'(\theta_{n})+\Delta_{n}$, $|\Delta_{n}|\leq
b\delta_{n}^{\beta}$, and $Var[h_{n}|\theta_{n}] \geq
c\delta_{n}^{\gamma}$, we have $$\lim_{n\rightarrow\infty}\inf
n^{2\sigma}E[(\theta_{n}-\theta^{*})^{2}]\geq C_{1}$$ where $C_{1}>0$ is a constant.
2. If $\alpha+\gamma\eta \geq 2\beta\eta$, $E[h_{n}|\theta_{n}]=
J'(\theta_{n})+b\delta_{n}^{\beta}(1+\varepsilon_{n})$, and $J'(\theta_{n})=(\theta_{n}-\theta^{*})(K_{3}+\tau_{n})$, $\varepsilon_{n}=o(1)$ and $\tau_{n}=o(1)$ uniformly as $n\rightarrow\infty$, $K_{3}=J''(\theta^{*})>0$, $\sigma < aK_{3}$, then $$\lim_{n\rightarrow\infty}\sup
n^{\sigma}E[\theta_{n}-\theta^{*}]\leq - C_{2}$$ where $C_{2}>0$ is a constant.
[Proof]{}. Let consider the first statement. For simplicity and without loss of generality, we assume $\theta^{*}=0$. Parallel to the derivation of (\[bound1\]) we have $$\begin{aligned}
E[\theta_{n+1}^{2}]
= E[\theta_{n}^{2}]-2a_{n}E[\theta_{n}(J'(\theta_{n})
+\Delta_{n})] +a_{n}^{2}\{(E[h_{n}])^{2}
+Var[h_{n}]\} \end{aligned}$$ which implies $$E[\theta_{n+1}^{2}]
\geq E[\theta_{n}^{2}]-2a_{n}E[\theta_{n}J'(\theta_{n})]
-2b a_{n}\delta_{n}^{\beta}E[|\theta_{n}|]
+c a_{n}^{2}\delta_{n}^{\gamma}.$$Assumption (A2) implies that $0\leq\theta_{n}J'(\theta_{n}) \leq
K_{2}\theta_{n}^{2}$ which, together with (\[bound\_th\]) where $K_{1}$ is replaced by $K_{2}$, shows that $$E[\theta_{n+1}^{2}]\geq E[\theta_{n}^{2}](1-(2-\epsilon)K_{2}a_{n})
+c a_{n}^{2}\delta_{n}^{\gamma}
-\frac{4b^{2}}{K_{2}\epsilon}a_{n}\delta_{n}^{2\beta}.
\label{bound_th3}$$
If $\alpha+\gamma\eta < 2\beta\eta$, there exists an $n_{0}>1$ such that when $n\geq n_{0}$ $$c a_{n}^{2}\delta_{n}^{\gamma}-\frac{4b^{2}}{K_{2}\epsilon}
a_{n}\delta_{n}^{2\beta}
\geq \frac{1}{2} c a_{n}^{2}\delta_{n}^{\gamma}.$$ Therefore, we know from (\[bound\_th3\]) that when $n\geq n_{0}$ $$E[\theta_{n+1}^{2}]\geq E[\theta_{n}^{2}](1-\frac{A_{n}}{n^{\alpha}})
+\frac{ca^{2}d^{\gamma}}{2n^{\alpha+2\sigma}}.$$ Since $aK_{2}>\sigma$, we can always choose $\epsilon>0$ so small that $(2-\epsilon)aK_{2}>2\sigma$. The first statement of the theorem follows from applying Lemma 1 with $C_{1}= ca^{2}d^{\gamma}/(2(2-\epsilon)aK_{2})$ if $\alpha<1$ and $C_{1}= ca^{2}d^{\gamma}/(2(2-\epsilon)aK_{2}-4\sigma)$ if $\alpha=1$.
If $\alpha+\gamma\eta \geq 2\beta\eta$, we know that $\sigma=\beta\eta>0$ and $$\begin{aligned}
E[\theta_{n+1}] &= & E[\theta_{n}] - a_{n}E[J'(\theta_{n})]-
b a_{n}\delta_{n}^{\beta}(1+\varepsilon_{n}) \nonumber \\
&=& E[\theta_{n}](1-K_{3}a_{n})+a_{n}E[\tau_{n}\theta_{n}]-
b a_{n}\delta_{n}^{\beta}(1+E[\varepsilon_{n}])
\label{eth1}\end{aligned}$$ Define $z_{n}=n^{\sigma}E[\theta_{n}]$. Then (\[eth1\]) shows that $$\begin{aligned}
z_{n+1} & = & n^{\sigma}(1+\frac{1}{n})^{n^{\sigma}}
[E[\theta_{n}](1-K_{3}a_{n})+a_{n}E[\tau_{n}\theta_{n}]
-b a_{n}\delta_{n}^{\beta}(1+\varepsilon_{n})] \nonumber \\
& = & z_{n}[1+\frac{\sigma}{n}-K_{3}a_{n}
-\frac{\sigma}{n}K_{3}a_{n}+O(\frac{1}{n^{2}})]
+(1+\frac{1}{n})^{\sigma} n^{\sigma}(a_{n}E[\tau_{n}\theta_{n}]
-ba_{n}\delta_{n}^{\beta}(1+E[\varepsilon_{n}])).\end{aligned}$$ Denote $$A_{n} = 1+\frac{\sigma}{n}-K_{3}a_{n}
-\frac{\sigma}{n}K_{3}a_{n}+O(\frac{1}{n^{2}}),$$ $$B_{n} = (1+\frac{1}{n})^{\sigma} n^{\sigma}(ba_{n}
\delta_{n}^{\beta}(1+E[\varepsilon_{n}])-a_{n}E[\tau_{n}\theta_{n}]).$$ Then $$z_{n+1}= A_{n}z_{n}-B_{n}.
\label{z1}$$ Note that $a_{n}=an^{-\alpha}, 0<\alpha\leq 1, \delta_{n}=dn^{-\eta},
aK_{3}>\sigma$. We may choose $\tilde{A}_{1}, \tilde{A}_{2}>0, n_{1}>1$ such that, for all $n \geq n_{1}$, $$0\leq 1-\frac{\tilde{A}_{1}}{n^{\alpha}}
\leq A_{n}=1+\frac{\sigma}{n}-\frac{aK_{3}}{n^{\alpha}}
-\frac{a\sigma K_{3}}{n^{1+\alpha}}+O(\frac{1}{n^{2}})
\leq 1-\frac{\tilde{A}_{2}}{n^{\alpha}}.
\label{A1}$$ Since Assumptions (A1)-(A3) in Theorem 1 are satisfied, $\lim_{n\rightarrow\infty}\sup n^{2\sigma}E[\theta_{n}^{2}]\leq C$ which implies that $$\lim_{n\rightarrow\infty}\sup n^{\sigma}|E[\theta_{n}]|
\leq \lim_{n\rightarrow\infty}\sup (n^{2\sigma}E[\theta_{n}^{2}])^{1/2}
\leq \sqrt{C}.$$ According to the assumptions that $\epsilon_{n}=o(1)$, $\tau_{n}=o(1)$ uniformly as $n\rightarrow\infty$, and $\delta_{n}^{\beta}=
d^{\beta}n^{-\sigma}$. There exists an $n_{2}> 1$ such that, when $n\geq n_{2}$, $$\begin{aligned}
B_{n}& = &(1+\frac{1}{n})^{\sigma} n^{\sigma}(ba_{n}\delta_{n}^{\beta}(1
+E[\varepsilon_{n}])-a_{n}E[\tau_{n}\theta_{n}]) \nonumber \\
& \geq &(1+\frac{1}{n})^{\sigma} n^{\sigma}(ba_{n}\delta_{n}^{\beta}(1
+E[\varepsilon_{n}])-a_{n}E[|\tau_{n}|]E[|\theta_{n}|]) \nonumber \\
& \geq & (1+\frac{1}{n})^{\sigma} n^{\sigma}
\frac{1}{2}ba_{n}\delta_{n}^{\beta}
\geq \frac{1}{2}n^{\sigma}ba_{n}\delta_{n}^{\beta}.
\label{Bn1}\end{aligned}$$ Let $n_{0}=\max\{n_{1},n_{2}\}$. Then, from (\[z1\]) we know that for all $n\geq n_{0}$ $$z_{n} = z_{n_{0}}\prod^{n}_{i=n_{0}}A_{i}
-\sum_{i=n_{0}}^{n-1}B_{i}\prod_{j=i+1}^{n}A_{j}-B_{n}.
\label{z2}$$ Since $0<\alpha\leq 1$, (\[A1\]) shows that $$0\leq \lim_{n\rightarrow\infty} \prod^{n}_{i=n_{0}}A_{i}
\leq \lim_{n\rightarrow\infty}
\prod^{n}_{i=n_{0}}(1-\frac{\tilde{A}_{2}}{i^{\alpha}}) = 0.
\label{limA}$$ Furthermore, $\lim_{n\rightarrow\infty}B_{n} = 0$, and (\[A1\]) and (\[Bn1\]) imply that $$\begin{aligned}
\sum_{i=n_{0}}^{n-1}B_{i}\prod_{j=i+1}^{n}A_{j}
\geq \sum_{i=n_{0}}^{n-1}\frac{abd^{\beta}}{2i^{\alpha}}
\prod_{j=i+1}^{n}A_{j}
\geq \frac{abd^{\beta}}{2n^{\alpha}}\sum_{i=n_{0}}^{n-1}A_{n}^{n-i}
\geq \frac{abd^{\beta}}{2n^{\alpha}}\sum_{i=n_{0}}^{n-1}
(1-\frac{\tilde{A}_{1}}{n^{\alpha}})^{n-i}.
\label{BA}\end{aligned}$$ On the other hand, $$\lim_{n\rightarrow\infty}\frac{1}{n^{\alpha}}\sum_{i=n_{0}}^{n-1}
(1-\frac{\tilde{A}_{1}}{n^{\alpha}})^{n-i}
=\left\{ \begin{array}{ll}
1, & \mbox{if }0<\alpha<1 \\
1-e^{-\tilde{A}_{1}}, & \mbox{if }\alpha=1.
\end{array} \right.$$ Substituting the preceding inequality, (\[limA\]) and (\[BA\]) into (\[z2\]), we see that $$\lim_{n\rightarrow\infty}\sup z_{n} \leq - C_{2}$$ with $C_{2}=(1/2)abd^{\beta}(1-e^{-\tilde{A}_{1}})>0$. This is exactly what we want to prove.
------------------------------------------------------------------------
Theorems 1 and 2 show that the convergence rate for $\{\theta_{n}\}$ is generally $n^{-\sigma}$. If we are free to choose the positive numbers $\alpha, \eta$, it follows directly from Theorem 1 that
[Corollary 1]{}. [*Assume that $h_{n}$ satisfies (\[h1\])-(\[h2\]) and $\gamma
\leq 0$. Under Assumptions (A2)-(A3) in Theorem 1, the best possible convergence rate for the KW algorithm (\[kw\]) is $n^{-\beta/(2\beta-\gamma)}$ which is achieved by setting $\alpha = 1$, $\eta= 1/(2\beta-\gamma)$, and by choosing appropriate $a, d>0$.*]{}
For the KW algorithm (\[kw\]) with $h_{n}$ defined by (\[kw\_h\]), assume that $J(\theta)$ is continuously differentiable of order up to three and the third order derivative $J'''(\theta)$ is uniformly bounded on $\Theta$, we have $$E[h_{n}|\theta_{n}] = \frac{J(\theta_{n}+\delta_{n}) -J(\theta_{n}-
\delta_{n})}{2\delta_{n}} = J'(\theta_{n})+
\frac{1}{6}J'''(\tilde{\theta}_{n})\delta_{n}^{2}
= J'(\theta_{n})+O(\delta_{n}^{2})$$ where $\tilde{\theta}_{n}\in[\theta_{n}-\delta_{n},\theta_{n}+\delta_{n}]$. In this case, $\beta=2$. If the assumptions (\[h2\]) and (A1)-(A3) are satisfied and if the positive number $a$ is chosen sufficiently large, we know from Theorems 1 and 2 that $$\sigma = \frac{1}{2}\min\{\alpha+\gamma\eta,4\eta\}.
\label{sigma1}$$ If we use the one-sided finite-difference approximation in (\[kw\]): $$h_{n}=\frac{L(X(\theta_{n}+\delta_{n},\xi_{1,n}))
-L(X(\theta_{n},\xi_{2,n}))}{\delta_{2,n}}
\label{kw_1sideh}$$ and if $J(\theta)$ is twice continuously differentiable and the second order derivative $J''(\theta)$ is bounded on $\Theta$, then for any $\theta_{n},
\delta_{n}$ there exists a $\hat{\theta}_{n}\in[\theta_{n},\theta_{n}
+\delta_{n}]$ such that $$E[h_{n}|\theta]=\frac{J(\theta_{n}+\delta_{n})-J(\theta_{n})}{\delta_{n}}
= J'(\theta_{n})+\frac{1}{2}J''(\hat{\theta}_{n})\delta_{n}
= J'(\theta_{n})+O(\delta_{n}).$$ Therefore, $\beta=1$. Under the same conditions as those in the previous case we know that $$\sigma = \frac{1}{2}\min\{\alpha+\gamma\eta,2\eta\}.
\label{sigma2}$$ It is clear from (\[sigma1\]) and (\[sigma2\]) that, under the same condition for the variance $Var[h_{n}|\theta_{n}]$, the convergence rate of the KW algorithm is faster when symmetric differences are used than that when one-sided differences are used. Corollary 1 shows that the best possible convergence rate depends on two factors—how fast the bias decreases to zero and how slow the variance increases to infinity. Using symmetric finite difference (\[kw\_h\]) instead of the one-sided finite difference (\[kw\_1sideh\]) can reduce the bias of $h_{n}$. To summarize, we have the following conclusion which will be used later.
[Corollary 2]{}.
*Suppose that (A1)-(A3) are satisfied. If*
- $J(\theta)$ is continuously differentiable of order up to three and the third order derivative $J'''(\theta)$ is bounded on $\Theta$,
then the best possible convergence rate for the KW algorithm (\[kw\]) with $h_{n}$ defined in (\[kw\_h\]) is $n^{-2/(4-\gamma)}$. If
- $J(\theta)$ is twice continuously differentiable and the second order derivative $J''(\theta)$ is bounded on $\Theta$,
then the best possible convergence rate for the KW algorithm (\[kw\]) with $h_{n}$ defined in (\[kw\_1sideh\]) is $n^{-1/(2-\gamma)}$.
[**3. Rates of convergence for the MD algorithm.**]{} The rate of convergence of the MD algorithms was established by Nemivoski et al. (2009) when the $h_n$ in (\[md\]) is an unbiased estimate of the derivative, and by Duchi et al. (2012, 2013) when the $h_n$ is approximated by the one-sided finite difference (\[kw\_1sideh\]). In this section, we examine the rate of convergence of the MD algorithm for general $h_n$. Again, we only assume that $h_n$ satisfies (\[h1\])-(\[h2\]). For notational consistence, the norm $||.||$ in (\[md\]) is taken as the $l_2$ norm. Its dual norm $||x||_{*}:=\sup_{||y||\leq 1}y^Tx$ is also the $l_2$ norm. Define $$\hat{\theta}_n = \frac{1}{n}\sum_{i=1}^n \theta_i.$$ We next examine the convergence of $J( \hat{\theta}_n)$.
[Theorem 3]{}.
*Assume that $\{\theta_{n}\}$ is determined by (\[md\]), and*
- $\psi(\theta)$ is strongly convex, $\Theta$ is compact and convex, and there exists $r>0$ such that $D(\theta^{*},\theta)\leq (1/2)r^2, r > 0$ for all $\theta\in\Theta$;
- $L(X)$ is closed convex, and there exist two constants $K_{1}, K_{2}, 0<K_{1}\leq K_{2}
<\infty$, such that for all $\theta\in \Theta$, $$K_{1}|\theta-\theta^{*}|\leq |J'(\theta)|\leq
K_{2}|\theta-\theta^{*}|;$$
- conditioned on $\theta_{n}$, $h_{n}$ at the $n$th iteration is independent of those at the other iterations.
Then $$E[J(\hat{\theta}_{n})-J(\theta^{*})] \leq \frac{C_1}{n a_n}+\frac{C_2}{n}\sum_{i=1}^n a_i \delta_i^\gamma
+\frac{C_3}{n}\sum_{i=1}^n a_i
+\frac{C_4}{n}\sum_{i=1}^n a_i \delta_i^{2\beta}
+ \frac{C_5}{n}\sum_{i=1}^n \delta_i^\beta,
\label{eqn_master}$$ where $$C_1 = \frac{r^2}{2},\;\; C_2 = \frac{c}{2\kappa}, \;\; C_3 =\frac{K_2^2r^2}{2\kappa^2},\;\; C_4=\frac{b^2}{\kappa},\;\; C_5= \frac{br}{\sqrt{2\kappa}}.$$
[Proof]{}. Under Assumptions (B1)-(B3), we know from Duchi et al. (2013), eqn. (13), that $$J(\hat{\theta}_n) - J(\theta^{*}) \leq \frac{r^2}{2n a_n}+\frac{1}{2n\kappa}\sum_{i=1}^n a_i h_i^2 - \frac{1}{n}\sum_{i=1}^n \Delta_i(\theta_i-\theta^{*}).
\label{eqn1}$$ Therefore, $$E[ J(\hat{\theta}_n) - J(\theta^{*})] \leq \frac{r^2}{2n a_n}+\frac{1}{2n\kappa}\sum_{i=1}^n a_i E[h_i^2] + \frac{1}{n}\sum_{i=1}^n E[|\Delta_i(\theta_i-\theta^{*})|].
\label{eqn2}$$ The assumptions (\[h1\])-(\[h2\]) give $$E[h_i^2] = Var[h_i] +(E[J'(\theta_i)+\Delta_i])^2 \leq c\delta_i^\gamma+2(E[J'(\theta_i)])^2+2b^2\delta_i^{2\beta}
\label{eqn_h1}$$ On the other hand, according to Assumptions (B1)-(B2), $$(E[J'(\theta_i)])^2 \leq (K_2E[|\theta_i-\theta^{*}|])^2 \leq \frac{K_2^2}{2\kappa}r^2
\label{eqn_h2}$$ and $$E[|\Delta_i||\theta_i-\theta^{*}|] \leq \frac{ b\delta_i^\beta r}{\sqrt{2\kappa}}.
\label{eqn_h3}$$ By combining (\[eqn\_h1\])-(\[eqn\_h3\]) with (\[eqn2\]), we obtain $$E[ J(\hat{\theta}_n) - J(\theta^{*})] \leq \frac{r^2}{2n a_n}+\frac{c}{2n\kappa}\sum_{i=1}^n a_i \delta_i^\gamma
+\frac{K_2^2r^2}{2n\kappa^2}\sum_{i=1}^n a_i
+\frac{b^2}{n\kappa}\sum_{i=1}^n a_i \delta_i^{2\beta}
+ \frac{br}{n\sqrt{2\kappa}}\sum_{i=1}^n \delta_i^\beta,$$ which is exactly (\[eqn\_master\]).
------------------------------------------------------------------------
A special case of interest is $\gamma=0$, which corresponds to bounded variance of derivative estimation. The following Corollary 3 provides a bound on the convergence of the MD algorithm for this case with properly chosen $\{a_n\}$, $\{\delta_n\}$.
[Corollary 3]{}.
*Suppose that (A4), (B1)-(B3) are satisfied. Let $\gamma=0$, $a_n= a n^{-1/2}$, $\delta_n = d n^{-1}$, $a>0, d>0$.*
\(a) For $h_n$ defined in (\[kw\_1sideh\]), $$E[J(\hat{\theta}_{n})-J(\theta^{*})] \leq (C_1+2C_2+2C_3)\frac{\max(a,a^{-1})}{\sqrt{n}}+(2.5C_4 d^{2})\frac{a}{n}+(C_5d)\frac{1+\log n}{n}.
\label{md_case0}$$
\(b) For $h_n$ defined in (\[kw\_h\]), $$E[J(\hat{\theta}_{n})-J(\theta^{*})] \leq (C_1+2C_2+2C_3)\frac{\max(a,a^{-1})}{\sqrt{n}}+(9C_4 d^{4}/7)\frac{a}{n}+(2C_5 d^{2})\frac{1}{n}.
\label{md_case1}$$
[Proof]{}. For $\gamma=0$, $a_n= a n^{-1/2}$, $\delta_n = d n^{-1}$, combining the first three terms in (\[eqn\_master\]) gives the first term in (\[md\_case0\]). For $h_n$ defined by the one-sided finite difference (\[kw\_1sideh\]), under Assumption (A4), we have $\beta=1$. Consequently, the fourth term in (\[eqn\_master\]) is $$\frac{C_4}{n}\sum_{i=1}^n a_i \delta_i^{2} = \frac{C_4 a d^{2}}{n}\sum_{i=1}^n i^{-2.5}
\leq \frac{C_4 a d^{2}2.5}{n}.$$ The last term is $$\frac{C_5}{n}\sum_{i=1}^n \delta_i = \frac{C_5d}{n}\sum_{i=1}^n i^{-1} \leq \frac{C_5d (1+\log n)}{n}.$$ Combing the previous two inequalities with (\[eqn\_master\]) gives (\[md\_case0\]).
For $h_n$ defined by the symmetric finite difference (\[kw\_h\]), under Assumption (A4), we have $\beta=2$. Consequently, the fourth term in (\[eqn\_master\]) is $$\frac{C_4}{n}\sum_{i=1}^n a_i \delta_i^{4} = \frac{C_4 a d^{4}}{n}\sum_{i=1}^n i^{-4.5}
\leq \frac{C_4 a d^{4}(9/7)}{n}.$$ The last term is $$\frac{C_5}{n}\sum_{i=1}^n \delta_i^2 = \frac{C_5d^2}{n}\sum_{i=1}^n i^{-1} \leq \frac{2C_5d^2}{n}.$$ Combing the previous two inequalities with (\[eqn\_master\]) gives (\[md\_case1\]).
------------------------------------------------------------------------
Duchi et al. (2013) investigated the convergence of the MD algorithm using the one-sided finite difference (\[kw\_1sideh\]) as an approximation to the derivative. The bound (\[md\_case0\]) is technically the same as that in Duchi et al. (2013). When the symmetric finite difference (\[kw\_h\]) is used, the $\log n$ factor disappears in the last term of (\[md\_case1\]), which indicates that the symmetric finite-difference approximation (\[kw\_h\]) leads to a tighter bound under similar assumptions, which is due to that the symmetric finite difference (\[kw\_h\]) typically provides more accurate estimate of the mean of the derivative than the one-sided ones do. Note that Duchi et al. (2012, 2013) implicitly assumes that CRN is used in calculating the finite difference (\[kw\_h\]) or (\[kw\_1sideh\]) that will be covered in Sections 4 and 5.
It’s worth of noting that the rate of convergence for the MD algorithm, as given by (\[eqn\_master\]), is $n^{-1/2}$ which is not affected by the choice of finite-difference approximation, either symmetric or one-sided, to the derivative. This is by design since the MD algorithm was originally proposed for improving the robustness in the choice of stepsizes at the cost of slower convergence.
When a finite difference is used to approximate the derivative, it is desirable to have $\delta_n\rightarrow 0$ as $n\rightarrow\infty$ to ensure asymptotically unbiased estimate of the derivative. In this case, it is possible (and likely in practice!) that the variance of the estimate goes to infinity. This is a special case of (\[eqn\_master\]) with $\gamma<0$. Therefore, Theorem 3 allows flexibility to cover general cases.
A special situation is when $L(X(\theta_{n}+\delta_{n},\xi_{1,n}))$ and $L(X(\theta_{n}-\delta_{n},\xi_{2,n}))$ or $L(X(\theta_{n},\xi_{2,n}))$ in (\[kw\_h\]) or (\[kw\_1sideh\]) are sampled independently. In this case, $\gamma=-2$. Assume further that $\{a_n\}$ and $\{\delta_n\}$ are specified as in Assumption (A1). Then the right hand side of (\[eqn\_master\]) becomes $$H(n) := \frac{C_1}{n a_n}+\frac{C_2}{n}\sum_{i=1}^n a_i \delta_i^\gamma
+\frac{C_3}{n}\sum_{i=1}^n a_i
+\frac{C_4}{n}\sum_{i=1}^n a_i \delta_i^{2\beta}
+ \frac{C_5}{n}\sum_{i=1}^n \delta_i^\beta$$ $$= \frac{C_1}{a n^{1-\alpha}}+\frac{C_2}{n}\sum_{i=1}^n a\delta^{-2} i^{-\alpha+2\eta}
+\frac{C_3}{n}\sum_{i=1}^n a i^{-\alpha}
+\frac{C_4}{n}\sum_{i=1}^n a \delta^{2\beta} i^{-\alpha-\beta\eta}
+ \frac{C_5}{n}\sum_{i=1}^n \delta i^{-\beta\eta}$$ $$= O(n^{-1+\alpha})+O( n^{-\alpha+2\eta})+O( n^{-\alpha})+O( n^{-\alpha-\beta\eta})+O(n^{-\beta\eta})$$ $$= O(n^{-\sigma}),$$ where $$\sigma = \min\{ 1-\alpha, \alpha-2\eta, \alpha, \alpha+2\beta \eta, \beta\eta\}
= \min\{ 1-\alpha, \alpha+2\eta, \beta\eta\}.$$ For the one-sided finite difference (\[kw\_1sideh\]), $\beta=1$. Then $$\sigma = \min\{ 1-\alpha, \alpha+2\eta, \eta\} \leq 1/4.$$ For the symmetric finite difference (\[kw\_h\]), $\beta=2$. Then $$\sigma = \min\{ 1-\alpha, \alpha+2\eta, 2\eta\} \leq 1/3.$$ The previous discussion can be summarized in the following Corollary 4.
[Corollary 4]{}.
*Assume that Assumptions (A1), (A4), (B1)-(B3) are satisfied, and that $L(X(\theta_{n}+\delta_{n},\xi_{1,n}))$ and $L(X(\theta_{n}-\delta_{n},\xi_{2,n}))$ in (\[kw\_h\]) (or $L(X(\theta_{n},\xi_{2,n}))$ in (\[kw\_1sideh\])) are independent. Then*
- the best possible rate of convergence for the upper bound $H(n)$ is $n^{-1/4}$ when the one-sided finite difference (\[kw\_1sideh\]) is used,
- the best possible rate of convergence for the upper bound $H(n)$ is $n^{-1/3}$ when the symmetric finite difference (\[kw\_h\]) is used.
Note that the rates of convergence are only upper bounds of $ E[J(\hat{\theta}_{n})]$. Such rates of convergence are consistent with those for $\{\theta_n\}$.
[**4. The KW algorithm with CRN.**]{} In this section, we will show how CRN can accelerate the convergence of the KW algorithm. For clarity and without getting trapped into unnecessary tediousness of details, we focus our attention on the case in which $\xi\in R$ is a real one-dimensional random variable. In Monte Carlo optimization, $\xi$ is usually a psuedo-random number generated by a computer. For most applications, such a pseudo-random number is sufficiently good to be regarded as a random number uniformly distributed on $[0, 1)$. In Section 6, we extend the results to general situations.
To avoid repetition, we only consider the $h_{n}$ defined as in (\[kw\_h\]) with $\xi_{1,n}=\xi_{2,n}=\xi_{n}$. The analysis is applicable to the one-sided finite-difference approximation (\[kw\_1sideh\]) without any difficult. For a given $\theta_{n}$, $h_{n}$ is a finite-difference approximation to the derivative $J'(\theta)$ at $\theta=\theta_{n}$. For simplicity, we omit the subscript $n$. Then $$h=\frac{L(X(\theta+\delta,\xi))-
L(X(\theta-\delta,\xi))}{2\delta}.
\label{h_generic}$$ The mean of $h$ is $$E[h] = \frac{J(\theta+\delta)-J(\theta-\delta)}{2\delta}$$ which is the same as that of (\[kw\_h\]) without the use of CRN. However, the variance of (\[h\_generic\]), as we will show, is generally smaller than that of (\[kw\_h\]) without the use of CRN when $\delta>0$ is sufficiently small. We will also show that the reduction in the variance of $h$ may have a significant impact on the convergence rate for the KW algorithm for Monte Carlo optimization. Toward that end, we need to specify the generation of the random variable $X(\theta,\xi)$ with a given distribution $F(\theta,x)$. Next we examine the variance of (\[h\_generic\]) for several popular random number generation methods. Note that $$Var[h]=\frac{1}{(2\delta)^{2}}
\{E[(L(X(\theta+\delta,\xi))- L(X(\theta-\delta,\xi)))^{2}]+
(J(\theta+\delta)- J(\theta-\delta))^{2}\}.$$ If $J(\theta)$ is continuously differentiable on $\Theta$ with bounded derivatives, then $$Var[h]=\frac{1}{(2\delta)^{2}}
E[(L(X(\theta+\delta,\xi))- L(X(\theta-\delta,\xi)))^{2}]+O(1).
\label{h_var}$$
[*4.1. Inversion method*]{}. Inversion is one of the most popular methods for random variable generation. Let $F(\theta,x)$ be the distribution function of $X(\theta,\xi)$. The inversion method generates the random variable $X(\theta,\xi)$ in the following way:
1. Generate a random number $\xi$ uniformly distributed on $[0, 1)$.
2. Set $X(\theta,\xi)=F^{-1}(\theta,\xi)$.
Then it is straightforward to verify that $X(\theta,\xi)$ has the desired distribution. Note that the mapping $F(\theta,x): R\rightarrow R$ is not one to one in general. To ensure its existence for general distribution functions, the inverse function is defined as $$F^{-1}(\theta,\xi)=\min\{x\;|\;F(\theta,x)> \xi,\;x\in R\}$$ which is different from the usual definition \[see Krantz (1991)\]. It coincides with the usual definition if $F(\theta,x)$ is continuous and strictly increasing. Such a definition of the inverse function covers both continuous and discrete random variables. For example, consider a discrete random variable $X(\theta,\xi)=x_{i}$ with probability $p_{i}(\theta)$. Define $\rho_{0}(\theta)=0, \rho_{i}(\theta)=\sum_{j=1}^{i}p_{j}(\theta)$ for $i\geq 1$. Let $\xi$ be uniformly distributed on $[0, 1)$. The inversion method gives $F^{-1}(\theta,\xi)=x_{i}$ if $\xi\in
[\rho_{i-1}(\theta), \rho_{i}(\theta))$. Then direct verification shows that $X(\theta,\xi)$ obeys the desired distribution. This is a discrete version of the inversion method.
In order to proceed with our discussion, let us first examine the properties of distribution functions. A distribution $F(\theta,x)$ is a nondecreasing and right-continuous function of $x$. $F(\theta,x)$ has at most countably many points of discontinuity on $R$ and all of the discontinuities are of the first kind — that is, for any $x\in R$, $F(\theta,x^{-})=\lim_{y\uparrow x}F(\theta,y)$ and $F(\theta,x^{+})=\lim_{y\downarrow x}F(\theta,y)$ exist and are finite \[e.g. Krantz (1991),149-150\]. Therefore, we can divide $R$ into $\bigcup_{i}B_{i}(\theta)=R$, where $B_{i}(\theta)=[b_{i}(\theta),
b_{i+1}(\theta))$, such that, for each $i$, $F(\theta,x)$ is continuous on $B_{i}(\theta)$, but jumps at $b_{i}(\theta)$. Assume that, for each $i$, $F(\theta,x)$ is piecewise differentiable on $B_{i}(\theta)$. Then $F'_{x}(\theta,x)>0$ whenever it exists. We further divide the interval $B_{i}(\theta)$ into subintervals according to whether the derivative of $F(\theta,x)$ with respect to $x$ is zero or not. For simplicity, we assume that $B_{i}(\theta)= B_{i}^{0}(\theta)\bigcup B_{i}^{+}(\theta)$ such that $F'_{x}(\theta,x)=0$ on $B_{i}^{0}(\theta)=[b_{i}(\theta),
c_{i}(\theta)]$ and $F(\theta,x)=F_{i}(\theta,x)$ is continuously differentiable with strictly positive derivatives on $B_{i}^{+}(\theta)=
(c_{i}(\theta), b_{i+1}(\theta))$. It is possible that $b_{i}(\theta)=
c_{i}(\theta)$. On $B_{i}^{0}(\theta)$, the derivatives $F'_{x}(\theta,x)$ should be understood as the right and the left derivatives at $b_{i}(\theta), c_{i}(\theta)$, respectively. It is possible that $F(\theta,x)$ is not differentiable at $c_{i}(\theta)$. The inverse $F_{i}^{-1}(\theta,\xi)$ is defined in the usual sense. It is continuous, strictly increasing, and differentiable on $(F(\theta,c_{i}(\theta)),
F(\theta,b_{i+1}^{-}(\theta)))$.
Under the preceding decomposition, $F(\theta,x)$ is discontinuous at $b_{i}(\theta)$, is a constant on $B_{i}^{0}(\theta)$, and is strictly increasing and differentiable on $B_{i}^{+}(\theta)$.
The following Lemma 2 follows directly from the definition of the inverse function and the decomposition of $F(\theta,x)$.
[Lemma 2]{}. [*Let $X(\theta,\xi)$ be defined by the inverse function $X(\theta,\xi)=F^{-1}(\theta,\xi)$. Let $\Xi_{i}(\theta)=
[F(\theta,b_{i}^{-}(\theta)), F(\theta,b_{i+1}^{-}(\theta)))$. Then $\Xi_{i}(\theta)\subset[0, 1)$ and for any $\xi\in \Xi_{i}(\theta)$ $$X(\theta,\xi)=\left\{ \begin{array}{ll}
b_{i}(\theta), & \mbox{if } \xi\in [F(\theta,b_{i}^{-}(\theta)),
F(\theta,c_{i}(\theta))), \\
c_{i}(\theta), & \mbox{if } \xi=F(\theta,c_{i}(\theta)), \\
F_{i}^{-1}(\theta,\xi), & \mbox{if } \xi\in (F(\theta,c_{i}(\theta)),
F(\theta,b_{i+1}^{-}(\theta))).
\end{array} \right.
\label{X1}$$*]{}
We need the following result.
[Lemma 3]{}.
*Assume that*
- $L(X)$ and $L'_{X}(X)$ are bounded, $J(\theta)$ is continuously differentiable on $\Theta$;
- for each $i$, $F_{i}(\theta,x)$ is continuously differentiable on $B_{i}^{+}(\theta)$ with strictly positive derivatives with respect to $x$, and $$\sum_{i}E[(\max_{\theta}(F'_{i\theta}(\theta,x))^{2}/
F'_{i x}(\theta,x))I_{B_{i}^{+}(\theta)}]<\infty;$$
- $b_{i}(\theta)$ is continuously differentiable in $\theta$, and $ \sum_{i}\max_{\theta}(b_{i}'(\theta))^{2}< \infty; $
- for each $i$, the functions $F(\theta,c_{i}(\theta))$ and $F(\theta,b_{i}^{-}(\theta))$ are continuously differentiable in $\theta$, and $ \sum_{i}\max_{\theta}|F'(\theta,c_{i}(\theta))|<\infty,$ $ \sum_{i}\max_{\theta}|F'(\theta,b_{i}^{-}(\theta))|<\infty,$
Define $ M_{1}(\theta) = 2\sum_{i}(L(c_{i}(\theta))-L(b_{i}(\theta)))^{2}
|F'(\theta,c_{i}(\theta))|. $ Then $M_{1}(\theta)\geq 0$ is bounded for all $\theta$. If $M_{1}(\theta)>0$, we have $$E[(L(X(\theta+\delta,\xi))- L(X(\theta-\delta,\xi)))^{2}]
=M_{1}(\theta)\delta+o(\delta)
\label{EFD}$$ as $\delta>0$ goes to zero
.
[Proof]{}. We calculate $$\begin{aligned}
\lefteqn{\lim_{\delta\rightarrow 0}
\frac{1}{\delta}E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}]} \\
& & = \lim_{\delta\rightarrow 0}
\frac{1}{\delta}\sum_{i}E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}I_{\Xi_{i}(\theta-\delta)}].\end{aligned}$$ Let $$R_{i}(\theta,\delta)=\frac{1}{\delta}E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}I_{\Xi_{i}(\theta-\delta)}].$$ Then $$\lim_{\delta\rightarrow 0}\frac{1}{\delta}E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}] = \lim_{\delta\rightarrow 0}
\sum_{i} R_{i}(\theta,\delta).
\label{fd1}$$ Next, we prove that the limit and the summation commute. Define $D_{i,1} = \Xi_{i}(\theta-\delta)\bigcap [0, F(\theta+\delta,
b_{i}^{-}(\theta+\delta)))$, $D_{i,2} = \Xi_{i}(\theta-\delta)\bigcap
\Xi_{i}(\theta+\delta)$, and $D_{i,3} = \Xi_{i}(\theta-\delta)\bigcap
[F(\theta+\delta,b_{i+1}^{-}(\theta+\delta)), 1)$. It is possible for each of $D_{i,j}, j=1, 2,3,$ to be empty. Then $\Xi_{i}(\theta-\delta)=\bigcup
D_{i,j}$ and $$R_{i}(\theta,\delta) = \sum_{j=1}^{3} R_{i,j}, \;\; R_{i,j}=
\frac{1}{\delta}E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}I_{D_{i,j}}].
\label{r1}$$ By Assumption (C1), there exist $N_{1}, N_{2}>0$ such that $|L(X)|\leq
N_{1}$, $|L'_{X}(X)|\leq N_{2}$. Therefore, $$\begin{aligned}
R_{i,1} & \leq & (2N_{1})^{2}|F(\theta+\delta,b_{i}^{-}(\theta+\delta))-
F(\theta-\delta,b_{i}^{-}(\theta-\delta))|/\delta \\
& \leq & 2(2N_{1})^{2}\max_{\theta}|F'(\theta,
b_{i}^{-}(\theta))|, \end{aligned}$$ $$\begin{aligned}
R_{i,3} & \leq & (2N_{1})^{2}|F(\theta+\delta,b_{i+1}^{-}(\theta+\delta))-
F(\theta-\delta,b_{i+1}^{-}(\theta-\delta))|/\delta \\
& \leq & 2(2N_{1})^{2}\max_{\theta}|F'(\theta,
b_{i+1}^{-}(\theta))|, \end{aligned}$$ Without loss of generality, assume that $F(\theta+\delta,b_{i}^{-}(\theta+
\delta)))\geq F(\theta-\delta,b_{i}^{-}(\theta-\delta)))$ and $F(\theta+\delta,
b_{i+1}^{-}(\theta+\delta)))\leq F(\theta-\delta,b_{i+1}^{-}(\theta-\delta)))$. If $F(\theta+\delta,c_{i}(\theta+\delta))
>F(\theta-\delta,c_{i}(\theta-\delta))$, $$\begin{aligned}
R_{i,2} & = & \frac{1}{\delta}\int_{F(\theta+\delta,b_{i}^{-}(\theta
+\delta))}^{F(\theta-\delta,c_{i}(\theta-\delta))}
(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}d\xi
\label{rr2}\end{aligned}$$ $$\begin{aligned}
& & \mbox{} + \frac{1}{\delta}\int_{F(\theta-\delta,c_{i}(\theta-\delta))}
^{F(\theta+\delta,c_{i}(\theta+\delta))}
(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}d\xi \nonumber \\
& & \mbox{} + \frac{1}{\delta}\int_{F(\theta+\delta,c_{i}(\theta+\delta))}
^{F(\theta+\delta,b_{i+1}^{-}(\theta+\delta))}
(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}d\xi \nonumber \\
& = & \frac{1}{\delta}\int_{F(\theta+\delta,b_{i}^{-}(\theta
+\delta)))}^{F(\theta-\delta,c_{i}(\theta-\delta))}
(L(b_{i}(\theta+\delta))-L(b_{i}(\theta-\delta)))^{2}d\xi \nonumber \\
& & \mbox{} + \frac{1}{\delta}\int_{F(\theta-\delta,c_{i}(\theta-\delta))}
^{F(\theta+\delta,c_{i}(\theta+\delta))} (L(b_{i}(\theta+\delta))
-L(F_{i}^{-1}(\theta-\delta,\xi)))^{2}d\xi \nonumber \\
& & \mbox{} + \frac{1}{\delta}\int_{F(\theta+\delta,c_{i}(\theta+\delta))}
^{F(\theta+\delta,b_{i+1}^{-}(\theta+\delta))}
(L(F_{i}^{-1}(\theta+\delta,\xi))
-L(F_{i}^{-1}(\theta-\delta,\xi)))^{2}d\xi % \nonumber
%\label{rr2}\end{aligned}$$ The first two terms of (\[rr2\]) are bounded respectively by $$4N_{2}^{2}\max_{\theta}(b'_{i}(\theta))^{2}\delta\;\;\mbox{ and }\;\;
2(2N_{1})^{2}\max_{\theta}|F'(\theta,c_{i}(\theta))|.$$ The third term of (\[rr2\]) can be rewritten as $$\frac{1}{\delta}\int_{F(\theta+\delta,c_{i}(\theta+\delta))}
^{F(\theta,c_{i}(\theta))}
(L(F_{i}^{-1}(\theta+\delta,\xi))
-L(F_{i}^{-1}(\theta-\delta,\xi)))^{2}d\xi$$ $$+ \frac{1}{\delta}\int_{F(\theta,b_{i+1}^{-}(\theta))}
^{F(\theta+\delta,b_{i+1}^{-}(\theta+\delta))}
(L(F_{i}^{-1}(\theta+\delta,\xi))
-L(F_{i}^{-1}(\theta-\delta,\xi)))^{2}d\xi$$ $$+ \frac{1}{\delta}\int_{F(\theta,c_{i}(\theta))}
^{F(\theta,b_{i+1}^{-}(\theta))}
(L(F_{i}^{-1}(\theta+\delta,\xi))
-L(F_{i}^{-1}(\theta-\delta,\xi)))^{2}d\xi$$ $$\leq 2(2N_{1})^{2}\max_{\theta}|F'(\theta,c_{i}(\theta))|
+2(2N_{1})^{2}\max_{\theta}|F'(\theta,b_{i+1}^{-}(\theta))|$$ $$+ 4N_{2}^{2}E[(\max_{\theta}(F'_{i\theta}(\theta,x))^{2}/
F'_{i x}(\theta,x))I_{B_{i}^{+}(\theta)}]\delta.$$ Therefore, $R_{i,2}$ is bounded by $$4N_{2}^{2}\max_{\theta}(b'_{i}(\theta))^{2}\delta+
4(2N_{1})^{2}\max_{\theta}|F'(\theta,c_{i}(\theta))|$$ $$+2(2N_{1})^{2}\max_{\theta}|F'(\theta,b_{i+1}^{-}(\theta))|
+ 4N_{2}^{2}E[(\max_{\theta}(F'_{i\theta}(\theta,x))^{2}/
F'_{i x}(\theta,x))I_{B_{i}^{+}(\theta)}]\delta$$ Similarly, we can prove that if $F(\theta+\delta,c_{i}(\theta+\delta))
\leq F(\theta-\delta,c_{i}(\theta-\delta))$, $$\begin{aligned}
R_{i,2} & = & \frac{1}{\delta}\int_{F(\theta+\delta,b_{i}^{-}(\theta
+\delta)))}^{F(\theta+\delta,c_{i}(\theta+\delta)}
(L(b_{i}(\theta+\delta))-L(b_{i}(\theta-\delta)))^{2}d\xi
\label{rr3}
\end{aligned}$$ $$\begin{aligned}
& & \mbox{} + \frac{1}{\delta}\int_{F(\theta+\delta,c_{i}(\theta+\delta))}
^{F(\theta-\delta,c_{i}(\theta-\delta))}
(L(F_{i}^{-1}(\theta+\delta,\xi))
-L(b_{i}(\theta-\delta)))^{2}d\xi \nonumber \\
& & \mbox{} + \frac{1}{\delta}\int_{F(\theta-\delta,c_{i}(\theta-\delta))}
^{F(\theta+\delta,b_{i+1}^{-}(\theta+\delta)))}
(L(F_{i}^{-1}(\theta+\delta,\xi))
-L(F_{i}^{-1}(\theta-\delta,\xi)))^{2}d\xi \nonumber \\
&\leq & 4N_{2}^{2}\max_{\theta}(b'_{i}(\theta))^{2}\delta
+4(2N_{1})^{2}\max_{\theta}|F'(\theta,c_{i}(\theta))| \\
& &\mbox{}+2(2N_{1})^{2}\max_{\theta}
|F'(\theta,b_{i+1}^{-}(\theta))|\nonumber \\
& & \mbox{} + 4N_{2}^{2}E[(\max_{\theta}(F'_{i\theta}(\theta,x))^{2}/
F'_{ix}(\theta,x))I_{B_{i}^{+}(\theta)}]\delta. \nonumber
%\label{rr3}\end{aligned}$$ Substituting the upper bounds for $R_{i,j}, j=1,2,3,$ into (\[fd1\]), also noting the assumptions (C2)-(C4), we see that $R_{i}(\theta,\delta)$ is uniformly bounded with respect to $\delta$. Therefore, $\sum_{i}R_{i}(\theta,\delta)$ converges uniformly in $(0,\delta_{0})$ for any $\delta_{0}>0$. By the Weierstrass M-test \[Krantz (1991),211\], we know that the limit and the summation commute. From (\[fd1\]), $$\lim_{\delta\rightarrow 0}\frac{1}{\delta}E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}] = \lim_{\delta\rightarrow 0}
\sum_{i} R_{i}(\theta,\delta)
= \sum_{i} \lim_{\delta\rightarrow 0}R_{i}(\theta,\delta).
\label{fd2}$$ We next calculate $\lim_{\delta\rightarrow 0}
R_{i}(\theta,\delta)$. For each $i$, there exists a $\delta_{i}>0$ such that for any $\delta\leq \delta_{i}$ $$\begin{aligned}
\lefteqn{ |F(\theta+\delta,b_{j}^{-}(\theta+\delta)))-
F(\theta-\delta,b_{j}^{-}(\theta-\delta)))|} \\
& & \leq \frac{1}{4}\min_{j=i-1,i,i+1,i+2}
\{F(\theta-\delta,b_{j+1}^{-}(\theta-\delta))
-F(\theta-\delta,b_{j}^{-}(\theta-\delta))\}. \end{aligned}$$ Note that $D_{i,1}=\Xi_{i}(\theta-
\delta)\bigcap\Xi_{i-1}(\theta+\delta)$ and $D_{i,3}=\Xi_{i}(\theta-
\delta)\bigcap\Xi_{i+1}(\theta+\delta)$ when $\delta\leq\delta_{i}$. Therefore, by taking into account that each of $D_{i,1}$ and $D_{i,3}$ may be empty, we have $$\begin{aligned}
R_{i,1} & \leq & \int^{F(\theta+\delta,b_{i}^{-}(\theta+\delta))}
_{F(\theta-\delta,b_{i}^{-}(\theta-\delta))}
(L(b_{i}(\theta+\delta))-L(F_{i-1}^{-1}(\theta
-\delta,\xi)))^{2}d\xi \\
& \leq & \max_{\theta}|F'(\theta,b_{i}^{-}(\theta))|
(L(b_{i}(\theta+\delta))-L(F_{i-1}^{-1}(\theta-\delta,\tilde{\xi})))^{2} \\
& = & o(1) \end{aligned}$$ where $\tilde{\xi}\in[F(\theta-\delta,b_{i}^{-}(\theta-\delta)),
F(\theta+\delta,b_{i}^{-}(\theta+\delta)))$. Similarly, $R_{i,3} = o(1)$. Hence, $\lim_{\delta \rightarrow 0} D_{i,j} = 0$ for $j=1, 3$. Also, the analysis of (\[rr2\]) and (\[rr3\]) shows that $$\lim_{\delta\rightarrow 0} D_{i,2}
= 2(L(c_{i}(\theta))-L(b_{i}(\theta)))^{2}
|F'(\theta,c_{i}(\theta))|
\label{fd3}$$ Substituting (\[fd3\]) into (\[fd2\]) we get $$\lim_{\delta\rightarrow 0}\frac{1}{\delta}E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}] = M_{1}(\theta)$$ which is exactly what we want to prove.
------------------------------------------------------------------------
The proof of Lemma 3 shows that Assumptions (C2) and (C3) guarantee that the inverse function $F_{i}^{-1}(\theta,\xi)$ is sufficiently smooth. Assumption (C4) ensures the existence of $M_{1}(\theta)$. Assumptions (C2)-(C4) are mild. Assumption (C1) guarantees the smoothness of the function $L(X)$. The boundedness of $L(X)$ and $L'_{X}(X)$ can be removed if there are only a finite number of sets of $B_{i}(\theta)$. The finiteness of $B_{i}(\theta)$ can also relax the assumptions (C2)-(C4).
The case of $M_{1}(\theta) = 0$ can only occur when either $b_{i}(\theta)=
c_{i}(\theta)$ or $F'(\theta,c_{i}(\theta)) =0$. The situation of $b_{i}(\theta)=c_{i}(\theta)$ (assuming that $L(X)$ is not a constant) happens when $F(\theta,x)$ is strictly increasing. A repetition of the proof of Lemma 3 yields that
[Corollary 5]{}. [*If Assumption (C1) is satisfied and $$E[(F'_{\theta}(\theta,x))^{2}/F'_{x}(\theta,x)]<\infty,
\label{clr5_1}$$ then $E[(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}]=
O(\delta^{2})$.*]{}
Corollary 5 recovers a result obtained by Glasserman and Yao (1992) under the assumption of Lipschitz continuity of $L(F^{-1}(\theta,\xi))$. When $F'(\theta,c_{i}(\theta)) =0$ for all $i$, using the same arguments as that of Corollary 5 we can establish that
[Corollary 6]{}. [*In addition to Assumptions (C1)-(C4), assume that $F(\theta,c_{i}(\theta))$ is continuously twice differentiable for all $i$ with $$0< \sum_{i}(L(c_{i}(\theta))-L(b_{i}(\theta)))^{2}
|F''(\theta,c_{i}(\theta))|<\infty.
\label{clr6_1}$$ Then, $E[(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}]=
O(\delta^{2})$.*]{}
The following Theorem 4 is the main conclusion of this subsection.
[Theorem 4]{}. [*Assume that Assumptions (A1)-(A4) and (C1)-(C4) are satisfied. If $M_{1}(\theta) > 0$ for all $\theta$, then the best convergence rate for the KW algorithm (\[kw\]) with $h_{n}$ defined by (\[h\_generic\]) is $n^{-2/5}$. This rate is attained by choosing $a_{n}=an^{-1}$, $a>2/(5K_{1})$, and $\delta_{n}=n^{-1/5}$.*]{}
[Proof]{}. Under Assumption of (C1)-(C4) and $M_{1}(\theta)>0$, we know from Lemma 3 that $E[(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}]
=M_{1}(\theta)\delta+o(\delta)$. According to (\[h\_var\]), the variance of $h_{n}$ is of order $Var[h_{n}|\theta_{n}]=M_{1}(\theta_{n})/\delta_{n}
+o(1/\delta_{n})$. Lemma 3 shows that $M_{1}(\theta)$ is bounded. Therefore, $\gamma=-1$ in (\[h2\]). Since (A1)-(A4) are satisfied, Corollary 2 shows that the best convergence rate is $n^{-2/(4-\gamma)}=n^{-2/5}$.
------------------------------------------------------------------------
The following Theorem 5 summerizes the rate of convergence of the KW algorithm (\[kw\]) when (\[kw\_h\]) is replaced with one-sided finite difference approximation with CRN. The proofs are omitted since they are very similar to that of Theorem 4.
[Theorem 5]{}.
*(I) Under the same conditions as those of Theorem 4 but the estimate $h$ is replaced by the following one-sided finite difference with the use of CRN $$h=\frac{L(X(\theta+\delta,\xi))- L(X(\theta,\xi))}{\delta},
\label{th5_1}$$ the best convergence rate is $n^{-1/3}$ which is achieved by setting $a_{n}=an^{-1}, a>1/3K_{1}$, and $\delta_{n}=d n^{-1/3}$.*
\(II) Assume all the assumptions of Theorem 4 except that $M_{1}(\theta)
= 0$. Then Corollaries 5 and 6 show that $E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}]= O(\delta^{2})$ if either of (\[clr5\_1\]) or (\[clr6\_1\]) holds. Hence, $Var[h_{n}|\theta_{n}]=O(1)$ for $h_{n}$ defined by either (\[h\_generic\]) or (\[th5\_1\]). The best convergence rate for the KW algorithm (\[kw\]) is $n^{-1/2}$. This rate can be attained by setting $a_{n}=an^{-1},
a>1/2K_{1}$, and $\delta_{n} = d n^{-\eta}, \eta\geq 1/2$.
We would like to emphasize that the assumptions in Corollaries 5 and 6 are satisfied for a broad class of stochastic optimization problems \[see Glasserman and Yao (1992) for a discussion\]. Theorems 4 and 5 state that, when the inversion method is used in the generation of random variables and when $h$ is defined by (\[h\_generic\]), the convergence rate for the KW algorithm with CRN is $n^{-2/5}$ in general and is $n^{-1/2}$ for a large class of problems that satisfy the assumptions in Corollaries 5 and 6. The improvement is signficant since the best possible rate for the same KW algorithm without CRN is $n^{-1/3}$.
[*4.2. Rejection method*]{}. Let $f(\theta,x)$ be the density function of $X(\theta,\xi)$. Assume that, for all $\theta\in \Theta$, $f(\theta,x)$ is zero outside a finite interval $[a, b]$ and is bounded by $0\leq f(\theta,x)
\leq c$, $c>0$ is a constant. The rejection method generates $X(\theta,\xi)$ according to the following three steps:
1. Generate $\xi_{1}$ uniformly distributed on $[a, b]$.
2. Generate $\xi_{2}$ uniformly distributed on $[0, c]$.
3. If $\xi_{2}\leq f(\theta,\xi_{1})$, then set $X(\theta,\xi)=\xi_{1}$; otherwise go to 1.
The rejection method uses at least two random numbers $\xi_{1}$ and $\xi_{2}$ to generate $X(\theta,\xi)$. The total number of random numbers $\xi_{1}, \xi_{2}$ required before outputing $X(\theta,\xi)$ is a random value. The rejection method does not accurately meet the CRN requirements since it is impossible to define $X(\theta+\delta,\xi)$ and $X(\theta-\delta,\xi)$ using a fixed set of uniform random numbers \[Bratley et al. (1983); Franta (1975)\]. Therefore, we modify the definition of CRN in the sense defined by the following procedure for the generation of a paired random variables:
[*Generation of $X(\theta+\delta,\xi)$ and $X(\theta-\delta,\xi)$*]{}:
1. Generate $\xi_{1}$ uniformly distributed on $[a, b]$.
2. Generate $\xi_{2}$ uniformly distributed on $[0, c]$.
3. If $\xi_{2}\leq f(\theta-\delta,\xi_{1})$ and $\xi_{2}\leq
f(\theta+\delta,\xi_{1})$, then set $X(\theta-\delta,\xi)=
X(\theta+\delta,\xi) =\xi_{1}$.
4. If $\xi_{2}\leq f(\theta-\delta,\xi_{1})$ and $\xi_{2}>
f(\theta+\delta,\xi_{1})$, then set $X(\theta-\delta,\xi)
=\xi_{1}$ and generate a $X(\theta+\delta,\xi)=\xi_{3}$ by the rejection method.
5. If $\xi_{2} > f(\theta-\delta,\xi_{1})$ and $\xi_{2} \leq
f(\theta+\delta,\xi_{1})$, then set $X(\theta+\delta,\xi)
=\xi_{1}$ and generate a $X(\theta-\delta,\xi)=\xi_{4}$ by the rejection method.
6. If $\xi_{2}> f(\theta-\delta,\xi_{1})$ and $\xi_{2}>
f(\theta+\delta,\xi_{1})$, go to 1.
This is essentially a coupling procedure \[see Devroye (1990) for a discussion on coupling\]. Such a modification is necessary to mimic the scheme of CRN using the rejection method. We will soon see that even such a loosely defined scheme can accelerate the convergence of the KW algorithm. Let $X(\theta-\delta,\xi), X(\theta+\delta,\xi)$ be generated by the preceding procedure. It is obvious that $E[h]$ for $h$ in (\[h\_generic\]) remains the same as that in the inversion method.
[Theorem 6]{}.
*Suppose that $f(\theta,x)$ is zero outside $[a, b]$, $0\leq f(\theta,x)\leq c$ for all $x\in [a, b]$, and $X(\theta-\delta,\xi),
X(\theta+\delta,\xi)$ are generated by the previously described procedure. Assume that*
- $Var[L(X(\theta,\xi))]$ is continuous in $\theta\in\Theta$;
- $f(\theta,x)$ is differentiable in $\theta$ for each $x\in [a, b]$, $f(\theta,x)$ satisfies the Lipschitz condition with respect to $\theta$, i.e., there is a $K(x)$ such that $|f(\theta+\delta,x)-f(\theta,x)|\leq
K(x)\delta$, and that $\int_{a}^{b}K(x)dx<\infty$.
Define $$M_{2}(\theta) = \frac{Var[L(X(\theta,\xi))]}{2c(b-a)}
\int_{a}^{b}|f'_{\theta}(\theta,x)|dx.$$ Then $0\leq M_{2}(\theta)<\infty$. If $M_{2}(\theta)>0$ for all $\theta$, $Var[L(X(\theta,\xi))]$ is bounded, $h$ is defined by (\[h\_generic\]), and the assumptions (A1)-(A4) are satisfied, then the convergence rate for the KW algorithm with CRN is $n^{-2/5}$
.
[Proof]{}. We see from the procedure of generating $X(\theta-\delta,\xi)$ and $X(\theta+\delta,\xi)$ that, conditioned on either $\xi_{2}\leq
f(\theta-\delta,\xi_{1})$ or $\xi_{2}\leq f(\theta+\delta,\xi_{1})$, $X(\theta+\delta,\xi)= X(\theta-\delta,\xi)=\xi_{1}$ when $\xi_{2}\leq
f(\theta-\delta,\xi_{1})$ and $\xi_{2}\leq f(\theta+\delta,\xi_{1})$; otherwise $X(\theta+\delta,\xi)=\xi_{3}$ and $X(\theta-\delta,\xi)=\xi_{4}$. Note that $\xi_{3}$ and $\xi_{4}$ are independent. Therefore, $$Var[h] = \frac{1}{4\delta^{2}}(Var[L(\xi_{3})]
+Var[L(\xi_{4})])
\frac{1}{c}E[|f(\theta+\delta,\xi_{1})-f(\theta-\delta,\xi_{1})|]
\label{th6_varh}$$ Under Assumption (H1), $Var[L(\xi_{3})]+Var[L(\xi_{4})]=
2Var[L(X(\theta,\xi))]+o(1)$. By Assumption (H2), $$\frac{1}{\delta}E[|f(\theta+\delta,\xi_{1})-f(\theta-\delta,\xi_{1})|]
\leq\frac{2}{b-a}\int_{a}^{b}K(x)dx$$ and $K(x)$ is integrable on $[a, b]$. According to the Weierstrass M-test, (\[th6\_varh\]) implies that $$\begin{aligned}
Var[h] &=& \frac{1}{2\delta^{2}}Var[L(X(\theta,\xi))]
\frac{1}{c}E[|f(\theta+\delta,\xi_{1})-f(\theta-\delta,\xi_{1})|]
+o(\frac{1}{\delta^{2}}) \nonumber \\
& = & \frac{1}{2\delta}Var[L(X(\theta,\xi))]
\frac{1}{c(b-a)}\int_{a}^{b}|
f'_{\theta}(\theta,x)|dx+o(\frac{1}{\delta}) \nonumber \\
& = &\frac{M_{2}(\theta)}{\delta}+o(\frac{1}{\delta}). \end{aligned}$$ Thus, we know from Corollary 2 where $\gamma=-1$ that the conclusion follows.
------------------------------------------------------------------------
For simplicity, we only consider the simplest form of the rejection method and the case in which $f(\theta,x)$ is continuous. An analysis similar to the one used in the proof of Theorem 6 shows that $Var[h]=O(1/\delta)$ remains valid in the following three situations: (i) The estimate $h$ is replaced by the one-sided finite difference (\[th5\_1\]); (ii) The density function $f(\theta,x)$ is piecewise differentiable; (iii) The rejection method is replaced by the following [*generalized rejection method*]{}. Assume that there exist a positive constant $A$ and a density function $g(x)$ such that $f(\theta,x)\leq A g(x)$ for all $\theta$ and for all $x\in [a, b]$. Then
1. generate $\xi_{1}$ with the density function $g(x)$;
2. generate $\xi_{2}$ uniformly distributed on $[0, A g(\xi_{1})]$;
3. if $\xi_{2}\leq f(\theta,\xi_{1})$, then set $X(\theta,\xi)=\xi_{1}$; otherwise go to 1.
It is easy to verify that $X(\theta,\xi)$ has the desired distribution. The density function $g(x)$ should be chosen such that it is easier to generate a random variable with $g(x)$ than those with $f(\theta,x)$.
Generally speaking, the convergence rates for the KW algorithm are the same when either the inversion method or the rejection method is used in the generation of the random variable $X(\theta,\xi)$. However, the rate corresponding to the use of the rejection method is universally true for any function: It can be seen from its definition that $M_{2}(\theta)$ is always positive except when $Var[L(X)]=0$ or when $f(\theta,x)$ is independent of $\theta$. Both cases are of little practical relevance. Furthermore, assume that the assumptions in Theorem 6 are satisfied and, in addition, $f(\theta,x)$ is strictly positive on $(a, b)$. Then the best possible convergence rate for the KW algorithm is $n^{-2/5}$ when the rejection method is used in generating $X(\theta,\xi)$. On the other hand, the distribution function $F(\theta,x)=\int_{a}^{x}f(\theta,u)du$ is continuously differentiable and strictly increasing on $[a, b]$. Theorem 5 shows that the convergence rate for the KW algorithm is $n^{-1/2}$ if the inversion method is used in generating $X(\theta,\xi)$. Therefore, as far as the convergence of the KW algorithm is concerned, the inversion method leads to faster convergence than the rejection method. This conclusion is in favor of the argument that the inversion method is superior to the rejection method \[c.f. Bratley et al. (1983), 141\].
[*4.3. Composition method*]{}. Assume that the distribution function $F(\theta,x)$ of $X(\theta,\xi)$ is of the form $$F(\theta,x) = \sum_{i=1}^{m} p_{i}(\theta)F_{i}(\theta,x)$$ where $p_{i}(\theta)>0, m\leq\infty, \sum_{i}p_{i}(\theta)=1$, and for each $i$, $F_{i}(\theta,x)$ is a distribution function. The composition method generates the random variable $X(\theta,\xi)$ in the following way:
1. Generate a random variable $Y$ with distribution $Prob\{Y=i\}
=p_{i}(\theta)$.
2. If $Y=i$, generate $X(\theta,\xi)$ according to distribution $F_{i}(\theta,x)$.
In the composition method, there is no specification on the method for the generation of random variables at each step. Any method such as inversion and rejection can be used. As an example, we consider the case in which random variables are generated using the inversion method which is superior to the rejection method, as we have argued in the previous subsection. Define $\rho_{0}(\theta)=0, \rho_{i}(\theta)
=\sum_{j=1}^{i}p_{j}(\theta)$ for $i\geq 1$. The following procedure is the actual composition method we are considering.
1. Generate a random number $\xi_{1}$ uniformly distributed on $[0, 1)$.
2. If $\xi_{1}\in[\rho_{i-1}(\theta),\rho_{i}(\theta))$, then generate a random number $\xi_{2}$ uniform on $[0, 1)$ and set $X(\theta,\xi)=
F_{i}^{-1}(\theta,\xi_{2})$.
In this algorithm, we need two uniform random numbers in the generation of $X(\theta,\xi)$. Actually we can do with only one random number by setting $\xi_{2}=(\xi_{1}-\rho_{i-1}(\theta))/p_{i}(\theta)$. Direct verification shows that, conditional on $\xi_{1}\in [\rho_{i-1}(\theta),\rho_{i}(\theta))$, $\xi_{2}$ is uniform on $[0, 1)$. In the composition method, we regard that $X(\theta-\delta,\xi)$ and $X(\theta+\delta,\xi)$ conform the CRN requirement if they are generated by the preceding procedure using the same $\xi=(\xi_{1},\xi_{2})$. We can prove that it can accelerate the convergence of the KW algorithm.
For simplicity, we assume that, for each $i$, $F_{i}(\theta,x)
=F_{i}(x)$ is independent of $\theta$, and the number of distribution component is finite, i.e., $m<\infty$. The case in which $F(\theta,x)$ is of general form can be treated parallel to the proof of Theorem 4. Our aim here is to find special features of the decomposition method rather than to develop the complete theory which is not difficult to derive. We first consider the situation where $\xi_{1}$ and $\xi_{2}$ are independent. We then examine the case where $\xi_{2}=
(\xi_{1}-\rho_{i-1}(\theta))/p_{i}(\theta)$.
[Theorem 7]{}. [*Suppose that $\xi_{1}$ and $\xi_{2}$ are independent , $p_{i}(\theta)$ is differentiable in $\theta$, and $$M_{3}(\theta) = \sum_{i=1}^{m}E[(L(F_{i+1}^{-1}(\xi))
-L(F_{i}^{-1}(\xi)))^{2}]|\rho'_{i}(\theta)|$$ exits and is finite. If $M_{3}(\theta)>0$ is bounded from above for all $\theta$, $h$ is defined by (\[h\_generic\]), and the assumptions (A1)-(A4) are satisfied, then the convergence rate for the KW algorithm is $n^{-2/5}$.*]{}
[Proof]{}. The proof is a simplified version of that of Lemma 3 since the assumptions here are stronger. According to the generation scheme of $X(\theta+\delta,\xi)$ and $X(\theta-\delta,\xi)$, we have $$\begin{aligned}
E[(L(X(\theta+\delta,\xi))
-L(X(\theta-\delta,\xi)))^{2}]
\label{th7_1}\end{aligned}$$ $$\begin{aligned}
& & =\sum_{i=1}^{m}\int_{\rho_{i-1}(\theta-\delta)}^{\rho_{i}(\theta-\delta)}
E_{\xi_{2}}[(L(X(\theta+\delta,\xi))-
L(X(\theta-\delta,\xi)))^{2}]d\xi_{1} \nonumber\end{aligned}$$ Note that $m$ is finite and, for each $i$, $\rho_{i}(\theta)$ is continuous. There exists a $\delta_{0}$ such that when $\delta\leq\delta_{0}$ $$|\rho_{i}(\theta+\delta)-\rho_{i}(\theta-\delta)|
\leq \frac{1}{4}\min_{j}\{\rho_{j+1}(\theta-\delta)
-\rho_{j}(\theta-\delta)\}, \;\;\mbox{ for all $i$.}
\label{th7_2}$$ Let us first consider the case in which $\rho_{i-1}(\theta+\delta)>
\rho_{i-1}(\theta-\delta)$ and $\rho_{i}(\theta+\delta)\leq
\rho_{i}(\theta-\delta)$. When $\delta\leq\delta_{0}$, (\[th7\_2\]) ensures that (\[th7\_1\]) can be rewritten as $$\sum_{i=1}^{m}\int_{\rho_{i-1}(\theta-\delta)}^{\rho_{i-1}(\theta+\delta)}
E_{\xi_{2}}[(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}]d\xi_{1}$$ $$+\sum_{i=1}^{m}\int_{\rho_{i-1}(\theta+\delta)}^{\rho_{i}(\theta+\delta)}
E_{\xi_{2}}[(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}]d\xi_{1}$$ $$+\sum_{i=1}^{m}\int_{\rho_{i}(\theta+\delta)}^{\rho_{i}(\theta-\delta)}
E_{\xi_{2}}[(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}]d\xi_{1}$$ $$=\sum_{i=1}^{m}\int_{\rho_{i-1}(\theta-\delta)}^{\rho_{i-1}(\theta+\delta)}
E_{\xi_{2}}[(L(F_{i-1}^{-1}(\xi_{2}))-L(F_{i}^{-1}(\xi_{2})))^{2}]d\xi_{1}$$ $$\;\;\;+\sum_{i=1}^{m}\int_{\rho_{i}(\theta+\delta)}^{\rho_{i}(\theta-\delta)}
E_{\xi_{2}}[(L(F_{i+1}^{-1}(\xi_{2}))-L(F_{i}^{-1}(\xi_{2})))^{2}]d\xi_{1}$$ $$=\sum_{i=1}^{m} E[(L(F_{i-1}^{-1}(\xi)-L(F_{i}^{-1}(\xi)))^{2}]
\rho_{i-1}'(\theta)2\delta$$ $$\;\;\; +\sum_{i=1}^{m}E[(L(F_{i+1}^{-1}(\xi))-L(F_{i}^{-1}(\xi)))^{2}]
\rho_{i}'(\theta)2\delta +o(\delta)$$ By considering every case of $\rho_{i-1}(\theta+\delta)\leq
\rho_{i-1}(\theta-\delta)$ and $\rho_{i}(\theta+\delta)\leq
\rho_{i}(\theta-\delta)$, $\rho_{i-1}(\theta+\delta)>
\rho_{i-1}(\theta-\delta)$ and $\rho_{i}(\theta+\delta)>
\rho_{i}(\theta-\delta)$, and $\rho_{i-1}(\theta+\delta)\leq
\rho_{i-1}(\theta-\delta)$ and $\rho_{i}(\theta+\delta)>
\rho_{i}(\theta-\delta)$, we obtain that $$E[(L(X(\theta+\delta,\xi))-L(X(\theta-\delta,\xi)))^{2}]
= M_{3}(\theta)\delta+o(\delta).$$ It follows from (\[h\_var\]) that $Var[h] = (1/4)M_{3}(\theta)/\delta+o(1/\delta)$. Applying Corollary 2, it is easy to see that the convergence rate for the KW algorithm is $n^{-2/5}$. We have thus completed the proof.
------------------------------------------------------------------------
Similarly, we can prove that the rate $n^{-2/5}$ remains valid for the case in which the random variable $X(\theta,\xi)=
F_{i}^{-1}(\theta,\xi_{2})$ is generated by setting $\xi_{2}=
(\xi_{1}-\rho_{i-1}(\theta))/p_{i}(\theta)$.
[Theorem 8]{}. [*Suppose that $p_{i}(\theta)>0$ is differentiable and the following function exists and is finite for all $\theta$: $$M_{4}(\theta) = \sum_{i=1}^{m}[L(F_{i+1}^{-1}(1^{-}))
-L(F_{i}^{-1}(0^{+}))]^{2}|\rho_{i}'(\theta)|,$$ where $F_{i+1}^{-1}(1^{-})=\lim_{\xi\uparrow 1}F_{i+1}^{-1}(\xi)$ and $F_{i}^{-1}(0^{+})=\lim_{\xi\downarrow 0}F_{i}^{-1}(\xi)$. If $M_{4}(\theta)>0$ is bounded for all $\theta$ and (A1)-(A4) are satisfied, then the convergence rate for the KW algorithm is $n^{-2/5}$*]{}.
The previous Theorems 7 and 8 show that the convergence rate for the KW algorithm is $n^{-2/5}$ when the composition method is used. This rate does not depend on how many random numbers are used in the generation of random variables. We would emphasize that it is unlikely for each of $M_{3}(\theta)$, $M_{4}(\theta)$ to be zero in practice.
[**5. The MD algorithm with CRN.**]{} In this section, we examine the rates of convergence for the MD algorithm under CRN. As shown in the previous section, the use of CRN largely affects $E[h_n^2]$ and thus the reduction of the variance $Var[h_n]$. The analysis in the previous Section 4 provides direct information on $E[h_n^2]$. Therefore, in this section we directly work $E[h_n^2]$ without going through $Var[h_n]$. We may represent $E[h_n^2]$ in the following form. $$E[h_n^2] \leq \tilde{c}\delta_n^\gamma.
\label{h3}$$ Recall that $\gamma=-2$ for independent samplings of $X(\theta,\xi)$ without CRN. With CRN, $\gamma=-1$ if $M_1(\theta)>0$ and $\gamma=0$ if $M_1(\theta)=0$. By following the same arguments as in the proof of Theorem 3 and applying (\[h3\]) directly for $E[h_i^2]$ in (\[eqn2\]), we obtain the following theorem.
[Theorem 9.]{} [*Assume (B1)-(B3) and (\[h3\]). Then we have $$E[J(\hat{\theta}_{n})-J(\theta^{*})] \leq \frac{C_1}{n a_n}+\frac{\tilde{C}_2}{n}\sum_{i=1}^n a_i \delta_i^\gamma
+ \frac{C_5}{n}\sum_{i=1}^n \delta_i^\beta,
\label{eqn_master2}$$ where $\tilde{C}_2 = \tilde{c}/(2\kappa)$, $C_1$ and $C_5$ are specified in Theorem 3.*]{}
The following Corollary 7 summarizes the rates of convergence for the MD algorithm with using CRN in calculating the finite difference (\[h\_generic\]) and (\[th5\_1\]).
[Corollary 7.]{}
*Assume (B1)-(B3) and (\[h3\]). Denote $$\tilde{H}_n = \frac{C_1}{n a_n}+\frac{\tilde{C}_2}{n}\sum_{i=1}^n a_i \delta_i^\gamma
+ \frac{C_5}{n}\sum_{i=1}^n \delta_i^\beta.$$ Then*
- if $\gamma=-1$, the best possible rate of convergence for the upper bound $H(n)$ is $n^{-1/3}$ when the one-sided finite difference (\[th5\_1\]) is used,
- if $\gamma=-1$, the best possible rate of convergence for the upper bound $H(n)$ is $n^{-2/5}$ when the symmetric finite difference (\[h\_generic\]) is used.
- if $\gamma=0$, the best possible rate of convergence for the upper bound $H(n)$ is $n^{-1/2}$ when either the one-sided finite difference (\[th5\_1\]) or the symmetric finite difference (\[h\_generic\]) is used,
[**6. Generalization and applications**]{}. In Sections 4-5, all the results are obtained for one dimensional random variables only. In this section, we extend the results to a case of multivariates, which is not difficult but very tedious. Assume that $J(\theta) = E_{\xi}[L(X(\theta,\xi))]$, where the multidimensional random variable $X(\theta,\xi)=
[X_{1}(\theta,\xi),X_{2}(\theta,\xi),...,X_{m}(\theta,\xi)]^{T} \in
R^{m}$. For each $i$, $X_{i}(\theta,\xi)=X_{i}(\theta,\xi_{i})\in R$, $\xi_{i}$ is uniform on $[0, 1)$. We only consider the case in which each $X_{i}(\theta,\xi_{i})$ is generated from $\xi_{i}$ using the inversion method. To avoid repetition, we list the result without proof which is very similar to that of Theorem 5.
Assume that $J(\theta)\in R$ and $\theta\in \Theta$. For each $i$, $1\leq i
\leq m<\infty$, let $F_{i}(\theta,x)$ be the distribution function of $X_{i}(\theta,\xi_{i})$ with the decomposition that $$\frac{dF_{i}(\theta,x)}{dx}=\left\{ \begin{array}{ll}
0, & \mbox{if }x\in B_{i,j}^{0}(\theta)=
[b_{i,j}(\theta),c_{i,j}(\theta)] \\
f_{i,j}(\theta,x), & \mbox{if }x\in B_{i,j}^{+}(\theta)=
(c_{i,j}(\theta),b_{i,j+1}(\theta)),
\end{array}\right.$$ where $\bigcup_{j}\{B_{i,j}^{0}(\theta)\bigcup B_{i,j}^{+}(\theta)\} = R$ for all $i$, $f_{i,j}(\theta,x) > 0$ for any $x\in B_{i,j}^{+}(\theta)$. It is possible that $F_{i}(\theta,x)$ is discontinuous at $b_{i,j}(\theta)$.
[Theorem 10]{}.
*Assume Assumptions (A1)-(A4) and, in addition,*
- $L(X)$ is continuously differentiable in $X$, $L(X)$ and $L'_{X_{i}}(X)$ are bounded for all $i$;
- for each $i$, $$\sum_{j}E[(\max_{\theta}\left(\frac{\partial F_{i,j}(\theta,x)}
{\partial\theta}\right)^{2}/\frac{\partial F_{i,j}(\theta,x)}
{\partial x})I_{B_{i,j}^{+}(\theta)}]<\infty;$$
- for all $i, j$, $b_{i,j}(\theta)$ is continuously differentiable in $\theta$, and $ \sum_{j}\max_{\theta}(b_{i,j}'(\theta))^{2}< \infty; $
- for all $i, j$, the functions $F_{i}(\theta,c_{i,j}(\theta))$ and $F_{i}(\theta,b_{i,j}^{-}(\theta))$ are continuously differentiable in $\theta$, and $ \sum_{j}\max_{\theta}|F'_{i}(\theta,c_{i,j}(\theta))|<\infty,$ $ \sum_{j}\max_{\theta}|F'_{i}(\theta,b_{i,j}^{-}(\theta))|<\infty,$
Define $ \tilde{M}_{1}(\theta) =
\sum_{i,j}(L(c_{i,j}(\theta))-L(b_{i,j}(\theta)))^{2}
|F'_{i}(\theta,c_{i,j}(\theta))|. $ Then $\tilde{M}_{1}(\theta)\geq 0$ is bounded. If $\tilde{M}_{1}(\theta)>0$ for all $\theta$, the best possible convergence rate for the KW algorithm (2) with $h_{n}$ defined by (24) is $n^{-2/5}$. This rate is attained by choosing $a_{n}=an^{-1}$, $a>2/(5K_{1})$, and $\delta_{n}=n^{-1/5}$
.
Similar results can be obtained if other methods are used in the generation of random variables or if $L(X)$ is a piecewise continuous function of $X$. The analysis can be applied to general problems such as Monte Carlo optimization of queueing systems and other general systems. Although such a generalization is not trivial, the basic idea is the same except that the analysis becomes tedious and lengthy. Next we illustrate an application of Theorem 10 to the optimization of queueing systems \[see, e.g. Kleinrock (1976)\].
[Example 1. GI/G/1 queue with single class of customers]{}. In a GI/G/1 queue, there is one server (such as a teller in a bank) and one queue. Upon its arrival, a customer enters the server for service if the server is free, otherwise it joins the queue and waits for its turn. The service discipline is first-come-first-serve. The server cannot be free if there is at least one customer waiting in the queue. Assume that the distribution of interarrival times is $G_{a}(t)$ and the distribution of service times is $G_{s}(\theta,t)= p(\theta)G_{s}^{1}(t)+
(1-p(\theta))G_{s}^{2}(t)$. For simplicity, we assume that $G_{a}(t)$, $G_{s}^{1}(t)$, and $G_{s}^{2}(t)$ are independent of $\theta$ and $\int t^{2}dG_{s}^{j}(t)<+\infty, j = 1, 2$, $p(\theta)$ is continuously differentiable in $\theta$, $G_{a}(t), G_{s}^{j}(t),
j=1,2,$ are strictly increasing and continuously differentiable in $t$. In queueing theory, the system time of a customer is defined as the time period from its arrival till departure. Let $L(X(\theta,\xi))$ be the average system time of the first N customers $$L(X(\theta,\xi)) = \frac{1}{N}\sum_{i=1}^{N}T_{i}(\theta,\xi),$$ where $T_{i}(\theta,\xi)$ is the system time of the $i$th customer. Then $J(\theta)=E[L(X(\theta,\xi))]$ is the mean system time of the first $N$ customers. We want to find the optimal parameter $\theta^{*}$ to minimize $J(\theta)$. It is known that the analytical form of $J(\theta)$ is not available for general $G_{a}(t)$, $G_{s}^{1}(t)$, and $G_{s}^{2}(t)$ \[e.g. Kleinrock (1976)\]. So we find $\theta^{*}$ via the KW algorithm. Assume that the queue is initially empty. According to Lindley’s equation \[e.g. Kleinrock (1976)\]: $$T_{i}(\theta,\xi) =
\max\{T_{i-1}(\theta,\xi)-A_{i},0\}
+S_{i},\;\; T_{0}(\theta,\xi) = 0,
\label{queue1}$$ where $A_{i}$ is the interarrival time between the $(i-1)$th and the $i$th customer, $S_{i}$ is the service time of the $i$th customer. The distributions of $A_{i}$ and $S_{i}$ are respectively $G_{a}(t)$ and $G_{s}(\theta,t)$. We consider two scenarios.
[*Case 1*]{}. We find $\theta^{*}$ through computer simulation. We write a program to simulate the GI/G/1 queue. At the $n$th iteration, we perform two experiments with the same $\xi_{n}$ to obtain a $h_{n}$ that is defined by (\[h\_generic\]). Consider that the inversion method is used in the generation of random variables $A_{i}=G_{a}^{-1}(u_{i}),
S_{i}=G_{s}^{-1}(\theta,v_{i}), i=1,2,..., N$. Define the random factor as $\xi=
[u_{1},u_{2},...,u_{N},v_{1},v_{2},...,v_{N}]^{T}$, $A(\xi)=
[A_{1},A_{2},...,A_{N}]$, $S(\theta,\xi)=[S_{1},S_{2},...,S_{N}]$, and $X(\theta,\xi)=[A(\xi), S(\theta,\xi)]^{T}$. Since the function $\max\{x,0\}$ is continuous in $x$, $L(X)$ is continuous in $X$. According to Theorem 10, we know that $\tilde{M}_{1}(\theta)=0$ since both $G_{a}(t)$ and $G_{s}(\theta,t)$ are strictly increasing and continuously differentiable in $t$. Note that $L(X)$ is not differentiable in $X$. However, $L(X)$ is left and right differentiable with bounded one-sided derivatives. A simple modification of the proof of Corollary 3 shows that $Var[h_{n}]=O(1)$. Therefore, the convergence rate for the KW algorithm is $n^{-1/2}$. If the composition method is used in the generation of $S(\theta,\xi)$ according to the distribution $G_{s}(\theta,t)$, then from Theorems 7 and 8 (which is applicale to the case of multivariates) we know that the rate of convergence is $n^{-2/5}$.
[*Case 2*]{}. Assume that this is a real system and we want to perform on-line parameter adjustment. Let visualize $n$ as the $n$th day of service. Suppose that the server serves more than $N$ customers each day. At the $n$th day, the server serves customers with parameter value $\theta_{n}$ and simultaneously collects information of $X(\theta_{n},\xi_{n})$ which simply is a record of interarrival times $\{A_{i}^{n}\}$ and service times $\{S_{i}^{n}\}$. At the end of the $n$th day, the server calculates $$v_{i}=G_{s}(\theta_{n},S_{i}^{n}), i=1,2,3, ..., N.$$ It is easy to verify that each $v_{i}$ is uniform on $[0, 1)$. Then the server defines $\xi_{n}$ from the preceding $v_{i},
i=1,2, ..., N$, takes a $\delta_{n}>0$, and $$S(\theta_{n}+\delta_{n},\xi_{n})=[S_{1}^{n,1},S_{2}^{n,1},...,S_{N}^{n,1}],
S_{i}^{n,1}
=G_{s}^{-1}(\theta_{n}+\delta_{n},G_{s}(\theta_{n},S_{i}^{n})),
\;\; i=1,2,..., N;$$ $$S(\theta_{n}-\delta_{n},\xi_{n})=[S_{1}^{n,2},S_{2}^{n,2},...,S_{N}^{n,2}],
S_{i}^{n,2}
=G_{s}^{-1}(\theta_{n}-\delta_{n},G_{s}(\theta_{n},S_{i}^{n})),
\;\; i=1,2, ..., N.$$ If $G_{s}(\theta,t)=1-e^{-t/\theta}$ is exponential, then $S_{i}^{n,1}=
(\theta_{n}+\delta_{n})S_{i}^{n}/\theta_{n}$, $S_{i}^{n,2}=
(\theta_{n}-\delta_{n})S_{i}^{n}/\theta_{n}$. With the values of $A(\xi_{n}), S(\theta_{n}+\delta_{n},\xi_{n}),
S(\theta_{n}+\delta_{n},\xi_{n})$, from (40) and the form of $L(X(\theta,\xi))$, the server computes $L(X(\theta_{n}+
\delta_{n},\xi_{n}))$ and $L(X(\theta_{n}-\delta_{n},\xi_{n}))$, which determines a $h_{n}$. With this $h_{n}$, the server updates the parameter $\theta_{n+1}$ according to the KW algorithm (\[kw\]) for the next $(n+1)$th day. In such a way, we have formulated an on-line optimization problem that mimics the Monte Carlo optimization. Its convergence can be analyzed similarly to that of Case 1. Our purpose here is simply to point out that the results of this paper are not restricted to Monte Carlo optimization.
[**7. Summary**]{}. So far, we have examined several variations of the KW algorithm and the MD algorithm under the symmetric finite difference, the one-sided finite difference, and the use of CRN when different methods are used in the generation of random variables. The results of this paper, together with previous results on the KW algorithm without the use of CRN \[c.f. Fabian (1971); Kushner and Clark (1978)\], provide a complete view toward the rates of convergence for the KW algorithm. For the ease of comparison, we summarize all the results in the following table.
Table I. Rates of convergence for the KW/MD algorithm
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\begin{array}{c} \mbox{with CRN}\\ \mbox{$h$(\ref{h_generic})} $\begin{array}{c} \mbox{with CRN}\\ $\begin{array}{c} $\begin{array}{c} \mbox{without CRN}\\
\end{array}$ \mbox{$h$(\ref{th5_1})} \end{array}$ \mbox{without CRN}\\ \mbox{$h$(\ref{kw_h})} \mbox{$h$(\ref{kw_1sideh})} \end{array}$
\end{array}$
------------------------------------------------------------------------- ----------------------------------------------------------------- -------------------------------------------- --------------------------------------------------- ------------------------------------------------
$\begin{array}{c} \mbox{inversion:} \\ M_{1}(\theta)\neq 0 \end{array}$ $n^{-2/5}$ $n^{-1/3}$ $n^{-1/3}$ $n^{-1/4}$
$\begin{array}{c} \mbox{inversion:} \\ M_{1}(\theta)= 0 \end{array}$ $n^{-1/2}$ $n^{-1/2}$ $n^{-1/3}$ $n^{-1/4}$
$\begin{array}{c} \mbox{rejection:} \\ \mbox{general} \end{array}$ $n^{-2/5}$ $n^{-1/3}$ $n^{-1/3}$ $n^{-1/4}$
$\begin{array}{c} \mbox{composition:} \\ \mbox{general} \end{array}$ $n^{-2/5}$ $n^{-1/3}$ $n^{-1/3}$ $n^{-1/4}$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In Table I, $h$(\[kw\_h\]), $h$(\[kw\_1sideh\]), $h$(\[h\_generic\]), and $h$(\[th5\_1\]) refer to the finite-difference approximation $h_{n}$ defined by (\[kw\_h\]), (\[kw\_1sideh\]), (\[h\_generic\]), and (\[th5\_1\]), respectively. The phrase “without CRN” refers to using independent samples in calculating the finite difference $\{h_n\}$, which excludes sampling schemes that may lead to correlations between the samples. In other words, “without CRN” simply means that $\xi_{1,n}$ and $\xi_{2,n}$ are independent in (\[kw\_h\]) and (\[kw\_1sideh\]). When the inversion method is used in the generation of random variables and when $M_{1}(\theta)=0$, we assume that Corollaries 3 and 4 are applicable. Results pertaining to the KW algorithm without the use of CRN can be found in, for example, Fabian (1971), and Kushner and Clark (1978).
Generally speaking, the use of CRN is always helpful in accelerating the convergence of the KW algorithm or the MD algorithm. In some cases, such as when $M_{1}(\theta)=0$ in Theorem 5, CRN helps a lot. In some of other cases, CRN may help less much. When the inversion method is used and when $M_{1}(\theta)=0$, the convergence rate can reach the best possible rate for the two types of stochastic approximation algorithms. The remark at the end of Subsection 3.2 shows that, as far as the convergence rate of the KW algorithm is concerned, the inversion method is superior to the rejection method. Note that inversion can also be used to generate random variables with distributions of the form $\sum_{i}p_{i}(\theta)F_{i}(\theta,x)$. A comparison of Theorem 4 and Theorems 7 and 8 shows that inversion is also superior to composition. When the distribution function $F(\theta,x)$ of $X(\theta,\xi)$ is strictly increasing and continuous, a close examination of the inversion, rejection, and composition methods shows that $X(\theta,\xi)$ is continuous in $\theta$ if it is generated from inversion. However, $X(\theta,\xi)$ is discontinuous in $\theta$ if it is generated from either rejection or composition. It is such a distinction of continuity that determines the rates of the convergence for the KW algorithm.
REFERENCES
1. [P. Bratley, B. Fox, and L. Schrage,]{} [*A Guide to Simulation*]{}, Springer-Verlag, New York, 1983
2. [D.L. Burkholder,]{} [*On a class of stochastic approximation processes,*]{} Annals of Mathematical Statistics, 27 (1956), pp. 1044-1059.
3. [S. Cambanis, G. Simons, and W. Stout,]{} [*Inequalities for $Ek(X,Y)$ when marginals are fixed,*]{} Z. Whar. Geb. 36 (1976), pp. 285-294.
4. [K.L. Chung,]{} [*On a stochastic approximation method,*]{} Annals of Mathematical Statistics, 25 (1954), pp. 463-483.
5. [R. W. Conway,]{} [*Some tactical problems in digital simulation,*]{} Management Science, 10 (1963), pp. 47-61.
6. [L. Devroye,]{} [*Coupled samples in simulation,*]{} Operations Research, 38 (1990), pp. 115-126.
7. [V. Dupa$\check{c}$,]{} [*On the Kiefer-Wolfowitz approximation method,*]{} Casopis Pest. Math, 82 (1957), pp. 47-75.
8. [J.C. Duchi, A. Agarwal, M. Johansson, and M.I. Jordan,]{} [*Ergodic Mirror Descent,*]{} SIAM Journal on Optimization, 22 (2012), pp. 1549-1578.
9. [J. Duchi, M.I. Jordan, M. Wainwright, and A. Wibisono,]{} [ *Finite sample convergence rates of zero-order stochastic optimization methods,*]{} In [*Advances in Neural Information Processing Systems (NIPS)*]{}, P. Bartlett, F. Pereira, L. Bottou and C. Burges (Eds.), 2013.
10. [A. Dvoretzky,]{} [*On stochastic approximation,*]{} Proc. Third Berkeley Symp. Math. Statis. Prob., 1 (1956), pp. 39-56.
11. [V. Fabian,]{} [*Stochastic approximation,*]{} In [*Optimizing Methods in Statistics*]{}, J.S. Rustagi (ed.), Academic Press, New York, 1971.
12. [G.S. Fishman,]{} [*Correlated simulation experiments,*]{} Simulation, 23 (1974), pp. 177-180.
13. [W.R. Franta,]{} [*The Process View of Simulation*]{}. North Holland, New York, 1975.
14. [P. Glasserman and D. Yao,]{} [*Some guidelines and guarantees for common random numbers,*]{} Management Sciences, 38(1992), pp. 884-908.
15. [J.M. Hammersley and D.C. Handscomb,]{} [*Monte Carlo Methods*]{}, Methuen, London, 1964.
16. [R.G. Heikes, D.C. Montogomery, and R.L. Rardin,]{} [*Using common random numbers in simulation experiments,*]{} Simulation, 27 (1976), pp. 81-85.
17. [Y.C. Ho and X.R. Cao,]{} [*Perturbation Analysis of Discrete Event Dynamic Systems,*]{} Kluwer Academic Publishers, Boston, 1991.
18. [H. Kesten,]{} [*Accelerated stochastic approximation,*]{} Annals of Mathematical Statistics, 29 (1958), pp. 41-59.
19. [J. Kiefer and J. Wolfowitz.]{} [*Stochastic estimation of the maximum of a regression function,*]{} Annals of Mathematical Statistics, 23 (1952), pp. 462-466.
20. [J.P.C. Kleijnen,]{} [*Statistical Techniques in Simulation,*]{} Marcel Dekker, New York, 1974.
21. [L. Kleinrock,]{} [*Queueing Systems*]{}, Vol.I, Wiley, New York, 1976.
22. [S.G. Krantz,]{} [*Real Analysis and Foundations*]{}, CRC Press, Bocan Raton, FL, 1991.
23. [H.J. Kushner and D.S. Clark,]{} [*Stochastic Approximation Methods for Constrained and Unconstrained Systems*]{}, Springer-Verlag. New York, 1978.
24. [A.M. Law and W.D. Kelton.]{}[*Simulation Modeling and Analysis*]{}, McGraw-Hill, New York, 1982.
25. [A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro,]{} [ *Robust stochastic approximation approach to stochastic programming,*]{} SIAM Journal on Optimization, 19 (2009), pp. 1574-1609.
26. [A. Nemirovski, and D. Yudin,]{} [*Problem Complexity and Method Efficiency in Optimization*]{}, Wiley, New York, 1983.
27. [H. Robbins, and S. Monro,]{} [*A stochastic approximation method,*]{} Annals of Mathematical Statistics, 22 (1951), pp. 400-407.
28. [J. Sacks,]{} [*Asymptotic distribution of stochastic approximation procedures,*]{} Annals of Mathematical Statistics, 29 (1958), pp. 373-405.
29. [A.N. Shiryayev,]{} [*Probability*]{}, Springer-Verlag, New York, 1984.
30. [M.T. Wasan,]{} [*Stochastic Approximation,*]{} Cambridge University Press, Cambridge, England, 1969.
31. [W. Whitt,]{} [*Bivariate distributions with given marginals,*]{} Ann. Math. Stat., 4 (1976), pp. 1280-1289.
[^1]: *This work is supported in part by the U.S. Army Research Office under agreement W911NF-04-D-0003.*
|
---
abstract: 'We examine the correlation of the limit price with the order book, when a limit order comes. We analyzed the Rebuild Order Book of Stock Exchange Electronic Trading Service, which is the centralized order book market of London Stock Exchange. As a result, the limit price is broadly distributed around the best price according to a power-law, and it isn’t randomly drawn from the distribution, but has a strong correlation with the size of cumulative unexecuted limit orders on the price. It was also found that the limit price, on the coarse-grained price scale, tends to gather around the price which has a large size of cumulative unexecuted limit orders.'
address:
- 'Department of Management Information, Fukuyama Heisei University, Fukuyama, Hiroshima 720-0001, Japan'
- 'NiCT, 2-2-2 Hikaridai Seiki-cho Soraku-gun, Kyoto 619-0288, Japan (ATR)'
author:
- 'Jun-ichi Maskawa'
title: Correlation of coming limit price with order book in stock markets
---
Limit order ,Order book ,Stock market ,Data analysis 07.05.Kf ,89.65.Gh
Introduction {#intro}
============
The fat tail of the price fluctuation and the long memory of the volatility are common features observed in financial markets employing the continuous double auction as the mechanism for price formation [@gopik1998; @plerou1999; @liu1999; @cont2001]. The origin of such features is an important problem to be solved in econophysics and finance, and still under debated. Many models that give rise to one of those or both features have been proposed [@bouchaud2003].
Among those models, Maslov has studied a simple model of markets driven by continuous double auctions [@maslov2000]. In his model, traders choose one from the two types of orders at random. One is a market order to sell or buy at the best available price at the time when the trader places the order. The other is a limit order to sell or buy with the specification of the limit price, which is the worst allowable price for the trader. A new limit order to sell (buy) is placed above (below) the current market price with a random offset drawn from a uniform distribution in a given interval. The size of order is fixed for simplicity. Numerical simulation has shown that the distribution of the price fluctuation generated by this model has a power-law tail, and the volatility has a long range correlation. Although his model has some unsatisfactory details about the statistics of the price fluctuation [@note1], the approach of microscopic market models to this problem is convincing, because they describe the actual process of price formation from the ultimate microscopic description level. Several authors have proposed the market models in the same class [@challet2003; @smith2003; @maskawa2006].
The way traders select a limit price is random and non-strategic in Maslov’s model. It is a simplification of the actual way of traders to select a limit order. In the paper [@maskawa2006], traders are assumed to be mimetic, and prefer the limit price which has a large stock of limit orders, when they place limit orders. Incorporating this tendency of coming limit price into the microscopic model of stock market, the author reproduces the price fluctuation with the more real statistics depending on the parameter of the model.
In this paper, we empirically examine the correlation of the limit price with the condition of order book at the time when a limit order comes. We analyze the Rebuild Order Book of Stock Exchange Electronic Trading Service (SETS), which is the centralized order book market of London Stock Exchange. As a result, the selected limit price is broadly distributed according to a power-law, and it isn’t randomly drawn from the distribution, but has a strong correlation with the size of cumulative unexecuted limit orders on the price. It was also found by the price scale transformation of the conditional probability that the limit price, on the coarse-grained price scale, tends to gather around the price which has a large size of cumulative unexecuted limit orders.
Rebuild Order Book and our data {#data}
===============================
We analyze the Rebuild Order Book of SETS. In this section, we give a brief description of the Rebuild Order Book and our data. SETS is fully automated order book trading service of London Stock Exchange [@lse]. The order book holds details of all orders. A coming new order to sell or buy will be fully or partially executed against existing orders on the order book, if both requirements agree. The unexecuted portion of the order will be stored on the order book. The Rebuild Order Book is composed of 3 files. Order detail file contains details of new orders, their prices, sizes and so on. Order history file contains a history of each order and the method by which it is removed, that is, deletion, expiry, partial match and full match. Trade report contains details of every trade, that is, the price, the size and so on. Merge of these files and sort of records by the time stamp enable us to pursue the occurrence and the change of each order.
We study the Rebuild Order Book of the 6 months since July to December in 2004. Our data contains all orders and transactions of the selected 13 stocks listed on SETS. The selected stocks are most actively traded in the period, which are the stocks of top 13 bargains in July 2004. There are millions of limit orders, cancellations and executions in our data. The stocks we analyzed are Astrazeneca (AZN), Barclays (BARCS), BT Group (BT), BP. (BP), Diageo (DGE), Glaxosmithkline (GSK), HBOS (HBOS), HSBA HLDGS (HSBA), Lloyds Tsb Group (LLOY), Shell Tranport & Trading Co. (SHEL), Tesco (TSCO), Royal Bank Scot (RBS) and Vodafone Group (VOD), belonging to various industries, that is, bank, beverages, oil & Gas, telecommunication services and so on.
Results of data analysis and discussions {#results}
========================================
The purpose of this paper is to examine the correlation of the limit price with the condition of order book at the time when a limit order comes. In this section, we report some results of analysis on the data described in the previous section.
First of all, we show that the coming limit price is broadly distributed around the best price at the time when the order comes. Figure \[fig1\] is the semi-log plot of the probability distribution function of relative limit price: $price-ask$ (for sell limit orders), $bid-price$ (for buy limit orders). The unit of price is the tick size of each stock. The inset is the log-log plot of the cumulative distribution function for limit orders placed on the book. The exponent of the power-law tail of the relative limit price for orders placed on the book is near -1.5, and the value is consistent with the previous work of this kind [@zovko].
Second, we examine the correlation of the selection of limit price with the cumulative size of unexecuted limit orders at each price. Let us consider a situation of the order book. There may be several prices at which unexecuted limit orders are stored. Each price may hold a fraction of whole unexecuted orders. Then, which price will be selected as a coming limit price? Figure \[fig2\] shows the conditional probability $P_f$ that a coming limit order is placed at the price holding a given fraction $f$ of all limit orders. In both sides of order, The conditional probability $P_f$ decreases with fraction $f$ while $f$ is smaller than about 0.4, and increases with $f$ while $f$ is smaller than about 0.6. The cumulative size of unexecuted limit orders at each price influences the probability of the selection of the price. However, the conditional probability is not a monotonically increasing function of $f$.
Next, we perform the scale transformation of price, and renormalize the conditional probability. We group the relative price $2i$ and $2i+1$ into the group $i$, where the relative price $i=...,-2,-1,0,1,2,...$ is defined by the equation $i=price -
ask$ and $i=bid - price$ for sell and buy orders respectively. On the coarse-grained price scale, we call the group $0$ the best price (ask/bid), and the group $i$ the relative price $i$ again. Repeating this procedure k-times, we have the scale transformation of the relative price, $i^{(k)}=2^k i$. When the stochastic variable $F^k_i$ denote the cumulative size of limit order at the price $i^{(k)}$ and the binary stochastic variable $S^k_i=0,1$ the selection of the price $i^{(k)}$, the renormalization of the conditional probability for each price is defined by the equation: $$\begin{aligned}
P(S^k_i=1|F^k_i=f)=\frac{P(S^k_i=1,F^k_i=f)}{P(F^k_i=f)}\nonumber\\
=\frac{\sum\limits_{f^{k-1}_{2i}+f^{k-1}_{2i+1}=f}P(S^{k-1}_{2i}=1
\cup S^{k-1}_{2i+1}=1,F^{k-1}_{2i}=f^{k-1}_{2i} \cap
F^{k-1}_{2i+1}=f^{k-1}_{2i+1})}{\sum\limits_{f^{k-1}_{2i}+f^{k-1}_{2i+1}=f}P(F^{k-1}_{2i}=f^{k-1}_{2i}
\cap F^{k-1}_{2i+1}=f^{k-1}_{2i+1})}.%\nonumber
\label{renormalization}\end{aligned}$$ The conditional probability $P^k_f$, on the coarse-grained price scale, is expressed by the weighted average $P^k_f=\sum\limits_i
P(F^k_i=f|\cup_j F^k_j=f)P(S^k_i=1|F^k_i=f)$. The results for $k=0$ to $5$ ($k=0$ means the original price scale) are shown in Fig. \[fig3\]. On the coarse-grained price scales $k \ge 1$, the conditional probabilities are monotonically increasing function of $f$. It means that the limit price, on the coarse-grained price scale, tends to gather around the price which has a large size of cumulative unexecuted limit orders.
Finally, we study the dependency of the conditional probability on the relative price $i^{(k)}$. We divide the whole prices into three groups, $i^{(k)}=0$ (the best price), $i^{(k)}>0$ (prices on the book) and $i^{(k)}<0$ (prices in the spread), and derive the corresponding conditional probabilities $P(S^k_0=1|F^k_0=f)$, $P(\cup_{i>0}S^k_i=1|\sum_{i>0}F^k_i=f)$ and $P(\cup_{i<0}S^k_i=1|F^k_0=f)$. The conditional probability $P(\cup_{i>0}S^k_i=1|\sum_{i>0}F^k_i=f)$ is derived from the equation $P(S^k_0=1|F^k_0=1-f)+
P(\cup_{i>0}S^k_i=1|\sum_{i>0}F^k_i=f)+P(\cup_{i<0}S^k_i=1|F^k_0=1-f)=1$. The results for sell and buy order on the price scale $i^{(5)}=2^5
i$ ($k=5$) are shown in Fig. \[fig4\]. The unconditional probability that traders select the group is also shown by the horizontal line. On this coarse-grained price scale, each price contains 32 original price levels. In the region of $f$ close to 1, the conditional probability that traders select the prices separate from the best price by over 32 original price levels is hundreds times as large as the unconditional probability, which is tiny. In the region $f<0.07$, the inequality $P(\cup_{i\le0}S^k_i=1|F^k_0=f)<P(\cup_{i>0}S^k_i=1|F^k_0=f)$ holds, which means the state such that the stored limit orders are sparse within the distance of 31 original price levels from the best price is stable. As is shown by the simulation in the paper [@maskawa2006], such unequal attractive power of prices has amplified the fluctuation of gaps between occupied price levels, and cause the power-law fluctuation of price changes and the long memory of the volatility. This result agrees with the statement by Farmer et al. in the paper [@farmer2004]. They has argued that large returns are not caused by large orders, while the large gaps between the occupied price levels on the order book lead to large price changes in each transaction.
Conclusions
===========
We have analyzed the Rebuild Order Book of SETS, which is fully automated order book trading service of London Stock Exchange in order to examine the correlation of the limit price with the condition of order book at the time when a limit order comes.
We have found that there is a obvious correlation. Especially on the coarse-grained price scale, our data clearly shows that the limit price tends to gather around the price at which a large size of unexecuted limit orders has been stored. Such unequal attractive power of prices is a promising candidate for the origin of the fat tail of the price fluctuation and the long memory of the volatility.
This work was supported by the Japan Society for the Promotion of Science under the Grant-in-Aid, No. 15201038.
[00]{}
P. Gopikrishnan, M. Meyer, L. A. N. Amaral and H. E. Stanley, Inverse Cubic Law for the Distribution of Stock Price Variations, Eur. Phys. J. B 3 (1998) 139. V. Plerou, P. Gopikrishnan, L. A. N. Amaral, M. Meyer, H. E. Stanley, Scaling of the distribution of price fluctuations of individual companies, Phys. Rev. E 60, (1999) 6519. Y. Liu, P. Gopikrishnan, Cizeau, Meyer, Peng, H. E. Stanley, Statistical properties of the volatility of price fluctuations, Phys. Rev. E 60, (1999) 1390. Cont, R., Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance 1 (2001) 223. J.-P. Bouchaud, M. Potters, [*Theory of Financial Risk and Derivtive Pricing*]{}, 2nd edition, Cambridge University Press, Cambridge, 2003 and references therein. S. Maslov, Simple model of a limit order-driven market, Physica A 278 (2000) 571. The exponent of the power-law tail of price fluctuations generated by Maslov’s model is inside the Levy stable region, while the observed value in actual markets is close to -3 [@gopik1998; @plerou1999]. In addition, the Hurst exponent of the price diffusion is H = 1/4 ( under diffusive) , regardless of the range of time window. In actual markets, we have H = 1/2 for long-term, which is the value for free diffusion. D. Challet, R. Stinchcombe, Non-constant rates and over-diffusive prices in a simple model of limit order market, Quantitative Finance 3 (2003) 155. E. Smith, J. D. Farmer, L. Gillemot, S. Krishnamurthy, Statistical theory of the continuous double auction, Quantitative Finance 3 (2003) 481. J. Maskawa, Stock price fluctuations and the mimetic behaviors of traders, 2006, http://arXiv.org/abs/physics/0607202. London Stock Exchange, Guide to Trading Services, 2006, available at http://www.londonstockexchange.com/. I. Zovko, J. D. Farmer, The power of patience: a behavioral regularity in limit-order placement, Quantitative Finance 2 (2002) 387. J. D. Farmer, L. Gillemot, F. Lillo, S. Mike, A. Sen, What really causes large price change?, Quantitative Finance 4 (2004) 383.
![Probability distribution function of the relative limit price. The unit of price is the tick size of each stock. The inset is the log-log plot of the cumulative distribution function for limit orders placed on the book. The dashed-line represents the power-law with the exponent -1.5. []{data-label="fig1"}](maskawa_fig1.eps){width="8cm"}
![Conditional probability that a coming limit order is placed at the price holding a given fraction $f$. The error bar represents 95 % confidential interval. (a)Sell order. (b)Buy order.[]{data-label="fig2"}](maskawa_fig2a.eps "fig:"){width="8cm"} ![Conditional probability that a coming limit order is placed at the price holding a given fraction $f$. The error bar represents 95 % confidential interval. (a)Sell order. (b)Buy order.[]{data-label="fig2"}](maskawa_fig2b.eps "fig:"){width="8cm"}
![Conditional probability on different price scales ($k=0$ to $5$). The parameter k is explained in text. (a)Sell order. (b)Buy order.[]{data-label="fig3"}](maskawa_fig3a.eps "fig:"){width="8cm"} ![Conditional probability on different price scales ($k=0$ to $5$). The parameter k is explained in text. (a)Sell order. (b)Buy order.[]{data-label="fig3"}](maskawa_fig3b.eps "fig:"){width="8cm"}
![Conditional probability on a coarse-grained price scale ($k=5$). Prices are divided into three groups, the best price, prices on the book and prices in the spread. The horizontal axis for the prices in the spread is the fraction $f$ of limit orders held at the best price. The horizontal line for each group represents the unconditional probability that traders select the group.[]{data-label="fig4"}](maskawa_fig4a.eps "fig:"){width="8cm"} ![Conditional probability on a coarse-grained price scale ($k=5$). Prices are divided into three groups, the best price, prices on the book and prices in the spread. The horizontal axis for the prices in the spread is the fraction $f$ of limit orders held at the best price. The horizontal line for each group represents the unconditional probability that traders select the group.[]{data-label="fig4"}](maskawa_fig4b.eps "fig:"){width="8cm"}
|
---
abstract: |
We explore potential strategies for testing General Relativity via the coherent motions of galaxies. Our position at $z=0$ provides the reference point for distance measures in cosmology. By contrast, the Cosmic Microwave Background at $z \simeq 1100$ acts as the point of reference for the growth of large scale structure. As a result, we find there is a lack of synergy between growth and distance measures. We show that when measuring the gravitational growth index $\gamma$ using redshift-space distortions, typically $80\%$ of the signal corresponds to the local growth rate at the galaxy bin location, while the remaining fraction is determined by its behaviour at higher redshifts.
In order to clarify whether modified gravity may be responsible for the dark energy phenomenon, the aim is to search for a modification to the growth of structure. One might expect the magnitude of this deviation to be commensurate with the apparent dark energy density $\Omega_\Lambda(z)$. This provides an incentive to study redshift-space distortions (RSD) at as *low* a redshift as is practical. Specifically, we find the region around $z = 0.5$ offers the optimal balance of available volume and signal strength.
author:
- Fergus Simpson
bibliography:
- 'M:/Routines/dis.bib'
date:
-
-
title: Redshift Sensitivity of the Kaiser Effect
---
Introduction
============
Dark energy is a low-redshift phenomenon, and in accordance with the standard $\Lambda CDM$ model, it appears to exert a rapidly decaying influence towards higher redshifts, at a rate approaching $(1+z)^{-4}$. Physical models of dark energy may be distinguished by an equation of state $w \neq -1$, while a break from general relativity would likely exhibit a distinctive structure formation history. The consequences of a modification to general relativity are somewhat speculative at this stage, but naturally one might expect any alteration of the growth rate to become particularly prominent at late times, in accordance with the observed change in global dynamics. For instance, the $f(R)$ models explored by Hu & Sawicki [@2007PhRvD..76j4043H] found this to be the case on large scales, as did He et al [@2009JCAP...07..030H] with their interacting model.
A number of probes are capable of measuring $w$ via its influence on the redshift-distance relation. It is rather more difficult to study the growth rate, due to our uncertainty in the behaviour of galaxy bias, but there are currently two promising avenues available for future exploration. Weak gravitational lensing provides a direct measurement of the dark matter distribution, and its evolution with redshift. However the focus of this work will be redshift-space distortions, which exploit the relationship between the large-scale coherent velocities of galaxies and the growth rate of perturbations.
Weight functions have previously been applied to the equation of state as a function of redshift $w(z)$ [@2003MNRAS.343..533D; @2005PhRvD..71h3501S; @2006PhRvD..73h3001S], and this work extends the concept to the growth index $\gamma$ [@2005PhRvD..72d3529L]. They are designed to illustrate the redshift sensitivity of a given survey, and this is often quite different from the source redshift distribution. Weight functions are closely related to, and may be derived from, the principal component approach. In related work, Zhao et al. [@2009arXiv0905.1326Z] recently explored the principal components of the metric ratio, a different quantity also often denoted by $\gamma$.
In §\[sec:wts\] we briefly review the concept of principal components and their application in deriving the weight function. §\[sec:rsd\] explores the RSD redshift sensitivity to the growth index $\gamma$, and compares these weight functions to those of the dark energy equation of state. §\[sec:opt\] addresses the question of which redshift is most efficient at measuring $\gamma$, while in §\[sec:gro\] we consider a change of parameterisation to the growth rate $f$.
Weight Functions {#sec:wts}
================
The main purpose of a “weight function" is to illustrate the redshift range over which we are most sensitive. This has previously been applied to the dark energy equation of state [@2003MNRAS.343..533D; @2005PhRvD..71h3501S; @2006PhRvD..73h3001S], where the best-fit constant equation of state $w^{fit}$ is expressed as a weighted integral over the true functional form
$$w^{fit} = \int \Phi_{w}(z) w(z) {\mathrm{d}}z .$$
Here we apply this approach to the growth index $\gamma$ [@2005PhRvD..72d3529L] (not to be confused with another modified gravity parameter, the metric ratio), defined by the approximate relation [@1998ApJ...508..483W]
$$f(z)\equiv\frac{{\mathrm{d}}\ln\delta}{{\mathrm{d}}\ln a}\simeq\Omega_{m}^{\gamma}(z) ,$$
where $f(z)$ is the logarithmic growth rate of linear perturbations. Ordinarily this is well approximated by $\Omega_m^{0.55}$, but in order to act as a phenomenological test of gravity the exponent is treated as a variable, denoted by $\gamma$. Much like the equation of state $w(z)$, an important aim for future cosmological surveys will be to test whether the growth index $\gamma$ matches the value corresponding to that expected from General Relativity, despite the time-dependence of any deviation remaining highly uncertain. Few feasible alternatives to GR are yet known, but DGP [@2000PhLB..485..208D] is one example where the growth index has been evaluated, with $\gamma_{DGP} = 0.69$ [@2007APh....28..481L]. Due to its scale-dependent growth factor, f(R) models are not so well described by this simple parameterisation. However it is worth noting that in general these models enhance linear growth and as such typically exhibit lower values at around $\gamma \sim 0.4$ [@2009PhRvD..80h4044T].
The main focus of this work is to identify the redshift at which we are observing $\gamma$. We reach a quantitative solution by considering the constant value of $\gamma$ one would infer from an arbitrary function $\gamma(z)$, as given by
$$\label{eq:gammawt}
\gamma^{fit} = \int \Phi_{\gamma}(z) \gamma(z) {\mathrm{d}}z .$$
In order to generate $\Phi(z)$, we begin by decomposing the function $\gamma(z)$ into orthonormal eigenfunctions $e_i(z)$ as given by
$$\gamma(z) = \sum_i \alpha_i e_i (z) .$$
As shown in [@2006PhRvD..73h3001S], the weight function may be expressed as a sum of the eigenmodes $e_i (z)$ weighted by the errors $\sigma_i$ associated with the eigenvalues $\alpha_i$, along with a second weighting term which penalises highly oscillatory modes, for they will contribute little when fitting a constant value.
$$\label{eq:phipca} \Phi(z)= \frac{\sum_i{e_i(z)\int e_i(z') {\mathrm{d}}z'/\sigma^2(\alpha_i)}}{\sum_j \left[\int e_j(z'') {\mathrm{d}}z''\right]^2
/\sigma^2(\alpha_j)} \,\,.$$
In general the form of the weight function is invariant to the absolute error of the survey, and for a given technique its strongest dependence lies with the source redshift distribution.
In recent work by Kitching & Amara [@2009arXiv0905.3383K], the authors highlight the potential dangers of evaluating the eigenmodes using a finite set of discretised redshift bins, given the variation which may arise when adopting different basis sets. While this may well prove problematic for studies which adopt fewer than $\sim 30$ bins, we ensure adequate convergence by dividing the scale factor into $300$ bins.
Since we are essentially trying to determine the Green’s function associated with the observable, we would ideally use Dirac’s $\delta$-functions to perturb the parameter in question, and in some cases such as supernovae this step may be performed analytically [@2003MNRAS.343..533D]. However, given that $\gamma$ enters as a raw exponent, as opposed to the integrated form that $w(z)$ appears in, we are restricted to working numerically.
![The redshift sensitivity of structure formation with linear redshift-space distortions. The solid line shows the weight function $\Phi_\gamma$, as defined in (\[eq:gammawt\]), where $\gamma$ is the gravitational growth index. The survey spans the redshift range $0.2<z<1.8$, with fixed solid angle and constant comoving number density, and is combined with Planck. The dashed line is the weight function for fitting a constant equation of state $\Phi_w$, for a typical high-redshift supernova and weak lensing survey, with sources spanning the same redshift range.[]{data-label="fig:rsd_wt"}](fig1){width="80mm"}
![A comparison of the weight functions for $\Phi_\gamma$ (solid) and $\Phi_w$ (dashed) using RSD and BAO from the same sample of galaxies at a mean redshift $z=1$ and with a bin width $\Delta z = 0.2$. This highlights the restricted range of redshift sensitivity for $\gamma(z)$, in particular it vanishes below the bin edge. There is also very little overlap between the two functions.[]{data-label="fig:rsd_wt_bin"}](fig2){width="80mm"}
Redshift-Space Distortions {#sec:rsd}
==========================
In redshift space the galaxy power spectrum is significantly anisotropic, and may be expressed as a function of the line-of-sight and tangential components of the Fourier modes, denoted $k_{\parallel}$ and $k_{\perp}$ respectively. On sufficiently large scales it is well described by
$$\label{eq:pkkaiser}
P(k_{\parallel},k_{\perp})=P(k)\left(1+\beta\mu^{2}\right)^{2},$$
where $\mu=k_\parallel / |\bf{k}|$, and $\beta = f/b$, with the local linear galaxy bias $b$ taken to be scale-independent. Combined with the CMB, the observable quantity is essentially given by [@2001MNRAS.327..689T; @2009MNRAS.393..297P; @simpsonp09]
$$\label{eq:fsig}
\frac{f(z) \sigma_8(z)}{\sigma_8(z_{CMB})} ,$$
and $\sigma_8(z)$ is the amplitude of dark matter perturbations at the redshift of the survey, $z$. Whilst this approach does assume that the background cosmology is already known, the form of the weight function is independent of the absolute error associated with the survey, and we find this description generates eigenmodes in very good agreement with those from the more rigorous approach presented in [@simpsonp09].
Figure \[fig:rsd\_wt\] demonstrates the redshift sensitivity which is achieved when studying the rate of structure formation with linear redshift-space distortions. Compared to that of the equation of state, a tendency towards substantially higher redshifts should not be too surprising given that the benchmark of structure growth is the cosmic microwave background. Any geometrical measure will rely on the behaviour of $w(z)$ all the way down to $z=0$, for a given $\Omega_{m}(z=0)$.
The difference becomes more striking when in Figure \[fig:rsd\_wt\_bin\] we compare these two sensitivities for sources in a limited redshift range $0.9<z<1.1$. A spectroscopic survey which generates will inevitably produce a simultaneous measure of the BAO at the same redshift. Much like lensing and supernovae, the BAO harbours most of its sensitivity at low redshift $z<0.5$, while the RSD measure $\gamma(z)$ in the bin itself ($\sim 80\%$), with the remaining weight all attributed to higher redshifts. This fraction remains fairly robust to a change in redshift.
The lack of overlap between the two weight functions seen in Figure \[fig:rsd\_wt\_bin\] is of concern, since even if a deviation of w=-1 was discovered, and $\gamma$ was in turn found to be perfectly consistent with General Relativity, this would leave us in a position where we are still unable to deduce the nature of the underlying physics. In other words, these observations have two equally plausible explanations - either (a) dark energy takes the form of a physical fluid with $w \neq -1$, within the framework of General Relativity, which recently took control of the cosmological expansion rate, or alternatively (b) cosmological dynamics are governed by a modified theory of gravity, generating an effective equation of state $w_{eff}(z)$ and a modified $\gamma(z)$, but which only became apparent at some lower redshift. Therefore given that the weight function $\Phi_\gamma$ is concentrated exclusively at $z>0.9$, we would be left unable to distinguish between the physics of these very different scenarios.
For redshift-space distortions, the spike in $\Phi_\gamma$ within the redshift bin is associated with the measure of $f(z)$ from \[eq:fsig\], whereas the non-zero sensitivity at higher redshift arises through the $\sigma_8(z)$ term, since this is a cumulative effect. It is typically found that almost $80\%$ of the weight lies within the redshift bin. As we shall see in §\[sec:gro\], this is partly due to the nature of our parameterisation.
![The precision with which the parameter $\gamma$ could be measured, as a function of the position of a single redshift bin of width $\Delta z = 0.2$. The dashed line corresponds to the simple case of a 20,000 square degree survey, with a fixed comoving number density at $n=10^{-3} h^{3}\mathrm{Mpc}^{-3}$, demonstrating the limitations of even the most ambitious high-redshift survey. For the solid line we impose an upper limit on the area density $N = 50 \, \rm{gals \, deg}^{-2}$. The dotted line illustrates the impact of changing the background cosmology such that $w=-0.9$. This leads to more dark energy at high redshift, pushing the peak out to $z \simeq 0.6$. []{data-label="fig:dgamma"}](3dgamma){width="80mm"}
![The sensitivity of redshift distortions to the modified growth rate $f(z)$, as given by (\[eq:phif\]) for a redshift bin of width $0.2$ at $z=1$. The dashed line is equivalent to the solid line in Figure \[fig:rsd\_wt\_bin\], but now recast in terms of the scale factor in accordance with (\[eq:scalefac\]).[]{data-label="fig:figf"}](4figf){width="80mm"}
Optimal Survey Strategies {#sec:opt}
=========================
One aspect the weight function does not address is the question of precision. Here we briefly review the impact that the position of a single galaxy redshift bin has on the absolute error on $\gamma$.
Whilst it will undoubtedly be of interest to explore the growth rate of structure across all observable redshifts, the priority must lie with the regime which maximises the opportunity of a significant result. We emphasise that at present dark energy is only known to be a low-redshift phenomenon; its influence at any other epoch remains highly speculative. As such it appears advantageous to study the growth of structure in regions of high $\Omega_\Lambda$, and the parameterisation of $\gamma$ offers a natural mechanism for this preferential weighting.
This raises the question, galaxies at which redshift are best suited to measuring $\gamma$? To help answer this, the plot in Figure \[fig:dgamma\] illustrates the precision with which $\gamma$ may be determined, as a function of redshift bin position. Rather than the simplified approach of treating $f \sigma_8$ as the observable, in this section we adopt the methodology outlined in the companion paper [@simpsonp09]. This essentially results in a simultaneous measure of the baryon acoustic oscillations, redshift space distortions, combined with the DETF Fisher matrix for Planck. To begin we consider the simplest case of the cosmic variance limit, such that the error on $P(k)$ is dictated by the available comoving volume at that redshift. We assume a sky coverage of $20,000$ square degrees, and marginalise over the parameters $[w_0, w_a, \Omega_\Lambda, \Omega_k, \Omega_m h^2, \Omega_b h^2, n_s, A_s, \beta, \gamma,]$ . However the emphasis here is not on the absolute value of the error itself, but how this changes as a function of the source redshift, which is relatively insensitive to our choice of parameter set.
At very low redshift the dominant factor is cosmic variance, due to the limited comoving volume available. Beyond the optimal peak close to $z=0.5$ there is simply a lack of “dark energy" at high redshift, which leads to $\Omega_m \simeq 1$ and thus $\sigma_\gamma$ diverges.
$$\delta\gamma=\frac{1}{\ln\left({\Omega_m}(z)\right)} \frac{\delta f}{f} .$$
A further contributing factor is the diminishing amplitude of the dark matter power spectrum at higher redshifts.
Note that the dashed line in Figure \[fig:dgamma\] assumes a constant comoving number density, neglecting the inevitable decline towards higher redshifts, and the associated increase in observing time due to the greater volume covered. These factors further disadvantage the prospects of high-redshift studies. Conversely, at low redshifts non-linearities in the power spectrum may prove problematic, as the simple form presented in (\[eq:pkkaiser\]) will likely prove inadequate.
At high redshift this corresponds to a prohibitively large number of sources. The solid line in Figure \[fig:dgamma\] reflects the error on $\gamma$ achievable with an upper bound on the number of sources per unit area ($N = 50 \, \rm{deg}^{-2}$). Whilst there are a variety of choices and assumptions that can be made regarding the survey parameters, most of these are only found to significantly affect the absolute error, leaving the peak position largely unaffected at $z \sim 0.5$. For the choice of fiducial background cosmology, changes in the abundance of dark energy will have some impact on the form of this function. For example, the dotted line illustrates the influence of setting $w=-0.9$, which corresponds to a greater $\Omega_\Lambda$ at high redshift and thus a stronger signal. Conversely, a value of $w=-1.1$ generates a stronger supression at high redshift. We also find that adding auxiliary information from geometric probes such as supernovae tends to preferentially strengthen the low-redshift measurements, shifting the peak down to $z \sim 0.3$.
Sensitivity to the Growth Rate {#sec:gro}
==============================
The high-redshift sensitivity diminishes in Figures \[fig:rsd\_wt\] and \[fig:rsd\_wt\_bin\] due to the nature of the parameter $\gamma$, such that at high redshift when $\Omega_m$ approaches unity, the value of $\partial f / \partial \gamma$ vanishes. As a comparison, it is of interest to consider $f(z)$ as the function to perturb instead of $\gamma$. To generate the appropriate weight function, the fiducial growth rate $f(z)$ is modulated by a prefactor $\alpha(z)=1$, such that
$$f(z) = \alpha(z) \Omega_m^{0.55}(z) ,$$
$$\label{eq:phif}
\alpha^{fit} = \int \Phi_{\alpha}(z) \alpha(z) {\mathrm{d}}z .$$
The corresponding weight function $\Phi_\alpha$ is shown in Figure \[fig:figf\]. Since a substantial degree of sensitivity now extends up to $z=1100$, it is more appropriate to view this in terms of the scale factor. Given the requirement $|\Phi(z) {\mathrm{d}}z| = |\Phi(a) {\mathrm{d}}a|$, the rescaling is simply given by
$$\label{eq:scalefac}
\Phi(a)=\Phi(z)(1+z)^2 .$$
Since the majority of fractional structure growth occurs at early times, a strong bias towards this epoch can be seen in Figure \[fig:figf\].
Conclusions
===========
At present there are two key approaches to study the evolution of dark matter perturbations, namely redshift-space distortions and weak gravitational lensing. They will provide a crucial piece of evidence in determining whether the phenomenon of dark energy may be attributed to a physical entity, or is simply due to a misunderstanding of the laws of gravity.
In this work we have quantified the epoch at which a galaxy redshift survey would be sensitive to the growth index $\gamma$. Specifically, given the absence of any weight at redshifts lower than that of the survey, low redshift surveys are left with the significant advantage of probing a broader behaviour $\gamma(z)$. Another interesting feature we have highlighted is that approximately $80 \%$ of the “weight" for $\gamma$ for any given redshift bin corresponds to the local value of $\gamma(z)$ at the location of the bin.
The main limitation of a low redshift survey would be the available comoving volume, along with the stronger prevalence of nonlinear perturbations which may hinder an accurate determination of $\beta$. However, for a given error on $\beta$, one reaches a much better determination of $\gamma$ compared to higher redshifts.
By contrast, a weak lensing survey with source galaxies at any redshift will still be sensitive to the value of $\gamma(z)$ across *all* redshifts $(0<z<1100)$. Although this may be restricted at very low redshift by the lensing efficiency.
For a purely cosmic-variance limited redshift survey, it appears the shell surrounding $z \sim 0.5$ is optimal in its efficiency. Incorporating more realistic constraints, such as limited observing time, will likely serve to lower this value further unless non-linearities prove highly problematic. This preference towards a low-redshift measurement reflects the increased likelihood of discovering a modified gravity signature when searching within the era of dark energy.
[**Acknowledgements**]{}\
The author would like to thank S. Bridle, A. Heavens, J. Peacock and P. Norberg for helpful comments. FS is supported by an STFC rolling grant, and acknowledges the hospitality of the Aspen Center for Physics where part of this work was undertaken.
|
---
abstract: |
In this paper the superheating of electron plasma by femtosecond laser pulses is investigated. With Heaviside thermal equation (*Lasers in Engineering*, **12,** (2002), p. 17) the generation of superhot electrons is described. It is shown that in hot electron plasma (i.e. with electron energies $>5$ MeV) the thermal shock waves can be generated.\
**Key words:** Femtosecond laser pulses; Hot electron plasma; Shock thermal waves.
---
**SUB- AND SUPERSONIC HEAT MOTION\
INDUCED BY\
FEMTOSECOND LASER PULSES**
*\**\
*Institute of Electron Technology*, *Al.* *Lotników 32-46*, *02-628 Warsaw Poland.*\
\
*Institute of Experimental Physics and Physics Teacher College*, *Warsaw University*, *Hoża 69*, *00-681 Warsaw*, *Poland.*\
to 1.5in[width1.5in height0.4pt depth0in]{}
*\*corresponding author*
Introduction
============
Recently it has become possible to produce MeV electrons with short-pulse multiteravat laser system [@1]. The fast ignitor concept [@2; @3] relevant to the inertial confinement fusion enhances the interest in this process. In an underdense plasma, electrons and ions tend to be expelled from the focal spot by the ponderomotive pressure of an intense laser pulse, and the formed channel [@4; @5] can act as a propagation guide for the laser beam. Depending on the quality of the laser beam, the cumulative effects of ponderomotive and relativistic self focusing [@5] can significantly increase the laser intensity. For these laser pulses, the laser electric and magnetic fields reach few hundreds of GV/m and megagauss, respectively, and quiver velocity in the laser field is closed to the light speed. The component of the resulting Lorentz force $(-ev\,x~\,\vec{B})$ accelerates electrons in the longitudinal direction, and energies of several tens of MeV can be achieved [@6]. Recently the spectra of hot electrons (i.e. with energy in MeV region) were investigated. In paper [@7] the interaction of 500 fs FWHM pulses with CH target was measured. The electrons with energy up to 20 MeV were observed. Moreover for electrons with energies higher than 5 MeV the change of electron temperature was observed: from 1 MeV (for energy of electrons $<$ 5 MeV) to 3 MeV (for energy of electrons $>5$ MeV). In this paper the interaction of femtosecond laser pulse with electron plasma will be investigated. Within the theoretical framework of Heaviside temperature wave equation, the heating process of the plasma will be described. It will be shown that in vicinity of energy of 5 MeV the sound velocity in plasma reaches the value $\frac{c}{\sqrt{3} } $ and is independent of the electron energy.
The model
=========
The mathematical form of the hyperbolic quantum heat transport was proposed in [@8] and [@9]. Under the absence of heat or mass sources the equations can be written as the Heaviside equations: $$\frac{1}{v_{\rho }^{2} } \frac{\partial ^{2} \rho }{\partial t^{2}
} +\frac{1}{D_{\rho } } \frac{\partial \rho }{\partial t}
=\frac{\partial ^{2} \rho }{\partial x^{2} }\label{eq1}$$ and $$\frac{1}{v_{T}^{2} } \frac{\partial ^{2} T}{\partial t^{2} }
+\frac{1}{D_{T} } \frac{\partial T}{\partial t} =\frac{\partial
^{2} T}{\partial x^{2} }\label{eq2}$$ for mass and thermal energy transport respectively. The discussion of the properties of Eq. (\[eq1\]) was performed in [@8] and Eq. (\[eq2\]) in [@9]. In Eq. (\[eq1\]) $v_{\rho } $ is the velocity of density wave, $D_{\rho } $ is the diffusion coefficient for mass transfer. In Eq. (\[eq2\]) $v_{T} $ is the velocity for thermal energy propagation and $D_{T} $ is the thermal diffusion coefficient.
In the subsequent we will discuss the complex transport phenomena, i.e. diffusion and convection in the external field. The current density in the case when the diffusion and convection are taken into account can be written as: $$j=-D_{\rho } \frac{\partial \rho }{\partial t} -\tau
\frac{\partial j}{\partial t} +\rho V.\label{eq3}$$ In equation (3) the first term describes the Fourier diffusion, the second term is the Maxwell-Cattaneo term and the third term describes the convection with velocity $V$. The continuity equation for the transport phenomena has the form: $$\frac{\partial j}{\partial x} +\frac{\partial \rho }{\partial t}
=0.\label{eq4}$$ Considering both equations (3) and (4) one obtains the transport equation: $$\frac{\partial \rho }{\partial t} =-\tau _{\rho } \frac{\partial^{2} \rho }{\partial t^{2} }
+D_{\rho } \frac{\partial ^{2} \rho}{\partial x^{2} } -V\frac{\partial \rho }{\partial x}.\label{eq5}$$ In equation (\[eq5\]) $\tau _{\rho } $ denotes the relaxation time for transport phenomena. Let us perform the Smoluchowski transformation for $\rho (x,t)$ $$\rho =\exp\left[ \frac{Vx}{2D} -\frac{V^{2} t}{4D} \right] \rho_{1} (x,t).\label{eq6}$$ After substituting $\rho (x,t)$ formula (6) to equation (5) one obtains for $\rho _{1} (x,t):$ $$\tau _{\rho } \frac{\partial ^{2} \rho _{1} }{\partial t^{2} }
+\left( 1-\tau _{\rho } \frac{V_{\rho }^{2} }{2D_{\rho } } \right)
\frac{\partial \rho _{1} }{\partial t} +\tau _{\rho } \frac{V^{4}}{16D_{\rho }^{2} } \rho _{1}
=D_{\rho } \frac{\partial ^{2} \rho_{1} }{\partial x^{2} }.\label{eq7}$$ Considering that $D_{\rho } =\tau _{\rho } v_{\rho }^{2} $ equation (\[eq7\]) can be written as $$\tau _{\rho } \frac{\partial ^{2} \rho _{1} }{\partial t^{2} }
+\left( 1-\frac{V_{\rho }^{2} }{2v_{\rho }^{2} } \right)
\frac{\partial \rho _{1} }{\partial t} +\frac{1}{16\tau _{\rho } }
\frac{V^{4} }{v_{\rho }^{4} } \rho _{1} =D_{\rho } \frac{\partial^{2} \rho _{1} }{\partial x^{2} }.\label{eq8}$$ In the same manner equation for the temperature field can be obtained: $$\tau _{T} \frac{\partial ^{2} T_{1} }{\partial t^{2} } +\left(
1-\frac{V_{T}^{2} }{2v_{T}^{2} } \right) \frac{\partial T_{1}}{\partial t}
+\frac{1}{16\tau _{T} } \frac{V_{T}^{4} }{v_{T}^{4}} T_{1}
=D_{T} \frac{\partial ^{2} T_{1} }{\partial x^{2} }.\label{eq9}$$ In equation (\[eq9\]) $\tau _{T} ,\;D_{T} ,\,\;V_{T} $ and $v_{T} $ are: relaxation time for heat transfer, diffusion coefficient, heat convection velocity and thermal wave velocity.
In this paper we will investigate the structure and solution of the equation (\[eq9\]). For the hyperbolic heat transport Eq. (\[eq9\]) we seek a solution of the form: $$T_{1} (x,t)=e^{-\frac{t}{2\tau _{T} } } u(x,t).\label{eq10}$$ After substitution of Eq. (\[eq10\]) into Eq. (\[eq9\]) one obtains: $$\begin{aligned}
\tau _{T} \frac{\partial ^{2} u(x,t)}{\partial t^{2} }&-&D_{T}
\frac{\partial ^{2} u(x,t)}{\partial x^{2} } +\left(
-\frac{1}{4\tau _{T} } +\frac{V_{T}^{2} }{4D_{T} } +\tau _{T}
\frac{V_{T}^{4} }{16D_{T}^{2} } \right) u(x,t)\nonumber\\
&-&\tau _{T}
\frac{V_{T}^{2} }{2D_{T} } \frac{\partial u(x,t)}{\partial t}
=0\label{eq11}
\end{aligned}$$ Considering that $D_{T} =\tau _{T} v_{T}^{2} $ Eq. (\[eq11\]) can be written as $$\tau _{T} \frac{\partial ^{2} u}{\partial t^{2} } -\tau _{T} v_{T}^{2}
\frac{\partial ^{2} u(x,t)}{\partial x^{2} } +\left( -\frac{1}{4\tau _{T}}
+\frac{V_{T}^{2} }{4\tau _{T} v_{T}^{2} } +\tau _{T} \frac{V_{T}^{4}}
{16\tau _{T}^{2} v_{T}^{4} } \right) u(x,t)-\frac{V_{T}^{2} }{2v_{T}^{2}}
\frac{\partial u}{\partial t} =0.\label{eq12}$$ After omitting the term $\frac{V_{T}^{4} }{v_{T}^{4} } $ in comparison to the term $\frac{V_{T}^{2} }{v_{T}^{2} } $ Eq. (\[eq12\]) takes the form: $$\frac{\partial ^{2} u}{\partial t^{2} } -v_{T}^{2} \frac{\partial^{2} u}
{\partial x^{2} } +\frac{1}{4\tau _{T}^{2} } \left(-1+\frac{V_{T}^{2} }{v_{T}^{2} } \right) u(x,t)
-\frac{V_{T}^{2}}{2v_{T}^{2} \tau _{T} } \frac{\partial u}{\partial t}
=0.\label{eq13}$$ Considering that $\tau _{T}^{-2} >>\tau _{T}^{-1} $ one obtains from Eq. (\[eq13\]) $$\frac{\partial ^{2} u}{\partial t^{2} } -v_{T}^{2} \frac{\partial^{2} u}{\partial x^{2} }
+\frac{1}{4\tau _{T}^{2} } \left(-1+\frac{V_{T}^{2} }{v_{T}^{2} } \right)
u=0.\label{eq14}$$ Equation (\[eq14\]) is the master equation for heat transfer induced by ultra-short laser pulses, i.e. when $\Delta t\approx
\tau _{T} $. In the following we will consider the Eq. (\[eq14\]) in the form: $$\frac{\partial^2 u}{\partial t^2} -v_T^2 \frac{\partial ^2u}{\partial x^2}-qu=0\label{eq15}$$ where $$q=\frac{1}{4\tau^2_T}\left(\frac{V^2_T}{v^2_T}-1\right).\label{eq16}$$ In equation (\[eq16\]) the ratio $$M_{T} =\frac{V_{T} }{v_{T} } =\frac{V_{T} }{v_{S}
}\label{eq17}$$ is the Mach number for thermal processes, for $v_{T} =v_{S} $ is the sound velocity in the gas of heat carriers [@10].
In monograph [@10] the structure of equation (15) was investigated. It was shown that for $q<0$, i.e. $V_{T} <v_{S} $, subsonic heat transfer is described by the modified telegrapher equation $$\frac{1}{v_{T}^{2} } \frac{\partial ^{2} u}{\partial t^{2} }
-\frac{\partial ^{2} u}{\partial x^{2} } +\frac{1}{4\tau _{T}^{2}
v_{T}^{2} } \left( \frac{V_{T}^{2} }{v_{S}^{2} } -1\right)
u=0.\label{eq18}$$ For $q>0,\,\;v_{S} <V_{T} ,$ i.e. for supersonic case heat transport is described by Klein-Gordon equation: $$\frac{1}{v_{T}^{2} } \frac{\partial ^{2} u}{\partial t^{2} }
-\frac{\partial ^{2} u}{\partial x^{2} } +\frac{1}{4\tau _{T}^{2}
v_{T}^{2} } \left( \frac{V_{T}^{2} }{v_{S}^{2} } -1\right)
\;u=0.\label{eq19}$$ The velocity of sound $v_{S} $ depends on the temperature of the heat carriers. The general formula for sound velocity reads [@11]: $$v_{S}^{2} =\left( zG-\frac{G}{z} \left( 1+\frac{5G}{z} -G^{2} \right)^{-1} \right)
^{-1}.\label{eq20}$$ In formula (\[eq20\]) $z=\frac{mc^{2} }{T} $ and *G* is of the form [@11]: $$G=\frac{K_{3} (z)}{K_{2} (z)}
,\label{eq21}$$ where $c$ is the light velocity, $m$ is the mass of heat carrier, $T$ is the temperature of the gas and $K_{3} (z),\;K_{2} (z)$ are modified Bessel functions of the second kind. In Fig. \[fig1\] the ratio of $\left( \frac{v_{S} }{c} \right) ^{2} $ was presented are the function of $\frac{T}{mc^{2} } $. Fig. \[fig1\]a presents the $\left(
\frac{u}{c} \right) ^{2} $ for $\frac{T}{mc^{2} } <1$ (nonrelativistic approximations) and Fig. \[fig1\]b presents the $\left( \frac{u}{c} \right) ^{2}$ for the very high temperature heat carriers gas, i.e. $T>mc^{2}$ (relativistic gas). It is interesting to observe that for nonrelativistic gas, $v_{S}^{2}$ is a linear function of temperature. From formula (\[eq21\]) can be concluded [@11] that for $T<mc^{2}$ one obtains $$\left( \frac{v_{S} }{c} \right) ^{2} =\left( \frac{5T}{3mc^{2} }
\right)\label{eq22}$$ i.e as for Maxwellian nonrelativistic gas. On the other hand for $T\gg mc^{2}$, $\left( \frac{v_{S} }{c} \right) ^{2} =\frac{1}{3}
$ and is independent of $T$ where $v_{S}^{2} =\frac{c^{2} }{3} $ denotes the sound velocity in the photon gas. In this paper we will study the heat transfer in the relativistic gas, i.e. when $\left( \frac{v_{S} }{c} \right) ^{2}$ is constant. In that case Eqs. (\[eq18\]) and (\[eq19\]) are the linear hyperbolic equations. In Fig. \[fig1\](c) it is shown that Maxwellian approximation is not valid for $\frac{T}{mc^{2} }
>0.05$ and moreover gives a wrong description of $v_{S} $ for $\frac{T}{mc^{2} } =0.6,$ for $v_{S} >c$ in complete disagreement with special relativity theory. In Fig. \[fig2\](a,b,c) the results of calculations of the sound velocity of electron gas and in Fig. \[fig3\](a,b,c) for proton gas are presented respectively. For the initial conditions $$u|_{t=0} =f(x),\quad \left. \frac{\partial u}{\partial t} \right| _{t=0}
=F(x).$$ Solutions of the equation can be find in [@10]: $$u(x,t)=\frac{f(x-v_{T} t)+f(x+v_{T} t)}{2} +\frac{1}{2}
\int\limits_{x-v_{T} t}^{x+v_{T} t}\Phi (x,t,z)dz,\label{eq23}$$ where $$\begin{aligned}
\Phi (x,t,z)=\frac{1}{v_{T} } F(z)J_{0} \left( \frac{\sqrt{q} }{v_{T} }
\sqrt{(z-x)^{2} -v_{T}^{2} t_{2} } \right)\\ +\sqrt{q} tf(z)\frac{J_{0}^{'}
\left( \frac{\sqrt{q} }{v_{T} } \sqrt{(z-x)^{2} -v_{T}^{2} t^{2} } \right)}
{\sqrt{(z-x)^{2} -v_{T}^{2} t^{2} } }
\end{aligned}$$ and $$q=\frac{1}{4\tau _{T}^{2} } \left( \frac{V_{T}^{2} }{vT2} -1\right) .$$ The general equation for complex heat transfer: diffusion plus convection can be written as: $$\frac{\partial T}{\partial t} =-\tau _{T} \frac{\partial ^{2} T}{\partial
t^{2} } +D_{T} \frac{\partial ^{2} T}{\partial x^{2} } -V_{T}
\frac{\partial T}{\partial x} .\label{eq24}$$ Considering Eqs. (\[eq6\]), (\[eq10\]) and (\[eq23\]) the solution of equation (\[eq24\]) is $$\begin{aligned}
T(x,t)&=&\exp\left[ \frac{V_{T} x}{2D} -\frac{V_{T}^{2} t}{4D} \right]
e^{-\frac{2}{2\tau _{T} } }
\cdot \bigg( \frac{f(x-v_{T} t)+f(x+v_{T}
t)}{2}\\ &&\mbox{} +\frac{1}{2} \int\limits_{x-v_{T} t}^{x+v_{T} t}\Phi (x,t,z)dz\bigg).
\end{aligned}$$ In Fig. \[fig4\] the comparison of the calculation of sound velocity for electron gas and the electron spectra \[7\] is presented. The change of the shape of the electron spectra in vicinity of 5 MeV can be easily seen. At the total energy of the 5 MeV the sound velocity in electron plasma is constant and independent of electron energy. Electrons with velocities greater than $\frac{1}{\sqrt{3} } c$ can generate the shock thermal waves which heats the plasma to higher temperature.
![(a) The ratio: sound velocity/light velocity $\left( \frac{u}{c}
\right) ^{2} $ as the function $\frac{T}{m} $ for cold heat carriers $\left( \frac{T}{m} \ll 1\right) $ ; (b) for hot heat carriers $\left( \frac{T}{m} \gg 1\right) $ ; (c) Comparison of the ratio $\left( \frac{u}{c} \right) ^{2} $ for hot carriers (width 1em height .8ex depth -.5ex), ultra-relativistic carriers (width 1em height 1ex depth -.5ex) and Maxwellian approximation (width 1em height .6ex depth -.5ex)[]{data-label="fig1"}](rys1a.eps){height="5.5cm"}
![(a) The ratio: sound velocity/light velocity $\left( \frac{u}{c}
\right) ^{2} $ as the function $\frac{T}{m} $ for cold heat carriers $\left( \frac{T}{m} \ll 1\right) $ ; (b) for hot heat carriers $\left( \frac{T}{m} \gg 1\right) $ ; (c) Comparison of the ratio $\left( \frac{u}{c} \right) ^{2} $ for hot carriers (width 1em height .8ex depth -.5ex), ultra-relativistic carriers (width 1em height 1ex depth -.5ex) and Maxwellian approximation (width 1em height .6ex depth -.5ex)[]{data-label="fig1"}](rys1b.eps){height="5.5cm"}
![(a) The ratio: sound velocity/light velocity $\left( \frac{u}{c}
\right) ^{2} $ as the function $\frac{T}{m} $ for cold heat carriers $\left( \frac{T}{m} \ll 1\right) $ ; (b) for hot heat carriers $\left( \frac{T}{m} \gg 1\right) $ ; (c) Comparison of the ratio $\left( \frac{u}{c} \right) ^{2} $ for hot carriers (width 1em height .8ex depth -.5ex), ultra-relativistic carriers (width 1em height 1ex depth -.5ex) and Maxwellian approximation (width 1em height .6ex depth -.5ex)[]{data-label="fig1"}](rys1c.eps){height="5.5cm"}
![The same as in Fig. \[fig1\], but for electrons with mass m=0.51 MeV/c$^{2}$[]{data-label="fig2"}](rys2a.eps){height="6cm"}
![The same as in Fig. \[fig1\], but for electrons with mass m=0.51 MeV/c$^{2}$[]{data-label="fig2"}](rys2b.eps){height="6cm"}
![The same as in Fig. \[fig1\], but for electrons with mass m=0.51 MeV/c$^{2}$[]{data-label="fig2"}](rys2c.eps){height="6cm"}
![The same as in Fig. \[fig2\] but for protons with mass m=0.98 GeV[]{data-label="fig3"}](rys3a.eps){height="6cm"}
![The same as in Fig. \[fig2\] but for protons with mass m=0.98 GeV[]{data-label="fig3"}](rys3b.eps){height="6cm"}
![The same as in Fig. \[fig2\] but for protons with mass m=0.98 GeV[]{data-label="fig3"}](rys3c.eps){height="6cm"}
![(a) The experimental data [@7] for the electron populations. Circles = data. The interaction beam intensity is $I_{\max } \approx 6\cdot 10^{18} \;$W/cm$^{2} $. (b) The ratio $\frac{u}{c} $ as the function of the electron temperature $T$ \[MeV\][]{data-label="fig4"}](rys4a.eps){height="5cm"}
![(a) The experimental data [@7] for the electron populations. Circles = data. The interaction beam intensity is $I_{\max } \approx 6\cdot 10^{18} \;$W/cm$^{2} $. (b) The ratio $\frac{u}{c} $ as the function of the electron temperature $T$ \[MeV\][]{data-label="fig4"}](rys4b.eps){height="5cm"}
Conclusions
===========
In paper the Heaviside equation for laser heating electron plasma was formulated and solved. It was shown that for high energy electrons, with energy $>5$ MeV the sound velocity in plasma is constant and equal $v_{S} =\frac{1}{\sqrt{3} } c.$ The superheating of plasma with electron energy $>~5$ MeV can be achieved by the generation of thermal shock waves.
[99]{} N. Blanchot et al., *Opt. Lett*., **20**, (1995) p. 395.\
M. Tabak et al., *Phys. Plasmas*, **1**, (1994), p. 1626.\
P. Monot et al., *Phys. Rev. Lett*., **74**, (1995) p. 2953.\
M. Borghesi et al., *Phys. Rev. Lett*. **78**, (1997), p. 879.\
A. Borisov et al., *Plasma Phys. Controlled Fusion*, **37**, (1995), p. 569.\
G. Malka et al., *Phys.* *Rev. Lett*., **78**, (1997), p. 3314.\
G. Malka et al., *Phys.* *Rev. Lett*. **79**, (1997), p. 2053.\
J. Marciak-Kozlowska, M. Kozlowski, *Lasers in Engineering*, **12**, (2002), p. 17.\
J. Marciak-Kozlowska, M. Kozlowski, *Lasers in Engineering*,\
(2003, to be published).\
M. Kozlowski, J. Marciak-Kozlowska, *From Quarks to Bulk Matter*,\
Hadronic Press, 2001, USA.\
J. L. Synge, *The Relativistic Gas*, North-Holland Publisher Company, 1957.
|
---
author:
- 'A. Gozar, Seiki Komiya, Yoichi Ando'
- 'G. Blumberg'
title: 'Magnetic and Charge Correlations in : Raman Scattering Study'
---
The Phase Diagram and Structural Properties of the High Temperature Superconductor
===================================================================================
is one of the most studied Cu-O based layered perovskites [@KastnerRMP98]. It exhibits some of the most important aspects related to the physics of strongly correlated electrons and, more important, is one of the compounds which belong to the family of high temperature superconducting cuprates. In fact the high T$_{c}$ superconductivity (SC) rush which began in 1986 started with a variant of , a Ba-La-Cu-O based compound [@BednorzZPB86], where the authors observed a highest onset SC temperature T$_{c}$ in the 30 K range.
The phase diagram of is shown in Fig. \[f11\], see also Ref. [@KeimerPRB92]. Several electronic ground states as well as structural phases evolve with Sr concentration. For $x$(Sr) $\leq 0.02$ the crystals have long range antiferromagnetic (AF) order and one can observe a very rapid suppression of the Néel ordering temperature T$_{N}$ with the amount of Sr. While for $x = 0$ the AF transition is slightly above room temperature, T$_{N}$ decreases in the 150 - 200 K range for $x = 0.01$ and it is completely suppressed above $x = 0.02$. The phase diagram shows also a SC dome starting at $x = 0.05$ and ending around $x~=~0.32$. The maximum T$_{c}$ of about 40 K is reached at the optimal doping $x = 0.2$. The highest SC temperature ($T_{c} = 51.5$ K) in the family was achieved in thin films under epitaxial strain [@BozovicPRL02]. There are also two structural phases of the this compound. One is tetragonal and the other one is orthorhombic, see Fig. \[f12\]. Sr substitution for La decreases the orthorhombicity and the crystal remains tetragonal at all temperatures at values of $x$(Sr) which correspond roughly to the region of maximum T$_{c}$. Other intervening phases shown in Fig. \[f11\], spin glass at low temperatures and low Sr concentration, Fermi or non-Fermi liquid behavior depending on if one is in the far right side of the phase diagram or not, are discussed in literature [@KastnerRMP98].
The crystal structure of is shown in Fig. \[f12\]. The occurrence of several structural phases is typical for perovskites and they generally happen as a result of the lattice strain between the rare-earth and the CuO$_{2}$ layers. The strain is often released by various bucklings of the transition metal - oxygen planes and this is also the case here. The HTT phase has flat CuO$_{2}$ planes and the transition to the LTO phase can be understood within a good approximation as a rigid rotation of the CuO$_{6}$ octahedra around an axis making 45$^{\circ}$ with respect to the orthorhombic axes. As a result, half of the O atoms will be situated above and the other half below the plane determined by the Cu atoms, see Fig. \[f12\]. The lattice constants of the LTO phase at low temperatures are $a = 5.354$ Å, $b = 5.401$ Å and $c = 13.153$ Å. So the orthorhombicity, defined by $2(a - b) / (a + b)$, is small, only of about 0.8%.
In the parent compound, , one has La$^{3+}$ and O$^{2-}$ non-magnetic ions so copper will be in a Cu$^{2+}$ oxidation state to insure neutrality. As a result, the last Cu $3d^{9}$ shell will contain a hole carrying a spin $S = 1/2$ which is responsible for the magnetic properties. Sr$^{2+}$ substitution for La leads to hole doping of the CuO$_{2}$ planes. It is believed that hole pairing and the acquirement of 3D coherence lead to the occurrence of superconductivity.
In this chapter we will also talk about certain properties of Nd doped and we mention here some well established effects associated with Nd substitution for La. One is that Nd in suppresses superconducting correlations. For instance magnetic susceptibility data in with $x = 0.2$ show that SC vanishes for values of $y$ greater than about 0.6 [@BuchnerPRL94]. Another effect is that this suppression of SC is accompanied by the enhancement of other types of correlations, the appearance of the so called ’stripes’ [@TranquadaNature96], which are proposed to be quasi-1D in plane charge and/or spin super-modulations. While the discussion above suggests that these two states act against each other, it is not clear at this moment if the stripes are helping or competing with SC. Another effect is related to changes in the crystal structure as a result of inter-layer chemical modifications. Nd doping brings in another phase, the low temperature tetragonal (LTT) structure, which can be imagined as a rigid CuO$_{6}$ octahedra tilt around the axis whose vector is defined by $1/\sqrt{2} (\hat{a} + \hat{b})$ where $a$ and $b$ are the orthorhombic axes of the LTO phase.
In the following we will discuss low energy magnetic properties of at light $x$(Sr) doping level. Although much is known about the physics of 2D $S = 1/2$ antiferromagnets, there are recent experiments which show surprising properties in macroscopically orthorhombic crystals in the presence of external magnetic fields. It is worth mentioning in this respect that recent neutron scattering in such crystals studies show that even the crystal structure has not rigorously been determined yet [@KeimerPrivate] although the deviations from the $Bmab$ symmetry may be very small. We will show later in this chapter, especially in connection to the phononic and electronic properties, that the effects of orthorhombicity are surprisingly large. In the following we discuss long wavelength spin-wave excitations as a function of temperature, doping and magnetic field. We show that the low energy spin dynamics allows us to observe a spin ordered state induced by magnetic fields, a state which persist up to quite high temperatures in crystals with long range AF order [@GozarPRL04]. It will be shown that although the orthorhombicity is small, there are dramatic anisotropy effects in the in plane electronic and phononic excitations. Our data indicate that at commensurate hole doping $x = 1/8$ and independent of Nd concentration there are local deviations in the crystal structure due to a spread in the CuO$_{6}$ tilt angle. We will discuss this behavior in connection with possible spin and charge modulations in the CuO$_{2}$ planes [@GozarPRB03].
Magnetic and Electronic Properties of Macroscopically Orthorhombic at Light Doping ($0 \leq x \leq 0.03$)
=========================================================================================================
Why is a Study of Low Energy Magnetism Interesting?
---------------------------------------------------
SC as well as the normal properties of 2D Mott-Hubbard systems have already triggered a lot of effort to understanding the evolution of the ground state and of the AF correlations as a function of doping. However, in spite of the small orthorhombicity, the impact of the low energy magnetism on the carrier and lattice dynamics in *detwinned* crystals has recently been shown to be significant and surprising new effects were found.
What does detwinned mean in the first place? On cooling from the HTT to the LTO phase the crystal develops orthorhombic domains, called twins, on the nanometer to micron scale. The sign of the orthorhombic distortions changes across the twin boundaries, as shown in Fig. \[f13\]a. Accordingly, for a macroscopic probe (and a Raman setup which uses a focussed laser spot larger than about several $\mu$ diameter is an example) the sample looks effectively tetragonal. If uniaxial pressure of about 15-30 MPa is applied while slowly cooling the crystal through the HTT-LTO phase transition, a detwinned, i.e. macroscopically orthorhombic, crystal can be grown [@LavrovPRL01].
This leads to non-trivial effects if one looks in Fig. \[f13\]b-c. The magnetization data shows two peaks at the Néel transition of around 300 K in and the magnetic anisotropy is preserved in a wide range of temperatures above T$_{N}$. The susceptibility along the $a$-axis, $\chi_{a} (T)$, is featureless showing that this axis is magnetically inert, at least at small fields. The structure with two peaks is due to the various spin anisotropy terms present in crystals, they will be discussed in more detail later in the section [@LavrovPRL01]. is an insulator, but small carrier concentrations in the CuO$_{2}$ planes give rise to metallic behavior of the resistivity at high temperatures[@AndoPRL01]. Moreover, the $dc$ resistivity shows also sizeable anisotropy if measured along the $a$ and $b$ orthorhombic axes. The relative anisotropy is almost 30% for x = 0.01 around the metal-insulator transition and goes beyond this value in x = 0.03 at low temperatures. One can also notice at high temperatures a decrease of the resistivity anisotropy with doping from x = 0.01 to x = 0.03 and that there is a sign change in this anisotropy around 170 K for x = 0.03 [@AndoPRL02]. The magnetoresistance can be very large (up to 80%) at low temperatures [@AndoPRL03].
One can conclude form Fig. \[f13\] that detwinned samples show non-negligible effects in transport and magnetization data. The Zeeman energy in finite external magnetic fields becomes comparable with the spin-anisotropy induced gaps and this will influence the low temperature thermodynamics. As for the intrinsic ground state properties at small dopings, inelastic neutron scattering (INS) argues that there are changes in the low frequency magnetic scattering (45$^{\circ}$ rotation in the $k$ space of low energy incommensurate magnetic peaks) when superconductivity occurs around $x~=~0.05$ in [@WakimotoPRB99] and also that macroscopic phase separation takes place below $x~=~0.02$ [@MatsudaPRB02].
All the above constitute general arguments for a detailed high energy resolution study of long wavelength spin excitations as a function of doping and temperature. Even more interesting is a recent magnetic field experiment done at room temperature in $x~=~0.01$ . The main result of the experiment is shown in Fig. \[f14\] and it says that the $b$ orthorhombic axis follows the direction of the applied field [@LavrovNature02]. So magnetic fields of about 10-14 T are able to produce structural changes and detwin the crystal. The switch of the crystallographic axes is reversible and can be monitored by using a regular optical microscope. It is worth noting that 300 K is roughly about 100 K above the 3D long range AF ordering temperature in 1% doped crystals. Two interesting points can be mentioned in this regard. One is that there is strong spin-lattice interaction in this material. The other one is related to the coupling of the spins to the external field. While magnetic field induced structural changes are easier to be understood in ferromagnetic crystals because the net magnetic moment can provide a substantial coupling to the external field, the fact that these effects take place in a AF system makes a unique compound. The rotation of the orthorhombic axes can be also observed in $dc$ resistivity or magnetic susceptibility measurements by monitoring the changes in the anisotropic properties shown in Fig. \[f13\] as a function of the direction of the applied external magnetic field $\vec{H}$.
These results highlight the importance of a magnetic field study of the low energy magnetism in low doped . We believe that the field induced spin ordering we observe at temperatures up to 300 K in samples displaying long range AF order is related to the effects shown in Fig. \[f14\].
Low Energy Magnetism in Detwinned with ($0 \leq x \ \leq 0.03$)
---------------------------------------------------------------
[**(A) The 2D Heisenberg antiferromagnets and effects of inter-layer coupling.**]{} CuO$_{2}$ planes form 2D square lattices and the $S = 1/2$ Cu spins interact antiferromagnetically *via* the intermediate O atoms. The nearest neighbor super-exchange $J$ takes place along the 180$^{\circ}$ Cu-O-Cu bonds and it has a value of approximately 140 meV [@ColdeaPRL01]. The inter-layer correlations are weak for two reasons: on one hand the spacing between the layers is large and on the other hand the magnetic interaction along this direction is frustrated. So in the first approximation the spin dynamics (especially in the paramagnetic phase) will be dominated by the properties of a 2D isotropic Heisenberg antiferromagnet. The starting Hamiltonian to characterize these systems is then: $$\hat{H}_{2D} = \sum_{<i,j>} J_{ij} \vec{S}_{i} \cdot \vec{S}_{j}
\label{e11}$$ where $\vec{S}_{i}$, $\vec{S}_{j}$ are spins on the sites $i$ and $j$ and $J_{ij} = J \approx 140$ meV when $<i,j>$ corresponds to a pair of nearest neighbor (NN) spins.
The spin-spin correlation function $\xi(T)$ is one of the fundamental parameters characterizing the paramagnetic state. This quantity is extracted from an equation relating the average staggered magnetization to the inter-site distance, of the type: $< \vec{S}_{i} \cdot \vec{S}_{j} > \ \ \propto \ \ \exp(- r_{ij} / \xi)$. Here $r_{ij}$ is the distance between the sites $i$ and $j$. Continuum field theory predicts in the paramagnetic phase a spin-spin correlation length given by: $\xi(T) \propto \frac{c}{2 \pi \rho_{s}} \exp[\frac{2 \pi \rho_{s}}{k_{B} T}]$ [@ChakravartyPRB89]. The parameters $c$ and $\rho_{s}$ are for the spin-wave velocity and spin stiffness respectively. This correlation length diverges as $T \rightarrow 0$ leading to true long range magnetic order only at zero temperature. This microscopic result is consistent with a theorem showing rigorously that at any finite temperature a 1D or 2D isotropic Heisenberg model with finite-range interactions can be neither ferromagnetic nor antiferromagnetic [@MerminPRL66]. However, in agreement with theoretical predictions, neutron scattering measurements show that the number of correlated spins within the 2D CuO$_{2}$ planes is substantial even at high temperatures. For example, at 500 K which is about 200 K above the 3D ordering temperature in , $\xi$ is of the order of 50 Å [@KeimerPRB92], approaching values of 200 - 300 lattice constants around T$_{N}$.
A small interlayer coupling $J_{\perp}$ pushes the Néel ordering temperature to finite values but does not affect significantly the 2D magnetic correlations. It is believed that the magnitude of the inter-layer exchange is very small, $J_{\perp} \approx 10^{-5} J < 0.02$ K [@KastnerRMP98; @ChakravartyPRB89; @ThioPRB88]. In spite of such a small perpendicular exchange, the AF ordering temperatures are quite high and this is due to the large in-plane correlation lengths. Agreement in terms of the order of magnitude for T$_{N}$ using the above value for $J_{\perp}$ can be obtained simply by comparing the thermal and magnetic energies in: $$k_{B} T_{N} \approx J_{\perp} (m_{s} S)^{2} \left[ \frac{\xi(T_{N})}{a}\right ]^{2}
\label{e12}$$ where $m_{s}$ is the sublattice magnetization in units of $g \mu_{B}$ and $[\xi(T) / a]^{2}$ is proportional to the number of ’ordered’ spins in each CuO$_{2}$ plane. It should be noted that in the HTT phase of every Cu atom has eight nearest neighbors in the adjacent planes (four above and four below its own CuO$_{2}$ plane). Due to symmetry, the super-exchange is almost exactly cancelled and the effective $J_{\perp}$ is even smaller than $10^{-5} J$. It is the distortion associated with the LTO phase, see Fig. \[f12\] which partially lifts this degeneracy giving rise to a reasonably sized, although very small, inter-layer exchange.
Eq. (\[e12\]) leaves an open question: how to reconcile similar 3D ordering temperatures (T$_{N}$’s typically in the range between 200 and 300 K) for various layered Cu-O based materials (examples are , , , Nd$_{2}$CuO$_{4}$ or Pr$_{2}$CuO$_{4}$, see Refs. [@KastnerRMP98; @MatsudaPRB90]) with rather different exchange paths and accordingly values of $J_{\perp}$ that can be quite far apart. For instance in the CuO$_{2}$ planes are exactly flat so it is expected that the cancellation of terms because of the inter-layer frustration would decrease $J_{\perp}$ by another few orders of magnitude, requiring anomalously high $\xi (T_{N})$ in order to satisfy Eq. (\[e12\]). It has been suggested that the 3D ordering temperature T$_{N}$ follows immediately after a 2D Kosterlitz-Thouless (KT) phase transition at T$_{KT}$ due to the in-plane spin anisotropy of the $XY$ type which characterizes all the above mentioned AF materials. It was found that T$_{KT}$ is appreciable, $\approx 0.25 J / k_{B}$, and quite insensitive to the magnitude of the in-plane anisotropy [@MatsudaPRB90]. This would explain the magnitude as well as the similarity between the measured T$_{N}$’s in various Cu-O based 2D AF’s.
What does the excitation spectrum of a 2D AF ordered square lattice look like? Within the spin-wave approximation the excitations are coherent transverse oscillations of the ordered moments. Taking into account only the nearest neighbor exchange $J$ the wavevector dependent spin-wave energies are given by $$\omega(k) = z S J \sqrt{1 - \gamma^{2}_{k}} \ \ \ \mathrm{with} \ \ \ \gamma_{k} = \frac{\cos(k_{x}) + \cos(k_{y})}{2}
\label{e13}$$ where $S = 1/2$ is the total spin and $z = 4$ is the number of nearest neighbors for the simple square lattice [@SandvikPRB98]. Note that in the 2D isotropic Heisenberg AF lattice there will be two degenerate acoustic spin-wave branches. In Fig. \[f15\] we show relatively recent INS results for the spin-wave dispersion up to high energies.
While the dispersion predicted by the nearest neighbor isotropic Hamiltonian reproduces qualitatively the experimental results, there are discrepancies at high energies. One can note that along the AF zone boundary we have $k_{x} + k_{y} = \pi$ and this implies that the spin-wave energy along this line is a constant given by $2 J$. The experimental data in Fig. \[f15\] shows that there is substantial dispersion for instance along the $(\pi,0)$ to $(3 \pi / 2, \pi / 2)$ line. The authors resolve this discrepancy by including higher order spin interactions. In particular, the most prominent term is due to $J_{ring}$ which corresponds to a spin exchange around a square plaquette as shown in Fig. \[f15\]a. Quantitatively, from the fit to the experimental data which includes quantum corrections [@SinghPRB89] (the solid line in Fig. \[f15\]b), this term turns out to be as high as 41% of the nearest neighbor $J$ at low temperatures, $J_{ring} \approx 61$ meV, and it is about twenty times larger than the second and third nearest neighbor exchanges [@ColdeaPRL01]! In support for such a claim we note that a large value of $J_{ring}$ was needed to explain the dispersion of the elementary triplet excitations in two-leg ladder materials. The same $J_{ring}$ seems also to improve the results concerning the large absorption frequency range in which the phonon induced two-magnon excitation is thought to be observed in 2D insulating cuprates [@LorenzanaPRL99].
[**(B) In-plane magnetic anisotropies.**]{} There are two dominant in-plane magnetic anisotropies characterizing each CuO$_{2}$ plane. In general these terms, arising as a result of spin-orbit coupling, connect the spin space to the real space and can be sometimes described in terms of effective magnetic interactions. One of these interactions is the $XY$ exchange anisotropy term mentioned above in connection to the 3D Néel ordering and its origin is in the layered structure of the cuprates, i.e. it has nothing to do with the buckling of the CuO$_{2}$ plane in the LTO phase. Due to to the $XY$ term the NN spin-exchange interaction in Eq. (\[e11\]) changes to $(J + \alpha) (S^{x}_{i} S^{x}_{j} + S^{y}_{i} S^{y}_{j}) + J S^{z}_{i} S^{z}_{j}$. Because $\alpha > 0$ the classical configuration giving the minimum energy is one with all the spins lying in the $(ab)$ plane. The other important anisotropy term is due to the antisymmetric Dzyaloshinskii-Moriya (DM) interaction and it has the form: $\vec{d}_{ij} \cdot (\vec{S}_{i} \times \vec{S}_{j})$ where $\vec{d}_{ij}$ is the DM vector [@DzyaloshinskiiJPCS58; @MoriyaPR60]. The two-spin classical ground state configuration for this interaction considered alone is one with $\vec{S}_{i} \perp \vec{S}_{j} \perp \vec{d}_{ij}$.
The balance of these terms determines the equilibrium position of the spins. These anisotropy terms are expected to be much smaller than $J$ and can be quantitatively determined from the energy of the spin-waves in the long wavelength limit as will be discussed in the next section. A few words are in order about the DM term. Due to the existence in the LTO phase of a $C_{2}$ (rotation by 180$^{\circ}$) symmetry axis which passes through in-plane O atoms and is perpendicular to the $(ab)$ surface, the DM vector between two adjacent Cu atoms has to satisfy $\vec{d} \perp \hat{c}$ [@MoriyaBook].
The symmetry elements of the $Bmab$ space group associated to the LTO phase allow the DM vectors $\vec{d}_{ij}$ to form the configurations shown in Fig. \[f16\]a. Once a convention is made that the order of spins in the vector product of the $\vec{d}_{ij} \cdot (\vec{S}_{i} \times \vec{S}_{j})$ term is always from a given sublattice to the other, it can be noted that there are two possible arrangements for the DM vectors: one involving $\vec{d} \parallel \hat{a}$ and the other one in which the DM vectors are parallel to the $b$-axis but have alternating signs.
The effective two-dimensional spin Hamiltonian and the associated free energy density at T = 0 K which takes the $XY$, DM terms as well as an external field into account can be written as: $$\begin{aligned}
\hat{H} = \sum_{<i,j>} [ (J + \alpha) (S^{x}_{i} S^{x}_{j} + S^{y}_{i} S^{y}_{j}) + J S^{z}_{i} S^{z}_{j} + \vec{d}_{ij} \cdot ({\bf S}_{i} \times {\bf S}_{j})] - \vec{H} \sum_{i} \vec{S}_{i}
\label{e14}
\\
f = z (J + \alpha) (M^{x}_{1} M^{x}_{2} + M^{y}_{1} M^{y}_{2}) + z J M^{z}_{1} M^{z}_{2} + z \vec{d} \cdot (\vec{M}_{1} \times \vec{M}_{2}) - \vec{H} (\vec{M}_{1} + \vec{M}_{2})
\label{e15}\end{aligned}$$ In Eq. (\[e14\]) the sum runs over the nearest neighbors. In Eq. (\[e15\]) $z$ is the number of nearest neighbors and $M_{1,2}$ are the sublattice magnetizations. The ’thermodynamic’ Dzyaloshinskii vector $\vec{d}$ in Eq. (\[e15\]) is given in terms of the ’microscopic’ Moriya terms $\vec{d}_{ij}$ in Eq. (\[e14\]) by $\vec{d} = (1 / z) \sum_{NN} \vec{d}_{ij}$ and from Fig. \[f16\] one can infer that $\vec{d} \parallel \hat{a}$ [@ShekhtmanPRL92]. The relative strength of the DM terms corresponding to the two configurations in Fig. \[f16\]a is determined by microscopic parameters. It is interesting to note here a point made by the authors of Ref. [@ShekhtmanPRL92], i.e. that the identification of $\vec{d}$ to the microscopic $\vec{d}_{ij}$ is a non-trivial problem in the sense that a difference between them is a necessary condition for the existence of an observable weak ferromagnetism (WF) with a specific value of the net WF moment. In other words, although $\vec{d}$ is parallel to the $a$-axis, it is required that the DM vectors in both configurations shown in Fig. \[f16\]a are finite. If on one hand only vectors $\vec{d}_{ij} \parallel \hat{b}$ are considered (frustrating interaction), then $\vec{d} \equiv 0$ and the spins order antiferromagnetically without any WF moment. On the other hand if one takes into account only vectors $\vec{d}_{ij} \parallel \hat{a}$ (non-frustrating interaction) the classical ground state cannot be characterized as ferromagnetic because it consists of a manifold of degenerate configurations having a net WF moment ranging continuously from zero to some finite value [@ShekhtmanPRL92].
The equilibrium position of the spins in zero external field is shown in Fig. \[f16\]b. For a 2D plane this can be obtained from the minimization of the free energy in Eq. (\[e15\]) with respect to the angles between the magnetizations and crystallographic axes with the constraint $m = |\vec{M}_{1}| = |\vec{M}_{2}|$. The canting angle is given by $\tan(2 \varphi) = 2 d / (2 J + \alpha)$ and since $d \ll J$ (in reality $\varphi < 0.5^{\circ}$) the net WF moment of each plane is approximately $M_{F} \approx 2 m \varphi = 2 d m / (2 J + \alpha)$. Here $m$ is the sublattice magnetization. The interaction $J_{\perp}$ does not significantly change this angle.
[**Long wavelength spin-wave excitations**]{} On general grounds, from Eq. (\[e15\]) one can say the following about the behavior the spin-wave modes in the long wavelength limit: (1) if $\alpha = d = 0$ there will be two acoustic modes; (2) if $\alpha \neq 0$ and $d = 0$ or $\alpha = 0$ and $d \neq 0$ there will be one acoustic and one gapped spin-wave branch; (3) if $\alpha \neq 0$ and $d \neq 0$ both spin-wave branches will be gapped. This is because unless we are in case (3), there is a global continuous symmetry which is broken at the AF transition due to the ordering of the magnetic moments (the gapless branches are typical Goldstone modes).
The situation described above is shown schematically in Fig. \[f17\]. One can intuitively understand how the spin gaps look like at a classical level by solving the equations of motion: $$\beta \frac{\partial \vec{M}_{j}}{\partial t} \ = \ \vec{M}_{j} \times \nabla_{\vec{M}_{j}} f \ \ \ \
j = 1,2
\label{e16}$$ where $\beta$ is a constant related to the Bohr magneton and $f$ is the free energy from Eq. (\[e15\]). The equilibrium condition $\nabla_{\vec{M}_{j}} f = 0$ gives the ground state shown in Fig. \[f16\]b. Linearizing the equations of motion from (\[e16\]) around equilibrium and choosing oscillatory solutions for the obtained set of homogeneous equations one can get (to first order in anisotropy terms) the following energies corresponding to the $XY$ and DM gap respectively: $$\omega_{XY} = z \ m \ \sqrt{2 \alpha J} \ \ \ \mathrm{and} \ \ \ \omega_{DM} = z \ m \ d
\label{e17}$$ With $z = 4$, $m = 1/2$ and taking $J = 145$ meV [@ColdeaPRL01] one can calculate from Eq. (\[e17\]) the anisotropy parameters $\alpha$ and $d$ if $\omega_{XY}$ and $\omega_{DM}$ are known. If the quantum corrections for the spin-wave velocity are taken into account [@SinghPRB89] the expressions for the gap energies become $\omega_{XY} = 2.34 \sqrt{2 \alpha J}$ and $\omega_{DM} = 2.34 d J$ [@KastnerRMP98]. The ellipses shown in Fig. \[f17\] are very elongated, the ratio of their small and big axes being essentially given by ratios of the anisotropy parameters with respect to the large super-exchange $J$. This is why in the literature the $XY$ mode (which corresponds to the precession of the net WF moment around the $c$-axis) is also called the out-of-plane gap while the DM mode (which corresponds to the $c$-axis oscillations of the WF moment) is called the in-plane gap.
In Fig. \[f17\] we also show the low energy INS measurements in of Peters *et al.* [@PetersPRB88]. The dots are the experimental data and the solid line is a fit using the spin-wave approximation convoluted with the experimental resolution. The energies of these two gaps are shown by arrows. The most direct way to check the magnetic nature of these modes is to apply an external magnetic field which has not been done so far. It is also desirable that such a study be performed with a higher energy resolution probe.
Magnetic Field, Temperature and Doping Dependence of the Dzyaloshinskii-Moriya Gap in ($0 \leq x \ \leq 0.03$)
--------------------------------------------------------------------------------------------------------------
[**(A) Field dependent Dzyaloshinskii-Moriya gap in .**]{} Fig. \[f18\] shows 10 K Raman spectra taken from 93% detwinned crystal using circular $(RL)$ polarization.
A sharp resonance is seen in zero field at 17 -1. This excitation disperses continuously upwards (downwards) for $\vec{H} \parallel \hat{a}$ ($\vec{H} \parallel \hat{b}$) axes. For $\vec{H} \parallel \hat{c}$, Fig. \[f18\]c, the mode disperses downwards until the magnetic field reaches the value $H_{WF} \approx 6$ T. At this point a transition to the WF state takes place. Initially observed in magnetic field dependent magnetization data [@ThioPRB88] this first order transition could be also studied by neutron scattering measurements [@KastnerPRB88] because due to the change in magnetic symmetry the scattering form factors allowed new Bragg peaks. In the 6 - 7 T range the resonance remains around 15 -1 but decreases in intensity and we observe a concomitant appearance of another feature around 21 -1. The 6.8 T spectrum in Fig. \[f18\]c shows clearly the coexistence of AF and WF states. Recent magnetization data [@AndoPRL03] and with a cartoon comparing the spin configuration in zero field and in the WF state are shown in Fig. \[f19\]. The hysteretic loops in the magnetic field dependent magnetization correspond to the hysteretic loops of the 15 and 21 -1 modes shown in Fig. \[f18\]b. This is in turn very similar to the behavior of the (100) and (201) magnetic Bragg peaks [@KastnerPRB88], reflecting the dynamics of magnetic domains in the presence of small crystalline imperfections.
The energies of the 17 -1 resonance as a function of magnetic field are plotted in Fig. \[f110\]a. The expressions for the fitting functions and the values for the fitting parameters used for the data in this figure are summarized in the following. If $\vec{H} \parallel \hat{a}$ and $\vec{H} \parallel \hat{b}$ $\ \ \ \Rightarrow \ \ \ \Delta_{H} = \sqrt{\Delta_{DM}^{2} + \gamma H^{2}}$ with $\Delta_{DM} = 17.35 \pm 0.25$ cm$^{-1}$ and $\gamma_{H \parallel a} = 0.96$ and $\gamma_{H \parallel b} = -1.65$ (cm T)$^{-2}$. If $\vec{H} \parallel \hat{c}$ and $H \geq H_{WF}$ $\ \ \ \Rightarrow \ \ \ \Delta_{H} = \sqrt{\Delta_{DM}^{2} + \beta H}$ with $\beta = 22.6$ cm$^{-2}$T$^{-1}$. The quadratic dependence in the first two cases can be understood because the spin re-arrangement in finite magnetic fields is independent of the directions parallel to the $a$ or $b$ axes along which the field is applied. This is not the case if $\vec{H} \parallel \hat{c}$ and the system is in the WF state, see Fig. \[f19\]c. Note that if $\vec{H} \parallel \hat{c}$ but $H \leq H_{WF}$ one observes again a quadratic dispersion with field. Moreover, the similar field dispersion for $\vec{H} \parallel \hat{b}$ versus $\vec{H} \parallel \hat{c}$ ($H < H_{WF}$) seen in Fig. \[f110\]a is intriguing because this degeneracy does not follow from the model of Eq. (\[e15\]) but it rather suggests rotational symmetry with respect to the $a$-axis.
Confirmation that the 17 -1 (in zero field) resonance observed in Fig. \[f18\] is the DM spin-wave gap comes from a 2D semiclassical spin-wave calculation. Assuming a fully ordered moment on Cu sites ($m = 1/2$), a zero field DM gap $\Delta_{DM} = 17$ -1 and minimizing Eq. (\[e15\]), one can obtain the dispersions of the $k = 0$ DM gap for the three directions of the applied field. The results are shown in Fig. \[f110\]b and the reasonable agreement with the experimental data allows one to assign this excitation to the DM interaction induced spin gap. Two comments on Fig. \[f110\]b. The first is that the 2D calculation can account only for the situation where the two sublattices in adjacent CuO$_{2}$ planes ’respond similarly’ to the external field. This is the case for $\vec{H} \parallel \hat{a}$, $\vec{H} \parallel \hat{b}$ and $\vec{H} \parallel \hat{c}$ with $H \geq H_{WF}$ and one can see that in all these cases the theoretical predictions agree with the experiment. If $\vec{H} \parallel \hat{c}$ and $H \leq H_{WF}$ the 2D approximation clearly breaks down and Eq. (\[e15\]) cannot be used in this region. The second comment is just a remark that it is surprising that a semi-classical spin-wave calculation as shown in Fig. \[f110\] is able to reproduce with relatively good accuracy the experimental data in a low spin system. This is in view of the expectation that such an approximation is valid to order $1 / S$ [@AndersonPR52] which is not a ’small’ number for $S = 1/2$. One may conclude from here that in order to explain the low energy spin dynamics in undoped 2D cuprates one does not need to go beyond a semiclassical approximation.
We believe that the magnetic field dependent data shown in Figs. \[f18\] and \[f110\] may also be relevant for a quantitative estimation of higher order spin interactions which are thought to be important in cuprates. One example is the ring exchange, see Fig.\[f15\], and it would be interesting to check the influence of $J_{ring}$ on the DM gap energy and possible renormalization effects on its magnetic field dependence. An example of a system where substantial effects of $J_{ring}$ on the spin-gap were pointed out is that of two-leg spin ladders. Using the expression $\Delta_{DM} = 2.34 d$ we can extract for the value $d = 0.92 \pm 0.013$ meV.
[**(B) Doping and temperature effects on the Dzyaloshinskii-Moriya gap.**]{} The results of doping and temperature on the DM gap are summarized in Fig. \[f111\]. The Néel temperatures for the x = 0 and 0.01 crystals studied here are 310 and 215 K respectively. The 1 and 3% Sr doped crystals were detwinned in proportion of 98 and 97%. In Fig. \[f111\]a we show the gap as a function of doping at 10 K. The 17 -1 resonance in the undoped crystal seen in the B$_{1g}$ orthorhombic channel becomes weaker in intensity, remains as sharp as in and softens to 12.5 -1 for x = 0.01, an energy 30% smaller compared to what we see for x = 0. We note also the absence of the DM mode in x = 0.02 and 0.03 crystals.
The doping dependence shows that the the DM mode is present at low temperatures only in the long range AF ordered region of the phase diagram (Fig. \[f11\]). This behavior is somehow surprising because one would expect to see for 2 or 3% doping at least a broadened feature in view of the large 2D magnetic correlations just outside the AF ordered phase. Such fluctuations are not observed in our data and this suggests that the presence of the long wavelength DM excitation is related to the existence of a true 3D order. A point discussed in the preceding section was related to the fact that it is the orthorhombicity which generates the DM interaction. The decrease of almost 30% in its energy from x = 0 to 0.01 is much more pronounced compared with the decrease in orthorhombicity and this relates this renormalization effects to a strong sensitivity on hole doping and points to a considerable renormalization of the ordered Cu moment at only 1% hole doping. Our data suggest that the antisymmetric interaction is strongly competing with other sources of disorder in the magnetic system and we suggest that this is most likely due to the frustration effects and the associated spin distortions induced by hole doping [@AharonyPRL90GoodingPRB97].
A last point we make regarding Fig. \[f111\]a is in regard to the macroscopic phase separation scenario proposed by the authors of Ref. [@MatsudaPRB02] to take place for $x \leq 0.02$. If this were true than one would observe in the $x~=~0.01$ crystal two features: one corresponding to the undoped region which would be found at 17 -1 and another one corresponding to the region with carrier concentration $c_{h} \approx 0.02$. The 30% decrease in energy observed in the $x~=~0.01$ crystal with respect to the undoped case seems to rule out the scenario proposed in Ref. [@MatsudaPRB02].
As a function of temperature what we see in Fig. \[f111\]b-c is that the DM gap softens with raising the temperature and disappears below 5 -1 as we approach the Néel temperature from below in both $x~=~0$ and 0.01 samples. The temperature dependence of the peak energies in the two crystals is shown in Fig. \[f111\]c to be similar and points towards a conventional soft mode behavior of this excitation, i.e. both its energy and its intensity approach zero in the limit $T \rightarrow T_{N}$, $T < T_{N}$. These spectra support the statement made in the previous paragraph that the DM induced gap exists in the narrow (T,$x$) region of the phase diagram from Fig. \[f11\] where the AF order is long ranged. It is possible that the reason it disappears at higher dopings is because the low energy magnetic fluctuations move away from the Brillouin zone center [@TranquadaNature04]. Interestingly, a broad peak at 300 K is seen around 15 -1 for $x~=~0$. This peak becomes a kink at 200 K and, as opposed to the conventional behavior of the DM gap, it disappears with further cooling. It is the purpose of the following section to investigate this excitation.
Magnetic Field Induced Spin Ordering in $x = 0$ and $0.01$
-----------------------------------------------------------
Magnetic field dependent $(RR)$ polarized Raman spectra in at several temperatures in $\vec{H} \parallel \hat{b}$ configuration are shown in Fig. \[f112\]a. At 10 K and in zero external field the Raman spectrum is featureless. For $H = 6$ T we see a sharp field induced mode (FIM) situated at 37.5 -1 which moves to slightly higher frequency (38.3 -1) for $H = 9$ T. The triangles in Fig. \[f112\]e show the relative energy of this excitation with respect to the value at 9 T. The extra data point corresponding to the 4.5 T Raman spectrum (not shown for clarity in panel (a)) marks the magnetic field at which the FIM starts to be seen. We remark only a small hardening (of about 4%) with magnetic field from 4.5 to 9 T. At 230 K the FIMs in 6 and 9 T fields are broader than at 10 K. However, with increasing field the FIM softens gaining spectral weight from the lower energy side.
At 300 K, as long as the field is less than about 6 T, we observe qualitatively similar behavior as at 230 K. For magnetic fields beyond that value we see the emergence of two independent peaks and both of them harden with further increasing the field. Fig. \[f112\]b we plot the total integrated intensity of the magnetic modes (for T = 300 K) at a given field, the data showing a maximum around $H = 7$ T, and in panel (c) the symbols denote the position of the FIMs as the magnetic field is swept from 0 to 11 T.
If $\vec{H} \parallel \hat{a}$ or $\vec{H} \parallel \hat{c}$ we do not observe any changes in the $(RR)$ polarized Raman spectra. Note that the spectra showing the DM gap in Fig. \[f18\] were taken in $(RL)$ polarization. Circular polarizations probe ’good’ symmetries if the crystal has a symmetry higher than tetragonal. Because the orthorhombicity in our samples is small it allows to separate the excitations appearing in these two geometries, but because it is finite we observe small ’leakage’ effects. Their magnitude can be estimated for instance by looking at the small feature corresponding to the DM gap which is found around 6-7 -1 in the H = 9 T and T = 230 K spectrum from Fig. \[f112\]a.
In the FIMs dynamics marks two events. The first seems to be a phase transition at 300 K and fields around 6 T. This is indeed the case because we know that the Néel temperature in is around 310 K and that the magnetic susceptibility $\chi_{b}$ shows T$_{N}$ decreasing at a rate of about 1 K/T if the magnetic field is applied parallel to the $b$-axis, as is the case in Fig. \[f112\]. Moreover, the narrow widths of the magnetic excitations above 6 T (2 -1 $\approx$ 0.25 meV) at temperatures more than two orders of magnitude higher (300 K $\approx$ 25 meV) argue strongly for the collective nature of these excitations which correspond to another magnetically ordered state with a well defined gap in the excitation spectrum. Such a transition is expected from the low temperature data shown in Figs. \[f18\] and \[f110\], more precisely from the behavior of the DM gap for $\vec{H} \parallel \hat{b}$. In this configuration we can fit the behavior of the DM gap by $\sqrt{\Delta_{DM}^{2} + \gamma_{b} H^{2}}$ with $\gamma_{b} < 0$. Extrapolating to higher fields would lead to a collapse of this gap marking a field induced transition. The second event, a crossover taking place between 230 and 10 K, is reflected in the opposite dispersion with field and different peak widths at these two temperatures.
As for the doping dependence, except for a much weaker intensity (see Fig. \[f113\]), we observed the same qualitative behavior in x = 0.01 . The FIM is not seen (in fields up to 9 T) at any temperature for $x \geq 0.02$. Accordingly, it seems that, like the DM gap, this feature is a characteristic of the phase diagram where long range AF order exists. As for the DM gap, the reason for its absence at higher dopings could be because the low energy magnetic excitations move away from $k = 0$ concomitant to the development of incommensurate magnetic excitations. In the following we try to identify the nature of the FIM and field induced transition by looking at the effects of the temperature on the Raman data in magnetic fields.
Fig. \[f113\] shows temperature dependent $(RR)$ polarized spectra in a 9 T field $\vec{H} \parallel \hat{b}$ for $x~=~0$ and 0.01. The data in panel (a) show that the crossover mentioned above (regarding the change in the FIM width and energy dispersion with field) takes place around 150 K. This is the temperature below which the FIM width narrows. Fig. \[f113\]c shows that the intensity of this excitation increases as we approach T$_{N}$ from below and that around 300 K we observe the splitting due to the occurrence of the field induced ordering. At this temperature the data for $x~=~0$ can be clearly fit with two peaks, see panel (a), and these two peaks correspond to those observed in Fig. \[f112\]a for T = 300 K and $H \geq 7$ T.
The temperature dependence of the FIM across the Néel boundary can be studied in the x = 0.01 crystal which has a lower T$_{N}$, see the panels (b) and (d) from Fig. \[f113\]. We mention here that for the $x~=~0.01$ crystal T$_{N}$ was measured (for fields lower than 7 T) to decrease on the average by almost 4 K/T for fields $\vec{H} \parallel \hat{b}$ axis. Given a $T_{N} (0 T) \approx 215$ K, in a 9 T field one expects that $T_{N} (9~T) \approx 180$ K. Indeed, at 9 T and below 180 K the behavior for x = 0.01 is very similar to that in the undoped crystal showing a softening of the FIM as we warm to T$_{N}$ but the situation changes with further warming. The 200 K data show that the FIM has, similarly to the 295 K data for x = 0, a low energy shoulder which is marked by an arrow. The data at 250 and 275 K can also be fitted by two Lorentzians. Along with the 200 K spectrum, these data seem to suggest the following picture: above 180 K we observe two features, one whose energy does not show significant magnetic field dependence and another one which softens from 20 to about 8 -1 with decreasing the temperature from 275 to 200 K. This latter excitation is marked by arrows in Fig. \[f113\]c and its energy is plotted in panel (d) by empty circles. The filled red circles in the same panel show the energy of the other mode (whose frequency is almost magnetic field independent above 180 K). The solid lines offer an explanation for the softening of the peak marked with arrows in panel (b): this is a magnetic soft mode corresponding to the field induce spin order taking place at $T \approx 180$ K in x = 0.01 for $\vec{H} \parallel \hat{b}$ and $H = 9$ T. Its energy approaches zero on cooling towards T$_{N}$ and we propose that it becomes the DM gap in the Néel phase. Fig. \[f113\]d also shows that the plot of the integrated intensities of the FIMs as a function of temperature is peaked at T$_{N}$. This points toward an unusual behavior in the sense that in a conventional picture the intensities of long wavelength gap modes scale with the AF order parameter, i.e. both of them vanish as T$_{N}$ is approached from below [@KeimerZP93].
What is the nature of the FIM within the AF phase? A possible explanation is its identification to the $XY$ gap. Support for this assignment is the presence of this mode only in x = 0 and 0.01 as well as the comparison to INS data [@KeimerPRB92; @KeimerZP93] which estimates $\Delta_{XY} \approx$ 40 -1 at 10 K in . The very small experimentally found hardening of the FIM with increasing field from 4.5 to 9 T at T = 10 K shown in Fig. \[f112\]e is consistent within 20% with the predictions of the spin-wave theory, which was found to describe fairly well the DM gap. This difference may be also accounted for if one invokes possible gap renormalization effects induced by higher order spin interactions [@ColdeaPRL01].
Regarding the nature of the magnetic field induced order we propose a state like the one depicted in Fig. \[f114\]c. This is suggested by the magnetic susceptibility data which shows that the moments on Cu sites remain confined in the $(bc)$ plane above T$_{N}$ [@LavrovPRL01] and also by recent magnetoresistance measurements [@AndoPRL03] which are consistent with a gradual rotation of the WF moments. In fact a departure from a two step transition [@ThioPRB90], involving a spin-flop process occurring between the states shown in Fig. \[f114\]b-c and which is characterized by a large component of the staggered magnetization along the $a$ orthorhombic axis, is expected. In a regular spin-flop transition a magnetic field applied parallel to the easy axis (which in our case is the $b$ orthorhombic axis, see Fig. \[f16\]b) will end up rotating the staggered magnetization along a direction perpendicular to this axis. The reason is that above some critical value of the field the magnetic anisotropy energy becomes smaller than the gain in magnetic energy due to the larger transverse susceptibility in the AF state [@KefferBook]. In the case, the situation seems not to be the same: because the transverse susceptibility, $\chi_{a}$, is the smallest below 300 K for x = 0 and 0.01 (see Fig. \[f13\]), the spins cannot partake of the field energy $- (\chi_{a} - \chi_{b}) H^{2} / 2$. Accordingly, a flop along the $a$-axis is not favorable from this point of view [@Ono04].
The identification of the FIM with the $XY$ gap can also explain other observed features. The crossover around 150 K shown in Figs. \[f112\] and \[f113\] may be understood as a departure of the direction of the WF moments from perpendicular to the $(ab)$ plane to a direction almost parallel to the $b$-axis (see Fig. \[f114\]) where the $XY$ anisotropy, weaker due to temperature fluctuations, ceases to play a decisive role. Physically, this corresponds to the fact that the conventional out-of-plane XY mode changes its nature as the WF moment rotates away from the $c$-axis. Prompted by this idea we calculated (solid lines in Fig. \[f112\]c) the spin-wave dispersions using Eq. (\[e15\]) and (\[e16\]) in the extreme case of $\alpha = 0$ and a small DM gap which still confines the moments in the $(bc)$ plane. Although finite temperature effects have to be taken into account, we note that this simple estimation reproduces, at least qualitatively, the experimental dispersions. We also comment on the possible relevance of our findings to the switch of orthorhombic axes in magnetic fields [@LavrovNature02]. If a state like Fig. \[f114\]c is realized (which is shown in Fig. \[f113\]b to persist to temperatures close to 300 K even for x = 0.01) then the magnetic force in an external field is significantly enhanced due to the net in-plane ferromagnetic moment. Still, the origin of the coupling between the spins and the tilt of the CuO$_{6}$ octahedra remains as a very interesting question.
The qualitative scenario we propose regarding the nature of the FIMs and the nature of the magnetic field induced order leaves several open questions. One of them is the following: if the FIM in the AF state is the XY gap, why is its spectral weight peaked at T$_{N}$, as shown in Fig. \[f114\]b for x = 0.01 ? A second question is related to the finite intensity of the FIMs only for magnetic fields $\vec{H} \parallel \hat{b}$-axis. On the other hand if we assume that the FIM mode is an excitation other than the XY gap, arising for instance as a result of the 4-sublattice structure, then the common interpretation of the excitation around 40 -1 found in several 2D layered AF’s has to be reconsidered.
One may wonder if up to now there are any other transport signatures of this magnetic field induced transition which could back up our spectroscopic conclusions. As for the undoped crystal, where the transition should be most prominent and which has strongly insulating behavior, to our knowledge there are no magnetoresistance measurements so far and in terms of magnetization it would be highly desirable to see measurements especially as a function of magnetic field at several temperatures down to 10 K. Higher fields than 7 T are needed though as the temperature is decreased below 300 K. However, relative magnetoresistance data in twinned samples of x = 0.01 (Fig. 2 in Ref. [@AndoPRL03]) show a ’peel off’ from a temperature independent curve. For a given magnetic field value, this phenomenon is seen to occur at temperatures where magnetization data indicate the transition outside the AF order. This shows that the $dc$ transport responds to field induced changes in the AF environment in the anticipated (H,T) parameter space. Supplementary magnetization, magnetoresistance and especially neutron scattering measurements in magnetic field would be necessary to verify our claims.
Phononic and Electronic Anisotropy in Detwinned
------------------------------------------------
We observed in the previous section that detwinned crystals revealed strong anisotropy effects in terms of the dynamics of long wavelength spin excitations. Here we show that the small lattice orthorhombicity has drastic effects also on the phononic and electronic Raman continuum. A summary of our results in this perspective is shown in Fig. \[f115\]. The left panel shows T = 10 K Raman data in taken in $(aa)$ and $(bb)$ polarizations. The axes notation is shown in the inset. Both these symmetries probe fully symmetric excitations and, in terms of phonons, there are 5 allowed in the LTO phase. Four of them, denoted by $A$, $B$, $C$ and $D$ are in the energy region below 300 -1. All the five A$_{g}$ modes and their atomic displacements will be discussed in more detail in the section devoted to Nd doped . For now we remark that while in the insulating the Raman continuum is, as expected, very weak at low temperatures, there is a tremendous intensity anisotropy in the $A$ and $B$ phonons. Mode $A$ is seen clearly in $(bb)$ polarization and has almost vanishing intensity in $(aa)$ polarization and the situation is reversed for mode B.
In the middle panel we observe that the anisotropy in these two modes is preserved. However one can note that we observe intense Raman backgrounds (shaded areas), quite different in intensity in $(aa)$ versus $(bb)$ polarizations. The relative intensities of the continua match the anisotropy in the $dc$ conductivity along the $a$ and $b$ axes [@AndoPRL02] shown in the inset. Looking at the $x~=~0.03$ data (right panel) one can notice that the sign of the low temperature resistivity anisotropy changes with respect to the $x~=~0.01$ case. Similarly, the Raman background in $(aa)$ polarization becomes stronger than in $(bb)$ polarization and this change is also accompanied by the reversal of the intensity anisotropy of the $A$ and $B$ phonons.
This switch is a remarkable effect. Could it be that it is induced by structural changes, in particular a 90$^{\circ}$ rotation of the CuO$_{6}$ octahedra between 1 and 3% Sr doping? X-ray data showed that this is not the case, suggesting that the reversal is due to the development of a new kind of anisotropy in the spin-charge dynamics at low doping, possibly occurring as the the system crosses at low temperatures the boundary of the long range AF order. Beyond this observed switch between 1 and 3% doping, the strong phononic anisotropy seen most clearly in data is an interesting problem by itself. Since it is determined by the CuO$_{6}$ octahedra tilt around the $a$-axis, one may suspect that the $p_{z}$ orbitals of the apical oxygens may be involved in the coupling process and its hybridization with in plane orbitals is not negligible. We remark one other point in regard to the phononic features shown in Fig. \[f115\]: While in the observed number of modes does not exceed the number predicted by group theory (we observe also the 5$^{th}$ mode in $(cc)$ polarization at 430 -1, see Fig \[f120\]), a much larger number of additional features sitting on top of the Raman continuum is seen for $x~=~0.01$ and 0.03. It is possible that this is connected to charge and/or spin supermodulation within the 2D CuO$_{2}$ planes. While not explained, the experimental observations in Fig. \[f115\] pose intriguing questions, some of them, like the possibility of 2D spin and/or charge order, being tied to problems actively scrutinized in relation to the occurrence of superconductivity in cuprates.
Spin and Lattice Dynamics at Commensurate $x~=~1/8$ Sr Doping in
=================================================================
Motivation: Intrinsic Spin/Charge Modulations in the CuO$_{2}$ planes?
----------------------------------------------------------------------
The origin of the interest in studying lattice and electronic dynamics in 2D cuprates at carrier concentrations commensurate with the lattice is essentially due to the increased tendency of the doped system to form real space patterns characterized by certain periodic modulations of the charge and spin density. Among correlated systems, this situation is not peculiar to high T$_{c}$’s but it has been discussed for instance in different type of materials like manganites or nickelates. Ground states in which charges self organize in quasi-1D ’rivers’ (called stripes) acting as AF domain walls were predicted at the mean field level as early as 1989 [@ZaanenPRB89] and later it has been proposed that the charge and/or spin ordering is not necessarily static, but the carriers could form electronic liquid-crystal like phases [@KivelsonNature98].
From the experimental point of view, one of the observed ’$x~=~1/8$’ effects, discovered initially in [@MoodenbaughPRB88] but also observed in Nd doped [@TranquadaPRL97], was a suppression of superconductivity manifested through a decrease of the transition temperature T$_{c}$. In fact a similar observation (but in terms of the *onset* of superconductivity as seen by magnetization measurements [@YamadaPRB98]) was made in Nd free at 1/8 Sr doping. The ’stripology’ in cuprates got a lot of momentum after the discovery of a constellation of neutron Bragg peaks in 0418. The data showed superlattice peaks associated with static spin and hole ordering, the magnetic moment modulation being characterized by a wavelength twice as big as the one observed for the charge [@TranquadaNature95]. Some of the effects discussed above are illustrated in Fig. \[f116\].
The almost complete suppression of T$_{c}$ in 0418 as well as the fact that in this compound neutron scattering sees long ranged charge and spin supermodulations suggested that the stripes may be the ’looked for’ competing state to superconductivity. The presence of the incommensurate magnetic peaks also in Nd free and their observation in *elastic* neutron scans at $x~=~1/8$ doping show that with $x~=~1/8$ are some of the most suitable 2D cuprate compounds to look for the effects of such modulations. It is important to note that while in Nd doped the charge ordering was also confirmed by X-rays [@ZimmermannEL98], this is not the case (yet) in Nd free samples. Raman spectroscopy can be a powerful technique in this respect because optical phonons can be used as local probes of fast changes in the charge distribution and magnetic Raman scattering provides information about local AF correlations. However, it can also provide information regarding the side effects of Sr substitution and what we argue in the study presented in the following is that some of those effects, structural distortions as well as the disorder introduced by Sr substitution, are important at $1 / 8$ doping in irrespective of Nd concentration [@GozarPRB03].
It was mentioned in the introduction that in the various changes in the crystal structure are due to the lattice mismatch between the cation and CuO$_{2}$ layers. Like , the 0418 compound undergoes a transition from the HTT to the LTO phase above room temperature. This transition is followed around T$_{LTT} = 70$ K by another structural change, from the LTO to the low temperature tetragonal (LTT) phase where the CuO$_{6}$ octahedra tilt around an axis parallel to the Cu-O-Cu bonds. The structural order parameter of these transitions is the libration of the CuO$_{6}$ octahedra shown in Fig. \[f117\]. It was noticed in 0418 that the LTO-LTT transition takes place over a range of temperatures and that disorder in the striped phase leads to a glassy nature of the ground state. Intermediate states characterized by a tilt angles in between those of the LTO and LTT phase have also been proposed [@BuchnerPRL94]. The coexistence in of several phases in a complex mixture was suggested by transmission electron microscopy [@HoribePRB00].
So there are interesting topics associated to the presence of Nd, but after all why are these structural effects, especially the ones related to the tilt of CuO$_{6}$ octahedra, relevant to the spin and charge dynamics? The importance of the local structural distortions for the superconducting properties characterized in cuprates by a short coherence length should not be ignored. The stabilization of the LTT phase was observed to trace the suppression of superconductivity in Nd doped [@CrawfordPRB91] and also in the related La$_{2-x}$Ba$_{x}$CuO$_{4}$ compound [@AxePRL89]. A critical value of the CuO$_{6}$ tilt was associated with the stabilization of magnetic against superconducting order, see Fig. \[f117\]b and Ref. [@BuchnerPRL94]. Rapid suppression of superconductivity, similar to that due to Cu replacement by non-magnetic impurities, was observed with increasing the cation radius variance [@McAllisterPRL99].
In this context our study provides direct spectroscopic information about the LTO-LTT transition in 0418 and local deviations from the average structure existent in Nd doped and Nd free structures. The persistent fluctuations of the structural order parameter down to T = 10 K reveal substantial disorder in the cation-oxygen layers. The distinct Raman signatures accompanying a transition to a state with deep spin/charge modulations are not observed in the temperature dependence of the two-magnon (2M) scattering around 2200 -1 and the $c$-axis polarized phonons below 500 -1 [@GozarPRB03].
Inhomogeneous CuO$_{6}$ Octahedra Distribution in $x = 1/8$
------------------------------------------------------------
In the following we will show Raman data from with the following doping concentrations: $x \approx 1/8$, $y = 0$; $x \approx 1/8$, $y = 0.4$; and $x = 0.01$, $y = 0$. The spectra were taken from the $(a'c)$ and $(ab)$ faces of the $x = 1/8$ samples and from the $(ac)$ surface of a $x = 0.01$ crystal as determined by X-ray diffraction. See Fig. \[f117\] for axes notations. Note that they are consistent with the ones in Fig. \[f12\], the primed letters corresponding to directions parallel to the Cu-O-Cu bonds The laser excitation energy used was $\omega_{in} = 1.92$ eV.
Raman spectra taken in the $(ca')$ geometry may provide direct information about tetragonal to orthorhombic distortions. In this polarization we probe phononic modes with B$_{2g}$ and B$_{3g}$ symmetries in the LTO phase which become degenerate with E$_{g}$ symmetry in the LTT phase. In Fig. \[f118\] we show the temperature dependence of the modes around 250 -1 corresponding to the apical O vibrations parallel to the CuO$_{2}$ plane in 0418 [@OhanaPRB89]. One can think about the spectral changes in analogy to the evolution with temperature of the orthorhombically split X-ray diffraction Bragg peaks [@CrawfordPRB91]. We observe a broad peak around 245 -1 at room temperature which, with cooling, becomes resolved into two components, one hardening and one softening. A new central peak can be seen at 50 K around 248 -1 which gains spectral weight as the temperature is decreased to 10 K. While the total integrated intensity of the modes remains constant, Fig.\[f118\]c, we observe a redistribution of spectral weight among the three modes as a function of temperature. The split components become weaker but can still be seen as ’orthorhombic satellites’ of the central peak down to 10 K. The coalescence of the features into the 248 -1 mode signals the recurrence of a phase with tetragonal symmetry which should be the expected LTT phase of 0418. However, the finite residual intensity of the satellites appearing on the tails of the broad central peak shows an incompletely developed LTT phase and that even at 10 K there exists about 7% LTO phase (determined from the relative ratio of phonon intensities). Note that the width of the main peak at T = 10 K is comparable to the widths of the components of the doublet seen at temperatures as high as 200 K.
Raman data in $(cc)$ polarization is well suited for the study of lattice dynamics due to weaker coupling to underlying electronic excitations. Temperature dependent Raman spectra in this scattering geometry are shown in Fig. \[f119\]b. Group theory predicts five fully symmetric modes at $k = 0$ in each of the LTO and LTT phases and only two for the HTT phase. The two fully symmetric modes of the HTT along with the additional three modes in the LTO phase are shown in Fig. \[f119\]a. In Fig. \[f119\]b we observe all the five phonons corresponding to the LTO and LTT phases and they are denoted by A, B, C, D and E. Four out these five modes can also be seen in Fig. \[f115\] where the same notation was used. Although every one of these excitations should be considered as linear combinations of all the A$_{g}$ movements depicted in Fig. \[f119\]a, one could roughly say that they are mainly composed of the following vibrations. The modes C and E (which at T = 10 K are found at 228 and 433 -1) are inherited from the HTT phase and they correspond to the $c$-axis vibrations of La/Sr/Nd and O atoms respectively, see the upper part of Fig. \[f119\]a. Mode A is the soft mode of the HTT-LTO transition (the CuO$_{6}$ octahedra tilt), mode B is mainly due to the vibration of La/Sr/Nd atoms in the direction imposed by the CuO$_{6}$ tilt and mode D consists of $c$-axis vibrations of the in-plane O atoms [@OhanaPRB89; @SugaiPRB89; @WeberPRB88]. The energies of the last three phonons at the lowest temperature are: 106 -1 (mode A), 156 -1 (mode B) and 275 -1 (mode D).
The above qualitative description indicates that we could expect a strong coupling between the lowest energy modes (A and B). These two excitations can be distinguished in Fig. \[f119\]b from the other ones because they remain much broader and look like composite features even at the lowest temperature in comparison with the the modes C, D and E which harden and sharpen smoothly through the LTO-LTT transition taking place around 70 K. As seen in Fig. \[f119\]b, the temperature variation of the intensities of the modes C and E inherited from the HTT phase is not as pronounced which is not surprising. A comparison of the temperature dependent energy and full width at half maximum (FWHM) of modes A and D is shown in Fig. \[f119\]d-e. The large variation in energy and width of mode A above the transition (see also the inset of Fig. \[f120\]), the softening below 70 K, as well as its energy around 110 -1 in agreement with neutron scattering studies [@ThurstonPRB89] show that this mode corresponding to the octahedra tilt is the soft mode of the structural changes [@SugaiPRB89]. The smooth decrease in the energy in the LTT phase is only apparent because this space group is not a subgroup of the LTO group and as a result a true LTO-LTT transition is expected to be of first order. Although unresolved due to broadening effects, the large width of mode A around 70 K shows the coexistence of the LTO and LTT tilts, the latter appearing as a result of folding of the LTO $Z$-point to the $\Gamma$-point of the LTT phase which was observed also in La$_{2}$NiO$_{4}$ [@BurnsPRB90].
We infer from our data that the large FWHM of mode A reflects the spatial distribution of the octahedra tilt. The simultaneous broadening of the mode B shows coupling between the OP and La/Nd/Sr vibrations and as a result the influence of the dynamics in the cation-O layers on the properties of CuO$_{2}$ planes. Both modes B and C involve cation displacements as discussed above, the former perpendicular and the latter parallel to the $c$ axis. However, at 10 K the FWHM of mode C is 8 -1, smaller compared to the FWHM of mode B which is around 20 -1. We conclude that the large observed widths of the modes A and B are mainly caused by the locally fluctuating OP and not due to the inhomogeneous broadening introduced by the simultaneous presence of La, Nd and Sr in the inter-layer composition which should have been reflected also in a large width of mode C.
It is interesting to compare the $(cc)$ polarized phononic spectra with those in which the polarization of the incoming photon field is parallel to the CuO$_{2}$ planes. Fig. \[f119\]c shows the temperature dependent Raman spectra in the $b(aa)\bar{b}$ geometry. Different coupling to the electronic degrees of freedom when the polarization of the incident field is parallel to the CuO$_{2}$ planes leads to a stronger intensity in the underlying Raman continuum relative to the phononic features and also a different intensity/shape of the fully symmetric features observed in $(cc)$ polarization. A continuous suppression with cooling of the electronic background is due to the opening of a pseudogap in 0418 [@DummPRL02]. The electron-phonon coupling allows the observation of additional peaks around 370 and 480 -1 evolving smoothly from 300 to 10 K, both of which allowing an interpretation in terms of two-phonon scattering if some anharmonic interaction is taken into account. Marked by an arrow in this panel is a B$_{1g}$ symmetric excitation in the LTO phase which shows a jump from 325 to 335 -1 as the crystal enters in the LTT phase.
In order to understand the surprising behavior of the tilt pattern as reflected in the phononic data from Fig. \[f119\] a comparison with different materials from the same class is useful. In Fig. \[f120\] we show the 10 K $(cc)$ polarized Raman spectra of three crystals: 0418, x = 0.01 and 0.12 . For $x = 0.01$ mode A has a FWHM of 2.5 -1 (Note in the inset the strongly temperature dependent intensity and width which is a characteristic of a soft mode). For $x \approx 1/8$ LSCO the same phonon is around 85 -1 and its FWHM of about 23.5 -1 is larger than the width of mode A in the Nd doped crystal where it is slightly below 20 -1. Comparison of these relative phononic widths confirms the conclusion discussed before that Nd doping of LSCO crystals and the closer proximity to the T$^{\prime}$ phase induced by Nd doping in the La$_{2}$CuO$_{4}$ structure [@ManthiramJSSC91] cannot be responsible for the large observed broadening effects. Intrinsic phonon anharmonicity would lead to a broad mode A in $x = 0.01$ LSCO which is not the case. Neither can the tilt disorder across twin domains be the cause of such dramatic effects because the volume fraction occupied by these boundaries is expected to be very small [@HoribePRB00]. The 7% relative ratio of the orthorhombic satellites to the central peak in Fig. \[f118\]b would rather be consistent with such a small contribution. If the satellites are indeed due to twinning effects the data show that at 10 K the larger LTT domains are separated by regions of pure LTO tilt. The absence of the broadening effects on the vibrations along the $c$-axis points towards an ’anisotropic’ disorder relating primarily to bond randomness along directions parallel to the CuO$_{2}$ planes.
Could the spin-lattice coupling or the interaction with the stripe-ordered carriers in CuO$_{2}$ planes be the main cause of broadening? Stripe correlations are enhanced in 0418 which displays however a smaller width of mode A. Also, it is not clear why only the modes A and B would be affected by this interaction. In this sense one expects the movements of the in-plane atoms to be more sensitive to stripe ordering but we see no similar effects on mode D. Although less probable, spin-lattice induced broadening cannot be completely ruled out and the answer to this question lies in a Sr doping dependence of the $(cc)$ polarized spectra. Our data can be reconciled however with recent studies of local structure in Nd free and Nd doped systems [@HaskelPRL96HanPRB02]. Model analysis of the pair distribution function from X-ray absorption fine structure suggests that in this material class the average structure determined by diffraction is different from the local pattern which is characterized by disorder in the CuO$_{6}$ tilt direction and magnitude [@HaskelPRL96HanPRB02]. The Raman data shown in Figs. \[f119\] and \[f120\] are spectroscopic evidences that the system is characterized by disorder in the cation layers and that the locally fluctuating octahedra tilt is responsible for the observed effects.
Information about the relative magnitude of charge disproportionation in can be gained by comparison with Raman spectra in compounds like the nickelates [@BlumbergPRL98; @PashkevichPRL00] or manganites [@AbrashevPRB01; @YoonPRL00] where charge and spin modulations are well established [@ChenPRL93ChenPRL96]. New Raman active modes have been observed below the charge ordering within the Mn-O layers in La$_{0.5}$Ca$_{0.5}$MnO$_{3}$ [@AbrashevPRB01] and also in Bi$_{1-x}$Ca$_{x}$MnO$_{3}$ [@YoonPRL00]. Conspicuous changes in the lattice dynamics have also been observed in x = 0.33 and 0.225 La$_{2-x}$Sr$_{x}$NiO$_{4}$ by Raman scattering [@BlumbergPRL98; @PashkevichPRL00]. Lowering of the crystal symmetry at the stripe ordering transition gives rise to folding of the Brillouin zone and the appearance of new $k = 0$ phononic modes. Charge localization creates non-equivalent Ni sites generating phonon ’splitting’. The $c$ axis stretching modes corresponding to La and apical oxygens split by 14 and 30 -1 respectively [@BlumbergPRL98]. Within about 3 -1 resolution imposed by the phononic widths we do not observe such splittings in our spectra. The ratio of the integrated intensities of the split oxygen modes in Ref. [@BlumbergPRL98] is about the same as the ratio of doped versus undoped Ni sites. If we assume the same relation to hold for the case of cuprates, a factor of 12% in split phononic intensity should have been seen in our spectra. However, the latter argument has to take into account that different electron-phonon coupling might change this proportionality relation. Last but not least is the observation that we see, at least in $(cc)$ polarized spectra only the phononic excitations predicted by group theory for the LTO/LTT phases. We conclude that any charge ordering taking place in our case is much weaker than in the related compounds referred to above. This is not contradicting X-ray diffraction data [@ZimmermannEL98] which estimated a factor of 10$^{2}$ between the relative magnitude of charge modulations in cuprates and nickelates.
Two-Magnon Raman Scattering in $x = 1/8$ and $x = 0 - 0.03$
------------------------------------------------------------
Two-magnon (2M) Raman scattering provides an additional way to look at the effects of stripe correlations on magnetic excitations. For 2D square lattices the 2M peak is predicted to be seen in the B$_{1g}$ channel, probed by $(ab)$ polarization [@FleuryPR68ShastryPRL90]. Fig. \[f121\] shows 2M scattering around 2200 -1 at 300 and 5 K taken with the resonant $\omega_{in}$ = 3.05 eV incident frequency. As in other tetragonal 2D AF’s [@SugaiPRB90] we observe the spin pair excitations in the expected scattering geometry. The $c(a'b'){\bar c}$ polarization shows a featureless background which probably has a contribution from luminescence.
In La$_{2-x}$Sr$_{x}$NiO$_{4}$ there is a clear signature of the effect of stripe ordering on the high energy spin pair excitations: in the undoped case (x = 0) the 2M Raman band is seen around 1650 -1 [@SugaiPRB90]. At 33% Sr doping this excitation is not present at that frequency but instead two peaks at lower energies, 720 and 1110 -1 [@BlumbergPRL98], are observed below the magnetic ordering temperature, see Fig. \[f121\]b. In Ref. [@BlumbergPRL98], assuming an unrenormalized value for the superexchange $J \approx 240$ -1 with respect to the undoped case, it is proposed that these peaks originate from the two spin exchange channels opened due to the stripe order, one of them within and the other one across the antiphase AF domains depicted in Fig. \[f116\]. A more recent neutron scattering study, whose authors are however in favor of a renormalization of the magnetic super-exchange in the stripe phase, is in support of this assignment regarding the higher frequency peak at 1110 -1 ($\approx 2 \cdot 70$ meV) by finding that the upper edge of the spin-wave dispersion branch is around 70 meV [@BoothroydPRB03]. Irrespective of the microscopic origin, the 2M scattering is definitely a good probe for the study of local effects induced by the stripe order. Comparison with our high energy Raman spectra shown in Fig. \[f121\]a shows, as in the case of phonons, that in we observe only slight changes from 300 to 10 K emphasizing weak local spin modulations in this compound.
The differences we observe between cuprates and nickelates can be related to the much stronger carrier self-confinement in the latter [@AnisimovPRL92]. It has also been shown [@McQueeneyPRL01] that anomalies in phonon dispersions occur in at points in the Brillouin zone commensurate with charge ordering wavevectors inferred from neutron scattering studies. But as discussed, the charge modulation in Nd doped , where the stripe correlations were shown to be stabilized, is too weak to produce observable changes in the lattice unit cell. The number of phononic modes we observe can be explained solely in terms of LTO/LTT distortions. Our data, however, do not contradict the possible existence of charge modulations in the CuO$_{2}$ plane. In fact, the dynamics in the cation-O layers and the magnitude of octahedra tilt disorder affects the carrier distribution and our Raman results impose constraints on the magnitude of the charge modulations.
In the context of sensitive short wavelength magnetic excitations to possible spin/charge modulations in the transition metal oxygen planes, see Fig. \[f121\], and also the strong anisotropy effects found in detwinned crystals in terms of both phononic and low lying electronic excitations, see Fig. \[f122\], it is interesting to take a look at the evolution with hole concentration of the 2M excitation in lightly doped crystals as well as to check whether one can find at high frequencies (2000 - 4000 -1) anisotropy effects similar to those observed in Fig. \[f115\]. Our Raman results regarding these problems are shown in Fig. \[f122\]. In panel (a) we show the 2M feature at low temperatures in the B$_{1g}$ channel (probed in $(ab)$ polarization, see the axes notations in Figs. \[f12\] and \[f117\]) for four dopings.
The length of the arrows roughly indicate the evolution of the scattering width with increasing $x$ from 0 to 0.03. We found quite a sizeable effect in terms of the 2M FWHM. In the undoped sample the FWHM is around 770 -1 and although the in plane correlation lengths remain large in the lightly doped regime [@KeimerPRB92], we observe an approximately linear increase of about 60% with doping from $x = 0$ to 0.03 (inset of Fig. \[f122\]b). One may try to correlate this to the increase in the low energy electronic Raman background seen in Fig. \[f115\] in parallel polarization. We remark that although this enhancement is obvious for $(aa)$ and $(bb)$ scattering geometries, we checked that in $(ab)$ configuration the intensity of the Raman background at T = 10 K is doping independent. It is easier to understand that the magnetic spin-wave gap excitations in the long wavelength limit disappear together with the disappearance of long range magnetic order, however, the drastic change in the two-magnon scattering, which in principle requires ’good’ AF correlations on a much smaller scale (about 4 lattice constants), is not so straightforward to grasp. For an explanation one may resort again to the argument that in the CuO$_{2}$ plane, a hole entering the O$2p$ bands is more delocalized and one hole breaks effectively more magnetic bonds than just one between a pair of nearest neighbor spins.
Regarding possible anisotropy effects, Fig. \[f122\]b shows that the $(aa)$ and $(bb)$ polarized spectra look very much alike at high frequencies and although there are differences, the second and third order phononic scattering is much less sensitive to the macroscopic orthorhombicity than the one phonon excitations in the 0 to 500 -1 region. Unfortunately, as predicted by the Fleury and Loudon (see Ref. [@FleuryPR68ShastryPRL90]) and seen in Fig. \[f122\], the strongest 2M scattering is supposed to be seen in B$_{1g}$ tetragonal channel which in not probed in $(aa)$ and $(bb)$ polarizations ($(aa)$ and $(bb)$ configurations probe A$_{1g}$ + B$_{2g}$ tetragonal symmetries) so we cannot directly probe the effects of the macroscopic lattice spin anisotropy on the strong 2M feature from Fig. \[f122\]a. What we can however say is that if the finite Raman background between 2000 and 4000 -1 has some fully symmetric (A$_{1g}$) magnetic contribution from higher order light scattering Hamiltonian, see Ref. [@SulewskyPRL91], the influence of the lattice orthorhombicity on that contribution is negligible.
Summary
=======
Two aspects in connection with the magnetic properties of single crystals were discussed in some detail. One of them was related to long wavelength magnetic excitations in $x~=~0$, 0.01, and 0.03 detwinned crystals as a function of doping, temperature and magnetic field. Two magnetic modes were observed within the AF region of the phase diagram. The one at lower energies was identified with the spin-wave gap induced by the antisymmetric DM interaction and its anisotropic properties in magnetic field could be well explained using a canonical form of the spin Hamiltonian. A new finding was a magnetic field induced mode whose dynamics allowed us to discover a spin ordered state outside the AF order which was shown to persist in a 9 T field as high as 100 K above the Néel temperature T$_{N}$ for x = 0.01. We proposed for the field induced magnetic order a state with a net WF moment in the CuO$_{2}$ plane and analyzed the field induced modes in the context of in-plane magnetic anisotropy. For these single magnon excitations we mapped out the Raman selection rules in magnetic fields and we also found that their temperature dependent spectral weight (in the presence of a constant external magnetic field) was peaked at the Néel temperature.
The second aspect was related to phononic and magnetic Raman scattering in La$_{2-x-y}$Nd$_{y}$Sr$_{x}$CuO$_{4}$ with three doping concentrations: $x \approx 1/8$, $y =
0$; $x \approx 1/8$, $y = 0.4$; and $x = 0.01$, $y = 0$. We observed that around $1/8$ Sr doping and independent of Nd concentration there exists substantial disorder in the tilt pattern of the CuO$_{6}$ octahedra in both the orthorhombic and tetragonal phases which persist down to 10 K and are coupled to bond disorder in the cation layers. The weak magnitude of existing charge/spin modulations in the Nd doped structure did not allow us to detect specific Raman signatures on lattice dynamics or two-magnon scattering around 2200 -1.
It is possible that the discovery of weak charge modulations in the hole doped 2D CuO$_{2}$ planes characteristic of high T$_{c}$ materials is just a matter of time. The problem of doped Mott-Hubbard insulators seems however to be one in which numerous possible ground states are allowed and the supremacy of any one of them could require really fine tuning of microscopic parameters. In this respect, even if such a charge density modulation were observed, the question whether it helps understanding the mechanism of superconductivity or not would still need to be answered.
[**Acknowledgments –**]{} We acknowledge discussions and collaborations with B. S. Dennis, M. V. Klein and A. N. Lavrov.
[99]{}
E-mail: [email protected]
M. A. Kastner [*et al.*]{}, Rev. Mod. Phys. [**70**]{} (1998) 897 and references therein.
J. G. Bednorz and K. A. Müller, Z. Phys. B - Condensed Matter [**64**]{} (1986) 189.
B. Keimer [*et al.*]{}, Phys. Rev. B [**46**]{} (1992) 14304.
I. Bozovic [*et al.*]{}, Phys. Rev. Lett. [**89**]{} (2002) 107001.
B. Büchner [*et al.*]{}, Phys. Rev. Lett. [**73**]{} (1994) 1841.
J. M. Tranquada [*et al.*]{}, Nature [**375**]{} (1996) 561.
B. Keimer and M. Reehuis, private communications.
A. Gozar [*et al.*]{}, Phys. Rev. Lett. [**93**]{} (2004) 027001.
A. Gozar [*et al.*]{}, Phys. Rev. B [**68**]{} (2003) 052511.
Y. Horibe, Y. Inoue and Y. Koyama, Phys. Rev. B [**61**]{} (2000) 11922; Y. Inoue, Y. Horibe, and Y. Koyama, Phys. Rev. B [**56**]{} (1997) 14176.
A. N. Lavrov [*et al.*]{}, Phys. Rev. Lett. [**87**]{} (2001) 017007.
Y. Ando [*et al.*]{}, Phys. Rev. Lett. [**87**]{} (2001) 017001.
Y. Ando [*et al.*]{}, Phys. Rev. Lett. [**88**]{} (2002) 137005.
Y. Ando, A.N. Lavrov, and S. Komiya, Phys. Rev. Lett. [**90**]{} (2003) 247003.
S. Wakimoto [*et al.*]{}, Phys. Rev. B [**60**]{} (1999) R769.
M. Matsuda [*et al.*]{}, Phys. Rev. B [**65**]{} (2002) 134515.
A.N. Lavrov, S. Komiya, and Y. Ando, Nature [**418**]{} (2003) 385.
R. Coldea [*et al.*]{}, Phys. Rev. Lett. [**86**]{} (2001) 5377.
S. Chakravarty, B. I. Halperin and D. R. Nelson, Phys. Rev. B [**39**]{} (1989) 2344.
N. D. Mermin and H. Wagner, Phys. Rev. Lett. [**17**]{} (1966) 1133.
T. Thio [*et al.*]{}, Phys. Rev. B [**38**]{} (1988) 905.
M. Matsuda [*et al.*]{}, Phys. Rev. B [**42**]{} (1990) 10098; H.-Q. Ding, Phys. Rev. Lett. [**68**]{} (1992) 1927.
A. W. Sandvik [*et al.*]{}, Phys. Rev. B [**57**]{} (1998) 8478.
Rajiv R. P. Singh, Phys. Rev. B [**39**]{} (1989) R9760.
J. Lorenzana, J. Eroles and S. Sorella, Phys. Rev. Lett. [**83**]{} (1999) 5122.
I. Dzyaloshinskii, J. Phys. Chem. Solids [**4**]{} (1958) 241.
T. Moriya, Phys. Rev. [**120**]{} (1960) 91.
T. Moriya, Weak Ferromagnetism, Ch. 3 in Magnetism, Vol. 1, ed Rado and Suhl, Academic Press (1963).
L. Shekhtman, A. Aharony and O. Entin-Wohlman, Phys. Rev. Lett. [**69**]{} (1992) 836; L. Shekhtman, O. Entin-Wohlman and A. Aharony Phys. Rev. Lett. [**47**]{} (1993) 174.
C. J. Peters [*et al.*]{}, Phys. Rev. B [**37**]{} (1988) 9761.
M A. Kastner [*et al.*]{}, Phys. Rev. B [**38**]{} (1988) 6636.
P. W. Anderson, Phys. Rev. [**86**]{} (1952) 694.
A. Aharony [*et al.*]{}, Phys. Rev. Lett. [**60**]{} (1990) 1330; R.J. Gooding [*et al.*]{}, Phys. Rev. B [**55**]{} (1997) 6360.
J. M. Tranquada [*et al.*]{}, Nature [**429**]{} (2004) 534 and references therein.
B. Keimer [*et al.*]{}, Z. Phys. [**91**]{} (1993) 373.
T. Thio [*et al.*]{}, Phys. Rev. B [**41**]{} (1990) 231.
F. Keffer, Handbuch der Physik, ed. S Flügge, Springer-Verlag, Berlin (1966).
S. Ono [*et al.*]{}, http://xxx.lanl.gov/abs/cond-mat/0408604 (2004) preprint.
J. Zaanen and O. Gunnarsson, Phys. Rev. B [**40**]{} (1989) R7391; M. Kato and K. Machida, J. Phys. Soc. Jpn. [**59**]{} (1990) 1047.
S. A. Kivelson, E. Fradkin and V. J. Emery, Nature [**393**]{} (1998) 550.
A. R. Moodenbaugh [*et al.*]{}, Phys. Rev. B [**38**]{} (1988) 4596.
J. M. Tranquada [*et al.*]{}, Phys. Rev. Lett. [**78**]{} (1997) 338.
K. Yamada [*et al.*]{}, Phys. Rev. B [**57**]{} (1998) 6165.
J. M. Tranquada [*et al.*]{}, Nature [**375**]{} (1995) 561; J. M. Tranquada [*et al.*]{}, Phys. Rev. B [**54**]{} (1996) 7489.
M.V. Zimmermann [*et al.*]{}, Europhys. Lett. [**41**]{} (1998) 629.
M.K. Crawford [*et al.*]{}, Phys. Rev. B. [**44**]{} (1991) R7749.
J.D. Axe [*et al.*]{}, Phys. Rev. Lett. [**62**]{} (1989) 2751.
Judith A. McAllister, and J. Paul Attfield, Phys. Rev. Lett. [**83**]{} (1999) 3289.
I. Ohana [*et al.*]{}, Phys. Rev. B [**39**]{} (1989) 2293.
S. Sugai [*et al.*]{}, Phys. Rev. B [**39**]{} (1989) 4306.
W.H. Weber [*et al.*]{}, Phys. Rev. B [**38**]{} (1988) 917.
T.R. Thusrton [*et al.*]{}, Phys. Rev. B [**39**]{} (1989) 4327.
G. Burns [*et al.*]{}, Phys. Rev. B [**42**]{} (1990) R10777.
M. Dumm [*et al.*]{}, Phys. Rev. Lett. [**88**]{} (2002) 147003.
A. Manthiram, and J.B. Goodenough, J. Solid State Chem. [**92**]{} (1991) 231.
D. Haskel [*et al.*]{}, Phys. Rev. Lett. [**76**]{} (1996) 439; S.-W. Han [*et al.*]{}, Phys. Rev. B [**66**]{} (2002) 094101.
G. Blumberg, M.V. Klein, and S-W. Cheong, Phys. Rev. Lett. [**80**]{} (1998) 564; K. Yamamoto [*et al.*]{}, Phys. Rev. Lett. [**80**]{} (1998) 1493.
Yu.G. Pashkevich [*et al.*]{}, Phys. Rev. Lett. [**84**]{} (2000) 3919.
M.V. Abrashev [*et al.*]{}, Phys. Rev. B [**64**]{} (2001) 144429.
S. Yoon [*et al.*]{}, Phys. Rev. Lett. [**85**]{} (2000 3297).
C.H. Chen, S-W. Cheong, and A.S. Cooper, Phys. Rev. Lett. [**71**]{} (1993) 2461; C.H. Chen, and S-W. Cheong, Phys. Rev. Lett. [**76**]{} (1996) 4042.
P. A. Fleury and R. Loudon, Phys. Rev. [**166**]{} (1968) 514; B. S. Shastry and B. I. Shraiman, Phys. Rev. Lett. [**65**]{} (1990) 1068.
S. Sugai [*et al.*]{}, Phys. Rev. B [**42**]{} (1990) 1045.
A. T. Boothroyd [*et al.*]{}, Phys. Rev. B [**67**]{} (2003) 100407.
V. I. Anisimov [*et al.*]{}, Phys. Rev. Lett. [**68**]{} (1992) 345.
R. J. McQueeney [*et al.*]{}, Phys. Rev. Lett. [**87**]{} (2001) 077001.
P. E. Sulewsky [*et al.*]{}, Phys. Rev. Lett. [**67**]{} (1991) 3864.
|
---
abstract: |
We describe here a new method to estimate copula measure. From $N$ observations of two variables $X$ and $Y$, we draw a huge number $m$ of subsamples (size $n<N$), and we compute the joint ranks in these subsamples. Then, for $p,q\leq n$, the density in $(p/n,q/n)$ is estimated as $ \frac{1}{mn } \sum_{s=1}^m{\sum_{i=1}^n{{\mathbf{1}_{\left\{ R_{i, s}=p, S_{i, s}=q \right\}}} } } $ where $R_{i, s}$ (respectively $S_{i, s}$) is the rank in $X$ (resp. $Y$) of the [$i^{\textrm{th}}$]{} observation of the [$s^{\textrm{th}}$]{} sample.\
The simulation study shows that this method seems to gives a better than the usual kernel method. The main advantage of this new method is then we do not need to choose and justify the kernel. In exchange, we have to choose a subsample size: this is in fact a problem very similar to the bandwidth choice. We have then reduced the overall difficulty.
author:
- 'Jérôme Collet, Électricité de France R&D division [^1]'
title: 'Estimating copula measure using ranks and subsampling: a simulation study'
---
Introduction
============
Copula estimation
-----------------
A first way is to estimate cumulated density function, for the bivariate observations, and for each marginal. In this way, we get Deheuvels test [@deheuvels1; @deheuvels2]. This estimation is consistent. Nevertheless, one state that, for example to test independence, this estimation has some drawbacks. Due to the form of the region we use to count the points, the power of the test is good if the dependence is monotonic (${\mathbb{E} \left( X|Y=y \right)}$ is a monotonic function of $y$), because the regions $\{R_i<p,S_i<q\}$ are clearly overloaded in case of an increasing dependence, and underloaded in the decreasing case. On the other hand, for a more complex dependence (for example $Y=a \cdot X^2 +\epsilon$), the over or under loads will not be clear, on these regions.\
A second way is to estimate copula density using kernels. In this way, we find [@jdf; @charp; @sca].\
A third way is to use a parametric model, and likelihood maximum. We will not take into account this possibility in the following.
Proposal
--------
The main idea is that the space of the bivariate ranks is finite: it consists in $N^2$ points for a $N$ sample. It is then possible to fill it, though avoiding smoothing. To do it, we [**subsample**]{}.\
For example, if we have a sample of 30 observations, we draw many subsamples of 5 observations. Each one of these subsamples will fill 5 points out of the 25 values grid where one puts the $R$ and the $S$. If we draw enough subsamples, all the points will be filled out, several times if necessary, which gives a density.\
More formally, one notes: $n$ size of the subsamples, $m$ the number of subsamples, $R_{i, s}$ (resp. $S_{i, s}$) rank of the first (resp. second) coordinate of the [$i^{\textrm{th}}$]{} observation of the [$s^{\textrm{th}}$]{} subsample and $\delta$ the Dirac measure. The [**probability measure**]{} of $(F_X(X),F_Y(Y))$ is estimated by: $$\hat{\gamma} = \sum_{p,q\leq n} \left(\delta\left(\frac{p}{n},\frac{q}{n}\right) \times
\frac{1}{mn} \sum_{s=1}^m{\sum_{i=1}^n{{\mathbf{1}_{\left\{ R_{i, s}=p, S_{i, s}=q \right\}}} } } \right)$$ An important point is that we accept to use a discretized representation of the copula. In other words, we get a density, which is continuous only with respect to a discrete measure.\
Since we have a density, we can draw it. In the following we simulate 30 observations, with a linear correlation equal to 0.5. In the left graph, the radius of each circle is proportional to the density in each point.\
----------------------------------------------------------- --------------------------------------------------------------
Estimated copula density Original data
{width="40.00000%" height="40.00000%"} {width="40.00000%" height="40.00000%"}
----------------------------------------------------------- --------------------------------------------------------------
#### How to handle the ties?
To use this estimation in real life, we have to handle the ties. This problem, probably, has no good solution. A rank test means we assume the margins are continuous. In such a case, there would be no tie. Since we have to get rid of this problem, we will suppress the subsamples with ties. Obviously, it is possible only if the number of ties in the sample is not to big, which is necessary if we use a rank test.
Measure convergence
-------------------
We know that, for every random variable $X$, the empirical measure $N^{-1} \times \sum_{i=1}^{N}{\delta(X_i)}$ converges weakly to the probability measure of $X$. Furthermore, the empirical distribution function converges uniformly to the theoretical distribution function.\
Then [@vdw1], the measure $\hat{\beta}=\sum_{p,q\leq N} \left(\delta\left(\frac{p}{N},\frac{q}{N}\right) \times \frac{1}{N} \sum_{i=1}^N{{\mathbf{1}_{\left\{ R_{i}=p, S_{i}=q \right\}}} } \right)$ converges weakly to the probability measure of $(F_X(X),F_Y(Y))$.\
If we choose the subsample size $n$ such that $n^2/N {{\stackunder{N \rightarrow \infty}{\longrightarrow}}} 0 $, then drawing the subsamples with or without replacement is equivalent. More precisely, the proportion of subsamples with an observation (of the sample) drawn twice tends to 0.\
Then, for [**each subsample**]{}, the $\hat{\beta}$ measure converges weakly to the same measure, which is the $\hat{\beta}$ measure for the sample. Since $\hat{\gamma}$ is the average of all these $\hat{\beta}$ measures, $\hat{\gamma}$ converges weakly to the probability measure of $(F_X(X),F_Y(Y))$.\
Unfortunately, we need to prove the density convergence, which seems to be more difficult. Proving the convergence of a density on a finite support is tedious. One reason is that many tools used to prove convergences are designed to manage in the same way discrete and continuous issues.
Goodness-of-fit test
====================
We follow here [@sca]: this article empirically studies the power of a test based on kernel copula density estimation. To empirically study the power of a test, one simulates a lot of samples where $H_0$ is false, and uses the test. Then, the power of the test is the proportion of samples with $H_0$ rejected.
Benchmark
---------
We study two cases.\
In the first one, we simulate a sample with the following distribution: each observation is drawn with probability 0.5 from a Frank copula with parameter $\theta$ , and in the remaining case from a Student copula with 4 degrees of freedom and a 0.95 correlation parameter. The test is done to decide whether this sample is drawn from a Frank copula or not. The parameter $\theta$ has the following values: 1, 2 and 3. The Frank copula is defined by: $$C(u,v)= \frac{1}{\theta} \log{\left(1+\frac{(e^{-\theta u} -1)(e^{-\theta v} -1)}{e^{-\theta} -1}\right) }$$ The second case is the same, but with the Gaussian copula replacing the Frank copula. The copula is such that, with gaussian margins, the Pearson correlation is 0.17, 0.32 or 0.47.
Results
-------
We test whether a sample is drawn from [**a**]{} Frank copula or not. The first step is to estimate, using maximum likelihood, the parameter of a candidate Frank copula given this sample. In the following, we note this copula LFC (for Likeliest Frank Copula). Then, testing the goodness-of-fit to LFC consists in:
1. \[distance\] defining and choosing a distance,
2. \[estnul\] estimate the distribution of the distances between LFC and samples drawn from LFC,
3. compute the distance between LFC and the original sample,
4. reject goodness-of-fit if this distance is larger than the $(1-\alpha)$ quantile of the distribution estimated in \[estnul\].
For point \[estnul\], we will use simulation (as in [@sca]). The difficulty arises about point \[distance\] since LFC and the samples have not the same nature.\
To be able to define a distance, we will compare the discrete densities derived from both LFC and the samples. Unfortunately, we have no numerical expression of the discrete density derived from a given theoretical copula (except for independent copula). That is why we use simulation: we compute the discrete density for a big (for example with $1000 \times N$ observations) sample drawn from LFC. Then, we only have to compare discrete densities, and we use the Kullback divergence.\
Obviously, we need to choose a subsample size. In the following, we tried some, and chose the best one (as in [@sca] for kernel bandwidth). The results are summarized in table \[puissances\_gof\].
Copula type parameter $N$ discretized copula subsample size kernel density
------------- ----------- ----- -------------------- ---------------- ----------------
Gaussian 0.17 50 0.56 15 0.39
Gaussian 0.32 50 0.38 13 0.34
Gaussian 0.47 50 0.21 10 0.21
Frank 1 50 0.45 15 0.26
Frank 2 50 0.28 14 0.21
Frank 3 50 0.33 12 0.13
Gaussian 0.17 200 1.00 13 1.00
Gaussian 0.32 200 0.98 12 0.99
Gaussian 0.47 200 0.86 11 0.96
Frank 1 200 1.00 13 0.96
Frank 2 200 0.96 12 0.83
Frank 3 200 0.91 11 0.60
: \[puissances\_gof\]Power with optimal choice of the subsample size
Using ranks and subsampling seems to be more efficient than using kernels, at least on this example. Furthermore, the power differences become smaller when the size of the sample increases: it would be interesting to know how would evolve the differences for smaller samples.
Independence test
=================
Benchmark
---------
We will study:
1. a monotonous dependence: a simple linear relation $y=a \cdot x + \epsilon$
2. a non-monotonous dependence: a quadratic relation $y=a \cdot x^2 + \epsilon$
3. an non-functional dependence $(x,y)= a \cdot (\cos(2\pi u),\sin(2\pi u))+(\epsilon_1,\epsilon_2)$ (called [“[donut]{}”]{} in the following because of the form of the point cloud).
4. a dependence modifying only the volatility $y= (1+a \cdot |x|)\cdot\epsilon$ (called [“[butterfly]{}”]{}).
The variables $\epsilon$ in dependences 1, 2, 3 and 4 and the variables $x$ in dependences 1, 2 and 4, are normally distributed (null mean and variance unity). The variable $u$ is uniformly distributed on $[0,1]$ in the dependence 3.\
We will test in the cases of 30 and 300 observations.\
We will compare the powers of the new test, Deheuvels test, and a [“[smart]{}”]{} test. This test uses an additional knowledge about the form of the dependence (but not on the value of the parameter $a$). For the dependence 1, this test will be the Pearson test on $y$ and $x$, for dependence 2 on $y$ and $x^2$. For the dependence 3, we will test use the Komogorov-Smirnov test to know whether $x^2+y^2$ is exponentially distributed (true if $a=0$). For the dependence 4, we will use Pearson test on $|x|$ and $|y|$.\
For Deheuvels test, we use the expression of the statistic given in [@GQR]. The significativity thresholds have been computed by simulation, but they can also be checked in table 1 of [@GR].\
The value of $a$ will be selected so that the [“[smart]{}”]{} test has a power of 0.5 or 0.9, with level 0.05. The powers are calculated on 1000 simulations.
Results
-------
The test statistic is the Kullback divergence between the estimated density and the uniform density. The significativity thresholds are computed by simulation (using 3000 simulations).\
This table \[puissances\_indep\] gives the powers obtained for three tests: new test (column [“[new]{}”]{}), Deheuvels test and [“[smart]{}”]{} test. The size of the subsample is selected to maximize the power, which is obviously possible only because we have a big number of samples. In a practical setting, such a choice would be impossible.\
So, it is obvious we need to address the subsample choice issue. We tried, [**on the 8 dependences simulated here (for each sample size)**]{}, a [**minimax-regret**]{} policy. In other words, if $P(s,d)$ denotes the test power for dependence $d$, and subsample size $s$, we sought the value of $s$ realizing: $$\min_s \max_d \left[\max_s(P(s,d))-P(s,d)\right]$$ The chosen size is 8 for 30 observations and 10 for 300. The powers are in the column [“[new minimax]{}”]{} of table \[puissances\_indep\].\
Dependence form $n$ $a$ [“[smart]{}”]{} Deheuvels new optimal size new minimax
----------------- ----- ------ ----------------- ----------- ------ -------------- -------------
Linear 30 0.38 0.5 0.40 0.42 2 0.31
Quadratic 30 0.29 0.5 0.08 0.21 10 0.20
Donut 30 2.90 0.5 0.04 0.08 15 0.05
Butterfly 30 0.87 0.5 0.07 0.22 15 0.18
Linear 30 0.67 0.9 0.82 0.86 2 0.76
Quadratic 30 0.57 0.9 0.16 0.58 9 0.57
Donut 30 3.76 0.9 0.04 0.21 13 0.13
Butterfly 30 4.9 0.9 0.11 0.60 14 0.48
Linear 300 0.11 0.5 0.50 0.44 2 0.34
Quadratic 300 0.08 0.5 0.10 0.20 17 0.17
Donut 300 1.53 0.5 0.06 0.18 19 0.14
Butterfly 300 0.16 0.5 0.07 0.22 20 0.15
Linear 300 0.19 0.9 0.86 0.85 4 0.78
Quadratic 300 0.14 0.9 0.20 0.55 17 0.52
Donut 300 1.79 0.9 0.07 0.40 19 0.07
Butterfly 300 0.32 0.9 0.09 0.55 21 0.45
: \[puissances\_indep\]Power with optimal choice of the subsample size
We have to emphasize that the relatively good power of the minimax-regret policy is not a conclusive result (oppositely, a small power would have been conclusive). It only shows that, if we are able to define it in a general and formal way, a minimax-regret policy could lead to good results.\
One important conclusion is that the cumulated density function is not the right tool to see a non-monotonic dependence. We need something different, for example a density (even a discrete one).\
Another important teaching (still not sure, since we did not take account of all dependencies in the minimax-regret policy) is that we can test independence, even with a very poor prior knowledge about the possible dependence. For example, the first line of the table shows that, if we assume that the dependence is linear, we are able to detect it in half of the cases. If we only have a minimax-regret policy, using density estimation, we detect it in a third of the cases. The power loss is not unbearable.
Conclusion and further work
===========================
The use of kernels for density estimation implies two choices: the kernel, the bandwidth. We have seen in this article that, in order to estimate copula measures, one can bypass the first one, using ranks and subsampling. In exchange, we have to choose a subsample size: this is in fact a problem very similar to the bandwidth choice. We have then reduced the overall difficulty.\
The `R` and `C` code used to this simulation study is available on demand, at anyone of the addresses given in first page.\
A theoretical article showing the convergence of the estimation is under work. Other further works would be:
- studying the limiting distribution of the statistic, or at least large deviations of this statistic;
- studying a minimax strategy to choose the subsample size.
A nonparametric test of independence; Paul Deheuvels; Publications de l’ISUP 26 (1981) A Kolmogorov-Smirnov type test for independence and multivariate samples; Paul Deheuvels; Rev. Roum. Math. Pures et Appl., Tome XXVI (1981) Goodness-of-fit tests for copulas; Jean-David Fermanian; Journal of Multivariate Analysis 95 (2005) Local efficiency of a Cramér-von Mises test of independence; Christian Genest, Jean-François Quessy, Bruno R[é]{}millard; Journal of Multivariate Analysis 97 (2006) Tests of Independence and Randomness Based on the Empirical Copula Process; Christian Genest, Bruno R[é]{}millard; Test 13 (2004) [Dependence structures and limiting results, with applications in finance and insurance; A. Charpentier; PhD thesis of Katholieke Universiteit Leuven (2006)]{} Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters; O. Scaillet; Journal of Multivariate Analysis 98 (2007) Weak Convergence and Empirical Processes; A.W. van der Waart, J. Wellner; Springer (1996)
[^1]: Jerome dot Collet at edf dot fr or Jerome dot Collet at ensae dot org
|
[**High-Energy Neutrino Astronomy:**]{} Christian Spiering$^{\dag}$\
[email:`[email protected]` ]{}
**Abstract**
[With the identification of a diffuse flux of astrophysical (“cosmic”) neutrinos in the TeV-PeV energy range, IceCube has opened a new window to the Universe. However, the corresponding cosmic landscape is still uncharted: so far, the observed flux does not show any clear association with known source classes. In the present talk, I sketch the way from Baikal-NT200 to IceCube and summarize IceCube’s recent astrophysics results. Finally, I describe the present projects to build even larger detectors: GVD in Lake Baikal, KM3NeT in the Mediterranean Sea and IceCube-Gen2 at the South Pole. These detectors will allow studying the high-energy neutrino sky in much more detail than the present arrays permit.]{} [PACS: 95.55Vj, 95.85Ry]{}
Introduction
============
The first conceptual ideas how to detect high energy neutrinos date back to the late fifties. The long evolution towards detectors with a realistic discovery potential started in the seventies, by the pioneering works in the Pacific Ocean close to Hawaii (DUMAND). The DUMAND 1978 design envisaged an array of about 20000 photomultipliers spread over a 1.26 cubic kilometer volume of water. This project was terminated in 1995, but the baton was taken by the projects NT200 in Lake Baikal, AMANDA at the South Pole, ANTARES in the Mediterranean Sea and, again at the South Pole, IceCube (see for detailed information on the history and on corresponding references [@Spiering-History]). But only now, half a century after the first concepts, a cubic kilometer detector is in operation: IceCube at the South Pole. With the discovery of a flux of high-energy neutrinos of astrophysical origin (“cosmic neutrinos”) in 2013 [@Science-2013], the IceCube Neutrino Observatory has opened a new window to the Universe of non-thermal cosmic processes. A next generation of arrays is under construction or planned: KM3NeT in the Mediterranean Sea [@KM3NeT], the Gigaton Volume Detector GVD in Lake Baikal [@Baikal], and IceCube-Gen2 [@Gen2].
The primary goal of these detectors is identifying the sources of high-energy cosmic rays. In contrast to charged particles, neutrinos are not deflected in cosmic magnetic fields and keep their direction; in contrast to gamma rays they provide a direct, water-tight prove for the acceleration of hadrons in the emitting sources. This makes them unique tracers of sources of cosmic rays. On the other hand, due to their small interaction cross section they are difficult to detect: The “neutrino effective area” of the 1 km$^3$ IceCube detector (essentially the geometrical area multiplied with the interaction probability, the trigger efficiency and the transparency to neutrinos of the Earth) is less than 1m$^2$ at 1TeV and of the order of 100m$^2$ at 100TeV [@point-sources]. It is therefore no surprise that it took several decades to detect cosmic neutrinos.
Neutrino telescopes are multi-purpose detectors. Apart from investigating cosmic neutrinos, they exploit atmospheric neutrinos to study neutrino oscillation, to search for sterile neutrinos or to test fundamental laws of physics. They are used to search for neutrinos from Dark Matter annihilations in the Sun or the Galactic halo, to search for exotic particles like magnetic monopoles, or to study muons from cosmic-ray induced air showers.
This paper focuses to the search for neutrinos from cosmic acceleration processes. I will further focus to the developments at Lake Baikal and at the South Pole where I have been involved myself over a long period.
From Baikal NT200 to IceCube
============================
NT200 and AMANDA
----------------
First test deployments in Lake Baikal started in 1981. The construction of the NT200 detector was started in 1993, about 30 km South-West from the outflow of Lake Baikal into the Angara river, at a distance of 3.6km to shore and at a depth of about 1.1km, and was completed in 1998. NT200 was an array of 192 optical modules (glass spheres containing large photomultipliers) at eight strings, 72m in height and 43m in diameter (see Fig.\[nt200\]). Actually, this is not much more than twice the size of Super-Kamiokande. First upward going muons (i.e.neutrino events) were found already with the 3-string version from 1993, and then with the 4-string version from 1996: the first “underwater neutrinos” ever! This was the first proof of principle to detect neutrinos in open media and a breakthrough for the field.
![: The Baikal Neutrino Telescope NT200. [*Right*]{}: One of the first upward moving muons from a neutrino interaction recorded with the 4-string stage of the detector in 1996 [@Baikal-atm-Balkanov-1999]. The Cherenkov light from the muon is recorded by 19 channels.[]{data-label="nt200"}](figures/nt200andgoldplated.pdf){width="0.80\linewidth"}
In 1988, a new, spectacular idea appeared on stage: to use Antarctic ice instead of water as target and as detector medium. The project was named AMANDA (Antarctic Muon And Neutrino Detection Array). AMANDA was deployed some hundred meters from the Amundsen-Scott station, first at a too shallow depth (where bubbles disturb light propagation), and then from 1996 – 2000 at depths between 1500 and 2000m. It consisted of 677 optical modules at 19 strings. AMANDA was switched off in April 2009, after more than 9 years of data taking in its full configuration and with 6959 neutrino events collected. Naturally this sample was dominated by atmospheric neutrinos. No indication of point sources was found, and no excess of high-energy events which might have pointed to an admixture of cosmic neutrinos. Figure \[A-B-skyplot\] shows – as a kind of sentimental reminiscence – a combination of NT200 and AMANDA data compiled in 2005 (including all NT200 data and 2 years AMANDA data).
![A skyplot in Galactic coordinates, combining neutrinos detected by NT200 (circles) and AMANDA (crosses). Compiled in 2005 [@Igor-Christian-skyplot]. []{data-label="A-B-skyplot"}](figures/Amanda+Baikal.png){width="0.65\linewidth"}
AMANDA provided record limits on fluxes for cosmic neutrinos, be it for diffuse fluxes (where the much smaller NT200 could compete for some years) and for point sources – steady as well as transient. AMANDA also extended the measured spectrum of atmospheric neutrinos by nearly two orders of magnitude, from a few TeV to 200TeV.
Mediterranean Projects
----------------------
First site studies in the Mediterranean Sea were performed in 1989, leading to the NESTOR project, with the goal to install towers of hexagonal floors close to Pylos in Greece. The first floor was deployed only 15 years later, in 2004. But the project was further and further delayed and the community split into a Greek project (NESTOR), an Italy-based project (NEMO) and a project in France (ANTARES). Only the French site made it to a working – and actually excellently working! – detector.
The construction of ANTARES started in 2002 with the deployment of a shore cable. The detector in its final 12-string configuration was installed in 2006–2008 and has been operational since then. The strings have lateral distances of 60–70m, and each of them carries 25triplets of optical modules at depths of 2.1–2.4km.
ANTARES has demonstrated that a stable operation of a deep-sea detector is possible. Similar in size to AMANDA, it has collected more than 8000 upward-going muon tracks over eight years of operation. With its excellent view of the Galactic plane and good angular resolution, the telescope could constrain the Galactic origin of the cosmic neutrino flux reported by IceCube. ANTARES has explored he Southern sky and in particular central regions of our Galaxy searching for point-like objects, for extended regions of emission, and for signals from transient objects selected through multi-messenger observations [@Antares-results].
IceCube
-------
Like AMANDA, the IceCube Observatory [@IceCube] is located at the geographical South Pole. It consists of the main IceCube array with its subarray DeepCore and the surface array IceTop. IceCube comprises 5160 digital optical modules (DOMs) installed on 86 strings at ice depths of 1450 to 2450m and covers 1 km$^3$ of ice. A string carries 60 DOMs. DeepCore, a high-density sub-array of eight strings at the center of IceCube, has smaller spacing and DOMs with more sensitive photomultipliers than IceCube and sits in the midst of the clearest ice layers. This results in a threshold of about 10GeV and opens a new venue for oscillation physics. The threshold of the full IceCube detector is about 100GeV. In its final configuration, IceCube takes data since spring 2011, with a duty cycle of more than 99%. It collects about $10^5$ clean neutrino events per year, with nearly 99.9% of them being of atmospheric origin.
Where do we stand?
==================
Diffuse Fluxes
--------------
It has been predicted since long that the first evidence for extragalactic cosmic neutrinos would be provided by a diffuse flux rather than by single-source signals [@Lipari]. The first tantalizing hint to cosmic neutrinos in IceCube came from two shower-like events with energies $\approx 1$ PeV, discovered in 2012 and dubbed “Ernie” and “Bert” [@Ernie]. A follow-up search of the same data (May 2010 to April 2012) with a lowered threshold (30 TeV) provided 25 additional events. This analysis used only events starting in a fiducial volume of about 0.4 km$^3$ (High Energy Starting Events, or “HESE”), using the other 60% of IceCube as veto against all sorts of background. Energy spectrum and zenith angle distribution of the 27 events excluded an only-atmospheric origin with $4.1\sigma$ and suggested that about 60% were of cosmic origin, at energies above 100 TeV even about 80% [@Science-2013]. A four-year data set with 54 neutrinos provided another shower-like PeV event (deposited energy $\approx$ 2 PeV) and confirmed a dominant cosmic contribution with nearly $6.5\sigma$. Very recently, the results from a six-year sample have been presented [@ICRC-Kopper], with 82 events above 30 TeV. Figure 3 shows the energies deposited by these events inside IceCube.
![ Distribution of the energy deposited by 82 events from the six-year HESE analysis. Backgrounds of atmospheric origin come from punch-through down-going muons and from atmospheric neutrinos. While the flux of neutrinos from $\pi$ and K decays is well known (blue region), the neutrino flux from charm decays in the atmosphere is uncertain and dominates the uncertainty of all background sources (gray region with 1$\sigma$ uncertainties). The best-fit astrophysical spectra are shown as gray lines, for a single power-law spectrum as solid line, for a two power-law model as dashed line. See [@ICRC-Kopper] for details. []{data-label="l-HESE-spectrum"}](figures/HESE-spectrum.pdf){width="50.00000%"}
A $5.6\sigma$ excess of high-energy cosmic neutrinos is also seen in the spectrum of secondary muons generated by neutrinos that have traversed the Earth, with a zenith angle less than 5 degrees above the horizon (“upward through-going muons”[@muon-7years]). Figure 4 shows the median neutrino energy. It is calculated for each energy deposited by the muon in the detector, assuming the best-fit spectrum. The highest energy muon has deposited $2.6\pm 0.3$ PeV inside the instrumented volume, which corresponds to a most probable neutrino energy of about 9 PeV.
![ Spectrum of the median neutrino energy derived from the energy deposit of through-going muons with zenith angles less than 5 degrees above horizon (8 years sample [@muon-8years]). []{data-label="l-muon-energy-spectrum"}](figures/muon-energy-spectrum.png){width="65.00000%"}
While both analyses (HESE and through-going muons) have reached a significance for a strong non-atmospheric contribution of more than 5$\sigma$, the spectral indices $\gamma$ of the astrophysical flux from both analyses disagree: $\gamma = 2.92 \pm 0.33/0.29$ for the HESE events (unbroken spectrum $E^{-\gamma}$) and $\gamma = 2.19 \pm 0.10$ [@muon-8years] for the throughgoing muons. Adding two more years to the HESE sample has resulted in an even softer energy spectrum since all events of the recent two years have energies below 200 TeV. Fig.\[l-spectrum-combined\]a shows the two fits under the assumptions of a single-power law. The possibility that all but the three PeV HESE events emerge from pion/Kaon/charm decays in the atmosphere is excluded by the zenith angle distribution.
In [@combined] the flavor ratio of the astrophysical neutrino flux has been investigated. It is consistent with an observed flavor ratio $\nu_e : \nu_{\mu} : \nu_{\tau} = 1 : 1 : 1$ and also with source neutrino ratios 1:2:0 (pion decay) and 0:1:0 (pion decay with suppressed muon decay) while largely excluding 1:0:0 (neutrinos from neutron decay).
The hope to see any clustering of the HESE and muon-track events at highest energies has not fulfilled. An initial indication of clustering of HESE events close to the Galactic center has vanished with more statistics. In addition, ANTARES has looked to a point source at the position of IceCube’s initial excess and could exclude that it is due to a point source, assuming that the extension of the source does not exceed 0.5 degrees and that the spectrum follows an $E^{-2}$ shape [@Antares-check].
A recent IceCube analysis has used 7 years of the medium-energy $\nu_{\mu}$ data (which are optimized to search for point sources, see next section) to set constraints on the diffuse emission of neutrinos from the Galactic plane [@ICRC-Galacticplane]. The resulting limits are shown in Fig.5b and compared to the flux of the HESE and highest-energy $\nu_{\mu}$ data. They exclude that more than 14% of the observed diffuse astrophysical flux come from the Galactic plane. However, the limit is not far from model predictions (gray band). Joining IceCube and ANTARES data and exploiting cascade-like events in addition to the $\nu_{\mu}$ sample may drive the sensitivity into the region predicted by KRA models.
![: Best-fit of the per-flavor neutrino fluxes as a function of energy. The black points with $1\sigma$ uncertainties are extracted from a combined likelihood fit of all background components together with an astrophysical flux component with an independent normalization in each band (assuming an $E^{-2}$ spectrum within each band and atmospheric neutrino and muon fluxes subtracted). The best-fit conventional flux and the upper limit for prompt neutrinos are shown separately, not taking into account the HESE self-veto which actually reduces their contribution. The blue band shows the $1\sigma$ uncertainties of a single power-law fit to the HESE data. The pink band shows the fit for the muon neutrino data, again with $1\sigma$ uncertainties. Its length indicates the approximate range providing 90% of the significance of this analysis [@ICRC-Kopper]. [*Bottom*]{}: Upper limits on the three flavor neutrino flux from the Galaxy with respect to KRA model predictions [@KRA] and the measured astrophysical flux [@ICRC-Galacticplane]. Dots, yellow and green bands have the same meaning as the bands in the top figure. []{data-label="l-spectrum-combined"}](figures/spectrum-combined.pdf "fig:"){width="70.00000%"} ![: Best-fit of the per-flavor neutrino fluxes as a function of energy. The black points with $1\sigma$ uncertainties are extracted from a combined likelihood fit of all background components together with an astrophysical flux component with an independent normalization in each band (assuming an $E^{-2}$ spectrum within each band and atmospheric neutrino and muon fluxes subtracted). The best-fit conventional flux and the upper limit for prompt neutrinos are shown separately, not taking into account the HESE self-veto which actually reduces their contribution. The blue band shows the $1\sigma$ uncertainties of a single power-law fit to the HESE data. The pink band shows the fit for the muon neutrino data, again with $1\sigma$ uncertainties. Its length indicates the approximate range providing 90% of the significance of this analysis [@ICRC-Kopper]. [*Bottom*]{}: Upper limits on the three flavor neutrino flux from the Galaxy with respect to KRA model predictions [@KRA] and the measured astrophysical flux [@ICRC-Galacticplane]. Dots, yellow and green bands have the same meaning as the bands in the top figure. []{data-label="l-spectrum-combined"}](figures/galplane-first.pdf "fig:"){width="70.00000%"}
Search for steady point sources
-------------------------------
For the standard steady-source search, a sample of through-going muons with good angular resolution (median error smaller $1\deg$) is selected. In the lower hemisphere, the Earth acts as filter against muons generated in the atmosphere. In the upper hemisphere, a radical energy cut removes most of the atmospheric muons which have a rather soft energy spectrum, but naturally also rejects all but the most energetic cosmic neutrinos. Therefore only hard-source spectra would result in a significant number of events from the upper hemisphere (for IceCube: South).
Figure \[l-skyplot7years\] shows the all-sky plot of seven years data, with 422 791 upward muons from neutrino interactions and 289 078 downward muons, the latter almost all from atmospheric showers. The downward sample contains also 961 tracks starting inside the detector, i.e. generated in neutrino interactions [@point-sources].
![ All-sky plot of seven years IceCube data in equatorial coordinates. Shown is the negative logarithm of the pre-trial p-value, assuming no clustering as null-hypothesis [@point-sources]. []{data-label="l-skyplot7years"}](figures/skyplot7years.png){width="85.00000%"}
No significant excess is found, resulting in the flux constraints show in Figure \[l-pt-limits\]. Apart from sensitivities and limits for selected sources, the discovery potential is shown, i.e. the flux that would lead to a $5\sigma$ discovery of a source in 50% of the cases.
One can then compare these values to predictions for selected sources. Fig.\[l-limitsblazars\] compares our sensitivities and the obtained 90% upper limits to predictions [@Petropoulou] for three blazars. The limits are within a factor 5 of the predictions, for Mkr 421 even slightly below predictions. Similar relations hold for the Crab nebula – always optimistically assuming that the gamma flux observed from these sources is basically due to $\pi^0$ decay and not to inverse Compton scattering.
![ Discovery potential and sensitivity (red solid and dashed, respectively) versus declination, assuming an unbroken $E^{-2}$ neutrino spectrum. Upper limits of 32 pre-selected source candidates are given as red crosses, the blue line represents the upper limit for the most significant spots in each half of the sky (actual positions of the spots are given by blue stars). The gray line shows the results from ANTARES. See [@point-sources] for details. []{data-label="l-pt-limits"}](figures/limitsblazars.png){width="80.00000%"}
![Differential energy spectra versus neutrino energy for blazars of the BL Lac type compared to model predictions [@Petropoulou]. Thick lines give the 90% upper limits from IceCube, thin lines represent the model. The sensitivities of the Icecube search are shown as dashed line. 90% upper limit and sensitivity are shown for the energy interval where 90% the events originate that are most signal-like [@point-sources]. []{data-label="l-limitsblazars"}](figures/limits-vs-declination.png){width="75.00000%"}
From these figures one could conclude that an improved angular reconstruction and twice more data could bring us close to discovery. For blazars, however, this hope is downsized by various blazar stacking analyses, none of them yielding an excess in the directions of blazars. The most recent one [@blazarstacking] indicates that only 4-6% of the observed diffuse astrophysical muon neutrino flux could come from blazars.
Search for transient sources
----------------------------
To improve the signal-to-background ratio one can search for transient signals, preferentially in coincidence with an observation in electromagnetic waves. Examples are flares of Actice Galactic Nuclei (AGN) or Gamma-Ray Bursts (GRB). GRBs are interesting objects since there are models which assume that they are the dominant source of the measured cosmic-ray flux at highest energies, either by neutron escape [@Ahlers] or by escape of both neutrons and protons [@WB-GRB] from the relativistic fireball. Naturally models where protons a kept in the acceleration region and only neutrons escape and constitute the observed cosmic ray flux give a higher neutrino/cosmic ray ratio.
All three collaborations – Baikal, ANTARES and IceCube – have searched for neutrinos in local and spatial coincidence with GRBs. In particular IceCube limits on neutrinos from GRBs have drastically improved over the recent years. A recent analysis has combined the searches for spatial and temporal coincidences of upward and downward tracks and cascade-type events with 1172 GRBs. No significant correlations between the gamma-ray signals and neutrinos have been observed. Figure \[l-GRB\] shows exclusion contours for double broken power-law spectra, with breaks from $E^{-1} \times \epsilon_b$ to $E^{-2}$ at energy $\epsilon_b$, and from $E^{-2}$ to $E^{-4} \times (10\epsilon_b)^2$ at an energy $10\epsilon_b$.
![Excluded regions for 99%, 90% and 68% confidence level of the generic double broken power law neutrino spectrum as a function of the first break energy $\epsilon_b$ and per-flavor quasi-diffuse flux normalization derived from upward and downward muon tracks and all-sky cascades. []{data-label="l-GRB"}](figures/GRB-limits-WB-1.pdf){width="65.00000%"}
Both models, those with cosmic ray escape via neutrons and those which allow additionally for cosmic ray escape via protons, are excluded at over 90% confidence level, with most of the model assumption phase space excluded at over the 99% confidence level, greatly constraining the hypothesis that GRBs are significant producers of ultra-high energy cosmic rays in the prompt GRB phase.
30 years after the discovery of supernova SN1987 it is worth to highlight that IceCube runs a supernova trigger, with a duty time of more than 99%. The trigger reacts to a collective rise in photomultiplier counting rates on top of the dark-noise rate. This rise would be due to the feeble signals from $\nu_e$ reactions close to a photomultiplier. A 1987A-type supernova at 30 kpc distance (edge of the Galaxy) would lead to an collective-rate enhancement with a significance of about 20 standard deviations, and even at distance of the Large Magellanic Cloud (50 kpc) the excess would reach $6-7\sigma$ [@SN]. IceCube is part of the SuperNova Early Warning System (SNEWS [@SNEWS]), together with the underground neutrino detectors Borexino, Super-Kamiokande, LVD and Kamland which, in the case of a significant coincidence from more than one of the detectors, would alarm the astronomers community. However, no significant neutrino signal has been recorded yet, neither with the analogous trigger of IceCube’s predecessor AMANDA nor with that of IceCube.
Real-time alert and follow-up programs
--------------------------------------
With no steady sources of high-energy neutrinos observed so far, neutrinos produced during transient astrophysical events are a viable alternative. High-energy neutrinos from the prompt phase of GRBs or MeV neutrinos from a supernova collapse as discussed in the previous section are just two examples. Coincident detections could enhance the significance of the IceCube observation and, more generally, contribute to the mosaic of informations from different messengers, providing a more complete picture of the source. Since IceCube and ANTARES have nearly $4\pi$ acceptance (depending on energy), they could could trigger detections with pointing devices like optical or gamma-ray telescopes, which otherwise would have been missed.
Both collaborations run a number of high-energy alert and follow-up programs [@alerts; @alertsAnt] which react to particular single events. In the case of IceCube, neutrino alert candidates are identified in real-time at the South Pole. A brief message sent to the North is automatically issued to the Gamma-Ray Coordinates Network (GCN [@AMON]) via the Astrophysical Multimessenger Observatory Network (AMON [@AMON]). In parallel, quality checks are applied and the directional and energy reconstruction refined. Results from that are completed within a few hours and lead to an updated alert notification in the form of a CGN circular. IceCube runs two of these alerts: a “HESE Track alert” which is issued if a track-like HESE event is recorded (4.8 events expected per year, with 1.1 being of astrophysical origin) and an “EHE Track Alert” which is based on a selection which originally targets cosmological neutrinos (10 PeV to 1 EeV) but here is modified to be sensitive down to 500 TeV (about 5 alerts per year).
Apart from these public alerts, IceCube also issues alerts to optical, X-ray and gamma-ray observatories which are based on neutrino [*multiplets*]{}. These alerts are based on individual agreements with these observatories. The multiplets can be due to phenomena on the second-to-minute scale (high-energy neutrinos from relativistic jets in SN or GRB), or to phenomena of the hour-to-week scale (like AGN flares). None of the alerts yet has led to a significant correlation, although at least two cases have generated some initial excitement. The one [@doublet] was a neutrino doublet detected in March 2012 which triggered follow-up observations by the Palomar Transient Factory (PTF). PTF found a Type IIn supernova within an error radius of $0.54\deg$ of the direction of the doublet. A Pan-STARRS1 survey, however, showed that its explosion time was at least 158 days before the neutrino alert, so that a causal connection is unlikely. The second case [@triplet] was the first triplet: three neutrinos arriving within 100 s of one another at February 17, 2016. Follow-up observations by SWIFT’s X-ray telescope, by ASAS-SN, LCO and MASTER at optical wavelengths, and by VERITAS in the very-high-energy gamma-ray regime did not detect any likely electromagnetic counterpart. In a refined reconstruction, the directions of the events changed slightly, so that the triplet turned to a double-doublet (error circle of the one event overlapping with those of the two others, but not all three with each other). Still, these two cases impressively illustrate the potential of and challenges for future follow-up campaigns. Although no significant correlations have been detected so far, the IceCube/ANTARES alerts and the triggered electromagnetic-domain observations herald the era of multi-messenger observation. This remark also applies to the follow-up programs where IceCube scrutinizes its own data to search for correlations with signals from Gravitational Waves [@GW].
Where do we go?
===============
Four years after the detection of cosmic neutrinos, we have learned a lot about their spectrum and flavor composition. We have learned that blazar jets and GRBs can contribute only a small fraction to the observed astrophysical neutrino flux. The spectral features of this flux (single power law or two power law) open new questions about the contributing source classes. No individual sources have been detected yet. The non-observation of neutrinos coinciding with GRBs strongly constrains models which attribute the highest-energy cosmic rays to GRBs. Neutrino events possibly related to supernova explosions have been observed, although with a non-negligible probability for a chance occurrence. No neutrinos have been observed that could be attributed to the GZK effect [@BZ], but the non-observation starts constraining evolution scenarios for ultra-high energy cosmic rays sources (not addressed in this report).
IceCube continues collecting data. A twofold statistics combined with improved directional precision, also for cascade-like events, and better understanding of systematics effects will considerably improve the understanding of what has been observed so far and may even provide first detection of individual (point-like or extended) sources. IceCube’s capabilities, however, are limited by its size, the chance to detect neutrinos from the central part of our Galaxy are constrained by its location at the South Pole.
The next important steps are being done at the Northern hemisphere: GVD in Lake Baikal and KM3NeT in the Mediterranean Sea.
Baikal-GVD
----------
Baikal-GVD (Gigaton Volume Detector) is configured in “clusters”, where each cluster consists of eight strings, instrumented over a length of 520m with 36 optical modules. The OMs house 10” Hamamatsu photomultipliers with a high-sensitive photocathode. After an extensive period of in-situ tests of single components and prototype strings, a first cluster with 24 OMs per string was deployed in Spring 2015. One year later, it was upgraded to a full cluster with 36 PMs per string, and in Spring 2017 a second cluster was added. First preliminary results of the 2016 cluster have been presented at the ICRC 2017 [@Baikal-ICRC].
Baikal-GVD will be built in two phases. Phase-1 will consist of eight clusters, each 120m in diameter, with lateral distances of 300m (see Fig.\[GVD\]). The effective volume for cascades in the 10-100 TeV energy range will be about 0.4km$^3$, the sensitivity to muons is negligible below 1TeV but rapidly raises in the multi-TeV range. Phase-1 is financed and planned to be completed in 2020/21. In a second phase, Baikal-GVD will be extended to 18 clusters and then surpass the cubic kilometer benchmark.
![Schematic view of phase-1 of Baikal-GVD, consisting of 8 clusters, each with 120m diameter and 520m height. A cluster consists of eight strings with 36 optical modules along each string. []{data-label="GVD"}](figures/GVD.pdf){width="80.00000%"}
KM3NeT
------
KM3NeT has two main, independent objectives: a) the discovery and subsequent observation of high-energy cosmic neutrino sources and b) precise oscillation measurements and the determination of the mass hierarchy of neutrinos. For these purposes the KM3NeT Collaboration plans to build an infrastructure distributed over three sites: off-shore Toulon (France), Capo Passero (Sicily, Italy) and Pylos (Peloponnese, Greece). In a configuration to be realized until 2020/22, KM3NeT will consist of three so-called building blocks (“KM3NeT Phase-2”). A building block comprises 115 strings, each string with 18 optical modules.Two building blocks will be sparsely configured to fully explore the IceCube signal with a comparable instrumented volume, different methodology, improved resolution and complementary field of view, including the Galactic plane. These two blocks will be deployed at the Capo Passero site and are referred to as ARCA: Astroparticle Research with Cosmics in the Abyss. The third building block will be densely configured to precisely measure atmospheric neutrino oscillations. This block, being deployed at the Toulon site, is referred to as ORCA: Oscillation Research with Cosmics in the Abyss (see Fig.\[KM3NeT\]).
![The two incarnations of KM3NeT. The two ARCA blocks (bottom) have diameters of 1km and a height of about 600m and focus to high-energy neutrino astronomy. ORCA (top) is a shrinked version of ARCA with only 200m diameter and 100m height. Both ARCA and ORCA have 115 strings with 18 optical modules (OMs) per string. Top left, a drawing of an OMs is shown. Each OM houses 31 small photomultipliers. []{data-label="KM3NeT"}](figures/KM3NeT.pdf){width="65.00000%"}
A novel concept has been chosen for the KM3NeT optical module: The 43cm glass spheres of the DOMs will be equipped with 31 PMTs of 7.5cm diameter, with the following advantages: a) The overall photocathode area exceeds that of a 25cm PMT by more than a factor three; b) The individual readout of the PMTs results in a very good separation between one- and two-photoelectron signals which is essential for online data filtering; c) some directional information is provided. This technical design has been validated with in situ prototypes. A cross-sectional view of this DOM is shown at the top of Fig.\[KM3NeT\].
With a fully equipped ARCA, IceCube’s cosmic neutrino flux could be detected with high-significance within one year of operation. In practise the detector will be deployed in stages allowing to reach the one-year sensitivity of two clusters much before the second cluster is fully installed. Actually the same is true for Baikal-GVD, with a good chance that GVD will find cosmic neutrinos before ARCA. ORCA could determine the neutrino mass hierarchy with at least 3$\sigma$ significance after three years of operation, i.e. as early as 2023.
IceCube-Gen2
------------
The progress from IceCube over the next decade is limited by the modest numbers of cosmic neutrinos measured, even in a cubic kilometer array. In [@Gen2] a vision for the next-generation IceCube neutrino observatory is presented. At its heart is an expanded array of optical modules with a volume of 7 to 10km$^3$. This high-energy array will mainly address the 100TeV to 100PeV scale. It has the potential to deliver first GZK neutrinos, of anti-electron neutrinos produced via the Glashow resonance, and of PeV tau neutrinos, where both particle showers associated with the production and decay of the tau are observed (“double bang events”).
Another possible component of IceCube-Gen2 is the PINGU sub-array. It targets – similar to ORCA – precision measurements of the atmospheric oscillation parameters and the determination of the neutrino mass hierarchy. The facility’s reach would further be enhanced by exploiting the air-shower measurement and vetoing capabilities of an extended surface array. Moreover, a radio array (“ARA”, for Askarian Radio Array) will achieve improved sensitivity to neutrinos in the $10^{16} - 10^{20}$ eV energy range, including GZK neutrinos. Figure \[Gen2\] sketches a possible design of IceCube-Gen2.
![Schematic view of IceCube Gen-2, comprising the existing IceCube array with its densely equipped inner region DeepCore, the high-energy array of Gen2, the super-densely equipped PINGU sub-detector, and an extended surface array. Not shown is the radio array ARA with its size exceeding that of the basic surface array. []{data-label="Gen2"}](figures/IceCube_Gen2.pdf){width="80.00000%"}
For point sources, the high-energy array will have five times better sensitivity than IceCube, and the rate for events at energies above a few hundred TeV will be ten times higher than for IceCube.
Expectations
------------
Personal expectations do not necessarily agree with the optimistic time schedules of the experiments. Still, one can certainly expect that at the early 2020s, the IceCube cosmic neutrino signal will be scrutinized by KM3NeT-ARCA and Baikal-GVD, with different experimental systematics and from the Northern hemisphere: what is the exact form of the spectrum, what is the flavor composition, what is the contribution from our Galaxy? The diffuse flux from the Galactic plane will be almost certainly discovered, at least if moderately conservative predictions are correct. The discovery of point or extended sources in our Galaxy is not guaranteed but seems likely. Some bet that the first discovery of individual sources will come from transient events correlated to electromagnetic or gravitational wave observations. Actually, within the next few years, this seems to be the most promising chance for IceCube.
Another aspect of increased discovery potential are analyses combining data from different detectors. Within the Global Neutrino Network GNN [@GNN], such analyses are being performed between IceCube and ANTARES [@common] and bear the chance to detect diffuse emission from the Galaxy rather soon. When Baikal-GVD and ARCA approach the cubic kilometer scale, these efforts will become even more important than now.
With the appearance of four or more ARCA blocks, with GVD Phase-2 and with IceCube Gen2 in the second half or at the end of the 2020s, one could have 3–5 km$^3$ instrumented volume in the North and 7–10 km$^3$ in the South. I seems unlikely that this coordinated attack to the high-energy neutrino frontier would fail to detect structures and individual sources. That, at the end, will allow charting the neutrino landscape to which IceCube has enabled a first glance!
[99]{}
C.Spiering, Eur.Phys.J. H37 (2012) 515.
M.G.Aartsen et al. (IceCube Coll.) Science 342 (2013) 1242856.
S. Adrián-Martínez et al. (KM3NeT Coll.), J.Phys. G43 (2016) 084001.
A.Avronin et al. (Baikal Coll.), Phys.Part.Nucl. 46 (2015) 211.
M.G.Aartsen et al. (IceCube Coll.), arXiv:1412.5106.
M.G.Aartsen et al. (IceCube Coll.) Astrophys.J. 835 (2017) no.2, 151.
R.Balkanov, et al. (Baikal Coll.), Astropart.Phys.12 (1997) 75.
I.Belolaptikov and C.Spiering, Baikal-Amanda Internal report, 2005.
P.Coyle and C.W.James, arXiv:1701.02144.
M.G.Aartsen et al. (IceCube Coll.), JINST 12,3 (2017) P03012.
P.Lipari, Nucl.Instrum. Meth. A, 567 (2006) 405.
M.G.Aartsen et al., Phys.Rev.Lett. 111 (2013) 021103.
C.Kopper for the IceCube Coll., ICRC 2017.
M.G.Aartsen et al. (IceCube Coll.) Astrophys.J.833(2016)3.
C.Haack and C.Wiebusch for the IceCube Coll., ICRC 2017.
M.G.Aartsen et al. (IceCube Coll.) Astrophys.J.808(2015)98.
S. Adrián-Martínez et al. (ANTARES Coll.) Astrophys.J.786 (2014) L5.
D.Gaggero, D.Grasso, A.Marinelli, A.Urbano and M.Valliet, Astrophys.J.815,2 (2015) L25.
C.Haack and J.Dumm for the IceCube collaboration, ICRC 2017.
M.Petropoulou, S.Dimitrakoudis, P.Padovani, A.Mastichiadis and\
E.Resconi, Mon.Not.Roy.Astron.Soc., 448 (2015) 2412.
M.Huber and K.Krings for the IceCube Collaboration, ICRC 2017.
M.G.Aartsen et al. (IceCube Coll.), Astrophys.J.843 (2017) no.2, 112.
M.Ahlers, M.Gonzalez-Garcia and F.Halzen, Astropart.Phys.35 (2011) 87.
E.Waxmann and J.Bahcall, Phys.Rev.Lett.78 (1997) 2292.
R.Abbasi et al. (IceCube Coll.) Astron.Astrophys. 535 (2011) A109.
E.Blaufuss for the IceCube collaboration, ICRC 2017, and M.G.Aartsen et al. (IceCube Coll.) Astrophys.J.811 (2015) no.1, 52.
S. Adrián-Martínez et al. (ANTARES, TAROT, ROTSE, Swift and Zadko Collaborations), JCAP 1602, 2 (2016) 062.
see [*https://gcngsfc.nasa.gov/*]{} and
M.G.Aartsen et al. (IceCube, PTF and SWIFT Coll.), Astrophys.J. 811 (2015) no.1, 52.
M.G.Aartsen et al. (IceCube Coll.) arXiv:1702.06131.
M.Albert et al. (ANTARES and IceCube Coll.) arXiv:1703.06298.
V.Berezinsky and G.Zatsepin, G., Yad.Fiz. 11 (1970) 200.
Zh. Djilkibaev for the Baikal Collaboration, ICRC 2017.
S. Adrián-Martínez et al., Astrophys.J. 823, 1 (2016) 65.
|
---
abstract: 'The Sculptor dwarf spheroidal galaxy has a giant branch with a significant spread in colour, symptomatic of an intrinsic age/metallicity spread. We present here a detailed study of the Sculptor giant branch and horizontal branch morphology, combining new near-infrared photometry from the Cambridge InfraRed Survey Instrument (CIRSI), with optical data from the ESO Wide Field Imager. For a Sculptor-like old and generally metal-poor system, the position of Red Giant Branch (RGB) and Asymptotic Giant Branch (AGB) stars on the colour-magnitude diagram (CMD) is mainly metallicity dependent. The advantage of using optical-near infrared colours is that the position of the RGB locus is much more sensitive to metallicity than with optical colours alone. In contrast the horizontal branch (HB) morphology is strongly dependent on both metallicity and age. Therefore a detailed study of both the RGB in optical-near infread colours and the HB can help break the age-metallicity degeneracy. Our measured photometric width of the Sculptor giant branch corresponds to a range in metallicity of 0.75 dex. We detect the RGB and AGB bumps in both the near-infrared and the optical luminosity functions, and derive from them a mean metallicity of \[M/H\]$ = -1.3 \pm 0.1$. From isochrone fitting we derive a mean metallicity of \[Fe/H\]$ = -1.42$ with a dispersion of 0.2 dex. These photometric estimators are for the first time consistent with individual metallicity measurements derived from spectroscopic observations. No spatial gradient is detected in the RGB morphology within a radius of 13 arcminutes, twice the core radius. On the other hand, a significant gradient is observed in the HB morphology index, confirming the ‘second parameter problem’ present in this galaxy. These observations are consistent with an early extended period of star formation continuing in time for a few Gyr.'
date: 'Accepted xxx. Received xxx'
title: 'A near-infrared and optical photometric study of the Sculptor dwarf spheroidal galaxy: implications for the metallicity spread'
---
\[firstpage\]
galaxies: individual: Sculptor dwarf spheroidal – galaxies: stellar content – Local Group – infrared: stars
Introduction
============
The dwarf galaxies of the Local Group offer a unique opportunity to quantify the evolution and interactions of low-mass galaxies. Dwarf spheroidals were originally thought to be very similar in their metallicity and star formation histories to the galactic globular clusters, but their star formation history is now known to be more complex. The Sculptor dwarf spheroidal, the first dSph discovered (Shapley 1938), has a population which is predominantly old and moderately metal poor (e.g. @DAC84, @MAT98, @MON99, @DOL02). However, the presence of an extended horizontal branch (e.g. @DAC84, @MAJ99) and the detection of associated neutral hydrogen (@CAR98, @BOU03) suggest the possibility of a more complex star formation history. A metallicity spread has been inferred in Sculptor from the large spread of its red giant branch (e.g. @DAC84, @SCH95, @KAL95, @MAJ99) and the period distribution of RR Lyrae (@KAL95, @KOV01). Metallicity gradients have been discovered in several dwarf spheroidals (@HAR01). [@MAJ99] and [@HUR99] showed that the red horizontal branch (RHB) of Sculptor is more strongly concentrated towards the centre than is the bluer population. [@MAJ99] suggest this and the detection of two bumps in the Sculptor red giant branch are direct evidence for a bimodality in Sculptor’s metallicity distribution. [@HUR99] find no radial gradient of the age or metallicity distribution within Sculptor, whereas using the same data, [@HAR01] find a metallicity gradient. Spectroscopic studies of a few stars in the central regions confirmed a wide abundance range, over 1 dex (@SHE03), while first results from an extensive spectroscopic study of [@TOL04] confirm the broad abundance distribution, with range -2.5$<$\[Fe/H\]$<$-1.5, and show that the central regions, where the RHB stars are found, also contains a significant subpopulation of stars as metal rich as -1 dex.
Photometric studies of the age distribution in Sculptor (@MON99, @DOL02 and @RIZ04) confirm that a significant metallicity gradient, with some relatively smaller age spread, are both required by the stellar colour-magnitude diagram, and are consistent with all other data. This predominately old age is puzzling, given that Sculptor is apparently unique among the dSph galaxies in having associated HI. Sculptor is further of interest in that it may belong to a common orbital plane with several other dSph (@LYN76).
In this paper we present the first near-infrared photometric study of Sculptor. The combination of our near-infrared data, obtained with the Cambridge InfraRed Survey Instrument (CIRSI) on the Las Campanas 2.5m duPont telescope, and optical data from the ESO 2.2m Wide Field Imager archive, has allowed us to undertake a broader waveband study of the metallicity spread in Sculptor. Observations and data reduction procedures are presented in Section 2. In Section 3 colour-magnitude diagrams of Sculptor are compared with theoretical isochrones and globular cluster data. In section 4 we present the detection of the RGB and AGB bumps. The metallicity distribution function is derived in section 5. Section 6 is devoted to the study of the variation of RGB and HB morphologies with radius. We conclude with a discussion of the main results.
The data
========
Photometric studies provide a valuable tool to determine internal metallicity spreads in dSph galaxies. Although individual spectroscopic measurements are more precise, as yet the numbers of stars with direct spectra is small. Most such photometric studies use optical photometry. The combination of optical and near-infrared data provides substantially enhanced information, reducing the effect of photometric errors, allowing colour-colour selection of sources, and providing in particular the V-K colour, which is a good indicator of the stellar effective temperature.
We have obtained wide-area near-infrared J and K-band photometry of the Sculptor dSph galaxy, complementing this with optical V and I-band data from the ESO archive.
Figure \[fov\] presents the Sculptor area observed, while table \[ObsLog\] summarises the observations.
Instrument Field Filter Date Exposure (secs) seeing ()
------------ --------------- -------- ------------ ----------------- ----------- --
CIRSI Sculptor-West J 2001-09-04 9 x 5 x 20 1.02-1.4
K$_s$ 2001-09-03 9 x 3 x 45 0.8-0.9
Sculptor-East J 2001-09-04 9 x 5 x 20 1.14-1.4
K$_s$ 2001-09-03 9 x 5 x 20 0.96-1.3
ESO WFI Sculptor V 1999-07-22 3 x 300, dit-3 1.00
I 1999-07-22 3 x 300 0.91
\[ObsLog\]
![Field of view of the CIRSI and ESO-WFI data. The contours are from Irwin & Hatzidimitriou (1995). Sculptor’s core and tidal radii are respectively 5.8 and 75 arcminutes.[]{data-label="fov"}](fig1.eps){width="8cm"}
The near-infrared data
----------------------
Near-infrared observations in the $J$(1.25$\mu$m) and $K_s$(2.15$\mu$m) bands were made with the Cambridge Infrared Survey Instrument (@BEC97, @MAC00) on the du Pont 2.5m at Las Campanas Observatory. CIRSI is a mosaic imager consisting of four Rockwell 1k x 1k detectors. The pixel scale is 0.2 arcsec pixel$^{-1}$. The gaps between the detectors being comparable to the detector size, four dither sets are needed to fill the mosaic image, leading to a field of view of about 13 x 13 arcmin$^2$. For each dither set, 9 dither frames are taken with offsets of about 10 arcsec. Each dither frame is observed in 5 loops of 20 seconds exposure time each. The total exposure time per mosaic is then 900s. Two mosaics were taken at the centre of the Sculptor dwarf spheroidal, as pictured in figure \[fov\].
### CIRSI data reduction
The data reduction was made using an updated version of the InfraRed Data Reduction (IRDR) software package, first developed by [@SAB01]. A summary of the full process is given here. The updated version of IRDR with its documentation are available at http://www.ast.cam.ac.uk/\~optics/cirsi/software. A fuller description is provided by [@BAB05], in their presentation of a CIRSI study of the Galactic bulge and bar.
First each image is corrected for non-linearity. The dark current for the relevant exposure time is then subtracted. The data are flatfield corrected using lamp-on domeflats subtracted with lamp-off domeflats and normalized to the first detector sensitivity. These flatfields are also used to detect bad pixels and create weight maps, used during the dither frame coaddition.
The sky is subtracted in two passes. A first pass sky image is made by median combining the nearest loop-combined frames of the dither set. After a first dither frame coaddition, object masks are produced using SExtractor source extraction (@BER96). A masked frame is created from this source-detection list, with an enlarged area around each detected source being used. This object-masked frame is used to make a second pass sky subtraction on each loop.
Spatial offsets between loop-combined dither frames are computed by cross-correlating object pixels mapped by SExtractor. Dither frames are then coadded using a weighted bi-linear interpolation, excluding bad pixels.
Finally, the astrometry is calibrated by correlating the SExtractor’s object catalogue with the USNO-A2 catalogue.
### CIRSI photometry
Once the data are reduced, we use the IRAF photometry routines. Source are detected using the IRAF [daofind]{} procedure, with a significance threshold set at five-sigma. Aperture photometry is then obtained using the IRAF [phot]{} task. The aperture radius is assigned for each dither frame to the measured PSF FWHM. Observations of standard stars from [@PER98] were obtained each night, and were used to derive the magnitude zero-point of each night. Standard star photometry used an aperture photometry radius of 20 pixels, equivalent to a diameter of 8 arcsec. The internal zero point dispersion derived from multiple observations of the standard stars during a night is 0.013 mag in J and 0.008 mag in K. The instrumental magnitudes derived using the psf-fwhm aperture are corrected for aperture effects using the curve-of-growth method (@STE90), implemented with the IRAF [mkapfile]{} task applied to selected bright isolated stars. Airmass corrections of $k_J$=0.1 and $k_K$=0.08 mag per air-mass (@PER98) are applied to the photometry. Finally, images detected near the borders of the images are eliminated, so that only stars observed in all the dither frames are kept. False detections located in the wings of highly saturated stars are manually deleted.
The photometric calibration was checked by correlating the brightest stars with the 2MASS catalogue. The CIRSI K$_s$ photometry is consistent with the 2MASS photometric system. However, an offset in the J-band photometry of $J_{(CIRSI)}-J_{(2MASS)} = -0.042 \pm 0.005$ is observed. This is due to the 2MASS J-band filter being more extended into the atmospheric water absorption features at around 1.1 and 1.4 $\mu$m (@CAR01) than is the filter used for CIRSI.
The 5-$\sigma$ magnitude completeness limits are about J$\sim$20 and K$_s$$\sim$18.8. The photometric errors are about 0.025 mag for J$<$17, 0.08 mag at J=20, 0.022 mag for K$_s<$16, and 0.08 mag at K$_s$=18.8.
The optical data
----------------
The optical data were obtained from the ESO 2.2m telescope Wide Field Imager archive. They consist of 3x300s dithered exposures in V and I. Each of the eight 4kx2k CCDs in the ESO 2.2m WFI covers around 8 arcmin x 16 arcmin of sky at a sampling of 0.238 arcsec per pixel, comparable with the CIRSI data. The total field of view of the WFI is about 33 arcmin x 33 arcmin, including small gaps of around 10 arcsec between CCDs. This field of view entirely covers the observed CIRSI fields (figure \[fov\]).
The WFI data were processed using a variant of the standard optical pipeline described by [@IRW01]. After trimming, bias-correcting, flatfielding and gain normalisation, the I-band exposures were additionally defringed using a I-band fringe frame derived from the 3 I-band science images. To generate the fringe frame, the 3 dithered I-band exposures were object masked, combined with rejection to remove residual artifacts, and then further filtered to improve the local signal-to-noise ratio.
Object catalogues were generated for each individual processed science frame and used to define, and refine, the World Coordinate System (WCS) for each frame, by comparison with the online APM plate catalogues. After updating the 2D images with the derived WCS, the 3 V and 3 I frames of Sculptor were stacked with cosmic ray rejection, using the WCS for coalignment, and confidence maps, derived from the flatfield frames and bad column lists for each CCD, to aid in rejection.
Final object catalogues were then derived from the stacked frames, and used both to update the WCS and to provide morphological classification information for each detected object. Object detection proceeds via a search for contiguous pixels at a threshold above the local sky; following detection a series of object parameters are derived. These latter are used to generate position, flux and shape information for use in later processing stages (@IRW85, @IRW97). The basic photometric measurement used is an aperture flux estimate with radius set to the average FWHM in the stacked frames. Additionally, all detected images have a series of fixed aperture measures produced (scaled to the basic aperture size used) in order to automatically derive aperture corrections for stellar images for each CCD.
Archive observations of [@LAN92] standard fields define native system zero-points in each passband using the colour equations for WFI available on the ESO web site. The gain-correction in the previous stage ensures that all CCDs are on the same internal system, normalised to CCD1. However scattered light leads to a variation of the photometric zero points across the mosaic (@MAN01). From the standard field observations, a correction term of 1.5r$^2$, with r being the distance in degrees from the optical axis, was found to be a good approximation of the effect of the scattered light. Overall, this provides a zero-point calibration with 1-2% accuracy.
The 5-$\sigma$ magnitude completeness limits are about V$\sim$24, I$\sim$22.5, so that the censorship on the data is due only to the CIRSI J and K$_s$ photometric limits. The errors are smaller than 0.01 mag in V and 0.02 in I for magnitudes brighter than V$\sim$21.5 and I$\sim$19.5.
Photometry of the Sculptor dSph galaxy
======================================
![The V-J vs J-K$_s$ colour-colour diagram of all point sources detected in V,I,J and K$_s$. The solid line shows the quasar selection criterion from the KX method (Warren et al. 2000).[]{data-label="colourtoclean"}](fig2.eps){width="8cm"}
Our next task is to generate a list of point-sources which are stellar members of the Sculptor dSph galaxy. Extended objects are readily eliminated from further consideration using the morphological flags from the optical pipeline.
Selection of Sculptor stars
---------------------------
The VJK colour-colour diagram, figure \[colourtoclean\], allows the detection of three other point-source populations unrelated to the Sculptor dSph galaxy. Red quasars are clearly seen with colours which are too red to correspond to any star. All sources with J-K$_s>$1 were eliminated. A stream of stars with J-K$_s$ about 0.8 mag and V-J colour redder than 3 mag can be seen. They are foreground low-mass dwarfs (e.g. @LEG92); these can be eliminated, without discarding any Sculptor stars, by excluding all stars redder than the colour of the tip of the Sculptor giant branch in all relevant colour-colour CMDs (e.g. V-I$>$1.9). A number of probable blue quasars can be seen bluer in V-J than the main stellar locus of Sculptor and foreground stars. These are selected and eliminated according to the KX method criterion (@WAR00), which is shown as the solid line in figure \[colourtoclean\]. Remaining foreground Galactic stars are minimised by our selection inside each colour-magnitude diagram of the location of Sculptor member stars, as described further below.
![Theoretical isochrones of the Padova group (a) AGB and RGB isochrones for an age of 14 Gyr and for metallicities \[Fe/H\]=-2.3,-1.7,-1.3,-0.7 dex from left to right. The data points are our photometry. (b) The RGB and main sequence turn-off for ages 14 and 8 Gyr and for the same four metallicities as in the top panel.[]{data-label="isochrones"}](fig3a.eps "fig:"){width="8cm"} ![Theoretical isochrones of the Padova group (a) AGB and RGB isochrones for an age of 14 Gyr and for metallicities \[Fe/H\]=-2.3,-1.7,-1.3,-0.7 dex from left to right. The data points are our photometry. (b) The RGB and main sequence turn-off for ages 14 and 8 Gyr and for the same four metallicities as in the top panel.[]{data-label="isochrones"}](fig3b.eps "fig:"){width="8cm"}
The Sculptor Colour-Magnitude diagrams
--------------------------------------
It is apparent from figure \[isochrones\]a. that there is a significant real width to the RGB of Sculptor in the (V-K$_s$,K$_s$) colour-magnitude diagram, confirming several previous studies of Sculptor CMD. The RGB photometric width is about $\Delta(V-K_s)=0.3$ at magnitude K$_s$=16, where the photometric measurement errors are 0.023 mag.
To determine the origin of this dispersion, theoretical isochrones from the Padova group, given in the ESO-WFI and 2MASS photometric systems by respectively [@GIR02] and [@BON04], have been overlaid on the Sculptor’s CMDs, using a distance modulus (m$-$M)$_0$=19.54 and an extinction E(B-V)=0.02 (@MAT98). The extinction is derived in the different photometric bands using the [@CAR89] extinction curve.
Figure \[isochrones\]b shows theoretical RGB isochrones for metallicities Z=0.0001, 0.0004, 0.001 and 0.004 with solar mixture, and ages 8 and 14 Gyr. It can be seen that the RGB is much more sensitive to metallicity than to age. The main sequence turn-off, which is more sensitive to age, was used by [@MON99] to derive an age of 15$\pm$2 Gyr for Sculptor. The results of [@DOL02] and [@RIZ04] confirm the predominance of old stars in Sculptor. We therefore adopted isochrones of age 14 Gyr for figure \[isochrones\]a. These isochrones clearly confirm the presence of a metallicity spread within the Sculptor RGB stars.
![Sculptor photometric data from this study (points), together with the fiducial RGB lines for the sample of Galactic globular clusters from Ferraro et al. (2000), with their global metallicities as defined by Ferraro et al. (1999) indicated.[]{data-label="ferraro"}](fig4.eps){width="8cm"}
The colour magnitude diagrams can also be compared directly to globular cluster observations. [@FER00] and [@SAV00] provide fiducial lines for the RGB of Galactic globular clusters for a wide range of metallicity, in the (V,J,K) and (V,I) photometric systems. Figure \[ferraro\] shows the fiducials of [@FER00] globular clusters on the (V-K$_s$,K$_s$) colour-magnitude diagram of Sculptor. The transformation from absolute to relative magnitudes is the same as the one applied for the theoretical isochrones. However the V and K magnitudes are not on the same photometric system as ours, leading to expected photometric differences of the order of 0.1 mag. Considering that globular clusters tend to have alpha-enhanced element ratios, whereas the Sculptor stars do not, we indicate their global metallicities as defined by Ferraro et al. (1999). The definition of this global metallicity scale and how it can be translated into \[Fe/H\] for Sculptor is discussed in section 4. Here again a metallicity spread is confirmed as being consistent with the width of the Sculptor RGB. Figure \[ferraro\] illustrates that all stars in Sculptor are more metal-poor than \[M/H\]$=-1.0$.
The spread of the RGB in the (J-K$_s$,K$_s$) and (V-I,I) CMDs is smaller and more sensitive to photometric errors than in (V-K$_s$,K$_s$). According to [@SAV00], a variation of metallicity from $-2.0$ to $-1.5$ dex results in a difference in V-I of 0.04 mag at I=$-2$ (one magnitude brighter than the RGB bump), while according to [@FER00], the same variation of metallicity at K=$-3$ results in a variation of 0.2 mag in V-K. Considering the relative photometric errors in V, I and K$_s$, this means that V-K$_s$ is 1.7 times as sensitive to metallicity as is V-I. We will therefore use preferentially the (V-K$_s$,K$_s$) CMD in the following to derive photometric metallicity indicators.
The RGB and AGB bumps
=====================
![Luminosity Function (LF) and cumulative LF (a) in the K$_s$ band for all stars, (b) in the V band separately for stars redder and bluer than the giant branch mean fiducial.[]{data-label="bump"}](fig5a.eps "fig:"){width="8cm"} ![Luminosity Function (LF) and cumulative LF (a) in the K$_s$ band for all stars, (b) in the V band separately for stars redder and bluer than the giant branch mean fiducial.[]{data-label="bump"}](fig5b.eps "fig:"){width="8cm"}
Local maxima are observed in the luminosity function of the giant branch of old metal-poor stellar populations. These RGB and AGB bumps are well known features of the colour-magnitude diagrams of globular clusters (e.g. @FER99). Those two bumps are detected in all our visible and near-infrared bands (table \[bumptab\]), as illustrated in figure \[bump\] for the V and K$_s$ bands. To allow us to compare the absolute magnitudes of these features in Sculptor with other studies, the ESO WFI V and I photometry was converted into the standard Johnson photometry using the [@GIR02] isochrones in those two filter systems. At the location of the bumps, $(V-I)_{J}=0.96$, a given simulated star of the isochrones present the colours $V_J-V_{WFI}=-0.06$ and $I_J-I_{WFI}=0.1$. As previously, the conversion to absolute magnitudes assumes a distance modulus of 19.54 mag and E(B-V)=0.02. Errors in the determination of the bump location are of about 0.1 mag.
---------- --------------- ---------------- ------- ----------- ----------- ----------- ------- -----------
$m_{V_{WFI}}$ $m_{I_{WFI}} $ $m_J$ $m_{K_s}$ $M_{V_J}$ $M_{I_J}$ $M_J$ $M_{K_s}$
RGB bump 19.92 18.83 18.23 17.61 0.26 -0.64 -1.33 -1.94
AGB bump 19.40 18.38 17.75 17.13 -0.26 -1.09 -1.81 -2.42
---------- --------------- ---------------- ------- ----------- ----------- ----------- ------- -----------
\[bumptab\]
[@MAJ99] also detected those two bumps within the central 10, but associated the second one with the RGB bump of a more metal-poor population. Indeed, the AGB bump of a population of metallicity \[Fe/H\]$\simeq -1.5$ is located at the same position on the CMD as is the RGB bump of a population of metallicity \[Fe/H\]$\simeq -2$. The V magnitude of the second bump is consistent with the value of $M_V(AGBbump) = -0.3 \pm 0.1$ used by [@FER99]. Its clear detection can be explained by the fact that the luminosity level of the AGB bump stays fairly constant with the cluster metallicity (e.g. @CAS91). The AGB bump being always bluer than the RGB, it explains the detection of this second bump on the blue side of the RGB by [@MAJ99]. Using, as they did, a division of the giant branch into a red and blue part (figure \[fiduc\]), we not only find as expected the second bump on the blue side of the giant branch, but also the RGB bump shifted by about 0.2 magnitudes (figure \[bump\]b), which is consistent with a variation in metallicity. If the second bump is to be due to a distinct metal-poor population, being clearly detected at all wavelengths, this second population should also show a clear imprint on the tip of the red giant branch. No such distinct second RGB can be detected (figures \[isochrones\]a. and \[ferraro\]). We then conclude that the second bump is the AGB bump.
The V magnitudes of the HB and the RGB bump are good indicators of the metallicity (e.g. @FER99). The peculiar shape of Sculptor’s HB however makes the determination of its mean V magnitude unreliable (figure \[hbindex\]). The V magnitude of the RGB bump leads to a global metallicity of \[M/H\]$=-1.30\pm$0.12 from the calibration of [@FER99]. Its K$_s$ magnitude leads to \[M/H\]$=-1.39\pm0.14$ according to [@CHO02], while it leads to \[M/H\]$=-1.19\pm0.12$ from the new calibration of [@VAL04]. Those calibrations were made using the relation of [@SAL93]: $$[M/H] = [Fe/H] + \log(0.638*10^{[\alpha/Fe]}+0.362)
\label{eq:sal93}$$ Sculptor does not seem to present a strong enhancement of alpha-elements: [@SHE03] measured for 5 stars the following values for \[$\alpha$/Fe\]: 0.18, 0.13,0.13,-0.01 and 0.23 (table 2 of @TOL03). For \[$\alpha$/Fe\]=0.13, \[M/H\]=-1.3 can be translated to \[Fe/H\]=-1.4 by equation \[eq:sal93\].
It should be stressed that the indicated errors do not take into account the uncertainty in the distance modulus, estimated to be 0.08 mag in [@MAT98]. Another metallicity indicator, independent of the distance modulus and of zero point calibration errors, is the difference between the AGB and RGB bump luminosities. From [@FER99], with ${\delta}V^{RGBbump}_{AGBbump}=0.52\pm0.14$, we can derive \[M/H\]=$-1.4\pm0.2$.
Those metallicity indicators agree with the previous comparisons of the RGB morphology with theoretical isochrones and globular clusters (figure \[isochrones\]a and figure \[ferraro\]).
The metallicity distribution function
=====================================
The mean fiducial of the Sculptor red giant branch was computed through a least squares fit to a second order polynomial on the (V-K$_s$,K$_s$) CMD. Horizontal branch stars and foreground stars have been eliminated by selecting only the stars 0.2 magnitudes away from the mean fiducial (figure \[fiduc\]). Only the giant branch stars brighter than K$_s$=18.7 and up to the RGB tip (K$_s$$>$13.8) have been selected. As the AGB stars occupy the same location on the CMD as the most metal poor RGB population, no attempt to discard AGB stars was made.
![Least square fit of a second order polynomial to the Sculptor giant branch. The dashed curves frames the stars selected for the giant branch studies.[]{data-label="fiduc"}](fig6.eps){width="8cm"}
![Sculptor (V-I,V) CMD. The boxes indicate how the HB index was computed, selecting the B,V and R stars.[]{data-label="hbindex"}](fig7.eps){width="8cm"}
We derive the metallicity distribution of those selected Sculptor giant branch stars using the Padova isochrones described previously. Each isochrone of figure \[isochrones\]a can be approximated by a second order polynomial: $$V-K_s = a_0 + a_1 K_s + a_2 {K_s}^2
\label{eq:fidiso}$$ A second order polynomial regression of those coefficients as a function of the metallicity of the isochrones is obtained: $$a_i = a_{(i,0)} + a_{(i,1)} [Fe/H] + a_{(i,2)} {[Fe/H]}^2
\label{eq:fidmet}$$ By inverting equation \[eq:fidiso\], each point in the (V-K$_s$,K$_s$) CMD can then be assigned an estimate of its metallicity. Taking into account only the uncertainty of the accuracy on the polynomical regression of equations \[eq:fidiso\] and \[eq:fidmet\] and the photometric errors, the typical uncertainty in the resulting measurement of the metallicity of a individual star is smaller than 0.04 dex.
The resulting metallicity distribution function is represented in figure \[fig:mdf\]. The secondary peak at \[Fe/H\]$= -2.2$ is an artefact due to AGB stars and should be ignored. The mean metallicity obtained is \[Fe/H\]$ = -1.42$ with a dispersion of 0.2 dex. This mean metallicity is in agreement with the value obtained from the RGB bump. A comparison of those metallicity estimates with the literature is given in the discussion section.
![Photometric metallicity distribution function as derived using theoretical isochrones.[]{data-label="fig:mdf"}](fig8.eps){width="8cm"}
Metallicity gradient indicators
===============================
The Giant Branch
----------------
Since the RGB in our photometric system provides a good indicator of metallicity, we studied its variation with radius as a test of a possible metallicity gradient in Sculptor.
The metallicity indicators derived in the previous section have been studied as a function of radius. The central 10 arcminutes of Sculptor has zero ellipticity (@IRW95), so as our data do not extend beyond a radius of 15 arcminutes, we use a simple circular annulus. Figure \[fig:metrad\] show that no metallicity gradient is detected within our data. This gives an upper limit of 0.03 dex for the metallicity gradient within twice the core radius of Sculptor.
![Mean metallicity, as estimated in section 5, as a function of radius. The vertical line show the measured dispersion around this value.[]{data-label="fig:metrad"}](fig9.eps){width="8cm"}
The Horizontal Branch
---------------------
The horizontal branch of Sculptor can be clearly divided into a blue (B) and a red (R) part, lying on either side of the instability strip (V). The ratio of the number counts of those different parts, quantified by the HB index (B-R)/(B+V+R), is dependent on the metallicity. But there is a well known ‘second parameter problem’, which could be age (e.g. @LEE94).
The HB index was computed from the (V-I,V) CMD, as illustrated in figure \[hbindex\], for the same radial annuli as used for the RGB study. Both the full ESO WFI field of view data and the sub-area in common with the CIRSI field of view are presented in figure \[hbrad\]. A K-S test for the hypothesis that the red and blue horizontal branch stars have the same radial distribution gives a significance level of 10$^{-9}$%. This confirms the HB gradient detected by [@HUR99] and [@MAJ99].
![The HB index against radius for all the ESO-WFI data and stars only within the CIRSI field of view (figure \[fov\]).[]{data-label="hbrad"}](fig10.eps){width="8cm"}
The [@LEE94] models give theoretical isochrones that show, for an HB index between -0.5 and 0.5 and a given age, a linear relation between the HB index and the metallicity: $$[Fe/H] \simeq -0.34 * HBindex + cte
\label{hbmet}$$ The observed gradient in HBindex of about 0.5 then corresponds to a gradient in metallicity of 0.17 dex. Considering the upper limit of 0.04 dex for a metallicity gradient derived previously from the RGB morphology, an age gradient is required to explain the observed HB gradient. As always when discussing HB morphology however, one must recall that the ‘second parameter problem’ is not yet solved and that another parameter may influence the HB morphology.
The simplest conclusion is that a small gradient in mean age is apparent in Sculptor and that an eventual small metallicity gradient associated with it would have a too small effect on the RGB compared to the large abundance dispersion to be detected.
Conclusions and discussion
==========================
The combination of near-infrared photometry from CIRSI with optical data from the ESO WFI, allowed a detailed study of the Sculptor dwarf spheroidal giant branch morphology. We confirm that the broad giant branch of Sculptor demonstrates an intrinsic metallicity spread (e.g. @DAC84, @SCH95, @KAL95, @MAJ99). From our photometric study we quantify this spread into a metallicity range of $\Delta[Fe/H]=0.75$ dex. The RGB and AGB bumps are detected in all the optical and near-infrared luminosity functions, excluding the substantial metal-poor contribution to Sculptor’s metallicity distribution proposed by [@MAJ99]. We derive a mean metallicity within two core radii in Sculptor of about \[Fe/H\]$=-1.4$ from both the RGB and AGB bumps magnitudes and isochrones fitting. Our mean metallicity and the metallicity range are higher than those derived in [@DAC84] from photometry of the RGB and in [@KAL95] from RR Lyrae stars: these results were summarized in the [@MAT98] review as a mean of \[Fe/H\]$=-1.8\pm$0.1 with a spread of 0.3 dex. However our photometry is in agreement with the metallicity estimations from Sculptor RR Lyrae stars of [@KOV01] and CaII triplet observations of 37 stars of [@TOL01]. It is in excellent agreement with the very recent spectroscopic survey of [@TOL04], whose derived metallicity distribution inside two core radii indicates a mean metallicity of $-1.4$ dex, and metallicity range of about 1 dex. Their data show the population structure is complex, with the more metal-poor part of the distribution function becoming dominant at radii beyond those we have studied here.
We do not detect a gradient in the RGB morphology within a radius of 13, 2.2 times the Sculptor core radius. Although [@HAR01] and [@TOL04] find a metallicity gradient in Sculptor, our result is in agreement with their data. Indeed figure 6 of [@HAR01] shows that the radial distribution of blue and red RGB stars begins to differ only after 13. Figure 3 of [@TOL04], based on spectroscopic data, indicates that a metallicity gradient is indeed visible only beyond this radius. On the other hand, we do detect at high significance a gradient in the horizontal branch (HB) morphology, confirming the results of [@HUR99] and [@MAJ99]. As this cannot be explained by metallicity, the most likely second parameter could be age. [@HUR99] did not find evidence for a gradient in the main-sequence turn off, leading to an upper limit of a 2 Gyr variation at constant metallicity. According to the models of [@LEE94] a small variation in age of even a few Gyr leads to a strong variation in the HB morphology. Age could then still be the second parameter. Moreover, \[Fe/H\] and age are not the only variables which can affect the HB morphology, but also other element abundances, in particular the \[O/Fe\] ratio (@LEE94). [@HUR99] detected another population presenting a gradient in Sculptor: a ‘spur’ of stars extending $\sim$0.7 mag above the old main sequence turn off. They conclude that it cannot be explained by the presence of an intermediate age population, preferring the interpretation of the spur as a binary sequence, and speculate that it could be related to the variation of the HB morphology.
No significant intermediate age population has been found in Sculptor, excluding star formation within the last 5 Gyr. The large metallicity spread requires extended star formation, while the evidence of low alpha-element enhancement implies extended star formation and self-enrichment over a period of at least 1 Gyr: quite long enough to affect the HB morphology. The HB morphology gradient implies that the most recent star formation episode occurred in the centre of the galaxy, consistent with naive expectation that gas is more easily retained deeper in the galaxy potential well. Indeed in most dwarf galaxies observed with sufficiently deep wide-field imaging, populations gradients have been found with the younger stars being more centrally concentrated (e.g. @SAV01 and references therein). Our lack of detection of a metallicity gradient may be explained by the age-metallicity degeneracy that would hide a small age and metallicity difference and by the stronger dependence of the HB on other parameters such as the oxygen abundance, stellar rotation or binarism, as already suggested by [@HUR99] and [@MAJ99].
All this could be consistent for Sculptor with a single period of star formation extended in time for of the order of a few Gyr. [@TOL03] conclude that their study of the element abundances are consistent with a closed-box chemical evolution scenario. The small dynamical mass of dwarf spheroidals such as Sculptor means that their binding energy is small compared to the energy released by several supernovae, which leads the high metallicity spread and relatively high mean metallicity derived for Sculptor puzzling: how did the gas stay bound long enough to have an extended star formation and gas enrichment? The star formation rate should be low to allow the chemical enrichment to proceed gradually. Hydrodynamical simulations are trying to answer this question (e.g. @CARR01, @CAR02).
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Jacco van Loon for his assistance with the CIRSI observations. We thank the anonymous referee for helpful comments. The development and construction of CIRSI was made possible by a grant from the Raymond and Beverly Sackler Foundation.
C., [Gilmore]{} G., 2005, , [in press]{}
M. G., [Mackay]{} C. D., [McMahon]{} R. G., [Parry]{} I. R., [Piche]{} F., [Ellis]{} R. S., 1997, in Proc. SPIE Vol. 2871, p. 1152-1159, Optical Telescopes of Today and Tomorrow, Arne L. Ardeberg; Ed. [CIRSI: the Cambridge Infrared Survey Instrument]{}. pp 1152–1159
E., [Arnouts]{} S., 1996, A&A Suppl. Ser., 117, 393+
C., [Bica]{} E., [Girardi]{} L., 2004, , 415, 571
A., [Carignan]{} C., [Mashchenko]{} S., 2003, , 126, 1295
J. A., [Clayton]{} G. C., [Mathis]{} J. S., 1989, , 345, 245
L., [Hernandez]{} X., [Gilmore]{} G., 2002, , 334, 117
C., [Beaulieu]{} S., [C[\^ o]{}t[' e]{}]{} S., [Demers]{} S., [Mateo]{} M., 1998, , 116, 1690
J., 2001, AJ, 121, 2851+
G., [Chiosi]{} C., [Girardi]{} L., [Lia]{} C., 2001, , 327, 69
V., [Chieffi]{} A., [Pulone]{} L., 1991, , 76, 911
D., [Lee]{} S., 2002, , 124, 977
G. S., 1984, , 285, 483
A. E., 2002, , 332, 91
F. R., [Messineo]{} M., [Fusi Pecci]{} F., [de Palo]{} M. A., [Straniero]{} O., [Chieffi]{} A., [Limongi]{} M., 1999, , 118, 1738
F. R., [Montegriffo]{} P., [Origlia]{} L., [Fusi Pecci]{} F., 2000, , 119, 1282
L., [Bertelli]{} G., [Bressan]{} A., [Chiosi]{} C., [Groenewegen]{} M. A. T., [Marigo]{} P., [Salasnich]{} B., [Weiss]{} A., 2002, , 391, 195
D., [Grebel]{} E. K., [Holtzman]{} J., [Guhathakurta]{} P., [Brandner]{} W., [Geisler]{} D., [Sarajedini]{} A., [Dolphin]{} A., [Hurley-Keller]{} D., [Mateo]{} M., 2001, , 122, 3092
D., [Mateo]{} M., [Grebel]{} E. K., 1999, , 523, L25
M., 1985, MNRAS, 214, 575
M., 1997, Instrumentation for Large Telescopes. Cambridge University Press
M., [Hatzidimitriou]{} D., 1995, , 277, 1354
M., [Lewis]{} J., 2001, New Astronomy Review, 45, 105
J., [Kubiak]{} M., [Szymanski]{} M., [Udalski]{} A., [Krzeminski]{} W., [Mateo]{} M., 1995, , 112, 407
G., 2001, , 375, 469
A. U., 1992, , 104, 340
Y., [Demarque]{} P., [Zinn]{} R., 1994, , 423, 248
S. K., 1992, , 82, 351
D., 1976, , 174, 695
C. D., [McMahon]{} R. G., [Beckett]{} M. G., [Gray]{} M., [Ellis]{} R. S., [Firth]{} A. E., [Hoenig]{} M., [Lewis]{} J. R., [Medlen]{} S. R., [Parry]{} I. R., [Pritchard]{} J. . M., [Sabbey]{} C. S., 2000, in Proc. SPIE Vol. 4008, p. 1317-1324, Optical and IR Telescope In strumentation and Detectors, Masanori Iye; Alan F. Moorwood; Eds. [CIRSI: the Cambridge infrared survey instrument for wide-field ast ronomy]{}. pp 1317–1324
S. R., [Siegel]{} M. H., [Patterson]{} R. J., [Rood]{} R. T., 1999, , 520, L33
J., [Selman]{} F., 2001, The Messenger, 104, 16
M. L., 1998, , 36, 435
J., [et al.]{} 1999, , 111, 1392
S. E., [Murphy]{} D. C., [Krzeminski]{} W., [Roth]{} M., [Rieke]{} M. J., 1998, AJ, 116, 2475
L., [Held]{} E. V., [Bertelli]{} G., [Saviane]{} I., 2004, Memorie della Societa Astronomica Italiana, 75, 110
C. N., [McMahon]{} R. G., [Lewis]{} J. R., [Irwin]{} M. J., 2001, in ASP Conf. Ser. 238: Astronomical Data Analysis Software and Systems X [Infrared Imaging Data Reduction Software and Techniques]{}. pp 317–+
M., [Chieffi]{} A., [Straniero]{} O., 1993, , 414, 580
I., [Rosenberg]{} A., [Piotto]{} G., [Aparicio]{} A., 2000, , 355, 966
I., [Held]{} E. V., [Momany]{} Y., [Rizzi]{} L., 2001, Memorie della Societa Astronomica Italiana, 72, 773
A. E., [Cudworth]{} K. M., [Majewski]{} S. R., [Suntzeff]{} N. B., 1995, , 110, 2747
M., [Venn]{} K. A., [Tolstoy]{} E., [Primas]{} F., [Hill]{} V., [Kaufer]{} A., 2003, , 125, 684
P. B., 1990, PASP, 102, 932+
E., [Irwin]{} M. J., [Helmi]{} A., [Battaglia]{} G., [Jablonka]{} P., [Hill]{} V., [Venn]{} K. A., [Shetrone]{} M. D., [Letarte]{} B., [Cole]{} A. A., [Primas]{} F., [Francois]{} P., [Arimoto]{} N., [Sadakane]{} K., [Kaufer]{} A., [Szeifert]{} T., [Abel]{} T., 2004, , 617, L119
E., [Irwin]{} M. J., [Cole]{} A. A., [Pasquini]{} L., [Gilmozzi]{} R., [Gallagher]{} J. S., 2001, , 327, 918
E., [Venn]{} K. A., [Shetrone]{} M., [Primas]{} F., [Hill]{} V., [Kaufer]{} A., [Szeifert]{} T., 2003, , 125, 707
E., [Ferraro]{} F. R., [Origlia]{} L., 2004, , 354, 815
S. J., [Hewett]{} P. C., [Foltz]{} C. B., 2000, , 312, 827
\[lastpage\]
|
---
abstract: 'We give an expression for number of points for the family of Dwork K3 surfaces $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene’s finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy’s $p$-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.'
address: 'Department of Mathematics, University of Minnesota'
author:
- Heidi Goodson
bibliography:
- 'DworkPaper.bib'
title: Hypergeometric Functions and Relations to Dwork Hypersurfaces
---
Introduction
============
The motivation for this work comes from a particular family of elliptic curves. For $\lambda\not= 0,1$ we define an elliptic curve in the Legendre family by $$E_\lambda : y^2 = x(x-1)(x-\lambda).$$
We compute a period integral associated to the Legendre elliptic curve given by integrating the nowhere vanishing holomorphic $1$-form $\omega=\frac{dx}{y}$ over a $1$-dimensional cycle containing $\lambda$. This period is a solution to a hypergeometric differential equation and can be expressed as the classical hypergeometric series $$\pi= \int_{0}^{\lambda} \frac{dx}{y}={}_2F_1\left(\left.\begin{array}{cc}
\frac12&\frac12 \\
{}&1
\end{array}\right|\lambda\right).$$ See the exposition in [@Clemens] for more details on this.\
We now specialize to the case where $\lambda\in\mathbb Q\setminus\{0,1\}$. Koike [@Koike1995 Section 4] showed that, for all odd primes $p$, the trace of Frobenius for curves in this family can be expressed in terms of Greene’s hypergeometric function $$a_{E_\lambda}(p)=-\phi(-1)p\cdot {}_2F_1\left(\left.\begin{array}{cc}
\phi&\phi\\
{}&\epsilon
\end{array}\right|\lambda\right)_{p},$$ where $\epsilon$ is the trivial character and $\phi$ is a quadratic character modulo $p$.\
Note the similarity between the period and trace of Frobenius expressions: the period is given by a classical hypergeometric series whose arguments are the fractions with denominator 2 and the trace of Frobenius is given by a finite field hypergeometric function whose arguments are characters of order 2. This similarity is to be expected for curves. Manin proved in [@Manin] that the rows of the Hasse-Witt matrix of an algebraic curve are solutions to the differential equations of the periods. In the case where the genus is 1, the Hasse-Witt matrix has a single entry: the trace of Frobenius. Igusa showed in [@Igusa1958] that the trace of Frobenius is congruent modulo $p$ to the classical hypergeometric expression $$(-1)^{\frac{p-1}{2}}{}_2F_1\left(\left.\begin{array}{cc}
\frac12&\frac12 \\
{}&1
\end{array}\right|\lambda\right)$$ for odd primes $p$ (see the exposition in Clemens’ book [@Clemens]). Furthermore, in Corollary \[cor:2F1ECcongruence\] we show that these classical and finite field ${}_2F_1$ hypergeometric expressions are congruent modulo $p$ for odd primes. This result would imply merely a congruence between the finite field hypergeometric function expression and the point count over $\mathbb F_p$. The fact that Koike showed that we actually have an equality is very intriguing and leads us to wonder for what other varieties this type of equality holds.\
Further examples of this correspondence have been observed for algebraic curves [@Swisher2015Arxiv; @Fuselier10; @Lennon1; @Mortenson2003a] and for particular Calabi-Yau threefolds [@AhlgrenOno00a; @McCarthy2012b]. For example, Fuselier [@Fuselier10] gave a finite field hypergeometric trace of Frobenius formula for elliptic curves with $j$-invariant $\frac{1728}{t}$, where $t\in \mathbb F_p \setminus \{0,1\}$. Lennon [@Lennon1] extended this by giving a hypergeometric trace of Frobenius formula that does not depend on the Weierstrass model chosen for the elliptic curve. In [@AhlgrenOno00a], Ahlgren and Ono gave a formula for the number of $\mathbb F_p$ points on a modular Calabi-Yau threefold. We extend these works to Dwork hypersurfaces, largely focusing on results that hold for Dwork K3 surfaces. Recall that the family of Dwork K3 surfaces is defined by $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4.$$ We show that the number of points on the family of Dwork K3 surfaces over finite fields can be expressed in terms of Greene’s finite field hypergeometric functions. The following is proved in Section \[sec:DworkSurfaces\].
\[thm:K3PointCount\] Let $q=p^e$ be a prime power such that $q\equiv 1\pmod 4$, $t=\frac{q-1}{4}$, and $T$ be a generator for $\widehat{\mathbb F_q^{\times}}$. When $\lambda^4=1$ we have $$\#X_{\lambda}^4(\mathbb F_q)=\frac{q^3-1}{q-1}+3qT^t(-1)+q^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|1\right)_q.$$ More generally, for $\lambda\not=0$, $$\begin{aligned}
\#X_{\lambda}^4(\mathbb F_q)&=\frac{q^3-1}{q-1}+12qT^t(-1)T^{2t}(1-\lambda^4)\\
&\hspace{.2in}+q^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_q+3q^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_q, \end{aligned}$$ for all prime powers $q\equiv 1\pmod 4$ away from the primes of $\lambda$.
Salerno [@Salerno2013] used a finite field hypergeometric function defined by Katz [@Katz1990] to develop a point count formula for a larger class of diagonal hypersurfaces that could specialize to Dwork K3 surfaces. Theorem 5.5 of Salerno’s paper gives a congruence between the number of points on diagonal surfaces and classical truncated hypergeometric series. This specializes to a single hypergeometric term in the Dwork K3 surface case (see Section 5.4 of Salerno’s paper). Our result in Theorem \[thm:K3PointCount\] gives an exact formula for the point count, not just a congruence. Furthermore, in Section \[sec:HGFCongruence\] of this paper we give a congruence between the ${}_3F_2$ finite field hypergeometric function of Theorem \[thm:K3PointCount\] and classical truncated hypergeometric series that appears in Section 5.4 of Salerno’s paper. We later use this congruence to prove a result that gives the relationship between the trace of Frobenius and certain periods associated to Dwork K3 surfaces, so for our purposes, Greene’s finite field hypergeometric function is a natural choice of functions to work with.
Note that the character $T^t$ is only defined over $\mathbb F_q$ when $q\equiv 1\pmod 4$. We would like to develop a point count formula to use for fields $\mathbb F_p$, where $p\equiv 3\pmod 4$ is prime. In this case, it seems unlikely that we will be able to develop a finite field hypergeometric formula. However, McCarthy [@McCarthy2013] defined a $p$-adic version of these hypergeometric functions which we will use to write a concise formula to calculate the number of points on Dwork K3 surfaces over these fields.
\[thm:K3PointCountnGn\] When $p\equiv 3 \pmod 4$ and $\lambda\not=0$, the point count is given by $$\begin{aligned}
\#X_{\lambda}^4(\mathbb F_p)&=\frac{p^3-1}{p-1}+{}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p -3p\hspace{.02in}{}_2G_2\left[\left.\begin{array}{ccc}
3/4&1/4\\
0&1/2
\end{array}\right|{\lambda^4}\right]_p,\end{aligned}$$ for all primes $p\equiv 1\pmod 4$ away from the primes of $\lambda$.
Though we have already developed a hypergeometric point count formula that holds for primes $p\equiv 1 \pmod 4$ in Theorem \[thm:K3PointCount\], we show that we can also give a $p$-adic hypergeometric formula for these primes. Note the similarity between this formula and that of Theorem \[thm:K3PointCountnGn\].
\[thm:K3PointCountnGn1\] When $p\equiv 1 \pmod 4$ and $\lambda\not=0$, the point count is given by $$\begin{aligned}
\#X_{\lambda}^4(\mathbb F_p)&=\frac{p^3-1}{p-1}+12pT^t(-1)T^{2t}(1-\lambda^4)\\
&\hspace{1in}+{}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p+3p{}_2G_2\left[\left.\begin{array}{cc}
3/4&1/4\\
0&2/4
\end{array}\right|\lambda^4\right]_p,\end{aligned}$$ for all primes $p\equiv 1\pmod 4$ away from the primes of $\lambda$.
We observe an interesting phenomenon with certain periods associated to the Dwork surfaces we have studied. For an $n$-dimensional Dwork hypersurface, we calculate a period integral, obtained by choosing dual bases of the space of holomorphic $(n,0)$-differentials and the space of cycles ${H^n(X_\lambda^{n+2},\mathscr O)}$ and integrating the differentials over each cycle. Note that the dimension of both spaces is 1 since Dwork hypersurfaces are Calabi-Yau manifolds and, therefore, have genus $g=1$. The natural choice for a basis of differentials would be the nowhere vanishing holomorphic $n$-form. These periods can be written in terms of classical hypergeometric series, a fact that was first noted by Dwork in [@Dwork1969]. Interestingly, the hypergeometric expressions for the periods and the point counts “match” in the sense that fractions with denominator $a$ in the classical series coincide with characters of order $a$ in the finite field hypergeometric functions. We use these matching expressions to prove the following result for Dwork K3 surfaces in Section \[sec:PeriodTrace\].
\[thm:K3PeriodTrace\] For the Dwork K3 surface $$X_{\lambda}^4:\hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4,$$ the trace of Frobenius over $\mathbb F_p$ and the period associated to the surface are congruent modulo $p$ when $p \equiv 1 \pmod 4$.
We conjecture an analogous result for higher dimensional Dwork hypersurfaces $$X_{\lambda}^d:\hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=d\lambda x_1x_2\cdots x_d$$ in Section \[sec:DworkHypersurface\].\
The remainder of this paper is organized as follows. In Section \[sec:background\] we give some necessary background information that will be used throughout the paper. Section \[sec:HGFCongruence\] gives some congruences between classical hypergeometric series and finite field hypergeometric functions that we will use in our period and trace of Frobenius results. We prove Theorem \[thm:K3PointCount\] in Section \[sec:DworkSurfaces\], using results that are proved in Section \[sec:Koblitz\]. In Section \[sec:K3PointCountnGn\] we prove the $p$-adic point count formulas of Theorems \[thm:K3PointCountnGn\] and \[thm:K3PointCountnGn1\]. We prove Theorem \[thm:K3PeriodTrace\] in Section \[sec:PeriodTrace\]. Finally, we extend these Dwork K3 surface results to higher dimensional hypersurfaces in Section \[sec:DworkHypersurface\].
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank her thesis advisor, Benjamin Brubaker, as well as Rupam Barman, Frits Beukers, Dermot McCarthy, and Steven Sperber for helpful conversations while working on these results.
Preliminaries {#sec:background}
=============
Hypergeometric Series and Functions {#sec:HGF}
-----------------------------------
We start by recalling the definition of the classical hypergeometric series $$\label{eqn:classicalHGF}
{}_{n+1}F_{n}\left(\left.\begin{array}{cccc}
a_0&a_1&\ldots& a_n\\
{}&b_1&\ldots,& b_n
\end{array}\right|x\right) = \displaystyle\sum_{k=0}^{\infty}\dfrac{(a_0)_k\ldots(a_n)_k}{(b_1)_k\ldots(b_n)_kk!}x^k,$$ where $(a)_0=1$ and $(a)_k=a(a+1)(a+2)\ldots(a+k-1)$. In Section \[sec:HGFCongruence\] we will also be interested in truncated hypergeometric series. For a positive integer, $m$, define the hypergeometric series truncated at $m$ to be
$$\label{eqn:truncHGF}
{}_{n+1}F_{n}\left(\left.\begin{array}{cccc}
a_0&a_1&\ldots& a_n\\
{}&b_1&\ldots,& b_n
\end{array}\right|x\right)_{\text{tr}(m)} = \displaystyle\sum_{k=0}^{m-1}\dfrac{(a_0)_k\ldots(a_n)_k}{k!(b_1)_k\ldots(b_n)_k}x^k.$$
In his 1987 paper [@Greene], Greene introduced a finite field, character sum analogue of classical hypergeometric series that satisfies similar summation and transformation properties. Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a power of an odd prime $p$. If $\chi$ is a multiplicative character of $\widehat{\mathbb F_q^{\times}}$, extend it to all of $\mathbb F_q$ by setting $\chi(0)=0$. For any two characters $A,B$ of $\widehat{\mathbb F_q^{\times}}$ we define the normalized Jacobi sum by
$$\label{eqn:normalizedjacobi}
\binom{A}{B}:=\frac{B(-1)}{q}\sum_{x\in\mathbb F_q} A(x)\overline B(1-x) = \frac{B(-1)}{q}J(A,\bar{B}),$$
where $J(A,B)=\sum_{x\in \mathbb F_q} A(x)B(1-x)$ is the usual Jacobi sum.\
For any positive integer $n$ and characters $A_0,\ldots, A_n,B_1,\ldots, B_n$ in $\widehat{\mathbb F_q^{\times}}$, Greene defined the finite field hypergeometric function ${}_{n+1}F_n$ over $\mathbb F_q$ by $$\label{eqn:HGFdef}
{}_{n+1}F_{n}\left(\left.\begin{array}{cccc}
A_0,&A_1,&\ldots,&A_n\\
{} &B_1,&\ldots,&B_n
\end{array}\right|x\right)_q = \displaystyle\frac{q}{q-1}\sum_{\chi}\binom{A_0\chi}{\chi}\binom{A_1\chi}{B_1\chi}\ldots\binom{A_n\chi}{B_n\chi}\chi(x).$$
In the case where $n=1$, an alternate definition, which is in fact Greene’s original definition, is given by $$\label{eqn:2F1def}
{}_{2}F_{1}\left(\left.\begin{array}{cc}
A&B\\
{} &C
\end{array}\right|x\right)_q = \epsilon(x)\frac{BC(-1)}{q}\sum_yB(y)\overline{B}C(1-y)\overline{A}(1-xy).$$
Gauss and Jacobi Sums {#sec:GaussJacobi}
---------------------
Unless otherwise stated, information in this section can be found in Ireland and Rosen’s text [@Ireland Chapter 8].\
We define the standard trace map $\text{tr}:\mathbb F_q\rightarrow \mathbb F_p$ by $$\text{tr}(x)=x+x^p+\ldots + x^{p^{e-1}}.$$ Let $\pi\in\mathbb C_p$ be a fixed root of $x^{p-1}+p=0$ and let $\zeta_p$ be the unique $p^\text{th}$ root of unity in $\mathbb C_p$ such that $\zeta_p \equiv 1+\pi \pmod{\pi^2}$. Then for $\chi \in\widehat{\mathbb F_q^{\times}}$ we define the Gauss sum $g(\chi)$ to be $$\label{eqn:GaussSum}
g(\chi):=\sum_{x\in\mathbb F_q} \chi(x)\theta(x),$$ where we define the additive character $\theta$ by $\theta(x)=\zeta_p^{\text{tr}(x)}$. Note that if $\chi$ is nontrivial then $g(\chi)g(\overline\chi)=\chi(-1)q$.\
We have the following connection between Gauss sums and Jacobi sums. For non-trivial characters $\chi$ and $\psi$ on $\mathbb F_q$ whose product is also non-trivial, $$J(\chi,\psi)=\frac{g(\chi)g(\psi)}{g(\chi\psi)}.$$ More generally, for non-trivial characters $\chi_1,\ldots,\chi_n$ on $\mathbb F_q$ whose product is also non-trivial, $$J(\chi_1,\ldots,\chi_n)=\frac{g(\chi_1)\cdots g(\chi_n)}{g(\chi_1\cdots \chi_n)}.$$
Another important product formula is the Hasse-Davenport formula.
\[thm:HasseDavenport\][@Lang1990 Theorem 10.1] Let $m$ be a positive integer and let $q$ be a prime power such that $q\equiv 1 \pmod m$. For characters $\chi,\psi\in\widehat{\mathbb F_q^{\times}}$ we have $$\prod_{i=0}^{m-1}g(\chi^i\psi)=-g(\psi^m)\psi^{-m}(m) \prod_{i=0}^{m-1}g(\chi^i).$$
In Section \[sec:Koblitz\] we will use the following specializations of this.
\[cor:HasseDavenport\] $$g(T^{4j})=\frac{\prod_{i=0}^3 g(T^{it+j})}{qT^{-4j}(4)T^t(-1)g(T^{2t})}.$$
This follows from Theorem \[thm:HasseDavenport\] using $m=4, \chi=T^t,$ and $\psi=T^j$.
\[cor:HasseDavenportGeneral\] More generally, $$g(T^{dj})=\frac{\prod_{i=0}^{d-1} g(T^{it+j})}{T^{-dj}(d)\prod_{i=1}^{d-1}g(T^{it})},$$ where $q\equiv 1\pmod{d}$, $t=\frac{q-1}{d}$, and $T$ is a generator for $\widehat{\mathbb F_q^{\times}}$.
Though there are other relations for Gauss sum expressions (see, for example, [@Evans1981; @Yamamoto1966]), the Hasse-Davenport formula is our main tool for simplifying the expressions that appear in this paper’s results. However, the following theorem of Helverson-Pasotto will be useful for rewriting an expression that appears in Section \[sec:Koblitz\].
\[thm:Helversen1978\][@Helversen1978 Theorem 2] $$\frac{1}{q-1}\sum_{\chi}g(A\chi)g(B\overline\chi)g(C\chi)g(D\overline\chi) = \frac{g(AB)g(AD)g(BC)g(CD)}{g(ABCD)}+q(q-1)AC(-1)\delta(ABCD),$$ where $\delta(ABCD)=1$ if $ABCD$ is the trivial character and $0$ otherwise.
The following is a new result about Gauss sums that is also helpful for simplifying formulas in Section \[sec:Koblitz\].
\[prop:gaussproduct\] Let $q$ be a prime power such that $q\equiv 1\pmod 4$, $t=\frac{q-1}{4}$, and $T$ be a generator for $\widehat{\mathbb F_q^{\times}}$. Let $a,b$ be multiples of $t$ satisfying $a+b=2t$. Then $$g(T^{2t})\displaystyle \sum_{j=0}^{q-2} g(T^{j+a})g(T^{-j+b})T^j(-1)T^{4j}(\lambda)=q(q-1)T^b(-1)T^{2t}(1-\lambda^4).$$
We start by using Equation \[eqn:GaussSum\] to write $$\begin{aligned}
\sum_{j=0}^{q-2} g(T^{j+a})g(T^{-j+b})T^j(-1)T^{4j}(\lambda) &= \sum_{j=0}^{q-2}T^j(-\lambda^4) \left(\sum_{x\in\mathbb F_q}T^{j+a}(x)\theta(x)\right) \left(\sum_{y\in\mathbb F_q}T^{-j+b}(y)\theta(y)\right)\\
&= \sum_{j=0}^{q-2}T^j(-\lambda^4) \sum_{x,y\in\mathbb F_q}T^{j+a}(x)T^{-j+b}(y)\theta(x+y)\\ %can assume x,y\not=0 since there sum=0 anyways
&= \sum_{j=0}^{q-2}T^j(-\lambda^4) \sum_{x,y\in\mathbb F_q^{\times}}T^{j}(x/y)T^{a}(x)T^{b}(y)\theta(x+y)\\
&= \sum_{x,y\in\mathbb F_q^{\times}}T^{a}(x)T^{b}(y)\theta(x+y)\sum_{j=0}^{q-2}T^j\left(-\tfrac{\lambda^4x}{y}\right).
\end{aligned}$$ Note that $\sum_{j=0}^{q-2}T^j\left(-\tfrac{\lambda^4x}{y}\right)=0$ unless $-\frac{\lambda^4x}{y}=1$, in which case the sum equals $q-1$. So, we let $x=-\frac{y}{\lambda^4}$ to get $$\begin{aligned}
\sum_{j=0}^{q-2} g(T^{j+a})g(T^{-j+b})T^j(-1)T^{4j}(\lambda) &= (q-1)\sum_{y\in\mathbb F_q^{\times}}T^{a}\left(-\tfrac{y}{\lambda^4}\right)T^{b}(y)\theta\left(-\tfrac{y}{\lambda^4}+y\right)\\
&= (q-1)\sum_{y\in\mathbb F_q^{\times}}T^{a}\left(-\tfrac{y}{\lambda^4}\right)T^{b}(y)\theta\left(y(-\tfrac{1}{\lambda^4}+1)\right).
\end{aligned}$$
We now consider two cases. First, suppose $\lambda^4=1$. Then we have $$\sum_{y\in\mathbb F_q^{\times}}T^{a}(-y)T^{b}(y)\theta(0) %&=T^a(-1)\sum_{y\in\mathbb F_q^{\times}}T^{a+b}(y)\\
=T^a(-1)\sum_{y\in\mathbb F_q^{\times}}T^{2t}(y)= 0$$ since $T^{2t}$ is not a trivial character. Now suppose $\lambda^4 \not=1$. Then we perform the change of variables $y\to y(-1/\lambda^4+1)^{-1}$ to get $$\begin{aligned}
\sum_{y\in\mathbb F_q^{\times}}T^{a}\left(\tfrac{-y}{\lambda^4-1}\right)T^{b}\left(\tfrac{y}{-1/\lambda^4+1}\right)\theta(y) &=T^{-a}(1-\lambda^4)T^{-b}\left(\tfrac{-1}{\lambda^4}+1\right)\sum_{y\in\mathbb F_q^{\times}}T^{a+b}(y)\theta(y)\\
&=T^{b-2t}(1-\lambda^4)T^{-b}\left(\tfrac{\lambda^4-1}{\lambda^4}\right)\sum_{y\in\mathbb F_q^{\times}}T^{2t}(y)\theta(y)\\
&=T^{b}(-\lambda^4)T^{-2t}(1-\lambda^4)g(T^{2t}).\end{aligned}$$ Note that if $\lambda^4=1$ then $T^{b}(-\lambda^4)T^{-2t}(1-\lambda^4)g(T^{2t})=0$, so we can use this expression for all $\lambda$. Hence, $$\begin{aligned}
g(T^{2t})\displaystyle \sum_{j=0}^{q-2} g(T^{j+a})g(T^{-j+b})T^j(-1)T^{4j}(\lambda)&= g(T^{2t})\cdot (q-1)T^{b}(-\lambda^4)T^{-2t}(1-\lambda^4)g(T^{2t})\\
&=qT^{2t}(-1)\cdot(q-1)T^{b}(-\lambda^4)T^{-2t}(1-\lambda^4)\\
%&=q(q-1)T^{b}(-\lambda^4)T^{2t}(1-\lambda^4)\\
&=q(q-1)T^{b}(-1)T^{2t}(1-\lambda^4),\end{aligned}$$ where the last equation holds because $b$ is a multiple of $t$ and $T^{4t}(\lambda)=1$.
$p$-adic Gamma Function {#sec:padicGamma}
-----------------------
Throughout this section let $q=p^e$ be a power of an odd prime $p$ and let $\mathbb Z_p$ denote the ring of $p$-adic integers. Also, let $\mathbb Q_p$ denote the field of $p$-adic numbers and $\mathbb C_p$ be its $p$-adic completion. The following facts can be found in, for example, [@AhlgrenOno00a], [@GrossKoblitz], and [@Mortenson2003a]. We define the $p$-adic Gamma function $\Gamma_p: \mathbb Z_p \rightarrow \mathbb Z_p^*$ by $$\label{eqn:padicdefinition}
\Gamma_p(n):= (-1)^n \prod_{j<n, p\nmid j} j$$ for numbers $n\in\mathbb N$. We extend this to all $x$ in $\mathbb Z_p$ by $$\Gamma_p(x):= \lim_{n\rightarrow x}\Gamma_p(n),$$ where in the limit we take any sequence of positive integers that approaches $x$ $p$-adically.
[@Mortenson2003a Proposition 4.2] \[prop:Gamma\_p\] If $p\geq5$ is prime, $x,y\in\mathbb Z_p$, and $z\in p\mathbb Z_p$, then
1. $\Gamma_p(x+1) =\left\{
\begin{array}{l l}
-x\Gamma_p(x) & \quad \text{if } x\in\mathbb Z_p^*,\\
-\Gamma_p(x) & \quad \text{if } x\in p\mathbb Z_p.
\end{array} \right.$
2. If $n\geq1$ and $x\equiv y \pmod{p^n}$ then $\Gamma_p(x)\equiv\Gamma_p(y) \pmod{p^n}.$
3. $\Gamma_p'(x+z)\equiv\Gamma_p'(x) \pmod p$.
4. $|\Gamma_p(x)|=1$.
5. Let $x_0\in \{1,\ldots, p\}$ be the constant term in the $p$-adic expansion of $x$. Then\
$\Gamma_p(x)\Gamma_p(1-x)=(-1)^{x_0}.$
The following proposition relates the $p$-adic Gamma function to the Pochhammer symbol.
\[prop:MortensonPadic\][@Mortenson2003a Proposition 5.1] Let $m$ and $d$ be integers with $1\leq m<d$. If $p\equiv 1\pmod d$ is a prime, then define $t$ such that $t=\tfrac{p-1}{d}$. If $0\leq j \leq mt$, then $$\frac{\Gamma_p\left(\tfrac{m}{d} +j\right)\Gamma_p\left(1-\tfrac{m}{d} +j\right)}{\Gamma_p\left(1+j\right)^2} = (-1)^{mt+1}\frac{\left(\tfrac{m}{d}\right)_j\left(\tfrac{d-m}{d}\right)_j}{j!^2}.$$
We will use this proposition in the proof of Theorem \[thm:2F1congruence\] in Section \[sec:HGFCongruence\].\
We now state a relationship between Gauss sums and the $p$-adic Gamma function. Recall that $\pi\in\mathbb C_p$ is the fixed root of $x^{p-1}+p=0$ given in Section \[sec:GaussJacobi\]. Define the Teichmüller character, $\omega$, to be the primitive character on $\mathbb F_p$ that is uniquely defined by the property $\omega(x)\equiv x \pmod p$ for all $x$ in $\{0,\ldots,p-1\}$. Then the Gross-Koblitz formula, specialized to the case where the field is $\mathbb F_p$, is the following.
[[@GrossKoblitz Theorem 1.7]]{}\[thm:GrossKoblitz\] For $j$ in $\{0,\ldots,p-1\}$, $$g(\overline{\omega}^j)=-\pi^j\Gamma_p\left(\frac{j}{p-1}\right).$$
$p$-adic Hypergeometric Function {#sec:padicHGF}
--------------------------------
In Section \[sec:HGF\] we defined finite field hypergeometric functions, which will be used in may of the results in this thesis. These hypergeometric functions, however, are limited to primes in which the characters are defined. For example, in Theorem \[thm:K3PointCount\] our point count result is limited to primes and prime powers congruent to 1 mod 4. In order to extend our results to primes in other congruence classes, we use a $p$-adic hypergeometric function developed by McCarthy in [@McCarthy2013].\
For $x\in \mathbb Q$ we let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$ and let $\langle x \rangle$ denote the fractional part of $x$, i.e. $x-\lfloor x\rfloor $.
\[def:padicHGF\][@McCarthy2013 Definition 1.1] Let $p$ be an odd prime and let $t\in\mathbb F_p$. For $n\in \mathbb Z^+$ and $1\leq i \leq n$, let $a_i,b_i\in \mathbb Q \cap \mathbb Z_p$. Then we define
$$\begin{aligned}
{}_nG_n&\left[\left.\begin{array}{cccc}
a_1&a_2&\ldots&a_n\\
b_1&b_2&\ldots&b_n
\end{array}\right|t\right]_p :=\frac{-1}{p-1}\sum_{j=0}^{p-2}(-1)^{jn}\overline\omega^j(t)\\
&\hspace{.2in}\times\prod_{i=1}^n\frac{\Gamma_p\left(\left\langle a_i-\frac{j}{p-1}\right\rangle\right)}{\Gamma_p\left(\langle a_i\rangle\right)}\frac{\Gamma_p\left(\left\langle -b_i+\frac{j}{p-1}\right\rangle\right)}{\Gamma_p\left(\langle -b_i\rangle\right)}(-p)^{-\lfloor\langle a_i\rangle-\frac{j}{p-1}\rfloor-\lfloor \langle-b_i\rangle +\frac{j}{p-1}\rfloor}.\end{aligned}$$
Hypergeometric Congruences {#sec:HGFCongruence}
==========================
In this section we prove congruences between classical truncated hypergeometric series and finite field hypergeometric functions. This builds on results of Mortenson [@Mortenson2003a; @Mortenson2005] by considering hypergeometric functions evaluated away from 1, though our results hold mod $p$ instead of $p^2$.\
The first result is for ${}_2F_1$ hypergeometric functions.
\[thm:2F1congruence\] Let $m$ and $d$ be integers with $1\leq m<d$. If $p\equiv 1\pmod d$ and $T$ is a generator for the character group $\widehat{\mathbb F_p^{\times}}$ then, for $x\not=0$, $${}_{2}F_{1}\left(\left.\begin{array}{cc}
\tfrac{m}{d}&\tfrac{d-m}{d}\\
{} &1
\end{array}\right|x\right)_{\text{tr}(p)} \equiv -p\hspace{.05in}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{mt}&\overline T^{mt}\\
{} &\epsilon
\end{array}\right|x\right)_p \pmod p,$$ where $t=\tfrac{p-1}{d}$.
The following corollary applies to the hypergeometric functions that appear in Clemens’ and Koike’s trace of Frobenius expressions for Legendre elliptic curves.
\[cor:2F1ECcongruence\] If $p$ is an odd prime and $\phi$ is a quadratic character in $\widehat{\mathbb F_p^{\times}}$, then, for $\lambda\not=0$, $$(-1)^{\frac{p-1}{2}}{}_{2}F_{1}\left(\left.\begin{array}{cc}
\tfrac12&\tfrac12\\
{} &1
\end{array}\right|\lambda\right)_{\text{tr}(p)} \equiv -\phi(-1)p\hspace{.05in}{}_{2}F_{1}\left(\left.\begin{array}{cc}
\phi&\phi\\
{} &\epsilon
\end{array}\right|\lambda\right)_p \pmod p.$$
We use Equation \[eqn:HGFdef\] to write $$\begin{aligned}
-p\hspace{.05in}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{mt}&\overline T^{mt}\\
{} &\epsilon
\end{array}\right|x\right)_p &=\frac{p^2}{1-p}\sum_{\chi}\binom{T^{mt}\chi}{\chi}\binom{\overline T^{mt}\chi}{\epsilon\chi}\chi(x)
\end{aligned}$$ Using properties of Gauss and Jacobi sums from Sections \[sec:HGF\] and \[sec:GaussJacobi\] we break this down to $$\begin{aligned}
\frac{p^2}{1-p}\sum_{\chi}\frac{\chi(-1)}{p}J(T^{mt}\chi,\overline\chi)&\frac{\chi(-1)}{p}J(\overline T^{mt}\chi,\overline\chi)\chi(x)\\
&=\frac{1}{1-p}\sum_{\chi}\frac{g(T^{mt}\chi)g(\overline\chi)}{g(T^{mt})}\frac{g(\overline T^{mt}\chi)g(\overline\chi)}{g(\overline T^{mt})}\chi(x)\\
&=\frac{1}{1-p}\sum_{\chi}\frac{g(T^{mt}\chi)g(\overline T^{mt}\chi)g(\overline\chi)^2}{T^{mt}(-1)p}\chi(x)\\
&=\frac{\overline T^{mt}(-1)}{p(1-p)}\sum_{\chi}g(T^{mt}\chi)g(\overline T^{mt}\chi)g(\overline\chi)^2\chi(x).\end{aligned}$$ We rewrite this in terms of the Teichmüller character $\omega$ defined in Section \[sec:padicGamma\] by letting $T=\overline\omega$ and $\chi=\overline\omega^{-j}$. Furthermore, we split up the sum as Mortenson does in [@Mortenson2003a]. This yields the expression $$\begin{aligned}
\frac{\omega^{mt}(-1)}{p(1-p)}\sum_{\chi}g(T^{mt}\chi)g(\overline\chi)^2\chi(x)&=\frac{\overline T^{mt}(-1)}{p(1-p)}\left[\sum_{j=0}^{mt}g(\overline\omega^{mt-j})g(\overline\omega^{p-1-mt-j})g(\overline\omega^{j})^2 \omega^{j}(x)\right.\\
&+\sum_{j=mt+1}^{p-1-mt}g(\overline\omega^{p-1+mt-j})g(\overline\omega^{p-1-mt-j})g(\overline\omega^{j})^2\omega^{j}(x)\\
&+\left.\sum_{j=p-mt}^{p-2}g(\overline\omega^{p-1+mt-j})g(\overline\omega^{2(p-1)-mt-j})g(\overline\omega^{j})^2 \omega^{j}(x)\right].\end{aligned}$$ We use the Gross Koblitz formula given in Theorem \[thm:GrossKoblitz\] to write this expression in terms of the $p$-adic Gamma function $$\begin{aligned}
\frac{\omega^{mt}(-1)}{p(1-p)}&\left[\sum_{j=0}^{mt}\pi^{p-1}\Gamma_p\left(\tfrac{mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 \omega^{j}(x)\right.\\
&+\sum_{j=mt+1}^{p-1-mt}\pi^{2(p-1)}\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 \omega^{j}(x)\\
&+\left.\sum_{j=p-mt}^{p-2}\pi^{3(p-1)}\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{2(p-1)-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 \omega^{j}(x)\right].\end{aligned}$$ Recalling that $\pi$ is a solution to $x^{p-1}+p=0$, we have that $\pi^{p-1}=-p$. We use this to rewrite the sum as $$\begin{aligned}
\frac{\omega^{mt}(-1)}{p(1-p)}&\left[\sum_{j=0}^{mt}-p\Gamma_p\left(\tfrac{mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 \omega^{j}(x)\right.\\
&+\sum_{j=mt+1}^{p-1-mt}p^2\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 \omega^{j}(x)\\
&+\left.\sum_{j=p-mt}^{p-2}-p^3\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{2(p-1)-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 \omega^{j}(x)\right].\end{aligned}$$ In Section \[sec:padicGamma\] we define the Teichmüller character by the property $\omega(x)\equiv x \pmod p$ for all $x$ in $\{0,\ldots,p-1\}$. Using this we prove that the above sum is congruent modulo $p$ to the expression $$\begin{aligned}
\frac{(-1)^{mt}}{p(1-p)}&\left[\sum_{j=0}^{mt}-p\Gamma_p\left(\tfrac{mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 x^{j}\right.\\
&+\sum_{j=mt+1}^{p-1-mt}p^2\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 x^{j}\\
&+\left.\sum_{j=p-mt}^{p-2}-p^3\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{2(p-1)-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 x^{j}\right].\end{aligned}$$ We simplify this expression to get that this is congruent modulo $p$ to the expression $$\begin{aligned}
\frac{(-1)^{mt}}{(1-p)}&\left[\sum_{j=0}^{mt}-\Gamma_p\left(\tfrac{mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 x^{j}\right.\\
&+\sum_{j=mt+1}^{p-1-mt}p\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 x^{j}\\
&+\left.\sum_{j=p-mt}^{p-2}-p^2\Gamma_p\left(\tfrac{p-1+mt-j}{p-1}\right)\Gamma_p\left(\tfrac{2(p-1)-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 x^{j}\right].
\end{aligned}$$ Note that since $\Gamma_p: \mathbb Z_p \rightarrow \mathbb Z_p^*$, the last two summands are congruent to 0 modulo $p$. Hence, we are left with $$\begin{aligned}
\frac{(-1)^{mt}}{(1-p)}\left[\sum_{j=0}^{mt}-\Gamma_p\left(\tfrac{mt-j}{p-1}\right)\Gamma_p\left(\tfrac{p-1-mt-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^2 x^{j}\right] \pmod p.\end{aligned}$$ This much simpler expression can be broken down even further. Note that $\tfrac{mt}{p-1}=\tfrac{m}{d}$. Then the above expression is congruent, modulo $p$, to $$\label{eqn:padicGammaSum}
-(-1)^{mt}\sum_{j=0}^{mt}\Gamma_p\left(\tfrac{m}{d} +j\right)\Gamma_p\left(1-\tfrac{m}{d} +j\right)\Gamma_p\left(-j\right)^2 x^{j}.$$ We then use part 5 of Proposition \[prop:Gamma\_p\] to get $$\Gamma_p(y)^2=\frac{1}{\Gamma_p(1-y)^2}.$$ We use this to write that Equation \[eqn:padicGammaSum\] is congruent modulo $p$ to the expression $$\begin{aligned}
(-1)^{mt+1}\sum_{j=0}^{mt}\frac{\Gamma_p\left(\tfrac{m}{d} +j\right)\Gamma_p\left(1-\tfrac{m}{d} +j\right)}{\Gamma_p\left(1+j\right)^2} x^{j}.\end{aligned}$$ We now use Proposition \[prop:MortensonPadic\] to write that Equation \[eqn:padicGammaSum\] is congruent modulo $p$ to the expression $$\begin{aligned}
\sum_{j=0}^{mt}\frac{\left(\tfrac{m}{d}\right)_j\left(\tfrac{d-m}{d}\right)_j}{j!^2} x^{j}.\end{aligned}$$ Note that we are summing to $mt=\tfrac{m(p-1)}{d}$. If $j>mt$ then the rising factorial $\left(\tfrac{m}{d}\right)_j$ has as a factor $$\begin{aligned}
\tfrac{m}{d}+\left(\tfrac{m(p-1)}{d}+ 1\right) -1 &= \tfrac{m}{d}+\tfrac{m(p-1)}{d}\\
%&=\tfrac{m}{d}(1+p-1)\\
&=p\cdot\tfrac{m}{d}.\end{aligned}$$ Thus, $$\begin{aligned}
\sum_{j=0}^{mt}\frac{\left(\tfrac{m}{d}\right)_j\left(\tfrac{d-m}{d}\right)_j}{j!^2} x^{j}\equiv \sum_{j=0}^{p-1}\frac{\left(\tfrac{m}{d}\right)_j\left(\tfrac{d-m}{d}\right)_j}{j!^2} x^{j} \pmod p,\end{aligned}$$ since each missing term in the summand on the left has a factor of $p$ in it.\
Noting that $(1)_j=j!$, we use Equation \[eqn:truncHGF\] to identify this as a truncated hypergeometric series $$\begin{aligned}
\sum_{j=0}^{p-1}\frac{\left(\tfrac{m}{d}\right)_j\left(\tfrac{d-m}{d}\right)_j}{j!^2} x^{j}&= {}_{2}F_{1}\left(\left.\begin{array}{cc}
\tfrac{m}{d}&\tfrac{d-m}{d}\\
{} &1
\end{array}\right|x\right)_{\text{tr}(p)}.\end{aligned}$$ Thus, we have proved that $$\begin{aligned}
{}_{2}F_{1}\left(\left.\begin{array}{cc}
\tfrac{m}{d}&\tfrac{d-m}{d}\\
{} &1
\end{array}\right|x\right)_{\text{tr}(p)}\equiv -p\hspace{.05in}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{mt}&\overline T^{mt}\\
{} &\epsilon
\end{array}\right|x\right)_p \pmod p.
\end{aligned}$$
Our next congruence result holds for hypergeometric functions that appear in our work with Dwork hypersurfaces. The proof is similar to that of Theorem \[thm:2F1congruence\] and so we omit some steps.
\[thm:dFdcongruence\] Let $p\equiv 1\pmod d$ and $T$ be a generator for the character group $\widehat{\mathbb F_p^{\times}}$ then, $$\begin{aligned}
p^{d-2}\hspace{.05in}{}_{d-1}F_{d-2}&\left(\left.\begin{array}{cccc}
T^{t}&T^{2t}&\ldots&T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|x\right)_p \\
&\equiv (-1)^d {}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
\tfrac{1}{d}&\tfrac{2}{d}&\ldots&\tfrac{d-1}{d}\\
{} &1&\ldots&1
\end{array}\right|x\right)_{\text{tr}(p)} \pmod p,
\end{aligned}$$ where $t=\tfrac{p-1}{d}$.
We will use the following corollary in our work with Dwork K3 surfaces in Section \[sec:PeriodTrace\].
\[cor:3F2congruence\] Let $p\equiv 1\pmod 4$ and $T$ be a generator for the character group $\widehat{\mathbb F_p^{\times}}$ then, $${}_{3}F_{2}\left(\left.\begin{array}{ccc}
\tfrac{1}{4}&\tfrac{2}{4}&\tfrac{3}{4}\\
{} &1&1
\end{array}\right|x\right)_{\text{tr}(p)} \equiv p^2\hspace{.05in}{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^{t}&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|x\right)_p \pmod p,$$ where $t=\tfrac{p-1}{4}$.
We start by using Equation \[eqn:HGFdef\] to write $$\begin{aligned}
p^{d-2}\hspace{.05in}{}_{d-1}F_{d-2}&\left(\left.\begin{array}{cccc}
T^{t}&T^{2t}&\ldots&T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|x\right)_p&=\frac{p^{d-1}}{p-1}\sum_{\chi}\binom{T^{t}\chi}{\chi}\binom{T^{2t}\chi}\cdots{\epsilon\chi}\binom{T^{(d-1)t}\chi}{\epsilon\chi}\chi(x).
\end{aligned}$$ Using properties of Gauss and Jacobi sums from Sections \[sec:HGF\] and \[sec:GaussJacobi\] we break this down to $$\begin{aligned}
&\frac{1}{p-1}\sum_{\chi}J(T^{t}\chi,\overline\chi)J(T^{2t}\chi,\overline\chi)\cdots J(T^{(d-1)t}\chi,\overline\chi)\chi((-1)^{d-1}x)\\
&\hspace{1in}=\frac{1}{p-1}\sum_{\chi}\frac{g(T^{t}\chi)g(T^{2t}\chi)\cdots g(T^{(d-1)t}\chi)g(\overline\chi)^{d-1}}{g(T^{t})g(T^{2t})\cdots g(T^{(d-1)t})}\chi((-1)^{d-1}x).\end{aligned}$$ We are able to simplify the denominator, but the result will be different depending on the parity of $d$. We split into two cases: $d$ even and $d$ odd.\
We start by assuming $d$ is even. In this case our expression becomes $$\begin{aligned}
=\frac{1}{p-1}\sum_{\chi}\frac{g(T^{t}\chi)g(T^{2t}\chi)\cdots g(T^{(d-1)t}\chi)g(\overline\chi)^{d-1}}{g(T^{t})g(T^{2t})\cdots g(T^{(d-1)t})}\chi(-x).\end{aligned}$$ Note that there are $d-1$ terms in the denominator, which is an odd number. We will have $\frac{d-2}{2}$ pairings of the form $g(T^{mt})g(T^{(d-m)t}) = pT^{mt}(-1)$. The remaining term that does not get paired off is $g(T^{td/2})$. Thus, our expression can be written as $$\begin{aligned}
= \frac{T^a(-1)}{p^{(d-2)/2}(p-1)g(T^{td/2})}\sum_{\chi}g(T^{t}\chi)g(T^{2t}\chi)\cdots g(T^{(d-1)t}\chi)g(\overline\chi)^{d-1}\chi(-x),\end{aligned}$$ where $a= t+2t+\ldots+\tfrac{d-2}{2}t$. We now rewrite this in terms of the Teichmüller character $\omega$ by letting $T=\overline\omega$ and $\chi=\overline\omega^{-j}$. Furthermore, we can split up the sum in a similar manner to how we did in the proof of Theorem \[thm:2F1congruence\]. We will focus on the first summand only, with $0\leq j \leq t$, since, as in the previous proof, the remaining summands are congruent to 0 modulo $p$. Thus, we have $$\begin{aligned}
\frac{\overline\omega^a(-1)}{p^{(d-2)/2}(p-1)g(\overline\omega^{td/2})}\sum_{j=0}^{t}g(\overline\omega^{t-j})g(\overline\omega^{2t-j})\ldots g(\overline\omega^{(d-1)t-j})g(\overline\omega^{j})^{d-1} \omega^{j}(-x).\end{aligned}$$ We use the Gross Koblitz formula to write this expression in terms of the $p$-adic Gamma function. Note that the resulting power of $\pi$ cancels with the $p^{(d-2)/2}$ in the denominator. $$\begin{aligned}
=\frac{-\overline\omega^a(-1)}{(-1)^{(d-2)/2}(p-1)\Gamma_p\left(\tfrac{td/2}{p-1}\right)}\sum_{j=0}^{t}\Gamma_p\left(\tfrac{t-j}{p-1}\right)\Gamma_p\left(\tfrac{2t-j}{p-1}\right)\ldots\Gamma_p\left(\tfrac{(d-1)t-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^{d-1} \omega^{j}(-x).\end{aligned}$$ We now reduce this sum modulo $p$ and simplify using the same techniques as in the proof of Theorem \[thm:2F1congruence\]. We recall that $\omega(x)\equiv x \pmod p$ for all $x$ in $\{0,\ldots,p-1\}$ and $p-1\equiv -1 \pmod p$. Furthermore, by Equation \[eqn:padicdefinition\] we have that $\Gamma_p(1+j)=(-1)^{1+j}j!$. We combine this with part 5 of Proposition \[prop:Gamma\_p\] to get $$\Gamma_p(-j)^{d-1}=\frac{1}{j!^{d-1}}.$$ Lastly, if $0\leq j\leq t$, then $\Gamma_p\left(\frac{m}{d}+j\right)=(-1)^j\left(\tfrac{m}{d}\right)_j\Gamma_p\left(\frac{m}{d}\right)$, for $m=1,2,\ldots,d-1$. Thus, our expression is congruent modulo $p$ to $$\begin{aligned}
\frac{(-1)^{a-(d-2)/2}}{\Gamma_p\left(\tfrac{d/2}{d}\right)}\left[\sum_{j=0}^{t}\frac{\left(\tfrac1d\right)_j\left(\tfrac2d\right)_j\cdots\left(\tfrac{d-1}{d}\right)_j\Gamma_p\left(\frac1d\right)\Gamma_p\left(\frac2d\right)\cdots\Gamma_p\left(\frac{d-1}{d}\right)}{j!^{d-1}}(x)^j\right].\end{aligned}$$ We use part 5 of Proposition \[prop:Gamma\_p\] to simplify the $p$-adic Gamma pairs $$\begin{aligned}
\Gamma_p\left(\frac{m}{d}\right)\Gamma_p\left(1- \frac{m}{d}\right)&=(-1)^{(d-m)t+1}.\end{aligned}$$ The resulting exponent of $-1$ from this will be $b+\tfrac{d-2}{2}$, where $b=(d-1)t+\ldots + \left(d-\tfrac{d-2}{2}\right)t$. Thus, our expression is congruent modulo $p$ to $$\begin{aligned}
\equiv (-1)^{a+b}\sum_{j=0}^{t}\frac{\left(\tfrac1d\right)_j\left(\tfrac2d\right)_j\cdots\left(\tfrac{d-1}{d}\right)_j}{j!^{d-1}}(x)^j \pmod p,\end{aligned}$$ Note that $$\begin{aligned}
a+b&=t+2t+\ldots+\tfrac{d-2}{2}t+(d-1)t+\ldots + \left(d-\tfrac{d-2}{2}\right)t\\
&=dt+\ldots+dt\\
&=dt\cdot \tfrac{d-2}{2}\\
&=(p-1)\cdot \tfrac{d-2}{2},\end{aligned}$$ which is even. Thus we can write that $(-1)^{a+b}=(-1)^{d}$.\
Now suppose that $d$ is odd. In this case our expression becomes $$\begin{aligned}
\frac{1}{p-1}\sum_{\chi}\frac{g(T^{t}\chi)g(T^{2t}\chi)\cdots g(T^{(d-1)t}\chi)g(\overline\chi)^{d-1}}{g(T^{t})g(T^{2t})\cdots g(T^{(d-1)t})}\chi(x).\end{aligned}$$
The rest of the proof is similar to the case when $d$ is even, but slightly easier. There are $d-1$ terms in the denominator, which is an even number. We will have $\frac{d-1}{2}$ pairings of the form $g(T^{mt})g(T^{(d-m)t}) = pT^{mt}(-1)$. Thus, our expression can be written as $$\begin{aligned}
\frac{T^{a'}(-1)}{p^{(d-1)/2}(p-1)}\sum_{\chi}g(T^{t}\chi)g(T^{2t}\chi)\cdots g(T^{(d-1)t}\chi)g(\overline\chi)^{d-1}\chi(x),\end{aligned}$$ where $a'= t+2t+\ldots+\tfrac{d-1}{2}t$.\
We now rewrite this in terms of the Teichmüller character $\omega$ by letting $T=\overline\omega$ and $\chi=\overline\omega^{-j}$. Furthermore, we can split up the sum in a similar manner to how we did when $d$ was even. We will focus on the first summand only, with $0\leq j \leq t$, since, as in the previous proof, the remaining summands are congruent to 0 modulo $p$. Thus, we have $$\begin{aligned}
\frac{\overline\omega^{a'}(-1)}{p^{(d-1)/2}(p-1)}\sum_{j=0}^{t}g(\overline\omega^{t-j})g(\overline\omega^{2t-j})\ldots g(\overline\omega^{(d-1)t-j})g(\overline\omega^{j})^{d-1} \omega^{j}(x).\end{aligned}$$ We use the Gross Koblitz formula to write this expression in terms of the $p$-adic Gamma function $$\begin{aligned}
=\frac{\overline\omega^{a'}(-1)}{(-1)^{(d-1)/2}(p-1)}\sum_{j=0}^{t}\Gamma_p\left(\tfrac{t-j}{p-1}\right)\Gamma_p\left(\tfrac{2t-j}{p-1}\right)\ldots\Gamma_p\left(\tfrac{(d-1)t-j}{p-1}\right)\Gamma_p\left(\tfrac{j}{p-1}\right)^{d-1} \omega^{j}(x).\end{aligned}$$ Note that the resulting power of $\pi$ canceled with the $p^{(d-1)/2}$ in the denominator. Furthermore we reduce the sum modulo $p$ and simplify using the same techniques as in the proof for $d$ even. Our expression is congruent modulo $p$ to $$\begin{aligned}
(-1)^{a'-(d-1)/2+1}\left[\sum_{j=0}^{t}\frac{\left(\tfrac1d\right)_j\left(\tfrac2d\right)_j\cdots\left(\tfrac{d-1}{d}\right)_j\Gamma_p\left(\frac1d\right)\Gamma_p\left(\frac2d\right)\cdots\Gamma_p\left(\frac{d-1}{d}\right)}{j!^{d-1}}(x)^j\right].\end{aligned}$$ We use part 5 of Proposition \[prop:Gamma\_p\] to simplify each term of the form $$\begin{aligned}
\Gamma_p\left(\frac{m}{d}\right)\Gamma_p\left(1- \frac{m}{d}\right)&=(-1)^{(d-m)t+1}.\end{aligned}$$ The resulting exponent of $-1$ from this will be $b'+\tfrac{d-2}{2}$, where $b'=(d-1)t+\ldots + \left(d-\tfrac{d-2}{2}\right)t$. Thus, our expression is congruent modulo $p$ to $$\begin{aligned}
(-1)^{a'+b'+1}\sum_{j=0}^{t}\frac{\left(\tfrac1d\right)_j\left(\tfrac2d\right)_j\cdots\left(\tfrac{d-1}{d}\right)_j}{j!^{d-1}}(x)^j.\end{aligned}$$ Note that $ a'+b'+1=\frac{d-1}{2}(p-1)+1$, which is an odd number. Thus, $(-1)^{a'+b'+1}=(-1)^{d}$, which matches the exponent of $-1$ in the case where $d$ was even.\
We now bring the two cases together. For both even and odd $d$ we have $$\begin{aligned}
p^{d-2}\hspace{.05in}{}_{d-1}F_{d-2}&\left(\left.\begin{array}{cccc}
T^{t}&T^{2t}&\ldots&T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|x\right)_p&\equiv(-1)^d \sum_{j=0}^{t}\frac{\left(\tfrac1d\right)_j\left(\tfrac2d\right)_j\cdots\left(\tfrac{d-1}{d}\right)_j}{j!^{d-1}}(x)^j \pmod p .
\end{aligned}$$ We use Equation \[eqn:truncHGF\] to identify this as a truncated hypergeometric series $$\begin{aligned}
\equiv (-1)^d {}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
\tfrac{1}{d}&\tfrac{2}{d}&\ldots&\tfrac{d-1}{d}\\
{} &1&\ldots&1
\end{array}\right|x\right)_{\text{tr}(p)} \pmod p ,\end{aligned}$$ where, as in the proof of Theorem \[thm:2F1congruence\], the terms with $j>t$ are congruent to 0 modulo $p$.\
Koblitz’s Point Count Formula {#sec:Koblitz}
=============================
Koblitz [@KoblitzHypersurface] developed a formula for the number of points on diagonal hypersurfaces in the Dwork family in terms of Gauss sums. We specialize this formula to the family of Dwork K3 surfaces, i.e. to the case when $d,n=4, h=1$.\
Let $W$ be the set of all 4-tuples $w=(w_1,w_2,w_3,w_4)$ satisfying $0\leq w_i<4$ and $\sum w_i\equiv 0 \pmod 4$. We denote the points on the diagonal hypersurface $$x_1^4+x_2^4+x_3^4+x_4^4=0$$ by $N_q(0):=\sum N_{q}(0,w)$, where $$N_{q}(0,w)=
\begin{cases}
0 &\text{if some but not all } w_i=0,\\
\frac{q^3-1}{q-1} &\text{if all } w_i=0,\\
-\frac1q J\left(T^{\tfrac{w_1}{4}},T^{\tfrac{w_2}{4}},T^{\tfrac{w_3}{4}},T^{\tfrac{w_4}{4}}\right) &\text{if all } w_i\not=0.\\
\end{cases}$$
[@KoblitzHypersurface Theorem 2]\[thm:KoblitzHypersurface\] The number of points on the Dwork K3 surface $X_{\lambda}^4$ is given by $$\#X_{\lambda}^4(\mathbb F_q)=N_q(0)+\frac1{q-1}\sum\frac{\prod_{i=1}^4g\left(T^{w_it+j}\right)}{g(T^{4j})}T^{4j}(4\lambda)$$ where the sum is taken over $j\in\{0,\ldots,q-2\}$ and $w=(w_1,w_2,w_3,w_4)$ in $W$.\
We wish to simplify this formula. We start by considering the term $N_q(0)$.
\[prop:N0\] $N_q(0)=q^2+7q+1 + \frac1q\sum_{i=1}^3g(T^{it})^4 +12qT^t(-1)$.
The list of possible 4-tuples (up to reordering since the order doesn’t matter in the Jacobi sum) is $$W^*=\{ (1,1,1,1)^1, (2,2,2,2)^1, (3,3,3,3)^1, (1,1,3,3)^6, (1,2,2,3)^{12}\},$$ $w=(0,0,0,0)$, and all of the 4-tuples where some, not all, of the $w_i=0$ (we exclude this list since $N_{q}(0,w)=0$ for these tuples). Letting $t=\frac{q-1}{4}$ we see that $$J\left(T^{\tfrac{w_1}{4}},T^{\tfrac{w_2}{4}},T^{\tfrac{w_3}{4}},T^{\tfrac{w_4}{4}}\right)= -\prod_i g(T^{w_it}).$$ Thus, we have $$\begin{aligned}
N_q(0) &= \sum_w N_q(0,w)\\
&=\frac{q^3-1}{q-1} +\sum_{w\in W^*}N_q(0,w)\\
&=q^2+q+1 + \frac1q\left[\sum_{i=1}^3g(T^{it})^4 +6g(T^t)^2g(T^{3t})^2+12g(T^t)g(T^{2t})^2g(T^{3t}) \right]\\
%&=q^2+q+1 + \frac1q\left[\sum_{i=1}^3g(T^{it})^4 +6q^2T^t(-1)T^{t}(-1)+12q^2T^t(-1)T^{2t}(-1) \right]\\
&=q^2+q+1 + \frac1q\left[\sum_{i=1}^3g(T^{it})^4 +6q^2+12q^2T^t(-1) \right]\\
&=q^2+7q+1 + \frac1q\sum_{i=1}^3g(T^{it})^4 +12qT^t(-1).\end{aligned}$$
Define the equivalence relation $\sim$ on $W$ by $w\sim w'$ if $w-w'$ is a multiple of $(1,1,1,1)$. Up to permutation, there are three cosets (up to permutation) in $W/\sim$: $$(0,0,0,0)^1, (0,1,1,2)^{12}, (0,0,2,2)^3.$$ The first coset contains the obvious four elements. The second set of 12 cosets is made up of permutations of the coset $\overline{(0,1,1,2)}=\{(0,1,1,2),(1,2,2,3), (2,3,3,0), (3,0,0,1)\}$. The third set of 3 cosets is made up of permutations of the coset $\overline{(0,0,2,2)}=\{(0,0,2,2),(1,1,3,3)\}$. This last set of cosets is different in that some permutations are in the same coset. For example, the element $(2,2,0,0)$ is a permutation of $(0,0,2,2)$ but they are also in the same coset.\
Let $$\label{eqn:DefS_w}
S_{[w]}=\frac{1}{q-1}\sum_{j=0}^{q-2}\frac{\prod_{i=1}^4g\left(T^{w_it +j}\right)}{g\left(T^{4j}\right)}T^{4j}(4\lambda)$$ denote the summands corresponding to all $w'\in[w]$. Our main tool for simplifying terms of this form is the Hasse-Davenport relation for Gauss sums.\
In the propositions that follow, we give explicit formulas for each $S_{[w]}$.
\[prop:0000\] Let $w=(0,0,0,0)$. Then $$S_{[w]}=-\frac{1}{q}\sum_{i=1}^3g\left(T^{it}\right)^4+q^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_q$$
Note that the term $-\frac{1}{q}\sum_{i=1}^3g\left(T^{it}\right)^4$ negates a term in the overall point count (see Theorem \[prop:N0\]).\
By Equation \[eqn:DefS\_w\] we have $$\begin{aligned}
S_{(0,0,0,0)}&=\frac{1}{q-1}\sum_{j=0}^{q-2}\frac{g\left(T^{j}\right)^4}{g\left(T^{4j}\right)}T^{4j}(4\lambda).\end{aligned}$$ If $t\mid j$, i.e. if $j=t,2t,3t$, then $$\begin{aligned}
\frac{g\left(T^{j}\right)^4}{g\left(T^{4j}\right)}T^{4j}(4\lambda)&=-g\left(T^{j}\right)^4.\end{aligned}$$ Thus, $$\begin{aligned}
S_{(0,0,0,0)}&=-\frac1{q-1}\sum_{i=1}^3g\left(T^{it}\right)^4 + \frac{1}{q-1}\sum_{j=0, t\nmid j}^{q-2}\frac{g\left(T^{j}\right)^4}{g\left(T^{4j}\right)}T^{4j}(4\lambda)\\
&= -\frac1{q-1}\sum_{i=1}^3g\left(T^{it}\right)^4 + \frac{1}{q-1}\sum_{j=0, t\nmid j}^{q-2}\frac{g\left(T^{j}\right)^4g\left(T^{-4j}\right)}{T^{4j}(-1)q}T^{4j}(4\lambda)\\
&= -\frac1{q-1}\sum_{i=1}^3g\left(T^{it}\right)^4 + \frac{1}{q(q-1)}\sum_{j=0, t\nmid j}^{q-2}g\left(T^{j}\right)^4g\left(T^{-4j}\right)T^{4j}(4\lambda).\end{aligned}$$ Note that if $t\mid j$ then $$\begin{aligned}
g\left(T^{j}\right)^4g\left(T^{-4j}\right)T^{4j}(4\lambda)&=-g\left(T^{j}\right)^4.\end{aligned}$$ Hence, $$\begin{aligned}
S_{(0,0,0,0)}&=-\frac1{q-1}\sum_{i=1}^3g(T^{it})^4 +\frac{1}{q(q-1)}\sum_{i=1}^3g(T^{it})^4+ \frac{1}{q(q-1)}\sum_{j=0}^{q-2}g(T^{j})^4g(T^{-4j})T^{4j}(4\lambda)\\
&=\frac{-q+1}{q(q-1)}\sum_{i=1}^3g(T^{it})^4 +\frac{1}{q(q-1)}\sum_{j=0}^{q-2}g(T^{j})^4g(T^{-4j})T^{4j}(4\lambda)\\
&=-\frac{1}{q}\sum_{i=1}^3g(T^{it})^4 +\frac{1}{q(q-1)}\sum_{j=0}^{q-2}g(T^{j})^4g(T^{-4j})T^{4j}(4\lambda).\end{aligned}$$ Working from the other direction we see that $$\begin{aligned}
{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_q &= \frac{q}{q-1}\sum_{\chi}\binom{T^t\chi}{\chi}\binom{T^{2t}\chi}{\epsilon\chi}\binom{T^{3t}\chi}{\epsilon\chi}\chi(1/{\lambda^4})\\
&=\frac{q}{q-1}\sum_{\chi} \left(\frac{\chi(-1)}{q}\right)^3 J(T^{t}\chi,\overline\chi)J(T^{2t}\chi,\overline\chi)J(T^{3t}\chi,\overline\chi)\overline\chi({\lambda^4})\\
&=\frac{1}{q^2(q-1)}\sum_{\chi} \frac{g(T^{t}\chi)g(T^{2t}\chi)g(T^{3t}\chi)g(\overline\chi)^3}{\prod_{i=1}^3 g(T^{it})}\overline\chi(-{\lambda^4}).
\end{aligned}$$ Use the Hasse-Davenport relation $$\prod_{i=1}^3\frac{g\left(T^{it}\chi\right)}{g\left(T^{it}\right)}=\frac{g(\chi^4)\chi^{-4}(4)}{g(\chi)}$$ to get $$\begin{aligned}
{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_q &= \frac{1}{q^2(q-1)}\sum_{\chi} \frac{g(\chi^4)\chi^{-4}(4)g(\overline\chi)^3}{g(\chi)}\overline\chi(-{\lambda^4})\\
&= \frac{1}{q^2(q-1)}\sum_{\chi} \frac{g(\chi^4)\chi^{-4}(4)g(\overline\chi)^4}{\chi(-1)q}\overline\chi(-{\lambda^4})\\
%&= \frac{1}{q^3(q-1)}\sum_{\chi} g(\chi^4)\chi^{-4}(4)g(\overline\chi)^4\overline\chi({\lambda^4})\\
&= \frac{1}{q^3(q-1)}\sum_{j=0}^{q-2}g(T^j)^4 g(T^{-4j})T^{4j}(4{\lambda}), \end{aligned}$$ which proves the desired result.
\[prop:0112\] Let $w=(0,1,1,2)$. Then $$S_{[w]}=12qT^t(-1)\left(T^{2t}(1-\lambda^4)-1\right).$$
Note that the term $-12qT^t(-1)$ negates a term in the overall point count (see Theorem \[prop:N0\]) Also note that $T^{2t}(1-\lambda^4)=0$ when $\lambda^4=1$, so the final expression for the point count when $\lambda^4=1$ is as simple as we might expect.
\[cor:0112b\] Let $w=(0,1,1,2)$ and $\lambda^4=1$. Then $$S_{[w]}=-12qT^t(-1).$$
By Equation \[eqn:DefS\_w\] we have $$\begin{aligned}
S_{(0,1,1,2)}&=\frac{12}{q-1}\sum_{j=0}^{q-2}\frac{g\left(T^{j}\right)g\left(T^{t+j}\right)^2g\left(T^{2t+j}\right)}{g\left(T^{4j}\right)}T^{4j}(4\lambda).
\end{aligned}$$ If $j=t$ then $$\begin{aligned}
\frac{g\left(T^{j}\right)g\left(T^{t+j}\right)^2g\left(T^{2t+j}\right)}{g\left(T^{4j}\right)}T^{4j}(4\lambda)&=\frac{g\left(T^{t}\right)g\left(T^{2t}\right)^2g\left(T^{3t}\right)}{g\left(T^{4t}\right)}T^{4t}(4\lambda)\\
&=\frac{T^t(-1)q\cdot T^{2t}(-1)q}{-1}\\
&=-T^t(-1)q^2.
\end{aligned}$$ For the terms with $j\not=t$ we use Corollary \[cor:HasseDavenport\] to write $$\begin{aligned}
\sum_{j=0}^{q-2}\frac{g\left(T^{j}\right)g\left(T^{t+j}\right)^2g\left(T^{2t+j}\right)}{g\left(T^{4j}\right)}T^{4j}(4\lambda)
&=g(T^{2t})T^t(-1)q\\
&\hspace{.2in}\times\sum_{j=0, j\not=t}^{q-2}\frac{g\left(T^{j}\right)g\left(T^{t+j}\right)^2g\left(T^{2t+j}\right)}{g\left(T^{j}\right)g\left(T^{t+j}\right)g\left(T^{2t+j}\right)g\left(T^{3t+j}\right)}T^{4j}(\lambda)\\
&=g(T^{2t})T^t(-1)q\sum_{j=0, j\not=t}^{q-2}\frac{g\left(T^{t+j}\right)}{g\left(T^{3t+j}\right)}T^{4j}(\lambda)\\
&=g(T^{2t})T^t(-1)q\sum_{j=0, j\not=t}^{q-2}\frac{g\left(T^{t+j}\right)g\left(T^{t-j}\right)}{T^{3t+j}(-1)q}T^{4j}(\lambda)\\
&=g(T^{2t})\sum_{j=0, j\not=t}^{q-2}g\left(T^{t+j}\right)g\left(T^{t-j}\right)T^j(-1)T^{4j}(\lambda).
\end{aligned}$$ Note that if $j=t$ then $$\begin{aligned}
g(T^{2t})\cdot g\left(T^{t+j}\right)g\left(T^{t-j}\right)T^j(-1)T^{4j}(\lambda)&=g\left(T^{2t}\right)^2g\left(T^{0}\right)T^t(-1)T^{4t}(\lambda)\\
&=-T^t(-1)q.
\end{aligned}$$ Thus, $$\begin{aligned}
&g(T^{2t})\sum_{j=0, j\not=t}^{q-2}g(T^{t+j})g(T^{t-j})T^j(-1)T^{4j}(\lambda)\\
&\hspace{1in}=g(T^{2t})\sum_{j=0}^{q-2}g(T^{t+j})g(T^{t-j})T^j(-1)T^{4j}(\lambda)+T^t(-1)q\\
&\hspace{1in}=q(q-1)T^t(-1)T^{2t}(1-\lambda^4)+T^t(-1)q\end{aligned}$$ by Proposition \[prop:gaussproduct\]. Hence, $$\begin{aligned}
S_{(0,1,1,2)}&=\frac{12}{q-1}\left(q(q-1)T^t(-1)T^{2t}(1-\lambda^4)+T^t(-1)q -T^t(-1)q^2\right)\\
&=\frac{12}{q-1}\left(q(q-1)T^t(-1)T^{2t}(1-\lambda^4)-T^t(-1)q(q-1)\right)\\
&=12qT^t(-1)\left(T^{2t}(1-\lambda^4)-1\right).\end{aligned}$$
The remaining piece of the point count formula for the Dwork K3 surface family is the term associated to $S_{[w]}$ with $w=(0,0,2,2)$. It’s interesting that this term can be expressed in terms of a ${}_2F_1$ hypergeometric function. Note that the ${}_2F_1$ that appears is of a different form than what has been observed in other point-count formulas [@AhlgrenOno00a; @Fuselier10; @Koike1995; @Lennon1; @McCarthy2012b; @Mortenson2003a; @Swisher2015Arxiv], where the lower characters are all the trivial character.
\[prop:0022\] Let $w=(0,0,2,2)$. For all $\lambda\not\equiv 0 \pmod q$, $$S_{[w]}=-6q+3q^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_q.$$
\[cor:0022\] As we will see in Equation \[eqn:S0022lambda1\], this expression simplifies nicely when $\lambda^4=1$. In this case, $$S_{[w]}=-6q+3qT^t(-1).$$
The proof starts out similar to the proof of Proposition \[prop:0112\]. By definition we have $$\label{eqn:S0022}
S_{(0,0,2,2)}=\frac{3}{q-1}\sum_{j=0}^{q-2}\frac{g\left(T^{j}\right)^2g\left(T^{2t+j}\right)^2}{g\left(T^{4j}\right)}T^{4j}(4\lambda).$$ We use Corollary \[cor:HasseDavenport\] to write $$\begin{aligned}
S_{(0,0,2,2)}&=\frac{3g(T^{2t})T^t(-1)q}{q-1}\sum_{j=0}^{q-2}\frac{g\left(T^{j}\right)^2g\left(T^{2t+j}\right)^2}{g\left(T^{j}\right)g\left(T^{t+j}\right)g\left(T^{2t+j}\right)g\left(T^{3t+j}\right)}T^{4j}(\lambda)\\
&=\frac{3g(T^{2t})T^t(-1)q}{q-1}\sum_{j=0}^{q-2}\frac{g\left(T^{j}\right)g\left(T^{2t+j}\right)}{g\left(T^{t+j}\right)g\left(T^{3t+j}\right)}T^{4j}(\lambda).
\end{aligned}$$ If $j=t$ then $$\begin{aligned}
\frac{g\left(T^{j}\right)g\left(T^{2t+j}\right)}{g\left(T^{t+j}\right)g\left(T^{3t+j}\right)}T^{4j}(\lambda)&=\frac{g\left(T^{t}\right)g\left(T^{3t}\right)}{g\left(T^{2t}\right)g\left(T^{4t}\right)}T^{4t}(\lambda)\\
&=\frac{T^t(-1)q}{-g(T^{2t})}.
\end{aligned}$$ Similarly, if $j=3t$ then $$\begin{aligned}
\frac{g\left(T^{j}\right)g\left(T^{2t+j}\right)}{g\left(T^{t+j}\right)g\left(T^{3t+j}\right)}T^{4j}(\lambda)%&=\frac{g\left(T^{3t}\right)g\left(T^{t}\right)}{g\left(T^{4t}\right)g\left(T^{2t}\right)}T^{12t}(\lambda)\\
&=\frac{T^t(-1)q}{-g(T^{2t})}.
\end{aligned}$$ For the terms with $j\not=t,3t$ we have $$\begin{aligned}
\sum_{j=0,j\not=t,3t}^{q-2}\frac{g\left(T^{j}\right)g\left(T^{2t+j}\right)}{g\left(T^{t+j}\right)g\left(T^{3t+j}\right)}T^{4j}(\lambda)&=\sum_{j=0,j\not=t,3t}^{q-2}\frac{g\left(T^{j}\right)g\left(T^{2t+j}\right)g\left(T^{t-j}\right)g\left(T^{3t-j}\right)}{T^{t+j}(-1)q\cdot T^{3t+j}(-1)q}T^{4j}(\lambda)\\
&=\frac1{q^2}\sum_{j=0,j\not=t,3t}^{q-2}g\left(T^{j}\right)g\left(T^{2t+j}\right)g\left(T^{t-j}\right)g\left(T^{3t-j}\right)T^{4j}(\lambda).\end{aligned}$$ Note that if $j=t$ then $$\begin{aligned}
\frac{1}{q^2}g\left(T^{j}\right)g\left(T^{2t+j}\right)g\left(T^{t-j}\right)g\left(T^{3t-j}\right)T^{4j}(\lambda)&=\frac{1}{q^2}g\left(T^{t}\right)g\left(T^{3t}\right)g\left(T^{0}\right)g\left(T^{2t}\right)T^{4t}(\lambda)\\
%&=-\frac{g(T^{2t})T^t(-1)q}{q^2}\\
&=-\frac{g(T^{2t})T^t(-1)}{q}.\end{aligned}$$ The same holds for $j=3t$. Hence, $$\begin{aligned}
S_{(0,0,2,2)}&=\frac{3g(T^{2t})T^t(-1)q}{q-1}\left[\frac1{q^2}\sum_{j=0}^{q-2}g(T^{j})g(T^{2t+j})g(T^{t-j})g(T^{3t-j})T^{4j}(\lambda)\right.\\
&\hspace{2in} \left.+\frac{2g(T^{2t})T^t(-1)}{q}+\frac{2T^t(-1)q}{-g(T^{2t})}\right].\end{aligned}$$ Observe that $$\begin{aligned}
\frac{3g(T^{2t})T^t(-1)q}{q-1}\left[\frac{2g(T^{2t})T^t(-1)}{q}+\frac{2T^t(-1)q}{-g(T^{2t})}\right]&=\frac{6g(T^{2t})}{q-1}\left[g(T^{2t})-\frac{q^2}{g(T^{2t})}\right]\\
&=\frac{6}{q-1}\left(T^{2t}(-1)q-{q^2}\right)\\
%&=\frac{6}{q-1}\left(q(1-q)\right)\\
&=-6q.\end{aligned}$$ To simplify $$\label{eqn:S0022a}
\frac{3g(T^{2t})T^t(-1)q}{q-1}\left[\frac1{q^2}\sum_{j=0}^{q-2}g(T^{j})g(T^{2t+j})g(T^{t-j})g(T^{3t-j})T^{4j}(\lambda)\right],$$ we consider two cases. We first restrict to the case where $\lambda^4=1$ and apply Theorem \[thm:Helversen1978\] to our formula to get $$\begin{aligned}
\frac1{q-1}\sum_{j=0}^{q-2}g(T^{j})g(T^{2t+j})g(T^{t-j})g(T^{3t-j})&=\frac{g(T^t)g(T^{3t})g(T^{3t})g(T^t)}{g(T^{2t})}\\
%&=\frac{T^t(-1)q\cdot T^t(-1)q}{g(T^{2t})}\\
&=\frac{q^2}{g(T^{2t})}.\end{aligned}$$ Hence, when $\lambda^4=1$ we have $$\begin{aligned}
\label{eqn:S0022lambda1}
\begin{split}
S_{(0,0,2,2)}&=-6q+\frac{3g(T^{2t})T^t(-1)q}{q^2}\left(\frac{q^2}{g(T^{2t})}\right)\\
&=-6q+3qT^t(-1).
\end{split}\end{aligned}$$ For all other $\lambda\not=0$ we proceed as we did in the proof of Proposition \[prop:gaussproduct\]. Recalling that $g(\chi)=\sum_x \chi(x)\theta(x)$ we can write $$\begin{aligned}
&\sum_{j=0}^{q-2}g(T^{j})g(T^{2t+j})g(T^{t-j})g(T^{3t-j})T^{4j}(\lambda)\\
&\hspace{1in}= \sum_{x,y,z,w}T^{2t}(y)T^t(z)T^{3t}(w)\theta(x+y+z+w)\sum_{j=0}^{q-2}T^j\left(\tfrac{xy\lambda^4}{zw}\right),\end{aligned}$$ where $x,y,z,w\not=0$. Note that $T^j\left(\tfrac{xy\lambda^4}{zw}\right)=q-1$ if $\tfrac{xy\lambda^4}{zw}=1$ and equals 0 otherwise. Hence, letting $x=\tfrac{zw}{y\lambda^4}$, the sum simplifies to $$\begin{aligned}
(q-1)\sum_{y,z,w}T^{2t}(y)T^t(z)T^{3t}(w)\theta\left(\tfrac{zw}{y\lambda^4}+y+z+w\right).\end{aligned}$$ Since $y$ and $\lambda$ are both nonzero, we can perform the change of variables $z\rightarrow zy\lambda^4$ and get $$\begin{aligned}
(q-1)\sum_{y,z,w}T^{2t}(y)T^t(zy\lambda^4)T^{3t}(w)\theta\left(zw+y+zy\lambda^4+w\right).\end{aligned}$$ Since $T^t(\lambda^4)=1$ we get $$\begin{aligned}
(q-1)\sum_{y,z,w}T^{3t}(y)T^t(z)T^{3t}(w)\theta\left(w(z+1)+y(1+z\lambda^4)\right).\end{aligned}$$ Note that if $z=-1$ then the above expression equals $$\begin{aligned}
(q-1)\sum_{y,w}T^{3t}(y)T^t(z)T^{3t}(w)\theta\left(y(1-\lambda^4)\right)&=(q-1)\sum_{y}T^{3t}(y)T^t(z)\theta\left(y(1-\lambda^4)\right)\sum_wT^{3t}(w),\end{aligned}$$ which equals 0 since $T^{3t}\not=\epsilon$ implies $\sum_wT^{3t}(w)=0$. Similarly, the expression equals 0 when $z=-\lambda^{-4}$.\
For $z\not=-1, -\lambda^{-4}$ we can perform the changes of variables $w\rightarrow\tfrac{w}{z+1}$ and $y\rightarrow\tfrac{y}{1+z\lambda^{4}}$. This portion of the sum then becomes $$\begin{aligned}
(q-1)\sum_{y,w}T^{3t}(y)T^{3t}(w)\theta(w+y)\sum_{z\not=-1,-\lambda^{-4}}T^t(z)T^{-3t}(1+z\lambda^{4})T^{-3t}(z+1),\end{aligned}$$ where $$\sum_{z\not=-1,-\lambda^{-4}}T^t(z)T^{-3t}(1+z\lambda^{4})T^{-3t}(z+1)=\sum_{z\not=-1,-\lambda^{-4}}T^t(z)T^{t}(1+z\lambda^{4})T^{t}(z+1).$$ Note that if $z=-1$ or $z=-\lambda^{-4}$ then $$T^t(z)T^{t}(1+z\lambda^{4})T^{t}(z+1)=0$$ so that we can include these $z-$values in the sum to get $$\begin{aligned}
(q-1)\sum_{y,w}T^{3t}(y)T^{3t}(w)\theta(w+y)\sum_{z}T^t(z)T^{t}(1+z\lambda^{4})T^{t}(z+1).\end{aligned}$$ This expression reduces further since $$\sum_{y,w}T^{3t}(y)T^{3t}(w)\theta(w+y)=g(T^{3t})g(T^{3t}).$$ Furthermore, using the change of variables $z\rightarrow -z$ we see that $$\begin{aligned}
\sum_{z}T^t(z)T^{t}(1+z\lambda^{4})T^{t}(z+1)&=\sum_{z}T^t(-z)T^{t}(1-z\lambda^{4})T^{t}(-z+1)\\
&=T^t(-1)\sum_{z}T^t(z)T^{t}(1-z\lambda^{4})T^{t}(1-z)\\
&=\frac{q}{\epsilon(\lambda^4)}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\lambda^4\right)_q\end{aligned}$$ where the last expression is obtained by using Equation \[eqn:2F1def\] with $A=T^{3t}, B=T^t$, and $C=T^{2t}$.\
Thus we have shown that the original Gauss sum expression can be written as $$q(q-1)g(T^{3t})^2\frac{q}{\epsilon(\lambda^4)}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\lambda^4\right)_q.$$
Putting all of this work together leads to $$\begin{aligned}
S_{(0,0,2,2)}&=-6q+\frac{3g(T^{2t})T^t(-1)q}{q-1}\cdot\frac1{q^2}\left[q(q-1)g(T^{3t})^2\frac{q}{\epsilon(\lambda^4)}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\lambda^4\right)_q\right]\\
&=-6q+3g(T^{2t})g(T^{3t})^2T^t(-1){}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\lambda^4\right)_q,\end{aligned}$$ for $\lambda^4\not=1$.\
We now show that this expression may also be used in the case where $\lambda^4=1$. We use Equation \[eqn:2F1def\] and properties of Gauss and Jacobi sums to get $$\begin{aligned}
&3g(T^{2t})g(T^{3t})^2T^t(-1){}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|1\right)_q\\
&\hspace{1in}= 3g(T^{2t})g(T^{3t})^2T^t(-1)\cdot\frac{T^{3t}(-1)}{q}\sum_y T^t(y)T^t(1-y)T^t(1-y)\\
&\hspace{1in}=\frac{3g(T^{2t})g(T^{3t})^2}{q}\sum_y T^t(y)T^{2t}(1-y)\\
&\hspace{1in}=\frac{3g(T^{2t})g(T^{3t})^2}{q}J(T^t,T^{2t})\\
&\hspace{1in}=\frac{3g(T^{2t})g(T^{3t})^2}{q}\frac{g(T^t)g(T^{2t})}{g(T^{3t})}\\
%&\hspace{1in}=\frac{3g(T^{2t})g(T^{3t})g(T^t)g(T^{2t})}{q}\\
% &\hspace{1in}=\frac{3q^2T^t(-1)}{q}\\
&\hspace{1in}=3qT^t(-1).\end{aligned}$$ Hence, for all $\lambda\not\equiv0 \pmod q$, $$S_{[w]}=-6q+3g(T^{2t})g(T^{3t})^2T^t(-1){}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\lambda^4\right)_q.$$ Note that $g(T^{3t})^2=g(T^{2t})J(T^{3t},T^{3t})$, hence $$\begin{aligned}
g(T^{2t})g(T^{3t})^2T^t(-1)&=g(T^{2t})^2J(T^{3t},T^{3t})T^t(-1)\\
&=qT^t(-1)J(T^{3t},T^{3t})\\
&=q^2\binom{T^{3t}}{T^t},\end{aligned}$$ where the last equality holds by Equation \[eqn:normalizedjacobi\]. Thus we can write $$S_{[w]}=-6q+3q^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\lambda^4\right)_q.$$
We finish the proof by using Theorem 4.2 of [@Greene] to rewrite the hypergeometric term. $$\begin{aligned}
{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\lambda^4\right)_q &= T^{2t}(-1)T^{3t}(\lambda^4){}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_q\\
&={}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_q\end{aligned}$$
Proof of Theorem \[thm:K3PointCount\] {#sec:DworkSurfaces}
=====================================
In Section \[sec:Koblitz\] we found that $$\#X_{\lambda}^4(\mathbb F_q)= N_q(0)+ S_{(0,0,0,0)}+S_{(0,0,2,2)}+S_{(0,1,1,2)}.$$ Combining the results of Propositions \[prop:N0\], \[prop:0000\], and \[prop:0022\] gives us $$\begin{aligned}
\#X_{\lambda}^4(\mathbb F_q)&=q^2+7q+1 + \frac1q\sum_{i=1}^3g(T^{it})^4 +12qT^t(-1)-\frac{1}{q}\sum_{i=1}^3g\left(T^{it}\right)^4\\
& \hspace{1in}+q^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_q \\
&\hspace{1in}-6q+3q^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_q\\
&\hspace{1in}+ 12qT^t(-1)\left(T^{2t}(1-\lambda^4)-1\right) \\\\
&=\frac{q^3-1}{q-1}+12qT^t(-1)T^{2t}(1-\lambda^4)\\
&\hspace{.2in}+q^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_q+3q^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_q.\\
\end{aligned}$$ In the case where $\lambda^4=1$, this, combined with Corollaries \[cor:0112b\] and \[cor:0022\], gives us $$\begin{aligned}
\#X_{\lambda}^4(\mathbb F_q)&=\frac{q^3-1}{q-1}+3qT^t(-1)+q^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|1\right)_q.\end{aligned}$$
Proof of Theorems \[thm:K3PointCountnGn\] and \[thm:K3PointCountnGn1\] {#sec:K3PointCountnGn}
======================================================================
In this section we prove our two $p-$adic hypergeometric point count formulas.
Let $N_{p}^A(\lambda)$ denote the number of points on the Dwork K3 surface in $\mathbb A^4(\mathbb F_p)$. Then $$\label{eqn:padicPointCont}
\#X_{\lambda}^4(\mathbb F_p)=\frac{N_{p}^A(\lambda)-1}{p-1}.$$ Letting $f(\overline x)=x_1^4+x_2^4+x_3^4+x_4^4-4\lambda x_1x_2x_3x_4$ we can write $$\begin{aligned}
pN_{p}^A(\lambda)&=p^4+\sum_{z\in\mathbb F_p^*}\sum_{x_1,x_2,x_3,x_4}\theta(zf(\overline x))\\
&=p^4+\sum_{z\in\mathbb F_p^*}\sum_{x_i\not=0}\theta(zf(\overline x))+\sum_{z\in\mathbb F_p^*}\sum_{x_i, some =0}\theta(zf(\overline x)).\end{aligned}$$
We will call the first summand A and the second B. We first work to rewrite B. We can have 1, 2, 3, or 4 of the $x_i$’s equal to zero, and there are 4, 6, 4, and 1 way, respectively, for this to occur. We will call these sums $B_1, B_2, B_3,$ and $B_4$, respectively. $B_4=p-1$ because $\theta$ is an additive character. We can simplify the others using basic facts about characters and Gauss sums $$\begin{aligned}
B_1 &=4\sum_{z\in\mathbb F_p^*}\sum_{x_i\not=0}\theta(zx_1^4)\theta(zx_2^4)\theta(zx_3^4)\\
&=\frac{4}{(p-1)^3}\sum_{x_i,z\in\mathbb F_p^*}\sum_{a,b,c=0}^{p-2} g(T^{-a})T^{4a}(x_1)g(T^{-b})T^{4b}(x_2)g(T^{-c})T^{4c}(x_3)T^{a+b+c}(z)\\
&=\frac{4}{(p-1)^3}\sum_{a,b,c=0}^{p-2}g(T^{-a})g(T^{-b})g(T^{-c})\sum_{x_1}T^{4a}(x_1)\sum_{x_2}T^{4b}(x_2)\sum_{x_3}T^{4c}(x_3)\sum_{z}T^{a+b+c}(z).\end{aligned}$$ This sum is non-zero only when the following congruences hold: $$4a,4b,4c\equiv 0 \pmod{p-1},\text{ and } a+b+c\equiv 0 \pmod{p-1}.$$ Since $p\not\equiv 1\pmod 4$, these congruences simultaneously hold only when 0 or 2 of $a,b,c$ are $\frac{p-1}{2}$ and the remaining terms are 0. In this case, each character sum is $p-1$. Note that there are 3 ways to have two of $a,b,c$ equal to zero. Thus, $$\begin{aligned}
B_1&=4(p-1)\left(g(T^0)^3+3g\left(T^{(p-1)/2}\right)^2g(T^0)\right)\\
&=4(p-1)\left(-1+3(-1p)(-1)\right)\\
&=4(p-1)(3p-1).\end{aligned}$$
Similarly, $$\begin{aligned}
B_2 &=6\sum_{z\in\mathbb F_p^*}\sum_{x_i\not=0}\theta(zx_1^4)\theta(zx_2^4)\\
&=\frac{6}{(p-1)^2}\sum_{a,b=0}^{p-2}g(T^{-a})g(T^{-b})\sum_{x_1}T^{4a}(x_1)\sum_{x_2}T^{4b}(x_2)\sum_{z}T^{a+b}(z).\end{aligned}$$ This sum is non-zero only when the following congruences hold: $$4a,4b\equiv 0 \pmod{p-1},\text{ and } a+b\equiv 0 \pmod{p-1}.$$ Since $p\not\equiv 1\pmod 4$, these congruences simultaneously hold only when 0 or 2 of $a,b$ are $\frac{p-1}{2}$. Thus, $$\begin{aligned}
B_2&=6(p-1)\left(g(T^0)^2+g(T^{(p-1)/2})^2\right)\\
&=-6(p-1)(p-1).\end{aligned}$$
Finally, $$\begin{aligned}
B_3 &=4\sum_{z\in\mathbb F_p^*}\sum_{x_1\not=0}\theta(zx_1^4)\\
&=\frac{4}{p-1}\sum_{a=0}^{p-2}g(T^{-a})\sum_{x_1}T^{4a}(x_1)\sum_{z}T^{a}(z).\end{aligned}$$ This sum is non-zero only when $a=0$. Thus, $$\begin{aligned}
B_2&=4(p-1)g(T^0)=-4(p-1).\end{aligned}$$
Putting this all together gives $$\begin{aligned}
B&=B_1+B_2+B_3+B_4\\
&=4(p-1)(3p-1)-6(p-1)(p-1)-4(p-1)+(p-1)\\
&=(p-1)(6p-1).\end{aligned}$$
Rewriting summand A requires more work and we will end up with $p-$adic hypergeomtric functions. $$\begin{aligned}
A&=\sum_{z\in\mathbb F_p^*}\sum_{x_i\not=0}\theta(zf(\overline x))\\
&=\sum_{z\in\mathbb F_p^*}\sum_{x_i\not=0}\theta(zx_1^4)\theta(zx_2^4)\theta(zx_3^4)\theta(zx_4^4)\theta(-4\lambda x_1x_2x_3x_4)\\
&=\frac{1}{(p-1)^5}\sum_{i,j,k,l,m=0}^{p-2} g(T^{-i})g(T^{-j})g(T^{-k})g(T^{-l})g(T^{-m})T^{m}(-4\lambda)\sum T^{4i+m}(x_1)\\
&\hspace{1.5in} \times \sum T^{4j+m}(x_2)\sum T^{4k+m}(x_3)\sum T^{4l+m}(x_4)\sum T^{i+j+k+l+m}(z).\end{aligned}$$ We consider congruences that must hold for $i,j,k,l,m$ as we did for summand B. This sum is non-zero only when the following congruences hold: $$4i+m,4j+m, 4k+m, 4l+m\equiv 0 \pmod{p-1},\text{ and } i+j+k+l+m\equiv 0 \pmod{p-1}.$$ There are two cases. In the first case, which we will denote by $A_1$, $i,j,k,l$ are equal in pairs. Here we need two equal to each other and the remaining two equal to that value plus $\frac{p-1}{2}$. For example, $i=j$ and $k=l=i+\frac{p-1}{2}$. In this case we have $m\equiv -4i \pmod{p-1}$. There are 3 ways for this to occur. In the second case we have $i=j=k=l$ and $m\equiv -4j \pmod{p-1}$. We will denote this case by $A_2$.\
We now work to rewrite $A_1$. We start with $$\begin{aligned}
A_1&=3\sum_{i=0}^{p-2} g(T^{-i})^2g(T^{-(i+(p-1)/2})^2g(T^{4i})T^{-4i}(-4\lambda).\end{aligned}$$ We use the Hasse-Davenport relation with $\chi^{(p-1)/2}$ and $\psi=T^{-i}$ to get $$\begin{aligned}
g(T^{-(i+(p-1)/2})^2&=\left(\frac{-g(T^{-2i})T^{2i}(2)g(T^0)g(T^{((p-1)/2})}{g(T^{-i})}\right)^2\\
&=\frac{-pg(T^{-2i})^2T^{2i}(4)}{g(T^{-i})^2}.\end{aligned}$$ Thus, $$\begin{aligned}
A_1&=-3p\sum_{i=0}^{p-2} g(T^{-2i})^2g(T^{4i})T^{2i}(4)T^{-4i}(-4\lambda)\\
&=6p-3p\sum_{i\not=0,(p-1)/2} g(T^{-2i})^2g(T^{4i})T^{2i}(4)T^{-4i}(-4\lambda)\end{aligned}$$ We multiply the summand by $g(T^{2i})/g(T^{2i})$ to get $$\begin{aligned}
A_1&=6p-3p^2\sum_{i\not=0,(p-1)/2} \frac{g(T^{-2i})g(T^{4i})T^{2i}(4)T^{-4i}(-4\lambda)}{g(T^{2i})}.\end{aligned}$$
As we have done in previous proofs, we let $T=\omega$, the Teichmüller character. Thus, $$\begin{aligned}
A_1&=6p-3p^2\sum_{i\not=0,(p-1)/2} \frac{g(\overline\omega^{2i})g(\overline\omega^{-4i})\overline\omega^{-2i}(4)\overline\omega^{4i}(4\lambda)}{g(\overline\omega^{-2i})}.\end{aligned}$$ We use the Gross-Koblitz formula to write this in terms of the $p-$adic Gamma function. $$\begin{aligned}
A_1&=6p+3p^2\sum_{i\not=0,(p-1)/2} \frac{(-p)^a\Gamma_p\left(\left\langle\frac{2i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{-4i}{p-1}\right\rangle\right)\overline\omega^{4i}(2\lambda)}{\Gamma_p\left(\left\langle\frac{-2i}{p-1}\right\rangle\right)},\end{aligned}$$ where $$\begin{aligned}
a&=\left\langle\frac{2i}{p-1}\right\rangle+\left\langle\frac{-4i}{p-1}\right\rangle-\left\langle\frac{-2i}{p-1}\right\rangle\\
&=\frac{2i}{p-1}-\left\lfloor\frac{2i}{p-1}\right\rfloor+\frac{-4i}{p-1}-\left\lfloor\frac{-4i}{p-1}\right\rfloor-\frac{-2i}{p-1}+\left\lfloor\frac{-2i}{p-1}\right\rfloor\\
&=\left\lfloor\frac{-2i}{p-1}\right\rfloor-\left\lfloor\frac{2i}{p-1}\right\rfloor-\left\lfloor\frac{-4i}{p-1}\right\rfloor.\end{aligned}$$ On page 232 of [@McCarthy2013] we see that $$\left\lfloor\frac{-2i}{p-1}\right\rfloor-\left\lfloor\frac{4i}{p-1}\right\rfloor= -\left\lfloor\frac34-\frac{i}{p-1}\right\rfloor-\left\lfloor\frac14-\frac{i}{p-1}\right\rfloor.$$ Furthermore, $$\left\lfloor\frac{2i}{p-1}\right\rfloor=\begin{cases}
0 & \text{if } i< \frac{p-1}{2},\\
1 & \text{if } i\geq \frac{p-1}{2},
\end{cases}$$ so that $\left\lfloor\frac{2i}{p-1}\right\rfloor=\left\lfloor\frac{i}{p-1}\right\rfloor+\left\lfloor\frac12+\frac{i}{p-1}\right\rfloor$. Thus, $$a=-\left\lfloor\frac34-\frac{i}{p-1}\right\rfloor-\left\lfloor\frac14-\frac{i}{p-1}\right\rfloor-\left\lfloor\frac{i}{p-1}\right\rfloor-\left\lfloor\frac12+\frac{i}{p-1}\right\rfloor.$$
We use Lemma 4.1 of [@McCarthy2013] to rewrite the $p-$adic Gamma functions that appear in the summand. This yields the following expression for $A_1$. $$\begin{aligned}
A_1&=6p+3p^2\sum_{i\not=0,(p-1)/2}(-p)^a\cdot \frac{\Gamma_p\left(\left\langle\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac12+\frac{i}{p-1}\right\rangle\right)\prod_{h=0}^3\Gamma_p\left(\left\langle\frac{1+h}{4}-\frac{i}{p-1}\right\rangle\right)\overline\omega^{i}(\lambda^4)}{\prod_{h=1}^3\Gamma_p\left(\frac{h}{4}\right)\cdot\Gamma_p\left(\left\langle\frac12-\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle1-\frac{i}{p-1}\right\rangle\right)}\\
&=6p+3p^2\sum_{i\not=0,(p-1)/2}(-p)^a\cdot \frac{\Gamma_p\left(\left\langle\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac12+\frac{i}{p-1}\right\rangle\right)}{\Gamma_p\left(\frac{1}{4}\right)\Gamma_p\left(\frac{2}{4}\right)\Gamma_p\left(\frac{3}{4}\right)}\\
&\hspace{2in}\times\Gamma_p\left(\left\langle\frac{1}{4}-\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{3}{4}-\frac{i}{p-1}\right\rangle\right)\overline\omega^{i}(\lambda^4).\end{aligned}$$ If $i=0$ or $(p-1)/2$, then $$\begin{aligned}
(-p)^a\cdot \frac{\Gamma_p\left(\left\langle\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac12+\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{1}{4}-\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{3}{4}-\frac{i}{p-1}\right\rangle\right)\overline\omega^{i}(\lambda^4)}{\Gamma_p\left(\frac{1}{4}\right)\Gamma_p\left(\frac{2}{4}\right)\Gamma_p\left(\frac{3}{4}\right)}&=(-p)^{0}=1.\end{aligned}$$
Thus, $$\begin{aligned}
A_1&=6p-6p^2+3p^2\sum_{i=0}^{p-2}(-p)^a\cdot\frac{\Gamma_p\left(\left\langle\frac{1}{4}-\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{3}{4}-\frac{i}{p-1}\right\rangle\right)}{\Gamma_p\left(\left\langle\frac{-1}{4}\right\rangle\right)\Gamma_p\left(\left\langle\frac{-3}{4}\right\rangle\right)}\frac{\Gamma_p\left(\left\langle\frac{i}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac12+\frac{i}{p-1}\right\rangle\right)}{\Gamma_p\left(0\right)\Gamma_p\left(\left\langle\frac12\right\rangle\right)}.\end{aligned}$$ We recognize this sum as the following $p-$adic hypergeometric function expression $$A_1=6p-6p^2-3(p-1)p^2{}_2G_2\left[\left.\begin{array}{ccc}
3/4&1/4\\
0&1/2
\end{array}\right|{\lambda^4}\right]_p.$$
We now work to rewrite $A_2$. In this case we have $$\begin{aligned}
A_2&=\sum_j g(T^{-j})^4g(T^{4j})T^{-4j}(-4\lambda).\end{aligned}$$ As we have done in previous proofs, we let $T=\omega$, the Teichmüller character. Then $$\begin{aligned}
A_2&=\sum_{j=0}^{p-2} g(\overline\omega^{j})^4g(\overline\omega^{-4j})\overline\omega^{4j}(4\lambda)\\
&=-1+\sum_{j=1}^{p-2} g(\overline\omega^{j})^4g(\overline\omega^{-4j})\overline\omega^{4j}(4\lambda)\\
&=-1+p\sum_{j=1}^{p-2} \frac{g(\overline\omega^{j})^3g(\overline\omega^{-4j})\overline\omega^{j}(-1)\overline\omega^{4j}(4\lambda)}{g(\overline\omega^{-j})}.\end{aligned}$$ We would like to rewrite the summand in terms of the $p$-adic Gamma function. To do this, we use the Gross-Koblitz formula. $$\begin{aligned}
A_2&=-1-p\sum_{j=1}^{p-2} (-p)^k\frac{\Gamma_p\left(\langle\frac{j}{p-1}\rangle\right)^3\Gamma_p\left(\langle\frac{-4j}{p-1}\rangle\right)}{\Gamma_p\left(\langle\frac{-j}{p-1}\rangle\right)}\overline\omega^{j}(-1)\overline\omega^{4j}(4\lambda),\end{aligned}$$ where $$\begin{aligned}
k&=3\left\langle\frac{j}{p-1}\right\rangle+\left\langle\frac{-4j}{p-1}\right\rangle-\left\langle\frac{-j}{p-1}\right\rangle\\
&=\frac{3j}{p-1}-3\left\lfloor\frac{j}{p-1}\right\rfloor+\frac{-4j}{p-1}-\left\lfloor\frac{-4j}{p-1}\right\rfloor-\frac{-j}{p-1}+\left\lfloor\frac{-j}{p-1}\right\rfloor\\
&=-\left\lfloor\frac{-4j}{p-1}\right\rfloor-1\\
&=\begin{cases}
0 & \text{if } 0< j\leq \frac{p-1}{4},\\
1 & \text{if } \frac{p-1}{4}< j\leq \frac{2(p-1)}{4},\\
2 & \text{if } \frac{2(p-1)}{4}< j\leq \frac{3(p-1)}{4},\\
3 & \text{if } \frac{3(p-1)}{4}< j\leq p-2.
\end{cases}\end{aligned}$$ We use Lemma 4.1 in [@McCarthy2013] to rewrite $\Gamma_p\left(\left\langle \frac{-j}{p-1}\right\rangle\right)$ and $ \Gamma_p\left(\left\langle \frac{-4j}{p-1}\right\rangle\right)$. This leads to $$\begin{aligned}
A_2&=-1-p\sum_{j=1}^{p-2} (-p)^k\frac{\Gamma_p\left(\langle\frac{j}{p-1}\rangle\right)^3\prod_{h=0}^{3}\Gamma_p\left(\left\langle\frac{1+h}{4}-\frac{j}{p-1}\right\rangle\right)}{\Gamma_p\left(\langle1-\frac{j}{p-1}\rangle\right)\prod_{h=1}^{3}\Gamma_p\left(\frac{h}{4}\right)}\overline\omega^{j}(-\lambda^4)\\
&=-1-p\sum_{j=1}^{p-2} (-p)^k\frac{\Gamma_p\left(\left\langle\frac{j}{p-1}\right\rangle\right)^3}{\Gamma_p(0)^3}\cdot\prod_{i=1}^3\frac{\Gamma_p\left(\left\langle\frac{i}{4}-\frac{j}{p-1}\right\rangle\right)}{\Gamma_p\left(\frac{i}{4}\right)}\overline\omega^{j}(-\lambda^4).\end{aligned}$$ This expression can be written in terms of a $p$-adic hypergeometric function. By Definition \[def:padicHGF\], we have $$\begin{aligned}
{}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p &:=\frac{-1}{p-1}\sum_{j=0}^{p-2}\overline\omega^j(-\lambda^4)\prod_{i=1}^3\frac{\Gamma_p\left(\left\langle\frac{i}{4}-\frac{j}{p-1}\right\rangle\right)}{\Gamma_p\left(\frac{i}{4}\right)}\cdot\frac{\Gamma_p\left(\left\langle\frac{j}{p-1}\right\rangle\right)^3}{\Gamma_p(0)^3}\\
&\hspace{1in}\cdot(-p)^{-\lfloor\frac14-\frac{j}{p-1}\rfloor-\lfloor\frac24-\frac{j}{p-1}\rfloor-\lfloor\frac34-\frac{j}{p-1}\rfloor-3\lfloor\frac{j}{p-1}\rfloor}.\end{aligned}$$ Note that when $j=0$, the summand is simply $-\frac{1}{p-1}$. Our main task now is to determine what the power of $-p$ is for other values of $j$. First note that since $0\leq j\leq p-2$, we have $\lfloor\frac{j}{p-1}\rfloor=0$. For $i=1,2,3$ we have $$\left\lfloor\frac{i}{4}-\frac{j}{p-1}\right \rfloor=\begin{cases}
0 & \text{if }\frac{j}{p-1}\leq \frac{i}{4},\\
-1 & \text{if }\frac{j}{p-1}> \frac{i}{4}.
\end{cases}$$ Thus, the exponent of $-p$ is $$-\lfloor\tfrac14-\tfrac{j}{p-1}\rfloor-\lfloor\tfrac24-\tfrac{j}{p-1}\rfloor-\lfloor\tfrac34-\tfrac{j}{p-1}\rfloor-3\lfloor\tfrac{j}{p-1}\rfloor= \begin{cases}
0 & \text{if } 0< j\leq \frac{p-1}{4},\\
1 & \text{if } \frac{p-1}{4}< j\leq \frac{2(p-1)}{4},\\
2 & \text{if } \frac{2(p-1)}{4}< j\leq \frac{3(p-1)}{4},\\
3 & \text{if } \frac{3(p-1)}{4}< j\leq p-2.
\end{cases}$$ Note how this coincides with the powers of $-p$ in the summand $A_2$. Thus, $$\begin{aligned}
{}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p&= \\
&\hspace{-.2in}\frac{-1}{p-1}\left[ 1+\sum_{j=1}^{p-2} (-p)^k\frac{\Gamma_p\left(\left\langle\frac{j}{p-1}\right\rangle\right)^3}{\Gamma_p(0)^3}\prod_{i=1}^3\frac{\Gamma_p\left(\left\langle\frac{i}{4}-\frac{j}{p-1}\right\rangle\right)}{\Gamma_p\left(\frac{i}{4}\right)}\overline\omega^{j}(-\lambda^4)\right].\end{aligned}$$ This allows us to conclude that $$\begin{aligned}
A_2=p(p-1) {}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p +p-1.\end{aligned}$$
We finish the proof by combining these results to get an expression for $ \#X_{\lambda}^4(\mathbb F_p)$. First note that $$\begin{aligned}
A+B&=B+A_1+A_2\\
&=(p-1)(6p-1)+6p-6p^2-3(p-1)p^2{}_2G_2\left[\left.\begin{array}{ccc}
3/4&1/4\\
0&1/2
\end{array}\right|{\lambda^4}\right]_p\\
&\hspace{1in}+p(p-1){}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p +p-1\\
&=-3(p-1)p^2{}_2G_2\left[\left.\begin{array}{ccc}
3/4&1/4\\
0&1/2
\end{array}\right|{\lambda^4}\right]_p+p(p-1){}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p.\end{aligned}$$ We now substitute this into Equation \[eqn:padicPointCont\]. $$\begin{aligned}
\#X_{\lambda}^4(\mathbb F_p)&=\frac{N_{p}^A(\lambda)-1}{p-1}\\
&=\frac{\frac1p(p^4+A+B)-1}{p-1}\\
&=\frac{p^3-1}{p-1}-3p\hspace{.02in}{}_2G_2\left[\left.\begin{array}{ccc}
3/4&1/4\\
0&1/2
\end{array}\right|{\lambda^4}\right]_p+{}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p\end{aligned}$$
We start with the point count result over $\mathbb F_p$ of Theorem \[thm:K3PointCount\] $$\begin{aligned}
\#X_{\lambda}^4(\mathbb F_p)&=p^2+p+12pT^t(-1)T^{2t}(1-\lambda^4)+1\\
&\hspace{.2in}+p^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_p+3p^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_p\end{aligned}$$
We use transformation properties from [@McCarthy2013; @McCarthy2012c] to rewrite the two finite field hypergeometric expressions in terms of McCarthy’s $p-$adic hypergeometric function. In [@McCarthy2012c], McCarthy defines a new finite field hypergeometric function, normalized to satisfy transformation properties based on summation properties of Gauss sums. He also gives the relationship between McCarthy’s hypergeometric function and Greene’s hypergeometric function in [@McCarthy2012c Prop. 2.5]. $$\begin{aligned}
\label{eqn:GreeneMcCarthyTrans}
{}_{n+1}F_{n}\left(\left.\begin{array}{cccc}
A_0,&A_1,&\ldots,&A_n\\
{} &B_1,&\ldots,&B_n
\end{array}\right|x\right)^M_p&= \prod_{i=1}^n\binom{A_i}{B_i}^{-1} {}_{n+1}F_{n}\left(\left.\begin{array}{cccc}
A_0,&A_1,&\ldots,&A_n\\
{} &B_1,&\ldots,&B_n
\end{array}\right|x\right)_p.
\end{aligned}$$ Furthermore, in [@McCarthy2013 Lemma 3.3] McCarthy gives the following relationship between his finite field hypergeometric function and his $p$-adic hypergeometric function. For a fixed odd prime $p$, let $A_i,B_k$ be given by $\overline\omega^{a_i(p-1)}$ and $\overline\omega^{b_k(p-1)}$ respectively. Then $$\begin{aligned}
\label{eqn:McCarthyTransnGn}
{}_{n+1}F_{n}\left(\left.\begin{array}{cccc}
A_0,&A_1,&\ldots,&A_n\\
{} &B_1,&\ldots,&B_n
\end{array}\right|x\right)^M_p&={}_{n+1}G_{n+1}\left[\left.\begin{array}{cccc}
a_0&a_1&\ldots&a_n\\
0&b_1&\ldots&b_n
\end{array}\right|x^{-1}\right]_p\end{aligned}$$ We use these two properties to rewrite the hypergeometric function expressions in Theorem \[thm:K3PointCount\]. We start with the ${}_3F_2$ term. Equation 2.12 in [@Greene] states that for a character $A\in\widehat{\mathbb F_q^{\times}}$ we have $$\binom{A}{\epsilon}=-\frac1p.$$ We use this and Equation \[eqn:GreeneMcCarthyTrans\] above to write $$\begin{aligned}
p^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_p &= p^2\binom{T^{2t}}{\epsilon}\binom{T^{3t}}{\epsilon} {}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_p^M\\
&= p^2\left(-\frac1p\right)\left(-\frac1p\right){}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_p^M\\
&= {}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_p^M.\end{aligned}$$ Equation \[eqn:McCarthyTransnGn\] then tells us that this is equal to $${}_3G_3\left[\left.\begin{array}{ccc}
1/4&2/4&3/4\\
0&0&0
\end{array}\right|{\lambda^4}\right]_p.$$
We now work to rewrite the ${}_3F_2$ term. First note that $$\begin{aligned}
\binom{T^{3t}}{T^t}\binom{T^t}{T^{2t}}&=\frac{g(T^{3t})g(T^{3t})T^t(-1)}{g(T^{2t}p}\cdot\frac{g(T^{t})g(T^{2t})T^{2t}(-1)}{g(T^{3t}p}\\
&=\frac{g(T^{3t})g(T^{t})T^t(-1)}{p^2}\\
% &=\frac{pT^t(-1)\cdot T^t(-1)}{p^2}\\
&=\frac1p.\end{aligned}$$ We use this and Equations \[eqn:GreeneMcCarthyTrans\] and \[eqn:McCarthyTransnGn\] to write $$\begin{aligned}
3p^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_p&=3p^2\binom{T^{3t}}{T^t}\binom{T^t}{T^{2t}}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_p^M\\
&=3p{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_p^M\\
&=3p{}_2G_2\left[\left.\begin{array}{cc}
3/4&1/4\\
0&2/4
\end{array}\right|\lambda^4\right]_p.\end{aligned}$$
We have rewritten each finite field hypergeometric expression as a $p$-adic hypergeometric expression, and so we have proved the desired result.
Dwork K3 Surface Period Integrals {#sec:DworkPeriod}
=================================
In this section we give a formula for certain period integrals associated to Dwork K3 surfaces. The periods we are interested in are obtained by choosing dual bases of the space of holomorphic differentials and the space of cycles ${H^2(X_\lambda^4,\mathscr O)}$ and integrating the differentials over each cycle. Note that since K3 surfaces have genus $g=1$, the dimension of both spaces is 1 and, so, we get a single period integral. The natural choice for a basis of differentials is the nowhere vanishing holomorphic $2$-form.\
The following theorem tells us that the period obtained in this way is a solution to a hypergeometric differential equation.
\[prop:K3picardfuchs\][@Nagura1995 Section 3.2] The Picard-Fuchs equation for the Dwork K3 surface is $$\label{eqn:K3picardfuchs}
\left(\vartheta^3-z(\vartheta+1/4)(\vartheta+2/4)(\vartheta+3/4)\right)\pi=0,$$ where $\vartheta=z\frac{d}{dz}$ and $z=\lambda^{-4}$.
The proposition below gives us a classical hypergeometric series expression for the period. The result is not original, but we prove it to review the procedure.
\[prop:K3period\] The solution to Equation \[eqn:K3picardfuchs\] that is bounded near $z=0$ is given by $$\label{eqn:K3period}
\pi = {}_{3}F_{2}\left(\left.\begin{array}{ccc}
1/4&2/4&3/4\\
{} &1&1
\end{array}\right|\frac{1}{\lambda^4}\right).$$
Define $D:=\vartheta^3-z(\vartheta+1/4)(\vartheta+2/4)(\vartheta+3/4)$. Let $f(z)=\sum_m a_mz^m$ be a solution to $Df=0$, normalized so that $f(0)=1$, and let $f':=\frac{df}{dz}$. We see that $$\begin{aligned}
\left(\vartheta^3\right)f&=\left(z\frac{d}{dz}\right)^3f\\
&=\left(z\frac{d}{dz}\right)^2(zf')\\
&=\left(z\frac{d}{dz}\right)(zf'+z^2f'')\\
%&=zf'+z^2f''+2z^2f''+z^3f'''\\
&=z^3f'''+3z^2f''+zf'.
\end{aligned}$$ Furthermore, $$\begin{aligned}
(\vartheta+1/4)(\vartheta+2/4)(\vartheta+3/4)f&=(\vartheta+1/4)(\vartheta+2/4)(zf'+\tfrac34f)\\
%&=(\vartheta+1/4)\left(\tfrac24zf'+\tfrac{6}{16}f+zf'+z^2f''+\tfrac34zf'\right)\\
&=(\vartheta+1/4)(z^2f''+\tfrac94zf'+\tfrac38f)\\
%&=\tfrac14z^2f''+\tfrac{9}{16}zf'+\tfrac{3}{32}f+2z^2f''+z^3f'''+\tfrac94zf'+\tfrac94z^2f''+\tfrac38zf'\\
&=z^3f'''+\tfrac92z^2f''+\tfrac{51}{16}zf'+\tfrac{3}{32}f.\end{aligned}$$ Hence, $$\begin{aligned}
Df&=z^3f'''+3z^2f''+zf'-z(z^3f'''+\tfrac92z^2f''+\tfrac{51}{16}zf'+\tfrac{3}{32}f)\\
&=z^3(1-z)f'''+z^2(3-\tfrac92z)f''+z(1-\tfrac{51}{16}z)f'-\tfrac{3}{32}zf.\end{aligned}$$ Taking derivatives of $f$ gives us $$\begin{aligned}
f'&=\sum_m (m+1)a_{m+1}z^m,\\
f''&=\sum_m (m+2)(m+1)a_{m+2}z^m,\\
f'''&=\sum_m (m+3)(m+2)(m+1)a_{m+3}z^m.\end{aligned}$$ We substitute these into the equation $Df=0$ to get $$\begin{aligned}
\sum_m \left(z^3(1-z)(m+3)(m+2)(m+1)a_{m+3}+z^2(3-\tfrac92z)(m+2)(m+1)a_{m+2}\right.&{}\\
\left.+z(1-\tfrac{51}{16}z)(m+1)a_{m+1}-\tfrac{3}{32}za_m\right)z^m&=0. \end{aligned}$$ From the expression on the left side of the above equation we identify the coefficients of $z^m$ to rewrite the equation as $$\begin{aligned}
%\sum_m \left(\left((m+1)+3m(m+1)+(m-1)(m)(m+1)\right)a_{m+1} - \left( \tfrac{3}{32}+\tfrac{51}{16}m+\tfrac92m(m-1)+m(m-1)(m-2)\right)a_m\right)z^m\\
\sum_m \left((m+1)^3a_{m+1}-\tfrac{1}{64}(4m+1)(4m+2)(4m+3)a_m\right)z^m =0\end{aligned}$$ This gives the relationship $$\begin{aligned}
a_{m+1}&=\frac{(4m+1)(4m+2)(4m+3)}{64(m+1)^3}a_m\\
&=\frac{(m+1/4)(m+2/4)(m+3/4)}{(m+1)^3}a_m.\end{aligned}$$ Given that $a_0=1$ (so that $f(0)=1$) we get that $$\begin{aligned}
a_m=\frac{\left(\frac14\right)_m\left(\frac24\right)_m\left(\frac34\right)_m}{(1)_m(1)_m m!}.\end{aligned}$$ We now use Equation \[eqn:classicalHGF\] to express the function $f$ as the hypergeometric function $$f= {}_{3}F_{2}\left(\left.\begin{array}{ccc}
1/4&2/4&3/4\\
{} &1&1
\end{array}\right|z\right).$$ We conclude the proof by setting $\pi=f$ and by recalling that $z=\frac{1}{\lambda^4}$.
Proof of Theorem \[thm:K3PeriodTrace\] {#sec:PeriodTrace}
--------------------------------------
Our final result for Dwork K3 surfaces relates the trace of Frobenius to the period associated to the surface.
We denote the trace of Frobenius of the Dwork K3 surface over $\mathbb F_p$ by $a_{X^4_{\lambda}}(p)$. Recall that the trace of Frobenius and the point count of Dwork K3 surface are related in the following way $$a_{X^4_{\lambda}}(p)= \#X_{\lambda}^4(\mathbb F_p)-p^2-1.$$ See [@Manin1986 Theorem 27.1] for a proof of this. Thus, we have that the trace is given by $$\begin{aligned}
a_{X^4_{\lambda}}(p)&=p+ 12pT^t(-1)T^{2t}(1-\lambda^4)\\
&\hspace{1in}+p^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_p
+3p^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|\frac{1}{\lambda^4}\right)_p.\end{aligned}$$ We will show that, modulo $p$, the only term that remains is the ${}_3F_2$ hypergeometric function. We have the following lemma for the ${}_2F_1$ term.
\[lem:K3PeriodTrace\]
$$\begin{aligned}
3p^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|x\right)_p \equiv 0 \pmod p.\end{aligned}$$
Recall from the proof of Proposition \[prop:0022\] that $$\begin{aligned}
p^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|x\right)_p&=g(T^{2t})g(T^{3t})^2T^t(-1){}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|x\right)_p. \end{aligned}$$
We rewrite the hypergeometric function as $$\begin{aligned}
{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|x\right)_p&= \frac{p}{p-1}\sum_{\chi}\binom{T^{3t}\chi}{\chi}\binom{T^{t}\chi}{ T^{2t}\chi}\chi(x)\\
&=\frac{1}{p(p-1)}\sum_{\chi} J(T^{3t}\chi,\overline\chi)J(T^{t}\chi,T^{2t}\overline\chi) \chi(-x)\\
&=\frac{1}{p(p-1)}\sum_{\chi} \frac{g(T^{3t}\chi)g(\overline\chi)}{g(T^{3t})}\frac{g(T^{t}\chi)g(T^{2t}\overline\chi)}{g(T^{3t})} \chi(-x).\end{aligned}$$ As we have done in previous proofs, we let $T=\overline\omega$ and $\chi=\overline\omega^{-j}$, where $\omega$ is the Teichmüller character and rewrite our Gauss sum expression to get $$\begin{aligned}
&p^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|x\right)_p\\
&\hspace{1in}=\frac{g(\overline\omega^{2t})g(\overline\omega^{3t})^2\overline\omega^t(-1)}{p(p-1)}\sum_{j=0}^{p-2} \frac{g(\overline\omega^{3t-j})g(\overline\omega^j)}{g(\overline\omega^{3t})}\frac{g(\overline\omega^{t-j})g(\overline\omega^{2t+j})}{g(\overline\omega^{3t})}\overline\omega^{-j}(-x)\\
&\hspace{1in}=\frac{g(\overline\omega^{2t})\overline\omega^t(-1)}{p(p-1)}\sum_{j=0}^{p-2} g(\overline\omega^{3t-j})g(\overline\omega^j)g(\overline\omega^{t-j})g(\overline\omega^{2t+j})\overline\omega^{-j}(-x).\end{aligned}$$ Using the Gross-Koblitz formula from Theorem \[thm:GrossKoblitz\] allows us to rewrite this as $$\begin{aligned}
\frac{-\pi^{2t}\overline\omega^t(-1)}{p(p-1)}\sum_{j=0}^{p-2} \pi^{6t} \Gamma_p\left(\frac{3t-j}{p-1}\right) \Gamma_p\left(\frac{j}{p-1}\right) \Gamma_p\left(\frac{t-j}{p-1}\right) \Gamma_p\left(\frac{2t+j}{p-1}\right) \Gamma_p\left(\frac{2t}{p-1}\right)\overline\omega^{-j}(-x).\end{aligned}$$ Since $\pi^{2t}\cdot\pi^{6t}=\pi^{8t}=p^2$, the above expression is equal to $$\begin{aligned}
\frac{-p\overline\omega^t(-1)}{p-1}\sum_{j=0}^{p-2} \Gamma_p\left(\frac{3t-j}{p-1}\right) \Gamma_p\left(\frac{j}{p-1}\right) \Gamma_p\left(\frac{t-j}{p-1}\right) \Gamma_p\left(\frac{2t+j}{p-1}\right) \Gamma_p\left(\frac{2t}{p-1}\right)\overline\omega^{-j}(-x).\end{aligned}$$ Recall that $\omega(x)\equiv x \pmod p$ for all $x$ in $\{0,\ldots,p-1\}$ and $p-1\equiv -1 \pmod p$. Thus, the above sum is congruent modulo $p$ to the expression $$\begin{aligned}
p(-1)^t\sum_{j=0}^{p-2} \Gamma_p\left(\frac34+j\right) \Gamma_p\left(-j\right) \Gamma_p\left(\frac14 +j\right) \Gamma_p\left(\frac24-j\right) \Gamma_p\left(\frac24\right)(-x)^{j}.\end{aligned}$$ This expression is congruent to 0 modulo $p$ since $\Gamma_p: \mathbb Z_p \rightarrow \mathbb Z_p^*$. Hence, $$\begin{aligned}
p^2\binom{T^{3t}}{T^t}{}_{2}F_{1}\left(\left.\begin{array}{cc}
T^{3t}&T^{t}\\
{} &T^{2t}
\end{array}\right|x\right)_p &\equiv 0 \pmod p.\end{aligned}$$
We can now write that $$\begin{aligned}
a_{X^4_{\lambda}}(p)&\equiv p^2{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^t&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|\frac{1}{\lambda^4}\right)_p \pmod p.\end{aligned}$$
Recall that in Corollary \[cor:3F2congruence\] we showed that $$\begin{aligned}
{}_{3}F_{2}\left(\left.\begin{array}{ccc}
\tfrac{1}{4}&\tfrac{2}{4}&\tfrac{3}{4}\\
{} &1&1
\end{array}\right|x\right)_{\text{tr}(p)} \equiv p^2\hspace{.05in}{}_{3}F_{2}\left(\left.\begin{array}{ccc}
T^{t}&T^{2t}&T^{3t}\\
{} &\epsilon&\epsilon
\end{array}\right|x\right)_p \pmod p.\end{aligned}$$
This truncated series is equal to $$\begin{aligned}
\sum_{j=0}^{t}\frac{\left(\tfrac14\right)_j\left(\tfrac24\right)_j\left(\tfrac34\right)_j}{j!^3}x^j.\end{aligned}$$
Note that for the truncated series and the classical series we have the congruence $$\begin{aligned}
\sum_{j=0}^{t}\frac{\left(\tfrac14\right)_j\left(\tfrac24\right)_j\left(\tfrac34\right)_j}{j!^3}x^j \equiv \sum_{j=0}^{\infty}\frac{\left(\tfrac14\right)_j\left(\tfrac24\right)_j\left(\tfrac34\right)_j}{j!^3}x^j \pmod p,\end{aligned}$$ since the terms with $j>t$ are congruent to 0 modulo $p$. Thus, we have $$\begin{aligned}
a_{X^4_{\lambda}}(p)&\equiv {}_{3}F_{2}\left(\left.\begin{array}{ccc}
1/4&2/4&3/4\\
{} &1&1
\end{array}\right|\frac{1}{\lambda^4}\right) \pmod p.\end{aligned}$$
The expression on the right side of the above congruence is the classical hypergeometric series that we used in Equation \[eqn:K3period\] to express the period of the K3 surface, so we have proved the result.
Higher Dimensional Dwork Hypersurfaces {#sec:DworkHypersurface}
======================================
We have made partial progress on similar point count and period results for higher dimensional Dwork hypersurfaces $$X_{\lambda}^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=d\lambda x_1x_2\cdots x_d.$$
Point Count for Prime Powers $q\equiv 1 \pmod d$
------------------------------------------------
We begin with a point count formula that holds for prime powers $q\equiv 1\pmod d$. This is a partial breakdown of Koblitz’s point count formula of Section \[sec:Koblitz\] and demonstrates that we should expect to be able to develop hypergeometric point count formulas for a large class of varieties.
\[thm:DworkHyp\] Let $q\equiv 1\pmod d$, $t=\frac{q-1}{d}$, and $T$ be a generator for $\widehat{\mathbb F_q^{\times}}$. The number of points over $\mathbb F_q$ on the Dwork hypersurface is given by $$\begin{aligned}
\#X_{\lambda}^d(\mathbb F_q)&=\frac{q^{d-1}-1}{q-1} + q^{d-2}{}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
T^t&T^{2t}&\ldots &T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|\frac{1}{\lambda^d}\right)_q\\
&\hspace{.5in} +\frac1q\sum_{\overline w\in W^{**}} \prod_i g(T^{w_it})+\frac1{q-1}\sum_{\overline w\not=\overline0}\sum_{j=0}^{p-2}\frac{\prod_{i=1}^dg\left(T^{w_it+j}\right)}{g(T^{dj})}T^{dj}(d\lambda),\end{aligned}$$ where the $d-$tuples $\overline w=(w_1,\ldots,w_d)$ satisfy $0\leq w_i<d$ and $\sum w_i\equiv 0 \pmod d$, and $W^{**}$ is the subset with $w_i\not=0$ and $w_i$ not all equal.
Note that in the case where $d=4$, i.e. for Dwork K3 surfaces, the point count formula had a main ${}_3F_2$ hypergeometric term, a ${}_2F_1$ term, and a sum of multiples of $q$. It seems likely that we will have a similar breakdown of terms in the higher dimensional case.
This proof follows our work in proving Theorem \[thm:K3PointCount\]. Let $W$ be the set of all $d$-tuples $w=(w_1,\ldots,w_d)$ satisfying $0\leq w_i<d$ and $\sum w_i\equiv 0 \pmod d$. We denote the points on the diagonal hypersurface $$x_1^d+\ldots+x_d^d=0$$ by $N_q(0):=\sum N_{q}(0,w)$, where $$N_{q}(0,w)=
\begin{cases}
0 &\text{if some but not all } w_i=0,\\
\frac{q^{d-1}-1}{q-1} &\text{if all } w_i=0,\\
-\frac1q J\left(T^{\tfrac{w_1}{d}},\ldots,T^{\tfrac{w_d}{d}}\right) &\text{if all } w_i\not=0.\\
\end{cases}$$ Koblitz’s formula in this general case is as follows. $$\#X_{\lambda}^d(\mathbb F_q)=N_q(0)+\frac1{q-1}\sum\frac{\prod_{i=1}^dg\left(T^{w_it+j}\right)}{g(T^{4j})}T^{dj}(d\lambda)$$ where the sum is taken over $j\in\{0,\ldots,q-2\}$ and $w\in W$.\
Letting $W^*$ be set of all $d-$tuples where no $w_i=0$, we can write $$\begin{aligned}
N_{q}(0,w)&=\frac{q^{d-1}-1}{q-1} + \frac1q\sum_{w\in W^*} \prod_i g(T^{w_it}).\end{aligned}$$
As in the Section \[sec:Koblitz\] we consider cosets of $W$ with respect to the equivalence relation $\sim$ on $W$ defined by $w\sim w'$ if $w-w'$ is a multiple of $(1,\ldots,1)$. In the case where $d=4$ we had three cosets and their permutations. For general $d$, we should expect many more cosets. Regardless of the value of $d$, however, one of the cosets will be the zero element $w=(0,\ldots,0)^1$. We show that the summand associated to this coset can be expressed as a finite field hypergeometric function.\
When $w=(0,\ldots,0)$ we have $$\begin{aligned}
S_{(0,0,0,0)}&=\frac{1}{q-1}\sum_{j=0}^{q-2}\frac{g\left(T^{j}\right)^d}{g\left(T^{dj}\right)}T^{dj}(d\lambda)\end{aligned}$$
If $t\mid j$, then $$\begin{aligned}
\frac{g\left(T^{j}\right)^d}{g\left(T^{dj}\right)}T^{dj}(d\lambda)&=-g\left(T^{j}\right)^d.\end{aligned}$$ Thus, $$\begin{aligned}
S_{(0,0,0,0)}&=-\frac1{q-1}\sum_{i=1}^{d-1}g\left(T^{it}\right)^d + \frac{1}{q-1}\sum_{j=0, t\nmid j}^{q-2}\frac{g\left(T^{j}\right)^d}{g\left(T^{dj}\right)}T^{dj}(d\lambda)\\
&= -\frac1{q-1}\sum_{i=1}^{d-1}g\left(T^{it}\right)^d + \frac{1}{q-1}\sum_{j=0, t\nmid j}^{q-2}\frac{g\left(T^{j}\right)^dg\left(T^{-dj}\right)}{T^{dj}(-1)q}T^{dj}(d\lambda)\\
&= -\frac1{q-1}\sum_{i=1}^{d-1}g\left(T^{it}\right)^d + \frac{1}{q(q-1)}\sum_{j=0, t\nmid j}^{q-2}g\left(T^{j}\right)^dg\left(T^{-dj}\right)T^{dj}(-d\lambda).\end{aligned}$$ Note that if $t\mid j$ then $$\begin{aligned}
g\left(T^{j}\right)^dg\left(T^{-dj}\right)T^{dj}(-d\lambda)&=-g\left(T^{j}\right)^d.\end{aligned}$$ Hence, $$\begin{aligned}
S_{(0,0,0,0)}&=-\frac1{q-1}\sum_{i=1}^{d-1}g\left(T^{it}\right)^d +\frac{1}{q(q-1)}\sum_{i=1}^{d-1}g(T^{it})^d+ \frac{1}{q(q-1)}\sum_{j=0}^{q-2}g\left(T^{j}\right)^dg\left(T^{-dj}\right)T^{dj}(-d\lambda)\\
&=-\frac{1}{q}\sum_{i=1}^{d-1}g\left(T^{it}\right)^d+ \frac{1}{q(q-1)}\sum_{j=0}^{q-2}g\left(T^{j}\right)^dg\left(T^{-dj}\right)T^{dj}(-d\lambda).\end{aligned}$$ We would like to express this is as a finite field hypergeometric function. Recall that the ${}_{d-1}F_{d-2}$ hypergeometric function is given by $$\begin{aligned}
{}_{d-1}F_{d-2}&\left(\left.\begin{array}{cccc}
T^t&T^{2t}&\ldots &T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|\frac{1}{\lambda^d}\right)_q\\
&= \frac{q}{q-1}\sum_{\chi}\binom{T^t\chi}{\chi}\binom{T^{2t}\chi}{\epsilon\chi}\cdots\binom{T^{(d-1)t}\chi}{\epsilon\chi}\chi(1/{\lambda^d}).
\end{aligned}$$ We rewrite this so that it is in terms of Gauss sums $$\begin{aligned}
&=\frac{q}{q-1}\sum_{\chi} \left(\frac{\chi(-1)}{q}\right)^{d-1} J(T^{t}\chi,\overline\chi)\cdots J(T^{(d-1)t}\chi,\overline\chi)\overline\chi({\lambda^d})\\
&=\frac{1}{q^{d-2}(q-1)}\sum_{\chi} \chi(-1)^{d-1} \frac{\prod_{i=1}^{d-1} g(T^{it}\chi)}{\prod_{i=1}^{d-1} g(T^{it})}\overline\chi({\lambda^d})\\\end{aligned}$$ Use the Hasse-Davenport formula of Theorem \[thm:HasseDavenport\] to get $$\begin{aligned}
{}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
T^t&T^{2t}&\ldots &T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|\frac{1}{\lambda^d}\right)_q
&= \frac{1}{q^{d-2}(q-1)}\sum_{\chi} \chi(-1)^{d-1} \frac{g(\chi^d)g(\overline\chi)^{d-1}}{\chi^d(d)g(\chi)}\overline\chi({\lambda^d})\\
%&= \frac{1}{q^{d-2}(q-1)}\sum_{\chi} \chi(-1)^{d-1} \frac{g(\chi^d)g(\overline\chi)^{d}}{\overline\chi(-1)q}\overline\chi^d(d\lambda)\\
%&= \frac{1}{q^{d-1}(q-1)}\sum_{\chi} \chi(-1)^{d} g(\chi^d)g(\overline\chi)^{d}\overline\chi^d(d\lambda)\\
&= \frac{1}{q^{d-1}(q-1)}\sum_{\chi}g(\chi^d)g(\overline\chi)^{d}\overline\chi^d(-d\lambda)\\
&= \frac{1}{q^{d-1}(q-1)}\sum_{j=0}^{q-2}g(T^{-dj})g(T^{j})^{d}T^{dj}(-d\lambda).\end{aligned}$$
Thus, $$\begin{aligned}
S_{(0,0,0,0)}&=-\frac{1}{q}\sum_{i=1}^{d-1}g\left(T^{it}\right)^d+ q^{d-2}{}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
T^t&T^{2t}&\ldots &T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|\frac{1}{\lambda^d}\right)_q\end{aligned}$$ We now combine our results to get $$\begin{aligned}
\#X_{\lambda}^d(\mathbb F_q)&=\frac{q^{d-1}-1}{q-1} + q^{d-2}{}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
T^t&T^{2t}&\ldots &T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|\frac{1}{\lambda^d}\right)_q\\
&\hspace{.5in} +\frac1q\sum_{w\in W^*} \prod_i g(T^{w_it})-\frac{1}{q}\sum_{i=1}^{d-1}g\left(T^{it}\right)^d +\frac1{q-1}\sum_{\overline w\not=\overline0}\sum_{j=0}^{p-2}\frac{\prod_{i=1}^dg\left(T^{w_it+j}\right)}{g(T^{dj})}T^{dj}(d\lambda)\\
&=\frac{q^{d-1}-1}{q-1} + q^{d-2}{}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
T^t&T^{2t}&\ldots &T^{(d-1)t}\\
{} &\epsilon&\ldots&\epsilon
\end{array}\right|\frac{1}{\lambda^d}\right)_q\\
&\hspace{.5in} +\frac1q\sum_{w\in W^{**}} \prod_i g(T^{w_it})+\frac1{q-1}\sum_{\overline w\not=\overline0}\sum_{j=0}^{p-2}\frac{\prod_{i=1}^dg\left(T^{w_it+j}\right)}{g(T^{dj})}T^{dj}(d\lambda),\end{aligned}$$ where $W^{**}$ is the set of $d-$tuples with $w_i\not=0$ and $w_i$ not all equal.
Point Count for Primes $p\not\equiv 1 \pmod d$
----------------------------------------------
In this section we discuss a conjecture for a point count formula that is written in terms of McCarthy’s $p$-adic hypergeometric function for primes $p\not\equiv 1 \pmod d$.
\[conjec:DworkHypersurfacePoint\] Let $d$ be an odd prime and $p$ a prime number such that $p\not\equiv 1 \pmod d$. The number of points over $\mathbb F_p$ on the Dwork hypersurface is given by $$\begin{aligned}
\#X_{\lambda}^d(\mathbb F_p)&=\frac{p^{d-1}-1}{p-1}+{}_{d-1}G_{d-1}\left[\left.\begin{array}{cccc}
1/d&2/d&\ldots&(d-1)/d\\
0&0&\ldots&0
\end{array}\right|{\lambda^d}\right]_p\end{aligned}$$
Currently our conjecture is somewhat limited, only applying to Dwork hypersurfaces with $d$ a prime. When $d$ is not prime we have found that the number of terms to consider and simplify grows rather large. This is because we have to consider various congruences as we did in the proofs of Theorems \[thm:K3PointCountnGn\] and \[thm:DworkHyp\], and the number of solutions to these is unwieldy when $d$ is not prime.\
After the initial submission of this paper to the Arxiv, Barman, Rahman, and Saikia have demonstrated the validity of this conjecture in [@Barman2015].
The start of this proof is identical to the start of the proof of Theorem \[thm:DworkHyp\] and, so, we start at the point when we consider the two sums A and B. Starting with B we have $$\begin{aligned}
B&=\frac{1}{(q-1)^d}\sum_{a_i=0}^{p-2}g(T^{-a_1})\cdots g(T^{-a_d})\\
&\hspace{1in} \times\sum_{x_1}T^{da_1}(x_1)\cdots \sum_{x_d}T^{da_d}(x_d)\sum_{z}T^{a_1+\ldots a_d}(z).\end{aligned}$$ This sum is non-zero only when the following congruences hold: $$da_1,\ldots , da_d\equiv 0 \pmod{p-1},\text{ and } \sum a_i\equiv 0 \pmod{p-1}.$$ Since $p\not\equiv 1\pmod d$ and $d$ is prime, these congruences simultaneously hold only when $a_1,\ldots, a_d=0$. Hence, $$B=-(p-1).$$
We now work to rewrite $A$. $$\begin{aligned}
A&=\frac{1}{(q-1)^{d+1}}\sum_{a_i=0}^{p-2}g(T^{-a_1})\cdots g(T^{-a_{d+1}})T^{d+1}(-d\lambda )\\
&\hspace{1in} \times\sum_{x_1}T^{da_1+a_{d+1}}(x_1)\cdots \sum_{x_d}T^{da_d+a_{d+1}}(x_d)\sum_{z}T^{a_1+\ldots a_{d+1}}(z).\end{aligned}$$ We consider congruences that must hold for the $a_i$. This sum is non-zero only when the following congruences hold: $$da_1+a_{d+1},\ldots, da_d+a_{d+1}\equiv 0 \pmod{p-1},\text{ and } a_1+\ldots+a_{d+1}\equiv 0 \pmod{p-1}.$$ As in the proof of Theorem \[thm:DworkHyp\] we first consider having the $a_i$ not all equal. Here we would have $a_i=\frac{j_i(q-1)}{d}$, where $0\leq j_i\leq d-1$, $\sum{j_i}\equiv 0 \pmod d$, and the $a_i$ are not all identical. However, since $d$ is prime, this is not possible. Thus, we must have all of the $a_i$ being equal. Thus we have $$\begin{aligned}
A_2&=\sum_{j=0}^{p-2} g(T^{-j})^dg(T^{dj})T^{-dj}(-d\lambda).\end{aligned}$$
We expect that this term can be expressed as a $p$-adic hypergeometric function of the form that we saw in Theorem \[thm:K3PointCountnGn\]. Our conjecture is that we have $$\begin{aligned}
A_2&=(-1)^dp(p-1){}_{d-1}G_{d-1}\left[\left.\begin{array}{cccc}
1/d&2/d&\ldots&(d-1)/d\\
0&0&\ldots&0
\end{array}\right|{\lambda^d}\right]_p\end{aligned}$$ plus a term to cancel with B. Assuming this is true, we now write a formula for the point count. $$\begin{aligned}
\#X_{\lambda}^d(\mathbb F_p)&=\frac{\frac1p(p^d+A-B)-1}{p-1}\\
&=\frac{p^{d-1}-1}{p-1}+\frac{\frac1p(A_1+A_2-B)}{p-1}\\
% &=\frac{p^{d-1}-1}{p-1}+\frac{\frac1p(A_1-B)}{p-1}+{}_{d-1}G_{d-1}\left[\left.\begin{array}{cccc}
% 1/d&2/d&\ldots&(d-1)/d\\
% 0&0&\ldots&0
% \end{array}\right|{\lambda^d}\right]_p\\
&=\frac{p^{d-1}-1}{p-1}-{}_{d-1}G_{d-1}\left[\left.\begin{array}{cccc}
1/d&2/d&\ldots&(d-1)/d\\
0&0&\ldots&0
\end{array}\right|{\lambda^d}\right]_p.\\\end{aligned}$$
Dwork Hypersurface Period Calculation
-------------------------------------
\[prop:Dworkpicardfuchs\][@Nagura1995 Section 3.2] The Picard-Fuchs equation for the Dwork hypersurface $$X_{\lambda}^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=d\lambda x_1x_2\cdots x_d$$ is given by $$\label{eqn:Dworkpicardfuchs}
\left(\vartheta^{d-1}-z(\vartheta+\tfrac{1}{d})\cdots(\vartheta+\tfrac{d-1}{d})\right)\pi=0$$ where $\vartheta=z\frac{d}{dz}$ and $z=\lambda^{-d}$.
The following is adapted from results in Section 46 of Rainville’s text [@Rainville1960].
\[prop:Dworkperiod\] The solution to Equation \[eqn:Dworkpicardfuchs\] in Proposition \[prop:Dworkpicardfuchs\] that is bounded near $z=0$ is given by $$\label{eqn:Dworkperiod}
\pi = {}_{d-1}F_{d-2}\left(\left.\begin{array}{cccc}
\tfrac1d&\tfrac2d&\cdots&\tfrac{d-1}{d}\\
{} &1&\cdots&1
\end{array}\right|\frac{1}{\lambda^d}\right).$$
We saw in Theorem \[thm:dFdcongruence\] that this is congruent (up to a sign) modulo $p$ to the matching finite field hypergeometric function that appears in the point count. This leads us to a conjecture that extends the congruence we saw in Theorem \[thm:K3PeriodTrace\]
\[conj:Dworkperiodpoint\] For the Dwork hypersurface $$X_{\lambda}^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=d\lambda x_1x_2\ldots x_d$$ we have that the trace of Frobenius over $\mathbb F_p$ and the period associated to the surface are congurent modulo $p$ when $p\equiv 1\pmod d$.
The conjecture here is that, as in the Dwork K3 surface case, either the remaining Gauss sum terms in the point count formula of Theorem \[thm:DworkHyp\] are congruent to 0 modulo $p$ or that these terms are canceled out in the trace of Frobenius – point count relationship. This relationship becomes more complicated for higher dimensional varieties.\
For example, consider the family of Dwork threefolds $$X^5_{\lambda}: \hspace{.1in} x_1^5+x_2^5+\ldots+x_5^5=5\lambda x_1x_2\cdots x_5.$$ The trace of Frobenius over $\mathbb F_p$ when $p\equiv 1 \pmod 5$ is given by $$a_{X^5_{\lambda}}(p)=p^3+25p^2-100p+1- \#X^5_{\lambda}(\mathbb F_p).$$ See [@Meyer2005 Section 3.1] for a proof of this. In the case where $\lambda=1$ and $p\equiv 1 \pmod 5$, Conjecture \[conj:Dworkperiodpoint\] follows from Theorem \[thm:dFdcongruence\], Proposition and the point count work of McCarthy in [@McCarthy2012b]. More generally, for $\lambda\not=1$, the formula for $\#X^5_{\lambda}(\mathbb F_p)$ has a main ${}_4F_3$ hypergeometric term and several terms made up of products of Gauss sums.\
A result relating the trace of Frobenius and the periods is expected for algebraic curves because of Manin’s work in [@Manin]. However, there is not a result of this sort that holds generally for higher dimensional algebraic varieties. We expect that it should be the case that the period and trace of Frobenius over $\mathbb F_p$ are congruent for a large class of varieties. In particular it would seem possible to show, at least by comparing explicit formulas, that the trace and the period are congruent when these expressions are both hypergeometric. Better yet, given that we expect there to be a congruence between these two quantities, it seems possible that it is exactly the varieties whose periods are solutions to hypergeometric differential equations that have a finite field hypergeometric point count.
|
---
abstract: 'We formulate a theory based on the time-dependent Ginzburg Landau (TDGL) theory and Newtonian vortex dynamics to study the transverse acousto-electric response of a type-II superconductor with Abrikosov vortex lattice. When exposed to a transverse acoustic wave, Cooper pairs emerge from the the moving atomic lattice and moving electrons. As in the Tolman-Stewart effect in a normal metal, an electromagnetic field is radiated from the superconductor. We adapt the equilibrium-based TDGL theory to this non-equilibrium system by using a floating condensation kernel. Due to the interaction between normal and superconducting components, the radiated electric field as a function of magnetic field attains a maximum value occurring below the upper critical magnetic field. This local increase in electric field has weak temperature dependence and is suppressed by the presence of impurities in the superconductor.'
address:
- 'Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague 2, Czech Republic'
- 'Institute of Physics, Academy of Sciences, Cukrovarnick[á]{} 10, 162 00 Prague 6, Czech Republic'
- 'Research Department, Universal Analytics Inc., Airdrie, AB T4B 2A4, Canada'
author:
- 'P. Lipavský'
- 'J. Kol[á]{}[č]{}ek'
- 'P.-J. Lin'
title: 'Transverse acousto-electric effect in superconductors'
---
superconductivity ,acousto-electric effect ,Abrikosov vortex ,Tolman-Stewart effect 74.20.De ,74.25.Ld ,72.50.+b
Introduction {#Introduction}
============
Superconductivity appears at low temperatures when materials are rigid and fragile so that for the majority of experiments made in cryostat it is not necessary to consider any motion of the crystal. The standard time-dependent Ginzburg-Landau theory (TDGL), which contains the assumption of local equilibrium, and is formulated in the laboratory frame, is a powerful tool to study phenomena in the vicinity of the superconducting-normal phase transition. To study gyroscopes [@Gaw93], gravitational wave antennae [@GFNNST13] and the interaction of the superconducting condensate with strong sound waves [@HL96; @Sonin96; @Gutlyanskii98; @FFAGJ05; @Fil06; @JH07; @HJ09; @HCMPK00; @Gutlyanskii02; @Gutlyanskii03; @AC08], where the atomic lattice is in motion, an extension to TDGL is needed to accommodate the dynamical system.
In the presence of a transverse acoustic wave, the condensate does not experience friction with the crystal and its imperfections as Cooper pairs do not scatter on the underlying crystal or its imperfections. The moving lattice, however, acts on the condensate by three nondissipative mechanisms: (i) Induction: The motion of ions creates an electric current which affects the electrons by magnetic induction. (ii) Entrainment: A Cooper pair of zero momentum in a moving crystal has nonzero velocity with respect to the crystal. This effect is particularly strong in dirty superconductors, where the mass of the Cooper pair is strongly renormalized, $m^*\gg 2\emas$. In the reference frame locally moving with the lattice, entrainment together with the fictitious force compose the inertial force causing the Tolman-Stewart effect. (iii) Deformation potentials: Deformations of the crystal lead to local changes of the chemical potential and material parameters which control superconductivity, for example its critical temperature.
Because of the induction the supercurrent tends to oppose the ionic current, but the compensation is not always complete. For example, in a steadily-rotating (but stationary) superconductor, currents near the surface are only partly compensated and the residual current produces a magnetic field known as the London moment [@L50]. In non-stationary (oscillating) systems the compensation is even less effective. In particular, under the influence of the ultrasound wave, the inertial motion of normal electrons as well as superconducting electrons leads to nonzero bulk currents via the Tolman-Stewart effect. The oscillating current radiates electromagnetic waves [@FFAGJ05; @Fil06] so that the superconductor exhibits nonzero acousto-electric effect.
Theoretical analysis of the acousto-electric field in superconductors has been performed assuming the fully superconducting state with no normal current [@Fil06]; the normal electrons are to the lattice [@Sonin96; @Gutlyanskii98; @Gutlyanskii02; @Gutlyanskii03]. Using this assumption, the TDGL theory of Verkin and Kulik [@VK72] (originally developed for the case of steady rotation when normal currents are absent) can be used to study the acousto-electric field.
Experimental studies on the acousto-electric field of hole-type metal (niobium) and electron-type borocarbide (Y$_{0.95}$Tb$_{0.05}$Ni$_2$B$_2$C) show a 10% increase of the radiated electric field as the material transforms into the superconducting state (Fig. 2 in Ref. [@Fil06]). This small change shows that, at least in the vicinity of the transition temperature, the normal and the superconducting currents are comparable, which is incompatible with the assumption of stationary () normal electrons. Our aim is to develop a theory which accounts for coexisting normal and superconducting currents.
Here we study theoretically the effect of a transverse sound wave on a superconducting system near the superconducting-normal phase-transition line, $B_{c2}$; the transverse wave propagates along the $z$ axis, and oscillates in the $x$ direction. In the transverse wave the material experiences shear stress only, without compression. According to the experimental studies by Fil [*et al*]{} [@Fil06], the changes of potential and material parameters caused by shear deformations can be disregarded. To accommodate this non-equilibrium system, we use a modified version of TDGL theory based on a microscopic derivation [@LBK13].
In we formulate the set of TDGL equations for the dynamical system with oscillating atomic lattice driven by an external transverse acoustic wave. Motion of normal electrons is allowed and this current treated in a self-consistent way, instead of assuming the normal electrons move together to the ions as in [@Fil06; @Sonin96]. As a result, the normal current is driven by inertial forces as in the normal-state Tolman-Stewart effect. Details of our derivation based on the Boltzmann equation are given in \[A1\]. In using Newtonian dynamics, we analyze the acousto-electric effect in the mixed state. Vortex dynamics in a steady state is deduced from the force balance on vortices. Magnus, pinning and transverse forces are considered, along with friction forces from the atomic lattice and from normal electrons [@KK76; @Iordanskii66]. The effective forces acting on the superconducting electrons are identified from the extended TDGL equation in . Next we discuss the skin effect and the matching of the internal field and the radiated electromagnetic wave at the surface. councludes with the resulting complete set of equations. contains numerical computations of the radiated electric field, using material parameters provided in [@Fil06].
TDGL theory {#GL}
===========
To study the electrons in an oscillating atomic lattice, it is advantageous to choose the moving background as the reference frame as Cooper pairs emerge from the moving electrons. Previous study [@LBK13] shows that to apply standard TDGL theory to a dynamical system, it is optimal to choose a condensation kernel floating with the background. Here we omit the details and write down the set of equations known as the floating-kernel time-dependent Ginzburg-Landau (FK-TDGL) equations.
Relaxation of the Ginzburg-Landau (GL) order parameter $\psi$ in the dynamical system is described by the FK-TDGL equation $$\begin{aligned}
\frac{1}{2m^*}(-i\hbar\nabla-e^*{\bf A}-\molf )^2\psi
&-&\alpha\psi+\beta|\psi|^2\psi \nonum\\
&=&\Gamma\bigg(\frac{\del}{\del t}-i\frac{2}{\hbar}\mu\bigg)\psi
\label{FK1}\end{aligned}$$ with the molecular field $$\molf =\chi^* m^*{\dot{\bf u}}+m^*\vn.
\label{molfield}$$ The first term of the molecular field is due to the entrainment effect caused by motion of the ionic lattice with velocity $\dot{\bf u}$; the mass of a Cooper pair is $m^*=m_0/(1+\chi^*)$, where $m_0=2\emas $ is twice the electron mass $\emas $. The corresponding superconducting current is $$\js={e^*\over m^*}\Re \big[
\bar\psi
(-i\hbar\nabla-e^*{\bf A}-\molf )\psi \big].
\label{FKcur}$$
The velocity $\vs$ of the condensate can be defined using $\js=e^*|\psi|^2\vs = e\ns\vs$. Because of the presence of the second term $m^*\vn$ in $\molf$, the operator $(1/m^*)(\!-i\hbar\nabla\!-\!e^*{\bf A}\!-\!\molf )$ gives the velocity of Cooper pairs with respect to normal electrons $\vn$. The current generated by the moving ions is $$\jl=-en\dot{\bf u},
\label{ioncur}$$ where ${\bf u}$ is the ion displacement caused by the transverse sound wave.
In our treatment we relax the requirement of Sonin [@Sonin96] that the electrons move with the same velocity $\dot{\bf u}$ as ions. Instead we assume that as with the Tolman-Stewart effect in normal conducting metals [@Davis88], the normal electrons lag behind ions and move with velocity $\vn$. The normal current $\jn = e n \vn$ can be obtained from the Boltzmann equations, shown in \[A1\]; the electric current is $$\jn + \jl=\frac{\sigma_{n}}{e}
\bigg(
{\bf F}'-\frac{\tau}{1+i\tau\omega}\frac{e}{m}
{\bf B}\times{\bf F}'
\bigg)-\nu\dot{\bf u}.
\label{Ohm}$$ The effective driving force $${\bf F}'=-e{\del{\bf A}\over\del t}-\nabla\mu+
e\dot{\bf u}\times{\bf B}-\emas \ddot{\bf u}
\label{efforceText}$$ includes the effective electric field (first and second terms), a part of the Lorentz force ${\bf F}_{L}=
e({\bf v}\rq + \dot{\bf u})\times{\bf B}$, where ${\bf v}\rq$ is electron velocity relative to the lattice, and the inertial force. These terms can be understood in the reference frame moving with the lattice, where the third term enters the electric field via a Lorentz transformation. The relaxation time $\tau$ comes from the normal conductivity $\sigma_{n}$, from . The last term in relation results from the diffusion of the transverse momentum [@Fil01]; this is similar to the mechanism causing the shear viscosity. Detail derivation of , analogous to Ohm’s law, from the Boltzmann equation can be found in \[A1\].
From the continuity equation $\nabla \cdot {\bf j}=0$ we can obtain for the chemical potential $$\nabla^2\mu={e\over\sigma_{n}}\nabla\cdot\js+
e\nabla\cdot(\dot{\bf u}\times{\bf B}),
\label{potential}$$ which is simplified by the transversality condition ${\bf q}\cdot{\bf u}=0$ for wave vector ${\bf q}$. The total force has zero divergence, so $\nabla\cdot\ddot{\bf u}=0$. We consider a system with homogeneous conductivity, $\nabla\sigma_{n}=0$.
The vector potential ${\bf A}$ can be obtained from the Maxwell equation $$\nabla^2{\bf A}=-
\mu_0 ( \js + \jn + \jl );
\label{Maxwell}$$ we use the Coulomb gauge $\nabla\cdot{\bf A}=0$. To obtain the radiated electromagnetic wave, we must evaluate skin vector potential and match internal and external fields. In , we will show that the skin effect is negligible if the wavelength of radiation is much larger than the skin depth.
We have a three-component system consisting of normal electrons, condensate, and electromagnetic field. Equations , , , and form a complete set of equations of motion. We are interested below in a case that the transverse sound wave interacts with a superconductor in the mixed state. Here we compare our theory with the TDGL theory of Verkin and Kulik [@Fil01; @VK72; @Sonin96] referred as VK-TDGL.
To make the comparison, we rewrite our equations in terms of the relative velocities with respect to the atomic lattice, that is, the relative velocity of normal electrons as $\vn'=\vn-\dot{\bf u}$ and the relative velocity of the condensate as $\vs'=\vs-\dot{\bf u}$.
In this notation, the molecular field is $$\molf =m_0{\dot{\bf u}}+m^*\vn'.
\label{molfieldVerkin}$$ The first term is the fictitious force obtained by Verkin and Kulik [@VK72]. The second term which is absent in [@VK72] is a correction due to non-zero velocity of normal electrons with respect to the ionic lattice. From Ohm’s law , and , we can see that the relative velocity is proportional to mean free path $\ell$. In the dirty limit ( $\ell\ll\xi_0$) $\vn'\to {\bf 0}$, hence the second term in can be ignored; our theory then reduces to VK-TDGL.
Vortex dynamics {#Vortex}
===============
Near the normal and superconducting phase transition, vortex motion is well described by TDGL theory. By solving the TDGL equation with the assumption of rigid Abrikosov vortex lattice, the TDGL equation can be represented in the form of force balance of Newtonian equations. Here we consider a superconductor which occupies $z<0$ with rigid Abrikosov vortices; each vortex has a fluxon $\Phi_0$ along the $z$-axis; the magnetic induction is ${\bf B}=(0,0,B)$, $B>B_{c1}$; thus the interspacing between vortices is $a\sim\sqrt{\Phi_0/B}$. The transverse acoustic wave propagates along the $z$-axis and oscillates in the $x$ direction; the atomic lattice deformation can be evaluated as the real part of the complex function ${\bf u}\equiv \exp(i\omega t)\cos(qz) (u,0,0)$.
This physical system contains variables at microscopic scale, such as parameters describing motion of electrons, and variables at mesoscopic scale, such as the wavelength of the acoustic wave. The typical wavelength of the acoustic wave is $\sim 100$ ${\rm \mu}$m, and that of the radiation is of the order of a metre, while spacing between vortices is $\sim 100$ nm.
Since we are interested in phenomena at mesoscopic scale, we can average a microscopic field $f$ locally to produce a mesoscopic field $\local{f}$, by writing $$\local{f}(t,{\bf r}) :=
(B/\Phi_0) \int_{C_{\bf r}} \td x'\td y' f(t,{\bf r}')$$ where the 2-D region $C_{\bf r}$ is the size and shape of an elementary cell, but with centroid ${\bf r}$ rather than being aligned with the lattice.
The acoustic wave acts on the superconductor in a similar manner as far-infrared (FIR) light; the condensate accelerates, so $\local{\vs}\neq0$. Following the idea by Sonin [@Sonin96], we will use the theory of vortex motion derived and experimentally tested for FIR response to study the interaction of acoustic waves with a superconductor in the mixed state.
Balance of forces on the condensate {#Bfc}
-----------------------------------
Under the influence of the transverse acoustic wave, the effective force driving Cooper pairs into motion can be identified as the time derivative of the effective vector potential ${\bf A}_\eff ={\bf A}+\molf /e^*$ in : $$\local{{\bf F}}=-\frac{\del}{\del t}
\local{e^*{\bf A}+\molf }.
\label{efFsup}$$ This averaged local force is balanced by $$\local{{\bf F}}=
-e^*\dot\uv\times\local{{\bf B}}+
m^*\frac{\del{\local{\vs}}}{\del t},
\label{Josephson}$$ where $\dot\uv$ is the velocity of Abrikosov vortices. The first term comes from induction and by itself would comprise the Josephson relation [@Josephson65]; the second term is essential in dynamical systems when $\dot{\local{\vs}} \neq 0$. The full is known as the inertial Josephson relation (IJR) which can be obtained either from the standard TDGL theory [@LLM12; @Matlock12] or hydrodynamic theory [@AKK65]. Gutlyanski[ǐ]{} uses an identical equation (Eqn. (1) of [@Gutlyanskii98]) which he calls the London equation because of its inertial term.
Balance of forces on vortices {#Vd}
-----------------------------
In a dynamical situation, a vortex experiences a number of forces [@Kop01]. The simplest equation of motion for a vortex is through the compensation of the Lorentz force and the Bardeen-Stephen friction force due to dissipative scattering of quasi-particles in the vortex-core region. In reality, impurity of a sample complicates the dynamics of vortices; it leads to such things as vortex pinning on the mesoscopic scale, and modified relaxation times of particles on the microscopic scale.
Here we adopt a widely-used equation from [@Kop01]. Considering all the forces on a vortex of unit length in the oscillating lattice, $$\begin{aligned}
e\ns(\dot{ \uv}-\local{\vs})\times {\bf z}&=&
\eta_{lat}(\dot\uv-\dot{\bf u}) \nonumber\\
&+&\eta_{qp}(\dot\uv-\local{\vn}) \nonumber\\
&+&\alpha_{L}(\uv-{\bf u}) \nonumber\\
&+&\alpha_{KK}(\dot\uv-\dot{\bf u})\times{\bf z} \nonumber\\
&-&\alpha_{I}(\dot\uv-\local{\vn})\times{\bf z},
\label{vortexmot}\end{aligned}$$ where we have defined ${\bf z} = \local{{\bf B}_\eff }/\local{B_\eff }$. The left side contains the hydrodynamic Lorentz and Magnus forces. The Magnus force can be obtained from the TDGL equation with the assumption of a rigid Abrikosov lattice. The Lorentz force comes from the effective magnetic field $\local{{\bf B}_\eff }=\curl\local{{\bf A}_\eff }$. In the linear response region $\dot\uv-\local{\vs}$ is small, therefore the effect of the acoustic wave on the magnetic field can be neglected, so $\local{{\bf B}_\eff } \approx \local{{\bf B}}$ and we may write ${\bf z} \approx\local{{\bf B}}/\local{B} = \zz$.
The forces on the right side of arise from a more detailed microscopic picture. The first two terms are frictional forces of vortex with the ionic lattice, and with normal electrons. The pinning force with the Labush parameter $\alpha_{L}$ is proportional to the relative displacement of the vortex from a pinning centre fixed to the ionic lattice. The transverse force of Kopnin and Kravtsov [@KK76], due to scattering on impurities, has coefficient $\alpha_{KK}$. Finally, the interaction of the vortex with quasiparticles has the transverse component of Iordanskii type [@Iordanskii66] with coefficient $\alpha_{I}$.
Rewriting in terms of relative velocities $\dot{\uv'}$, $\vs'$ and $\vn'$, we can separate the force imposed by the normal current, writing $$\begin{aligned}
-e\ns\local{\vs'}\times {\bf z}&=&
\eta\dot\uv\rq{}
+\alpha_{L}\uv\rq{}
-\alpha_{M}\dot\uv\rq{}\times{\bf z}
\nonumber\\
&&-\eta_{qp} \local{\vn'}+\alpha_{I} \local{\vn'}\times{\bf z},
\label{vortexmotrel}\end{aligned}$$ where $\eta=\eta_{lat} + \eta_{qp}$ is the total coefficient of friction, and $\alpha_{M}=e\ns-\alpha_{KK} + \alpha_{I}$ accounts for both corrections to the Magnus force.
In the limit $\vn' \to {\bf 0}$, corresponding to the electron model, coincides with the equation for forces on the vortex lattice used by Fil [*et al*]{} [@Fil06]. This limit is justified for dirty materials where $\tau\omega\ll1$ in Fil’s measurement ($\tau\sim 10^{-13}$ s and frequency $55$ MHz). The theory developed here without restriction to this limiting case is valid for moderately pure materials and higher frequencies near sub-gap frequencies.
Skin effect and the Maxwell equation {#Me}
------------------------------------
The vector potential inside the superconductor contains a large contribution ${\bf A}^0$ from the Abrikosov vortex lattice which satisfies $\del_t\local{{\bf A}^0}=0$, and a time-dependent perturbation which is the sum of the internal field with space dependence given by the acoustic wave $\local{{\bf A}'}= {\bf A}' \exp(i\omega t)\cos(qz)$ and the skin field $\local{{\bf A}''}={\bf A}''\exp(i\omega t+z/\lambda_{sk})$ where $\lambda_{sk}$ is the skin depth.
We solve for the surface fields by writing $\local{\vs'}(t,{\bf r}) = \vs' \exp(i\omega t) \cos(qz)$ (similarly for $\vn$ and other fields), shifting our notation so that from here onwards; $\vs'$ refers to the field at the surface at $t=0$.
The Maxwell equation gives $$q^2 {\bf A}'=
\mu_0e\left(\ns\vs'+i\omega \ns{\bf u}+n\vn'\right).
\label{Maxwellred}$$ $\vs'$ depends only on the oscillating transport current, as the strong static diamagnetic currents forming vortices average to zero over a cell.
At the surface, the matching of the internal electromagnetic wave ${\bf A}'(t, {\bf r})+{\bf A}''(t, {\bf r})$ and the outgoing radiation $\Aout \exp(i\omega(t-z/c))$ yields two conditions. The first condition, obtained from the Maxwell equation ${\bf E}=-\del_t {\bf A}$, is $$\begin{aligned}
\Aout&=&{\bf A}'+{\bf A}''.
\label{surfmatch1}\end{aligned}$$ The second condition, obtained from ${\bf B}=\curl{\bf A}$, is $$\begin{aligned}
-i\frac{\omega}{c}\Aout&=&\frac1{\lambda_{sk}}{\bf A}''.
\label{surfmatch2}\end{aligned}$$ In we used that the rotation of the field is proportional to $\sin(qz)$ which vanishes at the surface. Solving for the radiated field from and , we find $$\Aout=\frac1{1+i\frac{\omega}{c}\lambda_{sk}}
{\bf A}'.
\label{matching}$$ Since the wavelength of the radiation is $c/\omega\gg\lambda_{sk}$, we can approximate the radiated field by the internal one, so $\Aout={\bf A}'$.
Equations for surface fields {#Final}
----------------------------
The ionic displacement ${\bf u}$ is known. The vortex displacement $\uv'$, the condensate velocity $\vs'$, the normal velocity $\vn'$, and the vector potential ${\bf A}'$ are required. Here we rewrite equations in a convenient form.
The vortex displacement given by at frequency $\omega$ is $$\begin{aligned}
e \ns \zz\times\vs'&=&
( i\omega\eta + \alpha_L )\uv'
+ i\omega\alpha_{M}\zz\times\uv\rq{}\nonumber\\
&&-\eta_{qp}\vn'-\alpha_{I}\zz\times\vn'.
\label{vortexmotred}\end{aligned}$$
The condensate velocity is obtained from the IJR with the force $$2e{\bf A}'=-2m\omega_{c}\zz\times(\uv'+{\bf u})
-2i\omega \emas {\bf u}
-m^*(\vs'+\vn'+i\omega{\bf u}),
\label{Josephsonred}$$ where $\omega_{c}=eB/m$ is the cyclotron frequency.
The normal velocity is obtained from Ohm’s law ; the electric field in the force is needed. Using the periodicity of the Abrikosov vortex lattice, we obtained $\local{\nabla\mu} = 0$ in \[Chemical\]. Together with ${\bf B}\cdot{\bf u}=0$, the normal velocity is $$\begin{aligned}
m\vn'&=&-
\frac{i\tau\omega}{ 1+i\tau\omega}\bigg[
e{\bf A}'- \frac{\tau\omega_{c}}{ 1+i\tau\omega}
\,\zz\times e{\bf A}'+
\nonumber\\ &+&
\bigg(i\omega \emas +
m\omega_{c}\frac{\tau\omega_{c}}{ 1+i\tau\omega}+
\frac{e\nu}{\sigma_{n}}\bigg){\bf u}
\nonumber\\ &+&
\bigg( m\omega_{c}+
i\omega \emas
\frac{\tau\omega_{c}}{ 1+i\tau\omega}\bigg)
\,\zz\times{\bf u}\bigg].
\label{Ohmredfullred}\end{aligned}$$
We have considered radiation in response to a transverse acoustic wave; dependent on magnetic field, temperature and relaxation time, the radiation can now be evaluated by solving , -. With ${\bf B}$ and ${\bf q}$ along the $z$ axis, all of ${\bf u}$, $\uv'$, $\vs'$, $\vn'$ and ${\bf A}'$ have zero $z$-components. We have eight algebraic equations for $x$ and $y$ components of four unknown vectors.
Our model relaxes the assumption of VK-TDGL theory that normal electrons are stationary with respect to ions, but neither treatment takes into account effects of thermal fluctuations which are particularly strong at the superconducting and normal phase transition. The effect of thermal fluctuations can be included by the introduction of Langevin forces $\zeta({\bf r}, t)$ in the left side of the FK-TDGL equation , which equation describes the dynamics of the order parameter. In this paper, we will next restrict our discussion to the case of conventional superconductors where the macroscopic fluctuations are negligible.
Numerical predictions {#Numpred}
=====================
We study numerically the acousto-electric effect in the mixed state of a superconductor and in the region beyond the superconducting-normal phase transition line $B_{c2}$. In the normal state, our theory reproduces the Tolman-Stewart effect. The distance to the phase transition line is defined as $\delta b=(B-B_{c2}(T))/B_{c2}(0)$. The normal state corresponds to $\delta b>0$, the superconducting state to $\delta b<0$. Residual-resistance ratio (RRR) measured in the normal state is used to quantify the effect of imperfection of the atomic crystal. Here we focus on the case of niobium; necessary material parameters are taken from the measurement by Fil [*et al*]{} [@Fil06]. Parameters regarding the forces on vortices are specified in \[Foronvor\].
We first discuss the $\delta b$ dependence of the radiated electric field $\Ex$, parallel to the atomic displacement due to the transverse acoustic wave incident perpendicular to the surface. The radiated electric field is normalized by its magnitude at $B=0$ and $T=0$, where all electrons are in the condensate. Shown in is the radiation calculated from two different models, the FK-TDGL theory and the VK-TDGL theory, for a sample with an RRR value of $62$ at $T=0.75T_c$; this corresponds to Fil’s Fig. 3 of [@Fil06].
The overall radiation at $T=0.75T_c$ is smaller then the radiation at $T=0$ and $B=0$; the radiation increases when entering superconducting state and saturates as $B \rightarrow 0$. As expected, FK-TDGL theory gives a non-zero $\Ex$ in the normal state, consistent with the Tolman- Stewart value, but VK-TDGL theory which ignores the effect of normal current gives $\Ex=0$. Nevertheless, both theoretical curves show marked changes near the phase transition, and the two curves coincide at small magnetic field where superconductivity is robust.
Both of these models suggest that $\Im\Ex$ is negligible for very negative $\delta b$. Near the phase transition, the FK-TDGL model indicates a fundamental increase when approaching $\delta b=0$ and then remains constant. The VK-TDGL curve shows that $\Im\Ex$ remains negligible throughout the whole superconducting region.
$\Re \Ey$ is dominated by the normal current and has similar behaviour to $\Im \Ex$. In , $\Re \Ey$ in the FK-TDGL model increases with $\delta b$ before the abrupt change near $\delta b=0$, while $\Re \Ey$ remains negligible according to the VK-TDGL model. $\Im \Ey$ curves in the two models increase and coincide in the superconducting state; they separate when approaching $\delta b=0$. The VK-TDGL curve goes to zero at $\delta b=0$, while the FK-TDGL curve shows a continuous change through transition into the normal state.
-4mm ![Parallel acousto-electric coefficient as function of magnetic field for superconductor with an RRR of $62$: $\Ex$ as a function of $\delta b$ near the critical line. The dashed line corresponds to the VK-TDGL model, and is not defined for positive $\delta b$. The solid line shows the FK-TDGL result, and is continuous in $\delta b$. \[figEx12\]](Ex12.eps "fig:")
-4mm ![Transverse acousto-electric coefficient as a function of the $\delta b$ for RRR of $62$: $\Ey$, shown by the solid line, is continuous in FK-TDGL theory while the dashed line shows a ‘step’ appearing in the VK-TDGL model. \[figEy12\]](Ey12.eps "fig:")
According to the FK-TDGL model, imperfections of a superconductor influence radiation near the phase transition. The FK-TDGL model suggests that a cleaner superconductor with RRR of $620$ emits stronger radiation near the phase transition, shown in . The maximum $\Re \Ex$ is around three times larger than its value in the normal state, or its value in the purely-superconducting state at $T=0$ and $B=0$. The VK-TDGL plot shows the radiated electric field increasing with $\delta b$ as in the dirtier superconductor shown in ; the effect of impurities is negligible. The off-phase component $\Im \Ey$ in is suppressed in the superconducting state as in the dirtier superconductor discussed previously (with a different sign).
-4mm ![Parallel acousto-electric coefficient as a function of magnetic field for a superconductor with RRR of $620$ at $T=0.75
T_c$: The dot-dashed line shows the FK-TDGL result and the dotted line shows that of VK-TDGL; this convention is to facilitate comparison with the plots in . []{data-label="figEx34"}](Ex34.eps "fig:")
The enhancement of the radiation due to interaction between superconducting current and normal current is temperature dependent. $\Ex(\delta b)$ of the FK-TDGL model plotted at various temperatures is shown in . The location of the maximum gradually moves away from $\delta b=0$ as temperature decreases. While the peak is widened as temperature decreases, the magnitude of the peak changes little.
-4mm ![$\Ex$ as a function of the $\delta b$ for RRR of $620$: $t=0.1$ (solid line); $t=0.5$ (dashed line); $t=0.75$ (dot-dashed line); $t=0.99$ (dotted line). []{data-label="figExT"}](Ex5678.eps "fig:")
In contrast to the VK-TDGL model, FK-TDGL model accounts for the interaction between superconducting current and normal current. Our model provides a continuous description for a superconducting system in transition to normal state, and shows the Tolman-Stewart effect of normal metal in the normal state. When the superconducting system is away from the phase transition, the normal current contributes less and our model coincides with the VK-TDGL model. The VK-TDGL model is justified also in a dirty superconductor because the normal electrons scatter with impurities and thereby tend to move together with the lattice. However, in a clean superconductor the normal current contributes to the radiation; our model shows that radiation is enhanced due to the interaction between superconducting current and normal current. This enhancement occurs in superconducting state near the phase transition, and the field can reach three times that of radiation emitted in the normal state for certain values of $\delta b$.
Summary
=======
The acousto-electric effect has been shown in the vicinity of the critical magnetic field to reveal the interference of the superconducting and the normal response. To investigate this interference, we have employed the time-dependent Ginzburg-Landau theory, taking into account the effect of the normal current on the formation of the condensate. This formulation with normal current had been derived earlier from the microscopic approach within the framework of a floating nucleation kernel.
The Ginzburg-Landau theory with the inertial term of Verkin and Kulik provides reliable predictions, save for within a very narrow vicinity of the critical line between the normal and the superconducting state. This deficiency is emphasized in cleaner samples. The interference appearing in this narrow vicinity shows enhancement which we expect to be observable, in particular in the case of Niobium with a high RRR.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by M[Š]{}MT COST projects LD15062 and LD14060. P.-J. Lin acknowledges financial support from UA through SR-621-1207. Authors are grateful to V. D. Fil and D. V. Fil for many discussions on this subject and P. Matlock for critical reading.
Transverse acousto-electric effect in normal metal {#A1}
==================================================
In this appendix we derive the interaction of the transverse acoustic wave with the normal metal. The wave propagating in the $z$ direction, with wave vector ${\bf q} = (0,0,q)$, is described by the amplitude of lattice deviation in the $x$-direction, ${\bf u} = (u,0,0)$, with $u = ue^{i\omega t - i {\bf q}\cdot{\bf r} }$.
As the wave is transverse, the electric current generated by it is also transverse, therefore $\nabla\cdot{\bf j}=0$. From the equation of continuity follows that the charge density does not change $\del_t\rho=-\nabla\cdot{\bf j}=0$ so that we can set $\phi=0$. We note that this argument holds to the linear order in ${\bf u}$. At quadratic order, there is a small charge transfer along the $z$ axis e.g. due to the Bernoulli effect; we neglect quadratic effects.
The generated transverse electric field is covered by the Maxwell equation $-\nabla^2{\bf A} = \mu_0{\bf j}$. Using $-\del_t{\bf A}={\bf E}$ yields $$q^2{\bf E}=-i\omega\mu_0{\bf j}.
\label{MaxwellE}$$
We need to evaluate the current as a function of the electric field ${\bf E}$ and the deviation ${\bf u}$. To this end we use the Boltzmann equation in the relaxation time approximation $$\frac{\del f}{\del t}+{\bf v}\cdot\nabla f-
\nabla\varepsilon\cdot\frac{\del f}{\del {\bf k}}
=-{1\over\tau}\delta f,
\label{Boltzmann}$$ where $\delta f=f-{\bar f}$ is a deviation from local equilibrium. The local equilibrium distribution ${\bar f}$ represents electrons emitted from collisions with impurities and lattice vibrations. It has the same local density as the actual distribution $$2\int{\td{\bf k}\over (2\pi)^3}{\bar f}=
2\int{\td{\bf k}\over (2\pi)^3}f=n,
\label{relaxn}$$ where the factor of two accounts for the sum over spins. Assuming isotropic collisions, the mean velocity of electrons emitted from collisions equals the velocity of the lattice, $$2\int \frac{\td{\bf k}}{ (2\pi)^3}{\bar f}{\bf v}=n\dot{\bf u}.
\label{relaxvel}$$
The quasiparticle energy in the lattice moving with velocity $\dot{\bf u}$ is $$\varepsilon=\frac{|{\bf k}-e{\bf A}|^2}{ 2\emas }+
\chi\frac{|{\bf k}-e{\bf A}-\emas \dot{\bf u}|^2}{ 2\emas },
\label{quasienergy}$$ where $\chi$ measures the renormalization of the inverse mass $1/m=(1+\chi)/\emas $ in the normal state and the term proportional to it describes the normal entrainment. The corresponding quasiparticle velocity is $${\bf v}=\frac{\del\varepsilon}{\del{\bf k}}=
{{\bf k}-e{\bf A}-\chi m\dot{\bf u}\over m}.
\label{quasivelocity}$$
The local equilibrium distribution $\bar f$ is centered around the mean momentum ${\bar{\bf k}}$, $$2\int{\td{\bf k}\over (2\pi)^3}{\bar f}~{\bf k}=n{\bar{\bf k}}.
\label{relaxmom}$$ The condition then gives $${\bar{\bf k}}=e{\bf A}+\emas \dot{\bf u},
\label{relaxvel2}$$ where we have used $(1+\chi)m=\emas $. The local equilibrium is thus given by the Fermi-Dirac distribution $${\bar f}({\bf k},{\bf r},t)=f_{FD}(\bar\varepsilon)
\label{relaxdis}$$ with energy $\bar\varepsilon=|{\bf k}-{\bar{\bf k}}|^2/2m$ or $$\bar\varepsilon({\bf k},{\bf r},t)={|{\bf k}-e{\bf A}({\bf r},t)-
\emas \dot{\bf u}({\bf r},t)|^2\over 2m}.
\label{relaxen}$$
The total current is the sum of the ionic current $-en\dot{\bf u}$ and the electronic current $${\bf j}=-en\dot{\bf u}+2e\int{\td{\bf k}\over (2\pi)^3}f\,{\bf v}.
\label{curdef}$$ According to $$2e\int{\td{\bf k}\over (2\pi)^3}{\bar f}~{\bf v}=en\dot{\bf u}
\label{relaxcur}$$ which exactly cancels the ionic current. The total current due to the deviation from local equilibrium is thus $${\bf j}=2e\int{\td{\bf k}\over (2\pi)^3}{\delta f}\,{\bf v}.
\label{cur}$$
The distribution $\delta f$ we will find from the Boltzmann equation $$\left({1\over \tau}+
{\del\over\del t}+{\bf v}\cdot\nabla-
\nabla\varepsilon\cdot{\del\over\del {\bf k}}\right)\delta f
=-\bar I
\label{Boltzmannexp}$$ with the source term $$\bar I
={\del\bar f\over\del t}+{\bf v}\cdot\nabla\bar f-
{\del\bar f\over\del {\bf k}}\cdot\nabla\varepsilon.
\label{Boltzmannexpsource}$$ The local equilibrium depends on the time and space only via the central momentum $\bar{\bf k}$, therefore $$\begin{aligned}
{\del\bar f\over\del t}&=&{\del\bar f\over\del k_i}
\left(-e{\del A_i\over\del t}-\emas {\del\dot u_i\over\del t}
\right),
\label{Boltzmannexp1}\\
{\bf v}\cdot\nabla\bar f&=&{\del\bar f\over\del k_i}
\left(-ev_j\nabla_j A_i-\emas v_j\nabla_j\dot u_i\right),
\label{Boltzmannexp2}\\
-{\del\bar f\over\del {\bf k}}\cdot\nabla\varepsilon&=&
{\del\bar f\over\del k_i}
\left(ev_j\nabla_i A_j+(\emas -m)v_j'\nabla_i\dot u_j\right).
\nonumber\\
\label{Boltzmannexp3}\end{aligned}$$ We have used the velocity relative to the lattice $${\bf v}'={\del\bar\epsilon\over\del{\bf k}}=
{\bf v}-\dot{\bf u}.
\label{relvel}$$
Using relations $-\del_t{\bf A}=\bf E$, $v_j\nabla_i A_j-v_j\nabla_j A_i=[{\bf v}\times{\bf B}]_i$ and $\del_t\dot{\bf u}+\dot{\bf u}\cdot\nabla\dot{\bf u}=
\ddot{\bf u}$, the source term can be expressed as $$\begin{aligned}
\bar I&=&{\del\bar f\over\del{\bf k}}
\biggl(e{\bf E}+e{\bf v}\times{\bf B}-\emas \ddot{\bf u}+
\emas {\bf v}'\times[\curl\dot{\bf u}]\biggr)
\nonumber\\
&-&{\del\bar f\over\del k_i}m v_j'\nabla_i\dot u_j.
\label{Boltzmannexp4}\end{aligned}$$ As the local equilibrium distribution $\bar f$ depends only on $\bar\varepsilon$, the source term can be further simplified $$\bar I={\del\bar f\over\del\bar\varepsilon}{\bf v}'\cdot
\biggl(e{\bf E}+e\dot{\bf u}\times{\bf B}-\emas \ddot{\bf u}-
m({\bf v}'\cdot\nabla)\dot{\bf u}\biggr),
\label{Boltzmannexp6}$$ where we have used orthogonality ${\bf v}'\cdot({\bf v}'\times{\bf B})=0$ and ${\bf v}'\cdot\left({\bf v}'\times[\curl\dot{\bf u}]\right)=0$.
The current in terms of the relative velocity is $${\bf j}=2e\int{\td{\bf k}\over (2\pi)^3}\delta f{\bf v}'.
\label{curdev}$$ The term proportional to $\dot{\bf u}$ equals zero, because from follows $\int d{\bf k}\,\delta f=0$.
To evaluate the deviation to terms linear in $\dot{\bf u}$ we can neglect nonlinear terms in the left hand side of $$\left({1\over \tau}+i\omega+{\bf v}'\cdot\nabla-
\nabla\bar\varepsilon\cdot{\del\over\del {\bf k}}\right)
\delta f=-\bar I.
\label{Boltzmannlin}$$ The distribution $\delta f$ depends on ${\bf r}$ and ${\bf k}$ in two ways, via $\bar\varepsilon$ in $\bar f$, and via vectors ${\bf v}'$ and $\nabla\bar\varepsilon$. Dependence on $\bar\varepsilon$ can be eliminated. Let us write the derivatives as $$\begin{aligned}
\nabla\delta f&=&
{\del\delta f\over\del\bar\varepsilon}\nabla\bar\varepsilon+
\left({\del\delta f\over\del{\bf r}}\right)_{\bar\varepsilon},
\label{nablasplit}\\
{\del\delta f\over\del{\bf k}}&=&
{\del\delta f\over\del\bar\varepsilon}{\bf v}'+
\left({\del\delta f\over\del{\bf k}}\right)_{\bar\varepsilon}.
\label{derksplit}\end{aligned}$$ The energy derivative cancels, therefore $$\left({1\over \tau}+i\omega\right)\delta f
+{\bf v}'\cdot
\left({\del\delta f\over\del{\bf r}}\right)_{\bar\varepsilon}-
\nabla\bar\varepsilon\cdot
\left({\del\delta f\over\del{\bf k}}\right)_{\bar\varepsilon}=-
\bar I.
\label{Boltzmannlinsplit}$$
We will expand the solution in small $\tau/(1+i\tau\omega)$. The first order is $$\delta f_1=-{\tau\bar I\over 1+i\tau\omega},
\label{Boltzmannlin1}$$ and the second order is $$\begin{aligned}
\delta f_2&=&-{\tau\over 1+i\tau\omega}\left({\bf v}'\cdot
\left({\del\delta f_1\over\del{\bf r}}\right)_{\bar\varepsilon}-
\nabla\bar\varepsilon\cdot
\left({\del\delta f_1\over\del{\bf k}}\right)_{\bar\varepsilon}\right)
\nonumber\\
&=&\left({\tau\over 1+i\tau\omega}\right)^2\left({\bf v}'\cdot
\left({\del\bar I\over\del{\bf r}}\right)_{\bar\varepsilon}-
\nabla\bar\varepsilon\cdot
\left({\del\bar I\over\del{\bf k}}\right)_{\bar\varepsilon}\right)
\nonumber\\
&=&\left({\tau\over 1+i\tau\omega}\right)^2{\del\bar f\over\del\bar\varepsilon}
v'_j\biggl(\nabla_j\bigl(v'_i F'_i-mv'_iv'_k\nabla_k\dot u_i\bigr)
\nonumber\\
&&+
e(\nabla_l A_j)
{\del\over\del k_l}\bigl(v'_i F'_i-mv'_iv'_k\nabla_k\dot u_i\bigr)\biggr)
\nonumber\\
\label{Boltzmannlin2}\end{aligned}$$ with the force $${\bf F}'=e{\bf E}+e\dot{\bf u}\times{\bf B}-\emas \ddot{\bf u}.
\label{efforce}$$
In the linear response we can neglect $\dot{\bf u}$ in derivatives, $\nabla_jv'_i=-(e/m)\nabla_jA_i$ and $(\del v'_i/\del k_j)=
(1/m)\delta_{ij}$, therefore $$\begin{aligned}
\delta f_2
&=&\left({\tau\over 1+i\tau\omega}\right)^2{\del\bar f\over\del\bar\varepsilon}
\biggl({e\over m}v'_j F'_i(\nabla_i A_j-\nabla_jA_i)
\nonumber\\
&&~~~~~~-ev'_jv'_k(\nabla_i A_j-\nabla_jA_i)(\nabla_k\dot u_i+\nabla_i\dot u_k)
\nonumber\\
&&~~~~~~-
mv'_jv'_iv'_k\nabla_j\nabla_k\dot u_i+v'_jv'_i\nabla_j F'_i
\biggr)
\nonumber\\
&=&\left({\tau\over 1+i\tau\omega}\right)^2{\del\bar f\over\del\bar\varepsilon}
\biggl({e\over m}{\bf v}'\cdot[{\bf B}\times{\bf F}']
\nonumber\\
&&+m({\bf v}'\cdot{\bf q})^2({\bf v}'\cdot\dot{\bf u})-i({\bf v}'\cdot{\bf q})({\bf v}'\cdot{\bf F}')
\nonumber\\
&&+ie{\bf B}\cdot\Bigl([\dot{\bf u}\times{\bf v}']({\bf v}'\cdot{\bf q})+
[{\bf q}\times{\bf v}']({\bf v}'\cdot\dot{\bf u})\Bigr)\biggr).
\nonumber\\
\label{Boltzmannlin2a}\end{aligned}$$
The function $\delta f=\delta f_1+\delta f_2$ includes terms odd and even in the velocity ${\bf v}'$. We keep only the odd terms which contribute to the current, $$\begin{aligned}
&&\delta f_{odd}=-{\tau\over 1+i\tau\omega}
{\del\bar f\over\del\bar\varepsilon}
{\bf v}'\cdot{\bf F}' \nonumber\\
&&+\left({\tau\over 1+i\tau\omega}\right)^2{\del\bar f\over\del\bar\varepsilon}
\biggl({e\over m}{\bf v}'\cdot[{\bf B}\times{\bf F}']+m({\bf v}'\cdot{\bf q})^2({\bf v}'\cdot\dot{\bf u})\biggr).
\nonumber\\
\label{Boltzmannlinodd}\end{aligned}$$ The electric current is thus $${\bf j}={\sigma_{n}\over e}\left({\bf F}'-{\tau\over 1+i\tau\omega}{e\over m}{\bf B}\times{\bf F}'\right)-\nu\dot{\bf u},
\label{curdevfin}$$ where $$\sigma_{n}=-{2\tau e^2\over 1+i\tau\omega}{1\over 3}
\int{\td{\bf k}\over (2\pi)^3}
{\del\bar f\over\del\bar\varepsilon}v'^2=
{\tau e^2n\over m(1+i\tau\omega)}
\label{sigma}$$ is the usual conductivity in the absence of the magnetic field. The Hall component is implied by the force term ${\bf B}\times{\bf F}'$.
The last term in results from inhomogeneous velocity of the lattice, namely impurities and phonons. Its coefficient reminds the shear viscosity $$\begin{aligned}
\nu&=&-{2\tau^2 emq^2\over (1+i\tau\omega)^2}\int{\td{\bf k}\over (2\pi)^3}
{\del\bar f\over\del\bar\varepsilon}v_x^{\prime 2}v_z^{\prime 2}.
\label{shareviscgen}\end{aligned}$$ The integral over velocities in in the zero temperature limit is $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!-2\int{\td{\bf k}\over (2\pi)^3}
{\del\bar f\over\del\bar\varepsilon}v_x^{\prime 2}v_z^{\prime 2}
\nonumber\\
&=&
{2\over (2\pi)^3}\!\int\limits_{-1}^1\!dzz^2(1\!-\!z^2)\!\int\limits_{-\pi}^\pi\! d\varphi \sin^2\varphi
\nonumber\\
&&\times
\int\limits_0^\infty\! d\bar k \delta(\bar\varepsilon-E_{F}){\bar k^6\over m^4}
\nonumber\\
&=&{1\over 15\pi^2}{k^5_{F}\over m^3}
\nonumber\\
&=&{nv^2_{F}\over 5m},
\label{muint}\end{aligned}$$ where we have used the density $n=k_{F}^3/(3\pi^2)$ and the Fermi velocity $v_{F}=k_{F}/m$. Finally, we express the shear coefficient in terms of the mean free path $l=\tau v_{F}$ $$\begin{aligned}
\nu&=&{eq^2nl^2\over 5(1+i\tau\omega)^2}.
\label{sharevisc}\end{aligned}$$ For short lifetime $\tau\omega\to 0$, the coefficient $\nu$ agrees with the result of Fil [@Fil01].
Chemical potential {#Chemical}
==================
Here we show that the chemical potential can be excluded from assumptions dealing with the fields averaged over elementary cells of the Abrikosov vortex lattice.
Let us split the chemical potential as $\mu=\mu_{\bf j}+\mu_{\bf u}$, where the first term has the form standard in the TDGL theory $$\nabla^2\mu_{\bf j}={e\over\sigma_{n}}\nabla\cdot\js
\label{potentialj}$$ and the second term appears only in moving crystals and represents a change of the chemical potential due to the Lorentz force $$\nabla^2\mu_{\bf u}=e\nabla\cdot[\dot{\bf u}\times{\bf B}].
\label{efmusub}$$ Both potentials need boundary conditions which specify constant and linear terms. We use zero mean values, $\langle\mu_{\bf j}\rangle_s=0$ and $\langle\mu_{\bf u}\rangle_s=0$, where brackets denote average over sample volume. Since the system is periodic on the Abrikosov vortex lattice, this averaging is identical to averaging over single elementary cell and implies zero mean gradients $\langle\nabla\mu_{\bf j}\rangle={\bf 0}$ and $\langle\nabla\mu_{\bf u}\rangle={\bf 0}$.
It is necessary to show that the conditions $\langle\nabla\mu_{\bf j}\rangle={\bf 0}$ and $\langle\nabla\mu_{\bf u}\rangle={\bf 0}$ are not in conflict with equations and respectively. The source term in the right hand side of is a sum of the transport supercurrent $\langle\js\rangle$ and the circulating current due to the Abrikosov vortex lattice. In the homogeneous Abrikosov lattice the transport supercurrent has zero divergence $\nabla\cdot\langle\js\rangle=0$ because of the translation invariance. The circulating component has zero divergence in the approximation of rigidly moving Abrikosov lattice. Beyond this approximation one finds contributions that are nonzero but periodic on the Abrikosov lattice giving the zero mean value, $\langle\nabla\cdot\js
\rangle=0$. Zero mean value of the source term in is not in conflict with the boundary condition $\langle\mu_{\bf j}\rangle=0$.
The source term in the right hand side of the equation is rather complex. It simplifies in the linear approximation in $\bf u$ as $$\begin{aligned}
\nabla\cdot[\dot{\bf u}\times{\bf B}]&=&-
\dot{\bf u}\cdot[\curl{\bf B}]+{\bf B}\cdot[\curl\dot{\bf u}]
\nonumber\\
&=&-\mu_0\dot{\bf u}\cdot\left(\js+\jn+\jl\right)
\nonumber\\
&\approx&-\mu_0\dot{\bf u}\cdot\js^0,
\label{uB}\end{aligned}$$ where $\js^0$ is the supercurrent in the static Abrikosov lattice. We have used that the wave propagates along the magnetic field ${\bf B}\|{\bf q}$, therefore ${\bf B}\cdot[\curl\dot{\bf u}]=0$. In the last step we have neglected terms beyond the linear response. Since there is no transport current in the static Abrikosov lattice $\langle\js^0\rangle={\bf 0}$, the source term has zero mean value, $\langle\dot{\bf u}\cdot\js^0\rangle=
\dot{\bf u}\cdot\langle\js^0\rangle=0$. The boundary condition $\langle\mu_{\bf u}\rangle=0$ is thus not in conflict with the source term.
Parameters of niobium {#Filparam}
=====================
First we list characteristic values. At the upper critical field at zero temperature $B=B_{c2}=.49$ T, the cyclotron frequency $\omega_{c}=eB/m$ is $\omega_{c}=7.13\cdot 10^{10}$ s$^{-1}$, with the effective mass of niobium $m=1.2~\emas $. We note that niobium has a complicated Fermi surface appearing in the first, second and third Brillouin zone so that different effective masses appear, e.g. $m=3.2~\emas $ and $m=1.7~\emas $ from de Haas-van Alphen effect with and without phonon dressing [@KS70]. In all cases the cyclotron frequency is much higher than the frequency of applied sound $\omega=2\pi~5.5\cdot 10^7$ s$^{-1}$. The velocity of the transverse sound in niobium is $v_{s}=2100$ m/s. This gives the wave vector $q=\omega/v_{s}=1.6\cdot 10^5$/m and wave length $2\pi/q=3.8\cdot 10^{-5}$ m.
Because of the complicated energy band structure, it is preferable to use characteristics of the Fermi surface rather than the effective mass and electron density. The single-spin density of states is $N_0=5.7\cdot 10^{47}$/Jm$^3$, and the average of the Fermi velocity over the niobium Fermi surface is $v_{F}=0.59\cdot 10^6$ m/s, see Weber [*et al*]{} [@WSLSPJE91]. They enter the conductivity as $\sigma_{n}=
\frac23 e^2N_0v_{F}^2\tau$.
The relaxation time $\tau$ depends on impurities. The niobium sample measured by Fil [*et al*]{} [@Fil06] reveals a step of acousto-electric effect in the zero magnetic field. Going from the superconducting state to the normal one, the magnitude reduces by 10% and the phase increases by 7$^\circ$. Within the present theory it is reproduced by $\tau=1.2\cdot 10^{-13}$ s, which corresponds to the residual resistivity ratio $RRR=62$. This short relaxation time leads to rather small dimensionless numbers $\tau\omega_{c}=8.6\cdot 10^{-3}$ and $\tau\omega=4.1\cdot 10^{-5}$.
Different values one finds for pure samples. Weber [*et al*]{} [@WSLSPJE91] measured a sample of the residual resistivity ratio $RRR=2080$, giving the low temperature conductivity $\sigma_{n}=RRR/\rho_{n}=3\cdot 10^{13}/\Omega$m. This high conductivity corresponds to the relaxation time $\tau=8.9\cdot 10^{-9}$ s with the dimensionless number $\tau\omega=3.1$. The mean number of circulations between collisions is $\tau\omega_{c}=635$. Since the relaxation time $\tau$ is proportional to the $RRR$, it is possible to prepare samples with $\tau$ from $10^{-8}$ s to $10^{-14}$ s. Moreover, for a thin sample the magnetic field can be weak so that dimensionless numbers can have general values from small to values over unity.
The mean free path $l=v_{F}\tau$ spreads from $6\cdot 10^{-9}$ m to $6\cdot 10^{-3}$ m. For the mean free path exceeding the wave length the theory of the normal acousto-electric effect is not fully justified because it is based on local approach with the lowest order nonlocal correction $\nu$. To stay in the region of validity we assume $\tau\ll 10^{-11}$ s for which $l\ll 1/q$. The value RRR=620 used for demonstration corresponds to $\tau=1.2\cdot 10^{-12}$ s giving small dimensionless numbers $\tau\omega_{c}=8.6\cdot 10^{-2}$ and $\tau\omega=4.1\cdot 10^{-4}$.
Let us identify parameters for the superconducting state. For the dirty sample of Fil [*et al*]{}, the critical fields correspond to the GL parameter $\kappa=1.5$ given by the coherence length $\xi_0=2.6\cdot 10^{-8}$ m and the London penetration depth $\lambda=3.9\cdot 10^{-8}$ m. Here $\lambda=\sqrt{m^*/(2\mu_0e^2\ns)}$ with the Cooperon mass $m^*=2m\hbar v_{F}/(\pi\Delta_0 l_{free})=
5.52~\emas $. We have used the BCS gap $\Delta_0=1.76k_{B}T_{c}$. In the superconducting regime far from the critical temperature nonlocal contributions are negligible because $q\lambda=6.13\cdot 10^{-3}$.
Going to the clean limit there will be no dramatic changes. The Cooperon mass $m^*$ reaches the value of $2m$. The London penetration depth decreases to $\lambda=2.3\cdot 10^{-8}$ m and the GL parameter reduces close to the limiting value $\kappa\sim 1/\sqrt{2}$. Samples of width comparable to the wave length $\sim 10^{-5}$ m, but large in area $\sim 1$ cm$^2$, are penetrated by the magnetic field either in the form of Abrikosov vortices or in the form of slabs. We discuss the case in the vicinity of the critical temperature where sample becomes effectively thin as $\lambda$ is large so that vortices become preferable.
Forces on vortex {#Foronvor}
================
We take the friction according to Kopnin [@Kop01] (formula 12.38 with $B\to B_{c2}$ limit of $\sigma_f$ given by formula 12.35) $\eta=\eta_{lat}=\sigma_{n}B_{c2}=
1.8\cdot 10^6$ C/m$^3$. The quasiparticle friction is neglected in this approximation.
The coefficient of the Magnus-like force $\alpha_{M}=
e\ns+\alpha_{I}=3.5\cdot 10^9$ C/m$^3$ is dominated by the Iordanskii term $\alpha_{I}=e(n-\ns)$, see [@Kop01] formula below (14.97). We neglect the Kopnin-Kravtsov force.
The presented numerical results have been obtained with rather small Labusch coefficient $\alpha_{L}\ll e\ns/\omega$. We have found that acousto-electric effect remains the same within accuracy of figures even for values as large as $\alpha_{L}\sim 10^3
e\ns/\omega$.
[10]{} url \#1[`#1`]{}urlprefixhref \#1\#2[\#2]{} \#1[\#1]{}
E. T. Gawlinski, Phys. Rev. B 48 (1993) 351.
A. Gulian, J. Foreman, V. Nikoghosyan, S. Nussinov, L. Sica, J. Tollaksen, Journal of Physics: Conference Series 507 (4) (2014) 042013.
C. Hucho, M. Levy, Phys. Rev. Lett. 77 (1996) 1370–1373.
E. B. Sonin, Phys. Rev. Lett. 76 (1996) 2794–2797.
E. D. Gutlyanski[ǐ]{}, JETP Letters 67 (1998) 239.
V. D. Fil‘, D. V. Fil, Y. A. Avramenko, A. L. Gaiduk, W. L. Johnson, Phys. Rev. B 71 (2005) 092504.
V. D. Fil, D. V. Fil, A. N. Zholobenko, N. G. Burma, Y. A. Avramenko, J. D. Kim, S. M. Choi, S. I. Lee, Europhys. Lett. 76 (2006) 484.
F. Jachmann, C. Hucho, Solid State Communications 142 (2007) 212–216.
C. Hucho, F. Jachmann, Materials Science and Engineering: A 521–522 (2009) 307–309.
C. Hucho, J. M. Carter, V. M[ü]{}ller, A. Petrean, W. K. Kwok, Physica C 332 (2000) 370–373.
E. D. Gutlyanski[ǐ]{}, Phys. Rev. B 66 (2002) 052511.
E. D. Gutlyanski[ǐ]{}, Physics of the Solid State 45 (2003) 812–815.
J. Albert, E. M. Chudnovsky, Phys. Rev. B 77 (2008) 092506.
F. London, Superfluids, Vol. 1, Wiley, New York, 1950.
B. I. Verkin, I. O. Kulik, Sov. Phys. JETP 34 (1972) 1103.
P. Lipavsk[ý]{}, J. Bok, J. Kol[á]{}[č]{}ek, Physica C: Superconductivity 492 (0) (2013) 144 – 151.
N.-B. Kopnin, V. E. Kravtsov, JETP Lett. 23 (1976) 587.
S.-V. Iordanskii, Sov. Phys. JETP 22 (1966) 160.
T. J. Davis, G. I. Opat, Classical and Quantum Gravity 5 (7) (1988) 1011.
V. D. Fil, Low Temp. Phys. 27 (2001) 993.
B. Josephson, Physics Letters 16 (3) (1965) 242 – 243.
P.-J. Lin, P. Lipavsk[ý]{}, P. Matlock, Physics Letters A 376 (2012) 883–885.
P. Matlock, Physica C 476 (2012) 59–62.
A. Abrikosov, M. Kemoklidze, I. Khalatnikov, Sov. Phys. JETP 21 (1965) 506.
N. B. Kopnin, Theory of Nonequilibrium Superconductivity, Claredon Press, Oxford, 2001.
V. R. Karasik, I. Y. Shebalin, Soviet Physics JETP 30 (7) (1970) 1068–1075.
H. W. Weber, E. Seidl, C. Laa, E. Schachinger, M. Prohammer, A. Junod, D. Eckert, Phys. Rev. B 44 (1991) 7585–7600.
|
---
abstract: 'The theory presented here, cosmological general relativity, uses a Riemannian four-dimensional presentation of gravitation in which the coordinates are those of Hubble, i.e. distances and velocity rather than the traditional space and time. We solve the field equations and show that there are three possibilities for the Universe to expand. The theory describes the Universe as having a three-phase evolution with a decelerating expansion, followed by a constant and an accelerating expansion, and it predicts that the Universe is now in the latter phase. It is shown, assuming $\Omega_m=0.245$, that the time at which the Universe goes over from a decelerating to an accelerating expansion, i.e., the constant-expansion phase, occurs at 8.5 Gyr ago. Also, at that time the cosmic radiation temperature was 146K. Recent observations of distant supernovae imply, in defiance of expectations, that the Universe’s growth is accelerating, contrary to what has always been assumed, that the expansion is slowing down due to gravity. Our theory confirms these recent experimental results by showing that the Universe now is definitely in a stage of accelerating expansion. The theory predicts also that now there is a positive pressure, $p=0.034g/cm^2$, in the Universe. Although the theory has no cosmological constant, we extract from it its equivalence and show that $\Lambda=1.934\times 10^{-35}s^{-2}$. This value of $\Lambda$ is in excellent agreement with the measurements obtained by the [*High-Z Supernova Team*]{} and the [*Supernova Cosmology Project*]{}. It is also shown that the three-dimensional space of the Universe is Euclidean, as the Boomerang experiment shows. Comparison with general relativity theory is finally made and it is shown that the classical experiments as well as the gravitational radiation prediction follow from the present theory, too.'
---
[Accelerating Universe: Theory versus Experiment]{}
[Moshe Carmeli $^\star$]{}
[Department of Physics, Ben Gurion University, Beer Sheva 84105, Israel]{}
[PACS numbers: 04.50.+h, 11.25.Mj., 11.27.+d, 98.80Cq]{}
$^\star$Email: [email protected]
Preliminaries
=============
As in classical general relativity we start our discussion in flat spacevelocity which will then be generalized to curved space.
The flat-spacevelocity cosmological metric is given by $$ds^2=\tau^2dv^2-\left(dx^2+dy^2+dz^2\right).\eqno(1)$$ Here $\tau$ is Hubble’s time, the inverse of Hubble’s constant, as given by measurements in the limit of zero distances and thus zero gravity. As such, $\tau$ is a constant, in fact a universal constant (its numerical value is given in Section 8, $\tau=12.486$Gyr). Its role in cosmology theory resembles that of $c$, the speed of light in vacuum, in ordinary special relativity. The velocity $v$ is used here in the sense of cosmology, as in Hubble’s law, and is usually not the time-derivative of the distance.
The Universe expansion is obtained from the metric (1) as a null condition, $ds=0$. Using spherical coordinates $r$, $\theta$, $\phi$ for the metric (1), and the fact that the Universe is spherically symmetric ($d\theta=
d\phi=0$), the null condition then yields $dr/dv=\tau$, or upon integration and using appropriate initial conditions, gives $r=\tau v$ or $v=H_0r$, i.e. the Hubble law in the zero-gravity limit.
Based on the metric (1) a cosmological special relativity (CSR) was presented in the text \[1\] (see Chapter 2). In this theory the receding velocities of galaxies and the distances between them in the Hubble expansion are united into a four-dimensional pseudo-Euclidean manifold, similarly to space and time in ordinary special relativity. The Hubble law is assumed and is written in an invariant way that enables one to derive a four-dimensional transformation which is similar to the Lorentz transformation. The parameter in the new transformation is the ratio between the cosmic time to $\tau$ (in which the cosmic time is measured backward with respect to the present time). Accordingly, the new transformation relates physical quantities at different cosmic times in the limit of weak or negligible gravitation.
The transformation between the four variables $x$, $y$, $z$, $v$ and $x'$, $y'$, $z'$, $v'$ (assuming $y'=y$ and $z'=z$) is given by $$x'=\frac{x-tv}{\sqrt{1-t^2/{\tau}^2}}, \ \ \ \
v'=\frac{v-tx/{\tau}^2}{\sqrt{1-t^2/{\tau}^2}}, \ \ \ \
y'=y, \ z'=z.\eqno(2)$$ Equations (2) are the [*cosmological transformation*]{} and very much resemble the well-known Lorentz transformation. In CSR it is the relative cosmic time which takes the role of the relative velocity in Einstein’s special relativity. The transformation (2) leaves invariant the Hubble time $\tau$, just as the Lorentz transformation leaves invariant the speed of light in vacuum $c$.
Cosmology in spacevelocity
==========================
A cosmological general theory of relativity, suitable for the large-scale structure of the Universe, was subsequently developed \[2-5\]. In the framework of cosmological general relativity (CGR) gravitation is described by a curved four-dimensional Riemannian spacevelocity. CGR incorporates the Hubble constant $\tau$ at the outset. The Hubble law is assumed in CGR as a fundamental law. CGR, in essence, extends Hubble’s law so as to incorporate gravitation in it; it is actually a [*distribution theory*]{} that relates distances and velocities between galaxies. The theory involves only measured quantities and it takes a picture of the Universe as it is at any moment. The following is a brief review of CGR as was originally given by the author in 1996 in Ref. 2.
The foundations of any gravitational theory are based on the principle of equivalence and the principle of general covariance \[6\]. These two principles lead immediately to the realization that gravitation should be described by a four-dimensional curved spacetime, in our theory spacevelocity, and that the field equations and the equations of motion should be written in a generally covariant form. Hence these principles were adopted in CGR also. Use is made in a four-dimensional Riemannian manifold with a metric $g_{\mu\nu}$ and a line element $ds^2=g_{\mu\nu}dx^\mu dx^\nu$. The difference from Einstein’s general relativity is that our coordinates are: $x^0$ is a velocitylike coordinate (rather than a timelike coordinate), thus $x^0=\tau v$ where $\tau$ is the Hubble time in the zero-gravity limit and $v$ the velocity. The coordinate $x^0=\tau v$ is the comparable to $x^0=ct$ where $c$ is the speed of light and $t$ is the time in ordinary general relativity. The other three coordinates $x^k$, $k=1,2,3$, are spacelike, just as in general relativity theory.
An immediate consequence of the above choice of coordinates is that the null condition $ds=0$ describes the expansion of the Universe in the curved spacevelocity (generalized Hubble’s law with gravitation) as compared to the propagation of light in the curved spacetime in general relativity. This means one solves the field equations (to be given in the sequel) for the metric tensor, then from the null condition $ds=0$ one obtains immedialety the dependence of the relative distances between the galaxies on their relative velocities.
As usual in gravitational theories, one equates geometry to physics. The first is expressed by means of a combination of the Ricci tensor and the Ricci scalar, and follows to be naturally either the Ricci trace-free tensor or the Einstein tensor. The Ricci trace-free tensor does not fit gravitation in general, and the Einstein tensor is a natural candidate. The physical part is expressed by the energy-momentum tensor which now has a different physical meaning from that in Einstein’s theory. More important, the coupling constant that relates geometry to physics is now also [*different*]{}.
Accordingly the field equations are $$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa T_{\mu\nu},\eqno(3)$$ exactly as in Einstein’s theory, with $\kappa$ given by $\kappa=8\pi k/\tau^4$, (in general relativity it is given by $8\pi G/c^4$), where $k$ is given by $k=G\tau^2/c^2$, with $G$ being Newton’s gravitational constant, and $\tau$ the Hubble constant time. When the equations of motion will be written in terms of velocity instead of time, the constant $k$ will replace $G$. Using the above equations one then has $\kappa=8\pi G/c^2\tau^2$.
The energy-momentum tensor $T^{\mu\nu}$ is constructed, along the lines of general relativity theory, with the speed of light being replaced by the Hubble constant time. If $\rho$ is the average mass density of the Universe, then it will be assumed that $T^{\mu\nu}=\rho u^\mu u^\nu,$ where $u^\mu=
dx^\mu/ds$ is the four-velocity. In general relativity theory one takes $T_0^0=\rho$. In Newtonian gravity one has the Poisson equation $\nabla^2\phi=4\pi G\rho$. At points where $\rho=0$ one solves the vacuum Einstein field equations in general relativity and the Laplace equation $\nabla^2\phi=0$ in Newtonian gravity. In both theories a null (zero) solution is allowed as a trivial case. In cosmology, however, there exists no situation at which $\rho$ can be zero because the Universe is filled with matter. In order to be able to have zero on the right-hand side of Eq. (3) one takes $T_0^0$ not as equal to $\rho$, but to $\rho_{eff}=\rho-\rho_c$, where $\rho_c$ is the critical mass density, a [*constant*]{} in CGR given by $\rho_c=3/8\pi G\tau^2$, whose value is $\rho_c\approx 10^{-29}g/cm^3$, a few hydrogen atoms per cubic meter. Accordingly one takes $$T^{\mu\nu}=\rho_{eff}u^\mu u^\nu;\mbox{\hspace{5mm}}\rho_{eff}=\rho-\rho_c
\eqno(4)$$ for the energy-momentum tensor.
In the next sections we apply CGR to obtain the accelerating expanding Universe and related subjects.
Gravitational field equations
=============================
In the four-dimensional spacevelocity the spherically symmetric metric is given by $$ds^2=\tau^2dv^2-e^\mu dr^2-R^2\left(d\theta^2+\sin^2\theta d\phi^2\right),
\eqno(5)$$ where $\mu$ and $R$ are functions of $v$ and $r$ alone, and comoving coordinates $x^\mu=(x^0,x^1,x^2,x^3)=(\tau v,r,\theta,\phi)$ have been used. With the above choice of coordinates, the zero-component of the geodesic equation becomes an identity, and since $r$, $\theta$ and $\phi$ are constants along the geodesics, one has $dx^0=ds$ and therefore $$u^\alpha=u_\alpha=\left(1,0,0,0\right).\eqno(6)$$ The metric (5) shows that the area of the sphere $r=constant$ is given by $4\pi R^2$ and that $R$ should satisfy $R'=\partial R/\partial r>0$. The possibility that $R'=0$ at a point $r_0$ is excluded since it would allow the lines $r=constants$ at the neighboring points $r_0$ and $r_0+dr$ to coincide at $r_0$, thus creating a caustic surface at which the comoving coordinates break down.
As has been shown in the previous sections the Universe expands by the null condition $ds=0$, and if the expansion is spherically symmetric one has $d\theta=d\phi=0$. The metric (5) then yields $$\tau^2 dv^2-e^\mu dr^2=0,\eqno(7)$$ thus $$\frac{dr}{dv}=\tau e^{-\mu/2}.\eqno(8)$$ This is the differential equation that determines the Universe expansion. In the following we solve the gravitational field equations in order to find out the function $\mu\left(r.v\right)$.
The gravitational field equations (3), written in the form $$R_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right),\eqno(9)$$ where $$T_{\mu\nu}=\rho_{eff}u_\mu u_\nu+p\left(u_\mu u_\nu-g_{\mu\nu}\right),
\eqno(10)$$ with $\rho_{eff}=\rho-\rho_c$ and $T=T_{\mu\nu}g^{\mu\nu}$, are now solved. Using Eq. (6) one finds that the only nonvanishing components of $T_{\mu\nu}$ are $T_{00}=\tau^2\rho_{eff}$, $T_{11}=c^{-1}\tau pe^\mu$, $T_{22}=c^{-1}\tau
pR^2$ and $T_{33}=c^{-1}\tau pR^2\sin^2\theta$, and that $T=\tau^2\rho_{eff}-
3c^{-1}\tau p$.
The only nonvanishing components of the Ricci tensor yield (dots and primes denote differentiation with respect to $v$ and $r$, respectively), using Eq. (9), the following field equations: $$R_{00}=-\frac{1}{2}\ddot{\mu}-\frac{2}{R}\ddot{R}-\frac{1}{4}\dot{\mu}^2=
\frac{\kappa}{2}\left(\tau^2\rho_{eff}+3c^{-1}\tau p\right),\eqno(11a)$$ $$R_{01}=\frac{1}{R}R'\dot{\mu}-\frac{2}{R}\dot{R}'=0,\eqno(11b)$$ $$R_{11}=e^\mu\left(\frac{1}{2}\ddot{\mu}+\frac{1}{4}\dot{\mu}^2+\frac{1}{R}
\dot{\mu}\dot{R}\right)+\frac{1}{R}\left(\mu'R'-2R''\right)$$ $$=\frac{\kappa}{2}e^\mu\left(\tau^2\rho_{eff}-c^{-1}\tau p\right),
\eqno(11c)$$ $$R_{22}=R\ddot{R}+\frac{1}{2}R\dot{R}\dot{\mu}+\dot{R}^2+1-e^{-\mu}\left(
RR''-\frac{1}{2}RR'\mu'+R'^2\right)$$ $$=\frac{\kappa}{2}R^2\left(\tau^2\rho_{eff}-c^{-1}\tau p\right),\eqno(11d)$$ $$R_{33}=\sin^2\theta R_{22}=\frac{\kappa}{2}R^2\sin^2\theta\left(\tau^2
\rho_{eff}-c^{-1}\tau p\right).\eqno(11e)$$
The field equations obtained for the components 00, 01, 11, and 22 (the 33 component contributes no new information) are given by $$-\ddot{\mu}-\frac{4}{R}\ddot{R}-\frac{1}{2}\dot{\mu}^2=\kappa\left(
\tau^2\rho_{eff}+3c^{-1}\tau p\right),
\eqno(12)$$ $$2\dot{R}'-R'\dot{\mu}=0, \eqno(13)$$ $$\ddot{\mu}+\frac{1}{2}\dot{\mu}^2+\frac{2}{R}\dot{R}\dot{\mu}+e^{-\mu}\left(
\frac{2}{R}R'\mu'-\frac{4}{R}R''\right)=\kappa\left(\tau^2\rho_{eff}-c^{-1}\tau
p\right) \eqno(14)$$ $$\frac{2}{R}\ddot{R}+2\left(\frac{\dot{R}}{R}\right)^2+\frac{1}{R}\dot{R}
\dot{\mu}+\frac{2}{R^2}+e^{-\mu}\left[\frac{1}{R}R'\mu'-2\left(\frac{R'}{R}
\right)^2-\frac{2}{R}R''\right]$$ $$=\kappa\left(\tau^2\rho_{eff}-c^{-1}\tau p\right).\eqno(15)$$ It is convenient to eliminate the term with the second velocity-derivative of $\mu$ from the above equations. This can easily be done, and combinations of Eqs. (12)–(15) then give the following set of three independent field equations: $$e^\mu\left(2R\ddot{R}+\dot{R}^2+1\right)-R'^2=-\kappa\tau c^{-1} e^\mu R^2p,
\eqno(16)$$ $$2\dot{R}'-R'\dot{\mu}=0, \eqno(17)$$ $$e^{-\mu}\left[\frac{1}{R}R'\mu'-\left(\frac{R'}{R}\right)^2-\frac{2}{R}R''
\right]+\frac{1}{R}\dot{R}\dot{\mu}+\left(\frac{\dot{R}}{R}\right)^2+
\frac{1}{R^2}$$ $$=\kappa\tau^2\rho_{eff}, \eqno(18)$$ other equations being trivial combinations of (16)–(18).
Solution of the field equations
===============================
The solution of Eq. (17) satisfying the condition $R'>0$ is given by $$e^\mu=\frac{R'^2}{1+f\left(r\right)},\eqno(19)$$ where $f\left(r\right)$ is an arbitrary function of the coordinate $r$ and satisfies the condition $f\left(r\right)+1>0$. Substituting (19) in the other two field equations (16) and (18) then gives $$2R\ddot{R}+\dot{R}^2-f=-\kappa c^{-1}\tau R^2p, \eqno(20)$$ $$\frac{1}{RR'}\left(2\dot{R}\dot{R'}-f'\right)+\frac{1}{R^2}\left(\dot{R}^2-f
\right)=\kappa\tau^2\rho_{eff},\eqno(21)$$ respectively.
The simplest solution of the above two equations, which satisfies the condition $R'=1>0$, is given by $$R=r.\eqno(22)$$ Using Eq. (22) in Eqs. (20) and (21) gives $$f\left(r\right)=\kappa c^{-1}\tau pr^2,\eqno(23)$$ and $$f'+\frac{f}{r}=-\kappa\tau^2\rho_{eff}r,\eqno(24)$$ respectively. The solution of Eq. (24) is the sum of the solutions of the homogeneous equation $$f'+\frac{f}{r}=0,\eqno(25)$$ and a particular solution of Eq. (24). These are given by $$f_1=-\frac{2Gm}{c^2r},\eqno(26)$$ and $$f_2=-\frac{\kappa}{3}\tau^2\rho_{eff}r^2.\eqno(27)$$
The solution $f_1$ represents a particle at the origin of coordinates and as such is not relevant to our problem. We take, accordingly, $f_2$ as the general solution, $$f\left(r\right)=-\frac{\kappa}{3}\tau^2\rho_{eff}r^2=-\frac{\kappa}{3}\tau^2
\left(\rho-\rho_c\right)r^2$$ $$=-\frac{\kappa}{3}\tau^2\rho_c\left(
\frac{\rho}{\rho_c}-1\right)r^2.\eqno(28)$$ Using the values of $\kappa=8\pi G/c^2\tau^2$ and $\rho_c=3/8\pi G\tau^2$, we obtain $$f\left(r\right)=\frac{1-\Omega_m}{c^2\tau^2}r^2,\eqno(29)$$ where $\Omega_m=\rho/\rho_c$.
The two solutions given by Eqs. (23) and (29) for $f(r)$ can now be equated, giving $$p=\frac{1-\Omega_m}{\kappa c\tau^3}=\frac{c}{\tau}\frac{1-\Omega_m}{8\pi G}
=4.544\left(1-\Omega_m\right)\times 10^{-2} g/cm^2.\eqno(30)$$ Furthermore, from Eqs. (19) and (22) we find that $$e^{-\mu}=1+f\left(r\right)=1+\tau c^{-1}\kappa pr^2=1+
\frac{1-\Omega_m}{c^2\tau^2}r^2.\eqno(31)$$
It will be recalled that the Universe expansion is determined by Eq. (8), $dr/dv=\tau e^{-\mu/2}$. The only thing that is left to be determined is the signs of $(1-\Omega_m)$ or the pressure $p$.
Thus we have $$\frac{dr}{dv}=\tau\sqrt{1+\kappa\tau c^{-1}pr^2}=\tau\sqrt{1+
\frac{1-\Omega_m}{c^2\tau^2}r^2}.\eqno(32)$$ For simplicity we confine ourselves to the linear approximation, thus Eq. (32) yields $$\frac{dr}{dv}=\tau\left(1+\frac{\kappa}{2}\tau c^{-1}pr^2\right)=
\tau\left[1+\frac{1-\Omega_m}{2c^2\tau^2}r^2\right].\eqno(33)$$
Classification of universes
===========================
The second term in the square bracket in the above equation represents the deviation due to gravity from the standard Hubble law. For without that term, Eq. (33) reduces to $dr/dv=\tau$, thus $r=\tau v+const$. The constant can be taken zero if one assumes, as usual, that at $r=0$ the velocity should also vanish. Thus $r=\tau v$, or $v=H_0r$ (since $H_0\approx 1/\tau$). Accordingly, the equation of motion (33) describes the expansion of the Universe when $\Omega_m=1$, namely when $\rho=\rho_c$. The equation then coincides with the standard Hubble law.
The equation of motion (33) can easily be integrated exactly by the substitions $$\sin\chi =\sqrt{\frac{\left(\Omega_m-1\right)}{2}}\frac{r}{2c\tau};\hspace{5mm}
\Omega_m>1,\eqno(34a)$$ $$\sinh\chi =\sqrt{\frac{\left(1-\Omega_m\right)}{2}}\frac{r}{2c\tau};
\hspace{5mm}\Omega_m<1.\eqno(34b)$$ One then obtains, using Eqs. (33) and (34), $$dv=cd\chi/\left(\Omega_m-1\right)^{1/2}\cos\chi;\hspace{5mm}\Omega_m>1,\eqno(35a)$$ $$dv=cd\chi/\left(1-\Omega_m\right)^{1/2}\cosh\chi;\hspace{5mm}\Omega_m<1.\eqno(35b)$$
We give below the exact solutions for the expansion of the Universe for each of the cases, $\Omega_m>1$ and $\Omega_m<1$. As will be seen, the case of $\Omega_m=1$ can be obtained at the limit $\Omega_m\rightarrow 1$ from both cases.
[**The case $\Omega_m>1$.**]{} From Eq. (35a) we have $$\int dv=\frac{c}{\sqrt{\left(\Omega_m-1\right)/2}}\int\frac{d\chi}{\cos\chi},
\eqno(36)$$ where $\sin\chi=r/a$, and $a=c\tau\sqrt{\left(\Omega_m-1\right)/2}$. A simple calculation gives \[7\] $$\int\frac{d\chi}{\cos\chi}=\ln\left|\frac{1+\sin\chi}{\cos\chi}\right|.
\eqno(37)$$ A straightforward calculation then gives $$v=\frac{a}{2\tau}\ln\left|\frac{1+r/a}{1-r/a}\right|.\eqno(38)$$ As is seen, when $r\rightarrow 0$ then $v\rightarrow 0$ and using the L’Hospital lemma, $v\rightarrow r/\tau$ as $a\rightarrow 0$ (and thus $\Omega_m\rightarrow 1$).
[**The case $\Omega_m<1$.**]{} From Eq. (35b) we now have $$\int dv=\frac{c}{\sqrt{\left(1-\Omega_m\right)/2}}\int\frac{d\chi}{\cosh\chi},
\eqno(39)$$ where $\sinh\chi=r/b$, and $b=c\tau\sqrt{\left(1-\Omega_m\right)/2}$. A straightforward calculation then gives \[7\] $$\int\frac{d\chi}{\cosh\chi}=\arctan e^\chi.\eqno(40)$$ We then obtain $$\cosh\chi=\sqrt{1+\frac{r^2}{b^2}},\eqno(41)$$ $$e^\chi=\sinh\chi+\cosh\chi=\frac{r}{b}+\sqrt{1+\frac{r^2}{b^2}}.\eqno(42)$$ Equations (39) and (40) now give $$v=\frac{2c}{\sqrt{\left(1-\Omega_m\right)/2}}\arctan e^\chi +K,\eqno(43)$$ where $K$ is an integration constant which is determined by the requirement that at $r=0$ then $v$ should be zero. We obtain $$K=-\pi c/2\sqrt{\left(1-\Omega_m\right)/2},\eqno(44)$$ and thus $$v=\frac{2c}{\sqrt{\left(1-\Omega_m\right)/2}}\left(\arctan e^\chi-
\frac{\pi}{4}\right).\eqno(45)$$ A straightforward calculation then gives $$v=\frac{b}{\tau}\left\{2\arctan\left(\frac{r}{b}+\sqrt{1+\frac{r^2}{b^2}}
\right)-\frac{\pi}{2}\right\}.\eqno(46)$$ As for the case $\Omega_m>1$ one finds that $v\rightarrow 0$ when $r\rightarrow
0$, and again, using L’Hospital lemma, $r=\tau v$ when $b\rightarrow 0$ (and thus $\Omega_m\rightarrow 1$).
Physical meaning
================
To see the physical meaning of these solutions, however, one does not need the exact solutions. Rather, it is enough to write down the solutions in the lowest approximation in $\tau^{-1}$. One obtains, by differentiating Eq. (33) with respect to $v$, for $\Omega_m>1$, $$d^2r/dv^2=-kr;\mbox{\hspace{10mm}}k=\frac{\left(\Omega_m-1\right)}{2c^2},\eqno(47)$$ the solution of which is $$r\left(v\right)=A\sin\alpha\frac{v}{c}+B\cos\alpha\frac{v}{c},\eqno(48)$$ where $\alpha^2=(\Omega_m-1)/2$ and $A$ and $B$ are constants. The latter can be determined by the initial condition $r\left(0\right)=0=B$ and $dr\left(0
\right)/dv=\tau=A\alpha/c$, thus $$r\left(v\right)=\frac{c\tau}{\alpha}\sin\alpha\frac{v}{c}.\eqno(49)$$ This is obviously a closed Universe, and presents a decelerating expansion.
For $\Omega_m<1$ we have $$d^2r/dv^2=\frac{\left(1-\Omega_m\right)r}{2c^2},\eqno(50)$$ whose solution, using the same initial conditions, is $$r\left(v\right)=\frac{c\tau}{\beta}\sinh\beta\frac{v}{c},\eqno(51)$$ where $\beta^2=(1-\Omega_m)/2$. This is now an open accelerating Universe.
For $\Omega_m=1$ we have, of course, $r=\tau v$.
The accelerating universe
=========================
We finally determine which of the three cases of expansion is the one at present epoch of time. To this end we have to write the solutions (49) and (51) in ordinary Hubble’s law form $v=H_0r$. Expanding Eqs. (49) and (51) into power series in $v/c$ and keeping terms up to the second order, we obtain $$r=\tau v\left(1-\alpha^2v^2/6c^2\right), \eqno(52a)$$ $$r=\tau v\left(1+\beta^2v^2/6c^2\right), \eqno(52b)$$ for $\Omega_m>1$ and $\Omega_m<1$, respectively. Using now the expressions for $\alpha$ and $\beta$, Eqs. (52) then reduce into the single equation $$r=\tau v\left[1+\left(1-\Omega_m\right)v^2/6c^2\right].\eqno(53)$$ Inverting now this equation by writing it as $v=H_0r$, we obtain in the lowest approximation $$H_0=h\left[1-\left(1-\Omega_m\right)v^2/6c^2\right],\eqno(54)$$ where $h=\tau^{-1}$. To the same approximation one also obtains $$H_0=h\left[1-\left(1-\Omega_m\right)z^2/6\right]=h\left[1-\left(1-\Omega_m
\right)r^2/6c^2\tau^2\right],\eqno(55)$$ where $z$ is the redshift parameter. As is seen, and it is confirmed by experiments, $H_0$ depends on the distance it is being measured; it has physical meaning only at the zero-distance limit, namely when measured [*locally*]{}, in which case it becomes $h=1/\tau$.
It follows that the measured value of $H_0$ depends on the “short" and “long" distance scales \[8\]. The farther the distance $H_0$ is being measured, the lower the value for $H_0$ is obtained. By Eq. (55) this is possible only when $\Omega_m<1$, namely when the Universe is accelerating. By Eq. (30) we also find that the pressure is positive.
The possibility that the Universe expansion is accelerating was first predicted using CGR by the author in 1996 \[2\] before the supernovae experiments results became known.
It will be noted that the constant expansion is just a transition stage between the decelerating and the accelerating expansions as the Universe evolves toward its present situation.
Figure 1 describes the Hubble diagram of the above solutions for the three types of expansion for values of $\Omega_m$ from 100 to 0.245. The figure describes the three-phase evolution of the Universe. Curves (1)-(5) represent the stages of [*decelerating expansion*]{} according to Eq. (49). As the density of matter $\rho$ decreases, the Universe goes over from the lower curves to the upper ones, but it does not have enough time to close up to a big crunch. The Universe subsequently goes over to curve (6) with $\Omega_m=1$, at which time it has a constant expansion for a fraction of a second. This then followed by going to the upper curves (7) and (8) with $\Omega_m<1$, where the Universe expands with [*acceleration*]{} according to Eq. (51). Curve no. 8 fits the present situation of the Universe. For curves (1)-(4) in the diagram we use the cutoff when the curves were at their maximum. In Table 1 we present the cosmic times with respect to the big bang, the cosmic radiation temperature and the pressure for each of the curves in Fig. 1.
Figures 2 and 3 show the Hubble diagrams for the distance-redshift relationship predicted by the theory for the accelerating expanding Universe at the present time, and Figures 4 and 5 show the experimental results.
Our estimate for $h$, based on published data, is $h\approx 80$ km/sec-Mpc. Assuming $\tau^{-1}\approx 80$ km/sec-Mpc, Eq. (55) then gives $$H_0=h\left[1-1.3\times 10^{-4}\left(1-\Omega_m\right)r^2\right],\eqno(56)$$ where $r$ is in Mpc. A computer best-fit can then fix both $h$ and $\Omega_m$.
To summarize, a theory of cosmology has been presented in which the dynamical variables are those of Hubble, i.e. distances and velocities. The theory descirbes the Universe as having a three-phase evolution with a decelerating expansion, followed by a constant and an accelerating expansion, and it predicts that the Universe is now in the latter phase. As the density of matter decreases, while the Universe is at the decelerating phase, it does not have enough time to close up to a big crunch. Rather, it goes to the constant-expansion phase, and then to the accelerating stage. As we have seen, the equation obtained for the Universe expansion, Eq. (51), is very simple.
Theory versus experiment
========================
The Einstein gravitational field equations with the added cosmological term are \[9\]: $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu},
\eqno(57)$$ where $\Lambda$ is the cosmological constant, the value of which is supposed to be determined by experiment. In Eq. (57) $R_{\mu\nu}$ and $R$ are the Ricci tensor and scalar, respectively, $\kappa=8\pi G$, where $G$ is Newton’s constant and the speed of light is taken as unity.
Recently the two groups (the [*Supernovae Cosmology Project*]{} and the [ *High-Z Supernova Team*]{}) concluded that the expansion of the Universe is accelerating \[10-16\]. The two groups had discovered and measured moderately high redshift ($0.3<z<0.9$) supernovae, and found that they were fainter than what one would expect them to be if the cosmos expansion were slowing down or constant. Both teams obtained $$\Omega_m\approx 0.3,\hspace{5mm} \Omega_\Lambda\approx 0.7,\eqno(58)$$ and ruled out the traditional ($\Omega_m$, $\Omega_\Lambda$)=(1, 0) Universe. Their value of the density parameter $\Omega_\Lambda$ corresponds to a cosmological constant that is small but, nevertheless, nonzero and positive, $$\Lambda\approx 10^{-52}\mbox{\rm m}^{-2}\approx 10^{-35}\mbox{\rm s}^{-2}.
\eqno(59)$$
In previous sections a four-dimensional cosmological theory (CGR) was presented. Although the theory has no cosmological constant, it predicts that the Universe accelerates and hence it has the equivalence of a positive cosmological constant in Einstein’s general relativity. In the framework of this theory (see Section 2) the zero-zero component of the field equations (3) is written as $$R_0^0-\frac{1}{2}\delta_0^0R=\kappa\rho_{eff}=\kappa\left(\rho-\rho_c
\right),\eqno(60)$$ where $\rho_c=3/\kappa\tau^2$ is the critical mass density and $\tau$ is Hubble’s time in the zero-gravity limit.
Comparing Eq. (60) with the zero-zero component of Eq. (57), one obtains the expression for the cosmological constant of general relativity, $$\Lambda=\kappa\rho_c=3/\tau^2.\eqno(61)$$
To find out the numerical value of $\tau$ we use the relationship between $h=\tau^{-1}$ and $H_0$ given by Eq. (55) (CR denote values according to Cosmological Relativity): $$H_0=h\left[1-\left(1-\Omega_m^{CR}\right)z^2/6\right],\eqno(62)$$ where $z=v/c$ is the redshift and $\Omega_m^{CR}=\rho_m/\rho_c$ with $\rho_c=3h^2/8\pi G$. (Notice that our $\rho_c=1.194\times 10^{-29}g/cm^3$ is different from the standard $\rho_c$ defined with $H_0$.) The redshift parameter $z$ determines the distance at which $H_0$ is measured. We choose $z=1$ and take for $$\Omega_m^{CR}=0.245, \eqno(63)$$ its value at the present time (see Table 1) (corresponds to 0.32 in the standard theory), Eq. (62) then gives $$H_0=0.874h.\eqno(64)$$ At the value $z=1$ the corresponding Hubble parameter $H_0$ according to the latest results from HST can be taken \[17\] as $H_0=70$km/s-Mpc, thus $h=(70/0.874)$km/s-Mpc, or $$h=80.092\mbox{\rm km/s-Mpc},\eqno(65)$$ and $$\tau=12.486 Gyr=3.938\times 10^{17}s.\eqno(66)$$
What is left is to find the value of $\Omega_\Lambda^{CR}$. We have $\Omega_\Lambda^{CR}=\rho_c^{ST}/\rho_c$, where $\rho_c^{ST}=3H_0^2/8\pi
G$ and $\rho_c=3h^2/8\pi G$. Thus $\Omega_\Lambda^{CR}=(H_0/h)^2=0.874^2$, or $$\Omega_\Lambda^{CR}=0.764.\eqno(67)$$ As is seen from Eqs. (63) and (67) one has $$\Omega_T=\Omega_m^{CR}+\Omega_\Lambda^{CR}=0.245+0.764=1.009\approx 1,
\eqno(68)$$ which means the Universe is Euclidean.
As a final result we calculate the cosmological constant according to Eq. (61). One obtains $$\Lambda=3/\tau^2=1.934\times 10^{-35}s^{-2}.\eqno(69)$$
Our results confirm those of the supernovae experiments and indicate on the existance of the dark energy as has recently received confirmation from the Boomerang cosmic microwave background experiment \[18,19\], which showed that the Universe is Euclidean.
Some remarks
============
In this paper the cosmological general relativity, a relativistic theory in spacevelocity, has been presented and applied to the problem of the expansion of the Universe. The theory, which predicts a positive pressure for the Universe now, describes the Universe as having a three-phase evolution: decelerating, constant and accelerating expansion, but it is now in the latter stage. Furthermore, the cosmological constant that was extracted from the theory agrees with the experimental result. Finally, it has also been shown that the three-dimensional spatial space of the Universe is Euclidean, again in agreement with observations.
Recently \[20,21\], more confirmation to the Universe accelerating expansion came from the most distant supernova, SN 1997ff, that was recorded by the Hubble Space Telescope. As has been pointed out before, if we look back far enough, we should find a decelerating expansion (curves 1-5 in Figure 1). Beyond $z=1$ one should see an earlier time when the mass density was dominant. The measurements obtained from SN 1997ff’s redshift and brightness provide a direct proof for the transition from past decelerating to present accelerating expansion (see Figures 6 and 7). The measurements also exclude the possibility that the acceleration of the Universe is not real but is due to other astrophysical effects such as dust.
Table 2 gives some of the cosmological parameters obtained here and in the standard theory.
Comparison with general relativity
==================================
In order to compare the present theory with general relativity, we now add the time coordinate. We then have a time-space-velocity Universe with two time-like and three space-like coordinates, with signature $(+---+)$. We will be concerned with the classical experiments of general relativity and the gravitational waves predicted by that theory. In the following we show that all these results are also obtained from the present theory. To this end we proceed as follows.
We first find the cosmological-equivalent of the Schwarzschild spherically-symmetric solution in cosmology. It will be useful to change variables from the classical Schwarzschild metric to new variables as follows: $$\sin^2\chi=r_s/r,\mbox{\hspace{5mm}}
dr=-2r_s\sin^{-3}\chi\cos\chi d\chi,\eqno(70)$$ where $r_s=2GM/c^2$ is the Schwarzschild radius. We also change the time coordinate $cdt=r_sd\eta$, thus $\eta$ is a time parameter. The classical Schwarzschild solution will thus have the following form: $$ds^2=r_s^2\left[\cos^2\chi d\eta^2-4\sin^{-6}\chi d\chi^2-\sin^{-4}\chi
\left(d\theta^2+\sin^2\theta d\phi^2\right)\right].\eqno(71)$$ So far this is just the classical spherically symmetric solution of the Einstein field equations in four dimensions, though written in new variables. The non-zero Christoffel symbols are given by $$\Gamma^0_{01}=-\sin\chi\cos^{-1}\chi,\mbox{\hspace{5mm}}
\Gamma^1_{00}=-\frac{1}{4}\sin^7\chi\cos\chi,$$ $$\Gamma^1_{11}=-3\sin^{-1}\chi\cos\chi,\mbox{\hspace{5mm}}
\Gamma^1_{22}=\frac{1}{2}\sin\chi\cos\chi,$$ $$\Gamma^1_{33}=\frac{1}{2}\sin\chi\cos\chi\sin^2\theta,\mbox{\hspace{5mm}}
\Gamma^2_{12}=-2\sin^{-1}\chi\cos\chi,\eqno(72)$$ $$\Gamma^2_{33}=-\sin\theta\cos\theta,\mbox{\hspace{5mm}}
\Gamma^3_{13}=-2\sin^{-1}\chi\cos\chi,\mbox{\hspace{5mm}}
\Gamma^3_{23}=\sin^{-1}\theta\cos\theta.$$ It is very lengthy, but one can verify that all components of the Ricci tensor $R_{\alpha\beta}$ are equal to zero identically.
We now extend this solution to cosmology. In order to conform with the standard notation, the zero component will be chosen as the time parameter, followed by the three space-like coordinates and then the fourth coordinate representing the velocity $\tau dv$. We will make one more change by choosing $\tau dv=r_sdu$, thus $u$ is the velocity parameter. The simplest way to have a cosmological solution of the Einstein field equation is using the so-called co-moving coordinates in which: $$ds^2=r_s^2\left[\cos^2\chi d\eta^2-4\sin^{-6}\chi d\chi^2-\sin^{-4}\chi
\left(d\theta^2+\sin^2\theta d\phi^2\right)+du^2\right].\eqno(73)$$ The coordinates are now $x^0=\eta$, $x^1=\chi$, $x^2=\theta$, $x^3=\phi$, and $x^4=u$, and $r_s$ is now a function of the velocity $u$, $r_s=r_s(u)$ to be determined by the Einstein field equations in five dimensions. Accordingly we have the following form for the metric: $$g_{\mu\nu}=r_s^2\left(\begin{array}{ccccc}
\cos^2\chi&&&&0\\
&-4\sin^{-6}\chi&&&\\
&&-\sin^{-4}\chi&&\\
&&&-\sin^{-4}\chi\sin^2\theta&\\
0&&&&1\\\end{array}\right),\eqno(74a)$$ $$\sqrt{-g}=2r_s^5\sin^{-7}\chi\cos\chi\sin\theta.\eqno(74b)$$
The non-zero Christoffel symbols are given by $$\Gamma^0_{01}=-\sin\chi\cos^{-1}\chi, \mbox{\hspace{5mm}}
\Gamma^0_{04}=\dot{r_s}r_s^{-1},$$ $$\Gamma^1_{00}=-\frac{1}{4}\sin^7\chi\cos\chi,\mbox{\hspace{5mm}}
\Gamma^1_{11}=-3\sin^{-1}\chi\cos\chi,$$ $$\Gamma^1_{14}=\dot{r_s}r_s^{-1},\mbox{\hspace{5mm}}
\Gamma^1_{22}=\frac{1}{2}\sin\chi\cos\chi,\mbox{\hspace{5mm}}
\Gamma^1_{33}=\frac{1}{2}\sin\chi\cos\chi\sin^2\theta,$$ $$\Gamma^2_{12}=-2\sin^{-1}\chi\cos\chi,\mbox{\hspace{5mm}}
\Gamma^2_{24}=\dot{r_s}r_s^{-1},\mbox{\hspace{5mm}}
\Gamma^2_{33}=-\sin\theta\cos\theta,\eqno(75)$$ $$\Gamma^3_{13}=-2\sin^{-1}\chi\cos\chi,\mbox{\hspace{5mm}}
\Gamma^3_{23}=\sin^{-1}\theta\cos\theta,\mbox{\hspace{5mm}}
\Gamma^3_{34}=\dot{r_s}r_s^{-1},$$ $$\Gamma^4_{00}=-\dot{r_s}r_s^{-1}\cos^2\chi,\mbox{\hspace{5mm}}
\Gamma^4_{11}=4\dot{r_s}r_s^{-1}\sin^{-6}\chi,\mbox{\hspace{5mm}}
\Gamma^4_{22}=\dot{r_s}r_s^{-1}\sin^{-4}\chi,$$ $$\Gamma^4_{33}=\dot{r_s}r_s^{-1}\sin^{-4}\chi\sin^2\theta,\mbox{\hspace{5mm}}
\Gamma^4_{44}=\dot{r_s}r_s^{-1},$$ where the dots denote derivatives with respect to the velocity parameter $u$.
The Ricci tensor components after a lengthy but straightforward calculation, are given by: $$R_{00}=-\left(\ddot{r_s}r_s^{-1}+2\dot{r_s}^2r_s^{-2}\right)\cos^2\chi,$$ $$R_{11}=4\left(\ddot{r_s}r_s^{-1}+2\dot{r_s}^2r_s^{-2}\right)\sin^{-6}\chi,$$ $$R_{22}=\left(\ddot{r_s}r_s^{-1}+2\dot{r_s}^2r_s^{-2}\right)\sin^{-4}\chi,\eqno(76)$$ $$R_{33}=\left(\ddot{r_s}r_s^{-1}+2\dot{r_s}^2r_s^{-2}\right)\sin^{-4}\chi
\sin^2\theta,$$ $$R_{44}=-4\left(\ddot{r_s}r_s^{-1}-\dot{r_s}^2r_s^{-2}\right).$$ All other components are identically zero.
We are interested in vacuum solution of the Einstein field equations for the spherically symmetric metric (generalized Schwarzschild to cosmology), the right-hand sides of the above equations should be taken zero. A simple calculation then shows that $\dot{r_s}=0$, $\ddot{r_s}=0$. Accordingly the cosmological Schwarzschild metric is given by Eq. (74a) with a constant $r_s=2GM/c^2$. The metric (74a) can then be written, using the coordinate transformations (70), as $$g_{\mu\nu}=\left(\begin{array}{ccccc}
1-\frac{r_s}{r}&&&&0\\
&-\left(1-\frac{r_s}{r}\right)^{-1}&&&\\
&&-r^2&&\\
&&&-r^2\sin^2\theta&\\
0&&&&1\\\end{array}\right),\eqno(77)$$ where the coordinates are now $x^0=ct$, $x^1=r$, $x^2=\theta$, $x^3=\phi$, and $x^4=\tau v$.
We are now in a position to compare the present theory with general relativity.
Gravitational redshift
======================
We start with the simplest experiment of the gravitational redshift. Although this experiment is not considered as one of the proofs of general relativity (it can be derived from conservation laws and Newtonian theory).
Consider two clocks at rest at two points denoted by 1 and 2. The propagation of light is determined by $ds$ at each point. Since at these points all spatial infinitesimal displacements and change in velocities vanish, one has $ds^2=g_{00}c^2dt^2.$ Hence at the two points we have $$ds\left(1\right)=\left[g_{00}\left(1\right)\right]^{1/2}cdt,\eqno(78a)$$ $$ds\left(2\right)=\left[g_{00}\left(2\right)\right]^{1/2}cdt\eqno(78b)$$ for the proper time (see Fig. 8).
The ratio of the rates of similar clocks, located at different places in a gravitational field, is therefore given by $$ds\left(2\right)/ds\left(1\right)=
\left[g_{00}\left(2\right)/g_{00}\left(1\right)\right]^{1/2}.
\eqno(79)$$ The frequency $\nu_0$ of an atom located at point 1, when measured by an observer located at point 2, is therefore given by $$\nu=\nu_0\left[
g_{00}\left(1\right)/g_{00}\left(2\right)\right]^{1/2}.
\eqno(80)$$
If the gravitational field is produced by a spherically symmetric mass distribution, then we may use the generalized Schwarzschild metric given above to calculate the above ratio at the two points. In this case $g_{00}=1-2GM/c^2r$, and therefore $$\left[g_{00}\left(1\right)/g_{00}\left(2\right)\right]^{1/2}\approx
1+\left(GM/c^2\right)\left(1/r_2-1/r_1\right)$$ to first order in $GM/c^2r$. We thus obtain $$\Delta\nu/\nu_0=\left(\nu-\nu_0\right)/\nu_0\approx -\left(GM/c^2\right)
\left(1/r_1-1/r_2\right)$$ for the frequency shift per unit frequency. Taking now $r_1$ to be the observed radius of the Sun and $r_2$ the radius of the Earth’s orbit around the Sun, then we find that $$\Delta\nu/\nu_0\approx -GM_{Sun}/c^2r_{Sun},\eqno(81)$$ where $M_{Sun}$ and $r_{Sun}$ are the mass and radius of the Sun. Accordingly we obtain $\Delta\nu/\nu_0\approx -2.12\times 10^{-6}$ for the frequency shift per unit frequency of the light emitted from the Sun. The calculation made above amounts to neglecting completely the Earth’s gravitational field. The above result is the standard gravitational redshift (also known as the gravitational time dilation).
Motion in a centrally symmetric gravitational field
===================================================
We assume that small test particles move along geodesics in the gravitational field. We also assume that planets have small masses as compared with the mass of the Sun, to the extent that they can be considered as test particles moving in the gravitational field of the Sun. As a result of these assumptions, the geodesic equation in the cosmological Schwarzschild field will be taken to describe the equation of motion of a planet moving in the gravitational field of the Sun. In fact, we do not need the exact solution of the generalized Schwarzschild metric (77), but just its first approximation. We obtain in the first approximation the following expressions for the components of the metric tensor: $$g_{00}=1-r_s/r,\mbox{\hspace{5mm}}g_{0m}=0,\mbox{\hspace{5mm}}g_{04}=0,$$ $$g_{mn}=-\delta_{mn}-r_sx^mx^n/r^3,\mbox{\hspace{5mm}}
g_{m4}=0,\mbox{\hspace{5mm}}g_{44}=1.\eqno(82a)$$ The contravariant components of the metric tensor are consequently given, in the same approximation, by $$g^{00}=1+r_s/r,\mbox{\hspace{5mm}}g^{0m}=0,\mbox{\hspace{5mm}}g^{04}=0,$$ $$g^{mn}=-\delta^{mn}+r_sx^mx^n/r^3,\mbox{\hspace{5mm}}
g^{m4}=0,\mbox{\hspace{5mm}}g^{44}=1.\eqno(82b)$$ We may indeed verify that the relation $g_{\mu\lambda}g^{\lambda\nu}=
\delta_\mu^\nu$ between the contravariant and covariant components of the above approximate metric tensor is satisfied to orders of magnitude of the square of $r_s/r$. A straightforward calculation then gives the following expressions for the Christoffel symbols: $$\Gamma^0_{0n}=-\frac{r_s}{2}\frac{\partial}{\partial x^n}\left(\frac{1}{r}
\right),$$ $$\Gamma^k_{00}=--\frac{r_s}{2}\left(1-\frac{r_s}{r}\right)
\frac{\partial}{\partial x^k}\left(\frac{1}{r}\right),\eqno(83)$$ $$\Gamma^k_{mn}=r_s\frac{x^k}{r^3}\delta_{mn}-
\frac{3}{2}r_s\frac{x^kx^mx^n}{r^5}.$$ All other components vanish.
We now use these expressions for the Christoffel symbols in the geodesic equation $$\ddot{x^k}+\left(\Gamma^k_{\alpha\beta}-\Gamma^0_{\alpha\beta}\dot{x^k}
\right)\dot{x^\alpha}\dot{x^\beta}=0,\eqno(84)$$ where a dot denotes differentiation with respect to the time coordinate $x^0$. We obtain $$\Gamma^0_{\alpha\beta}\dot{x^\alpha}\dot{x^\beta}=\Gamma^0_{00}+
2\Gamma^0_{0n}\dot{x^n}+2\Gamma^0_{04}\dot{x^4}+
\Gamma^0_{mn}\dot{x^m}\dot{x^n}+2\Gamma^0_{m4}\dot{x^m}\dot{x^4}+
\Gamma^0_{44}\dot{x^4}\dot{x^4}$$ $$=-r_s\dot{x^n}
\frac{\partial}{\partial x^n}\left(\frac{1}{r}\right),\eqno(85a)$$ $$\Gamma^k_{\alpha\beta}\dot{x^\alpha}\dot{x^\beta}=\Gamma^k_{00}+
2\Gamma^k_{0l}\dot{x^l}+2\Gamma^k_{04}\dot{x^4}+\Gamma^k_{mn}\dot{x^m}\dot{x^n}
+2\Gamma^k_{m4}\dot{x^m}\dot{x^4}+\Gamma^k_{44}\dot{x^4}\dot{x^4}$$ $$=-\frac{r_s}{2}
\frac{\partial}{\partial x^k}\left(\frac{1}{r}\right)+
r_s\left[\frac{r_s}{2r}\frac{\partial}{\partial x^k}\left(\frac{1}{r}\right)
-\left(\dot{x^s}\dot{x^s}\right)\frac{\partial}{\partial x^k}
\left(\frac{1}{r}\right)-
\frac{3}{2r^5}\left(x^s\dot{x^s}\right)^2x^k\right].\eqno(85b)$$ Consequently we obtain from the geodesic equation (84) the following equation of motion for the planet: $$\ddot{x^k}-\frac{r_s}{2}
\frac{\partial}{\partial x^k}\left(\frac{1}{r}\right)$$ $$=r_s\left[\left(\dot{x^s}\dot{x^s}\right)\frac{\partial}{\partial x^k}
\left(\frac{1}{r}\right)-\frac{r_s}{2r}\frac{\partial}{\partial x^k}
\left(\frac{1}{r}\right)-\dot{x^n}\frac{\partial}{\partial x^n}
\left(\frac{1}{r}\right)\dot{x^k}+
\frac{3}{2r^5}\left(x^s\dot{x^s}\right)^2x^k\right].\eqno(86)$$ Replacing now the derivatives with respect to $x^0$ by those with respect to $t(\equiv x^0/c)$ in the latter equation, we obtain $$\ddot{\mbox{\bf x}}-GM\nabla\frac{1}{r}=
r_s\left[\left(\dot{\mbox{\bf x}}^2\right)\nabla
\left(\frac{1}{r}\right)-\frac{GM}{r}\nabla
\left(\frac{1}{r}\right)-\left(\dot{\mbox{\bf x}}\cdot\nabla
\frac{1}{r}\right)\dot{\mbox{\bf x}}+
\frac{3}{2r^5}\left(\mbox{\bf x}\cdot\dot{\mbox{\bf x}}\right)^2\mbox{\bf x}\right],
\eqno(87)$$ where use has been made of the three-dimensional notation.
Hence the equation of motion of the planet differs from the Newtonian one since the left-hand side of Eq. (87) is proportional to terms of order of magnitude $r_s$ instead of vanishing identically. This correction leads to a fundamental effect, namely, to a systematically secular change in the perihelion of the orbit of the planet.
To integrate the equation of motion (87) we multiply it vectorially by the radius vector [**x**]{}. We obtain $$\mbox{\bf x}\times\ddot{\mbox{\bf x}}=-r_s\left(\dot{\mbox{\bf x}}\cdot
\nabla\left(1/r\right)\right)
\left(\mbox{\bf x}\times\dot{\mbox{\bf x}}\right).\eqno(88)$$ All other terms in Eq. (87) are proportional to the radius vector [**x**]{} and thus contribute nothing. Equation (88) may be integrated to yield the first integral $$\mbox{\bf x}\times\dot{\mbox{\bf x}}=\mbox{\bf J}e^{-r_s/r}.\eqno(89)$$ Here [**J**]{} is a constant vector, the [*angular momentum*]{} per mass unit of the planet. One can easily check that the first integral (89) indeed leads back to Eq. (88) by taking the time derivatives of both sides of Eq. (89).
From Eq. (89) we see that the radius vector [**x**]{} moves in a plane perpendicular to the constant angular momentum vector [**J**]{}, thus the planet moves in a plane similar to the case in Newtonian mechanics. If we now introduce in this plane coordinates $r$ and $\phi$ to describe the motion of the planet, the equation of motion (87) consequently decomposes into two equations. Introducing now the new variable $u=1/r$, we can then rewrite the equations in terms of $u(\phi)$, using $$\dot{r}=-\frac{u'}{u^2}\dot{\phi},$$ $$\ddot{r}=\frac{2u'^2}{u^3}\dot{\phi}^2-\frac{u''}{u^2}\dot{\phi}^2-
\frac{u'}{u^2}\ddot{\phi},$$ where a prime denotes a differentiation with respect to the angle $\phi$. We subsequently obtain $$\ddot{\phi}=2\frac{u'}{u}\dot{\phi}^2-\frac{2GM}{c^2}u'\dot{\phi}^2.$$ A straightforward calculation then gives, using the expression for $\ddot{\phi}$, $$u''+u-GM\left(\frac{u^2}{\dot{\phi}}\right)^2=\frac{GM}{c^2}\left[2u^2-u'^2
-2GMu\left(\frac{u^2}{\dot{\phi}}\right)^2\right]\eqno(90).$$ The latter equation can be further simplified if we use the first integral $$r^2\dot{\phi}=Je^{-2GM/c^2r}.$$ We obtain $$\frac{u^2}{\dot{\phi}}=\frac{1}{J}e^{2GMu/c^2},$$ $$\left(\frac{u^2}{\dot{\phi}}\right)^2=\frac{1}{J^2}e^{4GMu/c^2}\approx
\frac{1}{J^2}\left(1+\frac{4GM}{c^2}u\right).$$ Hence, to an accuracy of $1/c^2$, Eq. (90) gives $$u''+u-\frac{GM}{J^2}=\frac{GM}{c^2}\left(2u^2-u'^2+2\frac{GM}{J^2}u\right).
\eqno(91)$$
Equation (91) can be used to determine the motion of the planet. The Newtonian equation of motion that corresponds to Eq. (91) is one whose left-hand side is identical to the above equation, but is equal to zero rather than to the terms on the right-hand side. This fact can easily be seen if one lets $GM/c^2$ go to zero in Eq. (91). Therefore in the Newtonian limit we have $$u''+u-\frac{GM}{J^2}\approx 0,\eqno(92)$$ whose solution can be written as $$u\approx u_0\left(1+\epsilon\cos\phi\right).\eqno(93)$$ Here $u_0$ is a constant, and $\epsilon$ is the eccentricity of the ellipse, $\epsilon=(1-b^2/a^2)^{1/2},$ where $a$ and $b$ are the semimajor and semiminor axes of the ellipse. Using the solution (93) in the Newtonian limit of the equation of motion (92) then determines the value of the constant $u_0$, as $u_0=GM/J^2$.
To solve the equation of motion (91), we therefore assume a solution of the form $$u=u_0\left(1+\epsilon\cos\alpha\phi\right),\eqno(94)$$ where $\alpha$ is some parameter to be determined, and whose value in the usual nonrelativistic mechanics is unity. The appearance of the parameter $\alpha\neq 1$ in our solution is an indication that the motion of the planet will no longer be a closed ellipse.
Using the above solution in Eq. (91), and equating coefficients of $\cos\alpha\phi$, then gives $$\alpha^2=1-\frac{2GM}{c^2}\left(2u_0+\frac{GM}{J^2}\right).$$ If we substitute for $GM/J^2$ in the above equation its nonrelativistic value $u_0$, then the error will be of a higher order. Hence the latter equation can be written as $$\alpha^2=1-\frac{6GM}{c^2}u_0$$ or $$\alpha=1-\frac{3GM}{c^2}u_0.\eqno(95)$$
Successive perihelia occur at two angles $\phi_1$ and $\phi_2$ when $\alpha
\phi_2-\alpha\phi_1=2\pi$. Since the parameter $\alpha$ is smaller than unity, we have $\phi_2-\phi_1=2\pi/\alpha>2\pi$. Hence we can write $\phi_2-\phi_1=
2\pi+\Delta\phi$, with $\Delta\phi>0$, or $$\alpha\left(\phi_2-\phi_1\right)=\alpha\left(2\pi+\Delta\phi\right)=\left(
1-\frac{3GM}{c^2}u_0\right)\left(2\pi+\Delta\phi\right)=2\pi.\eqno(96)$$ As a result there will be an [*advance*]{} in the perihelion of the orbit of the planet per revolution given by Eq. (96) or, to first order, by $$\Delta\phi=6\pi GMu_0/c^2.\eqno(97)$$
The constant $u_0$ can also be expressed in terms of the eccentricity, using the Newtonian approximation. Denoting the radial distances of the orbit, which correspond to the angles $\phi_2=0$ and $\phi_1=\pi$, by $r_2$ and $r_1$, respectively, we have from Eq. (93), $$1/r_2=u_0\left(1+\epsilon\right),\mbox{\hspace{5mm}}
1/r_1=u_0\left(1-\epsilon\right).$$ Hence since $r_1+r_2=2a$, we obtain (see Fig. 9) $$2a=r_1+r_2=2/u_0\left(1-\epsilon^2\right),$$ where $a$ is the semimajor axis of the orbit, and therefore $$u_0=1/a\left(1-\epsilon^2\right).$$ Using this value for $u_0$ in the expression (97) for $\Delta\phi$, we obtain for the perihelion advance the expression $$\Delta\phi=\frac{6\pi GM}{c^2a\left(1-\epsilon^2\right)}\eqno(98)$$ in radians per revolution (see Fig. 10). This is the standard general relativistic formula for the advance of the perihelion.
In the next section we discuss the deflection of a light ray moving in a gravitational field.
Deflection of light in a gravitational field
============================================
To discuss the effect of gravitation on the propagation of light signals we may use the geodesic equation, along with the null condition $ds=0$ at a fixed velocity. A light signal propagating in the gravitational field of the Sun, for instance, will thus be described by the null geodesics in the cosmological Schwarzschild field at $dv=0$.
Using the approximate solution for the cosmological Schwarzschild metric, given by Eq. (82a), we obtain $$g_{\mu\nu}dx^\mu dx^\nu=\left(1-\frac{2GM}{c^2r}\right)c^2dt^2-\left[dx^s
dx^s+\frac{2GM}{c^2}\frac{\left(x^sdx^s\right)^2}{r^3}\right]=0.\eqno(99)$$ Hence we have, to the first approximation in $GM/c^2$, the following equation of motion for the propagation of light in a gravitational field: $$\left(1+\frac{2GM}{c^2r}\right)\left[\left(\dot{x^s}\dot{x^s}\right)+
\frac{2GM}{c^2}\frac{\left(x^s\dot{x^s}\right)^2}{r^3}\right]=c^2,\eqno(100)$$ where a dot denotes differentiation with respect to the time coordinate $t(\equiv x^0/c)$.
Just as in the case of planetary motion (see previous section), the motion here also takes place in a plane. Hence in this plane we may introduce the polar coordinates $r$ and $\phi$. The equation of motion (100) then yields, to the first approximation in $GM/c^2$, the following equation in the polar coordinates: $$\left(\dot{r}^2+r^2\dot{\phi}^2\right)+\frac{4GM}{c^2}\frac{\dot{r}^2}{r}+
\frac{2GM}{c^2}r\dot{\phi}^2=c^2.\eqno(101)$$ Changing now variables from $r$ to $u(\phi)\equiv 1/r$, we obtain $$\left[u'^2+u^2+\frac{2GMu}{c^2}\left(2u'^2+u^2\right)\right]
\left(\frac{\dot{\phi}}{u^2}\right)^2=c^2,\eqno(102)$$ where a prime denotes differentiation with respectto the angle $\phi$.
Moreover we may use the first integral of the motion, $$r^2\dot{\phi}=Je^{-2GM/c^2r},\eqno(103)$$ in Eq. (102), thus getting $$u'^2+u^2+\frac{2GMu}{c^2}\left(2u'^2+u^2\right)=\left(\frac{c}{J}\right)^2
e^{4GMu/c^2}.\eqno(104)$$ Differentiation of this equation with respect to $\phi$ then gives $$u''+u+\frac{GM}{c^2}\left(2u'^2+4uu''+3u^2\right)=\frac{2GM}{J^2}.
\eqno(105)$$ In Eq. (105) terms have been kept to the first approximation in $GM/c^2$ only.
To solve Eq. (105) we notice that, in the lowest approximation, we have, from Eq. (104), $$u'^2\approx\left(\frac{c}{J}\right)^2-u^2,\eqno(106)$$ $$u''\approx-u.\eqno(107)$$ Hence using these approximate expressions in Eq. (105) gives $$u''+u=\frac{3GM}{c^2}u^2 \eqno(108)$$ for the equation of motion of the orbit of the light ray propagating in a spherically symmetric gravitational field.
In the lowest approximation, namely, when the gravitational field of the central body is completely neglected, the right-hand side of Eq. (108) can be taken as zero, and therefore $u$ satisfies the equation $u''+u=0$. The solution of this equation is a straight line given by $$u=\frac{1}{R}\sin\phi,\eqno(109)$$ where $R$ is a constant. This equation for the straight line shows that $r
\equiv 1/u$ has a minimum value $R$ at the angle $\phi=\pi/2$. If we denote $y=r\sin\phi$, the straight line (109) can then be described by $$y=r\sin\phi=R=\mbox{\rm constant} \eqno(110)$$ (see Fig. 11).
We now use the approximate value for $u$, Eq. (109), in the right-hand side of Eq. (108), since the error introduced in doing so is of higher order. We therefore obtain the following for the equation of motion of the orbit of the light ray: $$u''+u=\frac{3GM}{c^2R^2}\sin^2\phi.\eqno(111)$$ The solution of this equation is then given by $$u=\frac{1}{R}\sin\phi+\frac{GM}{c^2R^2}\left(1+\cos^2\phi\right).\eqno(112)$$
Introducing now the Cartesian coordinates $x=r\cos\phi$ and $y=r\sin\phi$, the above solution can then be written as $$y=R-\frac{GM}{c^2R}\frac{2x^2+y^2}{\left(x^2+y^2\right)^{1/2}}.\eqno(113)$$ We thus see that for large values of $|x|$ the above solution asymptotically approaches the following expression: $$y\approx R-\frac{2GM}{c^2R}|x|.\eqno(114)$$
As seen from Eq. (114), asymptotically, the orbit of the light ray is described by two straight lines in the spacetime. These straight lines make angles with respect to the $x$ axis given by $\tan\phi=\pm\left(2GM/c^2R
\right)$ (see Fig. 12). The angle of deflection $\Delta\phi$ between the two asymptotes is therefore given by $$\Delta\phi=\frac{4GM}{c^2R}.\eqno(115)$$
This is the angle of [*deflection*]{} of a light ray in passing through the gravitational field of a central body, described by the cosmological Schwarzschild metric. For a light ray just grazing the Sun, Eq. (115) gives the value $$\Delta\phi=\frac{4GM_{\mbox{\rm Sun}}}{c^2R_{\mbox{\rm Sun}}}=
1.75\mbox{\rm seconds}.$$ This is the standard general-relativistic formula. Observations indeed confirm this result. One of the latest measurements gives $1.75\pm 0.10$ seconds. It is worth mentioning that only general relativity theory and the present theory predict the correct factor of the deflection of light in the gravitational field.
In the next section the gravitational radiation prediction is considered.
Gravitational radiation
=======================
In the following we show that the present theory also predicts gravitational radiation, a distinguished result of classical general relativity theory. We will not develop a complete theory of gravitational radiation. Rather we will confine ourselves in showing that the present theory does predict the phenomenon. This is done in the weak field approximation, as is usually done in standard general relativity theory.
Linear approximation
--------------------
For convenience, the coordinate system to be used in the linearized theory will be Cartesian, and hence the Minkowskian metric will have the form $$\eta_{\mu\nu}=\eta^{\mu\nu}=(1,-1,-1,-1,1),\eqno(116)$$ when $c$ and $\tau$ are taken as unity. The gravitational field described by the metric tensor $g_{\mu\nu}$ is now called weak if it differs from the Minkowskian metric tensor by terms which are much smaller than unity, $$\left|g_{\mu\nu}-\eta_{\mu\nu}\right|\ll 1.\eqno(117)$$ The above condition need not be satisfied in the entire spacetime, and it could be valid at a region of it.
We now assume that the metric tensor can be expanded as an infinite series, $$g_{\mu\nu}=\eta_{\mu\nu}+\lambda{}_1g_{\mu\nu}+\lambda^2{}_2g_{\mu\nu}+
\cdots,\eqno(118)$$ where $\lambda$ is some small parameter, and we limit ourselves to the first-order term ${}_1g_{\mu\nu}$ alone. Hence we can write $$g_{\mu\nu}\approx \eta_{\mu\nu}+h_{\mu\nu},\eqno(119a)$$ where $h_{\mu\nu}=\lambda{}_1g_{\mu\nu}$. We also expand the contravariant components of the metric tensor, $$g^{\mu\nu}\approx \eta^{\mu\nu}+h^{\mu\nu}.\eqno(119b)$$ From the condition $g_{\mu\lambda}g^{\lambda\nu}=\delta_\mu^\nu$ one then is able to relate $h^{\mu\nu}$ to $h_{\mu\nu}$ (neglecting nonlinear terms), $$h^{\mu\nu}=-\eta^{\mu\rho}\eta^{\nu\sigma}h_{\sigma\rho}.\eqno(120)$$
The linearized Einstein equations
---------------------------------
We can now derive the linearized Einstein equations. To this end we have to find the first approximate value of the Einstein tensor, the Ricci tensor, the Ricci scalar, and the Christoffel symbols. A simple calculation then gives $$\Gamma^\mu_{\alpha\beta}\approx\frac{1}{2}\eta^{\mu\lambda}\left(
h_{\lambda\alpha,\beta}+h_{\lambda\beta,\alpha}-h_{\alpha\beta,\lambda}\right)
\eqno(121)$$ for the Christoffel symbols and $$R_{\alpha\beta\gamma\delta}\approx\frac{1}{2}\left(
h_{\alpha\delta,\beta\gamma}+h_{\beta\gamma,\alpha\delta}-
h_{\beta\delta,\alpha\gamma}-h_{\alpha\gamma,\beta\delta}\right)\eqno(122)$$ for the Riemann tensor. Accordingly we have the following expressions for the Ricci tensor, the Ricci scalar, and the Einstein tensor, respectively: $$R_{\beta\delta}\approx\frac{1}{2}\eta^{\alpha\gamma}\left(
h_{\alpha\delta,\beta\gamma}+h_{\beta\gamma,\alpha\delta}-
h_{\beta\delta,\alpha\gamma}-h_{\alpha\gamma,\beta\delta}\right)\eqno(123)$$ $$R\approx\eta^{\alpha\gamma}\eta^{\beta\delta}\left(
h_{\alpha\delta,\beta\gamma}-h_{\beta\delta,\alpha\gamma}\right)\eqno(124)$$ $$G_{\mu\nu}\approx-\frac{1}{2}\left[h_{,\mu\nu}+\eta^{\rho\sigma}\left(
h_{\mu\nu,\rho\sigma}-h_{\mu\rho,\nu\sigma}-h_{\nu\rho,\mu\sigma}\right)-
\eta_{\mu\nu}\eta^{\rho\sigma}\left(h_{,\rho\sigma}-\eta^{\alpha\beta}
h_{\rho\sigma,\alpha\beta}\right)\right],\eqno(125)$$ where $h=\eta^{\alpha\beta}h_{\alpha\beta}$.
A simplification in the linearized field equations occurs if we introduce the new variables $$\gamma_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h,\eqno(126)$$ from which one obtains $$h_{\mu\nu}=\gamma_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\gamma,\eqno(127)$$ with $\gamma=\eta^{\alpha\beta}\gamma_{\alpha\beta}$. Introducing the above expressions into the Einstein field equations we obtain $$\bigcirc\gamma_{\mu\nu}-\eta^{\alpha\beta}\left(\gamma_{\alpha\mu,\beta\nu}+
\gamma_{\alpha\nu,\beta\mu}\right)+\eta_{\mu\nu}\eta^{\lambda\rho}
\eta^{\alpha\beta}\gamma_{\lambda\alpha,\rho\beta}=-2\kappa T_{\mu\nu}
\eqno(128)$$ for the linearized gravitational field equations. In Eq. (128) the symbol $\bigcirc$ is the operator in flat space, $$\bigcirc f=\eta^{\alpha\beta}f_{,\alpha\beta}=\left(\frac{1}{c^2}
\frac{\partial^2}{\partial t^2}-\nabla^2+\frac{1}{\tau^2}
\frac{\partial^2}{\partial v^2}\right)f.\eqno(129)$$
We can simplify still further the above field equations by choosing coordinates in which $$\gamma_\mu=\eta^{\rho\sigma}\gamma_{\mu\rho,\sigma}=0.\eqno(130)$$ This is similar to choosing a gauge in solving the wave equation in electrodynamics. As a result we finally obtain for the linearized Einstein equations the following: $$\bigcirc\gamma_{\mu\nu}=-2\kappa T_{\mu\nu},\eqno(131)$$ along with the supplementary condition $$\eta^{\rho\sigma}\gamma_{\mu\rho,\sigma}=0,\eqno(132)$$ which solutions $\gamma_{\mu\nu}$ of Eq. (131) should satisfy. Finally we see from Eq. (131) that a necessary condition for Eq. (132) to be satisfied is that $$\eta^{\alpha\beta}T_{\mu\alpha,\beta}=0,\eqno(133)$$ which is an expression for the conservation of the energy and momentum without including gravitation.
Gravitational waves
-------------------
In vacuum, Eq. (131) reduces to $$\bigcirc\gamma_{\mu\nu}=0,\eqno(134)$$ or $$\left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)
\gamma_{\mu\nu}=\frac{1}{\tau^2}\frac{\partial^2\gamma_{\mu\nu}}{\partial v^2}.
\eqno(135)$$ Thus the gravitational field, like the electromagnetic field, propagates in vacuum with the speed of light. The above analysis also shows the existance of gravitational waves.
9 1. M. Carmeli, [*Cosmological Special Relativity: The Large-Scale Structure of Space, Time and Velocity*]{} (World Scientific, River Edge, N.J., 1997). 2. M. Carmeli, [*Commun. Theor. Phys.*]{} [**5**]{}, 159 (1996). 3. S. Behar and M. Carmeli, [*Intern. J. Theor. Phys.*]{} [**39**]{}, 1375 (2000). (astro-ph/0008352)4. M. Carmeli and S. Behar, Cosmological general relativity, pp. 5–26, in: [*Quest for Mathematical Physics*]{}, T.M. Karade, [*et al.*]{} Editors, New Delhi (2000). 5. M. Carmeli and S. Behar, Cosmological relativity: a general relativistic theory for the accelerating universe, Talk given at Dark Matter 2000, Los Angeles, February 2000, pp.182–191, in: [*Sources and Detection of Dark Matter/Energy in the Universe*]{}, D. Cline, Ed., Springer (2001).6. A. Einstein, [*The Meaning of Relativity*]{}, 5th Edition (Princeton Univ. Press, Princeton, 1955). 7. I.S. Gradshteyn and I.M. Ryshik, [*Table of Integrals, Series and Products*]{} (Academic Press, New York; 1980).8. P.J.E. Peebles, Status of the big bang cosmology, in:, C.W. Akerlof and M.A. Srednicki, Editors (New York Academy of Sciences, New York, 1993), p. 84. 9. M. Carmeli, [*Classical Fields: General Relativity and Gauge Theory*]{} (Wiley, New York, 1982).10. P.M. Garnavich [*et al.*]{}, [*Astrophys. J.*]{} [**493**]{}, L53 (1998). \[Hi-Z Supernova Team Collaboration (astro-ph/9710123)\].11. B.P. Schmidt [*et al.*]{}, [*Astrophys. J.*]{} [**507**]{}, 46 (1998). \[Hi-Z Supernova Team Collaboration (astro-ph/9805200)\].12. A.G. Riess [*et al.*]{}, [*Astronom. J.*]{} [**116**]{}, 1009 (1998). \[Hi-Z Supernova Team Collaboration (astro-ph/9805201)\].13. P.M. Garnavich [*et al.*]{}, [*Astrophys. J.*]{} [**509**]{}, 74 (1998). \[Hi-Z Supernova Team Collaboration (astro-ph/9806396)\].14. S. Perlmutter [*et al.*]{}, [*Astrophys. J.*]{} [**483**]{}, 565 (1997). \[Supernova Cosmology Project Collaboration (astro-ph/9608192)\].15. S. Perlmutter [*et al.*]{}, [*Nature*]{} [**391**]{}, 51 (1998). \[Supernova Cosmology Project Collaboration (astro-ph/9712212)\].16. S. Perlmutter [*et al.*]{}, [*Astrophys. J.*]{} [**517**]{}, 565 (1999). \[Supernova Cosmology Project Collaboration (astro-ph/9812133)\].17. W.L. Freedman [*et al.*]{}, Final results from the Hubble Space Telescope, Talk given at the 20th Texas Symposium on Relativistic Astrophysics, Austin, Texas 10-15 December 2000. (astro-ph/0012376)18. P. de Bernardis [*et al.*]{}, [*Nature*]{} [**404**]{}, 955 (2000). (astro-ph/0004404)19. J.R. Bond [*et al.*]{}, in [*Proc. IAU Symposium*]{} 201 (2000). (astro-ph/0011378) 20. A.V. Filippenko and A.G. Riess, p.227 in: [*Particle Physics and Cosmology: Second Tropical Workshop*]{}, J.F. Nieves, Editor (AIP, New York, 2000). 21. A.G. Riess [*et al.*]{}, [*Astrophys. J.*]{}, in press. (astro-ph/0104455)
Table 1: The Cosmic Times with respect to the Big Bang, the Cosmic Temperature and the Cosmic Pressure for each of the Curves in Fig. 1.
[c r@l c r@[.]{}l r@l r@l]{}\
Curve&&Time in Units && &\
No$^\star$.& & & of $\tau$ && &(K)&\
\
\
1&100& &$3.1\times 10^{-6}$&$3$&$87\times 10^{-5}$&1096&&-&4.499\
2&25& &$9.8\times 10^{-5}$&$1$&$22\times 10^{-3}$&195&.0&-&1.091\
3&10& &$3.0\times 10^{-4}$&$3$&$75\times 10^{-3}$&111&.5&-&0.409\
4&5& &$1.2\times 10^{-3}$&$1$&$50\times 10^{-2}$&58&.20&-&0.182\
5&1&.5&$1.3\times 10^{-2}$&$1$&$62\times 10^{-1}$&16&.43&-&0.023\
\
6&1& &$3.0\times 10^{-2}$&$3$&$75\times 10^{-1}$&11&.15&&0\
\
7&0&.5&$1.3\times 10^{-1}$&$1$&$62$&5&.538&+&0.023\
8&0&.245&$1.0$&$12$&$50$&2&.730&+&0.034\
\
$^\star$The calculations are made using Carmeli’s cosmological transformation, Eq. (2), that relates physical quantities at different cosmic times when gravity is extremely weak.
For example, we denote the temperature by $\theta$, and the temperature at the present time by $\theta_0$, we then have $$\theta=\frac{\theta_0}{\sqrt{1-\displaystyle\normalsize\frac{t^2}{\tau^2}}}=
\frac{\theta_0}{\sqrt{1-\displaystyle\normalsize\frac{\left(\tau-T\right)^2}
{\tau^2}}}=\frac{2.73K}{\sqrt{\displaystyle\normalsize\frac{2\tau T-T^2}
{\tau^2}}}=\frac{2.73K}
{\sqrt{\displaystyle\normalsize\frac{T}{\tau}\left(2-\frac{T}{\tau}\right)}},$$ where T is the time with respect to B.B.
The formula for the pressure is given by Eq. (30), $p=c(1-\Omega)/8\pi G\tau$. Using $c=3\times 10^{10}cm/s$, $\tau=3.938\times 10^{17}s$ and $G=6.67\times
10^{-8}cm^3/gs^2$, we obtain $$p=4.544\times 10^{-2}\left(1-\Omega\right)g/cm^2.$$
[Table 2: Cosmological parameters in cosmological general relativity and in standard theory]{}
[p[35mm]{}p[35mm]{}p[35mm]{}]{}\
&COSMOLOGICAL&STANDARD\
&RELATIVITY&THEORY\
\
Theory type&Spacevelocity&Spacetime\
Expansion&Tri-phase:&One phase\
type&decelerating, constant,&\
&accelerating&\
Present expansion&Accelerating&One of three\
&(predicted)&possibilities\
Pressure&$0.034g/cm^2$&Negative\
Cosmological constant&$1.934\times 10^{-35}s^{-2}$&Depends\
&(predicted)&\
$\Omega_T=\Omega_m+\Omega_\Lambda$&1.009&Depends\
Constant-expansion&8.5Gyr ago&No prediction\
occurs at&&\
Constant-expansion&Fraction of&Not known\
duration&second&\
Temperature at&146K&No prediction\
constant expansion&&\
[FIGURE CAPTIONS]{}
Fig. 1 Hubble’s diagram describing the three-phase evolution of the Universe according to cosmological general relativity theory. Curves (1) to (5) represent the stages of [*decelerating*]{} expansion according to $r(v)=
(c\tau/\alpha)\sin\alpha v/c$, where $\alpha^2=(\Omega-1)/2$, $\Omega=\rho/
\rho_c$, with $\rho_c$ a [*constant*]{}, $\rho_c=3/8\pi G\tau^2$, and $c$ and $\tau$ are the speed of light and the Hubble time in vacuum (both universal constants). As the density of matter $\rho$ decreases, the Universe goes over from the lower curves to the upper ones, but it does not have enough time to close up to a big crunch. The Universe subsequently goes to curve (6) with $\Omega=1$, at which time it has a [*constant*]{} expansion for a fraction of a second. This then followed by going to the upper curves (7), (8) with $\Omega<1$ where the Universe expands with [*acceleration*]{} according to $r(v)=(c\tau/\beta)\sinh\beta v/c$, where $\beta^2=(1-\Omega)/2$. Curve no. 8 fits the present situation of the Universe. (Source: S. Behar and M. Carmeli, Ref. 3)Fig. 2 Hubble’s diagram of the Universe at the present phase of evolution with accelerating expansion. (Source: S. Behar and M. Carmeli, Ref. 3) Fig. 3 Hubble’s diagram describing decelerating, constant and accelerating expansions in a logarithmic scale. (Source: S. Behar and M. Carmeli, Ref. 3) Fig. 4 Distance vs. redshift diagram showing the deviation from a constant toward an accelerating expansion. (Source: A. Riess [*et al.*]{}, Ref. 12) Fig. 5 Relative intensity of light and relative distance vs. redshift. (Source: A. Riess [*et al.*]{}, Ref. 12)Fig. 6 Hubble diagram of SNe Ia minus an empty (i.e., “empty" $\Omega=0$) Universe compared to cosmological and astrophysical models. The points are the redshift-binned data from the HZT (Riess [*et al.*]{} 1998) and the SCP (Perlmutter [*et al.*]{} 1999). The measurements of SN 1997ff are inconsistent with astrophysical effects which could mimic previous evidence for an accelerating Universe from SNe Ia at $z\approx 0.5$. (Source: A. Riess [*et al.*]{}, Ref. 21)Fig. 7 Same as Fig. 6 with the inclusion of a family of plausible, flat $\Omega_\Lambda$ cosmologies. The transition redshift (i.e., the coasting point) between the accelerating and decelerating phases is indicated and is given as $[2\Omega_\Lambda/\Omega_M]^{1/3}-1$. SN 1997ff is seen to lie within the epoch of deceleration. This conclusion is drawn from the result that the apparent brightness of SN 1997ff is inconsistent with values of $\Omega_\Lambda\geq 0.9$ and hence a transition redshift greater than that of SN 1997ff. (Source: A. Riess [*et al.*]{}, Ref. 21) Fig. 8 Propagation of light in curved spacetime.Fig. 9 Newtonian limit of planetary motion. The motion is described by a closed ellipse if the effect of other planets is completely neglected. Fig. 10 Planetary elliptic orbit with perihelion advance. The effect is a general relativistic one. The advance of the perihelion is given by $\Delta
\phi$ in radians per revolution, where $\Delta\phi=6\pi GM/c^2a(1-
\epsilon^2)$, with $M$ being the mass of the Sun, $a$ the semimajor axis, and $\epsilon$ the eccentricity of the orbit of the planet. Fig. 11 Light ray when the effect of the central body’s gravitational field is completely neglected. The light ray then moves along the straight line $y=r\sin\phi=R=\mbox{\rm constant}$, namely, $u=1/r=(1/R)\sin\phi$ Fig. 12 Bending of a light ray in the gravitational field of a spherically symmetric body. The angle of deflection $\Delta\phi=4GM/c^2R$, where $M$ is the mass of the central body and $R$ is the closest distance of the light ray from the center of the body.
| || | | |
| || | | |
| |
| | | |
| || | | |
| || | | |
|
---
abstract: 'Making use of coherence and entanglement as metrological quantum resources allows to improve the measurement precision from the shot-noise- or quantum limit to the Heisenberg limit. Quantum metrology then relies on the availability of quantum engineered systems that involve controllable quantum degrees of freedom which are sensitive to the measured quantity. Sensors operating in the qubit mode and exploiting their coherence in a phase-sensitive measurement have been shown to approach the Heisenberg scaling in precision. Here, we show that this result can be further improved by operating the quantum sensor in the qudit mode, i.e., by exploiting $d$ rather than 2 levels. Specifically, we describe the metrological algorithm for using a superconducting transmon device operating in a qutrit mode as a magnetometer. The algorithm is based on the base-3 semi-quantum Fourier transformation and enhances the quantum theoretical performance of the sensor by a factor 2. Even more, the practical gain of our qutrit-implementation is found in a reduction of the number of iteration steps of the quantum Fourier transformation by a factor $\log 2/\log 3 \approx 0.63$ as compared to the qubit mode. We show, that a two-tone capacitively coupled rf-signal is sufficient for the implementation of the algorithm.'
author:
- 'A.R. Shlyakhov'
- 'V.V. Zemlyanov'
- 'M.V. Suslov'
- 'A.V. Lebedev'
- 'G.S. Paraoanu'
- 'G.B. Lesovik'
- 'G. Blatter'
title: Quantum metrology with a transmon qutrit
---
The idea to boost metrological precision with the help of quantum resources underwent an impressive development during recent years [@Giovanetti:2011; @Degen:2017]. Both types of quantum resources, coherence and entanglement, are used in either sequential or parallel strategies, respectively [@Giovanetti:2006]. A key role in this endeavor is played by novel quantum algorithms, in particular, Kitaev’s phase estimation [@Kitaev:1995; @Cleve:1998] or the quantum Fourier transformation and its semi-classical variant [@Griffiths:1996], both exploiting phase coherence as their quantum resource and thus following the sequential strategy. Previous theoretical and experimental work in this direction has all routed in a base-2 computational scheme that exploits qubits as measuring devices. In this paper, we demonstrate that a superconducting transmon device[@Koch:2007] operated in a qutrit (or base-3) mode offers an enhanced performance as a magnetic-field sensor; we present a specific algorithm exploiting the semi-classical quantum Fourier transform as well as the required radio frequency (rf) voltage-pulses for its implementation.
The use of entanglement-free protocols in metrology has been developed in several steps, first addressing the problem of measuring the magnitude of classical fields [@Vaidman:2004], followed by a suggestion to quantify the mesoscopic magnetic field generated by an assembly of nuclear magnetic moments [@Giedke:2006], and a proposal to use nitrogen–vacancy (NV) centers in diamond for nanoscale magnetometry [@Twamley:2011]. The basic concept of this type of measurement was first used in a measurement of an optical phase through interferometry [@Higgins:2007] and followed by the implementation of a high-dynamic-range magnetic-field sensor in the form of a NV center [@Wrachtrup:2012]. An alternative route has been taken by starting from the statistical counting of charge in mesoscopic transport [@Levitov:1996] that required to include the measurement apparatus into the analysis. Originally, the latter has been described by a spin that interacts with the charge-transporting lead within a Gedankenexperiment. This idea has later been taken to a realistic setup with a measurement device in the form of a charge- or flux- qubit [@Hassler:2006; @Lebedev:2016]. Subsequently, the statistical counting has been refined to an algorithm that counts the number of charges traversing the lead by making full use of quantum engineering ideas in combination with the semi-classical Fourier transform [@Lesovik:2010]. While all of the above work is based on qubits or base-2 counting, the concept of quantum counting [@Lesovik:2010] suggests a natural generalization of such a scheme to qudits or base-$d$ counting [@Suslov:2011]. Here, we propose an application of the qudit counting algorithm to a metrological measurement scheme for a magnetometer that makes use of a superconducting transmon device [@Koch:2007] operated in a qutrit mode. Making efficient use of the larger Hilbert space of a transmon qutrit and its specific linear energy level dependence on the measured magnetic flux then allows for a faster acquisition of information and thereby a more efficient measurement.
The algorithmic use of quantum engineered devices requires a non-linear spectrum in order to address the quantum states individually. Superconducting circuit devices with Josephson junctions in loop geometries [@You:2011] can be viewed as artificial atoms; they exhibit energy spectra with unequal level spacings that can be designed on demand. Moreover, the position of energy levels in such devices is sensitive to the magnetic flux penetrating the SQUID loop of the artificial atom. In particular, the ultra-high sensitivity of the transition frequency of the flux qubit [@Mooij:1999] allows for its use as an ultra-high sensitive magnetic flux sensor [@Bal:2012]. Unfortunately, the high sensitivity of flux qubits to low-frequency noise [@Yoshihara:2006] reduces their coherence time, that makes them unfavorable for the implementation of quantum metrological procedures. On the other hand, the special design of the transmon atom [@Koch:2007] renders this device insensitive to the background charge noise, resulting in larger coherence times. Recently, the Kitaev- and Fourier-like phase estimation algorithms have been successfully implemented in a dc magnetic-flux measurement with a transmon qubit [@Danilin:2017], resulting in an algorithmically improved sensitivity at high dynamic range.
The spectrum of a transmon atom is characterized by a particularly simple form that corresponds to a harmonic oscillator with a weak non-linearity. Furthermore, the transmon-atom’s spectrum has a linear magnetic-flux dependence of the excited states with respect to the ground state (to leading order in the small ratio $E_{\rm \scriptscriptstyle C}/ E_{\rm
\scriptscriptstyle J}$ of charge- and Josephson energies of the transmon atom). Such a linear flux-dependence allows one to exploit several excited levels of the transmon atom and implement a Fourier phase-estimation algorithm operating in the qudit regime. On the other hand, the charge degree of freedom enters the spectrum in a non-linear way, allowing for the transmon’s manipulation via capacitively coupled rf-fields. Such coherent manipulation of a transmon atom operating in a qutrit regime has been demonstrated recently in several works[@Abdulmalikov:2013; @Kumar:2016]; below, we will show how to exploit the combination of these features in the effective operation of the transmon as a magnetometer approaching Heisenberg scaling over the coherence time of the device.
Operating the transmon in the Heisenberg limit, i.e., attaining a measurement precision that scales in the invested resources $R$ as $1/R$ rather than $1/\sqrt{R}$, requires a suitable metrological algorithm, in our case, the semi-classical Fourier transform. The basic step in this algorithm is a Ramsey cycle of duration $\tau$ that accumulates a phase $\phi = \mu H \tau /\hbar$ involving the magnetic field $H$ to be measured; here, the magnetic moment $\mu$ plays the role of a coupling constant. Subsequent Ramsey cycles with incrementally reduced times $\tau$, readout, and use of the result in the next step allows to implement the quantum Fourier transformation (QFT) that boosts the device performance to make it attain the Heisenberg limit. This QFT is usually implemented in a binary system exploiting qubit devices. The implementation of this algorithm in a ternary system with qutrits improves the performance by a factor of 2, i.e., using half the resources one arrives at the same precision as with the binary implementation. This information-theoretical result is further improved when considering the number of steps required for the same precision. Such a performance criterium makes sense as most of the time required for the measurement is invested in the preparation and readout of the transmon that has to be repeated in every step. It turns out that the number of steps is reduced by a factor $\ln(2)/\ln(3)
\approx 0.63$ when replacing the qubit-mode by a qutrit implementation, a considerable speedup of the measurement.
In the following, we briefly recapitulate the binary metrological algorithm (Sec. \[sec:qubit\]) and then extend the discussion to the ternary system that exploits qutrits, see Sec. \[sec:qutrit\]. The comparison between the two procedures in Sec. \[sec:comp\] leads us to the performance results discussed above. In Sec. \[sec:transmon\], we discuss the spectral properties of the transmon device and find the specific rf-pulses that prepare the qutrit for the measurement and for the readout: Essentially, these pulses generate a basis change (and back) from the computational basis (where the qutrit is operated) to the ‘counting’ or measurement basis where the field imposes its characteristic ‘rotation’ of the qutrit that encodes the unknown magnitude of the field. It turns out, that a two-tone pulse addressing the the first and second excited states is sufficient to perform this task. In Sec.\[sec:conclusion\], we summarize and conclude our discussion.
Qubit metrological procedure {#sec:qubit}
============================
We start with a brief summary of the standard qubit-based metrological procedure for the measurement of a constant magnetic field. The elementary step in this measurement scheme is a Ramsey interference experiment where the qubit (or equivalently a spin-1/2) is prepared in an equally weighted (or balanced) superposition of its $z$-polarized states $|\!\!\downarrow\rangle$ and $|\!\!\uparrow\rangle$, $|\psi_0\rangle = (|\!\!\downarrow\rangle
+|\!\!\uparrow\rangle)/\sqrt{2}$. The state $|\psi_0\rangle$ is located in the equatorial $xy$-plane of the Bloch sphere and can be prepared by performing a $\pi/2$ rotation of the $|\!\! \downarrow\rangle$ state around a $y$-axis, $|\psi_0\rangle = \hat{U}_y(\pi/2) |\!\!\downarrow\rangle$. Next, the state $|\psi_0\rangle$ is exposed during a time interval $\tau$ to a magnetic field $H$ directed along the $z$-axis, $|\psi_0\rangle \to
\exp(-i\hat\sigma_z \phi/2) |\psi_0\rangle \equiv |\psi_\phi\rangle$, with $\phi = \mu H \tau/\hbar$ and $\mu$ is the magnetic moment of the spin. As a result, the initial state $|\psi_0\rangle$ picks up the additional field-sensitive relative phase $\phi$, $|\psi_\phi\rangle = (e^{-i\phi/2}
|\!\!\downarrow\rangle + e^{i\phi/2} |\!\!\uparrow \rangle) /\sqrt{2}$. The last step of the Ramsey interference experiment is the readout procedure, where the information about the value of the magnetic field encoded in the qubit state is extracted through a projective measurement of the spin polarization along the $z$-axis. In order to make the outcome probabilities $P_{\uparrow, \downarrow}$ depend on the magnetic field $H$, the state $|\psi_\phi\rangle$ is first transformed by applying a unitary readout operation, which coincides with that for the preparation for the qubit case, $|\psi_\mathrm{out}\rangle\! = \hat{U}_y(\pi/2) |\psi_\phi\rangle =
-i\sin(\phi/2) {|\!\!\downarrow\rangle} + \cos(\phi/2)|\!\!\uparrow\rangle$. Repeating this Ramsey cycle several times, one can accumulate enough statistics and extract the value of the field $H$, see Refs..
Quite importantly, for the specific situation where the magnetic field $H$ can assume only two values $H = 0$ and $H = h$, one can unambiguously distinguish between these two possibilities during a single Ramsey cycle, i.e., a single-shot measurement. Indeed, adjusting the time delay $\tau$ such that $\phi = \mu h\tau/\hbar = \pi$, one has that either $|\psi_\phi\rangle =
|\psi_0\rangle$ or $|\psi_\phi\rangle = |\psi_1\rangle = (|\langle \downarrow
\rangle - |\uparrow\rangle)/\sqrt{2}$ and hence, $|\psi_\mathrm{out}\rangle =
|\uparrow\rangle$ or $|\psi_\mathrm{out}\rangle = |\downarrow\rangle$. This results in the probabilities $P_\uparrow = 1$ and $P_\downarrow = 0$ for $H=0$ or $P_\uparrow = 0$ and $P_\downarrow = 1$ for $H = h$, allowing for a single-shot distinction between the two field values. The basis states $|\uparrow \rangle$ and $|\downarrow \rangle$ then define the so-called computational basis, while the states $|\psi_0\rangle$ and $|\psi_1\rangle$ form the counting basis.
The above remarkable fact can be further exploited to distinguish between $2^K$ discrete magnetic-field values with only $K$ Ramsey experiments: Let the magnetic field $H \in [0,2h_0]$ assume only discrete values that correspond to an exact $K$-bit fractional binary representation of the form $$H = h_0\, \Bigl( \frac{b_0}{2^0} + \frac{b_1}{2^1} + \frac{b_2}{2^2} +
\dots + \frac{b_{K-1}}{2^{K-1}} \Bigr),
\label{eq:Hbinary}$$ where the amplitudes $b_n$, $n=0,\dots,K-1$, take binary values $0$ and $1$. Let us also choose an elementary time delay $\tau_0$ such that $\mu h_0
\tau_0/\hbar = \pi$. A Ramsey measurement with an enhanced time delay $\tau_{K-1} = 2^{K-1} \tau_0$ then accumulates a phase $\phi_{K-1} = \pi
b_{K-1} +2\pi n$, where the integer $n= 2^{K-1}b_0+\dots+ 2b_{K-2}$ is given by the previous bits $b_n$, $n=0,\dots,K-2$. Although this first Ramsey experiment provides the phase $\phi_{K-1}$ only modulo $2\pi$, the even or odd outcome for this modulo-$2\pi$ phase allows for an unambiguous identification of the last binary digit $b_{K-1}$.
In the next step, the time delay in the Ramsey experiment is twice reduced, $\tau_{K-2} = 2^{K-2}\tau_0$. The accumulated field-sensitive phase is now given by $\phi_{K-2} = \pi (b_{K-2} + b_{K-1}/2) \mod 2\pi$. Since we have already learned the value of the bit $b_{K-1}$ in the previous measurement, we can apply an additional rotation $\hat{U}_z(-\pi b_{K-1}/2)$ prior to the readout operation in order to compensate for the residual phase $\pi
b_{K-1}/2$. The subsequent readout operation and measurement along $z$ provides the next bit $b_{K-2}$ in a deterministic way. Proceeding analogously with gradually decreased time delays $2^{K-3}\tau_0, 2^{K-4}\tau_0,\dots,
\tau_0$ allows for an unambiguous determination of all bits $b_{K-1},
b_{K-2},\dots, b_0$ in the binary representation of the magnetic field $H$ and thus its precise selection out of the $2^K$ discrete allowed values, see Eq..
Qutrit metrology {#sec:qutrit}
================
Next, we consider the generalization of the above qubit-based metrological scheme to a qutrit, i.e., a quantum system that is endowed with a three-dimensional Hilbert space. Let the quantum system in question be a spin-1 system with (computational) basis states $|0\rangle$, $|1\rangle$, and $|2\rangle$ corresponding to the $m_z=-1, 0, 1$ angular-momentum polarizations along the $z$-axis. As with the qubit case, we prepare the qutrit in a *balanced* state $|\psi_0\rangle = (|0\rangle +|1\rangle
+|2\rangle)/\sqrt{3}$ and expose it to a constant magnetic field $H$ directed along $z$-axis during the time $\tau$. As a result, the information about the value of the field is encoded into the relative phases of the qutrit state, $$|\psi_\phi\rangle = \frac1{\sqrt{3}}\Bigl(|0\rangle + e^{i\phi} |1\rangle
+ e^{2i\phi} |2\rangle \Bigr),
\label{eq:exposedqtrit}$$ where $\phi = \mu H\tau/\hbar$ and we have omitted the overall phase factor $e^{-i\phi}$.
To start with, we consider a situation where the magnetic field assumes only one of three values $H \in \{0,h,2h\}$, $h > 0$; the task then is to unambiguously distinguish between these three alternatives via a single-shot measurement of the state (\[eq:exposedqtrit\]); such a one-shot discrimination is indeed possible as was shown in Ref.within the context of the quantum counting problem. We expose the initial balanced state $|\psi_0\rangle$ during a specific time interval $\tau_0$ to the field such that the phase $\phi = \mu h\tau_0/\hbar$ assumes the value $\phi = 2\pi/3$. As a result, the qutrit ends up in one of the counting states $|\psi_\phi \rangle =|\psi_0\rangle$, $|\psi_\phi \rangle
=|\psi_1\rangle = (|0\rangle + e^{2\pi\,i/3}|1\rangle +
e^{-2\pi\,i/3}|2\rangle)/\sqrt{3}$ or $|\psi_\phi \rangle =|\psi_2\rangle =
(|0\rangle + e^{-2\pi\,i/3}|1\rangle + e^{2\pi\,i/3}|2\rangle)/\sqrt{3}$, depending on the discrete field values $0,h$, or $2h$. Applying a base-$d$ quantum inverse Fourier transformation $\hat{F}^{-1}_d$, $$\hat{F}_d^{-1}|n\rangle = \frac1{\sqrt{d}} \sum_{k=0}^{d-1}
\mathrm{e}^{-2\pi i nk/d} \, |k\rangle,
\label{eq:invFourier}$$ with $d = 3$ to these counting states, one can check that the resulting state $|\psi_\mathrm{out}\rangle = \hat{F}^{-1}_3 |\psi_\phi\rangle$ coincides with one of the computational states $|0\rangle$, $|1\rangle$, or $|2\rangle$, depending on the magnetic field $H$ taking the values $0$, $h$ or $2h$, respectively. Therefore, measuring the polarization of the resulting state $|\psi_\mathrm{out}\rangle$ along $z$-axis allows for an unambiguous distinction between the three possible values of the magnetic field.
Next, consider the situation where the magnetic field $H$ has an exact ternary representation $$H = h_0\, \Bigl( \frac{t_0}{3^0}+ \frac{t_1}{3^1} + \dots
+ \frac{t_{K-1}}{3^{K-1}} \Bigr),
\label{eq:Htrinary}$$ where the amplitudes (or trits) $t_n$, $n=0,\dots,K-1$, can take only three discrete values $0,1$, and $2$. Then, similarly to the qubit case, one can successively determine all $K$ trinary digits starting from the least significant digit $t_{K-1}$ within $K$ separate preparation–exposure–readout steps. Indeed, in the first step, we prepare the qutrit in the balanced state $|\psi_0\rangle$ and expose it to the magnetic field $H$ during the time interval $\tau_{K-1} = 3^{K-1} \tau_0$, where $\tau_0$ is chosen to satisfy the relation $\mu h_0\tau_0/\hbar = 2\pi/3$. Then, the magnetic-field dependent phase $\phi_{K-1} = (2\pi/3)\,t_{K-1} \mod 2\pi$ can only take the three values $0, 2\pi/3$, and $4\pi/3$, which can be unambiguously distinguished by the inverse Fourier transform in the readout step described above. Next, the digit $t_{K-2}$ is determined by reducing the exposure time by one-third, $\tau_{K-2} = 3^{K-2} \, \tau_0$, that provides the field-dependent phase $\phi_{K-2} = (2\pi/3) (t_{K-2} +t_{K-1}/3)$. Making use of the digit $t_{K-1}$ found in the previous step, the residual phase $2\pi
t_{K-1}/9$ is compensated before readout through projection along $z$, that leads to the next trinary digit or trit $t_{K-2}$, and so on.
In the most general situation, the magnetic field $H$ assumes continuous values and has no exact finite representation as in Eqs. (\[eq:Hbinary\]) or (\[eq:Htrinary\]). Hence, the number of trinary (or binary) digits needed to describe it is infinite and one cannot measure the field $H$ exactly with a finite number of preparation–exposure–readout steps. Still, we can find an approximate value of the field by detecting the first $K$ digits of its numerical representation. E.g., using a base-3 representation with qutrits and applying the magnetic field $H$ during the time interval $\tau_{K-1}$ to the balanced state $|\psi_0\rangle$, the magnetic field induces the phase $\phi_{K-1} = (2\pi/3) t_{K-1} + 3^{K-1} \delta\phi$, where the residual phase $\delta\phi \in [0, \pi/3^K]$ as given by $$\delta\phi = \frac{2\pi}3\, \Bigl( \frac{t_K}{3^K}
+ \frac{t_{K+1}}{3^{K+1}}+ \dots \Bigr)$$ is unknown and cannot be compensated any more. As a consequence, rather than definitive outcomes, we have to find the probabilities $P_k$ to observe the qutrit in the states $|k\rangle$, $k=0,1,2$. Applying the inverse Fourier transform and analyzing the result, we find that the probabilities $$\begin{aligned}
\label{eq:P012}
&& P_0 = \frac19 \bigl[ 1 + 2\cos \bigl( {2\pi}\, t_{K-1}/3
+3^{K-1}\delta\phi\bigr) \bigr]^2, \\
&& P_1 = \frac19 \bigl[ 1 + 2\cos \bigl( {2\pi}\, (t_{K-1}-1)/3
+3^{K-1}\delta\phi \bigr) \bigl]^2, \nonumber \\
&& P_2 = \frac19 \bigl[ 1 + 2\cos \bigl( {2\pi}\, (t_{K-1}-2)/3
+ 3^{K-1}\delta\phi \bigr) \bigl]^2, \nonumber\end{aligned}$$ deviate from zero and unity due to the unknown phase $\delta \phi$, hence, we cannot any more distinguish between different $t_{K-1}$ unambiguously. Instead, we have to resort to a statistical analysis and select between the three alternatives $t_{K-1} = 0,1$ or $2$ by finding the maximum probability $P_k$, $k=0,1,2$. In practice, the weights of the three probabilities in Eq. are disjoint and a single measurement is sufficient to determine the trit’s value with good confidence; the overall success probabilities for the measurement schemes discussed below are calculated on the basis of this assumption. Repeating the procedure for the remaining $K-1$ trits including the required phase compensation prior to the readout, one arrives at a set of $K$ trinary digits $\vec{t} \equiv t_0,\dots, t_{K-1}$. The overall probability to observe the trinary string $\vec{t}$, given some unknown magnetic field $H$, is given by $$\begin{aligned}
P\bigl(\vec{t}\>|H) = \prod_{k=0}^{K-1}
\frac19\, \bigl[ 1+2\cos\bigl(3^k (\phi(H) - \tilde\phi_{\vec{t}}) \bigr)\bigr]^2,\end{aligned}$$ where the phase $\tilde\phi_{\vec{t}} = (2\pi/3)\sum_{k=0}^{K-1} t_k/3^k$ relates to the string $\vec{t}$ and $\phi(H) = \mu H \tau_0/\hbar$ is the true field-induced phase; for an exact trinary value of $H$ and its associated string $\vec{t}$, we have $P\bigl(\vec{t}\>|H) = 1$.
In a next step, we make use of Bayes’ theorem and infer the probability $P(\phi(H)|\vec{t}\>)$ for the accumulated phase $\phi(H)$, provided we have observed a string $\vec{t}$, $P(\phi(H)|\vec{t}\>) \propto P(\vec{t}\>|H)$. Making use of the trigonometric identity $\sin(3\alpha) = \sin(\alpha)[3-4\sin^2(\alpha)]$, one finds that, $$P\bigl(\phi(H)|\vec{t}\>) = \frac1{2\pi}\,\frac{\sin^2[3^K (\phi(H) -
\tilde\phi_{\vec{t}})/2]}{3^K\sin^2[(\phi(h)-\phi_{\vec{t}})/2]}.
\label{eq:qutritprob}$$ A similar analysis provides the result $$P\bigl(\phi(H)|\vec{b}\>) = \frac1{2\pi}\,\frac{\sin^2[2^K (\phi(H)
-\tilde\phi_{\vec{b}})/2]}{2^K\sin^2[(\phi(H)-\phi_{\vec{b}})/2]}
\label{eq:qubitprob}$$ for the qubit-based protocol, where $\vec{b}$ is the $K$-bit string learned during the $K$-step measurement process and $\tilde\phi_{\vec{b}} = \pi
\sum_{k=0}^{K-1} b_k/2^k$.
The above qubit and qutrit Fourier metrological schemes can be generalized to a setup with qudits that are endowed with a $d$-dimensional Hilbert space. The qudit then is prepared in the balanced state $|\psi_0\rangle = (1/\sqrt{d})
\sum_{j=0}^{d-1} |j\rangle$ and subsequently experiences a magnetic-phase accumulation $|\psi_0\rangle \to |\psi_\phi\rangle = (1/\sqrt{d})
\sum_{j=0}^{d-1} e^{ij\phi}|j\rangle$, followed by a $d$-base inverse Fourier readout measurement. The posterior probability density for the measured phase $\phi(H)$ is given by $$\label{eq:Px}
P(\phi(H)|\vec{x}\>) = \frac1{2\pi} \, \frac{\sin^2[d^K (\phi(H)
-\tilde\phi_{\vec{x}})/2]}{d^K\sin^2[(\phi(H)-\phi_{\vec{x}})/2]},$$ where $\vec{x}$ is a string of $K$ base-$d$ digits. Using the relation $\lim_{\gamma\to\infty} [\sin^2(\gamma x)/\pi\gamma x^2] = \delta(x)$ one easily checks that both results approach the limit of a $\delta$-function $P(\phi(H)|\vec{x}\>) \to \delta (\phi(H)-\tilde\phi_{\vec{x}})$ when $K \to \infty$.
Next, we discuss the impact of a false digit assignment on the measurement outcome of the Fourier metrological scheme. This follows from the probability density plot $P(\phi|\vec{x}\>) \equiv P(\delta\phi)$ evaluated as a function of the estimation error $\delta\phi = \phi - \tilde\phi_{\vec{x}}$ which is shown in Fig. \[fig:density\] for the qubit and qutrit based algorithms.
![The probability density plots $P(\phi-\tilde\phi_{\vec{x}})$ for the $K=3$ step Fourier metrological procedure operated in the qutrit (thick line) and qubit (thin line) regimes.[]{data-label="fig:density"}](density_plots){width="8truecm"}
Both plots show a sharp central peak $\delta\phi \in[-2\pi/d^K,2\pi/d^K]$, $d=2$ or $3$, and a number of decaying satellite peaks. These satellite peaks derive from a wrong assignment of the binary $b_i$ or trinary $t_i$ digit during the measurement run. The first satelites correspond to a false assignment of the least significant digit in the first step of the procedure, while the far weaker satellites further out correspond to assignment errors of subsequent readouts. This analysis shows that the Fourier metrological procedure is stable with respect to the assignment errors: the probability to determine a false digit decreases for each next step of the procedure, resulting in a confidence level that is highest for the most significant digits and decreases for the measurement of the less significant digits. The probability that the observed value of the phase $\phi$ lies within the region of the central peak, and hence no error has been made in the assignment of digits, is given by $$P\Bigl( \delta \phi \in \Bigl[ -\frac{2\pi}{d^K},
\frac{2\pi}{d^K}\Bigr] \Bigr) \approx \frac1\pi\int\limits_{-\pi}^\pi dy
\frac{\sin^2y}{y^2} \approx 0.903,$$ where we have assumed a large $K$: the error probability saturates and does not depend on the number of steps $K$ (or, equivalently, the number of digits), manifesting the stability of the Fourier procedure.
Comparing the qubit- and qutrit procedures {#sec:comp}
==========================================
The posterior distribution functions for the phase $\phi(H)$ in the qutrit- and qubit metrological procedures, see Eqs. (\[eq:qutritprob\]) and (\[eq:qubitprob\]), allows us to compare the efficiency of the two schemes quantitatively and reveal the advantage of using a higher-dimensional quantum system for metrological purposes. Let the unknown magnetic field $H$ be located somewhere within the continuous interval $H \in [0,H_0]$. Then, as follows from the Eqs. (\[eq:Hbinary\]) and (\[eq:Htrinary\]), the field scales for the qubit- and qutrit-based metrology are chosen as $h_0^\mathrm{qb} = H_0/2$ and $h_0^\mathrm{qt} = H_0/3$ and the corresponding minimal Ramsey delays are given by $\tau_0 = 2\pi\hbar/\mu H_0$. During the $K$ steps, the qubit and qutrit metrological procedures learn about the magnetic field to a precision $(\delta H)_\mathrm{qb} = H_0/2^K$ and $(\delta
H)_\mathrm{qt} = H_0/3^K$. These $K$-step precision boundaries have to be related to the amount of quantum resources required to achieve them. The quantum resource exploited in our metrological procedures is the coherence of the quantum devices, which can be quantified by the net coherence (or phase-accumulation) time accumulated during the $K$ steps [@Giovanetti:2004]. For each of the two protocols, this time is given by, $$\begin{aligned}
&&T_\mathrm{qb} = \tau_0 \sum_{k=0}^{K-1} 2^k
\approx \frac{2\pi \hbar}{\mu H_0}\, 2^K, \\
&&T_\mathrm{qt} = \tau_0 \sum_{k=0}^{K-1} 3^k
\approx \frac{2\pi \hbar}{\mu H_0}\, \frac{3^K}2.\end{aligned}$$ Expressing the $K$-step precisions $\delta H$ through the net coherence time, $$\begin{aligned}
(\delta H)_\mathrm{qb} = \frac{2\pi \hbar}{\mu}\, \frac1{T_\mathrm{qb}},
\qquad (\delta H)_\mathrm{qt} = \frac{\pi \hbar}{\mu}\, \frac1{T_\mathrm{qt}},
\label{eq:precision}\end{aligned}$$ one notes that both procedures attain the Heisenberg limit, with the precision $\delta H$ scaling as the inverse of the total coherence time, but the qutrit procedure has a twice better prefactor. For the general qudit metrological scheme, the combination of $(\delta H)_\mathrm{qd} = H_0/d^K$, $T_\mathrm{qd}
= \tau_0 d^K/(d-1)$, and $\tau_0 =2\pi\hbar/\mu H_0$ produces the sensitivity, $$(\delta H)_\mathrm{qd} = \frac{2\pi\hbar}{\mu}\, \frac{1}{(d-1) T_\mathrm{qd}}.
\label{eq:precision2}$$ In practice, however, the phase accumulation time is the minimal time used in a Ramsey experiment. Most of the overall measurement time is spent on the measurement and re-initialization of the quantum devices. Hence, in practice, the speed of the metrological procedure is mostly defined by the number $K$ of steps required to achieve a given precision. Here, the qutrit-based procedure has a clear advantage: in order to achieve a relative precision $\delta H/H_0$ it requires approximately $K \sim -\log_3(\delta H/H_0)$ steps, that is $\ln(2)/\ln(3) \approx 0.63$ fewer than the number of steps $K \sim
-\log_2(\delta H/H_0)$ needed for the qubit based scheme.
Metrology with a transmon device {#sec:transmon}
================================
The working principle of a transmon qubit as a magnetic-flux sensor has been recently demonstrated in Ref. \[\]; here, we describe how the qutrit metrological protocol can be realized with a transmon device. The superconducting transmon [@Koch:2007] involves a capacitively shunted SQUID loop and constitutes an excellent candidate to implement qudit metrological algorithms. Its dynamics is described by the Hamiltonian $$\hat{H} = 4E_{\rm\scriptscriptstyle C}(\hat{n}-n_g)^2
- E_{\rm\scriptscriptstyle J}(\Phi) \cos(\hat\varphi),
\label{eq:transmon_ham}$$ where $\hat{n}$ is the number of Cooper pairs transmitted between superconducting islands with a charging energy $E_{\rm \scriptscriptstyle C}$ and relative phase $\hat\varphi$ and $n_g$ is the charge bias. The Josephson energy $E_{\rm\scriptscriptstyle J}(\Phi)$ of the SQUID loop depends on the flux $\Phi$ through the loop, $$E_{\rm\scriptscriptstyle J}(\Phi) = E_{{\rm \scriptscriptstyle J}
{\scriptscriptstyle \Sigma}}
\sqrt{\cos^2\bigl( \pi {\Phi}/{\Phi_0}\bigr)
+ a^2 \sin^2\bigl( \pi {\Phi}/{\Phi_0}\bigr)}.
\label{eq:jjenergy}$$ Here, $E_{{\rm \scriptscriptstyle J} {\scriptscriptstyle \Sigma}} =
E_{\rm\scriptscriptstyle J1} + E_{\rm\scriptscriptstyle J2}$ is the total energy of the two Josephson junctions in the SQUID loop, $a =
(E_{\rm\scriptscriptstyle J1}-E_{\rm\scriptscriptstyle J2})/E_{{\rm
\scriptscriptstyle J} {\scriptscriptstyle \Sigma}}$ is the junctions’ asymmetry, and $\Phi_0$ is the magnetic flux quantum.
The transmon atom is operated in the limit where $E_{\rm\scriptscriptstyle
J}/E_{\rm\scriptscriptstyle C} \sim 80 -200$. Its energy spectrum comprises a discrete set of non-equidistant energy levels with positions that depend on the magnetic flux $\Phi$ penetrating the SQUID loop of the device. Expanding the $\cos(\hat\varphi)$ term in the Hamiltonian Eq. (\[eq:transmon\_ham\]) and treating the term quartic in $\hat\varphi$ as a perturbation, we obtain (to leading order) the transmon’s energy spectrum in the form $$E_n \approx \sqrt{8E_{\rm\scriptscriptstyle C} E_{\rm\scriptscriptstyle J}(\Phi)}
\Bigl( n + \frac12 \Bigr) - E_{\rm\scriptscriptstyle J}(\Phi)
-\frac{E_{\rm\scriptscriptstyle C}}{12}\bigl(6n^2+6n+3\bigr),
\label{eq:spectrum}$$ The nonlinearity of the spectrum $E_{n+1}-E_n = -E_{\rm\scriptscriptstyle
C}(n+1) + \sqrt{8E_{\rm\scriptscriptstyle C} E_{\rm\scriptscriptstyle
J}(\Phi)}$ allows for the individual addressing of the transmon’s quantum states through application of pulses of electromagnetic radiation with specific frequencies. On the other hand, to leading order, the dependence of the spectrum on the magnetic flux $\Phi$ is linear in $n$; while second-order corrections $\propto E_{\rm\scriptscriptstyle C} (E_{\rm\scriptscriptstyle
J}/E_{\rm\scriptscriptstyle C})^{-1/2}$ do modify this result, these corrections are small and we neglect them in the following.
In order to manipulate the first two excited states of our transmon atom, we consider a two-tone rf-pulse that generates a time-varying electric potential difference of the form $$V(t) = \Omega(t) \bigl( V_1 \cos(\omega_1 t) + V_2 \cos(\omega_2 t) \bigr),$$ at the transmon’s capacitor, where $\Omega(t)$ is the pulse envelope and $V_{1(2)}$ are the amplitudes of the pulse components with tone frequencies $\omega_{1(2)}$. The transmon evolution under the pulse $V(t)$ then is described by the Hamiltonian $$\begin{aligned}
\hat{H} = \sum_{n=0}^\infty E_n |n\rangle \langle n|
+ \bigl( \hbar g_{n,n+1}(t) |n\rangle \langle n\!+\!1| +
\mathrm{h.c.} \bigr),\end{aligned}$$ with the transition amplitudes $\hbar g_{n,n+1}(t) = 2\beta eV(t)$ $\langle
n|\hat{N}| n\!+\!1\rangle$; here, $\hat{N}$ is the number operator of Cooper pairs transferred between the transmon’s capacitor plates and $\beta$ is a geometrical factor which quantifies the coupling between the transmon’s capacitor and the rf-field, see Ref. \[\].
Let the tone frequencies $\omega_1$ and $\omega_2$ be near the transition frequencies $\omega_{01} = (E_1-E_0)/\hbar$ and $\omega_{12} =
(E_2-E_1)/\hbar$ of the first two pairs of levels, such that only the three lowest energy levels are affected by the rf-field. We work in the rotating frame (or interaction representation) with respect to the ‘free’ Hamiltonian $\hat{H}_0 = \hbar\omega_1 |1\rangle \langle 1| + \hbar(\omega_1+\omega_2)
|2\rangle \langle 2|$. A quantum state of this effective three-level system can be represented as $|\Psi(t)\rangle = a_0(t) |0\rangle + a_1(t)|1\rangle
+a_2(t) |2\rangle$. The time-dependent amplitudes $\vec{a}(t) =
[a_0(t),a_1(t),a_2(t)]$ obey the Schrödinger equation $i\hbar\partial_t
\vec{a}(t) = \hat{H}(t) \vec{a}(t)$, with the Hamiltonian assuming the following form in the rotating wave approximation, $$\hat{H}(t) = \hbar \left[ \begin{array}{ccc}
0& \Omega(t) \Delta_1& 0\\
\!\! \Omega(t)\Delta_1& \omega_{01}\!-\omega_1& \Omega(t)\Delta_2\\
0&\Omega(t)\Delta_2& \omega_{01}\!+\omega_{12}\!-\! \omega_1\!
-\!\omega_2\!\!\end{array}\right],
\label{eq:hamiltonian}$$ where $\Delta_1 = \beta e V_1 \langle 0|\hat{N}|1\rangle/\hbar$ and $\Delta_2
= \beta e V_2 \langle 1|\hat{N}|2\rangle/\hbar$ are effective transition amplitudes.
Qutrit metrological protocol {#sec:protocol}
----------------------------
Our qutrit metrological scheme involves three steps that have to be implemented with the help of proper manipulation signals $V(t)$. In the first step, we prepare the transmon in a balanced superposition of the form $$|\Psi_0\rangle \!=\! \frac1{\sqrt{3}} \bigl( e^{i\varphi_0} |0\rangle
+ e^{i\varphi_1}|1\rangle + e^{i\varphi_2} |2\rangle \bigr)
\!\equiv \! \frac1{\sqrt{3}} \left[ \begin{array}{c} \!\!e^{i\varphi_1}\!\!
\\ \!\!e^{i\varphi_2}\!\! \\ \!\!e^{i\varphi_3}\!\!\end{array} \right]\!.$$ In the second step, the free evolution $\hat{U}(\Phi)$ of the transmon generates the additional phase factors $|1\rangle \to e^{i\phi}|1\rangle$ and $|2\rangle \to e^{i\phi^\prime} |2\rangle$ with $\phi = [\omega_{01}(\Phi) -
\omega_1]\tau$ and $\phi^\prime = [\omega_{01}(\Phi) + \omega_{12}(\Phi) -
\omega_1 -\omega_2]\tau$. The remarkable property of the transmon is that its level separations scale equally in magnetic flux, see Eq.(\[eq:spectrum\]). Starting from the reference magnetic flux $\Phi_c$, we set the frequencies $\omega_1 = \omega_{01}(\Phi_c)$ and $\omega_2=
\omega_{12}(\Phi_c)$. Then, $\phi^\prime = 2\phi \equiv 2
[\omega_{01}(\Phi)-\omega_{01}(\Phi_c)]\tau$ and the state $|\Psi_0\rangle$ evolves to the new balanced state $$|\Psi_\phi\rangle = \frac1{\sqrt{3}} \left[ \begin{array}{c}
\!\!e^{i\varphi_0}\!\!\\
\!\!e^{i\varphi_1+i\phi}\!\!\\
\!\!e^{i\varphi_2+2i\phi}\!\!
\end{array} \right].
\label{eq:phitau}$$ Finally, in the third step of our qutrit metrological procedure, we need to construct a unitary readout operator of the form
$$\hat{U}_r = \frac1{\sqrt{3}} \left[ \begin{array}{ccc}
\!\! e^{i\chi_0}&0&0 \!\!\\
\!\! 0&e^{i\chi_1}&0 \!\!\\
\!\! 0&0&e^{i\chi_2} \!\!
\end{array}\right]
\left[ \begin{array}{ccc}
\!\! 1&1&1 \!\!\\
\!\! 1&e^{-2\pi i/3}&e^{+2\pi i/3} \!\!\\
\!\! 1&e^{2\pi i/3}&e^{-2\pi i/3}
\end{array}\right]
\left[ \begin{array}{ccc}
\!\! e^{-i\varphi_0}&0&0 \!\!\\
\!\! 0&e^{-i\varphi_1}&0 \!\!\\
\!\! 0&0&e^{-i\varphi_2} \!\!
\end{array}\right],
\label{eq:readout}$$
that represents a generalized base-3 inverse Fourier transform.
For the specific situation where the accumulated phase $\phi$ can only assume the three values $0, 2\pi/3$, and $4\pi/3 \leftrightarrow -2\pi/3$, this measurement scheme is deterministic and the transmon will always be found in one of the pure states $|0\rangle$, $|1\rangle$, or $|2\rangle$. Indeed, for $\phi=2\pi/3$ or $\phi=4\pi/3$, the state $|\Psi_0\rangle$ transforms into states $$|\Psi_1\rangle = \frac1{\sqrt{3}} \left[ \begin{array}{c}
\!\!e^{i\varphi_0}\!\!\\
\!\!e^{i\varphi_1+i \frac{2\pi}{3}}\!\!\\
\!\!e^{i\varphi_2+i \frac{4\pi}{3}}\!\!
\end{array} \right],\>
|\Psi_2\rangle = \frac1{\sqrt{3}} \left[ \begin{array}{c}
\!\!e^{i\varphi_0}\!\!\\
\!\!e^{i\varphi_1+i \frac{4\pi}{3}}\!\!\\
\!\!e^{i\varphi_2+i \frac{2\pi}{3}}\!\!
\end{array} \right],$$ which together with $|\Psi_0\rangle$ form the (orthonormal) computational basis. Then, the readout operation $\hat{U}_r$ provides a deterministic outcome, $$\hat{U}_r|\Psi_j\rangle = e^{i\chi_j}|j\rangle, \quad j=0,1,2.$$ For an arbitrary accumulated phase, the scheme is probabilistic and provides the probabilities $$\begin{aligned}
\label{eq:prob}
P_j(\phi) &=& |\langle j| \hat{U}_r \hat{U}(\Phi) \hat{U}_p|0\rangle|^2 \\
&=& \frac19 \bigl[ 1 + 2\cos\bigl(\phi(\Phi) - 2\pi\,j/3 \bigr)
\bigr]^2,
\quad j = 0,1,2,
\nonumber\end{aligned}$$ to observe the transmon in the state $|j\rangle$. The possible phase values $\phi$ then are divided into the three sectors $S_0=[-{\pi}/{3},{\pi}/{3}]$, $S_1 = [\pi/3,\pi]$ and $S_2 = [\pi,{5\pi}/3]$ with the maximal probability $P_j$ telling that $\phi \in S_j$.
rf-pulses for metrological protocol
-----------------------------------
In order to implement our metrological protocol, we have to find appropriate rf-pulses $V_p(t)$ and $V_r(t)$ that prepare the transmon in a balanced state and readout the state after its free evolution in the magnetic field to be measured. It turns out to be convenient to reverse the order and first find the readout pulse.
Hence, our next goal is to find an rf-pulse $V_r(t)$ that generates a unitary readout operation of the form (\[eq:readout\]), a task that we tackle in three steps: i) We show that any $3\times3$ unitary with equal-modulus matrix elements $|\hat{U}_{ij}| = {1}/{\sqrt{3}}$ is either of the form $\hat{U}_r$ or $\hat{U}_r^{-1}$. ii) We find the unitary associated with a rectangular two-tone rf-pulse of finite duration $\tau_p$. iii) We determine the constraints on the rf-pulse required for the readout action.
Starting with i), we consider an arbitrary $3\times3$ unitary matrix $\hat{U}$ with all matrix elements of modulus $1/\sqrt{3}$. After multiplication with suitable diagonal phase matrices from the left and right, we can arrive at the form $$\hat{U} \to \hat{U}' = \frac1{\sqrt{3}} \left[ \begin{array}{ccc}
1&1&1\\
1&e^{i\alpha_1}&e^{i\beta_1}\\
1&e^{i\alpha_2}&e^{i\beta_2} \end{array}\right],$$ where the remaining four phases have to satisfy the orthogonality constraints between columns, $1 + e^{i\alpha_1}+e^{i\alpha_2} = 0$, $1 + e^{i\beta_1}
+e^{i\beta_2} =0$ and $1 +e^{i(\alpha_1-\beta_1)}+e^{i(\alpha_2 - \beta_2)} =
0$. It follows that the solutions of these constraints define either the Fourier transform $\hat{U}' = \hat{F}_3$ or its inverse $\hat{U}' =
\hat{F}_3^{-1}$, see Eq. , that proves our statement.
Next, in step ii), we consider a two-tone rf-pulse with a rectangular shape of duration $\tau_p$ and frequencies $\omega_1 = \omega_{01}(\Phi_c) -
2\delta\omega$ and $\omega_2 = \omega_{12}(\Phi_c)+2\delta\omega$. Such a pulse generates a unitary rotation of the qutrit $\hat{U} =
\exp(-i\hat{H}\tau_p/\hbar) \equiv \exp(-i\hat{K})$, where $$\hat{K} = \left[ \begin{array}{ccc}
0&\Delta_1&0\\
\Delta_1&2\epsilon&\Delta_2\\
0&\Delta_2&0 \end{array} \right],
\quad \epsilon = \delta\omega \,\tau_p,$$ and $\Delta_{1,2}$ are effective transition amplitudes, see Eq.. The resulting unitary transformation has the form,
$$\hat{U} = \left[ \begin{array}{ccc}
\frac{\Delta_2^2}{\Delta_1^2+\Delta_2^2}&0&
-\frac{\Delta_1\Delta_2}{\Delta_1^2+\Delta_2^2}\\
0&0&0\\
-\frac{\Delta_1\Delta_2}{\Delta_1^2 +\Delta_2^2}&0&
\frac{\Delta_1^2}{\Delta_1^2+\Delta_2^2}
\end{array}\right]
+e^{-i\epsilon} \cos(\xi) \left[ \begin{array}{ccc}
\frac{\Delta_1^2}{\Delta_1^2+\Delta_2^2}&0&
\frac{\Delta_1\Delta_2}{\Delta_1^2 +\Delta_2^2}\\
0&1&0\\
\frac{\Delta_1\Delta_2}{\Delta_1^2 +\Delta_2^2}&0&
\frac{\Delta_2^2}{\Delta_1^2+\Delta_2^2}
\end{array}\right]
+\frac{ie^{-i\epsilon}\sin(\xi)}{\xi} \left[ \begin{array}{ccc}
\frac{\epsilon\Delta_1^2}{\Delta_1^2+\Delta_2^2}&-\Delta_1&
\frac{\epsilon\Delta_1\Delta_2}{\Delta_1^2 +\Delta_2^2}\\
-\Delta_1&-\epsilon&-\Delta_2\\
\frac{\epsilon\Delta_1\Delta_2}{\Delta_1^2 +\Delta_2^2}&-\Delta_2&
\frac{\epsilon\Delta_2^2}{\Delta_1^2+\Delta_2^2}
\end{array}\right],$$
where $\xi = \sqrt{\epsilon^2+ \Delta_1^2 +\Delta_2^2}$.
Let us then, iii), determine the parameters that generate the readout matrix $\hat{U}_r$. Following i), we have to require that $|[U_r]_{ij}|^2 = 1/3$. The conditions $|U_{12}|^2 = |U_{21}|^2 =1/3$ and $|U_{23}|^2 = |U_{32}|^2 = 1/3$ imply that $$\frac{\sin^2(\xi)}{\xi^2}\, \Delta_1^2 = \frac13, \quad
\frac{\sin^2(\xi)}{\xi^2}\, \Delta_2^2 = \frac13,$$ that gives $\Delta_1^2 = \Delta_2^2 \equiv \Delta^2$. Accounting for the remaining conditions $|U_{11}|^2=|U_{13}|^2=1/3$ and using the relation $2\Delta^2 = \xi^2-\epsilon^2$, we arrive at the following system of transzendental equations $$\begin{aligned}
&&\epsilon^2 = \xi^2 \Bigl( 1- \frac{2}{3\sin^2(\xi)} \Bigr),
\label{eq:sys1}
\\
&& \cos(\epsilon)\cos(\xi)+ \frac{\epsilon}{\xi} \sin(\epsilon)\sin(\xi) = 0.
\label{eq:sys2}\end{aligned}$$ Among its solutions, we choose the one with the minimal $\epsilon$ as it corresponds to the shortest rf-pulse for a given detuning $\delta\omega$ and find the numerical values $\epsilon_0 \approx 0.8525$, $\xi_0 \approx 2.0205$, and hence $\Delta_0 \approx 1.2953$. We note, that the system of Eqs.(\[eq:sys1\]) and (\[eq:sys2\]) remains unchanged under a sign-change of the parameters $\epsilon$, $\Delta_1$, and $\Delta_2$ characterizing the Hamiltonian.
Let us consider the specific solution with $\epsilon = -\epsilon_0$ and $\Delta_1 = \Delta_2 = \Delta_0$. The corresponding pulse then generates the readout unitary transformation $$\begin{aligned}
&&\hat{U}_r = \frac1{\sqrt{3}} \left[ \begin{array}{ccc}
e^{-i\frac{\pi}{6}}&-ie^{i\epsilon_0}&e^{-i\frac{5\pi}{6}}\\
-ie^{i\epsilon_0}&ie^{i2\epsilon_0}&-ie^{i\epsilon_0}\\
e^{-i\frac{5\pi}{6}}&-ie^{i\epsilon_0}&e^{-i\frac{\pi}{6}}
\end{array}\right]
\label{eq:Ur} \\
&&\equiv \left[ \begin{array}{ccc}
1&0&0\\
0&e^{i\epsilon_0+i\frac{5\pi}{6}}&0\\
0&0&e^{i\frac{4\pi}{3}}
\end{array}\right]
\hat{F}_3^{-1} \left[ \begin{array}{ccc}
e^{-i\frac{\pi}{6}}&0&0\\
0&-ie^{i\epsilon_0}&0\\
0&0&e^{-i\frac{5\pi}{6}}
\end{array}\right],
\nonumber\end{aligned}$$ that provides the desired inverse generalized Fourier transform.
As a preparation operator $\hat{U}_p$, we choose a unitary rotation generated by a rf-pulse with $\epsilon = +\epsilon_0$ and $\Delta_1 = \Delta_2 =
-\Delta_0$, i.e., the preparation pulse generates the inverse of the readout operator, $\hat{U}_p = \hat{U}_r^\dagger$.
The sign change in $\epsilon$ is trivially realized by inverting the detuning $\delta\omega \to - \delta\omega$. In order to change the sign (or more generally the phase) of the effective transition amplitudes $\Delta_{1,2}$, one can proceed with an appropriate modulation of the voltage signal $V(t)$. Making use of a standard IQ (In-phase and Quadrature)-mixing scheme, an incoming high-frequency signal $\cos(\omega_{\rm \scriptscriptstyle
LO}t)$ with the (local oscillator) frequency $\omega_{\rm\scriptscriptstyle
LO} = \frac12 \bigl[ \omega_{01}(\Phi_c) + \omega_{12}(\Phi_c)\bigr]$ is first physically split (and partly phase shifted) into two separate signals $\cos(\omega_{\rm \scriptscriptstyle LO}t) \to \frac12 \bigl[\cos(\omega_{\rm
\scriptscriptstyle LO}t) + \sin(\omega_{\rm \scriptscriptstyle LO}t)\bigr]$. These are independently mixed with the intermediate-frequency signals $A_1\Omega(t) \cos(\omega_{\rm \scriptscriptstyle IF}t+\varphi)$ and $A_2
\Omega(t) \sin(\omega_{\rm \scriptscriptstyle IF}t +\varphi)$ generated by an arbitrary-waveform generator. Finally, the signals are recombined and the resulting output signal sent to the transmon is given by $$\begin{aligned}
&&V(t) = \frac{\Omega(t)}4 \Bigl[(A_1-A_2)
\cos\bigl[ (\omega_{\rm\scriptscriptstyle LO}
+\omega_{\rm\scriptscriptstyle IF})t+\varphi\bigr]
\\
&&\qquad\qquad\quad + (A_1+A_2)
\cos\bigl[(\omega_{\rm\scriptscriptstyle LO}
-\omega_{\rm\scriptscriptstyle IF})t-\varphi\bigr]\Bigr]. \quad
\nonumber\end{aligned}$$ Choosing amplitudes $A_1 = 2(V_1+V_2)$, $A_2 = 2 (V_2-V_1)$, the frequency $\omega_{\rm\scriptscriptstyle IF} = \frac12 \bigl[\omega_{01} (\Phi_c) -
\omega_{12}(\Phi_c) \bigr] - 2\delta\omega$, and the phase $\varphi = 0$, one can generate the readout pulse. On the other hand, choosing the frequency $\omega_{\rm\scriptscriptstyle IF} = \frac12 \bigl[\omega_{01}(\Phi_c) -
\omega_{12}(\Phi_c) \bigr] + 2 \delta \omega$ and the phase $\varphi = \pi$ together with the sign inversion of the detuning $\delta\omega$ reverses the sign of all three parameters $\epsilon$, $\Delta_1$, and $\Delta_2$ and hence produces the preparation pulse.
Optimizing the transmon sensitivity
-----------------------------------
As follows from Eq. (\[eq:precision2\]), the measurement precision that can be attained by the qudit metrological protocol depends on two factors, the magnetic moment $\mu$ of the transmon device and the longest phase coherence time $d^{K-1}\tau_0$ required for the longest run of the metrological protocol. The magnetic moment of the transmon can be obtained via the curvature of its transition frequency, $$\mu = \hbar A \frac{\partial \omega_{01}(\Phi_c)}{\partial \Phi},$$ where $A$ is the area of the SQUID loop. Indeed, the relative phase $\phi =
\bigl[ \omega_{01}(\Phi) - \omega_{01}(\Phi_c) \bigr]\tau$ accumulated by the transmon’s wavefunction is given by, $$\begin{aligned}
\phi \approx \tau \frac{\partial \omega_{01}(\Phi_c)}{\partial \Phi}
(\Phi - \Phi_c) = \frac{\mu \delta H \tau}{\hbar},\end{aligned}$$ where $\delta H = H - H_c$ is the magnetic field measured relative to the reference magnetic field $H_c$, $\Phi_c = A H_c$. In order to attain a better sensitivity, one has to deviate from the ’sweet spot’, the upper maximum of the transmon spectrum $\omega_{01}(\Phi)$ where $\mu = 0$, and take the device to a (locally) linear regime with a lower transition frequency. In the limit of an almost symmetric Josephson junction loop with $a \to 0$, the maximal value of the transmon’s magnetic moment is given by (see Eqs.(\[eq:spectrum\]) and (\[eq:jjenergy\])) $$\label{eq:mu}
\mu = \pi \frac{A}{\Phi_0} \, \sqrt{\frac{8E_{\rm\scriptscriptstyle C}
E_{{\rm \scriptscriptstyle J}
{\scriptscriptstyle \Sigma}}}{a}},$$ which occurs near the bottom of the transmon spectrum where $\tan^2(\pi
\Phi_c/\Phi_0) = 1/a$. The result Eq. applies to the transmon limit $E_{\rm \scriptscriptstyle J} \gg E_{\rm\scriptscriptstyle C}$; for a symmetric device $a \to 0$, the largest moment $\mu$ appears near the point of maximal frustration $\Phi_c = \Phi_0/2$ where $E_{\rm \scriptscriptstyle J}$ becomes small and the approximation breaks down.
In our discussion above, we have implicitly assumed that the entire preparation–phase-accumulation–readout sequence involves a total time that is much below the coherence- ($T_2$) and relaxation ($T_1$) times of the transmon device. In a realistic situation, when operating the transmon away form the ’sweet spot’ the $T_2$-time gets reduced and so is the number $K$ of available steps for the metrological procedure. For a given qudit coherence time $T_2$, the delay time of the longest Ramsey sequence cannot exceed the $T_2$ time. Hence, the maximum number of steps in the Fourier procedure is limited by the condition $\tau_0 d^{K-1} = T_2$ that gives $K = 1 +
\log_d(T_2/\tau_0)$ steps, where $\tau_0$ is the minimum duration of a Ramsey sequence. Thus, the total amount of coherence time spent for the signal sensing is given by $T_\mathrm{qd} = \tau_0 \sum_{k=0}^{K-1} d^k = \tau_0
(d^{K-1}-1)/(d-1) \approx [d/(d-1)] T_2$ for $K \gg 1$. Then, according to the Eq. (\[eq:precision2\]), the best attainable field-resolution can be estimated as $$[\delta H]_{T_2} \approx \frac{2\pi \hbar}{\mu\, d\, T_2}.$$ Hence, we have to optimize the product $\mu(\Phi_c)T_2(\Phi_c)$ for the best flux bias $\Phi_c$, compromising between two opposing trends, a magnetic moment $\mu(\Phi_c)$ increasing and the coherence time $T_2(\Phi_c)$ decreasing away from the sweet spot. The magnetic moment of a transmon atom can attain a value of $10^5\mu_{\rm \scriptscriptstyle B}$, see Ref.. Assuming a coherence time $T_2 \sim 1~\mu$s, one can estimate that a magnetic-field precision of order $\delta H \sim 0.1$ nT can be achieved.
A further improvement of the field resolution is possible only within a standard statistical measurement scheme[@Sekatski:2017] by repeating the longest Ramsey measurement with $\tau \sim T_2$ a large number $N\gg 1$ of times. This leads to a standard scaling of the further field resolution with time duration $t$ of the experiment: with $N = t/T_2$ we obtain a precision $$[\delta H]_{t\gg T_2} \approx \frac{2\pi \hbar}{\mu\, d\, T_2\sqrt{t/T_2}}
= \frac{2\pi \hbar}{\mu\, d\, \sqrt{T_2 t}}.
\label{eq:standard}$$ Therefore, the Heisenberg scaling is limited to measurement times shorter then the coherence time $T_2$ of the device, with the standard quantum limit restored for larger measurement times. The standard scaling Eq.(\[eq:standard\]) can be achieved by a conventional scheme where one always measures the transmon state at the longest possible delay $T_2$ of the Ramsey sequence. However, such a conventional scheme has a limited measurement range $\Delta H$ for the field $H$ that results from the $2\pi$-periodicity of the accumulated phase $\phi \sim \mu \Delta H T_2/\hbar$, i.e., $\Delta H \sim
2\pi\hbar/{\mu T_2}$. In contrast, the quantum procedure does not suffer from this limitation: its measurement range is defined only by the minimal time duration $\tau_0$ of the Ramsey sequence that is limited in practice by the time duration of the controlling rf-pulses.
Making use of the above ideas in an experiment, a further practical restriction has to be considered besides the finite coherence time of the transmon device. First, the total duration of the experiment has to include the additional time spent for the measurement and reset of the transmon state. In fact, in the sensing experiment of Ref. using the transmon qubit-mode, most of the time of a single Ramsey measurement $T_\mathrm{rep}$ has been spent on the measurement and reset of the qubit $T_\mathrm{rep} \gg T_2$. In this limit, the long-time sensitivity of the sensor is reduced and given by $$[\delta H]_{t\gg T_\mathrm{rep}\gg T_2} \approx \frac{2\pi \hbar}
{\mu\, d\, T_2\, \sqrt{t/T_\mathrm{rep}}}.
\label{eq:standard_rep}$$ Second, the higher-excited states of the transmon have larger dipole matrix elements and hence are more sensitive to the external electromagnetic environment[@Koch:2007]. Therefore, a transmon atom has specific $T_1$ and $T_2$ times for each excited state and the number of levels which can be used is naturally limited by the coherence time of the highest-energy state involved. In practice, one can start the Fourier metrological procedure in a qubit measurement mode at Ramsey delays corresponding to the largest $T_2$ time belonging to the first excited level and then continue in a qutrit mode when the Ramsey delays have dropped below the $T_2$ time of the second excited level. Finally, operating a transmon atom in a qudit regime requires a more involved characterization of its spectrum and calibration of the corresponding rf-control pulses. At the same time, operating a transmon in the qutrit regime is a well established experimental procedure [@Abdulmalikov:2013; @Kumar:2016], what motivates work directed at the experimental implementation of our qutrit metrological procedure.
Summary and conclusion {#sec:conclusion}
======================
We have presented a variant of the standard quantum Fourier metrological procedure that replaces the usual qubit elements by qutrits and, more general, by qudits. While, all of these algorithms exploit phase coherence as their quantum resource, allowing them to reach the Heisenberg precision scaling, the use of higher-dimensional Hilbert spaces in the qutrit and qudit versions improves on the prefactor of this scaling. Even more, going to qudit devices reduces the number of iteration steps in the Fourier procedure, thus providing a marked improvement of the measurement algorithm. As a specific example, we have discussed the use of a superconducting transmon device operated in the qutrit mode that serves as an ideal resource for the measurement of dc and low-frequency magnetic fields, a consequence of the linear field-dependence of the transmon’s spectrum. It turns out, that a simple two-tone rf voltage signal in combination with a standard IQ-mixing scheme suffices to produce the appropriate preparation and readout pulses for the Fourier metrological algorithm.
The scheme presented in this paper relies on the assumption that the longest measurement providing the least relevant but precision-limiting digit can be performed within the coherence- or $T_2$-time of the transmon device; a further increase in precision proceeds via a conventional measurement procedure and follows the standard (shot-noise- or quantum-limited) scaling in precision. Furthermore, given a finite coherence time $T_2$, the algorithm can be further optimized to deal with this situation: Besides properly tuning the qutrit’s reference flux or working point $\Phi_c$ as discussed above, other elements of the algorithm can be improved. E.g., a finite $T_2$-time may require more than a single Ramsey measurement at each time delay $\tau_k$, $k=1,\dots K$, what modifies the probability Eq. to arrive at the correct sequence $\vec{x}$ of digits. In addition, the time delays $\tau_k$ appearing in the metrological protocol are subject to optimization, i.e., they must be chosen differently from the ideal case. The situation with finite $T_1$- and $T_2$-times then requires a separate study that will be the topic of a future analysis.
We acknowledge discussions with Pertti Hakonen and Vladimir Manucharyan. The research was supported by Government of the Russian Federation (Agreement 05.Y09.21.0018), by the RFBR Grants No. 17- 02-00396A and 18-02-00642A, Foundation for the Advancement of Theoretical Physics ”BASIS”, the Ministry of Education and Science of the Russian Federation 16.7162.2017/8.9 (A.V.L., A.R.S and V.V.Z. in part related to Section IV), the Swiss National Foundation via the National Centre of Competence in Research in Quantum Science and Technology (NCCR QSIT), the Pauli Center for Theoretical Physics, Academy of Finland Centers of Excellence “Low Temperature Quantum Phenomena and Devices” (project 250280) and “Quantum Technology Finland (QTF)” (project 312296), and the Center for Quantum Engineering at Aalto University. V.Z. and A.S. acknowledge the support from the 5-top 100 programm via the Laboratory of quantum information theory (MIPT).
[99]{}
V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photon. [**5**]{}, 222 (2011).
C. L. Degen, F. Reinhard, P. Cappellaro, Rev. Mod. Phys. [**89**]{}, 035002 (2017).
V. Giovannetti, S. Lloyd, and L. Maccone, Phys.Rev. Lett. [**96**]{}, 010401 (2006).
A.Yu. Kitaev, Quantum measurements and the Abelian Stabilizer Problem, e-print arXiv:quant-ph/9511026 (1995).
R. Cleve, A. Ekert, C. Macchiavello, M. Mosca, Proc.R. Soc. Lond. A [**454**]{}, 339 (1998).
R.B. Griffiths and C.-S. Niu, Phys. Rev. Lett., 3228 (1996).
J. Koch, T. M. Yu, J. Gambetta, A.A. Houck, D.I.Schuster, J. Majer, A. Blais, M.H. Devoret, S.M. Girvin, and R.J.Schoelkopf, Phys. Rev. A [**76**]{}, 042319 (2007).
L. Vaidman and Z. Mitrani, Phys. Rev. Lett. [**92**]{}, 217902 (2004).
G. Giedke, J.M. Taylor, D. D’Alessandro, M.D. Lukin, and A. Imamoglu, Phys. Rev. A [**74**]{}, 032316 (2006).
R.S. Said, D.W. Berry, and J. Twamley, Phys. Rev. B [**83**]{}, 125410 (2011).
B.L. Higgins, D.W. Berry , S.D. Bartlett, H.M.Wiseman, and G.J. Pryde, Nature [**450**]{}, 393 (2007).
G. Waldherr, J. Beck, P. Neumann, R.S. Said, M.Nitsche, M.L. Markham, D.J. Twitchen, J. Twamley, F. Jelezko, and J.Wrachtrup, Nature Nanotechnology [**7**]{}, 105 (2012).
L.S. Levitov, H.W. Lee, and G.B. Lesovik, J. Math.Phys. [**37**]{}, 4845 (1996).
G.B. Lesovik, F. Hassler, and G. Blatter, Phys.Rev. Lett. [**96**]{}, 106801 (2006).
A.V. Lebedev, G.B. Lesovik, and G. Blatter, Phys. Rev. B [**93**]{}, 115140 (2016).
G.B. Lesovik, M.V. Suslov, and G. Blatter, Phys.Rev. A [**82**]{}, 012316 (2010).
M.V. Suslov, G.B. Lesovik, and G. Blatter, Phys.Rev. A [**83**]{}, 052317 (2011).
J. Q. You and F. Nori, Nature [**474**]{}, 589 (2011).
J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, S. Lloyd, Science [**285**]{}, 1036 (1999).
M. Bal, C. Deng, J.-L. Orgiazzi, F. R. Ong, A.Lupascu, Nature Comm. [**3**]{}, 1324 (2012).
F. Yoshihara, K. Harrabi, A. O. Niskanen, Y.Nakamura, and J. S. Tsai, Phys. Rev. Lett. [**97**]{}, 167001 (2006).
S. Danilin, A.V. Lebedev, A. Vepsalainen, G.B.Lesovik, G. Blatter, G.S. Paraoanu, arXiv:[**1801.02230**]{} (2018).
A. A. Abdumalikov Jr, J. M. Fink, K.Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, Nature [**496**]{}, 482 (2013).
K. S. Kumar, A. Vepsäläinen, S. Danilin, and G.S. Paraoanu, Nature Comm. [**7**]{}, 10628 (2016).
V. Giovannetti, S. Lloyd, and L. Maccone, Science [**306**]{}, 1330 (2004).
P. Sekatski, M. Skotiniotis, J. Kolodynski, and W.Dür, Quantum [**1**]{}, 27 (2017).
|
---
abstract: |
Let $\mathscr{M}_{(2,1)}(N)$ denotes the infimum of the size of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there are a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+\varepsilon)N$ for any $\varepsilon>0$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $\omega(N)$ is still widely open.
In this paper, we study the analogue conjecture on $(k,\l)$-sum free sets and restricted $(k,\l)$-sum free sets. We determine the constant $c(k,\l)$ for every $(k,\l)$, and confirm the conjecture for infinitely many $(k,\l)$.
address: 'Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana IL, USA'
author:
- Yifan Jing
bibliography:
- 'ref.bib'
title: 'The largest $(k,\ell)$-sum free subsets'
---
Introduction
============
In 1965, Erdős asked the following question [@Erdos65]. Given an arbitrary sequence $A$ of $N$ different positive integers, what is the size of the largest sum-free subsequence of $A$? By *sum-free* we mean that if $x,y,z\in A$, then $x+y\neq z$. Let $$\M_{(2,1)}(N)=\inf_{\substack{A\subseteq\NN^{>0}\\|A|=N}}\max
_{\substack{S\subseteq A\\S\text{ is sum-free}}}|S|.$$ Using a beautiful probabilistic argument, Erdős showed that every $N$-element set $A\subseteq \NN^{>0}$ contains a sum-free subset of size at least $N/3$, in other words, $\M_{(2,1)}(N)\geq N/3$. It turns out that it is surprisingly hard to improve upon this bound. The result was later improved by Alon and Kleitman [@AK90], they showed that $\M_{(2,1)}(N)\geq (N+1)/3$. Bourgain [@Bourgain97], using an entirely different Fourier analytic argument, showed that $\M_{(2,1)}(N)\geq(N+2)/3$, which is the best lower bound on $\M_{(2,1)}(N)$ to date. In particular, the following conjecture has been made in a series of papers, see [@Erdos65; @Bourgain97; @EGM; @TV17] for example.
\[conj:main\] There is a function $\omega(N)\to\infty$ as $N\to\infty$, such that $$\M_{(2,1)}(N)> \frac{N}{3}+\omega(N).$$
On the other hand, a recent breakthrough by Eberhard, Green, and Manners [@EGM] proved that $\M_{(2,1)}(N)= (1/3+o(1))N$. More precisely, they showed that for every $\varepsilon>0$, when $N$ is large enough, there is a set $A\subseteq \NN^{>0}$ of size $N$, such that every subset of $A$ of size at least $(\varepsilon+1/3)N$ contains $x,y,z$ with $x+y=z$. The result was later generalized by Eberhard [@Eberhard15] to *$k$-sum-free*. A set $A$ is $k$-sum-free if for every $y,x_1,\dots,x_k\in A$, $y\neq \sum_{i=1}^k x_i$. Eberhard proved that for every $\varepsilon>0$, there is a set $A\subseteq \NN^{>0}$ of size $N$, such that every subset of $A$ of size at least $(\varepsilon+1/(k+1))N$ contains a $k$-sum. For more background we refer to the survey [@TV17].
In this paper, we study the analogue of the Erdős sum-free set problem on the $(k,\l)$-sum free sets. Given two positive integers $k,\l$ with $k>\l$, a set $A$ is *$(k,\l)$-sum free* if for every $x_1,\dots,x_{k},y_1,\dots,y_\l\in A$, $\sum_{i=1}^kx_k\neq\sum_{j=1}^\l y_j$. For example, using the notation of $(k,\l)$-sum free, sum-free is $(2,1)$-sum free; $k$-sum-free is $(k,1)$-sum free.
Let $$\M_{(k,\l)}(N)=\inf_{\substack{A\subseteq\NN^{>0}\\|A|=N}}\max
_{\substack{S\subseteq A\\S\text{ is } (k,\l)-\text{sum free}}}|S|.$$ Thus, to determine $\M_{(k,\l)}(N)$ is a natural task. We can also make the following conjecture for $(k,\l)$-sum free set, which is an analogue of Conjecture \[conj:main\].
\[conj:kl\] Let $k>\l>0$. There is a constant $c=c(k,\l)>0$, and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $$cN+\omega(N)< \M_{(k,\l)}(N)< (c+\varepsilon)N,$$ for every $\varepsilon>0$.
As we mentioned above, the constant $c(k,\l)$ in Conjecture \[conj:kl\] for $(k,\l)=(2,1)$ is determined by Eberhard, Green, and Manners [@EGM], and for $(k,\l)=(k,1)$ is determined by Eberhard [@Eberhard15]. The conjecture for $(k,\l)=(3,1)$ is confirmed by Bourgain [@Bourgain97].
Our first result determines the constant $c(k,\l)$ in Conjecture \[conj:kl\] for every $(k,\l)$, and confirms Conjecture \[conj:kl\] for infinitely many $(k,\l)$.
\[thm:one\] Let $k,\l$ be positive integers and $k>\l$. Then the followings hold:
*(i)* for every $k,\l$, we have $\M_{(k,\l)}(N)\geq\frac{N}{k+\l}$.
*(ii)* when $k\geq\l+2$, and either $k+\l$ is a prime, or $p=\frac{k+\l}{\gcd(k+\l,k-\l)}$ is a prime such that $p\mid k-\l$. Then $$\M_{(k,\l)}(N)\geq\frac{N}{k+\l}+c\frac{\log N}{\log\log N},$$ where $c>0$ is an absolute constant.
*(iii)* for every $k,\l$, we have $\M_{(k,\l)}(N)=\big(\frac{1}{k+\l}+o(1)\big)N$.
The upper bound construction given by Eberhard, Green, and Manners [@EGM] for the $(2,1)$-sum free set actually works in a more general setting: the restricted $(2,1)$-sum-free set. A set $A$ is *restricted $(k,\l)$-sum free* if for every $k$ distinct elements $a_1,\dots,a_k$ in $A$, and $\l$ distinct elements $b_1,\dots,b_\l$ in $A$, we have $\sum_{i=1}^ka_i\neq\sum_{j=1}^\l b_j$. Let $$\widehat{\M}_{(k,\l)}(N)=\inf_{\substack{A\subseteq\NN^{>0}\\|A|=N}}\max
_{\substack{S\subseteq A\\S\text{ is restricted } (k,\l)-\text{sum free}}}|S|.$$ Clearly, we have that $\M_{(k,\l)}(N)\leq \widehat{\M}_{(k,\l)}(N)$. Our next theorem gives us an upper bound on $\widehat{\M}_{(k,\l)}(N)$ when $k\leq2\l+1$.
\[thm:1.2\] Let $k,\l$ be positive integers, and $k\leq2\l+1$. Then $$\widehat{\M}_{(k,\l)}(N)=\Big(\frac{1}{k+\l}+o(1)\Big)N.$$
Overview {#overview .unnumbered}
--------
The paper is organized as follows. In the next section, we provide some basic definitions and properties in additive combinatorics, harmonic analysis, and model theory (or more precisely, nonstandard analysis) used later in the proof. Theorem \[thm:one\] (i) is proved by using the probabilistic argument introduced by Erdős, and some structural results for the $(k,\l)$-sum free open set on the torus. We will prove it in Section 3. Next, we consider Theorem \[thm:one\] (ii). The special case for $(3,1)$-sum free set is proved by Bourgain [@Bourgain97], but his argument relies heavily on the fact that a certain term of the Fourier coefficient of the characteristic function is multiplicative, which is not true for the other $(k,\l)$. Here we introduce a different and more involved sieve function, which can sieve out integers containing only small prime factors. We will discuss it in details in Section 4. In Sections 5 and 6, we prove Theorem \[thm:one\] (iii). The proof goes by showing that the constructions given by Eberhard [@Eberhard15] for $(k,1)$-sum free sets, the Følner sequence, is still the correct construction for the other $(k,\l)$-sum free sets. The new ingredients contain structural results for the large infinite $(k,\l)$-sum free sets, which can be viewed as a generalization of the Łuczak–Schoen Theorem [@LS97]. We will prove Theorem \[thm:1.2\] in Section 7. In Section 8, we make some concluding remarks, and pose some open problems.
Preliminaries
=============
We use standard definitions and notation in additive combinatorics as given in [@additive]. Throughout the paper, let $p$ be a prime, and let $m,n,N$ ranging over positive integers. Given $a,b,N\in \NN$ and $a<b$, let $[a,b]:=[a,b]\cap\NN$, and let $[N]:=[1,N]$. We use the standard Vinogradov notation. That is, $f\ll g$ means $f=O(g)$, and $f\asymp g$ if $f\ll g$ and $f\gg g$. Given $A,B\subseteq \ZZ$, we write $$A+B:=\{a+b\mid a\in A,b\in B\}, \quad\text{and}\quad AB:=\{ab\mid a\in A,b\in B\}.$$ When $A=\{x\}$, we simply write $x+B:=\{x\}+B$ and $x\cdot B:=\{x\}B$. Given $A\subseteq \ZZ$, let $$kA:=\{a_1+\dots +a_k\mid a_1,\dots,a_k\in A\},$$ for integer $k\geq2$. For example, $2\cdot\NN$ denotes the set of even natural numbers, while $2\NN$ denotes $\NN+\NN$ which is still $\NN$. Using this notation, a set $A$ is *$(k,\l)$-sum free* if $kA\cap \l A=\varnothing$.
We also define the restricted sums. Let $$\begin{aligned}
&A\widehat{+}B:=\{a+b\mid a\in A,b\in B, a\neq b\},\\
&\widehat{kA}:=\{a_1+\dots +a_k\mid a_1,\dots,a_k\in A, \text{ all of them are distinct}\}.\end{aligned}$$ Thus a set $A$ is *restricted $(k,\l)$-sum free* if $\widehat{kA}\cap \widehat{\l A}=\varnothing$.
Let $f:\ZZ\to\mathbb{C}$ be a function. Define $\widehat{f}:\TT\to\mathbb{C}$, where $\TT=\RR/\ZZ$ is a torus, and for every $r\in\TT$, $$\widehat{f}(r)=\sum_{x}f(x)e(-rx),$$ where $e(\theta)=e^{2\pi i\theta}$. By Fourier Inversion, for every $x\in\ZZ$, $$f(x)=\int_\TT \widehat{f}(r)e(rx) dr.$$
Let $\mu:\NN^{>0}\to\mathbb{C}$ be the *Möbius function*. Recall that $\mu$ is supported on the square-free integers, and $\mu(n)=(-1)^{\omega(n)}$ when $n$ is square-free, where $\omega(n)$ counts the number of distinct prime factors of $n$. By Inclusive-Exclusive Principle, $$\sum_{d\mid n}\mu(d)=\begin{cases}
0\quad &\text{ if }n>1,\\
1&\text{ if }n=1.
\end{cases}$$
We also make use of the weak Littlewood conjecture. The Littlewood problem [@Littlewood] is to ask that, what is $$I(N):=\min_{A\subseteq\mathbb{Z}, |A|=N}\int_{\TT}\Big|\sum_{n\in A}e^{inx}\Big|d\mu(x)?$$ The strong Littlewood conjecture asserts that the minimum occurs when $A$ is an arithmetic progression. The conjecture is still widely open. However, the weak Littlewood conjecture, $I(N)\gg\log N$, is resolved by McGehee, Pigno, and Smith [@MPS], and independently by Konyagin [@K81]. The analogues question in discrete setting is also well studied, we refer to [@Green; @Sanders; @S17] for the interested readers. In this paper, we use the following variant of the weak Littlewood conjecture proved by Bourgain [@Bourgain97].
\[thm:littlewood\] Let $\Lambda\subseteq \NN^{>0}$ be a finite set, and let $P\geq(\log |\Lambda|)^{100}$. Let $|a_n|,|b_n|\leq 1$ for all $n\in\NN^{>0}$. Then there exists $c\in\RR^{>0}$ such that $$\Bigg\Vert \,\sum_{m\in \Lambda}e^{imkx}+ \sum_{m\in\Lambda}\sum_{\substack{n\in\N}} \frac{1}{n}(a_ne^{imnx}+b_ne^{-imnx})\Bigg\Vert_{L^1(\TT)}\geq c\log|\Lambda|,$$ where $\N$ denotes set of positive integers $m$ such that every prime factor of $m$ is at least $P$.
Next, we give some basic definitions in nonstandard analysis which will be used later in the proof. For a more systematic accounts we refer to [@BT14; @non]. Let $S$ denote an infinite set. An *ultrafilter* $\mathscr{U}$ on $S$ is a collection of subsets of $S$, such that the characteristic function $\mathbbm{1}_{\mathscr{U}}:2^S\to \{0,1\}$ is a finitely additive $\{0,1\}$-valued probability measure on $S$. An ultrafilter is *principal* if it consists of all sets containing some element $s\in S$. Let $\beta S$ denotes the collection of all ultrafilters. One can embed $S$ into $\beta S$, by mapping $x\in S$ to the principal ultrafilter generated by $x$. By Zorn’s Lemma, $\beta S\setminus S$ is non-empty.
Fix $\mathscr{U}\in \beta \NN\setminus \NN$, and let $M_n$ be a structure for each $n\in \NN$. The *ultraproduct* $\prod_{n\to\mathscr{U}}M_n$ is a space consists of all ultralimits $\lim_{n\to\mathscr{U}}x_n$ of sequences $x_n$ defined in $M_n$, with $\lim_{n\to\mathscr{U}}x_n=\lim_{n\to\mathscr{U}}y_n$ if two sequences $\{x_n\}$ and $\{y_n\}$ agree on a set in $\mathscr{U}$. Let ${^{*}\RR}:=\prod_{n\to \mathscr{U}}\RR$ be the hyperreal field. Every finite hyperreal number $\xi\in{^{*}\RR}$ is infinitely close to a unique real number $r\in\RR$, called the standard part of $\xi$. In this case, we use the notation $r=\mathrm{st}(\xi)$.
Given a sequence of finite non-empty sets $F_n$, let $\mu_n(X)=|X\cap F_n|/|F_n|$ be the uniform probability measure. Let $F=\prod_{n\to\mathscr{U}}F_n$ be an ultraproduct. We define the *Loeb measure* [@Loeb] $\mu_L$ on $F$ to be the unique probability measure on the $\sigma$-algebra generated by the Boolean algebra of internal subsets of $F$, such that when $X=\prod_{n\to\mathscr{U}}X_n$ is an internal subset of $F$, we have $$\mu_L(X)=\mathrm{st}\Big(\lim_{n\to\mathscr{U}}\mu_{n}(X_n)\Big).$$
$(k,\l)$-sum free open sets in the torus
========================================
Let $\TT=\RR/\ZZ$ be the 1-dimensional torus. In this section, we use $\mu_H$ as the Haar probability measure on $\TT$.
\[prop:T1\] Let $A\subseteq \TT$ be a $(k,\l)$-sum free open set. Then $\mu_H(A)\leq\frac{1}{k+\ell}$.
Since $A$ is $(k,\l)$-sum free, we have $kA\cap \l A=\varnothing$. In particular, $\mu_H(kA)+\mu_H(\l A)\leq 1$. By Kneser’s inequality [@kenser], $$(k+\l)\mu_H(A)\leq \mu_H(kA)+\mu_H(\l A)\leq 1,$$ which implies that $\mu_H(A)\leq 1/(k+\l)$.
Next, we construct the largest $(k,\l)$-sum free open sets in $\TT$. When $k-\l\geq2$, our construction is asymmetric: this will help us to get a better lower bound on $\M_{(k,\l)}(N)$. We will discuss it in details in the next section.
\[lem:T\] Let $k,\l$ be positive integers and $k>\l$.\
*(i)* when $k-\l=1$, set $\Omega_1=\big(\frac{1}{2}-\frac{1}{2(k+\ell)},\frac{1}{2}+\frac{1}{2(k+\ell)}\big)$.\
*(ii)* when $k-\l\geq2$, set $\Omega_t=\big(\frac{t-1}{k-\l}+\frac{\ell}{k^2-\ell^2},\frac{t-1}{k-\l}+\frac{k}{k^2-\ell^2}\big)$ for every integer $t\in [k-\l]$.\
Then $\Omega_t$ is $(k,\l)$-sum free for every $t$.
Lemma \[lem:T\] is easy to verify, and we omit the details here.
First consider $k=\ell+1$, and let $\Omega_1$ be as in (i). For every $x_1,\dots,x_k,y_1,\dots,y_\ell$ in $\Omega$, we have $$\begin{aligned}
&\sum_{i=1}^kx_i\in \Big(\frac{\ell}{2}+\frac{\ell}{2(2\ell+1)},\frac{\ell}{2}-\frac{\ell}{2(2\ell+1)}\Big),\\
&\sum_{i=1}^\ell y_i\in\Big(\frac{\ell}{2}-\frac{\ell}{2(2\ell+1)},\frac{\ell}{2}+\frac{\ell}{2(2\ell+1)}\Big).\end{aligned}$$ Note that $\l/2\in\{1/2,1\}$. Thus $\sum_{i=1}^kx_i\neq\sum_{i=1}^\ell y_i$.
Now assume $k-\ell\geq2$. Similarly, for every $1\leq t\leq k-\l$ and for every $x_1,\dots,x_k,y_1,\dots,y_\ell$, we have $$\begin{aligned}
&\sum_{i=1}^kx_i\in\Big(\frac{(t-1)k}{k-\l}+\frac{k\l}{k^2-\l^2},\frac{(t-1)k}{k-\l}+\frac{k^2}{k^2-\l^2}\Big),\\
&\sum_{i=1}^\ell y_i\in\Big(\frac{(t-1)\l}{k-\l}+\frac{\l^2}{k^2-\l^2},\frac{(t-1)\l}{k-\l}+\frac{k\l}{k^2-\l^2}\Big).\end{aligned}$$ Hence $\Omega_t$ is $(k,\l)$-sum free for each $t\in[k-\l]$.
When $k=\l+1$, the following observation shows that all the possible $(k,\l)$-sum free open sets with maximum measure are symmetric. Thus one cannot apply the method used in the next section to improve the lower bound for the case $k=\l+1$.
Let $k=\l+1$. Suppose $A\subseteq \TT$ is a maximum $(k,\l)$-sum free open set. Then $A$ is symmetric.
Since $k=\l+1$, $A$ is $(k,\l)$-sum free implies that $(\l A-\l A)\cap A=\varnothing$. Hence $A\subseteq \TT\setminus (\l A-\l A)$. By Kneser’s inequality, $$\mu_H(\TT\setminus (\l A-\l A))\leq 1-2\l\mu_H(A).$$ By Proposition \[prop:T1\], $\mu_H(A)=\frac{1}{2\l+1}$. Thus $A=\TT\setminus(\l A-\l A)$, and this implies that $A$ is symmetric.
Using the argument by Erdős [@Erdos65], Lemma \[lem:T\] is able to give us the following lower bound on the maximum $(k,\l)$-sum free subsets of any set of $N$ integers, which proves Theorem \[thm:one\] (i).
Let $k,\l$ be positive integers and $k>\l$. Then for every $A\subseteq \NN^{>0}$ of size $N$, $A$ contains a $(k,\l)$-sum free subsets of size at least $\frac{1}{k+\ell}N$.
Let $\Omega_t$ be as in Lemma \[lem:T\], and let $\e_\Omega$ be the characteristic function of $\Omega$ in $\TT$. Thus by Fubini’s Theorem, $$\int_\TT\sum_{n\in A}\e_\Omega(nx)d\mu_H(x)=\sum_{n\in A}\int_\TT\e_\Omega(nx)d\mu_H(x)=\frac{N}{k+\ell}.$$ Therefore, by Pigeonhole principle, there exists $x\in\TT$ such that $$|\{n\in A\mid nx\in \Omega\}|\geq\frac{N}{k+\ell},$$ finishes the proof.
Lower Bound when $k-\l\geq2$
============================
Let $k,\l$ be positive integers, and $k-\l\geq2$. Let $I=\{1,\dots,k-\l\}$ be the index set. Set $$\Omega_t=\Big(\frac{t-1}{k-\l}+\frac{\ell}{k^2-\ell^2},\frac{t-1}{k-\l}+\frac{k}{k^2-\ell^2}\Big),$$ for every $t\in I$. Let $\ee$ be the indicator function of $\Omega_t$. Given $A\subseteq \NN^{>0}$ of size $N$. Let $\M(A)$ be the size of the maximum $(k,\l)$-sum free subset of $A$. We have $$\begin{aligned}
\label{eq:MA}
\M(A)\geq\max_{x\in\TT}\sum_{n\in A}\ee(nx), \end{aligned}$$ since $\Omega_t$ is $(k,\l)$-sum free for every $t$. Then $$\begin{aligned}
\label{eq:main}
&\max_{x\in\TT}\sum_{n\in A}\e_{\Omega_t}(nx)=\frac{N}{k+\l}+\max_{x\in\TT}\sum_{n\in A}\Big(\e_{\Omega_t}-\frac{1}{k+\l}\Big)(nx),\end{aligned}$$ for every $t\in I$. We introduce the balanced function $f_t:\TT\to\CC$ defined by $f_t=\e_{\Omega_t}-\frac{1}{k+\l}$. By orthogonality of characters we have $$\widehat{f_t}(x)=
\begin{cases}
0\quad & \text{ if } x=0,\\
\widehat{\e_{\Omega_t}}(x) & \text{ else}.
\end{cases}$$
By Fourier inversion, when $n>0$, $$\begin{aligned}
\widehat{f_t}(n)&=\int_{\TT}\ee(x)e(-nx)d\mu(x)\\
&=\frac{1}{2\pi in}\Big(-e\big(-\frac{(t-1)n}{k-\l}-\frac{nk}{k^2-\ell^2}\big)+e\big(-\frac{(t-1)n}{k-\l}-\frac{n\l}{k^2-\l^2}\big)\Big).\end{aligned}$$ Therefore, $$\begin{aligned}
\widehat{f_t}(n)&=\frac{1}{2\pi n} e\Big(\frac{(2t-1)n}{2(k-\l)}\Big)\Big(\sin(\frac{2k n\pi}{k^2-\l^2}-\frac{\pi n}{k-\l})-\sin(\frac{2\l n\pi}{k^2-\l^2}-\frac{\pi n}{k-\l})\Big)\\
&=\frac{1}{\pi n}e\Big(\frac{(2t-1)n}{2(k-\l)}\Big)\sin\Big(\frac{n\pi}{k+\l}\Big).\end{aligned}$$ Hence, for every $t\in I$ we have $$\begin{aligned}
&f_t(x)=\sum_{n\neq0}\widehat{f_1}(n)e(nx)=\sum_{n\neq0}\frac{1}{\pi n}e\Big(\frac{(2t-1)n}{2(k-\l)}\Big)\sin\Big(\frac{n\pi}{k+\l}\Big)e(nx).\end{aligned}$$
Let $F:=\sum_{t\in I}f_t$. Observe that $$\sum_{t=1}^{k-\l}\sin\Big(\frac{(2t-1)n\pi}{k-\l}\Big)=0.$$ Thus, let $M_1=k-\l$, we have $$\begin{aligned}
F(x)&=\frac{2M_1}{\pi }\sum_{n\geq1}\frac{1}{ n}\sin\Big(\frac{n\pi}{k+\l}\Big)\alpha(n)\cos(n x),\end{aligned}$$ where $$\alpha(n)=\frac{1}{M_1}\sum_{t\in I}\cos\Big(\frac{(2t-1)n\pi}{k-\l}\Big)=
\begin{cases}
0 \quad&\text{ when } (k-\l)\nmid n,\\
(-1)^{s} &\text{ when } n=(k-\l)s.
\end{cases}$$ Therefore, we get $$F(x)=\frac{2M_1}{\pi}\sum_{n\geq1}\frac{(-1)^{n-k+\l}}{n}\sin\Big(\frac{(k-\l)n\pi}{k+\l}\Big)\cos((k-\l)n x).$$ Let $\Phi(n)=(-1)^{n-k+\l}\sin(\frac{(k-\l)n\pi}{k+\l})$, we have that $\Vert\Phi\Vert_\infty\leq1$, and $\Phi$ is a periodic function with period $(k+\l)/\gcd(k+\l,k-\l)$. Set $R_F=(k+\l)/\gcd(k+\l,k-\l)$.
Let $I_1=\{1,\dots,\lfloor\frac{k-\l}{2}\rfloor\}$ and $I_1=\{\lceil\frac{k-\l}{2}\rceil+1,\dots,k-\l\}$. Now we consider two cases.
[**Case 1.**]{} *$k-\l$ is prime.*
Define $G:=\sum_{t\in I_1}f_t-\sum_{t\in I_2}f_t$. Similarly, observe that $$\sum_{t\in I_1}\cos\Big(\frac{(2t-1)n\pi}{k-\l}\Big)-\sum_{t\in I_2}\cos\Big(\frac{(2t-1)n\pi}{k-\l}\Big)=0.$$ Let $M_2=2\lfloor\frac{k-\l}{2}\rfloor$, we have $$\begin{aligned}
G(x)&=\frac{2M_2}{\pi}\sum_{n\geq1}\frac{1}{ n}\sin\Big(\frac{n\pi}{k+\l}\Big)\beta(n)\sin(n x),\end{aligned}$$ where $$\beta(n)=\frac{1}{M_2}\Big(\sum_{t\in I_2}\sin\Big(\frac{(2t-1)n\pi}{k-\l}\Big)-\sum_{t\in I_1}\sin\Big(\frac{(2t-1)n\pi}{k-\l}\Big)\Big).$$ Note that $\beta(n)$ is a periodic function with period $2(k-\l)$, $0<\Vert \beta\Vert_{\infty}\leq1$, $\beta(k-\l)=0$, and $\beta(x)=-\beta(x+k-\l)$.
[**Case 2.**]{} *$k-\l$ is composite.*
Let $p_1<\dots<p_s$ be all the prime factors of $k-\l$. Note that for every factor $d$ of $k-\l$, we always have $d\in I_1$. Observe that for every $t_1,t_2\in I_1$, if $t_1\neq t_2$, then $$\sin\Big(\frac{(2t_1-1)n\pi}{k-\l}\Big)-\sin\Big(\frac{(2t_2-1)n\pi}{k-\l}\Big)\neq0$$ for some $n\in\NN^{>0}$. Let $$M=\Big\{m\in\NN^{>0}\ \Big|\ m\leq \frac{k-\l}{2}, \text{ and } m \text{ is a multiple of some }p_i\Big\}.$$ Thus $|M|\leq \frac{(k-\l)(1-\prod_{i=1}^s(1-1/p_i))}{2}<|I_1|$.
Now we consider a system of linear equations $\mathcal{L}$ with $|M|$ equations and $|I_1|$ variables $x_1,\dots,x_{\lfloor (k-\l)/2\rfloor}$, such that $\mathcal{L}:\sum_{t\in I_1}x_t\sin (\frac{(2t-1)m\pi}{k-\l})=0$ for every $m\in M$. Since the rank of the system of linear equations is less than the number of variables, there are $\lambda_1,\dots,\lambda_{\lfloor (k-\l)/2\rfloor}\in\RR$, such that $|\lambda_t|\leq1$ for every $t\in I_2$, not all $\lambda_t$ are $0$, and $$\label{eq:lambda}
\sum_{t\in I_1}\lambda_t\sin\Big(\frac{(2t-1)n\pi}{k-\l}\Big)=0$$ For every $j\in I_2$, let $\lambda_j=\lambda_{k-\l+1-j}$. Define $G:=\sum_{t\in I_1}\lambda_tf_t-\sum_{j\in I_2}\lambda_jf_j$. Thus we obtain $$\begin{aligned}
G(x)&=\frac{2M_2}{\pi}\sum_{n\geq1}\frac{1}{ n}\sin\Big(\frac{n\pi}{k+\l}\Big)\beta(n)\sin(n x),\end{aligned}$$ where $$\beta(n)=\frac{1}{M_2}\Big(\sum_{t\in I_2}\lambda_t\sin\Big(\frac{(2t-1)n\pi}{k-\l}\Big)-\sum_{t\in I_1}\lambda_t\sin\Big(\frac{(2t-1)n\pi}{k-\l}\Big)\Big).$$ Note that $0<\Vert\beta\Vert_\infty\leq1$, $\beta(n)$ is a periodic function with period $2(k-\l)$, and by (\[eq:lambda\]), $\beta(m)=0$ when $m\in M$. By the symmetric property of the $\sin$ function, this implies $\beta(m)=0$ whenever $m$ is a multiple of some $p_i$ for $i\in[s]$.
In either case, let $\Psi(n)=\beta(n)\sin(\frac{n\pi}{k+\l})$, and set $R_G:=(k-\l)(k+\l)$. Thus $\Vert\Psi\Vert_\infty\leq1$, and $\Psi$ is a periodic function with period $R_G$ since $k+\l\equiv k-\l\pmod2$.
Next we are going to construct the sieve function, which is a crucial ingredient of our argument. Let $R$ be either $R_F$ or $R_G$, and let $\mathcal{S}=\{1,\dots,R-1\}$. For every $k\in \mathcal{S}$, let $a_1=1$, and $a_k\in \NN^{>0}$ be the smallest prime $p$ such that $p\equiv k^{-1}\pmod R$ and $\gcd(p,R)=1$. If such $a_k$ does not exist (when $\gcd(k,R)>1$), we just simply define $a_k=\infty$. Let $Q$ be the product of all finite $a_k$, and let $q$ be a prime such that $\gcd(q,Q)=1$ and $q\equiv Q^{-1}\pmod R$. Let $\MM$ be the set containing the square-free integers generated by all the finite $a_k$ and $q$. For every $x\in\NN^{>0}$, let $\Vert x\Vert_R\in \mathcal{S}\cup\{0\}$ such that $\Vert x\Vert_R\equiv x\pmod R$. Let $P$ be a prime, and $P\sim(\log N)^{100}$. Let $\mu$ be the Möbius function.
We are now going to analyze the property of $F$ and $G$. The computation contains three steps. In the first step, we sieve out integers which do not contain $(Qq)^{\phi(R)-2}$ as a factor, where $\phi$ is the Euler totient function. In step two, we sieve out integers containing prime factors from $\MM$ with multiplicity at least $\phi(R)-1$. Now, we may still have some integers containing only small prime factors, all of which have the form $u(qQ)^{\phi(R)-2}$, and $\gcd(u,qQ)=1$. In the final step, we sieve out integers containing small prime factors which are coprime with $qQR$. Note that for integers of the form $u(qQ)^{\phi(R)-2}$, we may have $u\not\equiv1\pmod R$, which will spoil the computation. In order to fix this, for each $u$ with $\gcd(u,R)=1$, we pair with an integer $a_{\Vert u\Vert_R}$. An important fact is, all the remaining integers of the form $u(qQ)^{\phi(R)-2}$ are divisible by some $ua_{\Vert u\Vert_R}$ except for the first term $(qQ)^{\phi(R)-2}$, and this is the reason why we cannot simply apply step three without the first two steps. Now, except for the term $(qQ)^{\phi(R)-2}$, all the remaining integers either containing at least one large prime factor, or only containing prime factors which are also factors of $R$.
Note that in the computation, we actually require that $\phi(R)-2\geq1$, which is not always true when $R_F$ or $R_G$ is small. In that case, we just simply let $R=R_F^j$ for some positive integer $j$ such that $\phi(R)$ is large, and all the computations still work. Hence, we may assume $\phi(R_F)$ and $\phi(R_G)$ are not small for convenience.
More precisely, let $R=R_F$, for $F(x)$ we have the following. $$\begin{aligned}
&\,\sum_{\substack{t\mid P!\\\gcd(t,RQq)=1}}\frac{\mu(t)}{ta_{\Vert t\Vert_R}}\sum_{k\in \MM}\frac{\mu(k)}{k^{\phi(R)-1}}\frac{1}{(Qq)^{\phi(R)-2}}\sum_{m\in A}F(ta_{\Vert t\Vert_R}k^{\phi(R)-1}(Qq)^{\phi(R)-2}mx)\nonumber\\ =&\,\frac{2M_1}{\pi}\sum_{\substack{t\mid P!\\\gcd(t,RQq)=1}}\frac{\mu(t)}{ta_{\Vert t\Vert_R}}\sum_{k\in \MM}\frac{\mu(k)}{k^{\phi(R)-1}}\sum_{m\in A}\sum_{(Qq)^{\phi(R)-2}\mid n}\frac{\Phi(n)}{n}\cos((k-\l)ta_{\Vert t\Vert_R}k^{\phi(R)-1}mnx)\nonumber\\ =&\,\frac{2M_1}{\pi}\sum_{\substack{t\mid P!\\\gcd(t,RQq)=1}}\frac{\mu(t)}{ta_{\Vert t\Vert_R}}\sum_{m\in A}\sum_{(Qq)^{\phi(R)-2}\mid n}\frac{\Phi(n)}{n}\cos((k-\l)ta_{\Vert t\Vert_R}mnx)\sum_{\substack{k^{\phi(R)-1}\mid n\\ k\in\MM}}\mu(k)\label{eq:1}\\ =&\,\frac{2M_1}{\pi}\sum_{\substack{t\mid P!\\\gcd(t,RQq)=1}}\frac{\mu(t)}{ta_{\Vert t\Vert_R}}\sum_{m\in A}\sum_{\substack{(Qq)^{\phi(R)-2}\mid n\\\forall p\in\MM\ p^{\phi(R)-1}\nmid n}}\frac{\Phi(n)}{n}\cos((k-\l)ta_{\Vert t\Vert_R}mnx)\label{eq:2}\\ =&\,\frac{2M_1}{\pi}\sum_{m\in A}\sum_{\substack{(Qq)^{\phi(R)-2}\mid n\\n\in\N'}}\frac{\Phi(n)}{n}\cos((k-\l)mnx)\sum_{\substack{t\mid P!\\\gcd(t,RQq)=1\\ta_{\Vert t\Vert_R}\mid n}}\mu(t)\label{eq:3}\\ =&\,\frac{2M_1}{\pi}\sum_{m\in A}\sum_{n\in\N}\frac{\Phi(n)}{n}\cos((k-\l)mnx)+\frac{2M_1\Phi(1)}{\pi (qQ)^{\phi(R)-2}}\sum_{m\in A}\cos((k-\l)(qQ)^{\phi(R)-2}mx).\label{eq:4}$$ Equality (\[eq:1\]) follows from the fact that $k^{\phi(R)-1}\equiv 1\pmod R$, and equality (\[eq:2\]) follows from the basic property of the Möbius function. The set $\N'$ in (\[eq:3\]) satisfies the property that for every $n\in\N'$ and every square-free $t\mid P!$ such that $\gcd(t,RQq)=1$, if $t\mid n$, we also have $a_{\Vert t\Vert_R}\mid n$. Equality (\[eq:4\]) follows from $a_1=1$, and $\N$ denotes the set of integers $n>(qQ)^{\phi(R)-2}$ such that either all the prime factors of $n$ are at least $P$, or the only prime factors of $n$ that smaller than $P$ are the prime factors of $R$. Since $R=R_F=\frac{k+\l}{\gcd(k+\l,k-\l)}$ is prime, and $\Phi(n)=0$ when $R\mid n$. Thus we can let $$\N=\{n: \forall {p\,|\, n}, \text{ if }p\neq1, \text{ then } p>P\}.$$ and (\[eq:4\]) still holds.
We now use the similar argument on $G(x)$. Let $R=R_G$, and we define $Q'$, $q'$ and $\MM_G$ in a similar way as we did for $F(x)$. We have $$\begin{aligned}
&\,\sum_{\substack{t\mid P!\\\gcd(t,RQ'q')=1}}\frac{\mu(t)}{ta_{\Vert t\Vert_R}}\sum_{k\in \MM_G}\frac{\mu(k)}{k^{\phi(R)-1}}\frac{1}{(Q'q')^{\phi(R)-2}}\sum_{m\in A}G(ta_{\Vert t\Vert_R}k^{\phi(R)-1}(Q'q')^{\phi(R)-2}mx)\nonumber\\
=&\,\frac{2M_2}{\pi}\sum_{m\in A}\sum_{n\in\N_G}\frac{\Psi(n)}{n}\sin(mnx)+\frac{2M_2\Psi(1)}{\pi (q'Q')^{\phi(R)-2}}\sum_{m\in A}\sin((q'Q')^{\phi(R)-2}mx),\label{eq:GX}\end{aligned}$$ where $\N_G$ denotes the set of integers $n>(q'Q')^{\phi(R)-2}$, such that either all the prime factors of $n$ are at least $P$, or the only prime factors of $n$ that smaller than $P$ are the prime factors of $R$. Since $R_G=(k+\l)(k-\l)$, and by the assumption, all the prime factors of $k+\l$ are also the prime factors of $k-\l$, and the fact that for every prime $t\mid k-\l$, $\Psi(m)=0$ whenever $t\mid m$, we can conclude that if we let $\N_G=\N$, (\[eq:GX\]) still holds.
Let $$\eta=(Qq)^{\phi(R_F)-2}, \quad \xi=(Q'q')^{\phi(R_G)-2}.$$ Let $K=\frac{\eta\xi}{\gcd(\eta,\xi)}$. Now we apply Theorem \[thm:littlewood\] to bound the exponential sums. Therefore, $$\begin{aligned}
&\,\log N\\
\ll&\,\Bigg\Vert \sum_{m\in A}\big(\cos((k-\l)Kmx)+i\sin((k-\l)Kmx)\big)\\
&\,+\sum_{m\in A}\sum_{n\in\N}\frac{1}{n}\Big(\frac{\Phi(n)\eta}{\Phi(1)}\cos((k-\l)Kmnx)+i\frac{\Psi(n)\xi}{\Psi(1)}\sin((k-\l)Kmnx)\Big)\Bigg\Vert_{L^1(\TT)}\\
\ll&\,\Bigg\Vert \sum_{\substack{t\mid P!\\\gcd(t,R_FQq)=1}}\frac{\mu(t)}{ta_{\Vert t\Vert_{R_F}}}\sum_{k\in \MM}\frac{\mu(k)}{k^{\phi(R_F)-1}}\frac{1}{\eta}\sum_{m\in A}F(ta_{\Vert t\Vert_{R_F}}k^{\phi(R_F)-1}Kmx)\Bigg\Vert_{L^1(\TT)}\\
&\,+\Bigg\Vert \sum_{\substack{t\mid P!\\\gcd(t,R_GQ'q')=1}}\frac{\mu(t)}{ta_{\Vert t\Vert_{R_G}}}\sum_{k\in \MM_G}\frac{\mu(k)}{k^{\phi(R_G)-1}}\frac{1}{\eta}\sum_{m\in A}G(ta_{\Vert t\Vert_{R_G}}k^{\phi(R_G)-1}Kmx)\Bigg\Vert_{L^1(\TT)}\\
\ll&\,\sum_{t\mid P!}\bigg|\frac{\mu(t)}{t}\bigg|\Bigg(\bigg\Vert \sum_{m\in A}F(ta_{\Vert t\Vert_{R_F}}k^{\phi(R_F)-1}Kmx)\bigg\Vert_{L^1(\TT)}+\bigg\Vert \sum_{m\in A}G(ta_{\Vert t\Vert_{R_G}}k^{\phi(R_G)-1}Kmx)\bigg\Vert_{L^1(\TT)}\Bigg)\\
\ll&\,\prod_{p\leq P}\Big(1+\frac{1}{p}\Big)\bigg(\sum_{t=1}^{k-\l}\bigg\Vert \sum_{m\in A}f_t(mx)\bigg\Vert_{L^1(\TT)}+\sum_{t\in I_1\cup I_2}\lambda_t\bigg\Vert \sum_{m\in A}f_t(mx)\bigg\Vert_{L^1(\TT)}\bigg).\end{aligned}$$ By Merterns’ estimates we get $$\prod_{p\leq P}\Big(1+\frac{1}{p}\Big)\ll\log P\asymp \log\log N.$$ Hence there is $t\in I$ such that $\big\Vert \sum_{m\in A}f_t(mx)\big\Vert_{L^1(\TT)}\gg\frac{\log N}{\log\log N}$.
Note that for every $t\in I$, $$\int_{\TT}\sum_{n\in A}f_t(nx)dx=0.$$ Thus we have $$\max_{x\in\TT}\sum_{n\in A}f_t(nx)\geq\frac{1}{2}\bigg\Vert \sum_{n\in A}f_t(nx)\bigg\Vert_{L^1(\TT)}.$$ Together with (\[eq:MA\]) and (\[eq:main\]), we get $$\M(A)-\frac{N}{k+\l}\gg\frac{\log N}{\log\log N},$$ and this proves Theorem \[thm:one\] (ii).
Structures of the infinite $(k,\l)$-sum free sets
=================================================
Given $A\subseteq \NN^{>0}$, the *upper density* of $A$ is $$\overline{d}(A)=\limsup_{N\to\infty}\frac{|A\cap[N]|}{N}.$$ We also define the *upper density on multiples* of $A$ by $$\widetilde{d}(A)=\limsup_{N\to\infty}\limsup_{n\to\infty}\frac{|A\cap (N!\cdot [n])|}{n}.$$ In this section, we will prove the following theorem.
\[thm:periodic\] Suppose $A\subseteq\NN^{>0}$, and $A$ is $(k,\l)$-sum free. Then $\widetilde{d}(A)\leq \frac{1}{k+\l}$.
For this, we will need three lemmas. The first lemma says that if a $(k,\l)$-sum free set $A$ contains a certain long arithmetic progression, then the upper density of $A$ is bounded.
\[lem:5.1\] Let $A\subseteq \NN^{>0}$ be a $(k,\l)$-sum free set. Let $x,s,d,m$ be positive integers, such that $s\in \l A-(k-1) A$, $x+d\cdot [m]\subseteq A$, and $s$ is in the coset $x+d\cdot\ZZ$. Then $$\overline{d}(A)\leq\frac{m+k+\l-2}{(k+\l)m+2(k+\l-2)}.$$
Since $s\in\l A-(k-1)A$ and $A$ is $(k,\l)$-sum free, we have $s\notin A$. We will only consider $s\leq x$, and the case when $s\geq x+m$ follows from the same proof. Since $x+d\cdot [m]\subseteq A$, then $\big(x+d\cdot [m]\big)\cap \big(\l A-(k-1)A\big)=\varnothing.$ Thus, there is $s_0\in x+d\cdot \ZZ$, such that $s_0\in\l A-(k-1)A$, and $$\label{eq:s0}
\big(s_0+d\cdot [m]\big)\cap \big(\l A-(k-1)A\big)=\varnothing.$$ Let $s_0=\sum_{i=1}^\l a_i-\sum_{j=1}^{k-1}b_j$, where $a_i,b_j\in A$ for every $1\leq i\leq \l$ and $1\leq j\leq k-1$.
Let $B\subseteq A$ such that $$B:=\{b\in A\mid \big(b+d\cdot[m]\big)\cap A\neq\varnothing\}.$$ Set $a_0=b_0=0$. Given integers $1\leq u\leq k-1$ and $2\leq v \leq \l$, let $$\C(u)=B+\sum_{j=1}^{k-u}b_j+(u-1)a_1,\quad\quad \D(v)=B+\sum_{j=0}^{\l-v}a_j+\sum_{i=0}^{v-1}b_i,$$ and $\C(k)=A+(k-1)a_1$, $\D(1)=A+\sum_{j=1}^{\l-1}a_j$. Let $\mathscr{F}=\{\C(u),\D(v)\mid u\in[k],v\in [\l]\}$ be the collection of all $\C(u)$ and $\D(v)$.
\[cm:1\] Elements in $\mathscr{F}$ are pairwise disjoint.
*Proof of Claim \[cm:1\].* Observe that for every $u\in[k]$ and $v\in[\l]$, $\C(u)\cap \D(v)=\varnothing$. Otherwise, we will get $kA\cap \l A\neq\varnothing$, contradicts that $A$ is $(k,\l)$-sum free. Let $u_1,u_2\in[k]$ and $u_1< u_2$. Suppose that $\C(u_1)\cap \C(u_2)\neq\varnothing$. Then there exist $y_1\in B$ and $y_2\in A$, such that $$y_1+\sum_{j=k-u_2+1}^{k-u_1} b_j=y_2+(u_2-u_1)a_1.$$ Then $$\begin{aligned}
s_0&=\sum_{i=1}^\l a_i-\sum_{j=1}^{k-1}b_j\\
&=y_1+\sum_{i=2}^\l a_i-y_2-(u_2-u_1-1)a_1-\sum_{j\in [1,k-u_2]\cup[k-u_1+1,k-1]}b_j.\end{aligned}$$ Since $y_1\in B$, thus there is $r\in [m]$ such that $y_1+rd\in A$. This implies $s_0+rd\in \l A-(k-1)A$, contradicts (\[eq:s0\]).
Suppose $\D(v_1)\cap \D(v_2)\neq\varnothing$ for some $v_1,v_2\in[\l]$ and $v_1<v_2$. Similarly, there exist $y_1\in A$ and $y_2\in B$, such that $$y_1+\sum_{j=\l-v_2+1}^{\l-v_1}a_j=y_2+\sum_{i=v_1}^{v_2-1}b_i.$$ Let $c_0=0$, and let $c_1,\dots,c_{v_2-v_1-1}\in A$ if $v_2>v_1+1$. Therefore $$\begin{aligned}
s_0=y_2+\sum_{j\in[0,\l-v_2]\cup[\l-v_1+1,\l]}a_j+\sum_{t=0}^{v_2-v_1-1}c_t-y_1-\sum_{i\in[0,v_1-1]\cup[v_2,k-1]}b_i-\sum_{t=0}^{v_2-v_1-1}c_t.\end{aligned}$$ Observe $y_2\in B$ implies that there is $r\in[m]$, such that $y_2+rd\in A$. Hence $s_0+rd\in \l A-(k-1)A$, which contradicts (\[eq:s0\]).
By Claim \[cm:1\], we obtain $$\label{eq:B1}
(k+\l-2)\overline{d}(B)+2\overline{d}(A)\leq 1.$$ On the other hand, let $\N(t)=A\setminus B+td$ for every $t\in[m]$, and let $$\mathscr{G}=\Big\{A,A-(k-1)x+\sum_{i=1}^{\l-1}a_i,\N(t)\ \Big|\ t\in[m]\Big\}.$$
\[cm:2\] Elements in $\mathscr{G}$ are pairwise disjoint.
*Proof of Claim \[cm:2\].* Suppose there are $u,v\in [m]$, $u<v$, such that $\N(u)\cap \N(v)\neq\varnothing$. Thus we have $c\in A\setminus B$ such that $c_1+(u-v)d\in A$, and this contradicts the assumption of $B$. Same conclusion holds if $A\cap \N(u)\neq\varnothing$. Observe that if $A\cap (A-(k-1)x+\sum_{i=1}^{\l-1}a_i)\neq\varnothing$, it will contradict that $A$ is $(k,\l)$-sum free. Finally, we assume that there are $c_1,c_2\in A$, $u\in [m]$ such that $$c_1+ud=c_2-(k-1)x+\sum_{i=1}^{\l-1}a_i.$$ Thus, $c_1+x+ud+(k-2)x=c_2+\sum_{i=1}^{\l-1}a_i$. Since $x+d\cdot[m]\subseteq A$, this contradicts $A$ is $(k,\l)$-sum free.
Thus, by Claim \[cm:2\], we get $$(m+2)\overline{d}(A)-m\overline{d}(B)\leq 1.$$ Together with (\[eq:B1\]), this finishes the proof.
The next lemma is a finite version of the Szemerédi Theorem [@Sz], and we will use it to find the arithmetic progression in Lemma \[lem:5.1\].
\[lem:Sz\] For every $\varepsilon>0$ and $m\in\NN^{>0}$, there is $L=L(\varepsilon,m)>0$ such that every set $A\subseteq\mathbb{N}^{>0}$ with $\overline{d}(A)>\varepsilon$, there exist $x\in\NN$, $d<L$, and $x+d\cdot [m]\subseteq A$.
Our final lemma says that a $(k,\l)$-sum free set $A$ with large upper density should be periodic. This structural result can be viewed as a generalization of the Łuczak–Schoen Theorem [@LS97].
\[lem:5.2\] Let $\varepsilon>0$. Then there is $D>0$ such that the following holds. Let $A\subseteq\NN^{>0}$ be a $(k,\l)$-sum free set, and $\overline{d}(A)>\frac{1}{k+\l}+\varepsilon$. Then $A$ is contained in a periodic $(k,\l)$-sum free set with period $D$.
We pick $m\in\NN^{>0}$ such that $$\label{eq:density}
\frac{m+k+\l-2}{(k+\l)m+2(k+\l-2)}<\frac{1}{k+\l}+\varepsilon.$$ Let $L=L(\varepsilon,m)$ be as in Lemma \[lem:Sz\]. Let $D=L!$. Suppose the lemma fails. Let $C\subseteq\NN^{>0}$ be a periodic set with period $D$, consists of all positive integers in every coset $a+D\cdot\ZZ$ for $a\in A$. Thus $C$ is not $(k,\l)$-sum free. This means, there are $a_1,\dots,a_\l$ and $b_1,\dots,b_k$ in $C$ such that $\sum_{i=1}^\l a_i=\sum_{j=1}^k b_j$. Let $P$ be the “$(k,\l)$-sum free” part of $C$. That is, $$P=C\setminus \big(\l C-(k-1)C\big).$$ Set $a_0=b_0=0$. For every $u\in[k]$ and $v\in[\l]$, let $$\MM(u)=P+\sum_{j=0}^{k-u}b_j+(u-1)a_1,\qquad \N(v)=P+\sum_{i=0}^{\l-v}a_i+(v-1)b_1.$$ Let $\mathscr{F}$ be the collection of all $\MM(u)$ and $\N(v)$.
\[cm:3\] Elements in $\mathscr{F}$ are pairwise disjoint.
*Proof of Claim \[cm:3\].* Observe that for every $u\in [k]$ and $v\in[\l]$, $\MM(u)\cap \N(v)=\varnothing$. Otherwise there are $p_1,p_2\in P$, such that $$p_1=p_2+\sum_{i=0}^{\l-v}a_1+(v-1)b-\sum_{j=0}^{k-u}b_j-(u-1)a_1\in \l C-(k-1)C,$$ contradicts the assumption of $P$. Now, suppose $u_1,u_2\in [k]$, $u_1<u_2$, such that $\MM(u_1)\cap \MM(u_2)\neq\varnothing$. The case that $\N(v_1)\cap \N(v_2)\neq\varnothing$ can be proved in the same way. Thus, there exist $p_1,p_2\in P$, such that $$p_1+\sum_{j=k-u_2+1}^{k-u_1}b_j=p_2+(u_2-u_1)a_1.$$ This implies $$0=\sum_{j=1}^k b_j-\sum_{i=1}^\l a_i=p_2+(u_2-u_1-1)a_1+\sum_{j\in[0,k-u_2]\cup [k-u_1+1,k]}b_j-\sum_{i=2}^\l a_i-p_1,$$ hence $P\cap(\l C-(k-1)C)\neq\varnothing$, contradiction.
By Claim \[cm:3\], we obtain that $\overline{d}(P)\leq\frac{1}{k+\l}$. This means, $\overline{d}(A\setminus P)\geq\varepsilon$. By Lemma \[lem:Sz\], $A\setminus P$ contains a progression $x+d\cdot [m]$, and $d<L$. By the way we construct $P$, there are $s_1,\dots,s_\l$ and $t_1,\dots,t_{k-1}$ in $C$ such that $$x+dm=\sum_{i=1}^\l s_i-\sum_{j=1}^{k-1}t_j.$$ Hence there are $e_1,\dots,e_\l$ and $f_1,\dots,f_{k-1}$ in $A$, such that for every $i\in[\l]$ and $j\in[k-1]$, we have that $e_i\in s_i+D\cdot \ZZ$, and $f_j\in t_j+D\cdot \ZZ$. Let $s=\sum_{i=1}^\l e_i-\sum_{j=1}^{k-1}f_j$, thus $s\in \l A-(k-1)A$, and $s\in x+D\cdot \ZZ$. Since $d\mid D$, we have $s\in x+d\cdot \ZZ$. By Lemma \[lem:5.1\], we have that $$\overline{d}(A)\leq\frac{m+k+\l-2}{(k+\l)m+2(k+\l-2)},$$ and this contradicts (\[eq:density\]).
Now we can prove the main result of this section.
Let $A/N!:=\{a\mid aN!\in A\}$. Thus $\widetilde{d}(A)>0$ implies that $A/N!$ contains a multiple of every natural number. In particular, $A/N!$ is not contained in a periodic $(k,\l)$-sum free set. By Lemma \[lem:5.2\], $\overline{d}(A/N!)\leq\frac{1}{k+\l}$. Observe that $\widetilde{d}(A)=\limsup_{N\to\infty}\overline{d}(A/N!)$, thus $\widetilde{d}(A)\leq\frac{1}{k+\l}$.
Upper bound constructions
=========================
Recall a *Følner sequence* in $(\NN,\cdot)$ is any sequence $\Phi:m\mapsto\Phi_m$ of finite non-empty subsets of $\NN$, such that for every $a\in\NN$, $$\lim_{m\to\infty}\frac{|\Phi_m\triangle (a\cdot \Phi_m)|}{|\Phi_m|}=0.$$
Følner sequence has been used as some good constructions in many additive combinatorics problems, see [@A; @B] for example. In this section, we will show that the sets in Følner sequence will never have large $(k,\l)$-sum free subsets. In fact, we will prove the following theorem.
\[thm:folner\] Let $\Phi=\{\Phi_m\}$ be a Følner sequence in $(\NN,\cdot)$. Suppose there are infinitely many $m$ such that $\Phi_m$ has a $(k,\l)$-sum free set of size at least $\delta|\Phi_m|$ for some positive real number $\delta\leq 1$. Then there exists a $(k,\l)$-sum free set $A\subseteq\NN$ such that $\widetilde{d}(A)\geq\delta$.
Theorem \[thm:one\] (iii) follows easily from Theorem \[thm:folner\] and Theorem \[thm:periodic\].
By passing to a subsequence, we may assume for every $\Phi_m\in\Phi$, there is a $(k,\l)$-sum free set $\phi_m\subseteq \Phi_m$, such that $|\phi_m|/|\Phi_m|\geq\delta$. Let $\beta\NN$ be the collection of ultrafilters, and let $\mathscr{U}\in\beta\NN\setminus\NN$ be a non-principal ultrafilter. Let $^{*}\ZZ=\prod_{m\to\mathscr{U}}\ZZ$ be the ultrapower of $\ZZ$. Let $\Sigma$ be the Loeb $\sigma$-algebra on $^{*}\ZZ$. Let $\mu_L$ be the Loeb measure induced by $\mu_m$, where $\mu_m(X)=|X\cap \Phi_m|/|\Phi_m|$ for every $X\subseteq\ZZ$. Let $\phi=\prod_{m\to\mathscr{U}}\phi_m$ be the internal set. Then by Łoś’s Theorem, $\phi$ is $(k,\l)$-sum free, and $$\mu_L(\phi)=\mathrm{st}\left(\lim_{m\to\mathscr{U}}\mu_m(\phi_m)\right)\geq\delta.$$
\[cm:4\] For every $a\in\NN$, the map $x\mapsto ax$ is $\Sigma$-measurable and $\mu_L$-preserving.
*Proof of Claim \[cm:4\].* Note that $x\mapsto ax$ sends internal sets to internal sets, thus it is $\Sigma$-measurable. For every $X\subseteq\ZZ$, since $$\mu_m(X)-\mu_m(a\cdot X)=\frac{|X\cap \Phi_m|-|(a\cdot X)\cap \Phi_m|}{|\Phi_m|}\leq \frac{|(a\cdot \Phi_m)\triangle \Phi_m|}{|\Phi_m|}\to 0$$ as $m\to \infty$, it preserves the Loeb measure $\mu_L$.
Now we are able to apply the probabilistic argument used in the proof of Proposition \[prop:T1\] on the set $\phi$. For every $x\in {^{*}\ZZ}$, let $A_x:=\{a\in\NN\mid ax\in \phi\}$. Thus $A_x$ is $(k,\l)$-sum free. By Claim \[cm:4\], $\widetilde{d}(A_x)$ is $\Sigma$-measurable on $x$. Suppose $x$ is chosen uniformly at random with respect to the measure $\mu_L$. By Fatou’s Lemma, $$\begin{aligned}
\mathbb{E}(\widetilde{d}(A_x))&\geq\limsup_{N\to\infty}\limsup_{n\to\infty} \mathbb{E}\left(\frac{|A_x\cap(N!\cdot[n])|}{n}\right)\\
&=\limsup_{N\to\infty}\limsup_{n\to\infty}\frac{1}{n}\sum_{j=1}^n\mathbb{P}(jN!x\in \phi).\end{aligned}$$ By Claim \[cm:4\], we have $$\mathbb{E}(\widetilde{d}(A_x))\geq \limsup_{N\to\infty}\limsup_{n\to\infty}\frac{1}{n}\sum_{j=1}^n\mathbb{P}(x\in \phi)=\mu_L(\phi)\geq\delta.$$ Thus by Pigeonhole Principle, there exists a set $A_x\subseteq\NN$ for some $x\in{^*\ZZ}$ such that $\widetilde{d}(A_x)\geq\delta$.
Restricted $(k,\l)$-sum free sets
=================================
In this section, we prove Theorem \[thm:1.2\]. Since restricted $(k,\l)$-sum free can be expressed by first order formula, once we prove the conclusion in Theorem \[thm:periodic\] also works for restricted $(k,\l)$-sum free sets, Theorem \[thm:1.2\] follows by using the same proof in Theorem \[thm:folner\]. We first consider the analogue of Lemma \[lem:5.1\] for restricted $(k,\l)$-sum free sets. The similar argument also works here, with a different and more involved constructions of sets $\C(u)$, $\D(v)$, and $\N(t)$, and a more careful analysis. These new constructions will lead a different structure for the large infinite restricted $(k,\l)$-sum free sets in Lemma \[lem:7.2\], compared to the non-restricted setting.
Let $k,\l$ be positive integers, and $\l<k\leq2\l+1$. Suppose $A\subseteq \NN^{>0}$ be a restricted $(k,\l)$-sum free set. Define $W\subseteq\NN^{>0}$, satisfies that for every $w\in W$, there are $\l$ distinct elements $y_1,\dots,y_\l\in A$, and $k-1$ distinct elements $z_1,\dots,z_{k-1}\in A$, such that $w\neq z_i$ for $i\in[k-1]$, and $w=\sum_{j=1}^\l y_j-\sum_{i=1}^{k-1}z_i$. Let $x,s,d,m$ be integers, such that $s\in W$, $m>k+\l$, $x+d\cdot [m]\subseteq A$, and $s$ is in the coset $x+d\cdot\ZZ$. Then $$\overline{d}(A)\leq\frac{m-2}{(k+\l)(m-k-\l)+2(k+\l-2)}.$$
$s\in W$ implies that $s\notin A$ since $A$ is restricted $(k,\l)$-sum free. We only consider the case when $s<x$. Since $A\cap W=\varnothing$, there is $s_0\in x+d\cdot\ZZ$ such that $s_0\in W$ and $(s_0+d\cdot[m])\cap W=\varnothing$. Thus there are $\l$ distinct elements $a_1,\dots,a_l\in A$, and $k-1$ distinct elements $b_1,\dots,b_{k-1}\in A$, $s_0\neq b_j$ for every $j\in[k-1]$, and $s_0=\sum_{i=1}^\l a_i-\sum_{j=1}^{k-1}b_j$. Let $\mathcal{E}$ consists of $k-1$ distinct elements $e_1,\dots,e_{k-1}\in A$, and all of them are disjoint from $\{a_i\}_{i=1}^\l$, $\{b_j\}_{j=1}^{k-1}$, $\{s_0\}$ and $s_0+d\cdot[m]$. Let $$\label{eq:A'}
A'=A\setminus\Big(\bigcup_{i=1}^\l\{a_i\}\cup\bigcup_{j=1}^{k-1}\{b_j\}\cup \mathcal{E}\cup\{s_0\}\cup (s_0+d\cdot [m])\Big).$$ Observe that $$\label{eq:b_j}
(s_0+d\cdot[m])\cap\{b_j\}_{j=1}^{k-1}=\varnothing,$$since $b_j\in W$ for every $j\in[k-1]$. Let $m'=m-k-\l$, we claim that $$\label{eq:a_i}
(s_0+d\cdot[m'])\cap\{a_i\}_{i=1}^{\l}=\varnothing.$$ Otherwise, suppose there is $r\in[m']$ such that $s_0+rd=a_t$ for some $t\in[\l]$. Then $$x'+\sum_{j=1}^{k-1}b_j=x'+rd+\sum_{j=1,j\neq t}^\l a_j.$$ By taking $x'\in x+d\cdot[0,m-r]$, then both $x'$ and $x'+rd$ are in $A$. Since $m-r\geq k+\l$, there is $\alpha\in[0,m-r]$ such that $x+\alpha d\notin\{b_j\}_{j=1}^{k-1}$, and $x+(\alpha+r)d\notin \{a_i\}_{i=1}^{\l}$. This contradicts that $A$ is restricted $(k,\l)$-sum free.
Let $B=\{b\in A'\mid (b+d\cdot[m'])\cap A\neq\varnothing\},$ and let $$B'=B\setminus\left(\Big(\bigcup_{i=1}^\l\{a_i\}\cup\mathcal{E}\Big)-d\cdot[m']\right).$$ Let $c_0=0$, $c_i=a_i$ when $i\in[\l]$, and $c_j=a_{j-\l}$ when $j\in[\l+1,k-1]$. For $u\in[k-1]$ and $v\in [2,\l]$, let $$\C(u)=B'+\sum_{j=1}^{k-u}b_j+\sum_{i=0}^{u-1}c_i,\quad \D(v)=B'+\sum_{i=0}^{\l-v}a_i+\sum_{j=0}^{v-1}b_j,$$ and $\C(k)=A'+\sum_{i=0}^{k-1}c_i$, $\D(1)=A'+\sum_{i=1}^{\l-1}a_i$. Let $\mathscr{F}$ consists of all $\C(u)$ and $\D(v)$, then Claim \[cm:1\] still holds. In fact, suppose there are $u_1,u_2\in [k]$, $u_1<u_2$ such that $\C(u_1)\cap \C(u_2)\neq\varnothing$ (the case when $\D(v_1)\cap \D(v_2)\neq\varnothing$ is simpler). Then there exist $y_1\in B'$, $y_2\in A'$ such that $$y_1+\sum_{j=k-u_2+1}^{k-u_1}b_j=y_2+\sum_{i=u_1}^{u_2-1}c_i.$$ Let $e_0=0$, and $e_1,\dots,e_{u_2-u_1-1}\in \mathcal{E}$ if $u_2>u_1+1$. If $u_2\leq\l$, we have $$s_0=y_1+\sum_{i\in[0,u_1-1]\cup[u_2,\l]}a_i+\sum_{t=0}^{u_2-u_1-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k-1]}b_j-\sum_{t=0}^{u_2-u_1-1}e_t.$$ If $u_1\geq \l+1$, we get $$s_0=y_1+\sum_{i\in[0,u_1-1-\l]\cup[u_2-\l,\l]}a_i+\sum_{t=0}^{u_2-u_1-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k-1]}b_j-\sum_{t=0}^{u_2-u_1-1}e_t.$$ If $u_1\leq\l$, $u_2\geq\l+1$, and $u_2-u_1+1\leq \l$, $$s_0=y_1+\sum_{i\in[u_2-\l,u_1-1]}a_i+\sum_{t=0}^{u_2-u_1-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k-1]}b_j-\sum_{t=0}^{u_2-u_1-1}e_t.$$ If $u_1\leq\l$, $u_2\geq\l+1$, and $u_2-u_1\geq \l$. Let $e_0=0$, $e_1,\dots,e_{\l-1}\in \mathcal{E}$ if $\l>1$. Thus $$s_0=y_1+\sum_{t=0}^{\l-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k-1]}b_j-\sum_{t=0}^{\l-1}e_t-\sum_{i=u_1}^{u_2-1-\l}a_i.$$ Note that $k\leq 2\l+1$ implies $u_2-1-\l\leq\l$.
In any case, since $y_1\in B$, by (\[eq:A’\]), (\[eq:b\_j\]), and (\[eq:a\_i\]), there is $r\in[m']$ such that $s_0+rd\in W$, which contradicts the assumption of $s_0$. Therefore, $$\label{eq:7.1.1}
(k+\l-2)\overline{d}(B)+2\overline{d}(A)\leq 1,$$ since $\overline{d}(A')=\overline{d}(A)$ and $\overline{d}(B')=\overline{d}(B)$.
We also modify the construction of $\N(t)$ in a similar way. For every $t\in[m']$, let $\N(t)=A'\setminus B+td$. Let $e_0=0$, and $e_1,\dots,e_{k-2}\in \mathcal{E}$ if $k\geq3$. Let $A''=A'\setminus(x+d\cdot[m'])$. Define $$\mathscr{G}=\Big\{\N(t),A',A''+\sum_{i=1}^{\l-1}a_i-x-\sum_{j=0}^{k-2}e_j\ \Big|\ t\in[m']\Big\}$$ Then by using the similar argument, it is easy to see that Claim \[cm:2\] still holds. We omit the details here. We have $$(m-k-\l+2)\overline{d}(A)-(m-k-\l)\overline{d}(B)\leq 1,$$ since $\overline{d}(A'')=\overline{d}(A)$. Together with (\[eq:7.1.1\]), finishes the proof.
Next, we consider the analogue of Lemma \[lem:5.2\] for restricted $(k,\l)$-sum free sets. The structure here is slightly different from the $(k,\l)$-sum free sets.
\[lem:7.2\] Let $\varepsilon>0$ and let $k,\l$ be positive integers with $\l<k\leq2\l+1$. Then there is $D>0$ such that the following holds. Let $A\subseteq\NN^{>0}$ be a restricted $(k,\l)$-sum free set, and $\overline{d}(A)>\frac{1}{k+\l}+\varepsilon$. Then after removing at most $D(2k+\l)$ elements from $A$, it is contained in a periodic restricted $(k,\l)$-sum free set with period $D$.
We pick $m>k+\l$ such that $$\label{eq:7.2.1}
\frac{m-2}{(k+\l)(m-k-\l)+2(k+\l-2)}<\frac{1}{k+\l}+\varepsilon.$$ Let $L=L(\varepsilon,m)$ be as in Lemma \[lem:Sz\], and let $D=L!$. We consider the partition of $\NN$ into cosets: $$\NN=\bigcup_{x\in[D]}x+D\cdot\NN.$$ For every $x\in[D]$, let $\NN_x=x+D\cdot\NN$, and $A_x=A\cap \NN_x$. Let $A'$ be a subset of $A$, obtained by removing $A_x$ from $A$ when $|A_x|<2k+\l$. Hence $\overline{d}(A')=\overline{d}(A)$. Next, we are going to show that $A'$ is contained in a periodic restricted $(k,\l)$-sum free set with period $D$. Suppose this is not the case. Let $$C=\Big(\bigcup_{a\in A'} a+D\cdot\ZZ\Big)\cap\NN^{>0}.$$ Thus $C$ is not restricted $(k,\l)$-sum free. This means, there are $\l$ distinct elements $a_1,\dots,a_\l\in C$ and $k$ distinct elements $b_1,\dots,b_k\in C$, such that $\sum_{i=1}^\l a_i=\sum_{j=1}^k b_j$. Let $P$ be the “$(k,\l)$-sum free” part of $C$, that for every $w\in P$, every $k-1$ distinct elements $y_1,\dots,y_{k-1}\in C\setminus\{w\}$, and every $\l$ distinct elements $z_1,\dots,z_\l\in C$, we have $w+\sum_{i=1}^{k-1}y_i\neq\sum_{j=1}^\l z_j$. Let $e_0=0$, and let $\mathcal{E}$ consists of $k-1$ distinct elements $e_1,\dots,e_{k-1}\in C$, such that $\mathcal{E}$ is disjoint from $\{a_i\}_{i=1}^\l$ and $\{b_j\}_{j=1}^k$. $$P'=P\setminus\Big(\bigcup_{i=1}^\l\{a_i\}\cup\bigcup_{j=1}^k\{b_j\}\cup\mathcal{E}\Big).$$
Set $a_0=b_0=c_0=0$. Let $c_t=a_t$ when $t\in[\l]$, and $c_t=a_{t-\l}$ when $t\in[\l+1,k-1]$. For every $u\in[k]$ and $v\in[\l]$, let $$\MM(u)=P'+\sum_{j=0}^{k-u}b_j+\sum_{t=0}^{u-1}c_t,\qquad \N(v)=P'+\sum_{i=0}^{\l-v}a_i+\sum_{t=0}^{v-1}b_t.$$ Let $\mathscr{F}$ be the collection of all $\MM(u)$ and $\N(v)$. Then elements in $\mathscr{F}$ are pairwise disjoint. Otherwise, suppose there are $u_1,u_2\in[k]$, $u_1<u_2$ such that $\MM(u_1)\cap \MM(u_2)\neq\varnothing$ (the case when $\N(v_1)\cap \N(v_2)\neq\varnothing$ is simpler). Thus, there are $y_1,y_2\in P'$, such that $$y_1+\sum_{k-u_2+1}^{k-u_1}b_j=y_2+\sum_{t=u_1}^{u_2-1}c_t.$$ Let $e_1,\dots,e_{u_2-u_1-1}\in\mathcal{E}$ if $u_2>u_1+1$. If $u_2\leq\l$, we have $$\begin{aligned}
0&=\sum_{i=1}^\l a_i-\sum_{j=1}^k b_j\\
&=y_1+\sum_{i\in[0,u_1-1]\cup[u_2,\l]}a_i+\sum_{t=0}^{u_2-u_1-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k]}b_j-\sum_{t=0}^{u_2-u_1-1}e_t. \end{aligned}$$ If $u_1\geq\l+1$, we have $$0=y_1+\sum_{i\in[0,u_1-1-\l]\cup[u_2-\l,\l]}a_i+\sum_{t=0}^{u_2-u_1-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k]}b_j-\sum_{t=0}^{u_2-u_1-1}e_t.$$ If $u_1\leq\l$, $u_2\geq\l+1$, and $\l\geq u_2-u_1$, we get $$0=y_1+\sum_{i=u_2-\l}^{u_1-1}a_i+\sum_{t=0}^{u_2-u_1-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k]}b_j-\sum_{t=0}^{u_2-u_1-1}e_t.$$ If $u_1\leq\l$, $u_2\geq\l+1$, and $\l<u_2-u_1$. Let $e_1,\dots,e_{\l-1}\in\mathcal{E}$ if $\l>1$, we get $$0=y_1+\sum_{t=0}^{\l-1}e_t-y_2-\sum_{j\in[0,k-u_2]\cup[k-u_1+1,k]}b_j-\sum_{i=u_1}^{u_2-1-\l} a_i-\sum_{t=0}^{\l-1}e_t.$$
In any case, we get a contradiction with the assumption of $P'$ and the fact that $y_2\in P'$. Therefore, $$\overline{d}(P)\leq\frac{1}{k+\l},$$ since $\overline{d}(P')=\overline{d}(P)$. This means, $\overline{d}(A'\setminus P)\geq\varepsilon$. By Lemma \[lem:Sz\], $A'\setminus P$ contains a progression $x+d\cdot [m]$, and $d<L$. By the way we construct $P$, there are $\l$ distinct elements $s_1,\dots,s_\l\in C$ and $k-1$ distinct elements $t_1,\dots,t_{k-1}$ in $C\setminus\{x+m\}$ such that $$x+m=\sum_{i=1}^\l s_i-\sum_{j=1}^{k-1}t_j.$$ By the way we construct $A'$, for every $r\in[D]$, if $|A'\cap\NN_r|>0$, then $|A'\cap\NN_r|\geq 2k+\l$. Thus, there are $\l$ distinct elements $\alpha_1,\dots,\alpha_\l\in A'$ and $k-1$ distinct elements $\beta_1,\dots,\beta_{k-1}\in A'$, such that for every $i\in[\l]$ and $j\in[k-1]$, we have that $\alpha_i\in s_i+D\cdot \ZZ$, and $\beta_j\in t_j+D\cdot \ZZ$. Let $s=\sum_{i=1}^\l \alpha_i-\sum_{j=1}^{k-1}\beta_j$. Note that $|A'\cap\NN_r|\geq 2k+\l$ also implies that there is $r'\in[\l]$, and $M\subseteq \NN^{>0}$, $|M|\geq k$, such that $$\alpha_{r'}+D\cdot M\subseteq A',\quad (\alpha_{r'}+D\cdot M)\cap\bigcup_{i=1}^{\l}\{\alpha_i\}=\varnothing.$$ Thus if $s\cap\{\beta_j\}_{j=1}^{k-1}\neq\varnothing$, then by changing $\alpha_{r'}$ by $\alpha_{r'}+nD$ for some $n\in M$, one can make $s+nD\cap\{\beta_j\}_{j=1}^{k-1}=\varnothing$. Since $d\mid D$, we have $s\in x+d\cdot \ZZ$. By Lemma \[lem:5.1\], we have that $$\overline{d}(A)\leq\frac{m-2}{(k+\l)(m-k-\l)+2(k+\l-2)},$$ and this contradicts (\[eq:7.2.1\]).
Let $A$ be a restricted $(k,\l)$-sum free set, and let $A'$ be a subset of $A$ obtained by removing finitely many elements from $A$. Observe that, if $A'$ is contained in a periodic restricted $(k,\l)$-sum free set, then $A$ cannot contain a multiple of every natural number. Thus, using the same proof in Theorem \[thm:periodic\], we conclude that $\widetilde{d}(A)\leq\frac{1}{k+\l}$ if $A$ is restricted $(k,\l)$-sum free.
Concluding Remarks
==================
In this paper, we study $\M_{(k,\l)}(N)$ and $\widehat{\M}_{(k,\l)}(N)$. In particular, we prove that Conjecture \[conj:kl\] is true for infinitely many $(k,\l)$. While solving Conjecture \[conj:kl\] might not be a realistic target at the moment, the following conjecture for the case when $k-\l\geq2$ might be feasible.
Let $k,\l$ be positive integers and $k\geq\l+2$. Then there is a function $\omega(N)\to\infty$ as $N\to\infty$, such that $$\M_{(k,\l)}(N)\geq \frac{N}{k+\l}+\omega(N).$$
A $(k,\l)$-sum free set is a set forbidding a linear equation $\sum_{i=1}^\l x_i=\sum_{j=1}^k y_j$. Another interesting direction is to consider the analogue problem on sets forbidding a system of linear equations. One of the most interesting problems along this line might be forbidding the projective cubes. Given a multiset $S=\{s_1,\dots,s_d\}$, a *$d$-dimensional projective cube* generated by $S$ is $$\square^d (S):=\Big\{\sum_{i\in I}s_i\ \Big|\ \varnothing\neq I\subseteq [d]\Big\}.$$ A set is *$\square^d$-free* if it does not contain any $d$-dimensional projective cubes as its subsets. Extremal properties of projective cubes have a vast literature, see e.g. [@AF88; @EF90; @GR98; @LW19]. The problem on forbidding $d$-dimensional projective cubes can be viewed as a generalization of sum-free sets in another direction, since a sum-free set is also a $\square^2$-free set. Thus, the following problem is worthwhile to pursue.
Let $d\geq3$ be an integer. Define $$\M_{\square^d}(N):=\inf_{\substack{A\subseteq\NN^{>0}\\|A|=N}}\max_{\substack{B\subseteq A\\B\text{ \emph{is} }\square^d\text{\emph{-free}}}}|B|.$$ Determine $\M_{\square^d}(N)$.
|
---
abstract: 'Forming a three dimensional view of the Universe is a long-standing goal of astronomical observations, and one that becomes increasingly difficult at high redshift. In this paper we discuss how tomography of the intergalactic medium (IGM) at $z\simeq 2.5$ can be used to estimate the redshifts of massive galaxies in a large volume of the Universe based on spectra of galaxies in their background. Our method is based on the fact that hierarchical structure formation leads to a strong dependence of the halo density on large-scale environment. A map of the latter can thus be used to refine our knowledge of the redshifts of halos and the galaxies and AGN which they host. We show that tomographic maps of the IGM at a resolution of $2.5\,h^{-1}$Mpc can determine the redshifts of more than 90 per cent of massive galaxies with redshift uncertainty $\Delta z/(1+z)=0.01$. Higher resolution maps allow such redshift estimation for lower mass galaxies and halos.'
author:
- |
Marcel Schmittfull$^{1}$, Martin White$^{1, 2, 3}$\
$^1$ Berkeley Center for Cosmological Physics, University of California, Berkeley, CA 94720, USA\
$^2$ Department of Astronomy and Department of Physics, University of California, Berkeley, CA 94720, USA\
$^3$ Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 93720, USA
bibliography:
- 'ms.bib'
title: 'Improving photometric redshifts with Ly$\alpha$ tomography'
---
\[firstpage\]
gravitation; large-scale structure of Universe
Introduction {#se:intro}
============
Measuring accurate distance for extragalactic objects is one of the most challenging problems in observational cosmology. Distances are needed in order to properly map large-scale structure, to convert observed into intrinsic properties and to correctly situate objects within the cosmic web. The highest quality distance estimates for high redshift objects come from spectroscopy, which can allow accurate measurements of redshift if suitably high signal-to-noise spectra are available. An estimate of the redshift can also be obtained directly from the photometry [a “photo-$z$”; e.g. see @Hil10; @Dah13; @San14; @Rau15 for recent reviews], though typically with lower precision and a higher catastrophic error rate.
A particularly relevant example is situating objects within the COSMOS field [@Sco07; @Cap07], which has become a premier extragalactic field for a wide variety of studies. Fully exploiting this deep sky map requires information on the redshifts of the objects, and the wide wavelength coverage and numerous bands available in COSMOS leads to very good photo-$z$ performance [near $0.03-0.06$ in $\Delta z/(1+z)$ for bright galaxies at $z=2.5$; @Ilb13; @Lai16]. Even so the implied line-of-sight resolution is relatively poor[^1] ($>100\,h^{-1}$Mpc at $1\,\sigma$ for $z\simeq 2.5$) and accurate redshifts are easier to obtain for some types of galaxies than others.
In this paper we discuss how knowledge of the large-scale environment of galaxies, as traced by fluctuations in neutral hydrogen absorption by the intergalactic medium, can be used to improve photo-$z$ performance. In particular we address how Ly$\alpha$ forest tomography [@Cau08; @Cai14; @Lee14a; @Lee14b] can be used to improve the redshift estimates of massive galaxies, in many cases significantly. We shall take as a test case simulated data such as would be returned by the “COSMOS Lyman-Alpha Mapping And Tomography Observations” (CLAMATO) survey, which will cover 1 deg$^2$ within the COSMOS field. By sampling the IGM absorption along and across sightlines with Mpc spacing, CLAMATO allows tomographic reconstruction of the 3D Ly$\alpha$ forest flux field. These tomographic maps have a [line of sight]{} resolution similar to the average transverse sightline spacing and naturally avoid projection effects or redshift errors. The final CLAMATO survey would provide a tomographic map with a volume of roughly $70\times 70\times 230\,h^{-1}$Mpc at $2.5\,h^{-1}$Mpc resolution. In such a map one can easily locate large overdensities [@Sta15a], voids [@Sta15b] and see the cosmic web [including its filaments and sheets]{} [@LeeWhi16].
[In structure formation from gravitational instability]{} most of the volume of the Universe is underdense, while the halos which host galaxies and AGN preferentially live in overdense regions, with the tendency [being]{} the strongest for the most massive halos. This simple fact allows us to significantly improve the performance of photo-$z$s given knowledge of the large-scale density field as traced by the Ly$\alpha$ forest. Though our method is different in detail, the end goal and basic insight is similar to @Kov10, [@Rak11,]{} @JasWan12 and in particular @Ara15.
The outline of the paper is as follows. In §\[sec:sim\] we introduce the N-body simulation which we use to test and illustrate our method, which we motivate and describe in §\[sec:method\]. Our main results are presented in §\[sec:results\], while §\[sec:future\] describes directions for future research which could further improve the performance of redshift determination using IGM tomography. Finally we conclude in §\[sec:conclusions\].
Simulations {#sec:sim}
===========
In order to demonstrate the [potential]{} of our method we make use of a set of mock catalogs based on N-body simulations. The simulations are described in more detail in @Sta15a [@Sta15b; @Lee16]. Briefly, the mock catalogs are generated from a high-resolution N-body simulation which employed $2560^3$ equal mass ($8.6 \times 10^7 \, h^{-1} M_\odot$) particles in a $256\,h^{-1}$Mpc periodic, cubical box leading to a mean inter-particle spacing of $100\,h^{-1}$kpc. The assumed cosmology was of the flat $\Lambda$CDM family, with $\Omega_{\rm m}\simeq 0.31$, $\Omega_{\rm b} h^2\simeq 0.022$, $h=0.6777$, $n_s=0.9611$, and $\sigma_8 = 0.83$, in agreement with @Planck2013. We shall work throughout with the $z=2.5$ output of the simulation, for which we have halo catalogs and mock Ly$\alpha$ forest data. The Ly$\alpha$ forest was simulated using the fluctuating Gunn-Peterson approximation which is sufficient to model the the large-scale features in the IGM at $z \simeq 2-3$ [see e.g. @MeiWhi01; @Ror13]. [We model the impact of observational noise by smoothing the simulated flux field, noting that larger noise leads to lower resolution of the reconstructed flux maps.]{} [Throughout our paper we account for redshift space distortions by moving halos by their line-of-sight velocity and by using the redshift space Ly$\alpha$ flux generated from the simulations of @Sta15a [@Sta15b; @Lee16]. ]{}
It is not our intention to provide a detailed modeling of galaxy formation within the simulation, but we anticipate that massive galaxies at $z\simeq 2.5$ should live at the centers of the most massive dark matter halos at the same era. [We choose a stellar mass limit of $3\times 10^{10}\,M_\odot$ based on highly complete samples of galaxies in COSMOS [see e.g. @Muz13 Fig. 5]. Using the conversion of @Mos13 to halo mass at $z\simeq 2.5$ and taking into account scatter in the stellar-mass–halo-mass relation suggests taking halos more massive than $10^{12}\,h^{-1}M_\odot$ as a proxy for “massive” galaxies in the COSMOS field.]{}
Method {#sec:method}
======
Hierarchical structure formation in cold dark matter models leads to a strong dependence of the halo mass function upon the large-scale density. In regions where the density is larger than average the number density of massive halos is increased, while in regions where it is smaller than average the number density of massive halos is decreased – often quite dramatically [@ColKai89; @MoWhi96; @TinCon09].
It is easy to see this within the Press-Schechter formalism [@PreSch74; @BBKS; @Pea99] where the number of halos is related to the number of peaks in the smoothed initial density field which exceed a threshold, $\delta_c$. In the presence of a long-wavelength perturbation the small-scale fluctuations need an amplitude of $\delta_c-\delta_{\rm long}$ in order to form a halo. This is more common in overdense regions and less common in underdense regions, with the amplitude of the effect being larger for more massive halos. Indeed, within this peak-background split argument, the large-scale bias of halos is the fractional change in the number density of halos per infinitesimal change in $\delta_{\rm long}$. For the Press-Schechter mass function this bias is $1+(\nu^2-1)/\delta_c$, where $\nu=\delta_c/\sigma(M)$ is the number of $\sigma$ a fluctuation has to be in order to cross $\delta_c$. Since larger halos correspond to a larger smoothing scale and smaller $\sigma$, the more massive halos are more biased and their number density is more sensitive to being in an overdense or underdense region. The halos of interest to us here are all on the exponential tail of the mass function ($\nu\gg 1$) and are highly biased [tracers of the dark matter, typically in clusters.]{}
Since the Ly$\alpha$ flux tracks the large-scale density we expect that massive halos will preferentially reside in regions of negative $\delta_F$, as we see in Table \[tab:fractions\]. Since such extrema occupy only a small fraction of the volume (much of the volume is occupied by voids) we can limit the positions of halos [see also @Kov10; @JasWan12; @Ara15 for related ideas].
Within the Press-Schechter formalism with the peak-background split, the number density of rare, highly biased halos is a Gaussian in the threshold, $\delta_c$. If the flux overdensity is a linearly biased tracer of the matter field, we thus expect the number density of halos to scale as $\exp[-a\delta_F-b\delta_F^2]$.
---------------------- ------- ------- ------- ------- -------
$\delta_F^{\rm lim}$ Vol $ 11$ $ 12$ $ 13$ $ 14$
0.00 44.45 91.9 99.5 100 100
-0.15 6.64 39.0 67.2 99.7 100
-0.30 0.58 6.7 17.1 72.3 100
-0.45 0.03 0.4 1.4 12.9 100
---------------------- ------- ------- ------- ------- -------
: The fraction (in per cent) of the volume and of the halos more massive than $M_{\rm min}$ (in $h^{-1}M_\odot$) that lie in regions of the simulation with $\delta_F$ (smoothed with a Gaussian of $2.5h^{-1}$Mpc) less than $\delta_F^{\rm lim}$. We see that more massive halos live preferentially in regions of lower $\delta_F$, even though those regions occupy a very small fraction of the total volume.[]{data-label="tab:fractions"}
We have estimated the conditional probability of seeing a halo (more massive than $10^{12}\,h^{-1}M_\odot$) given the smoothed Ly$\alpha$ flux directly from the simulations. Along any line of sight we have the smoothed Ly$\alpha$ flux field, $\delta_F$. To estimate the probability that a given galaxy (halo) will lie at a given redshift we use Bayes theorem. Specifically we know the distribution of $\delta_F$ in the simulation, and we know the distribution of $\delta_F$ at the halo locations. Then $$\label{eq:HistRatio}
P({\rm halo}|\delta_F) = \frac{P(\delta_F|{\rm halo})}{P(\delta_F)}$$ In the simulations we can estimate this as the ratio of two histograms: [First, the histogram of the flux density in the vicinity of massive halos; second, the histogram of the flux density over the whole simulated volume.]{} The result is a monotonically decreasing function of $\delta_F$ which can be well fit by a Gaussian (as expected from the arguments above) $$\label{eq:pcondfit}
P({\rm halo}|\delta_F)\propto e^{-a\delta_F-b\delta_F^2}$$ with two parameters ($a$ and $b$) aside from the normalization. To obtain this fit we compute the histogram ratio [(\[eq:HistRatio\])]{} from our simulations using all halos more massive than $10^{12}\,h^{-1}M_\odot$ and the flux density interpolated to a $320^3$ grid and smoothed with a Gaussian kernel $W_R(r)=\exp[-r^2/2R^2]$ with smoothing scale $R=2.5\,h^{-1}$Mpc. This gives $a=32.3$ and $b=37.1$ if we restrict the fit to smoothed flux values $\delta_F>-0.6$ that are present in the simulation.
Our procedure to predict the redshift PDF for galaxies given the flux along their lines of sight is thus as follows. Estimate the smoothed Ly$\alpha$ flux field, as described in @Lee14b [@Sta15a]. Along the line of sight to each galaxy, compute $p(z)=P\left({\rm halo}|\delta_F(z)\right)$ using [Eq. (\[eq:pcondfit\])]{} with the best-fit values for $a$ and $b$ quoted above. This redshift PDF along the line of sight of each galaxy is the final output of our method.
Results {#sec:results}
=======
Fig. \[fig:skewers\] gives three examples of redshift PDFs obtained from the simulation. The three skewers were chosen to be the lines of sight to the most massive, $100^{\rm th}$ most massive and $10,000^{\rm th}$ most massive halos in the simulation, with masses ranging from $10^{14}\,h^{-1}M_\odot$ to $10^{12}\,h^{-1}M_\odot$. The dashed red line shows the flux while the solid black line shows $p(z)$ (normalized to peak at unity in the top panel). The squares and circles show the positions of halos within $0.5$ and $1\,h^{-1}$Mpc (comoving) of the line of sight, with vertical position indicating their mass. The dotted line shows a Gaussian (normalized to peak at unity) centered at the true position of the halo with width $\Delta z/(1+z)=0.03$, comparable to a good photo-$z$. Clearly including prior photo-$z$ information could serve to eliminate possible peaks in the distribution, corresponding to matter overdensities, which are far from the photo-$z$-determined position.
The upper panel shows that massive halos are very well localized by this technique. Such halos are likely to be tracing protocluster regions at $z=2.5$. The middle panel shows that lines of sight can cross multiple overdensities (e.g. crossing multiple filaments or protocluster regions along the line of sight) and thus $p(z)$ can have more than one peak. This is different from the very low $z$ case [@Ara15], where there are very few filaments or overdense regions within the survey to cause confusion. In the lower panel we show the difficulties in finding lower mass halos. As Table \[tab:fractions\] shows, the abundance of lower mass halos is less sensitive to overdensities measured on $2.5\,h^{-1}$Mpc scales and such halos do not produce a large (smoothed) flux decrement. Thus only a very weak peak is seen at the true location of the halo, and similar peaks are seen at many other locations.
In Fig. \[fig:quantiles\] we show a summary statistic for the error rate of the predicted photo-$z$s for all of the lines of sight to halos above $2\times 10^{12}\,h^{-1}M_\odot$ in the simulation at $z=2.5$. The plot shows the fraction of halos whose redshift is correctly predicted, i.e. the fraction of halos whose true redshift is within the credible redshift region determined from the redshift PDF $p(z)$ computed with [Eq. (\[eq:pcondfit\])]{} (corresponding to grey regions in Fig. \[fig:skewers\]). This is shown as a function of the confidence level used to predict these credible redshift regions. As we can see, for halos above $10^{13}\,h^{-1}M_\odot$, 90 per cent of the halos lie within the 50-per-cent-confidence-level credible region (i.e. smaller than $1\,\sigma$ for a Gaussian), while only 3 per cent of randomly placed objects do. The situation is less good for lower mass halos, but still far better than random. To further decrease the error rate, we can choose a more conservative confidence level. For example, if we choose to believe the 90-per-cent-confidence-level credible regions, 99.5 per cent of the halos above $10^{13}\,h^{-1}M_\odot$, 98.5 per cent of the halos above $5\times
10^{12}\,h^{-1}M_\odot$, and 92 per cent of the halos above $2\times
10^{12}\,h^{-1}M_\odot$ are predicted correctly. By selecting regions more likely to host massive halos the flux field is drastically improving photo-$z$s.
Although a more conservative confidence level lowers the error rate of predicted halo redshifts, it comes at the cost of increasing the redshift uncertainty by broadening the credible redshift regions. This can be seen in Fig. \[fig:skewers\], where the conservative 90-per-cent-confidence-level credible regions (light grey) are broader than those for the less conservative 68 per cent confidence level (dark grey). For our fiducial flux smoothing scale of $R=2.5\,h^{-1}$Mpc we find that the total width of the 90 per cent credible region is $30-40\,h^{-1}$Mpc depending on halo mass. [This is roughly the scale of large voids at the high redshifts we are probing.]{} [This redshift uncertainty]{} corresponds to $\Delta
z/(1+\bar z)\simeq 0.01$ at $\bar z=2.5$, which is significantly better than typical photo-$z$ uncertainties (see Section \[se:intro\]).
Further improvements are possible if we had access to the flux smoothed on smaller scales. We demonstrate this in Fig. \[fig:credible\_R\], which shows the redshift uncertainty given by the total width of the credible region as a function of the flux smoothing scale $R$. With high-resolution flux fields smoothed on $R\sim
1\,h^{-1}$Mpc the redshift uncertainty could be reduced to $10\,h^{-1}$Mpc, while the redshift error rate is essentially independent of the flux smoothing scale (not shown). Longer and more expensive observations to obtain such high-resolution flux fields could thus tighten redshift credible regions by a factor of a few, without increasing the error rate of the redshift predictions [see @Lee14a for a discussion of the observational requirements]. [This would be important for mapping the cosmic web and for environmental studies.]{}
A potential caveat of our method is that the Ly$\alpha$ forest only traces about $400\,$Mpc along the line of sight, so that a galaxy could have higher (lower) redshift than the highest (lowest) redshift for which we have any flux information. Our method might then make a false positive mistake by erroneously assigning a redshift inside the region traced by the Ly$\alpha$ forest. In our simulation, fewer than $10$ per cent of the galaxies that are located at higher or lower redshift than the region traced by the Ly$\alpha$ forest would be (erroneously) given a significant redshift PDF inside the region traced by the Ly$\alpha$ forest. Of course this can be reduced further by trusting only the very highest peaks of the redshift PDF.
Future directions {#sec:future}
=================
There are numerous improvements that one could imagine to our simple method. First, we have treated all halos (above $10^{12}\,h^{-1}M_\odot$) in the same manner. If we knew in advance that a galaxy or its hosting halo was particularly massive we could tune our selection to further improve the performance by focusing on the most extremely overdense regions. We tested this in simulations by calibrating the conditional probability to find a halo given some flux using only halos above $10^{12.5}\,h^{-1}M_\odot$ or $10^{13}\,h^{-1}M_\odot$. For the same halos, this tightens the total width of credible redshift intervals, but it also leads to a higher fraction of objects outside the high-confidence interval. Usually the redshift error remains small, and it is outside the confidence interval because that interval shrinks so much. Whether this is an improvement depends on the ultimate application and should be studied more quantitatively in the future.
Second, we have not included information about the shape of the cosmic web in our analysis. @LeeWhi16 showed that Ly$\alpha$ forest tomography is capable of classifying the observed volume into voids, sheets, filaments and knots with high fidelity. This could improve our recovery further, [e.g. by assigning massive galaxies a higher probability to reside in knots rather than filaments or sheets.]{} As a simple first step in that direction, we [characterize different structures of the cosmic web by smoothing the flux on two different smoothing scales, noting that generically both smoothed flux fields should peak for clusters, whereas only the high-resolution flux field might peak if a filament or sheet crosses the sightline. Based on this intuition we generalize our algorithm to ]{} use the conditional probability of finding a halo given the flux smoothed on two different smoothing scales, $\delta_F$ and $\delta_F'$: $$P({\rm halo}|\delta_F,\delta_F') = \frac{P(\delta_F,\delta_F'|{\rm halo})}
{P(\delta_F,\delta_F')}.$$ We compute this in simulations as a ratio of 2d histograms and fit this with a multivariate Gaussian of the form $$P({\rm halo}|\delta_F,\delta_F') \propto
e^{-a\delta_F-a'\delta_F'-b\delta_F^2-b'\delta_F'^2
-c\delta_F\delta_F'}.$$ We find that this slightly improves the total widths of credible redshift intervals and the error rate, at the cost of making the method somewhat more complicated. It is possible that including the measured shear could further improve the performance of the method, though properly characterizing the multivariate probability distribution becomes more difficult. [Another possibility for improvement might be to increase the integration time of spectra, which would give higher resolution along the line of sight at the expense of observing fewer sightlines at fixed total observation time, effectively reducing the resolution perpendicular to the line of sight.]{} We have not attempted to implement these ideas here, as the simplest method is already performing quite well.
Finally, we have treated each galaxy separately whereas one could imagine a simultaneous recovery of the $p(z)$ for the entire population. Such a procedure is closer in spirit to the one in @JasWan12 and may yield dividends. Such a forward modeling approach would allow us to take into account the effects of redshift space distortions, bias and the cosmic web using theoretical expectations of how galaxies and the IGM behave in a model based on gravitational instability. [This method can also be combined with traditional photo-z’s and alternative redshift estimation techniques like the ones presented in @Ara15 and in our paper. ]{}
Another interesting question is related to observational programs for photo-$z$ calibration. Typically, accurate spectra are measured for all galaxies within the field of interest. In contrast, with our method one could imagine measuring only spectra for the brightest background galaxies, and then using tomographic information from the Ly$\alpha$ forest along the line of sight to calibrate photo-$z$s for multiple galaxies along the line of sight – independent of the galaxy spectral type or whether it has prominent spectral lines. From a single background galaxy spectrum one could then calibrate photo-$z$s in the whole $400$Mpc region along the line of sight. [Based on the halo mass dependence of our results this should work best with very massive background halos. ]{}
Conclusions {#sec:conclusions}
===========
Determining the redshift of large numbers of cosmological objects is one of the most difficult problems in observational cosmology. In this paper we have shown that a smoothed map of the intergalactic medium, obtained from spectra of distant galaxies, can be used to improve the redshift accuracy of galaxies and AGN within hundreds of Mpc of the source. This method works because in hierarchical structure formation the halos which host galaxies and AGN preferentially live in the overdense regions with small volume filling fraction.
Our method is extremely simple, once a map of the IGM has been obtained. We use a simple Gaussian form for $p({\rm halo}|\delta_F)$ to transform the observed flux perturbation, $\delta_F$, into a redshift PDF along the line-of-sight to any galaxy. This process can be repeated galaxy by galaxy. Since the mapping between $\delta_F$ and $p(z)$ is monotonic, it is easy to account for errors in the IGM map.
In the form presented herein, the method works best for the most massive galaxies which live in the most massive halos, which tend to form in rare regions of very negative $\delta_F$. For such massive halos the redshift accuracy and error rate are excellent: at our fiducial $2.5\,h^{-1}$Mpc smoothing $\Delta z/(1+z)\simeq 0.01$ and 90 per cent of halos above $10^{13}\,h^{-1}M_\odot$ lie within the 68 per cent credible region. Increasing the resolution of the IGM map reduces the redshift uncertainty, while decreasing the resolution increases the uncertainty. Lower mass halos demand a higher signal-to-noise, less smoothed map of the IGM. This is observationally more challenging.
Our method is extremely straightforward, but does not exhaust the information available in IGM tomography. By using more of the available information about the cosmic web, and by performing a global reconstruction rather than by analyzing galaxies one at a time, we expect to be able to work to lower masses and further improve redshift performance. We defer further developments of a global analysis to future work.
We thank Brice Menard and KG Lee for useful discussions. The simulation, mock surveys, and reconstructions discussed in this work were performed at the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research has made use of NASA’s Astrophysics Data System and of the astro-ph preprint archive at arXiv.org.
\[lastpage\]
[^1]: An fractional redshift uncertainty of $\Delta z/(1+z)$ translates into a distance uncertainty of $\delta\chi=[c(1+z)/H(z)]\ \Delta z/(1+z)$ with $c(1+z)/H(z)\simeq 4\,$Gpc at $z=2.5$.
|
---
abstract: |
Given a family of graphs $\mathcal{F}$, we consider the $\mathcal{F}$-saturation game. In this game, two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph that lies in $\mathcal{F}$. The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We let $\textrm{sat}_g(\mathcal{F};n)$ denote the number of edges that are in the final graph when both players play optimally.
The $\{C_3\}$-saturation game was the first saturation game to be considered, but as of now the order of magnitude of $\textrm{sat}_g(\{C_3\},n)$ remains unknown. We consider a variation of this game. Let $\mathcal{C}_{2k+1}:=\{C_3,\ C_5,\ldots,C_{2k+1}\}$. We prove that $\textrm{sat}_g(\mathcal{C}_{2k+1};n)\ge(\frac{1}{4}-\epsilon_k)n^2+o(n^2)$ for all $k\ge 2$ and that $\textrm{sat}_g(\mathcal{C}_{2k+1};n)\le (\frac{1}{4}-\epsilon'_k)n^2+o(n^2)$ for all $k\ge 4$, with $\epsilon_k<\frac{1}{4}$ and $\epsilon'_k>0$ constants tending to 0 as $k\to \infty$. In addition to this we prove $\textrm{sat}_g(\{C_{2k+1}\};n)\le \frac{4}{27}n^2+o(n^2)$ for all $k\ge 2$, and $\textrm{sat}_g(\mathcal{C}_\infty\setminus C_3;n)\le 2n-2$, where $\mathcal{C}_\infty$ denotes the set of all odd cycles.
author:
- Sam Spiro
bibliography:
- 'Hajnal.bib'
title: Saturation Games for Odd Cycles
---
Introduction
============
Hajnal proposed the following game. Initially $G$ is an empty graph on $n$ vertices. Two players alternate turns adding edges to $G$, with the only restriction being that neither player is allowed to add an edge that would create a triangle. The last player to add an edge wins the game, and the central question is which player wins this game as a function of $n$.
The answer to this problem is known only for small values of $n$, the most recent result being $n=16$ by Gordinowicz and Prałat [@pralat]. A variation of this game was considered by Füredi, Reimer, and Seress [@furedi]. In the modified version of the game, there are two players, Max and Mini, who alternate turns adding edges to an initially empty graph on $n$ vertices with the same rules as in Hajnal’s original triangle-free game. The main difference is that once no more edges can be added to $G$, Max receives a point for every edge in the graph and Mini loses a point for every edge in the graph, with both players trying maximize the number of points they receive at the end. The question is now to figure out how many edges are at the end of the game when both players play optimally.
This game can be generalized. For a family of graphs ${\mathcal{F}}$, we say that a graph $G$ is ${\mathcal{F}}$-saturated if $G$ contains no graph of ${\mathcal{F}}$ as a subgraph, but adding any edge to $G$ would create a subgraph of ${\mathcal{F}}$. The ${\mathcal{F}}$-saturation game consists of two players, Max and Mini, who alternate turns adding edges to an initially empty graph $G$ on $n$ vertices, with the only restriction being that $G$ is never allowed to contain a subgraph that lies in ${\mathcal{F}}$. The game ends when $G$ is ${\mathcal{F}}$-saturated. The payoff for Max is the number of edges in $G$ when the game ends, and Mini’s payoff is the opposite of this. We let ${\mathrm{sat}}_g({\mathcal{F}};n)$ denote the number of edges that the graph in the ${\mathcal{F}}$-saturation game ends with when both players play optimally, and we call this quantity the game ${\mathcal{F}}$-saturation number.
We note that this game, and hence the value of ${\mathrm{sat}}_g({\mathcal{F}};n)$, depends on whether Max or Mini makes the first move of the game, and in general this choice can significantly affect the value of ${\mathrm{sat}}_g({\mathcal{F}};n)$, as is illustrated in [@hefetz]. For simplicity we will only consider the game where Max makes the first move, though we claim that all of our results continue to hold when Mini makes the first move by making small adjustments to our current proofs.
Let $C_k$ denote the cycle of length $k$. The $\{C_3\}$-saturation game was the original saturation game studied in [@furedi], where they proved what is still the best known lower bound of ${\frac{1}{2}}n\log n+o(n\log n)$ for ${\mathrm{sat}}_g(\{C_3\};n)$. Erdős claimed to have proved an upper bound of $n^2/5$ for this game, but this proof has been lost. Recently, Bir[ó]{}, Horn, and Wildstrom [@horn] published the first non-trivial asymptotic upper bound of ${\frac{26}{121}}n^2+o(n^2)$ for ${\mathrm{sat}}_g(\{C_3\};n)$. A number of other results have been obtained for specific choices of ${\mathcal{F}}$, see for example [@westSurv], [@westMatch], and [@lee1]. In addition to this, saturation games have recently been generalized to directed graphs [@lee2], hypergraphs [@patkos], and to avoiding more general graph properties such as $k$-colorability in [@hefetz] and [@keusch].
Main Results
------------
Let ${\mathcal{C}}_{2k+1}:=\{C_3,C_5,\ldots,C_{2k+1}\}$, and let ${\mathcal{C}}_\infty$ denote the set of all odd cycles. Most of this paper will be focused on studying the ${\mathcal{C}}_{2k+1}$-saturation games for $k\ge 2$. The key idea with these games is that by forbidding either player from making $C_{5}$’s, both players can utilize a strategy that keeps the graph essentially bipartite throughout the game. This makes it significantly easier to analyze the correctness of our proposed strategies, and to bound the number of edges that are in the final graph. Our main result is the following upper and lower bound for ${\mathrm{sat}}_g({\mathcal{C}}_{2k+1};n)$ and most values of $k$.
\[T-gen\] For $k\ge 4$, $$\left({\frac{1}{4}}-{\frac{1}{5k^2}}\right)n^2+o(n^2) \le {\mathrm{sat}}_g({\mathcal{C}}_{2k+1};n)\le \left({\frac{1}{4}}-{\frac{1}{ 20^6k^4}}{\right})n^2+o(n^2).$$
We can also obtain a quadratic lower bound for smaller values of $k$.
\[T-5\] For $k\ge 2$, $${\mathrm{sat}}_g({\mathcal{C}}_{2k+1};n)\ge {\frac{6}{25}}n^2+o(n^2).$$
We emphasize that these results do not imply a quadratic lower bound for the triangle-free game. We consider two more saturation games. The first is the game where only one odd cycle is forbidden.
\[T-OneCyc\] For $k\ge 2$, $${\mathrm{sat}}_g(\{C_{2k+1}\};n)\le {\frac{1}{12}}\left(1+{\frac{1}{\ell}}\right)^2n^2+o(n^2),$$ where $\ell=\max(3,{\lfloor \sqrt{2k}\rfloor})$. In particular, ${\mathrm{sat}}_g(\{C_{2k+1}\};n)\le {\frac{4}{27}}n^2+o(n^2)$ for all $k\ge 2$.
We also consider the “complement” of the $\{C_3\}$-saturation game where every odd cycle except $C_3$ is forbidden. It turns out that in this setting the game saturation number is linear.
\[T-Mod3\] $${\frac{5}{4}}n-2\le {\mathrm{sat}}_g({\mathcal{C}}_\infty\setminus \{C_3\};n)\le 2n-2.$$
This result is in sharp contrast to the fact that ${\mathrm{sat}}_g({\mathcal{C}}_\infty;n)={\lfloor {\frac{1}{4}}n^2\rfloor}$, see [@westSurv].
**Notation**. Throughout the paper we let $G^t$ denote the graph in the relevant saturation game after $t$ edges have been added, and we let $e^t$ denote the edge of $G^t$ that is not in $G^{t-1}$. We let $N^t(x)$ denote the neighborhood of $x$ in $G^t$ and let $d^t(x,y)$ denote the distance between $x$ and $y$ in $G^t$. We let $t=\infty$ correspond to the point in time when the graph has become ${\mathcal{F}}$-saturated. If $X^t$ is a real number depending on $t$, we define $\Delta(X^t)=X^t-X^{t-2}$. We let $E(G)$ denote the set of edges of the graph $G$ and let $e(G)=|E(G)|$. We write $G-X$ when $X$ is a vertex, edge, or set of vertices and edges to denote the graph obtained by deleting these vertices and edges from $G$. We omit floor and ceiling signs throughout whenever these are not crucial.
**Organization**. In Section 2 we present a strategy for Max that guarantees that the game ends with at least as many edges as stated in Theorem \[T-5\], which will work the same way for all $k\ge 2$. In Section 3 we modify this strategy to take into account the choice of $k$, and from this we obtain the lower bound of Theorem \[T-gen\]. In Section 4 we present a strategy for Mini that guarantees that the game ends with at most as many edges as the upper bound of Theorem \[T-gen\]. Theorem \[T-OneCyc\] is proven in Section 5. Theorem \[T-Mod3\] is proven in Section 6. We end with some concluding remarks in Section 7.
Proof of Theorem \[T-5\] {#S-Low1}
========================
We wish to construct a strategy for Max in the ${\mathcal{C}}_{2k+1}$-saturation game for $k\ge 2$ such that at the end of each of Max’s turns, $G^t$ is bipartite with parts of roughly the same size. To this end, let $uv$ denote the edge of $G^1$. Let $1<\gamma\le 2$ and $\delta={\frac{1}{\gamma-1}}$. We say that $G^t$ is $\gamma$-good if it satisfies the following four conditions.
- $G^t$ contains exactly one non-trivial connected component, and this component is bipartite with parts $U^t\ni u$ and $V^t\ni v$.
Let $U_0^t=N^t(v)$ (the good vertices), and $U_1^t=U^t\setminus U_0^t$ (the bad vertices). Define an analogous partition for $V^t$.
- Every vertex of $U^t\cup V^t$ is adjacent to a vertex in $U_0^t\cup V_0^t$.
- $b_U^t:=|V_1^t|+(|U^t|-\gamma|V^t|-\delta)\le0$ and $b_V^t:=|U_1^t|+(|V^t|-\gamma|U^t|-\delta)\le 0$.
- $b_U^t+b_V^t\iffalse=(|U_1^t|+|V_1^t|)-(\gamma-1) (|U^t|+|V^t|)-2\delta\fi \le -2$.
We note that $b_U^t\le 0$ implies that, up to an additive constant factor, $|U^t|$ is larger than $|V^t|$ by a multiplicative factor of at most ${\gamma}$. Moreover, if $|U^t|\approx {\gamma}|V^t|$, then $b_U^t\le 0$ guarantees that there are few vertices in $V_1^t$. We note that (2\*) and (4\*) are trivially satisfied if $U_1^t=V_1^t=\emptyset$. An important consequence of being ${\gamma}$-good is the following.
\[L-Bipartite\] Let $t$ be such that $G^{t}$ satisfies (1\*) and (2\*). Then $U^{t+1}$ and $V^{t+1}$ are independent sets for any valid choice of $e^{t+1}$ in the ${\mathcal{C}}_{2k+1}$-saturation game for $k\ge 2$..
$U^{t}$ and $V^{t}$ are independent sets since $G^{t}$ satisfies (1\*). If $v',v''\in V^{t}$, let $u',u''\in U_0^{t}$ be neighbors of $v'$ and $v''$ respectively, noting that such vertices exist since $G^{t}$ satisfies (2\*). Then $$d^{t}(v',v'')\le d^{t}(v',u')+d^{t}(u',u)+d^{t}(u,u'')+d^{t}(u'',v'')=4.$$ Thus having $e^{t+1}=v'v''$ would create either a $C_3$ or a $C_5$ since $d^{t}(v',v'')$ is even, which is forbidden in the ${\mathcal{C}}_{2k+1}$-saturation game for $k\ge 2$. The analysis for $U^{t+1}$ is similar.
We prove Theorem \[T-5\] by first proving the following.
\[P-algLow\] There exists a strategy for Max in the ${\mathcal{C}}_{2k+1}$-saturation game when $k\ge 2$ such that for all odd $t$, whenever $G^{t-1}$ contains an isolated vertex, Max can choose $e^t$ so that $G^{t}$ is ${\frac{3}{2}}$-good.
It is not difficult to see that $G^1$ is ${\frac{3}{2}}$-good. Assume Max has been able to play so that $G^{t-2}$ is ${\frac{3}{2}}$-good with $t$ odd. If $G^{t-1}$ contains no isolated vertices then we’re done, so assume that there exists an isolated vertex $z$ in $G^{t-1}$. Let $e^{t-1}=xy$. We will say that $e^{t-1}$ is an $I$ (Internal) move if $x\in U^{t-2},\ y\in V^{t-2}$, an $O$ (Outside) move if $x,y\notin U^{t-2}\cup V^{t-2}$, an $AU$ (Add to $U$) move if $x\in V^{t-2},\ y\notin U^{t-2}\cup V^{t-2}$, and an $AV$ (Add to $V$) move if $x\in U^{t-2},\ y\notin U^{t-2}\cup V^{t-2}$. Note that an $AU$ move causes $y$ to be added to $U^{t-1}$. Lemma \[L-Bipartite\] shows that $e^{t-1}$ must be one of the four types of moves discussed above (possibly after relabeling $x$ and $y$), so it is enough to show how Max reacts to each of these types of moves.
We note that if we assume that $G^{t-2}$ satisfies (1\*), any vertex not in $U^{t-2}\cup V^{t-2}$ must be isolated. When Max plays, it will always be obvious that (1\*) is maintained, so we will not verify this condition in our analysis. Throughout the rest of this section we write ${\gamma}$ instead of ${\frac{3}{2}}$ whenever our argument continues to hold when $G^{t-2}$ is assumed to be ${\gamma}$-good for any $1<{\gamma}\le 2$, and we will emphasize whenever we need to use ${\gamma}={\frac{3}{2}}$ in our proofs. This will make proving the lower bound of Theorem \[T-gen\] somewhat simpler.
\[Cl-I\] If $e^{t-1}$ is an $I$ move, then Max can play so that $G^t$ is $\gamma$-good.
If there exists $u'\in U^{t-1},\ v'\in V^{t-1}$ with $u'v'\notin G^{t-1}$, then Max adds the edge $u'v'$, and it is not hard to see that in this case $G^t$ is $\gamma$-good. If no such pair of vertices exists, then $U^{t-1}\cup V^{t-1}$ is a complete bipartite graph with, say, $|U^{t-1}|\le |V^{t-1}|$, in which case Max adds the edge $zv$. This gives $\Delta(|U^t|)=1$ and $\Delta(|X^t|)=0$ for every other set of interest. Since $U^{t-1}\cup V^{t-1}$ is a complete bipartite graph (and since Max added no vertex to $U_1^{t-1}\cup V_1^{t-1}$), $U_1^t=V_1^t=\emptyset$, so (2\*) and (4\*) hold. We have $\Delta(b_V^t)= -\gamma\le 0$, and hence $b_V^t\le 0$. If $|U^{t-1}|< \delta= {\frac{1}{\gamma-1}}$, we automatically have $b_U^t\le 0$. Otherwise $|V^{t-1}|\ge |U^{t-1}|\ge{\frac{1}{\gamma-1}}$, which implies $$|U^t|=|U^{t-1}|+1\le |V^{t-1}|+(\gamma-1){\frac{1}{\gamma-1}}\le |V^{t-1}|+(\gamma-1)|V^{t-1}|=\gamma |V^{t-1}|=\gamma|V^t|,$$ so $b_U^t\le 0$ and (3\*) holds, so $G^t$ is $\gamma$-good.
\[Cl-O\] If $e^{t-1}$ is an $O$ move, then Max can play so that $G^t$ is ${\frac{3}{2}}$-good.
Since $b_U^{t-2}+b_V^{t-2}\le -2$, one of $b_U^{t-2}$ or $b_V^{t-2}$ must be at most $-1 $, say $b_U^{t-2}\le -1\le -{\frac{1}{2}}$. In this case, Max adds the edge $xv$ (otherwise Max adds the edge $xu$), which leads to $\Delta(|U^t|)=\Delta(|V^t|)=\Delta(|V_1^t|)=1,\ \Delta(|U_1^t|)=0$. $x$ and $y$ satisfy (2\*), so this continues to hold. We have $\Delta(b_V^t)=1-\gamma\le 0$ and $\Delta(b_U^t)=2-\gamma={\frac{1}{2}}$ when ${\gamma}={\frac{3}{2}}$, so $b_U^t\le 0$ since we assumed $b_U^{t-2}\le -{\frac{1}{2}}$, and thus (3\*) holds. We have $\Delta(b_U^t)+\Delta(b_V^t)=3-2\gamma =0$ since ${\gamma}={\frac{3}{2}}$, so (4\*) holds and $G^t$ is ${\frac{3}{2}}$-good.
In response to $AU$ and $AV$ type moves, Max has to consider the overall “State” of $G^{t-1}$ in order to make his move. To this end, we make the following observations.
\[Cl-Overflow\]
- If $|U^{t-1}|>\gamma |V^{t-1}|+\delta$, then $e^{t-1}$ is an $AU$ move, $V_1^{t-2}=V_1^{t-1}=\emptyset$, and $b_V^{t-2}\le -1$.
- If $|V^{t-1}|>\gamma |U^{t-1}|+\delta$, then $e^{t-1}$ is an $AV$ move, $U_1^{t-2}=U_1^{t-1}=\emptyset$, and $b_U^{t-2}\le -1$.
For (a), assume that $|U^{t-1}|>\gamma |V^{t-1}|+\delta$. Since we assumed that $b_U^{t-2}\le 0$, and in particular that $|U^{t-2}|\le \gamma|V^{t-2}|+\delta$ since $|V_1^{t-2}|$ is non-negative, it must be that $e^{t-1}$ is an $AU$ move, meaning $b_U^{t-1}=b_U^{t-2}+1\le 1$. Thus $|V_1^{t-1}|=(-|U^{t-1}|+\gamma|V^{t-1}|+\delta)+b_U^{t-1}<1$, which implies that $|V_1^{t-1}|=0$ since $|V_1^{t-1}|$ is a non-negative integer, and thus $|V_1^{t-2}|=0$ as nothing is removed from $V_1^{t-2}$ by an $AU$ move. Lastly, $b_U^{t-2}+b_V^{t-2}\le -2$ by (4\*) and $b_U^{t-2}+1=b_U^{t-1}>0$, so $$b_V^{t-2}< b_V^{t-2}+b_U^{t-1}= b_V^{t-2}+b_U^{t-2}+1\le-1.$$ This proves (a), and the analysis for (b) is similar.
\[Cl-States\] If $e^{t-1}$ is an $AU$ move, then the game must be in precisely one of the following three States.
- State N (Nice): $U_1^{t-1}= V_1^{t-1}=\emptyset$, $|U^{t-1}|\le \gamma|V^{t-1}|+\delta$, and $|V^{t-1}|\le \gamma|U^{t-1}|+\delta$.
- State OU (Overflow $U$): $|U^{t-1}|>\gamma|V^{t-1}|+\delta$ and $V_1^{t-1}=\emptyset$.
- State C (Clean-Up): $|U_1^{t-1}\cup V_1^{t-1}|\ne 0$, $|U^{t-1}|\le \gamma|V^{t-1}|+\delta$, and $|V^{t-1}|\le \gamma|U^{t-1}|+\delta$.
observe that we always have $|V^{t-1}|\le \gamma|U^{t-1}|+\delta$ by Claim \[Cl-Overflow\] since we assume that $e^{t-1}$ is not an $AV$ move. Assume that the game is not in State $N$. If $|U^{t-1}|>{\gamma}|V^{t-1}|+\delta$, then Claim \[Cl-Overflow\] shows that $V_1^{t-1}=\emptyset$. If $|U^{t-1}|\le {\gamma}|V^{t-1}|+\delta$, then by assumption of the game not being in State N, we must have $|U_1^{t-1}\cup V_1^{t-1}|\ne 0$, and hence the game is in State C.
By Claim \[Cl-States\], in order to show how Max reacts to an $AU$ move, it is enough to show how he reacts to $AU$ moves that put the game into each of the States defined above.
If $e^{t-1}$ is an $AU$ move putting the game into State N, then Max can play so that $G^t$ is $\gamma$-good.
Max adds the edge $zu$. With this we maintain that $U_1^t=V_1^t=\emptyset$, so (2\*) and (4\*) are satisfied, and we have that $\Delta(|U^t|)=\Delta(|V^t|)=1$, from which one can deduce that (3\*) is satisfied.
\[Cl-OU\] If $e^{t-1}$ is an $AU$ move putting the game into State OU, then Max can play so that $G^t$ is ${\frac{3}{2}}$-good.
Max adds the edge $zu$. This gives $\Delta(|U^t|)=\Delta(|V^t|)=1,\ \Delta(|V_1^t|)=0,\ \Delta(|U_1^t|)\le 1$. Clearly $z$ satisfies condition (2\*), and $y$ does as well since $V_1^{t-1}=V_1^t=\emptyset$ by virtue of the game being in State OU, so (2\*) is maintained. We have $\Delta(b_U^t)=1-\gamma\le 0$ and $\Delta(b_V^t)\le 2-\gamma\le 1$, so $b_V^t\le 0$ by Claim \[Cl-Overflow\], and thus (3\*) is maintained. Lastly, $\Delta(b_U^t)+\Delta(b_V^t)\le 3-2\gamma=0$ since ${\gamma}={\frac{3}{2}}$, so (4\*) is maintained and $G^t$ is ${\frac{3}{2}}$-good.
\[Cl-C\] If $e^{t-1}$ is an $AU$ move putting the game into State C, then Max can play so that $G^t$ is $\gamma$-good.
First assume that $U_1^{t-1}\ne \emptyset$. If $x\in U_1^{t-1}$, then Max adds the edge $xv$, otherwise Max picks an arbitrary $u'\in U_1^{t-1}$ and adds the edge $u'v$. After this we have $\Delta(|V^t|) =1,\ \Delta(|U^t|)=0,\ \Delta(|U_1^t|)=-1,\ \Delta(|V_1^t|)\le 1$. (2\*) is maintained since we made sure that $y$’s neighbor $x$ was in $U_0$. We have $\Delta(b_V^t)=0$ and $\Delta(b_U^t)\le 1-\gamma\le 0$, so (3\*) is maintained. $\Delta(b_V^t)+\Delta(b_U^t)\le 1-\gamma\le 0$, so (4\*) is maintained and $G^t$ is $\gamma$-good.
Now assume $U_1^{t-1}=\emptyset$. In this case Max arbitrarily picks a $v'\in V_1^{t-1}$ and adds the edge $v'u$, giving $\Delta(|V^t|)=1,\ \Delta(|U^t|)=\Delta(|U_1^t|)=0,\ \Delta(|V_1^t|)\le 0$. (2\*) is maintained since $y$ has a neighbor in $U^t=U_0^t$. We automatically have $b_V^t\le 0$ by assumption of us not being in State OV and having $|U_1^t|=|U_1^{t-1}|=0$, and $\Delta(b_U^t)\le -\gamma\le 0$, so (3\*) is maintained. Lastly, $\Delta(b_U^t)+\Delta(b_V^t)\le 1-\gamma\le 0$, so (4\*) is maintained and $G^t$ is $\gamma$-good.
Thus regardless of the State, Max can react as desired to $AU$ moves, and an analogous argument works for $AV$ moves. Thus regardless of what $e^{t-1}$ is, Max can play so that $G^t$ is ${\frac{3}{2}}$-good whenever $G^{t-1}$ contains an isolated vertex.
We emphasize for later that the only two places in this subsection where we explicitly used ${\gamma}={\frac{3}{2}}$ was in the proof of Claim \[Cl-O\] and in verifying (4\*) in the proof of Claim \[Cl-OU\].
Max uses the strategy given by Proposition \[P-algLow\] as long as $G^{t-1}$ contains isolated vertices. Let $T$ denote the smallest even number such that $G^{T}$ contains no isolated vertices. We claim that Max can choose $e^{T+1}$ so that $G^{T+1}$ satisfies (1\*) and (2\*). To this end, let $S$ denote the set of vertices of $G^{T-1}$ that are isolated, noting that $|S|\le 2$. If $G^{T-1}$ does not satisfy (1\*) and (2\*), then there must exist some $x\in S$ such that $G^{T}$ does not contain either $xu$ or $xv$. Since $G^T$ is bipartite by Lemma \[L-Bipartite\], Max can choose $e^{T+1}$ to be one of these edges, causing $G^{T+1}$ to satisfy (1\*) and (2\*). Thus Max can play so that $G^{T+1}$, and hence $G^\infty$, satisfies (1\*) and (2\*). After adding this edge, Max plays arbitrarily.
By Lemma \[L-Bipartite\], $G^\infty$ contains all edges between $U^\infty$ and $V^\infty$. Since $|U^\infty|+|V^\infty|=n$, the product $|U^\infty||V^\infty|$ will be minimized when $||U^\infty|-|V^\infty||$ is as large as possible. Thus in the extremal case we have, say, $|U^\infty|={\frac{3}{2}}|V^\infty|+\delta+2$ since $G^{T-1}$ satisfied (3\*). Then $$n=|U^\infty|+|V^\infty|= {\frac{5}{2}}|V^\infty|+\delta+2\implies |V^\infty|= {\frac{2}{5}}(n-2-\delta),$$ and hence there are at least $|U^\infty||V^\infty|={\frac{6}{25}}(n-2-\delta)^2+\delta |V^\infty|$ edges in $G^\infty$ as desired.
The Lower Bound of Theorem \[T-gen\] {#S-Low2}
====================================
When $k$ is large we can improve upon the strategy of Proposition \[P-algLow\]. Namely, Max will be able to maintain that $G^t$ is ${\gamma}_k$-good with ${\gamma}_k$ such that $\lim_{k\to \infty}{\gamma}_k\to 1$, increasing the total number of edges guaranteed at the end of the game for large $k$. The strategy will be essentially the same as before but with two key differences. First, we will identify our “bad vertices” as those that are distance roughly $k$ from $u$ or $v$, instead of those that are not adjacent to $u$ or $v$. Second, in the proof of Claim \[Cl-C\] we “fixed” a bad vertex $x$ by adding the edge, say, $xv$. In this setting we will instead fix this vertex by adding an edge $zv$ for some “representative” $z$ that lies along a shortest path from $x$ to $v$. The idea with this is that a given $z$ could represent multiple bad vertices, so adding this edge has the potential to make multiple bad vertices sufficiently close to $v$.
We now proceed with our proof of the lower bound of Theorem \[T-gen\]. Throughout this section we fix some $k\ge 3$. Our notation for the proof of Proposition \[P-lowAlg2\] will be very similar to the notation of the proof of Proposition \[P-algLow\], but we emphasize that a large portion of the notation used here will differ somewhat from how it was used before.
Let $uv$ denote the first edge of $G^t$. When the connected component containing $u$ and $v$ in $G^t$ is bipartite, we let $U^t\ni u$ and $V^t\ni v$ denote the parts of this bipartition. Let $\ell=k$ if $k$ is even and $\ell=k+1$ if $k$ is odd. Define $U_0^t=\{u'\in U^t: d^t(u',u)<\ell\}$ and $\tilde{U}_1^t=U^t\setminus U_0^t$. Arbitrarily assign a linear ordering to the vertices of $U^t$. We will say that a vertex $x\in U^t$ is the representative for $u'\in \tilde{U}_1^t$ if
1. $d^t(x,u)=4$.
2. $x$ lies along a shortest path from $u'$ to $u$.
3. $x$ is the minimal vertex (with respect to the ordering of $U^t$) satisfying these properties.
We note that since $k\ge 3$, we have $d^t(x,u)\le d^t(u',u)$, so every $u'\in \tilde{U}_1^t$ has a representative. Define $U_1^t$ to be the set of vertices that are representatives for some vertex of $\tilde{U}_1^t$. Note that $|\tilde{U}_1^t|=0$ iff $|U_1^t|=0$. We similarly define $V_0^t,\ \tilde{V}_1^t$, and $V_1^t$.
For $1<{\gamma}\le 2$ we let $\delta={\frac{1}{{\gamma}-1}}$, and we now say that $G^t$ is ${\gamma}$-good if it satisfies the same four conditions as we had before but with our new definitions for the sets $U_0^t,\ U_1^t,\ V_0^t$, and $V_1^t$ being used. We first prove an analog of Lemma \[L-Bipartite\] using our new definitions.
\[L-AltPart\] Let $t$ be such that $G^{t}$ satisfies (1\*) and (2\*). Then $U^{t+1}$ and $V^{t+1}$ are both independent sets for any valid choice of $e^{t+1}$ in the ${\mathcal{C}}_{2k+1}$-saturation game for $k\ge 3$.
Let $v',v''\in V^t$, and let $u',u''\in U_0^t$ be neighbors of $v'$ and $v''$ respectively, noting that such vertices exist by (2\*). We then have $$d^t(v',v'')\le d^t(v',u')+d^t(u',u)+d^t(u,u'')+d^t(u'',v'')\le 1+(\ell-2)+(\ell-2)+1\le 2k,$$
where we used that $d^t(u',u)<\ell$ is even and that $\ell\le k+1$. Since $d^t(v',v'')$ is even, having $e^{t+1}=v'v''$ would create a $C_{2k'+1}$ for some $k'\le k$, which is forbidden in the ${\mathcal{C}}_{2k+1}$-saturation game. The analysis for $U^t$ is similar.
\[P-lowAlg2\] There exists a strategy for Max in the ${\mathcal{C}}_{2k+1}$-saturation game when $k\ge 3$ such that for all odd $t$, whenever $G^{t-1}$ contains an isolated vertex, Max can add an edge so that $G^{t}$ is ${\gamma}_k$-good with $${\gamma}_k={\frac{4k^{-1}+\sqrt{16k^{-2}+4}}{2}}.$$
$G^1$ is ${\gamma}_k$-good, so inductively assume that Max has been able to play so that $G^{t-2}$ is ${\gamma}_k$-good with $t$ odd. Before describing the strategy, we first make an observation about $\tilde{U}_1^{t-2}$ and $\tilde{V}_1^{t-2}$.
\[Cl-Dist\] $\tilde{U}_1^{t-2}=\{u':d^{t-2}(u,u')=\ell\}$ and $\tilde{V}_1^{t-2}=\{v':d^{t-2}(v,v')=\ell\}$.
Let $u'\in U_0^{t-2}$ be a neighbor of $v'\in \tilde{V}_1^{t-2}$, which exists by (2\*). Then $$d^{t-2}(v',v)\le d^{t-2}(v',u')+d^{t-2}(u',u)+d^{t-2}(u,v)\le 1+(\ell-2)+1=\ell,$$ with the $\ell-2$ term coming from the fact that $d^{t-2}(u',u)$ is even and less than $\ell$. Since $v'\in \tilde{V}_1^{t-2}$ implies $d^{t-2}(v',v)\ge \ell$, the distance must be exactly $\ell$. The analysis for $\tilde{U}_1^{t-2}$ is similar.
We are now ready to describe the strategy we wish to use to prove Proposition \[P-lowAlg2\]. We define $I,\ O,\ AU,$ and $AV$ moves, as well as the States N, OU, and C exactly as we had written before, but we now use our new definitions for the relevant sets. The strategy for Proposition \[P-lowAlg2\] is almost the same strategy as that of Proposition \[P-algLow\] after using our new definitions for the relevant sets, move types, and States. The only change we make is how Max responds to an AU or AV move that brings the game into State C. Namely, before if Mini added the edge $e^{t}=xy$ with $y$ an isolated vertex in $G^{t-1}$, we checked to see if $x$ was in, say, $U_1^t$, in which case we added the edge $xv$. We now instead check if $x\in \tilde{U}_1^t$, and if it is we add the edge $zv$ where $z$ is the representative for $x$. Adding this edge makes $d^t(x,u)$ strictly smaller than $d^{t-2}(x,u)$, so by Claim \[Cl-Dist\] we will have $d^t(x,u)<\ell$ and $y$ will have a neighbor in $U_0^t$, so (2\*) will still hold.
One can verify that with this slight modification all of the previous analysis we did in proving the claims within the proof of Proposition \[P-algLow\] continues to hold. It remains to address the two points in the proof of Proposition \[P-algLow\] where we required $\gamma={\frac{3}{2}}$, namely Claim \[Cl-O\] and Claim \[Cl-OU\].
\[Cl-O’\] If $e^{t-1}$ is an $O$ move, then Max can play so that $G^t$ is ${\gamma}$-good.
Max reacts as he did in Claim \[Cl-O\]. Observe that no vertices are added to $\tilde{U}_1^{t-1}$ or $\tilde{V}_1^{t-1}$. Indeed, the new vertices are within distance $2<\ell$ of $u$ and $v$, so they will both be added to $U_0^{t-1}\cup V_0^{t-1}$. In particular, $\Delta(|U_1^{t}|)=\Delta(|V_1^t|)=0$, and the remaining analysis is straightforward.
In order to deal with $AU$ moves putting the game into State OU, we’ll need the following result showing that $|U_1^t|$ is small.
\[Cl-Rep\] For $k\ge 3$, $|U_1^t|\le 4k^{-1}|U^t|$.
For each $x\in U_1^t$, let $u_x$ denote a vertex that $x$ is the representative for, and let $P_x$ denote the set of vertices that make up a shortest path from $x$ to $u_x$. We claim that $P_x$ and $P_y$ are disjoint if $x\ne y$. Indeed, let $w\in P_x\cap P_y$. If $d^t(w,x)<d^t(w,y)$, then $$d^t(u_y,w)+d^t(w,y)+d^t(y,u)=d^t(u_y,u)< d^t(u_y,w)+d^t(w,x)+d^t(x,u).$$ Since $d^t(y,u)=d^t(x,u)=4$ this implies that $d^t(w,y)<d^t(w,x)$, a contradiction. By using a symmetric argument we see that we must have $d^t(w,x)=d^t(w,y)$. If we had, say, $x<y$ in the ordering of $U^t$, then $y$ could not be the representative for $u_y$ since $u_x$ also satisfies properties (1) and (2) for being a representative for $y$, meaning that $u_y$ is not the minimal vertex satisfying these properties. A similar result occurs if $x>y$. We conclude that the only way $P_x\cap P_y$ could be non-empty is if $x=y$.
For each $x\in U_1^t$, we observe that the number of vertices in $P_x\cap U^t$ is $${\frac{d^t(u_x,x)}{2}}+1={\frac{\ell-4}{2}}+1={\frac{\ell}{2}}-1\ge {\frac{k}{4}},$$ and none of these vertices appear in any other $P_y$ for $x\ne y\in U_1^t$. Since we can associate to each $x\in U_1^t$ a set of at least $k/4$ elements of $U^t$ without any element of $U^t$ appearing in more than one set, we must have $|U_1^t|\le 4k^{-1} |U^t|$.
\[Cl-OU’\] If $e^{t-1}$ is an $AU$ move putting the game into State OU, then Max can play so that $G^t$ is ${\gamma}_k$-good.
Max reacts as in Claim \[Cl-OU\]. As mentioned before, essentially the same proof used to prove Claim \[Cl-OU\] can be used here to show that $G^t$ satisfies (1\*), (2\*) and (3\*), so it remains to verify (4\*). By definition of State OU, we have $V_1^t=\emptyset$ and $|U^{t-1}|>{\gamma}_k |V^{t-1}|+\delta$. The latter implies that $$|V^t|=|V^{t-1}|+1< {\frac{1}{{\gamma}_k}}|U^{t-1}|-{\frac{1}{{\gamma}_k}}\delta+1={\frac{1}{{\gamma}_k}}|U^t|-{\frac{1}{{\gamma}_k}}\delta+1.$$ Combining these observations with Claim \[Cl-Rep\] gives $$\begin{aligned}
b_U^t+b_V^t&=|U_1^t|+(1-{\gamma}_k)(|U^t|+|V^t|)-2\delta \\
&\le 4k^{-1} |U^t|+(1-{\gamma}_k)\left(|U^t|+{\frac{1}{{\gamma}_k}}|U^t|-{\frac{1}{{\gamma}_k}}\delta+1\right)-2\delta\\ &=\left(-{\gamma}_k+4k^{-1}+{\frac{1}{{\gamma}_k}}{\right})|U^t|-\left(1+{\frac{1}{{\gamma}_k}}\right)\delta+(1-{\gamma}_k)\\
&=-\left(1+{\frac{1}{{\gamma}_k}}\right)\delta+(1-{\gamma}_k),\end{aligned}$$ with the last equality coming from the fact that ${\gamma}_k$ is a root of $-x^2+4k^{-1} x+1$. It is not too hard to see that the remaining value is at most $-2$. This shows that (4\*) is maintained in this case, completing the proof.
To finish the proof, observe that Lemma \[L-AltPart\] implies that $e^{t-1}$ must be an $I,\ O,\ AU$, or $AV$ type move. The claims we have proven here together with our work from Proposition \[P-algLow\] shows, provided $G^{t-1}$ contains an isolated vertex, that Max can play so that $G^t$ is ${\gamma}_k$-good regardless of the what type of move $e^{t-1}$ is, proving the statement.
Max follows the strategy of Proposition \[P-lowAlg2\] until there are no isolated vertices remaining, at which point he follows essentially the same strategy as described in the proof of Theorem \[T-5\]. Using the same sort of reasoning, one can show $e(G^\infty)\ge{\frac{{\gamma}_k}{(1+{\gamma}_k)^2}}n^2+o(n^2)$. Note that $${\frac{{\gamma}_k}{(1+{\gamma}_k)^2}}={\frac{k}{8}}\left(\sqrt{k^2+4}-k{\right})\le \left({\frac{1}{4}}-{\frac{1}{5k^2}}\right)$$ when $k\ge 4$. This gives the desired result.
The Upper Bound of Theorem \[T-gen\] {#S-Up}
====================================
Throughout this section we will consider the ${\mathcal{C}}_{2k-1}$-saturation game, so that the smallest odd cycle that can be made is a $C_{2k+1}$, and we will always have $k\ge 5$. We again let $e^1=uv$.
The proof of Theorem \[T-5\] shows that Mini has no strategy in the ${\mathcal{C}}_{2k-1}$-saturation game that guarantees the creation of any odd cycles. On the other hand, the fact that ${\mathrm{sat}}({\mathcal{C}}_\infty;n)={\lfloor {\frac{1}{4}}n^2\rfloor}$ shows that if Mini does not attempt to make any odd cycles, then Max can force the game to end with a complete balanced bipartite graph. Thus in order to get any non-trivial upper bound on ${\mathrm{sat}}({\mathcal{C}}_{2k-1};n)$, Mini must attempt to construct odd cycles, while also making sure that the game does not end as a balanced bipartite graph if she fails to construct any odd cycles.
Because of the obstructions mentioned above, the proof of the upper bound of Theorem \[T-gen\] needs some preparations. The main idea of the proof is that Mini will maintain a number of long, disjoint paths in $G^t$, and then eventually either Mini will be able to join these paths together and create many odd cycles, or the graph will be “almost” bipartite with one side significantly larger than the other.
Paths
-----
We wish to define a special set of disjoint paths $P^t$ in $G^t$, with each path having $v$ as one of its endpoints. To construct this set, we first need to establish some notation. We say that $xy$ is an isolated edge if $d(x)=d(y)=1$. Let $C_x^t$ denote the connected component containing $x$ in $G^t$. We say that $C_x^t$ is good if $v\notin C_x^t$ and if $C_x^t$ is either an isolated vertex or an isolated edge. If $p$ is a path in $G^t$, let ${\overline{C}}_p^t$ denote the connected component in $G^t-\{v\}$ containing $p-\{v\}$.
We start with $P^1=\{uv\}$, and inductively we define $P^t$ based on the following procedure.
- Set $P^{t}:=P^{t-1}$.
- If Mini adds the edge $e^t=xv$ with $C_x^{t-1}$ good, set $P^t:=P^t\cup \{xv\}$.
- If Mini adds the edge $e^t=xw$ with $C_x^{t-1}$ good, and if there exists some $p\in P^{t-1}$ with $p=w\cdots v$, set $P^t:=(P^t\setminus\{p\})\cup \{xp\}$.
- While there exists $p\in P^{t}$ and $x\in {\overline{C}}_p^t$ such that there does not exist a unique path from $x$ to $v$ in $G^t$, set $P^t:=P^t{\setminus}\{p\}$.
We first observe that the ${\overline{C}}_p^t$ components are disjoint from one another.
\[L-S3\] If $p,p'\in P^t$ and $p\ne p'$, then ${\overline{C}}_p^t\ne {\overline{C}}_{p'}^t$.
This certainly holds for $t=1$, so assume this holds up to time $t$. Assume for contradiction that there exists $p,p'\in P^t$ such that $p\ne p$ and ${\overline{C}}_p^t={\overline{C}}_{p'}^t$. If $p,p'\in P^{t-1}$, then we had ${\overline{C}}_p^{t-1}\ne {\overline{C}}_{p'}^{t-1}$ by our inductive hypothesis, which means we must have $e^t=xy$ with $x\in {\overline{C}}_p^{t-1},\ y\in {\overline{C}}_{p'}^{t-1}$. But this implies that $x\in {\overline{C}}_p^t$ has two paths from itself to $v$, namely the path it had in $G^{t-1}$ and the path from $y$ to $v$ in $G^{t-1}$ together with the edge $xy$. Thus $p\notin P^t$ by Step 3, a contradiction.
Thus we must have, say, $p\notin P^{t-1}$. The only way we could have $p\in P^t$ then is if $e^t=xw$ with $C_x^{t-1}$ good and either $w=v$ or $w$ the endpoint of some $p''\in P^{t-1}$. Observe in either case that $p$ is the only new path added to $P^t$ (Step 2 can not be applied twice since our inductive hypothesis shows that there exists at most one path with $w$ as an endpoint). Thus we must have $p'\in P^{t-1}$. Note that ${\overline{C}}_{p'}^{t-1}\ne {\overline{C}}_{p''}^{t-1}$ by our inductive hypothesis, and that ${\overline{C}}_{p'}^{t-1}\ne C_x^{t-1}$ since $p'$ contains $v$ while $C_x^{t-1}$ does not since it is good. Thus $e^t$ does not involve any vertex of ${\overline{C}}_{p'}^{t-1}$, and in particular ${\overline{C}}_{p'}^t={\overline{C}}_{p'}^{t-1}\ne {\overline{C}}_{p''}^{t-1}\cup C_x^{t-1}\cup\{xw\}={\overline{C}}_{p}^t$ as desired.
Another key point with this procedure is that Step 3 does not “interfere” with Steps 1 and 2.
\[L-Interfere\] For any $t$, if Step 1 or 2 adds the path $p$ to $P^t$, then $p\in P^t$. That is, $p$ is not removed by Step 3.
If Step 1 or 2 added the path $p$, then we must have $e^t=xw$ with $C_x^{t-1}$ good. First consider the case $w\ne v$, which means that $p=xp'$ with $p'=w\cdots v$ a path in $P^{t-1}$. Observe that $C_x^{t-1}$ and ${\overline{C}}_{p'}^{t-1}$ are disjoint since $p'$ contains $v$ while $C_x^{t-1}$ does not, and hence ${\overline{C}}_{p}^t={\overline{C}}_{p'}^{t-1}\cup C_x^{t-1}\cup \{xw\}$.
Since $p'$ was not deleted by Step 3, every vertex of ${\overline{C}}_{p'}^{t-1}$ contains a unique path to $v$ in $G^{t-1}$. Since no vertex of $C_x^{t-1}$ contains a path to $v$, every vertex of ${\overline{C}}_{p'}^t$ continues to have a unique path to $v$ in $G^t$. Moreover, every path from $y\in C_x^{t-1}$ to $v$ in $G^t$ must consist of a path from $y$ to $x$ (possibly the empty path) followed by the unique path from $w$ to $v$. Since $C_x^{t-1}$ is acyclic, the path from $y$ to $x$ is unique and we conclude that $p$ is not deleted by Step 3. The proof for the case $w=v$ is similar, and we omit the details.
We will be primarily interested in the endpoints of the paths of $P^t$. To this end, let $D_\ell^t$ denote the set of vertices other than $v$ that are the endpoint of a path of length $\ell$ in $P^t$.
\[L-DL\] Let $\ell,t\ge 1$ and assume $x\in D_\ell^t$.
1. $d^t(x,v)=\ell$.
2. $|D_\ell^{t}|-|D_\ell^{t-1}|\ge -2$, and this difference is 0 whenever Max adds an edge to $G^{t-1}$ that involves an isolated vertex of $G^{t-1}$.
3. For any $t\ge 0$, if $w_1\ne w_2$ are two vertices of $D_k^t$, then choosing $e^{t+1}=w_1w_2$ is a legal move in the ${\mathcal{C}}_{2k-1}$-saturation game.
For (a), by definition of $D_\ell^t$ there exists a path of length $\ell$ from $x$ to $v$, and this is the only path from $x$ to $v$ by Step 3.
For (b), observe that $e^t$ can involve at most two components of $G^{t-1}-\{v\}$, and each component contains at most one path of $P^{t-1}$ by Lemma \[L-S3\]. Any path not in these components will not be modified or deleted by the Steps. Hence $|D_\ell^t|-|D_\ell^{t-1}|\ge -2$. If Max makes an edge involving an isolated vertex, then none of the Steps for modifying $P^t$ apply and we have $D_\ell^t=D_\ell^{t-1}$.
For (c), let $p_i$ denote the path for which $w_i$ is an endpoint for, noting that $p_1\ne p_2$ since $w_1\ne w_2$ and neither are equal to $v$. Since $p_1\ne p_2$, $w_1$ and $w_2$ lie in different components of $G^t-\{v\}$ by Lemma \[L-S3\]. Thus if the edge $w_1w_2$ created a forbidden cycle it would have to involve the vertex $v$. Since $d^t(w_i,v)=k$ for $i=1,2$ by part (a), the smallest cycle that could be formed is a $C_{2k+1}$, which is allowed.
Phases and Phase Transitions
----------------------------
For the rest of the section we assume that $t$ is even. We wish to describe each $G^t$ as belonging to a certain “Phase” which will determine how Mini will play. To do this we will need some additional definitions.
Set $U^0:=\emptyset$ and $V^0:=\{v\}$. Assume $e^t=xy$. If $x\in V^{t-1}$ and $y$ is an isolated vertex, then $U^t:=U^{t-1}\cup \{y\},\ V^t:=V^{t-1}$. If $x\in V^{t-1}$ and $y$ is in an isolated edge $yz$ in $G^{t-1}$, then $U^t:=U^{t-1}\cup \{y\},\ V^t:=V^{t-1}\cup \{z\}$. If $x\in U^{t-1}$ we define $U^t$ and $V^t$ analogously. For any other case of $e^t$, set $U^t:=U^{t-1},\ V^t:=V^{t-1}$.
The idea behind these definitions is that for most of the game, Mini will try and make it so that $G^t$ consists only of isolated vertices, isolated edges, and a bipartite component containing $v$. If she achieves this for all $t\le t'$, then $U^{t'}\cup V^{t'}$ defines a bipartition for the component containing $v$. However, at some point the graph will likely cease to have these properties, in which case $U^t\cup V^t$ will serve as an “approximate” bipartition. Another feature of these sets is that they are compatible with the $D_\ell^t$ sets.
\[L-PV\] If $x\in D_\ell^t$ for some $t$ with $\ell$ odd, then $x\in U^{t'}$ for all $t'\ge t$. If $x\in D_\ell^t$ for some $t$ with $\ell$ even, then $x\in V^{t'}$ for all $t'\ge t$.
Note that the sets $U^t$ and $V^t$ never lose elements, so it is enough to consider the case $t'=t$ and $t$ chosen to be the minimal value such that $x\in D_\ell^t$. First consider the case $\ell=1$. By the Steps used to define $P^t$, having $x\in D_1^t$ and $t$ minimal implies that $e^t=xv$ with $x$ an isolated vertex or isolated edge of $G^{t-1}$. Since $v\in V^{t-1}$, this implies that $x\in U^t$, proving the statement for $\ell=1$.
Now inductively assume that we have proven the statement up to $\ell>1$, and for concreteness we will assume that $\ell$ is odd. Again by the Steps, having $x\in D_\ell^t$ implies that $e^t=xy$ for some $y\in D_{\ell-1}^{t-1}$ and that $x$ is an isolated vertex or isolated edge of $G^{t-1}$. By our inductive hypothesis we have $y\in V^{t-1}$, and hence $x\in U^t$ as desired.
Let $i^t$ denote the number of isolated vertices in $G^t$ and let $c=(1000k^2)^{-1}$. We will say that $G^t$ is in Phase $\ell$ for some $-1\le \ell\le k$ based on the following set of rules.
- $G^0$ is in Phase 0.
- If $G^{t-2}$ is in Phase 0, $|D_1^t|\ge ({\frac{1}{9}} +9c)n$, $|D_2^t|=0$, $||U^t|-|V^t||< cn$, and ${\frac{1}{2}}n\le i^t\le ({\frac{1}{2}}+4c)n$, then $G^t$ is in Phase 1.
- If $G^{t-2}$ is in Phase 1, $|D_2^t|\ge {\frac{1}{9}} n$, $||U^t|-|V^t||< cn$, and $i^t\ge ({\frac{5}{18}}-9c)n$, then $G^t$ is in Phase 2.
- For $2\le \ell<k$, if $G^{t-2}$ is in Phase $\ell$, $|D_{\ell+1}^t|\ge(9(k-\ell-1)+4)cn$, $||U^t|-|V^t||< cn$, and $i^t\ge (8(k-\ell-1)+\sum_{j=\ell+1}^k27(k-j))cn$, then $G^t$ is in Phase $\ell+1$.
- If $G^{t-2}$ is in Phase $\ell$ with $\ell<k$ and $||U^t|-|V^t||\ge cn$, then $G^{t}$ is in Phase $-1$.
- If $G^{t-2}$ is in Phase $\ell$ and if $G^t$ satisfies none of the above situations, then $G^t$ is in Phase $\ell$.
Intuitively, these rules say that if we ever have $||U^t|-|V^t||$ large, then the game enters Phase $-1$ and never leaves it. Otherwise the game goes from Phase $\ell$ to $\ell+1$ if $G^{t}$ contains many isolated vertices and many paths of length $\ell+1$ with $v$ as an endpoint. We note that to leave Phase 0 we additionally require there to not be too many isolated vertices in $G^t$. Our goal will be to show that Mini can play so that the game eventually enters either Phase $-1$ or Phase $k$, and that once the game reaches one of these Phases that Mini can play so that $e(G^\infty)$ is small.
The Beginning of the Game
-------------------------
We will say that a path in $U^t\cup V^t$ is alternating if the vertices in the path alternate being in $U^t$ and $V^t$, and we define $d_a^t(x,y)$ for $x,y\in U^t\cup V^t$ to be the length of the shortest alternating path in $U^t\cup V^t$ from $x$ to $y$, with this value being infinite if no such path exists. We record some observations about this definition as a lemma.
\[L-Alt\] Let $t,\ell\ge 1$.
1. $d_a^t(x,v)$ is even if $x\in V^t$ and odd if $x\in U^t$.
2. If $x\in D_\ell^t$, then $d_a^t(x,v)=d^t(x,v)$.
The proof of (a) is immediate from our definitions. For (b), every $x\in D_\ell^t$ has a unique path (of length $\ell$) from itself to $v$ by Step 3. Further, this path is alternating by Lemma \[L-PV\], so we conclude the result.
We now describe the kind of structure that Mini tries to preserve during the beginning of the game. If $t$ is even and $G^t$ is in Phase $\ell$ with $0\le \ell<k$, we say that $G^t$ is $\ell$-nice if it satisfies the following three conditions.
- $G^t$ contains exactly one non-trivial connected component whose vertices are $U^t\cup V^t$.
- $d_a^t(x,v)\le \ell+2$ for all $x\in U^t\cup V^t$.
- $i^t\ge 3$, and if $\ell\ne 0$ then $D_\ell^t\ge 3$.
\[P-Beg\] Let $k\ge 5$. Mini can play in the ${\mathcal{C}}_{2k-1}$-saturation game so that, whenever $G^{t}$ is in Phase $\ell$ for some even $t\ge 0$ and $0\le \ell<k$, $G^t$ is $\ell$-nice.
For any given $t$ we say that the edge $e^{t-1}$ is of type $I$ if it involves two vertices of $U^{t-2}\cup V^{t-2}$, of type $O$ if it involves two isolated vertices of $G^{t-2}$, of type $AU$ if it involves one isolated vertex of $G^{t-2}$ and one vertex of $V^{t-2}$, of type $AV$ if it involves one isolated vertex of $G^{t-2}$ and one vertex of $U^{t-2}$, and of type $X$ if it is not any of the four types mentioned above. Mini’s strategy is as follows, where we define $D_0^t=\{v\}$ for all $t$ to deal with the case $\ell=0$.
\[SS-Strat\] Let $t$ be such that $G^{t-2}$ is in Phase $\ell$ with $0\le \ell<k$, and assume that Max has just played $e^{t-1}$. If $|D_\ell^{t-1}|=0$, $i^{t-1}=0$, or if $e^{t-1}$ is an $X$ move, Mini plays arbitrarily, and in this case we will say that Mini has forfeited the game.
If Mini does not forfeit, let $y\in D_\ell^{t-1}$ and let $z$ be an isolated vertex of $G^{t-1}$. If $\ell$ is even, Mini plays as follows.
- If $e^{t-1}$ is an $I$ move, Mini plays $yz$.
- If $e^{t-1}=xw$ is an $O$ move, Mini plays $xy$.
- If $e^{t-1}=xu'$ is an $AV$ move with $u'\in U^{t-2}$ and $x\notin U^{t-2}\cup V^{t-2}$, Mini plays $yz$.
- Assume $e^{t-1}=xv'$ is an $AU$ move with $v'\in V^{t-2}$ and $x\notin U^{t-2}\cup V^{t-2}$. If $d_a^{t-1}(v',v)\le \ell$, Mini plays $zv$ (essentially skipping her turn). If $d_a^{t-1}(v',v)>\ell$, let $v'x'v''\cdots v$ be a shortest alternating path from $v'$ to $v$. Then Mini adds the edge $xv''$ if this is a legal move, otherwise she forfeits and plays arbitrarily.
The strategy for $\ell$ odd is exactly the same as the strategy for $\ell$ even, except that the roles of $U^t$ and $V^t$ are reversed throughout and that Mini plays $zu$ in order to “skip her turn” instead of $zv$.
We note that in the $AU$ case with $\ell$ even and with $d_a^{t-1}(v',v)>\ell\ge 0$, the vertex $v''$ always exists. Indeed, $d_a^{t-1}(v',v)$ is even by Lemma \[L-Alt\] and hence $d_a^{t-1}(v',v)\ge 2$, so $v''$ exists. A similar result holds with $\ell$ odd. With this in mind, it is not difficult to see that Mini can always follow Strategy \[SS-Strat\] in the ${\mathcal{C}}_{2k-1}$-saturation game. We would like to further argue that Mini never has to forfeit the game.
\[Cl-F\] If $G^{t-2}$ is in Phase $\ell$ with $0\le \ell<k$ and is $\ell$-nice, then Mini does not forfeit when using Strategy \[SS-Strat\] at time $t$.
We only prove this when $\ell$ is even, the case when $\ell$ is odd being essentially the same. Condition (1-$\ell$) guarantees that $e^{t-1}$ is not of type $X$, and (3-$\ell$) together with Lemma \[L-DL\] guarantees that $|D_\ell^{t-1}|,i^{t-1}\ne 0$ (for $\ell=0$ we use that $|D_0^{t-1}|=1$ for all $t$). Thus the only potential issue is when, say, $e^{t-1}=xv'$ is an $AU$ move with $d_a^{t-1}(v',v)>\ell$. Assume that this is the case.
Let $\tilde{G}^t=G^{t-1}\cup \{xv''\}$, and assume that $\tilde{G}^t$ contains an odd cycle $C$ of length less than $2k+1$. Note that $C$ is not contained in $G^{t-1}$ since we assume that $G^{t-1}$ is a legal state in the ${\mathcal{C}}_{2k-1}$-saturation game, so $xv''$ must be an edge of $C$. This implies that $xv'$ is also an edge of $C$ since $x$ has degree two in $\tilde{G}^t$. If $x'$ is not a vertex of $C$, then let $C'$ be $C$ after replacing the edges $xv'$ and $xv''$ with $x'v'$ and $x'v''$. Then $C'$ is an odd cycle of length less than $2k+1$ in $G^{t-1}$, a contradiction.
Thus $x'$ must be a vertex of $C$, which means that there exists in $G^{t-1}$ a path from $x'$ to $v'$ and a path from $x'$ to $v''$ that lie in $C$, and exactly one of these paths is of even length. Any path of even length from $x'$ to $v'$ in $G^{t-1}$ has length at least $2k$, since otherwise this path together with the edge $x'v'$ would create a forbidden odd cycle in $G^{t-1}$. A similar observation holds for paths of even length from $x'$ to $v''$. We conclude that $C$ has length at least $2k+3$, which is allowed.
With this established, we will prove our result by induction. $G^0$ is in Phase 0 and is 0-nice. From now on we inductively assume that $G^{t-2}$ is in Phase $\ell$ for some $0\le \ell<k$, that $G^{t-2}$ is $\ell$-nice, and further that Mini played according to Strategy \[SS-Strat\] for all even $t'<t$ without forfeiting, which we can assume by Claim \[Cl-F\]. Assume that Mini chooses $e^t$ as prescribed by Strategy \[SS-Strat\]. We note that $G^t$ may not be in Phase $\ell$.
\[Cl-2’\] $G^{t}$ satisfies (1-$\ell$) and (2-$\ell$).
It is not difficult to see that (1-$\ell$) is maintained. In verifying (2-$\ell$), we only consider the case $\ell$ even, the analysis for the odd case being exactly the same but with the roles of $U^t$ and $V^t$ reversed throughout. Observe that in this case we have $y\in V^{t-1}$ by Lemma \[L-PV\] and that $d_a^t(y,v)=\ell$ by Lemma \[L-Alt\].
If $e^{t-1}$ is an $I$ move, then $d_a^t(z,v)=\ell+1$ since $d_a^t(y,v)=\ell$, and no other distances increase, so (2-$\ell$) is maintained.
If $e^{t-1}$ is an $O$ move, observe that $xw$ is an isolated edge in $G^{t-1}$ since $G^{t-2}$ satisfies (1-$\ell$). Because of this and the fact that $y\in V^{t-1}$, we have $x\in U^t$ and $w\in V^t$ by how these sets are defined. We then have $d_a^t(x,v)=\ell+1$ and $d_a^t(w,v)=\ell+2$, so (2-$\ell$) is maintained.
If $e^t$ is an $AV$ move, note that $d_a^{t-1}(u',v)$ is odd by Lemma \[L-Alt\], and hence at most $\ell+1$ by (2-$\ell$) and the fact that $\ell+2$ is even. Thus $d_a^t(x,v)\le \ell+2$, and we also have $d_a^t(z,v)=\ell+1$, so (2-$\ell$) is maintained.
If $e^t$ is an $AU$ move with $d_a^{t-1}(v',v)\le \ell$ then it is not hard to see that (2-$\ell$) is maintained. Otherwise we have $d_a^t(x,v)=\ell+1$ and (2-$\ell$) is maintained.
Let $t_\ell$ denote the smallest even value such that $G^{t_\ell}$ is in Phase $\ell$. Observe that the game can not leave Phase $\ell$ and come back to it at a later time, which means that $G^{t'}$ is in Phase $\ell$ for all even $t'$ with $t_\ell\le t'<t$. For any even $t'$ with $t_\ell\le t'\le t$, define $g^{t'}=|V^{t'}|-|U^{t'}|$ if $\ell$ is even and $g^{t'}=|U^{t'}|-|V^{t'}|$ if $\ell$ is odd. Recall that $\Delta(X^{t'})=X^{t'}-X^{t'-2}$ and that we assumed that Mini used Strategy \[SS-Strat\] without forfeiting for all even $t'$ with $t_\ell<t'\le t$.
\[Cl-Change\] Let $t'$ be even with $t_\ell< t'\le t$. If $\ell>0$ is even, then the following hold.
- If $e^{t'-1}$ is of type $I$: $\Delta(|D_{\ell+1}^{t'}|)\ge -1,\ \Delta(|D_{\ell}^{t'}|)\ge -3,\ \Delta(g^{t'})=-1,\ \Delta(i^{t'})=-1$.
- If $e^{t'-1}$ is of type $O$: $\Delta(|D_{\ell+1}^{t'}|)=1,\ \Delta(|D_{\ell}^{t'}|)=-1,\ \Delta(g^{t'})=0$, $\Delta(i^{t'})=-2$.
- If $e^{t'-1}$ is of type $AV$: $\Delta(|D_{\ell+1}^{t'}|)= 1,\ \Delta(|D_{\ell}^{t'}|)= -1$, $\Delta(g^{t'})= 0$, $\Delta(i^{t'})=-2$.
- If $e^{t'-1}$ is of type $AU$: $\Delta(|D_{\ell+1}^{t'}|)\ge 0,\ \Delta(|D_{\ell}^{t'}|)=0,\ \Delta(g^{t'})\le -1,\ -1\ge \Delta(i^{t'})\ge -2$.
The same results hold for $\ell=0$ when one ignores the $\Delta(|D_{\ell}^{t'}|)$ terms. Analogous results hold for $\ell$ odd by switching the results for $AU$ with those of $AV$.
This is not particularly difficult to verify. In particular one uses Lemma \[L-Interfere\] to show that that $|D_{\ell+1}^{t'-1}|$ increases after Mini responds to $I,\ O$, and $AU$ moves, and Lemma \[L-DL\] to bound the changes in $|D_{\ell}^{t'-2}|$ and $|D_{\ell+1}^{t'-2}|$ after Max plays. We omit the details.
In order to show that $G^t$ satisfies (3-$\ell$), we will use Claim \[Cl-Change\] together with the following claim which shows that $|D_\ell^{t_\ell}|$ and $i^{t_\ell}$ are large.
\[Cl-Init\] We have the following.
- For $\ell\ge 2$ we have $|D_\ell^{t_\ell}|\ge (9(k-\ell)+4)cn$ and $i^{t_\ell}\ge (8(k-\ell)+\sum_{j=\ell}^k 27(k-j))cn$.
- For any $\ell$ with $0\le \ell<k$, $G^{t_\ell}$ satisfies (3-$\ell$) when $n$ is sufficiently large.
Part (a) is immediate for $\ell>2$ by how we defined our Phases. To deal with the case $\ell=2$, recall that $c=(1000k^2)^{-1}$. We have $$|D_{2}^{t_2}|\ge {\frac{1}{9}} n\ge {\frac{9}{1000k}} n\ge {\frac{9k-14}{1000k^2}}n=(9(k-2)+4)cn,$$ $$i^{t_2}\ge \left({\frac{5}{18}}-9c{\right})n \ge {\frac{35}{1000}}n\ge {\frac{8k+27k^2}{1000k^2}}n\ge (8(k-2)+\sum_{j=2}^k 27(k-j))cn.$$
Part (b) for $\ell\ge 2$ follows from (a). The statement is true for $\ell=0$ when $n\ge 3$ and is true for $\ell=1$ when $n\ge 27$ since $|D_{2}^{t_2}|,i^{t_2}\ge {\frac{1}{9}}n$.
\[Cl-Turn\] Let $\ell'$ denote the Phase of $G^t$. If $n$ is sufficiently large, then either $\ell'=-1$, $\ell'=k$, or $G^t$ satisfies (3-$\ell'$).
Note that by the way our Phases were defined, we must have $\ell'=-1,\ \ell$, or $\ell+1$. There’s nothing to prove if $\ell'=-1$, so we can assume that $\ell'\ne -1$. Consider the case that $\ell'=\ell+1$. If $\ell=k-1$ then there’s nothing to prove, and otherwise we have $t=t_{\ell'}$ since $G^{t-2}$ was in Phase $\ell$, which means that $G^t$ satisfies (3-$\ell'$) by Claim \[Cl-Init\]. We can thus assume that $\ell'=\ell$.
Assume first that $\ell$ is even. For any even $t'$ with $t_\ell<t'\le t$, let $r_1^{t'}$ denote the number of even $t''$ with $0<t''\le t'$ such that $e^{t''-1}$ was of type $I$ or $AU$, and similarly define $r_2^{t'}$ to correspond to the number of $O$ and $AV$ moves. Observe that we always have $r_1^{t'}+r_2^{t'}={\frac{1}{2}}(t'-t_\ell)$ since we assume that Mini has used Strategy \[SS-Strat\] up to time $t'$, and hence Max never played an $X$ move by Claim \[Cl-F\]. By Claim \[Cl-Change\] we have that $g^{t'}\le g^{t_\ell} -r_1^{t'}$. Note that $g^{t_\ell}< cn$ and $g^{t'}>-cn$, as otherwise we would have either $G^{t_\ell}$ or $G^{t'}$ in Phase $-1$. Thus we can assume that $r_1^{t'}\le 2cn$, and hence $r_2^{t'}={\frac{1}{2}}(t'-t_\ell)-r_1^{t'}\ge {\frac{1}{2}}(t'-t_\ell)-2cn$. By using this, Claim \[Cl-Change\], and $r_1^{t'}+r_2^{t'}={\frac{1}{2}}(t'-t_\ell)$, we conclude that $$\label{E-D1}
|D_\ell^{t'}|\ge |D_{\ell}^{t_\ell}|-3r_1^{t'}-r_2^{t'}\ge |D_{\ell}^{t_\ell}|-{\frac{1}{2}}(t'-t_\ell)-4cn,$$ $$\label{E-D2}
|D_{\ell+1}^{t'}|\ge |D_{\ell+1}^{t_\ell}|-r_1^{t'}+r_2^{t'}\ge |D_{\ell+1}^{t_\ell}|+{\frac{1}{2}}(t'-t_\ell)-4cn,$$ $$\label{E-I}
i^{t'}\ge i^{t_\ell}-2r_1^{t'}-2r_2^{t'}=i^{t_\ell}-(t'-t_\ell),$$ $$\label{E-I2}
i^{t'}\le i^{t_\ell}-r_1^{t'}-2r_2^{t'}\le i^{t_\ell}-(t'-t_\ell)+4cn.$$
First consider the case $\ell=0$, which implies that $t_0=0,\ |D_1^{t_0}|=0$, and $i^{t_0}=n$. Observe that using Strategy \[SS-Strat\] we have $|D_2^{t'}|=0$ for any $t'$ with $G^{t'}$ in Phase 0. Let $\tilde{t}$ be an even integer such that $({\frac{2}{9}}+26c)n\le \tilde{t}\le {\frac{1}{2}}n$, which exists when $n$ is sufficiently large since ${\frac{1}{2}}-({\frac{2}{9}}+26c)>0$. If $({\frac{2}{9}}+26c)n\le t'\le \tilde{t}\le {\frac{1}{2}}n$, then Equations , , and imply that $|D_1^t|\ge ({\frac{1}{9}}+9c)n$ and ${\frac{1}{2}}n\le i^t\le ({\frac{1}{2}}n+4c)n$. This implies that $G^{t'}$ is in Phase 1 since $G^{t'-2}$ was in Phase 0, but we assumed that $G^{t'}$ was in Phase 0, a contradiction. Thus there exists no even $t'\le t$ in this range, and in particular we must have $t<({\frac{2}{9}}+26c)n\le {\frac{1}{2}}n$. Equation \[E-I\] then implies that $i^t\ge {\frac{1}{2}}n\ge 3$ for $n\ge 6$, so $G^t$ satisfies (3-0).
Now assume $\ell>0$ is even. Let $\tilde{t}$ be such that $\tilde{t}\ge (16+18(k-\ell-1))cn+t_\ell$ with $\tilde{t}$ the smallest even integer with this property. If $\tilde{t}\le t$, then Claim \[Cl-Init\] together with Equations \[E-D2\] and \[E-I\] imply that $$|D_{\ell+1}^{\tilde{t}}|\ge {\frac{1}{2}}(16+18(k-\ell-1))cn-4cn= (4+9(k-\ell-1))cn,$$ and $$\begin{aligned}
i^{\tilde{t}}&\ge (8(k-\ell)+\sum_{j=\ell}^k 27(k-j))cn-(16+18(k-\ell-1))cn-2\\ &= (8(k-\ell-1)+\sum_{j=\ell+1}^k 27(k-j)+9(k-\ell)-8)cn-2\\ &\ge (8(k-\ell-1)+\sum_{j=\ell+1}^k 27(k-j))cn+3,
\end{aligned}$$ for $n$ such that $cn\ge 5$. Thus $G^{\tilde{t}}$ is in Phase $\ell+1$, a contradiction, so $t\le (16+18(k-\ell-1))cn+t_\ell$. Assuming this, one can go through the same calculations as above and conclude that $i^t\ge 3$, and we also have $$|D_\ell^t|\ge (9(k-\ell)+4)cn-{\frac{1}{2}}\cdot (16+18(k-\ell-1))cn-4cn=cn\ge 3$$ when $n$ is sufficiently large.
The analysis for $\ell\ge 3$ odd is essentially the same as above after switching the roles of $AU$ and $AV$ when defining $r_1^{t'}$ and $r_2^{t'}$. The analysis is almost the same for $\ell=1$, except we use $|D_1^{t_1}|\ge {\frac{1}{9}} n+9cn$, $i^{t_1}\ge {\frac{1}{2}}n$, and $\tilde{t}\ge ({\frac{2}{9}}+8c)n+t_1$.
By Claim \[Cl-Turn\] and Claim \[Cl-2’\], if $G^t$ is in Phase $\ell'$ with $\ell'\ne -1,k$, then $G^t$ satisfies (1-$\ell$), (2-$\ell$), and (3-$\ell'$). Since (1-$\ell$) and (2-$\ell$) imply (1-$\ell'$) and (2-$\ell'$) when we have $\ell'=\ell$ or $\ell'=\ell+1$, $G^t$ is $\ell'$-nice and we conclude the result by induction.
Endgame
-------
It remains to describe Mini’s strategy in Phases $-1$ and $k$, and to argue that with this strategy $G^\infty$ will have few edges.
\[P-k\] Assume that $G^{t_k}$ is in Phase $k$ in the ${\mathcal{C}}_{2k-1}$-saturation game with $k\ge 5$ and $t_k$ the minimum value for which this holds. Then Mini can play so that $e(G^{\infty})\le {\frac{1}{4}}(1-c^2)n^2+o(n^2)$.
Mini’s strategy is as follows. Let $t\ge t_k$ be even. If $|D_k^{t-1}|\ge 2$, Mini plays $e^t=xy$ with $x,y\in D_k^{t-1}$, which is a legal move by Lemma \[L-DL\]. Otherwise Mini plays arbitrarily.
Note that $\Delta(|D_k^t|)\ge -4$ by Lemma \[L-DL\]. Since $|D_k^{t_k}|\ge 4cn$, $G^\infty$ will contain at least $cn-1$ $C_{2k+1}$’s which all share a common vertex $v$ and with no other vertices shared between the cycles. Let ${\mathcal{C}}$ denote a set of $cn-1$ such cycles. We wish to show that there are few edges involving vertices of ${\mathcal{C}}$.
\[Cl-cycneigh\] If $C$ is a $C_{2k+1}$ in $G^\infty$, then every vertex of $G^\infty$ has at most two neighbors in $C$.
Let $v'$ be a vertex with neighbors $v_1,v_2,v_3\in C$. Let $d_C(x,y)$ denote the length of the shortest path from $x$ to $y$ using only edges of $C$. First assume $v'\in C$, and without loss of generality that $d_C(v',v_1)=d_C(v',v_2)=1$. Thus $d_C(v',v_3)=\ell$ for some $2\le \ell \le k$. In this case $G^\infty$ contains cycles of length $\ell+1$ and $2k+1-\ell+1$, one of which must be an odd number that is at most $2k-1$ since $k\ge 1$, a contradiction.
Now assume $v'\notin C$. Then $G^\infty$ contains cycles of length $2+d_C(v_i,v_j)$ and $2k+1-d_C(v_i,v_j)+2$ for each $i\ne j$. The only way these values can both be either even or at least $2k+1$ is if $d_C(v_i,v_j)=2$ for all $i\ne j$. This is impossible since $C$ is not a $C_6$.
By Claim \[Cl-cycneigh\], each cycle of ${\mathcal{C}}$ is involved in at most $2n$ edges, so the number of edges involving some vertex of ${\mathcal{C}}$ is at most $2n(cn-1)$. By Mantel’s theorem, the number of edges involving vertices that are not in ${\mathcal{C}}$ is at most
$${\frac{1}{4}}(n-2k(cn-1)-1)^2=\left({\frac{1}{4}}-kc+k^2c^2{\right})n^2+o(n^2).$$
In total then the number of edges in $G^\infty$ is at most $$\left({\frac{1}{4}}-(k-2)c+k^2c^2{\right})n^2+o(n^2).$$ One can verify that $(k-2)c-k^2c^2\ge c^2/4$ for $k\ge5$, from which the result follows.
We now deal with Phase $-1$.
\[P-1\] Assume that $G^{t_{-1}}$ is in Phase $-1$ in the ${\mathcal{C}}_{2k-1}$-saturation game with $k\ge 5$ and $t_{-1}$ the minimum value for which this holds. Then Mini can play so that $e(G^{\infty})\le {\frac{1}{4}}(1-c^2)n^2+o(n^2)$.
Let $\ell<k$ be the number such that $G^{t_{-1}-2}$ was in Phase $\ell$. Mini’s strategy is as follows. If $G^{t-1}$ is connected, Mini plays arbitrarily. Otherwise Mini plays almost the same same way as in Strategy \[SS-Strat\] with parameter $\ell$, with the only change being that Mini does not forfeit if $|D_\ell^t|=0$ and we replace $y$ with $v$ if $\ell$ is even, and we replace $y$ with $u$ if $\ell$ is odd. If one goes back through the analysis of Proposition \[P-Beg\] one can verify that with this strategy, for all even $t\ge t_{-1}$, $G^t$ satisfies (1-$\ell$), (2-$\ell$), and that $||U^{t}|-|V^t||\ge cn-1$.
Let $D'_\ell$ denote the set of vertices in $G^\infty$ that were in $D_\ell^t$ for some $t$. Let $U'=\{u'\in U^\infty: \exists u''\in U^\infty,\ u'u''\in E(G^\infty)\}$ and $V'=\{v'\in V^\infty:\exists v''\in V^\infty,\ v'v''\in E(G^\infty)\}$.
\[Cl-Adj\] No vertex $w\in U'\cup V'$ has $d_a^\infty(w,v)<k-1$. No vertex of $U'$ is adjacent to any vertex of $D_2'$, and no vertex of $V'$ is adjacent to any vertex of $D_1'$.
Let $u_1,u_2\in U'$ be such that $u_1u_2$ is an edge in $G^\infty$, and assume $d_a^\infty(u_1,v)<k-1$. Let $p_i$ denote a shortest alternating paths from $u_i$ to $v$, and let $w$ be the vertex that is in both $p_1$ and $p_2$ and with $d_a^{\infty}(w,v)$ maximal. Observe that the parity of $d_a^\infty(u_i,w)$ is independent of $i$ and that $d_a^\infty(u_2,w)\le d_a^\infty(u_2,v)\le k+1$ since $G^\infty$ satisfies (2-$\ell$). Thus $G^\infty$ contains an odd cycle of length $d_a^\infty(u_1,w)+d_a^\infty(w,u_2)+d(u_1,u_2)<2k+1$, a contradiction.
If $u'v'$ were an edge in $G^\infty$ for some $u'\in U'$ and $v'\in D_2'$, then we would have $d_a^\infty(u_1,v)\le 3<k-1$ when $k\ge 5$, a contradiction to what we have just proven. The proof for $V'$ is analogous.
We emphasize that the only place we truly require $k\ge 5$ is in proving the second part of the above claim.
With the above claim in hand, assume first that $G^\infty$ is not bipartite, or equivalently that $U'\cup V'$ is non-empty since $G^\infty$ satisfies (1-$\ell$). If $t$ is such that $G^t$ was in Phase 1, then Proposition \[P-Beg\] shows that any $x$ with $d_a^t(x,v)>3$ must be isolated in $G^t$. In particular, since $w\in U'\cup V'$ has $d_a^t(w,v)\ge d_a^\infty(w,v)\ge k-1\ge 4$ by Claim \[Cl-Adj\], every such $w$ was isolated during all of Phase 1. Further, $d_a^\infty(w,v)\ge 4$ implies that $U'\cup V'$ will be empty unless $\ell\ge 2$ since Mini maintains (2-$\ell$). In particular, there exists a smallest even number $t_2$ such that $G^{t_2}$ is in Phase 2 with $|D_2^{t_2}|\ge {\frac{1}{9}} n$. Observe then that $G^{t_2-2}$ is in Phase 1, $|D_2^{t_2-2}|\ge {\frac{1}{9}}n-1$, none of the vertices of $D_2^{t_2-2}$ are isolated at time $t_2-2$, and all of these vertices were isolated at the beginning of Phase 1 since we require $|D_2^t|=0$ in order to transition to Phase 1. Since Phase 1 starts with at most $({\frac{1}{2}}+4c)n$ isolated vertices, we conclude that $s:=|U'\cup V'|\le ({\frac{1}{2}}+4c)n-({\frac{1}{9}} n-1)$.
Let $G'$ be the complete bipartite graph with bipartition $U^\infty\cup V^\infty$, where we note that we have $G'=G^\infty$ if $s=0$. The only edges of $G^\infty$ that are not in $G'$ are those contained in $U'\cup V'$, and there are at most ${\frac{1}{4}}s^2+1$ such edges by Mantel’s theorem. However, $G'$ contains all of the edges from $D_2'$ to $U'$ and $D_1'$ to $V'$, and none of these edges are in $G^\infty$ by Claim \[Cl-Adj\]. There are at least $|D_2'||U'|+|D_1'||V'|\ge {\frac{1}{9}} ns$ edges of this kind, so in total $G'$ contains at least ${\frac{1}{9}} ns-{\frac{1}{4}}s^2-1$ more edges than $G^\infty$ does. One can verify that this number is non-negative if $s\ne 0$ and if $n$ is sufficiently large by our bound on $s$ and how we defined $c$. Thus it is enough to give an upper bound for $e(G')=|U^\infty||V^\infty|$. Since $||U^\infty|-|V^\infty||\ge cn-1$, we have $$|U^\infty||V^\infty|\le \left({\frac{1}{2}}n-{\frac{1}{2}}(cn-1){\right})\left({\frac{1}{2}}n+{\frac{1}{2}}(cn-1){\right})={\frac{1}{4}}\left(1-c^2{\right})n^2+o(n^2).$$
We are now ready to finish our proof of Theorem \[T-gen\].
Recall that Theorem \[T-gen\] is stated in terms of the ${\mathcal{C}}_{2k+1}$-saturation game as opposed to the ${\mathcal{C}}_{2k-1}$-saturation game. In particular, $c=(1000(k+1)^2)^{-1}$ and $G^t$ can be in Phase $\ell$ for any $-1\le \ell\le k+1$.
In the ${\mathcal{C}}_{2k+1}$-saturation game, Mini plays as in Proposition \[P-Beg\] as long as $G^t$ is not in Phase $-1$ or $k+1$. If the game ever enters Phase $-1$ or $k+1$, then she plays as in Proposition \[P-1\] or Proposition \[P-k\], respectively.
By Proposition \[P-Beg\], $i^t\ge 3$ whenever $G^t$ is not in Phase $-1$ or $k$. Since $i^\infty=0$, we must have $G^\infty$ be in Phase $-1$ or $k+1$, and in particular Mini must have played according to either Proposition \[P-1\] or Proposition \[P-k\]. By these propositions, $G^\infty$ contains at most $({\frac{1}{4}}-{\frac{1}{4}}c^2)n^2+o(n^2)$ edges. Plugging in $c$ and using $k+1\le 2k$ for $k\ge 4$ gives the result.
Proof of Theorem \[T-OneCyc\]
=============================
In order to prove Theorem \[T-OneCyc\] we need to argue that Mini can create a certain subgraph in $G^t$.
\[L-Sub\] Let $k\ge2$ and $\ell=\max(3,{\lfloor \sqrt{2k}\rfloor})$. There exists a constant $t_0$ such that, for $n$ sufficiently large, Mini can play in the $\{C_{2k+1}\}$-saturation game such that $G^{t_0}$ contains a clique on the vertex set $U=\{u_1,\ldots,u_\ell\}$, and such that there exist $\ell$ vertex disjoint paths of length $k-2$, each with a distinct $u_i$ as its endpoint.
Mini will use the following strategy.
\[SS-Strat2\]
- If there exists some $u_i,u_j$ such that $G^{t-1}$ does not contain the edge $u_iu_j$, Mini plays $e^t=u_iu_j$.
- Let $t'$ be the smallest even value such that $G^{t'-1}$ contains every edge of the form $u_iu_j$. Mini plays $e^{t'}=u_1x_1,\ e^{t'+2}=x_1x_2,\ldots,\ e^{t'+2k-6}= x_{k-3}x_{k-2}$, where $x_i$ is some isolated vertex in $G^{t'+2i-3}$.
- Mini plays $e^{t'+2k-4}=u_2y_1,\ e^{t'+2k-2}=y_1y_2,\ldots,\ e^{t'+4k-12}=y_{k-3}y_{k-2}$, where $y_i$ is some isolated vertex in $G^{t'+2k+2i-7}$.
One defines Step $i$ for all $3\le i\le \ell$ in essentially the same way as Steps 1 and 2.
Observe that if Mini can use the above strategy, then she finishes at time at most $t_0:=2{\ell \choose 2}+2\ell(k-2)$ with the desired structure. Thus its enough to argue that she can indeed use this strategy when $n$ is sufficiently large.
Since $t_0$ is a constant, for $n$ sufficiently large there will always exist an isolated vertex in $G^{t-1}$ for $t\le t_0$, so Mini can play as prescribed by Steps 1 through $\ell$ if the game reaches this point. It remains to argue that Mini can plays as prescribed by Step 0.
If $\ell=3$ then its not too difficult to see that Mini can play as prescribed by Step 0 regardless of what Max does, so assume $\ell=\sqrt{2k}\ge 3$. For any $t\le 2{\ell \choose 2}$, we claim that any choice of $e^t$ is a legal move. Indeed, for any such $t$, $G^{t-1}$ will contain at most $2{\ell \choose 2}-1\le 2\ell^2-1\le 2k-1$ edges, and hence any choice of $e^{t+1}$ will not create a $C_{2k+1}$ in $G^{t+1}$. Thus Mini can play according to Strategy \[SS-Strat2\] and we conclude the result.
Mini first uses the strategy in Lemma \[L-Sub\], making sure that $G^{t_0}$ contains a clique on $U=\{u_1,\ldots,u_\ell\}$ and vertex disjoint paths $\{p_1,\ldots,p_\ell\}$, each of length $k-2$ with $p_j$ starting at $u_j$ and ending at, say, $v_j$. Let $V=\{v_1,\ldots,v_\ell\}$, and let $v^t$ denote a $v_j$ with minimal degree in $G^t$. Let $i^t$ denote the number of isolated vertices of $G^t$.
For all even $t>t_0$, Mini uses the following strategy. If $i^{t-1}=0$, Mini plays arbitrarily. Otherwise if Max plays $xy$ with $x,y$ isolated vertices of $G^{t-1}$, Mini plays $xv^t$ (which is a legal move since this does not create a cycle). Otherwise Mini plays $xv^t$ with $x$ an isolated vertex of $G^{t-1}$.
We wish to bound the number of edges of $G^\infty$ when Mini uses the above strategy. To this end, let $P$ denote the vertices that belong to some $p_j$ (including $u_j$ and $v_j$), let $V_j=N^\infty(v_j)$, let $V=\bigcup V_j$, and let $W=V(G^\infty)\setminus (P\cup V)$. Let $p'_j$ denote $p_j$ but with $p'_j$ treated as a path from $v_j$ to $u_j$. Lastly, for $X,Y{\subseteq}V(G^\infty)$, let $e(X,Y)$ denote the number of edges in $G^\infty$ where one vertex lies in $X$ and the other in $Y$.
\[Cl-Edges\] The following bounds hold.
- $e(P,V(G^\infty))\le \ell(k-1)n=o(n^2)$.
- $e(V,V)\le k\cdot {\frac{2k-1}{2}}n=o(n^2)$.
- $e(W,V)\le ({\frac{1}{2}}(1+{\frac{1}{\ell}})n-|W|)|W|+o(n^2)$.
\(a) follows from $|P|= \ell(k-1)$.
For (b), we first claim that $e(V_j,V_j)\le {\frac{2k-1}{2}}n$. Indeed if this were not the case, then by the Erdős-Gallai Theorem there would exist a path of length $2k$ in $V_j$. Since $v_j$ is adjacent to the two endpoints of this path, this would imply that $G^\infty$ contains a $C_{2k+1}$, a contradiction. We also claim that $e(V_j,V_{j'})=0$ whenever $j\ne j'$. Indeed assume that $G^{\infty}$ contained the edge $w_jw_{j'}$ with $w_j\in V_j,\ w_{j'}\in V_{j'}$. Then for any $r\ne j,j'$ (and such an $r$ exists since $\ell\ge 3$), $G^\infty$ would contain the cycle $w_jp_j' u_{r} p_{j'} w_{j'}$. But this is a $C_{2k+1}$, a contradiction. We conclude that (b) holds.
For (c), we claim that, for any $w\in W$, we have $e(\{w\},V_j)\ne 0$ for at most one $i$. Indeed assume $G^{\infty}$ contained the edges $ww_j$ and $ww_{j'}$ with $w_j\in V_j,\ w_{j'}\in V_{j'},\ j\ne j'$. Then $G^\infty$ would contain the cycle $ww_jp_j'p_{j'}w_{j'}$, which is a $C_{2k+1}$, a contradiction. It follows that $e(W,V)\le |W|\max(|V_j|)$, so it will be enough to bound $\max(|V_j|)$.
Note that $i^{t_0}\ge n-2t_0$ and $\Delta(i^{t})\ge -2$ for all even $t\ge t_0+2$ by the way the strategy was constructed. It follows that there are at least $n/2+O(1)$ values of $t$ with $i^{t-1}\ne 0$, and hence Mini adds an edge of the form $xv^t$ for at least this many values of $t$. Thus Mini ensures that each of the $\ell$ vertices $v_j$ have at least $ {\frac{n}{2\ell}}+O(1)$ neighbors in $G^\infty$, and hence $|V_j|\ge {\frac{n}{2\ell}}+O(1)$ for all $i$. Thus $$|V_j|=n-\sum_{j'\ne i} |V_{j'}|-|W|-|P|\le \left(1-{\frac{\ell-1}{2\ell}}{\right})n-|W|+O(1)$$ for all $j$. Plugging this bound into $e(W,\bigcup V_j)\le |W|\max(|V_j|)$ and using $(1-{\frac{\ell-1}{2\ell}})={\frac{1}{2}}(1+{\frac{1}{\ell}})$ shows that (c) holds.
By Claim \[Cl-Edges\] and Mantel’s theorem, we have $$\begin{aligned}
e(G^\infty)&\le e(W,W)+e(W,V)+o(n^2)\\ &\le {\frac{1}{4}}|W|^2+\left({\frac{1}{2}}(1+{\frac{1}{\ell}})n-|W|\right)|W|+o(n^2).
\end{aligned}$$ This value is maximized when $|W|={\frac{1}{3}}(1+{\frac{1}{\ell}})n$, giving an upper bound of ${\frac{1}{12}}(1+{\frac{1}{\ell}})^2n^2+o(n^2)$ as desired.
Proof of Theorem \[T-Mod3\]
===========================
We will say that a vertex $v$ is good if all but at most one edge incident to $v$ is contained in a triangle. We will say that a graph $G$ is $k$-good if there exists a set of edges $B(G)$ with $|B(G)|\le k$ such that every vertex of $G-B(G)$ is good. Observe that if $G$ is $k$-good and $G'$ is $G$ plus an edge, then $G'$ is $(k+1)$-good.
\[P-3Alg\] Let M denote either Mini or Max. We will say that $t$ is M-parity if either $t$ is even and M is Mini, or if $t$ is odd and M is Max. There exists a strategy for M in the $({\mathcal{C}}_{\infty}{\setminus}\{C_3\})$-saturation game such that for all M-parity $t$, either $G^{t-1}$ is $({\mathcal{C}}_{\infty}{\setminus}\{C_3\})$-saturated or $G^t$ is 1-good.
To prove this we will need the following lemma concerning the structure of 2-good graphs.
\[L-Good\] Let $G$ be a 2-good graph that contains no $C_{2k+1}$ for any $k\ge 2$. Then $G$ contains no $C_{\ell}$ for any $\ell \ge 5$.
Assume for contradiction that ther exists an even cycle $C$ in $G$ of length $2k$ with $k\ge 3$ on the vertex set $\{v_1,\ldots,v_{2k}\}$, and let $C'=C- B(G)$. Since $k\ge 3$, there exists an $i$ such that $C'$ contains the edges $v_{i-1}v_i$ and $v_iv_{i+1}$. Since these edges are in $G-B(G)$, at least one of these edges is in a triangle, say $v_iv_{i+1}w$ is a triangle in $G$. If $w$ is not in $C$, then $v_1v_2\cdots v_i wv_{i+1}\cdots v_{2k}$ is a $C_{2k+1}$ in $G$, a contradiction. Thus $w=v_j$ for some $j\ne i,i+1$.
Note that $v_j\ne i+2,i+3$. Indeed if, say, $j=i+2$, then $v_1v_2\cdots v_i v_{i+2}\cdots v_{2k}$ would be a $C_{2k-1}$ in $G$, a contradiction. A similar result holds if $j=i+3$. Observe that $G$ contains the cycles $v_iv_{i+1}\cdots v_j$ and $v_{i+1}v_{i+2}\cdots v_j$. One of these cycles must have odd parity with length at least 5 since $j\ne i+2,i+3$, a contradiction. We conclude that $G$ contains no $C_{2k}$ with $k\ge 3$, proving the result.
$G^0$ and $G^1$ are 1-good, so inductively assume that M has been able to play so that $G^{t-2}$ is 1-good where $t$ is M-parity. If $G^{t-1}$ is saturated then the game is over and M does not play anything, so assume this is not the case. If $G^{t-1}$ is 0-good, then M plays $e^t$ arbitrarily and $G^t$ will be 1-good.
Now assume that $G^{t-1}$ is not 0-good. That is, there exist edges $v_1x$ and $v_2x$ with $v_1\ne v_2$ such that neither of these edges are contained in triangles. We claim that adding $e^t=v_1v_2$ is a legal move. If it were not, then there must exist a path $P$ of length $2k$ with $k\ge2$ from $v_1$ to $v_2$ in $G^{t-1}$. If $x$ is not a vertex of $P$, then $G^{t-1}$ contains the cycle formed by taking $P$ and adding the edges $xv_1$ and $xv_2$, which is a $C_{2k+2}$. Since inductively $G^{t-2}$ is 1-good, $G^{t-1}$ is 2-good, and hence does not contain such a $C_{2k+2}$ by Lemma \[L-Good\]. Thus $x$ must be a vertex of $P$. Let $P_i$ denote the path from $v_i$ to $x$ in $P$, and let $k_i$ denote the length of $P_i$.
$G^{t-1}$ contains a $C_{k_i+1}$, namely by taking $P_i$ together with the edge $xv_i$. Thus $k_i\le 3$ by Lemma \[L-Good\]. Also $k_i\ne 2$, since this would contradict $xv_i$ not being contained in a triangle. Since $k_1+k_2=2k\ge 4$, we must have, say, $k_1=3$. Let $C=v_1abx$ be the 4-cycle formed from $P_1$ and $xv_1$. If, say, $ab$ were contained in a triangle $abc$, then we must have $c=v_1$ or $c=x$, as otherwise $v_1acbx$ defines a $C_5$ in $G^{t-1}$. But if $c=v_1$ or $x$, then $v_1x$ is contained in a triangle, a contradiction. A similar analysis shows that no edge of $C$ is contained in a triangle. This is only possible if $B(G^{t-1})$ consists of two edges of $C$ that are not both incident to $x$, as otherwise one of $ab$ and $v_1a$ would be contained in a triangle. In particular, two of the edges $\{xv_1,xv_2,xb\}$ are not in $B(G^{t-1})$, and we conclude that at least one of these edges must be contained in a triangle. But we have assumed that none of these edges are in triangles, a contradiction. We conclude that $v_1v_2$ is a legal move to play.
Note that at least one of the edges $xv_1$ and $xv_2$ must be in $B(G^{t-1})$, as otherwise $G^{t-1}- B(G^{t-1})$ would not have all good vertices (namely, $x$ would not be a good vertex). Since $v_1x$, $v_2x$, and the new edge $v_1v_2$ are contained in a triangle of $G^t$, the set $B(G^t):=B(G^{t-1}){\setminus}\{v_1x,v_2x\}$ shows that $G^t$ is 1-good as desired.
It remains to bound how many edges $G^\infty$ will have after a player uses the strategy of Proposition \[P-3Alg\].
\[L-3Up\] $\operatorname*{ex}(n,\{C_5,C_6,\ldots\})\le 2n-2$, and this is sharp whenever $n=3k+1$ for some $k\ge 1$.
The statement is trivially true for $n\le 4$, so assume we have proven the statement up to $n>4$. Let $G$ be an extremal $n$-vertex graph and assume that $e(G)\ge 2n-1$. If $G$ contains a vertex $x$ with $d(x)\le 2$, then $G'=G-\{x\}$ is an $(n-1)$-vertex graph with $e(G')\ge 2(n-1)-1$. By our inductive hypothesis, $G'$ contains a large cycle, and hence the same is true for $G$. Thus we can assume that every vertex of $G$ has degree at least 3.
We can assume that $G$ is connected, as adding an edge between two components of $G$ would increase $e(G)$ without creating any cycles. Let $T$ be a depth-first-search tree of $G$. For any $x\in T$, let $x_1$ denote the parent of $x$ in $T$, and recursively define $x_i=(x_{i-1})_1$ for $i\ge 2$. Observe that if $xy$ is an edge in $G$, then either $y=x_i$ or $x=y_i$ with $i=1,\ 2$, or 3. Indeed, assume without loss of generality that $y$ was discovered before $x$ when constructing $T$. Observe that the subtree of $T$ with $y$ as a root will contain every neighbor of $y$ that has not been discovered before $y$. In particular, this subtree will contain $x$, and we will have $y=x_i$ where $i$ is the depth of $x$ in this subtree. Further, we must have $i=1,\ 2,$ or 3, as otherwise $G$ would contain a $C_k$ with $k\ge 5$.
Let $x$ be a vertex of maximum depth in $T$, which in particular means that $x$ is a leaf in $T$. We wish to show that $\{x,x_1,x_2,x_3\}$ induces a $K_4$ in $G$ and that $d(x)=d(x_1)=d(x_2)=3$. To this end, we will say that a vertex $y$ has label $i$ with $i>1$ if $y_iy$ is an edge in $G$, and we let $S(y)$ denote the set of vertices $z\ne y$ with $z_1=y_1$. Note that by our above argument, no vertex can have label $i>3$.
Since we assumed $d(x)\ge 3$ and since $x$ only has one neighbor in $T$, $x$ must have both label 2 and 3. We claim that $S(x)=\emptyset$. Indeed, if there exists some $y\in S(x)$, then $y$ would also be a leaf (since $x$ is a vertex of maximum depth), so it would also have to have label 3, but then $G$ would contain the cycle $xx_2x_3yx_1$, a contradiction. Thus we must have $S(x)=\emptyset$. Since $d(x_1)\ge 3$, and since $S(x)=\emptyset$, $x_1$ must have at least one of label 2 or 3, but if it had label 3 then $G$ would contain the cycle $xx_2x_3x_4x_1$, so $x_1$ only has label 2. If $x_2$ had label 2, then $G$ would contain the cycle $xx_1x_2x_4x_3$, and a similar result holds if $x_2$ had label 3. Thus $x_2$ does not have any label. We claim that $S(x_1)=\emptyset$. Indeed if we had $y\in S(x_1)$ with $y$ a leaf, then $y$ must have label 2 and $G$ would contain the cycle $yx_3xx_1x_2$. Otherwise, $y$ would have a child $z$ which is a leaf (since it is at the same depth as $x$) and hence must have label 3, which means that $G$ contains the cycle $zx_3xx_1x_2y$. Thus $S(x_1)=\emptyset$, proving our claims about the vertices $x,x_1,x_2,x_3$.
Let $G'=G-\{x,x_1,x_2\}$. By our above analysis, $G'$ is an $(n-3)$-vertex graph with $e(G')=e(G)-6\ge=2(n-3)-1$, so by the induction hypothesis $G'$, and hence $G$, contains a large cycle, proving the desired bound.
To see that this bound is sharp, let $G$ be the graph obtained by taking $k$ disjoint triangles and then adding an additional vertex which is made adjacent to every other vertex. Then $G$ has $3k+1$ vertices, $6k$ edges, and contains no cycle larger than a $C_4$.
We note that the above proof can easily be modified to characterize all extremal graphs, though we have no need for this here.
\[L-3Low\] Let $G$ be an $n$-vertex graph which is 2-good and $({\mathcal{C}}_\infty{\setminus}\{C_3\})$-saturated. We have $e(G)\ge {\frac{5}{4}}n-3$.
Let $G$ be as in the hypothesis. $G$ being $({\mathcal{C}}_\infty{\setminus}\{C_3\})$-saturated in particular implies that $G$ is connected, and we let $T$ denote a spanning tree of $G$. Let $R$ denote the set of edges in $G-B(G)$ that are not contained in a triangle. Since $G-B(G)$ is 0-good, $|R|\le n/2$ (if $xy\in R$, then this is the only edge of $R$ involving $x$ or $y$). Let $T'=T-R-B(G)$, noting that $e(T')\ge n/2-3$ since $|B(G)|\le 2$.
By construction, $E(T')\cap R=\emptyset$, and hence for each $e\in E(T')$ there exists some $e'\in E(G)$ such that $e$ and $e'$ are contained in the same triangle. Moreover, $e'$ can be chosen such that $e'\notin E(T)$ since $T$ contained no triangles. After choosing such an $e'$ for each $e$, define $E'=\{e':e\in E(T')\}$, noting that $E'\cap E(T)=\emptyset$ by construction. Observe that if $e=xy$ and $e'=yz$, then $e'=f'$ for some $f\in E(T'){\setminus}\{e\}$ only if $f=xz$, and hence $2|E'|\ge |E(T')|\ge n/2-3$. In total we conclude that $$e(G)\ge |E'|+|E(T)|\ge {\frac{5}{4}}n-3.$$
We do not believe that the above lemma is sharp, and we suspect that the true bound should be something closer to ${\frac{3}{2}}n$. This is motivated by the following two examples.
- Let $G_1$ denote the graph with vertex set $\{u,v,x_1,\ldots,x_k,y_1,\ldots,y_k\}$ and with the edges $uv,\ x_iu,\ x_iv,$ and $x_iy_i$ for all $i$. Then $G_1$ is 0-good, contains no odd cycle larger than a $C_3$, has $2k+2$ vertices, and has $3k+1$ edges.
- Let $H_i$ denote a $K_4$ on the vertex set $\{x_i,y_i,z_i,w_i\}$. Let $G_2'$ be the graph obtained by taking the disjoint union of $H_1$ through $H_k$ and adding all edges of the form $x_iy_{i+1}$. Let $G_2$ be $G_2'$ after attaching a leaf to every vertex of $G_2'$ with degree 3.
$G_2$ is 0-good, contains no odd cycle larger than a $C_3$, has $4k+(2k+2)=6k+2$ vertices, and has $6k+(k-1)+(2k+2)=9k+1$ edges.
Because of how different $G_1$ and $G_2$ look, proving a sharp version of Lemma \[L-3Low\] seems difficult, though we note that any improvement to the bound of Lemma \[L-3Low\] would give a corresponding improvement to the bound of Theorem \[T-Mod3\].
For the upper bound, Mini uses the strategy of Proposition \[P-3Alg\]. This implies that $G^\infty$ is 2-good with no $C_{2k+1}$ for any $k\ge 2$, and hence contains no $C_k$ for any $k\ge 5$. Lemma \[L-3Up\] then implies that $e(G^\infty)\le 2n-2$. For the lower bound, Max also uses the strategy of Proposition \[P-3Alg\]. This implies that $G^\infty$ is 2-good and $({\mathcal{C}}_\infty{\setminus}\{C_3\})$-saturated, so Lemma \[L-3Low\] implies that we have $e(G^\infty)\ge {\frac{5}{4}}n-3$.
Concluding Remarks {#S-Con}
==================
Determining the exact constant for the ${\mathcal{C}}_{2k+1}$-saturation game seems rather difficult, but this may be possible for the $({\mathcal{C}}_{\infty}{\setminus}\{C_3\})$-saturation game.
Can one determine the constant of ${\mathrm{sat}}_g({\mathcal{C}}_{\infty}{\setminus}\{C_3\};n)$? Further, can one determine the exact value of ${\mathrm{sat}}_g({\mathcal{C}}_{\infty}{\setminus}\{C_3\};n)$?
We claim that by analyzing the proof of Theorem \[T-gen\], one can conclude that ${\mathrm{sat}}_g({\mathcal{C}}_5;n)$ and ${\mathrm{sat}}_g({\mathcal{C}}_7;n)$ are strictly less than ${\lfloor {\frac{1}{4}}n^2\rfloor}$, which, as we mention in the beginning of Section \[S-Up\], is a non-trivial result. We suspect that stronger bounds exist.
For all $k\ge 1$ there exists a $c_k>0$ such that $${\mathrm{sat}}({\mathcal{C}}_{2k+1};n)\le \left({\frac{1}{4}}-c_k\right)n^2+o(n^2).$$
In fact, we believe that a stronger statement is true. As a consequence of the bounds of Theorem \[T-gen\], we know that ${\mathrm{sat}}_g({\mathcal{C}}_{2k+1};n)\le {\mathrm{sat}}_g({\mathcal{C}}_{2k'+1};n)$ when $k'$ is sufficiently larger than $k$ and $n$ is sufficiently large. We conjecture that this remains true when $k'=k+1$.
\[C-Main\] For all $k\ge 2$, $${\mathrm{sat}}_g({\mathcal{C}}_{2k-1};n)\le {\mathrm{sat}}_g({\mathcal{C}}_{2k+1};n)$$ for $n$ sufficiently large.
Note that the bound ${\mathrm{sat}}_g(\{C_3\};n)\le {\frac{26}{121}}n^2+o(n^2)$ of [@horn] together with Theorem \[T-5\] shows that the conjecture is true for $k=2$, and moreover that ${\mathrm{sat}}_g({\mathcal{C}}_3;n)\le {\mathrm{sat}}_g({\mathcal{C}}_{2k+1};n)$ for all $k\ge 2$ and $n$ sufficiently large.
One can verify that the proof of Theorem \[T-5\] generalize to bounding ${\mathrm{sat}}_g({\mathcal{F}};n)$ when ${\mathcal{F}}$ is any set of odd cycles containing ${\mathcal{C}}_{5}$. In particular this shows that ${\mathrm{sat}}_g({\mathcal{C}}_\infty{\setminus}\{C_{2k+1}\};n)$ is quadratic for all $k\ge 3$. Theorem \[T-Mod3\] shows that ${\mathrm{sat}}_g({\mathcal{C}}_\infty{\setminus}\{C_{3}\};n)$ is linear. Given this, it is natural to ask about the order of magnitude of ${\mathrm{sat}}_g({\mathcal{C}}_\infty{\setminus}\{C_{5}\};n)$.
What is the order of magnitude of ${\mathrm{sat}}_g({\mathcal{C}}_\infty {\setminus}\{C_5\};n)$? In particular, is this value linear, quadratic or something else?
Acknowledgments
===============
The author would like to thank Jacques Verstraete for suggesting this research topic, as well as his assistance with the general structure of the paper. The author would also like to thank the anonymous referee, whose comments greatly improved the structure and readability of this paper.
|
---
abstract: 'In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. Notable examples include Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries equation. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. For aforementioned classical numerical methods, the choice of trail and test functions is important for stability and accuracy issues in many cases. MIM shares this property when DNNs are employed to approximate unknowns functions in the first-order system. In one case, we use nearly the same DNN to approximate all unknown functions and in the other case, we use totally different DNNs for different unknown functions. Numerous results of MIM with different loss functions and different choice of DNNs are given for four types of PDEs. In most cases, MIM provides better approximations (not only for high-derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. When different DNNs are used, in many cases, MIM provides even better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Numerical observations also imply a successive improvement of approximation accuracy when the problem dimension increases and interesting connections between MIM and classical numerical methods. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis.'
address:
- 'School of Mathematical Sciences, Soochow University, Suzhou, 215006, China'
- 'CW Chu College, Soochow University, Suzhou, 215006, China'
- 'Nanjing TBS Information Technology Co. Ltd, Nanjing, 210000, China'
- 'Mathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China'
author:
- Liyao Lyu
- Zhen Zhang
- Minxin Chen
- Jingrun Chen
bibliography:
- 'ref.bib'
title: 'MIM: A deep mixed residual method for solving high-order partial differential equations'
---
Introduction {#sec:Intro}
============
Solving partial differential equations (PDEs) has been the most ubiquitous tool to simulate complicated phenomena in applied sciences and engineering problems. Classical numerical methods include finite difference method [@leveque_finite_2007], finite element method (FEM) [@Elman2014], discontinuous Galerkin method [@Cockburn2000], and spectral method [@shen2011spectral], which are typically designed for low dimensional PDEs and are well understood in terms of stability and accuracy. However, there are high dimensional PDEs such as Schrödinger equation in the quantum many-body problem [@dirac1981principles], Hamilton-Jacobi-Bellman equation in stochastic optimal control [@bardi2008optimal], and nonlinear Black-Scholes equation for pricing financial derivatives [@hull2009options]. Solving these equations is far out of the capability of classical numerical methods due to the curse of dimensionality, i.e., the number of unknowns grows exponentially fast as the dimension increases.
Until very recently, deep-learning based methods have been developed to solving these high-dimensional PDEs; see [@Weinan2017349; @Carleo2017; @E2018Mar; @Jiequn2018; @raissi2018deep; @sirignano2018dgm; @Hutzenthaler2019Jan; @raissi2019physics-informed; @Beck20191563; @GonzlezCervera2019; @fan2019multiscale; @khoo2019solving; @Beck2020Mar; @wang2020deep; @zang2020weak; @discacciati2020controlling] for examples. Typically, there are three main ingredients (stages) of a deep-learning method for solving PDEs: (1) modeling: the loss (objective) function to be optimized; (2) architecture: the deep neural network (DNN) for function approximation; (3) optimization: the optimal set of parameters in the DNN which minimizes the loss function. By design, the number of parameters in DNNs grows at most polynomially in terms of dimension. Meanwhile, possibly high-dimensional integrals in the loss function are approximated by Monte-Carlo method. Therefore, by design, deep learning overcomes the curse of dimensionality. In practice, deep learning performs well for Schrödinger equation [@Carleo2017; @Han2019], Hamilton-Jacobi-Bellman equation [@Jiequn2018; @Weinan2017349], and nonlinear Black-Scholes equation [@Beck20191563; @GonzlezCervera2019].
Typically, deep learning solves a PDE in the following way. For the given PDE, the loss function is modeled as the equation residual in the least-squares sense [@sirignano2018dgm] or the variational form if exists [@E2018Mar]. ResNet is often used as the network architecture [@He2015], which was tested to overcome the notorious problem of vanishing/exploding gradient. Afterwards, stochastic gradient descent method is used to find the optimal set of parameters in ResNet which minimizes the loss function. ResNet with the optimal set of parameters gives an approximation of the PDE solution.
In this work, we propose a deep mixed residual method (MIM) for solving high-order PDEs. In the modeling stage, by rewriting a given PDE into a first-order system, we obtain a larger problem in the sense that both the PDE solution and its high-order derivatives are unknown functions to be approximated. This has analogs in classical numerical methods, such as local discontinuous Galerkin method [@Cockburn2000] and mixed finite element method [@Boffi2013]. Compared to DGM, there are two more degrees of freedom in MIM:
- In the loss function stage, one can choose different high-order derivatives into the set of unknown functions. Take biharmonic equation as an example. The set of unknown functions can include the PDE solution and its derivatives up to the third order, or only contain the PDE solution and its second-order derivatives, and both choices have analogs in discontinuous Galerkin method [@Yan2002Dec; @cockburn2009a]. We then write the loss function as the sum of equation residuals in the least-squares sense, very much in the same spirit as the least-squares finite element method [@Bochev2015].
- In the architecture stage, one can choose the number of networks to approximate the set of unknown functions. In one case, one DNN is used to approximate the PDE solution and other DNNs are used to approximate its high-order derivatives; in the other case, the PDE solution and its derivatives share nearly the same DNN.
These two degrees of freedom allow MIM to produce better approximations over DGM in all examples, including Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries (KdV) equation. In particular, MIM provides better approximations not only for the high-order derivatives but also for the PDE solution itself. It is worth mentioning that the usage of mixed residual in deep learning was first introduced for surrogate modeling and uncertainty quantification of a second-order elliptic equation [@Zhu2019Oct] and was later adopted in a deep domain decomposition method [@Li2019Dec].
The paper is organized as follows. In Section \[sec:mixed residual\], we introduce MIM and DGM (for comparison purpose). In Section \[sec:Numerical result\], numerical results for four types of high-order PDEs are provided. Conclusions and discussions are drawn in Section \[sec:conclusion\].
Deep mixed residual method {#sec:mixed residual}
==========================
In this section, we introduce MIM and discuss its difference with DGM in terms of loss function and neural network structure.
Loss function {#subsection:Loss function}
-------------
Consider a potentially time-dependent nonlinear PDE over a bounded domain $\Omega\subset\mathbb{R}^d$ $$\left\{
\begin{aligned}
\label{equ:gen time_dependent pde example}
&\partial_t u + \mathcal{L} u = 0 & (t,x)\in (0,T]\times\Omega,\\
&u(0,x)= u_0(x) & x\in \Omega, \\
&u(t,x) = g(x) & (t,x)\in [0,T]\times \partial \Omega,
\end{aligned}
\right.$$ where $\partial \Omega$ denotes the boundary of $\Omega$. In DGM, the loss function is defined as the PDE residual in the least-squares sense $$\label{eqn:lossdgm}
\begin{aligned}
L(u) = \|\partial_t u + \mathcal{L} u\|^2_{2,[0,T]\times\Omega}
+ \lambda_1\|u(0,x)-u_0\|^2_{2,\Omega}
+ \lambda_2\|u - g\|^2_{2, [0,T]\times \partial \Omega},
\end{aligned}$$ where $\lambda_1$ and $\lambda_2$ are penalty parameters given *a priori*. These three terms in measure how well the approximate solution satisfies the PDE, the initial condition and the boundary condition, respectively.
In the absence of temporal derivatives, reduces to $$\left\{
\begin{aligned}
&\mathcal{L} u = 0 & x\in \Omega,\\
& u(x)= g(x) & x\in \partial \Omega,
\end{aligned}
\right.$$ and the corresponding loss function in DGM becomes $$\label{equ:loss function}
\begin{aligned}
L(u) = \|\mathcal{L}u\|^2_{2,\Omega} + \lambda \|u-g\|^2_{2,\partial \Omega}.
\end{aligned}$$
Table \[tbl:loss DGM\] lists four PDEs with their corresponding loss functions in DGM and Table \[tbl:loss BC\] lists different boundary conditions, the initial condition and their contributions to loss functions in DGM and MIM. More boundary conditions can be treated in this way. Interested readers may refer to [@chen2020bc] for details.
----------------------------------------------------------------------------------------------------------------
Equation Explicit form Loss function $L(u)$
-------------- ------------------------------------------------- -----------------------------------------------
Poisson $ $\|\Delta u + f(x) \|_{2,\Omega}^2$
-\Delta u = f(x)
$
Monge-Ampére $\det(\nabla^2 u) = f(x)$ $\|\det(\nabla^2u) - f(x)\|_{2,\Omega}^2 $
Biharmonic $ $\|
-\Delta^2 u = f(u,x) \Delta^2 u + f(u,x)
$ \|_{2,\Omega}^2
$
KdV $ $
u_t + \sum_{i=1}^{d}u_{x_ix_ix_i} = f(x) $ \|
u_t + \sum_{i=1}^{d}u_{x_ix_ix_i} - f(x)
\|_{2,\Omega}^2
$
----------------------------------------------------------------------------------------------------------------
: Loss functions for four types of PDEs in the deep Galerkin method.[]{data-label="tbl:loss DGM"}
Condition Explicit form Contribution to the loss function
----------- ------------------------------------- ------------------------------------------------------------------------------------------------------------------------
Dirichlet $u(x) = g$ $\|u-g\|_{2,[0,T]\times\partial\Omega}^2$
Neumann $\frac{\partial u}{\partial n} = g$ $\|\frac{\partial u}{\partial n} - g\|_{2,[0,T]\times\partial\Omega}^2$ or $\|p - g\|_{2,[0,T]\times\partial\Omega}^2$
Initial $u(0,x) = u_0(x)$ $\|u-u_0\|^2_{2,\Omega}$
: Contributions to the loss function for the initial condition and different types of boundary conditions used in the deep Galerkin method and the deep mixed residual method.[]{data-label="tbl:loss BC"}
In MIM, we first rewrite high-order derivatives into low-order ones using auxiliary variables. For notational convenience, auxiliary variables $p,q,w$ represent $$\label{eqn:auxiliary}
\begin{aligned}
p &= \nabla u, \\
q &= \nabla\cdot p = \Delta u, \\
w &= \nabla q = \nabla (\Delta u).
\end{aligned}$$ For KdV equation, we have $q=\mathrm{diag}(\nabla p)$ instead of the second formula in . With these auxiliary variables, we define loss functions for four types of PDEs in Table \[tbl:loss MIM\]. Since one can choose a subset of high-order derivatives into the set of unknown functions, there are more than one loss function in MIM. For biharmonic equation, there are two commonly used sets of auxiliary variables in local discontinuous Galerkin method and weak Galerkin finite element method: one with all high-order derivatives [@Yan2002Dec] and the other with part of high-order derivatives [@cockburn2009a; @Mu2015Sep]. Correspondingly, if all high-order derivatives are used, we denote MIM by MIM$_{a}$, and if only part of high-order derivatives are used, we denote MIM by MIM$_{p}$. In Section \[subsec:neural network\], we will discuss how to equip different loss functions with different DNNs. In short, if only one DNN is used to approximate the PDE solution and its derivatives, we denote MIM by MIM$^1$, and if multiple DNNs are used, we denote MIM by MIM$^2$. In Section \[sec:Numerical result\], different loss functions listed in Table \[tbl:loss DGM\], Table \[tbl:loss BC\] and Table \[tbl:loss MIM\] will be tested and discussed. By default, all the penalty parameters are set to be $1$.
Neural network architecture {#subsec:neural network}
---------------------------
ResNet [@He2015] is used to approximate the PDE solution and its high-order derivatives. It consists of $m$ blocks in the following form $$\label{equ:resnet}
s_k =\sigma(W_{2,k}\sigma(W_{1,k}s_{k-1}+b_{1,k})+b_{2,k}) +s_{k-1}, \quad k=1,2,\cdots,m.$$ Here $s_k, b_{1,k}, b_{2,k} \in \mathbb{R}^n$, $W_{1,k}, W_{2,k}\in \mathbb{R}^{n\times n}$. $m$ is the depth of network, $n$ is the width of network, and $\sigma$ is the (scalar) activation function. Explicit formulas of activation functions used in this work are given in Table \[tbl:activaction function\]. The last term on the right-hand side of is called the shortcut connection or residual connection. Each block has two linear transforms, two activation functions, and one shortcut; see Figure \[fig:resnet\] for demonstration. Such a structure can automatically solve the notorious problem of vanishing/exploding gradient [@DBLP:journals/corr/HeZRS15].
Activation function Formula
--------------------- -------------------
Square $x^2$
ReLU $\max\{x,0\}$
ReQU $(\max\{x,0\})^2$
ReCU $(\max\{x,0\})^3$
: Activation functions used in numerical tests.[]{data-label="tbl:activaction function"}
![One block of ResNet. A deep neural network contains a sequence of blocks, each of which consists of two fully-connected layers and one shortcut connection.[]{data-label="fig:resnet"}](Oneblock-eps-converted-to.pdf){width="30.00000%"}
Since $x$ is in $\mathbb{R}^d$ rather than $\mathbb{R}^n$, we can pad $x$ by a zero vector to get the network input $s_0$. A linear transform can be used as well without much difference. Meanwhile, $s_m$ has $n$ outputs which cannot be directly used for the PDE solution and its derivatives employed in the loss function. Therefore, a linear transform $T$ is applied to $s_m$ to transform it into a suitable dimension. Let $\{\theta\}$ be the whole set of parameters which include parameters in ResNet ($\left\{W_{1,k},b_{1,k},W_{2,k},b_{2,k}\right\}_{k=1}^m$) and parameters in the linear transform $T$. Note that the output dimension in MIM depends on both the PDE problem and the mixed residual loss. We illustrate network structures for biharmonic equation as an example in Figure \[fig:network for biharmonic\].
![Network structures for biharmonic equation with deep Galerkin method and deep mixed residual method. DGM only approximates solution $u$. MIM$^1_p$ approximate solution $u$ and $\Delta u$. MIM$^1_a$ approximates solution $u$ and all of its derivatives used in the equation $\nabla u, \Delta u , \nabla (\Delta u)$. MIM$^2_a$ uses four networks to approximate $ u , \nabla u, \Delta u , \nabla (\Delta u)$ and MIM$^2_p$ uses two networks to approximate $u,\Delta u$. Each network has a similar structure with different output dimensions.[]{data-label="fig:network for biharmonic"}](structure_Bi-eps-converted-to.pdf){width="\textwidth"}
From Figure \[fig:network for biharmonic\], we see that DGM has only $1$ output, MIM$^1_a$ has $2d+2$ outputs, and MIM$^1_{p}$ has $2$ outputs. In Figure \[fig:neural network for poisson equation MIM\], we illustrate networks structures of MIM$^1$ and MIM$^2$ for Poisson equation. In MIM$^2$, two DNNs are used: one to approximate the solution and the other one to approximate its derivatives. It is clear from Figure \[fig:network for biharmonic\] that network structures in DGM and MIM$^1$ only differ in the output layer and thus they have comparable numbers of parameters to be optimized. To be precise, we calculate their numbers of parameters in Table \[tbl:param num\], from which one can see the number of parameters in DGM and MIM$^1$ is close. The number of parameters in MIM$^2$ is nearly double for Poisson equation, Monge-Ampére equation and biharmonic equation (MIM$^2_p$), tripled for KdV equation, and quadrupled for biharmonic equation (MIM$^2_a$), respectively. In Section \[sec:Numerical result\], from numerical results, we observe a better performance of MIM$^1$ for all four equations, not only for derivatives of the PDE solution, but also for the solution itself.
Method Equation Size of the parameter set
--------------- ------------------------ ---------------------------------------
DGM Four equations $(2m-1)n^2 + (2m+d+1)n +1$
\*[MIM$^1$]{} Poisson \*[$(2m-1)n^2 + (2m+2d+1)n +d + 1$]{}
Monge-Ampére
Biharmonic (MIM$^1_a$) $(2m-1)n^2 + (2m+3d+2)n +2d + 2$
Biharmonic (MIM$^1_p$) $(2m-1)n^2 + (2m+d+2)n + 2$
KdV $(2m-1)n^2 + (2m+3d+1)n +2d + 1$
\*[MIM$^2$]{} Poisson \*[$(4m-2)n^2 + (4m+3d+1)n +d + 1$]{}
Monge-Ampére
Biharmonic (MIM$^2_a$) $(8m-4)n^2 + (8m+6d+2)n +2d + 2$
Biharmonic (MIM$^2_p$) $(4m-2)n^2 + (4m+2d+2)n + 2$
KdV $(6m-3)n^2 + (6m+5d+1)n +2d + 1$
: Number of parameters for different network structures used for different equations and different loss functions. $n$, $m$, and $d$ are the network width, the network depth, and the problem dimension, respectively. It is observed that the number of parameters in DGM and MIM$^1$ is close, and the number of parameters in MIM$^2$ is nearly double for Poisson equation, Monge-Ampére equation and biharmonic equation (MIM$^2_p$), tripled for KdV equation, and quadrupled for biharmonic equation (MIM$^2_a$), respectively.[]{data-label="tbl:param num"}
Stochastic Gradient Descent
---------------------------
For completeness, we also briefly introduce stochastic gradient descent method. For the loss function defined in , we generate two sets of points uniformly distributed over $\Omega$ and $\partial\Omega$: $\{\mathbf{x}_i\}^{N}_{i=1}$ in $\Omega$ and $\{\mathbf{\hat{x}}_j\}^{M}_{j=1}$ on $\partial \Omega$. $$\theta^{k+1} = \theta^{k} - \alpha \nabla_{\theta} \frac{|\Omega|}{N} \sum_{i=1}^N [\mathcal{L}u_{\theta}(\mathbf{x}_i;\theta^{k})]^2 + \lambda \alpha \nabla_{\theta} \frac{|\partial \Omega|}{M} \sum_{j=1}^{M} [u_{\theta}(\mathbf{\hat{x}}_j;\theta^{k})-g(\mathbf{\hat{x}}_j)]^2,$$ where $\alpha$ is the learning rate chosen to be $1e-3$ here. $|\Omega|$ and $|\partial \Omega|$ are measures of $\Omega$ and $\partial\Omega$, respectively. $u_{\theta}$ is the DNN approximation of PDE solution parameterized by $\{\theta\}$. Sampling points $\{\mathbf{x}_i\}^{N}_{i=1}$ and $\{\mathbf{\hat{x}}_j\}^{M}_{j=1}$ are updated at each iteration. In implementation, we use ADAM optimizer [@kingma2015adam] and automatic differentiation [@Paszke2017Oct] for derivatives in PyTorch.
Numerical Result {#sec:Numerical result}
================
In this section, we show numerical results of MIM for four types of equations. We use relative $L^2$ errors of $u$, $\nabla u$, $\Delta u$, and $\nabla (\Delta u)$ defined in Table \[tbl:R2error\] for comparison. In all figures, relative $L^2$ errors are in $\log _{10}$ scale.
Quantity DGM MIM
------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------
$u$ $\frac{\int_{\Omega} ( u_{\theta} - u ) ^2 dx}{\int_{\Omega} u ^2 dx}$ $\frac{\int_{\Omega} (u_{\theta} - u ) ^2 dx}{\int_{\Omega} u ^2 dx}$
$\nabla u $ $\frac{\int_{\Omega} (\nabla u_{\theta} - \nabla u ) ^2 dx}{\int_{\Omega} (\nabla u) ^2 dx}$ $\frac{\int_{\Omega} (p_{\theta} - \nabla u ) ^2 dx}{\int_{\Omega} (\nabla u) ^2 dx}$
$\Delta u $ $\frac{\int_{\Omega} (\Delta u_{\theta} - \Delta u ) ^2 dx}{\int_{\Omega} (\Delta u) ^2 dx}$ $\frac{\int_{\Omega} (q_{\theta} - \Delta u ) ^2 dx}{\int_{\Omega} (\Delta u) ^2 dx}$
$\nabla \Delta u $ $\frac{\int_{\Omega} \left(\nabla (\Delta u_{\theta}) - \nabla (\Delta u)\right) ^2 dx}{\int_{\Omega} \left(\nabla (\Delta u)\right) ^2 dx}$ $\frac{\int_{\Omega} \left(w_{\theta} - \nabla (\Delta u) \right) ^2 dx}{\int_{\Omega} \left(\nabla (\Delta u)\right) ^2 dx}$
$\mathrm{diag}(\nabla^2 u) $ $\frac{\int_{\Omega} (\mathrm{diag}(\nabla^2 u_{\theta} )- \mathrm{diag}(\nabla^2 u) ) ^2 dx}{\int_{\Omega} \left(\mathrm{diag}(\nabla^2 u)\right) ^2 dx}$ $\frac{\int_{\Omega} (q_{\theta} - \mathrm{diag}(\nabla^2 u) ) ^2 dx}{\int_{\Omega} \left(\mathrm{diag}(\nabla^2 u)\right) ^2 dx}$
: Relative $L^2$ errors used in deep Galerkin method and deep mixed residual method.[]{data-label="tbl:R2error"}
Poisson Equation
----------------
Consider the following Neumann problem $$\label{eq:case_Neumann Boundary condition}
\left\{
\begin{aligned}
&-\Delta u + \pi^2 u = 2 \pi^2 \sum_{k=1}^{d} \cos(\pi x_k) & x\in \Omega=[0,1]^d\\
&\frac{\partial u}{\partial n} = 0 & x\in \partial \Omega
\end{aligned}\right.$$ with the exact solution $u(x)=\sum_{k=1}^{d}\cos(\pi x_k)$. The neural network structure in DGM is the same as that for biharmonic equation shown in Figure \[fig:network for biharmonic\]. Following Table \[tbl:loss DGM\] and Table \[tbl:loss BC\], we use the loss function for $$\begin{aligned}
L(u) = &\|-\Delta u + \pi^2 u - 2 \pi^2 \sum_{k=1}^{d} \cos(\pi x_k)\|_{2,\Omega}^2 +\lambda \|\frac{\partial u}{\partial n} \|_{2,\partial \Omega}^2.
\end{aligned}$$
Since both $u$ and $p$ are explicitly used, one more advantage of MIM is the enforcement of boundary conditions. For , we multiply $p_i,\;i=1,\cdots,d$ by $x_i(1-x_i)$ to satisfy the Neumann boundary condition automatically; see Figure \[fig:neural network for poisson equation MIM\]. DGM only has $u$ as its unknown function, and thus it is unclear that how the exact Neumann boundary condition can be imposed.
Therefore, for DNNs in Figure \[fig:neural network for poisson equation MIM\], the loss function in MIM can be simplified as $$L(u,p) = \|p- \nabla u\|^2_{2,\Omega} + \|-\nabla\cdot p+\pi^2u-2\pi^2\sum_{k=1}^{d}\cos(\pi x_k)\|^2_{2,\Omega}.$$ We emphasize that Dirichlet boundary condition can be exactly imposed in DGM [@berg_unified_2018] and no penalty term is needed. For Neumann boundary condition, mixed boundary condition, and Robin boundary condition, however, it is difficult to build up a DNN representation which satisfies the exact boundary condition. Building up a DNN approximation which satisfies the exact boundary condition can have a couple of advantages [@chen2020bc]: 1) make ease of the training process by avoiding unnecessary divergence; 2) improve the approximation accuracy; 3) save the execution time. In MIM, however, we have the direct access to both $u$ and $p$. Therefore, all these boundary conditions can be imposed exactly in principle. This will be presented in a subsequent work [@lyu2020].
For , average errors of $u$ and $\nabla u$ over the last $100$ iterations are recorded in Table \[tbl:error of NBC\]. The network depth $m=2$ and the activation function $x^2$ is used. Network widths are $5,10,15,20$ for $2,4,8,16$ dimensional problems, respectively. Time is recorded as the average CPU time per iteration. It is not surprising that MIM$^1$ costs less time than DGM since the DNN approximation in MIM satisfies the Neumann boundary condition automatically and both methods have similar network structures. It is surprising that MIM$^2$ costs less time than DGM since the number of parameters in MIM$^2$ is about twice of that in DGM. In terms of execution time, MIM$^1 < $ MIM$^2 < $ DGM.
---------- -------------- -------- ---------------- ---------
\*[d]{} \*[Method]{} \*[Time (s)]{}
$u$ $\nabla u$
\*[2]{} DGM 0.3676 0.3714 0.04374
MIM$^1$ 0.2941 0.1639 0.02925
MIM$^2$ 0.0565 0.0236 0.03514
\*[4]{} DGM 1.0022 1.3272 0.07455
MIM$^1$ 0.3751 0.3290 0.03603
MIM$^2$ 0.2294 0.0690 0.04141
\*[8]{} DGM 2.0022 2.6551 0.13081
MIM$^1$ 0.9049 0.6423 0.06642
MIM$^2$ 0.7261 0.1499 0.08716
\*[16]{} DGM 3.9796 5.0803 0.25621
MIM$^1$ 1.7631 1.0041 0.11082
MIM$^2$ 0.0787 0.0236 0.15125
---------- -------------- -------- ---------------- ---------
: Relative errors for $u$ and $\nabla u$ in DGM and MIM for Poisson equation defined in .[]{data-label="tbl:error of NBC"}
Figure \[fig:NBCL2u\] and Figure \[fig:NBCL2du\] plot training processes of DGM and MIM in terms of relative $L^2$ errors for $u$ and $\nabla u $. Generally speaking, in terms of approximation error, MIM$^2 < $ MIM$^1 < $ DGM as expected. Therefore, MIM provides a better strategy over DGM. MIM provides better approximations in terms of relative $L^2$ errors for both $u$ and $\nabla u$. For $\nabla u$, the improvement of MIM$^1$ over DGM is about several times and that of MIM$^2$ over MIM$^1$ is about one order of magnitude. For $u$, the improvement is about several times. Moreover, a dimensional dependence is observed for both $u$ and $\nabla u$. The higher the dimension is, the better the approximation is.
Table \[tbl:indep of D and A\] records approximation errors of MIM and DGM in terms of activation function and network depth when $d=4$. MIM provides better approximations for both $\nabla u$ and $u$. It is not surprising that ReLU is not a suitable function for DGM due to high-order derivatives, but is suitable in MIM since only first-order derivatives are present in MIM.
---------------- ----------- -------- ------------ -------- ------------ -------- ------------
\*[$\sigma$]{} \*[$m$]{}
$u$ $\nabla u$ $u$ $\nabla u$ $u$ $\nabla u$
\*[ReLU]{} 1 0.9197 0.9259 0.0890 0.0444 0.0264 0.0080
2 0.9210 0.9230 0.0245 0.0104 0.0265 0.0068
3 0.9208 0.9216 0.0258 0.0113 0.0258 0.0084
\*[ReQU]{} 1 0.0684 0.1003 0.0182 0.0127 0.0107 0.0042
2 0.0057 0.0118 0.0113 0.0047 0.0049 0.0017
3 0.0124 0.0140 0.0040 0.0029 0.0042 0.0031
\*[ReCU]{} 1 0.4642 0.4644 0.0288 0.0159 0.0100 0.0033
2 0.0281 0.0170 0.0071 0.0055 0.0048 0.0013
3 0.0028 0.0031 0.0049 0.0036 0.0049 0.0013
---------------- ----------- -------- ------------ -------- ------------ -------- ------------
: Performance of MIM and DGM with respect to network depth and activation function for Poisson equation when $d=4$ . Network width is fixed to be $10$.[]{data-label="tbl:indep of D and A"}
Monge-Ampére equation
---------------------
Consider the nonlinear Monge-Ampére equation $$\label{equ:MA}
\left\{\begin{aligned}
&\det(\nabla^2 u) = f(x) & x\in \Omega = [-1,1]^d\\
&u(x) = g(x) & x\in \partial \Omega
\end{aligned} \right.$$ with the exact solution defined as $u(x)= e^{1/d(\sum_{i=1}^d x_i^2)}$. Following Table \[tbl:loss DGM\], \[tbl:loss MIM\] and \[tbl:loss BC\], we have the loss function in DGM $$L(u)= \|\det(\nabla^2u) - f\|_{2,\Omega}^2 + \lambda \|u-g\|^2_{2,\partial \Omega},$$ and the loss function in MIM $$L(u,p) = \|p - \nabla u \|_{2,\Omega}^2 + \|\det(\nabla p) - f\|_{2,\Omega}^2 + \lambda \|u-g\|^2_{2,\partial \Omega},$$ respectively. For , the Dirichlet boundary condition can be enforced for both DGM and MIM. For comparison purpose, instead, we have the penalty term in both DGM and MIM. However, imposing exact boundary conditions is always encouraged in practice.
In this example, we fix the network depth $m = 2$ and the activation function as $\sigma(x) = \mathrm{ReQU}(x)$. Relative $L^2$ errors in the last $1000$ iterations with respect to the network width in different dimensions are recorded in Table \[tab:MA\]. Figure \[fig:indep of W and D in MA\] plots errors in terms of network width for different dimensions. The advantage of MIM is obvious from these results.
--------- ----------- -------- ------------ -------- ------------ -------- ------------
\*[d]{} \*[$n$]{}
$u$ $\nabla u$ $u$ $\nabla u$ $u$ $\nabla u$
\*[2]{} 10 0.1236 0.7430 0.1023 0.3433 0.1251 0.5218
20 1.1100 3.1940 0.0922 0.3804 0.0784 0.0221
30 0.0913 0.5656 0.0522 0.1740 0.1075 0.0219
\*[4]{} 20 0.0981 0.7764 0.1095 0.6359 0.1230 0.3977
30 0.0921 0.7731 0.0903 0.4399 0.1063 0.2802
40 0.0943 0.6174 0.0636 0.3127 0.1287 0.2480
\*[8]{} 30 0.3584 3.3902 0.1435 1.6318 0.1155 0.5170
40 0.1179 1.4663 0.1344 1.0721 0.1330 0.4873
50 0.0997 1.2483 0.0977 0.8289 0.0917 0.4174
--------- ----------- -------- ------------ -------- ------------ -------- ------------
: Relative $L^2$ errors in the last $1000$ iterations with respect to the network width for Monge-Ampére equation defined in for different dimensions. The network depth is fixed to be $m = 2$ and the activation function is fixed to be $\sigma(x) = \mathrm{ReQU}(x)$. []{data-label="tab:MA"}
Biharmonic equation
-------------------
Consider the biharmonic equation $$\label{equ:biharmonic equation}
\left\{\begin{aligned}
&\Delta^2 u = \frac{\pi^4}{16} \sum_{k=1}^{d} \sin (\frac{\pi}{2}x) & x\in\Omega\\
& u(x) = \sum_{k=1}^{d}\sin(\frac{\pi x}{2}) & x\in\partial \Omega \\
& \frac{\partial u}{\partial n} = 0 & x\in\partial \Omega
\end{aligned}\right.$$ with the exact solution $u(x) = \sum_{k=1}^{d}\sin(\frac{\pi x}{2})$ over $\Omega = [-1,1]^d$. The loss function in DGM is $$L(u) = \|\Delta^2 u - \frac{\pi^4}{16} \sum_{k=1}^{d} \sin (\frac{\pi}{2}x) \|_{2,\Omega}^2 + \lambda_1 \| u - \sum_{k=1}^{d}\sin(\frac{\pi x}{2})\|_{2,\partial \Omega}^2 + \lambda_2 \| \frac{\partial u}{\partial n} \|_{2,\partial \Omega}^2.$$ The loss function in MIM$_a$ is $$\begin{gathered}
\label{equ:loss for biharmonic scheme1}
L(u,p,q,w) = \|p - \nabla u \|_{2,\Omega}^2 + \|q - \nabla \cdot p \|_{2,\Omega}^2 + \|w - \nabla q \|_{2,\Omega}^2 \\
+ \| \nabla \cdot w - \frac{\pi^4}{16} \sum_{k=1}^{d} \sin (\frac{\pi}{2}x) \|_{2,\Omega}^2
+ \lambda_1\| u - \sum_{k=1}^{d}\sin(\frac{\pi x}{2})\|_{2,\partial \Omega}^2 + \lambda_2\| p \|_{2,\partial \Omega}^2 ,
\end{gathered}$$ and the loss function in MIM$_p$ is $$\begin{gathered}
\label{equ:loss for biharmonic scheme2}
L(u,q) = \|q- \Delta u\|_{2,\Omega}^2 + \| \Delta q - \frac{\pi^4}{16} \sum_{k=1}^{d} \sin (\frac{\pi}{2}x) \|_{2,\Omega}^2 \\
+ \lambda_1\| u - \sum_{k=1}^{d}\sin(\frac{\pi x}{2})\|_{2,\partial \Omega}^2 + \lambda_2\| \frac{\partial u}{\partial n} \|_{2,\partial \Omega}^2.
\end{gathered}$$ Again, we can enforce the exact boundary condition in MIM but cannot enforce it in DGM. For comparison purpose, we use penalty terms in both methods.
Set $m=2$ and $n=8,10,20$ when $d=2,4,8$, respectively. Table \[tbl:error of biharmonic\] records averaged errors in the last 1000 iterations.
--------- -------------- -------- ---------------- ------------ --------------------- --------
\*[d]{} \*[Method]{} \*[Time (s)]{}
$u$ $\nabla u$ $\Delta u$ $\nabla (\Delta u)$
\*[2]{} DGM 0.1656 0.6454 1.2333 8.8001 0.1034
MIM$^1_a$ 0.1501 0.1929 0.1564 0.3067 0.1219
MIM$^1_p$ 0.0769 0.1155 0.1504 0.4984 0.1636
MIM$^2_a$ 0.0526 0.2066 0.2937 1.6821 0.1393
MIM$^2_p$ 0.0424 0.1417 0.3625 2.2231 0.2164
\*[4]{} DGM 0.1330 0.6454 1.2333 8.8008 0.3292
MIM$^1_a$ 0.4117 0.1929 0.1563 0.3066 0.2784
MIM$^1_p$ 0.0845 0.1155 0.1504 0.4984 0.4692
MIM$^2_a$ 0.1039 0.2066 0.2937 1.6821 0.2883
MIM$^2_p$ 0.1111 0.1417 0.3625 2.2301 0.5919
\*[8]{} DGM 0.2488 1.0514 1.4594 13.4003 0.3292
MIM$^1_a$ 0.3719 2.3855 0.6797 3.1015 0.2784
MIM$^1_p$ 0.1856 0.6909 0.7840 4.7209 0.4692
MIM$^2_a$ 0.1475 1.6657 1.2922 6.9594 0.8051
MIM$^2_p$ 0.2881 0.9223 0.9981 6.4658 6.5148
--------- -------------- -------- ---------------- ------------ --------------------- --------
: Relative errors for biharmonic equation defined in . MIM$_a$ and MIM$_b$ represent MIM with loss functions defined in and , respectively.[]{data-label="tbl:error of biharmonic"}
Relative $L^2$ errors for $u$ , $\nabla u$, $\Delta u$ and $\nabla (\Delta u)$ in terms of iteration number are plotted in Figure \[fig:biharmonic 2D\] when $d=2$.
Generally speaking, MIM provides better approximations for $u$, $\nabla u$, $\Delta u$, and $\nabla (\Delta u)$ than DGM. For MIM$_a$ and MIM$_p$, MIM$_p$ has a slightly better approximation accuracy comparable to that of MIM$_a$, although MIM$_a$ has $2d+2$ more outputs. These results are of interests since they are connected with results of local discontinuous Galerkin method that the formulation with a subset of derivatives has a better numerical performance [@Yan2002Dec; @cockburn2009a]. We point out that MIM$_a$ has the advantage that the exact boundary condition can be enforced, although we use penalty terms for this example.
KdV equation
------------
Consider a time-dependent linear KdV-type equation $$\label{eq:kdv}
\left\{
\begin{aligned}
&u_t + \sum_{k=1}^d u_{x_kx_kx_k} =0
& (t,x)\in[0,T]\times\Omega\\
&u(0,x) = u_0(x) = \sin(\sum_{k=1}^d x_k)
& (t,x)\in[0]\times\Omega\\
&u(t,x) \mathrm{\; is\; periodic\; in\; } x \\
\end{aligned}\right.$$ defined over $\Omega=[0,2\pi]^d$, where the exact solution $u(t,x)=\sin(\sum_{k=1}^d x_k+d t)$. We first rewrite it into the first-order system $$%\label{equ:LDG formula for kdv function}
\begin{aligned}
&p = \nabla u,\\
&q = \mathrm{diag}(\nabla p ),\\
&u_t + \nabla \cdot q = 0.
\end{aligned}$$ The loss function in DGM is $$\begin{aligned}
L(u) &= \|u_t+\sum_{k=1}^du_{x_k x_k x_k}\|^2_{2,[0,1]\times\Omega} + \lambda_1 \|u-\sin(\sum_{k=1}^d x_k+d t)\|^2_{2,[0,1]\times\partial\Omega} \\\
& \quad + \lambda_2 \left(\sum_{k=1}^d\|u(x,t)-u(x\pm 2\pi e_k,t)\|^2_{2,\Omega}\right) \\
& \quad + \lambda_3 \left(\sum_{k=1}^d\|\nabla u(x,t)-\nabla u(x\pm2\pi e_k,t)\|^2_{2,\Omega}\right).
\end{aligned}$$ Here $\{e_k\}_{k=1}^d$ is the standard basis set of $\mathbb{R}^d$. The loss function in MIM is $$\begin{aligned}
L(u,p,q) & = \|p - \nabla u\|^2_{2,[0,1]\times\Omega} + \|q- \mathrm{diag}(\nabla p)\|^2_{2,[0,1]\times\Omega} \\
& \quad + \|u_t+\nabla \cdot q\|^2_{2,[0,1]\times\Omega} + \lambda_1 \|u-\sin(\sum_{k=1}^d x_k+d t)\|^2_{2,[0,1]\times\partial\Omega} \\
& \quad + \lambda_2 \left(\sum_{k=1}^d\|u(x,t)-u(x\pm2\pi e_k,t)\|^2_{2,\Omega}\right) \\
& \quad + \lambda_3\left(\sum_{k=1}^d\|p(x,t)-p(x\pm2\pi e_k,t)\|^2_{2,\Omega}\right).
\end{aligned}$$ Relative $L^2$ errors of $u$, $\nabla u$, and $\mathrm{diag}(\nabla^2 u)$ are recorded in Table \[tbl:error of kdv\]. Again, as shown in previous examples, MIM provides better results compared to DGM, especially for ReQU activation function. No obvious improvement of MIM$^2$ over MIM$^1$ is observed.
----------- ---------------- -------------- ---------- ------------ -----------------------------
\*[$d$]{} \*[$\sigma$]{} \*[Method]{}
$u$ $\nabla u$ $\mathrm{diag}(\nabla^2 u)$
\*[1]{} \*[ReQU]{} DGM 34.9171 20.6788 34.3661
MIM$^1$ 0.5705 5.3709 0.5369
MIM$^2$ 1.2920 0.8129 1.9244
\*[ReCU]{} DGM 0.7603 0.4785 0.5977
MIM$^1$ 0.0991 0.7313 0.0128
MIM$^2$ 0.5035 0.5804 0.1229
\*[2]{} \*[ReQU]{} DGM 84.8708 85.8114 85.8954
MIM$^1$ 2.9393 1.9996 2.9443
MIM$^2$ 2.1820 2.5591 2.1383
\*[ReCU]{} DGM 2.5483 2.1856 2.4431
MIM$^1$ 1.5410 2.3865 1.5645
MIM$^2$ 5.5900 5.7440 5.8957
\*[3]{} \*[ReQU]{} DGM 168.1755 168.1697 169.3528
MIM$^1$ 4.0421 4.0987 3.8496
MIM$^2$ 7.7027 8.8787 9.1058
\*[ReCU]{} DGM 1.9132 1.4846 1.7970
MIM$^1$ 1.5410 2.3865 1.5645
MIM$^2$ 5.5900 5.7440 5.8957
----------- ---------------- -------------- ---------- ------------ -----------------------------
: Relative $L^2$ errors for KdV equation defined in .[]{data-label="tbl:error of kdv"}
Conclusion and Discussion {#sec:conclusion}
=========================
Motivated by classical numerical methods such as local discontinuous Galerkin method, mixed finite element method, and least-squares finite element method, we develop a deep mixed residual method to solve high-order PDEs in this paper. The deep mixed residual method inherits several advantages of classical numerical methods:
- Flexibility for the choice of loss function;
- Larger solution space with flexible choice of deep neural networks;
- Enforcement of exact boundary conditions;
- Better approximations of high-order derivations with almost the same cost.
Meanwhile, the deep mixed residual method also provides a better approximation for the PDE solution itself. These features make deep mixed residual method suitable for solving high-order PDEs in high dimensions.
Boundary condition is another issue which is important for solving PDEs by DNNs. Enforcement of exact boundary conditions not only makes the training process easier, but also improves the approximation accuracy; see [@berg_unified_2018; @chen2020bc] for examples. The deep mixed residual method has the potential for imposing exact boundary conditions such as Neumann boundary condition, mixed boundary condition, and Robin boundary condition. All these conditions cannot be enforced exactly in deep Galerkin method. This shall be investigated in a subsequent work [@lyu2020].
So far, in the deep mixed residual method, only experiences from classical numerical methods at the basic level are transferred into deep learning. We have seen its obvious advantages. To further improve the deep mixed residual method, we need to transfer our experiences from classical numerical analysis at a deeper level. For example, the choice of solution space relies heavily on the choice of residual in order to maximize the performance of least-squares finite element method [@Bochev2015]. Many other connections exist in discontinuous Galerkin method [@Cockburn2000] and mixed finite element method [@Boffi2013]. For examples, since only first-order derivatives appear in the deep mixed residual method, ReLU works well for all time-independent equations we have tested but does not work well for KdV equation. Therefore, it deserves a theoretical understanding of the proposed method in the language of linear finite element method [@he2018relu]. Another possible connection is to use the weak formulation of the mixed residual instead of least-squares loss, as done in deep learning by [@zang2020weak] and in discontinuous Galerkin method by [@Cockburn2000]. Realizing these connections in the deep mixed residual method will allow for a systematic way to understand and improve deep learning for solving PDEs.
Acknowledgments
===============
This work was supported by National Key R&D Program of China (No. 2018YFB0204404) and National Natural Science Foundation of China via grant 11971021. We thank Qifeng Liao and Xiang Zhou for helpful discussions.
|
---
abstract: 'We apply the recently developed notion of complexity for field theory to a quantum quench through a critical point in $1\!+\!1$ dimensions. We begin with a toy model consisting of a quantum harmonic oscillator, and show that complexity exhibits universal scalings in both the slow and fast quench regimes. We then generalize our results to a 1-dimensional harmonic chain, and show that preservation of these scaling behaviours in free field theory depends on the choice of norm. Applying our set-up to the case of two oscillators, we quantify the complexity of purification associated to a subregion, and demonstrate that complexity is capable of probing features to which the entanglement entropy is insensitive. We find that the complexity of subregions is subadditive, and comment on potential implications for holography.'
author:
- 'Hugo A. Camargo'
- Pawel Caputa
- Diptarka Das
- 'Michal P. Heller'
- Ro Jefferson
bibliography:
- 'biblio.bib'
title: 'Complexity as a novel probe of quantum quenches: universal scalings and purifications'
---
**INTRODUCTION**
Among the most exciting developments in theoretical physics is the confluence of ideas from quantum many-body systems, quantum information theory, and gravitational physics. Recent progress in this vein includes the development of tensor network methods for simulating quantum many-body systems (see, e.g., [@Orus:2013kga]), proofs of irreversibility of RG flows using quantum information techniques [@Casini:2012ei; @Casini:2004bw; @Myers:2010xs; @Myers:2010tj; @Casini:2015woa; @Casini:2017vbe], and the illumination of the role of codimension-2 extremal surfaces in the emergence of holographic spacetime (see, e.g., [@Rangamani:2016dms]). The central technical tool in these ground-breaking results is the reduced density matrix for a spatial subregion, and the associated von Neumann entropy, cf. [@Ryu:2006bv; @Hubeny:2007xt].
However, insights from black hole physics [@Susskind:2014rva; @Susskind:2014moa; @Brown:2015bva; @Brown:2015lvg] suggest that certain codimension-0 and codimension-1 surfaces may also play an important role in reconstructing bulk spacetime in holography, since these capture information beyond that which is accessible to the aforementioned codimension-2 surfaces—that is, beyond entanglement entropy. These geometric objects are conjectured to be dual to the “complexity” of the boundary field theory, according to the competing “complexity=volume” (CV) [@Susskind:2014rva; @Susskind:2014moa] and “complexity=action” (CA) proposals [@Brown:2015bva; @Brown:2015lvg].
Drawing on earlier developments [@Nielsen:2005mn1; @Nielsen:2006mn2; @Nielsen:2007mn3; @Haegeman:2011uy; @Nozaki:2012zj], [@Jefferson:2017sdb] and [@Chapman:2017rqy] sought to make the above conjectures more precise by defining the notion of complexity in (free, bosonic) quantum field theory (this idea was subsequently extended to fermionic theories in [@Hackl:2018ptj], see also [@Khan:2018rzm; @Reynolds:2017jfs]; for alternative approaches to defining complexity in field theories, see [@Caputa:2017urj; @Caputa:2017yrh; @Czech:2017ryf; @Bhattacharyya:2018wym; @Magan:2018nmu; @Caputa:2018kdj; @Yang:2018nda]). In light of the successes born of entanglement entropy mentioned above, understanding complexity in quantum field theory represents a very promising research direction. Particularly interesting open questions include the time-dependence of complexity, and the interplay between complexity and entanglement entropy in non-equilibrium systems. It is therefore of value to have a tractable system in which these ideas can be concretely explored.
To that end, one of the most active areas of research into non-equilibrium quantum dynamics is the study of quantum quenches [@Gring1318; @Polkovnikov:2010yn], in which remarkable progress has been made in understanding the mechanisms underlying thermalization encoded in the reduced density matrix [@Dunjko]. Theoretical studies within the scope of experimental verification have revealed that smooth quenches through a critical point exhibit universal signatures via scalings. The Kibble-Zurek (KZ) scaling [@Kibble:1976sj; @Zurek:1985qw] is the most well-known example of this behaviour, and has received a great deal of attention in recent years [@Chandran_etal; @Mondal_etal; @Gritsev:2009wt; @Dziarmaga; @Lamacraft]. In this case, the state is evolved adiabatically until very close to the critical point, and hence the regime of KZ can be characterized as “slow”. Recent studies in holography [@Buchel:2012gw; @Buchel:2013lla], free field theory [@Buchel:2013gba; @Das:2014jna; @Das:2014hqa], and lattice spin models [@Das:2017sgp] have also revealed new scaling behaviours in a “fast” (non-adiabatic) regime. This fast scaling behaviour appears to be a universal feature of any interacting theory which flows from a CFT in the ultraviolet (UV) [@Das:2016lla; @Goykhman:2018ihr; @Dymarsky:2017awt]. At a technical level, previous studies have mainly focused on the scalings of a restricted set of one- and two-point functions, and recently on entanglement [@Caputa:2017ixa]. However, as we shall argue below, the latter probes at most only a spatial subsystem, while complexity is a property of the entire wavefunction. Hence complexity represents a means of probing features of quench dynamics to which entanglement entropy is insensitive. Initial steps towards applying complexity to quenches were taken in [@Alves:2018qfv], for a quench which monotonically interpolates between two massive theories.
Motivated by these scaling phenomena, we explore the complexity of exact *critical* quench solutions for free scalar theories, and find evidence for universal scaling behaviour. Our primary model will consist of a bosonic oscillator whose frequency varies smoothly with time, and asymptotes to a finite constant in both the far future and past. We first define complexity for a single mode, and then generalize our results to a 1-dimensional harmonic chain. However, we find that a judicious definition of complexity is required in order to make the scaling expectations for free field theory manifest. Utilizing this set-up, we contrast the complexity and the entanglement entropy for a fixed bipartition of the Hilbert space of two coupled harmonic oscillators. This model enables us to quantify the notion of “complexity of purification” recently introduced in [@Agon:2018zso], which allows one to associate a complexity to subregions (i.e., mixed states). We find that the complexity of subregions is subadditive, which may have interesting implications for the CV vs. CA proposals above.
**COMPLEXITY OF QUANTUM QUENCHES**
**Quench model**
We shall begin with the following simple Hamiltonian describing a free bosonic oscillator: H(t) = P\^2 + M\^2 X\^2 ,\[eq:Hphys\] where $M$ is the mass of the oscillator, $\tilde\omega(t/\delta t)$ is some time-dependent frequency profile with an intrinsic scale set by the parameter $\delta t$, and the canonical position and momentum operators satisfy $[X,P]=i$. However, for reasons that will become apparent below, it is preferable to work with the dimensionless variables $x\!\equiv\!\varpi X$, $p\!\equiv\! P/\varpi$, $\omega\!\equiv\!\tilde\omega/\varpi$, where $\varpi$ is some new mass scale, which will be given an interpretation as the gate scale when we introduce our quantum circuit (see appendix A). Setting $\varpi=M$ for simplicity, becomes H(t) = M + \^[2]{}, \[eq:H\] where the quantities appearing in the parentheses are all dimensionless, and we shall henceforth set $M=1$. The time-evolved initial ground-state wavefunction at time $t$ for the Hamiltonian takes the form \_0(x,t)=(x\^2),\[eq:ground\] where $\mathcal{N}\equiv\lp2\pi f^*f\rp^{-1/4}$, and $f(t/\delta t)$ is the solution to the equation +\^2 f=0 .\[eq:eqf\] Now, we desire a quench profile $\omega^2(t/\delta t)$ which admits an exact solution to this equation, and which asymptotes to a constant at both early and late times, with changes occurring in the time-window $[-\delta t,\delta t]$. One of the most common profile used in the literature (see, e.g., [@Caputa:2017ixa]) is \^[2]{}(t/t) = \_[0]{}\^2 ( 1- ) .\[eq:profile\] Here $\omega_{0}$ is a free parameter, but will gain an interpretation as the dimensionless reference-state frequency below. This profile has the property that the system is initially gapped at $t=-\infty$, but becomes gapless at $t=0$, corresponding to oscillator excitations above the ground state as the system evolves via . In this case, the function $f(t)$ can be written explicitly in terms of hypergeometric functions—see [@Caputa:2017ixa].
Our interest in this set-up is due to the fact that it can also be used to study the ground state of two (or more) harmonic oscillators with a time-dependent coupling. The same model was considered in [@Bombelli:1986rw; @Srednicki:1993im; @Caputa:2017ixa] for investigating entanglement entropy during a quench. Explicitly, the Hamiltonian for two oscillators is given by H= . \[eq:H2\] In the normal-mode basis $x_\pm=(x_1\pm x_2)/\sqrt{2}$, this Hamiltonian takes the decoupled form H(t)=H\_[+]{}(t)+H\_[-]{}(t)\[eq:2sho\] where the subscript denotes the use of the $\pm$ mode in , with $\omega^2_+ = \omega(t)^2$ and $\omega^2_- = \omega(t)^2 + 4 \Omega(t)^2$. The corresponding wavefunction is then given by (x\_[+]{},x\_[-]{},t) =\_0(x\_+,t)\_0(x\_-,t) ,\[eq:psiProd\] with $\psi_0$ given by . Note that this construction naturally generalizes to an $N$-oscillator harmonic chain, which we will consider after introducing complexity below.
**Circuit complexity**
To evaluate the complexity of the target state , we shall apply the circuit complexity approach of [@Jefferson:2017sdb], adapted at the level of covariance matrices as in [@TFD]. The reader is referred to these works for details. In brief, a circuit $U$ is a unitary operator whose action on some reference state $\psi_\mr{R}$ produces the desired target state $\psi_\mr{T}$, =U .\[eq:evo\] In analogy with quantum circuits, $U$ can be thought of as a sequence of fundamental gates, each of which effects an infinitesimal change to the state. The complexity of the target state is then defined as the length of the optimum circuit according to some suitably chosen depth function (e.g., the number of gates). Note the keyword “optimal”: there may be arbitrarily many different circuits which satisfy . Hence the central feature of [@Jefferson:2017sdb] was to use the geometric approach of Nielsen and collaborators [@Nielsen:2005mn1; @Nielsen:2006mn2; @Nielsen:2007mn3] to convert the problem of finding the optimum circuit into that of identifying the minimum geodesic in the geometry generated by the algebra of gates.
Given the form of , it is sufficient to begin with a single oscillator. Hence we are interested in target states of the form \_(x,t)=\^[1/4]{}{-a+ibx\^2} ,\[eq:psiT\] where $a(t),b(t)\in\reals$ are the real and imaginary parts of the frequency $i\dot{f}^*/f^*$ in , and we have suppressed the time-dependence for compactness. Note that $a\!>\!0$ (one can verify that the solutions to indeed satisfy this normalizability constraint), while $b$ may take any sign. Our reference state will be provided by the ground state of our time-dependent Hamiltonian at $t\!=\!-\infty$, \_(x)=\^[1/4]{}{-x\^2} ,\[eq:psiR\] where $0\!<\!\omega_\mr{R}\!\in\!\reals$. Our task is now to construct a circuit $U$ satisfying according to the geometric approach outlined above.
The details of our complexity calculation are given in appendix A. The key point is that we may view $U$ as a matrix which acts at the level of covariance matrices, so that becomes G\_=UG\_U\^T ,\[eq:evoG\] where the matrix elements of $G$ are given by G\^[ab]{}=\^a\^b+\^b\^a , where $\xi^a\!\equiv\!\{x^1,p^1,\ldots x^N,p^N\}$ are the dimensionless phase-space operators for $N$ oscillators. The covariance matrix is an equivalent representation of the wavefunction, which has the advantage of making the explicit choice of gates more transparent. In particular, we seek the minimal set of gates necessary to effect the desired transformation. As explained in appendix A, this naturally leads to hyperbolic space, with the metric s\^2= , and therefore the complexity of the target state is given by the well-known geodesic distance formula on $\mathbb{H}^2$ (cf. appendix B), which admits a particularly compact expression in terms of the squeezed target-state covariance matrix $\tilde G_\mr{T}=SG_\mr{T}S^T$: =+ , G\_ ,\[eq:C\] where $S$ is the squeezing operator defined such that $SG_\mr{R}S^T=\mathbbm{1}$. This result immediately generalizes to the case of $N$ oscillators: since $\tilde G_\mr{T}$ is block-diagonal in an appropriate basis, the geometry factorizes into $N$ independent copies of $\mathbb{H}^2$. Hence the complexity of a 1-dimensional lattice of oscillators is =.\[eq:Cfield\] Note that in this expression, we have added the complexities in the $L_2$-norm; we shall comment on the use of other norms in appendix D. By taking the continuum limit of such a lattice, we obtain the complexity for a bosonic system in $1\!+\!1$ dimensions. Specifically, we consider the harmonic chain whose Hamiltonian is given by \[eq:Hfield\] H= \_[n=1]{}\^[N]{} ( \_n\^2 + ( \_[n+1]{} - \_n )\^2 + m\^2(t) \_n\^2 ) , where $(\phi_n, \Pi_n)$ are mutually conjugate scalar field variables. Since we work with dimensionless variables, we shall set the lattice spacing (i.e., the UV-cutoff) to unity. In momentum space, each mode then satisfies + ( 4 \^2 + m\^2(t) ) \_k = 0 , \[eq:mode\] where we have imposed periodic boundary conditions $k\!=\!k\!+\!2\pi$, and the quench profile is given by $m(t)\!=\!\omega(t/\delta t)$ in . The reference state, $\ket{\psi_R}$ is given by the ground state of the Hamiltonian at $t=-\infty$ when $m(t) = \omega_0$. Integrating over momentum modes, the continuum limit of is simply (t)=.\[eq:Cfield2\] where $\chi_k(t)$ is given in with the covariance matrix corresponding to the $k^{\mathrm{th}}$ oscillator.
![Log-log plot of complexity of the $(1\!+\!1)$-dimensional free field theory at the critical point $t\!=\!0$ vs. the quench rate $\delta t$ (measured in units of the lattice spacing), with $\omega_0\!=\!0.005$. The straight-line fit (blue) reveals linear scaling in the fast regime.\[fig:QC-field\]](scalingsFT.pdf){width="0.95\columnwidth"}
Since we are interested in the behaviour of complexity as the system passes through the critical point of the quench, it is sufficient to evaluate this function at $t=0$; see Fig. \[fig:QC-field\]. This then allows us to extract the universal scaling behaviours, which we examine in more detail in the next section.
**Universal scalings in complexity**
We now wish to examine the presence of universal scalings of the critical complexity with respect to the quench rate. In particular, the contributions from individual momentum modes to $\CC(0)$ in are plotted in Fig. \[fig:QC-mode\]. We find that all modes go to zero in the sudden-quench limit $\delta t\rightarrow0$, which is consistent with results for instantaneous quenches. For all $k>0$, we observe mode-dependent saturation in the slow regime $\delta t\rightarrow\infty$, consistent with what one expects from KZ. In the adiabatic approximation, the KZ scale arises from the Landau criterion for the breakdown of adiabaticity, \_[t\_ ]{} = 1 , \[eq:KZ-cond\] where $t_\mr{KZ}$ is the Kibble-Zurek time and $E$ is the time-dependent mass gap from criticality. For the profile , one finds $t_\mr{KZ}\!\approx\!\sqrt{ \delta t/ \omega_0}$, at which time the frequency is $\omega_\mr{KZ}(k)\!=\!\sqrt{ 4 \sin^2 \tfrac{k}{2} + m^2(t_\mr{KZ}) }
%= \sqrt{ 4 \sin^2 \tfrac{k}{2} + \tfrac{t_{KZ}^2}{\delta t^2} }
\!\approx\!\sqrt{ 4 \sin^2 \tfrac{k}{2} + \tfrac{\omega_0}{\delta t} }$, where we have used the fact that $m^2(t_\mr{KZ})\!\sim\! \omega_0^2 t_\mr{KZ}^2/\delta t^2$, since in the slow regime $\delta t\!>\!t_\mr{KZ}$ by definition. Hence the KZ scaling for the $k^\mathrm{th}$ mode may be extracted by calculating the complexity at this frequency. One finds logarithmic KZ scaling in the slow regime for $\delta t\!<\!\frac{\omega_0}{4}\csc^2 \frac{k}{2}$. As soon as $\delta t$ exceeds this value, we observe saturation in the frequency (to $2 \sin \frac{k}{2}$), and hence also in complexity to \_\^k = ( ) .\[eq:Csat\] The KZ approximation is superimposed on the exact results in Fig. \[fig:QC-mode\], which clearly shows agreement with the saturation value in the large-$\delta t$ limit.
![Single-mode contributions to the complexity at the critical point $t\!=\!0$ for $\omega_0\!=\!0.005$ and $k\!=\!\{0.006, 0.111, 0.216, 0.320, 0.425 \}$ (resp. red, orange, yellow, green, blue). For large $\delta t$, the exact solutions (dotted) agree with the saturation values predicted from KZ (solid).\[fig:QC-mode\]](scalingsk2.pdf){width="0.95\columnwidth"}
The critical complexity of the zero mode $k\!=\!0$ exhibits universal scalings in both the slow and fast regimes. Indeed, this same behaviour is exhibited by the single quantum oscillator we initially introduced upon sending the frequency to zero (i.e., we take the $\omega_-$ solution for the two-oscillator case above). Unlike higher modes, the zero mode does not saturate at large $\delta t$ since the logarithmic scaling is always present. From the KZ analysis above, we can derive the universal coefficient of the log as $\frac{1}{4}$, which is confirmed by fitting the exact solution, as shown in Fig. \[fig:QC-scalings\]. We note that the KZ scaling exhibited by entanglement entropy under a critical quantum quench has the same form, but with a $\frac{1}{6}$ coefficient instead [@Caputa:2017ixa; @Nishida:2017hqd]. Meanwhile in the fast regime ($\delta t<1$ in lattice units), the complexity grows linearly with $\delta t$. While these scalings are present for higher modes as well, they are confined to increasingly narrow regions of $\delta t$ for larger values of $k$.
![Zero-mode contribution to $\CC(0)$ as a function of the quench rate $\omega_0\delta t$, with $\omega_0\!=\!0.005$. The complexity scales linearly in the fast regime ($\ln C/\ln \delta t=1$, blue), and smoothly transitions to a logarithmic scaling $\tfrac{1}{4}\log \delta t$ in the slow regime (red). The transition to KZ occurs at $\omega_0\delta t\!\sim\!1$ which in this case is $\delta t\!\sim\!200$ in lattice units.\[fig:QC-scalings\]](scalings0-v3.pdf){width="0.95\columnwidth"}
**COMPLEXITY VS. ENTANGLEMENT**
One of the main motivations for the holographic complexity proposals was the observation that the information contained in the reduced density matrix of any spatial bipartition of the CFT Hilbert space, as encoded in the entanglement entropy, is generally insufficient to determine the entire bulk geometry [@Susskind:2014moa] (see also [@Freivogel:2014lja] and references therein). One can then ask whether complexity provides another take on the information contained in reduced density matrices. Indeed, recent proposals for the complexity of subregions in holography – that is, on the bulk side – have been made in [@Carmi:2016wjl; @Alishahiha:2015rta; @Ben-Ami:2016qex; @Abt:2017pmf; @Abt:2018ywl]. However, since the field-theoretic notion of complexity above is defined for pure states, it is not *a priori* clear how to define complexity for the reduced density matrix corresponding to some spatial subregion.
A particularly natural extension of existing pure-state definitions to this case is the *complexity of purification*, recently outlined in [@Agon:2018zso], in which the complexity of the subsystem is defined by minimizing over the complexities of all possible purifications (see also [@Stoltenberg:2018ink]). Applying our quench set-up above to the case of two oscillators allows us to quantify this proposal, by considering the reduced density matrix corresponding to a single oscillator, say $x_1$, and purifying within the original Hilbert space of Gaussian states (i.e., without ancilla). The total wavefunction depends on six real parameters, three of which we fix by our knowledge of the covariance matrix for oscillator $x_1$. Minimizing over the remaining three parameters then gives the complexity of purification for the subsystem, which we shall denote $\CC_A$ in reference to a generic subsystem $A$ and its complement $\bar{A}$.
![Comparison of the complexity as a function of time $t$ of the original target state (solid) and the optimum purification (dashed) for $\delta t=10$ (blue) and $\delta t=1$ (red), with $\omega_\mr{R}=0.5$ for both oscillators. Note that the latter never exceeds the former, and is always greater than $\CC/2$; that is, the complexity of purification appears to satisfy superadditivity . We have tested this conjecture numerically for $\sim\!70,\!000$ cases.\[fig:purification\]](Purification_w1w2.pdf){width="0.95\columnwidth"}
As observed in Fig. \[fig:purification\], the complexity of purification satisfies $\CC/2\leq\CC_A\leq\CC$, which we have verified numerically for a wide range of values in the six-parameter landscape spanned by the components of the covariance matrix. The upper inequality is saturated iff the original target state happens to be the least complex state among all possible purifications. Meanwhile, the lower inequality is saturated iff the original target state is a product state with respect to the chosen bipartition; i.e., subsystem $A$ describes a pure state, $S_A=0$. This can be understood from the fact that the purification process seeks to produce a state which is as close to the reference state as possible, since the latter has minimum complexity by fiat. Since in this case the reference state is an unentangled product state, the minimum purification is one in which the complement $\bar A$ is also an unentangled product state—but this is only possible if the original state is a tensor product of the form $\HH_A\otimes\HH_{\bar A}$, otherwise the entanglement across the bipartition prevents one from obtaining the reference state in the restriction $\bar A$. While one should exercise caution in blithely generalizing from this simple two-oscillator case, the above leads us to suggest that the complexity of subsystems is *subadditive*: \_A+\_[|A]{} .\[eq:super\] As observed in [@Agon:2018zso], this agrees with the holographic CA proposal, but not with the CV proposal, which is superadditive.\
**OUTLOOK**
Quenches represent tractable models of dynamical quantum systems in which complexity can be better understood, as well as yield new physical insights; e.g., we have found that complexity can be used to extract universal scalings. We have also examined the complexity of subregions (i.e., mixed states) via their purifications. Since complexity encodes global information about the state, it is sensitive to features to which entanglement is blind. We find that subregion complexity appears to satisfy superadditivity , which is consistent with the CA proposal. While it would be premature to take definitive lessons for holography from such simple free field models, this may provide further hints as to the proper notion of complexity in holographic field theories, and thereby shed light on ongoing efforts to reconstruct bulk spacetime in AdS/CFT.
**ACKNOWLEDGMENTS**
We thank E. M. Brehm, S. R. Das, D. A. Galante, E. López, J. Magan, T. Takayanagi, M. Walter, and the co-authors of [@TFD] for helpful conversations. We would especially like to thank H. Marrochio, R. C. Myers, and M. Smolkin for comments on a draft of this manuscript. HC, DD, MH and RJ acknowledge the hospitality of MITP during the “Modern Techniques for AdS and CFT” program during the completion of this project. HC is partially supported by the Konrad-Adenauer-Stiftung through their Sponsorship Program for Foreign Students. PC is supported by the Simons Foundation through the “It from Qubit” collaboration and by the JSPS starting grant KAKENHI 17H06787. The Gravity, Quantum Fields and Information group at AEI is generously supported by the Alexander von Humboldt Foundation and the Federal Ministry for Education and Research through the Sofja Kovalevskaja Award.
**SUPPLEMENTARY MATERIAL (APPENDICES)**
A. Circuit complexity for $\mathbb{H}^2$ {#appx:complexity}
========================================
In this appendix, we explain the calculation of the complexity of the state . Our approach closely follows that of [@Jefferson:2017sdb], but adapted at the level of covariance matrices as in [@TFD]; the reader is referred to these works for more details. As stated in the main text, one seeks the optimum circuit $U$ which acts on the reference state to produce the target state according to . Identifying, and associating a well-defined length to, this optimum circuit requires geometrizing the problem [@Nielsen:2005mn1; @Nielsen:2006mn2; @Nielsen:2007mn3]. To proceed, one represents the circuit as a path-ordered exponential, U(s)= ,\[eq:Um\] where the path parameter $s$ is chosen to run from 0 at the reference state to 1 at the target state (i.e., $s\!=\!1$ in ). The matrix generators $M_I$ represent the algebra of gates, while the parameters $Y^I$ can the thought of as turning these gates on and off at specified points along the path (these will be given a precise interpretation below as the components of the frame bundle that we use to construct the inner product, and thereby the geometry). However, in order to obtain a state-independent representation of our gate set, we shall instead work at the level of the covariance matrices, which completely characterize the Gaussian state. Hence one views the circuit as a matrix acting on the covariance matrix representation of the states in the usual manner; i.e., the evolution equation becomes G\_=UG\_U\^T ,\[eq:evoGapx\] where the matrix elements of $G$ are given by G\^[ab]{}=\^a\^b+\^b\^a ,\[eq:Gbasis\] where $\xi^a\equiv\{x^1,p^1,\ldots x^N,p^N\}$ are the dimensionless phase-space operators for $N$ oscillators; cf. in the main text.
Now, one has the freedom to choose the gate set generated by $M_I$ with which to build the circuit. Since the gate set ultimately determines the geometry, one will obtain different geodesics – and hence different complexities – for different choices. Provided the algebra that generates the set of gates is sufficient to produce the target state however, there are no rules for how to select it, and indeed several different choices have been analyzed in the literature. For example, [@Jefferson:2017sdb] chose a set of scaling and entangling gates that formed a representation of $\mathrm{GL}(N,\reals)$, while [@Chapman:2017rqy; @Alves:2018qfv] worked instead with $\mathrm{SU}(1,1)$. In order to study the time-dependence of the TFD, [@TFD] chose a representation of $\Sp(2N,\reals)$, on the basis that this is the general group of transformations that preserves the canonical commutation relations.
Given this freedom, we will be motivated by physical considerations in selecting our gates, namely, we wish to use the minimum set of gates sufficient to effect our particular class of target states. Since we shall work in the normal-mode basis, where the ground state factorizes as in , it suffices to compute the complexity for a single mode (the generalization to $N$ oscillators is then straightforward, and is given in the main text). For the reference and target states under consideration, we have G\_=
& 0\
0 & \_
,G\_=
& -\
- &
.\[eq:Gs\] Hence we wish to select a set of gates that allows us to interpolate between these two states. This can be done with a subalgebra of $\frak{sp}(2,\reals)$. We shall denote the generators in the full algebra by W=xp+px , V=x\^2 , Z=p\^2 ,\[eq:full\] where $x,p$ are the dimensionless variables introduced below . One can check that these generators satisfy the algebra =2V ,=-2Z ,=-2W .Thus one sees that the $W$ and $V$ gates close to form a subalgebra. Hence, in contrast to [@TFD], which analyzed circuits using the full algebra of $\frak{sp}(2,\reals)$, we shall restrict our circuits to the submanifold corresponding to the group elements X\_W=e\^[W]{} ,X\_V=e\^[V]{} ,\[eq:Xgates\] where $0<\eps\in\reals$. The corresponding matrix generators are then M\_1=
-1 & 0\
0 & 1
,M\_2=
0 & 0\
& 0
,\[eq:M\] where $M_1$ corresponds to $W$ and $M_2$ corresponds to $V$. Note that these are not orthonormal, but instead satisfy M\_I\^TM\_J=2\_[IJ]{} .\[eq:norm\] The gates corresponding to are then obtained by exponentiating these matrix elements: Q\_1=e\^[M\_1]{}=
e\^[-]{} & 0\
0 & e\^
, Q\_2=e\^[M\_2]{}=
1 & 0\
& 1
.\[eq:gates\] Let us pause briefly to remark on the scale $\varpi$, introduced in the dimensionless variables $x\!=\!\varpi X$, $p\!=\!P/\varpi$. We refer to this as the gate scale because, had we written the generators in terms of the dimensionful variables $X,P$, then both $V$ and $Z$ would contain factors of $\varpi^2$ in order to make the exponents in dimensionless. Setting $\varpi=M$ is then a natural choice in order to avoid introducing an auxiliary scale into the problem, and we shall do so henceforth.
Now, to find the geometry in which the circuit lives, we must find a suitable parametrization of the general group element with which to construct the metric. As shown in appendix B, a convenient choice is U=
z & 0\
& 1
.\[eq:Uxz\] The reason for this choice is that, upon isolating the components of the frame bundle in the usual manner [@Jefferson:2017sdb], Y\^I= (where the factor of $\frac{1}{2}$ is due to the normalization of the generators ), up to an unimportant overall normalization, the most general positive definite line element is s\^2&=g\_[IJ]{}Y\^IY\^J = , where we have taken g\_[IJ]{}=
1 & -A\
-A& A\^2
. \[eq:gPen\] In this expression, $A$ and $\sigma$ are penalty factors which account for different weighting of different gates. In contrast to metrics obtained in [@Jefferson:2017sdb; @TFD], their presence does not prevent us from having a closed form expression for geodesic length between two points $(z_0,y_0)$ and $(z_1,y_1)$, since we always deal with the metric on hyperbolic disc for which the distance function is known. However, for reasons which will become apparent below, we shall focus on the special case $\sigma\!=\!0$ and $A\!=\!2^{-1/2}$ for the remainder of this text. In this case, the distance function on the hyperbolic disc, \_[01]{}=X\_[01]{}+,\[eq:dist\] is evaluated with X\_[01]{} .\[eq:X01\] Now, the optimum circuit is the minimum geodesic on $\mathbb{H}^2$ that connects the reference $G_\mr{R}$ and target $G_\mr{T}$ states, and the complexity of the latter is given by the length of this circuit, . It therefore remains simply to express the initial and final coordinates, $(z_0,y_0)$ and $(z_1,y_1)$, in terms of the physical parameters of the problem at hand, i.e., the frequencies of the reference and target states.
However, one important caveat is in order: the generalization of this prescription to $N$ oscillators relies on the fact that for our quench solution, both the reference and target states remain block diagonal. When considering purifications however, we begin with a dense covariance matrix representing the target state, and must first bring it to block-diagonal form prior to computing the complexity. This can be done by defining a squeezing operator S={,,…,,} , where $\omega_1$ through $\omega_N$ are the reference state frequencies for each of the $N$ oscillators. The definition of $S$ is such that its action on the reference state produces the identity matrix, SG\_S\^T= . Applying this operator to the target state then allows us to block-diagonalize the latter without introducing off-diagonal terms in the reference state. Therefore in order to compute complexity for such states, we will evaluate the distance between the identity and the squeezed target state G\_SG\_S\^T=
& -\
- &
.
To proceed, we impose boundary conditions on the circuit $U$ , where $z(s), y(s)$ are functions of the path parameter $s\in[0,1]$ (that is, is merely the coordinate representation of ). At $s\!=\!0$, the circuit has not produced any change in the reference state, and hence the initial condition is U(0)G\_U(0)\^T=G\_U(0)= , which holds for $G_\mr{R}$ in for arbitrary $\omega_\mr{R}$. Conversely, at $s\!=\!1$ the circuit should satisfy , with $G_\mr{R}\!\rightarrow\!\mathbbm{1}$ and $G_\mr{T}\!\rightarrow\!\tilde G_\mr{T}$, which enables one to solve for the final coordinates in terms of the physical frequencies; one finds: z\_0, y\_0=1,0 ,z\_1, y\_1=,- . Substituting these into the expression for the geodesic length , we obtain . Due to the ability to simultaneously diagonalize $G_\mr{R}=\mathbbm{1}$ and $\tilde G_\mr{T}$, the result for $N$ oscillators is simply $N$ copies of the hyperbolic disc.
As alluded above, one can also explicitly solve the geometry in the presence of the general penalty factors in . These were originally introduced in [@Jefferson:2017sdb] in order to impose a notion of locality on the circuit, without which the gates are non-local; the interested reader is referred to the discussion therein for more details. In this case, the result is rather unwieldy and unilluminating, so we refrain from writing it out here. However, it is worth noting that the scaling behaviours observed in the main text are insensitive to the choice of penalty factors. The reason is two-fold: first, it is a general fact that one can absorb the penalty factor $A$ by rescaling the imaginary component of the target state $b\rightarrow b/A$. Second, it turns out that when specifying to the quench solution considered in the main text, this imaginary component vanishes for the full parameter range. Therefore the complexity only depends on the penalty factors through $\sin\sigma$, and one can furthermore show that the unique range of this parameter is $\sigma\in[0,\pi/2)$ (the complexity diverges at precisely $\sigma\!=\!\pi/2$). For values of $\sigma$ within this range, the effect is merely a constant shift in Fig. \[fig:QC-scalings\], and a slightly more gradual transition between the linear and logarithmic $\tfrac{1}{4}\ln\delta t$ regimes. The value of the coefficient in the latter may be obtained by numerical fitting, and is therefore sensitive to the width of this transition zone, but remains largely unchanged.
B. Geodesics on $\mathbb{H}^2$ {#appx:geometry}
==============================
In this appendix, we show that the geometry corresponding to the use of the two gates can be derived as an embedding in AdS, which enables us to readily obtain the geodesic distance . We begin by finding a suitable parametrization for the general group element, which will represent the circuit $U(s)$ . Of course, the most naïve way to express the generic group element generated by is U={M\_1+M\_2} =
e\^[-]{} & 0\
& e\^
where $\mu,\nu\in\reals$. However, by making the change of variables =-z ,=z , this becomes U=
z & 0\
& 1
. which is . As mentioned above, the reason for this choice is that the resulting metric becomes that of the hyperbolic plane. In particular, choosing $\sigma=0$, $A=2^{-1/2}$ yields s\^2=\^2 ,\[eq:ds\] where $\ell\equiv1/2$. One can reproduce $\mathbb{H}^2$ via the standard embedding of the pseudosphere in $\mathbb{R}^{d,1}$: X\^0\^2-X\^1\^2+X\^2\^2=-\^2 ,\[eq:embed\] where the metric of the embedding space is s\^2=-\_[MN]{}X\^MX\^N ,\_[MN]{}={-1,1,-1} .\[eq:dsMN\] One can check that the constraint allows the following choice of parameters: X\^0= , X\^1= , X\^2= , upon which reproduces . Then, for points $(z_0,y_0)$ and $(z_1,y_1)$, \^2X\_[01]{}\_[MN]{}X\^M(z\_0,y\_0)X\^N(z\_1,y\_1) , yields precisely , with the geodesic distance given by .
C. Complexity of purification {#appx:cpurification}
=============================
In this appendix, we explain how to compute the complexity of purification for the two-oscillator system considered in the main text, i.e., to associate a complexity to the subsystem consisting of a single oscillator. The position-space wavefunction for the total system is x\_1,x\_2=(-x\^2\_1-x\^2\_2-\_3x\_1x\_2) ,\[eq:totgauss\] where $\omega_{i}\!\equiv\! a_{i}\!+\!ib_{i}$ with $a_i,b_i\!\in\!\mathbb{R}$, and the normalization is $\mathcal{N}\!=\! (a_{1}a_{2}\!-\!a_{3}^2)^{1/4}/\sqrt{\pi}$, cf. . Note that we must have $a_i>0$ and $a_1a_2>a_3^2$ in order for the wavefunction to be well-defined. The subsystem corresponding to the first oscillator is described by the reduced density matrix \_[1]{}(x\_[1]{},y\_[1]{})=\^2e\^[-(x\^2\_1+y\^2\_1)+x\_1 y\_1]{}e\^[-(x\^2\_1-y\^2\_1)]{} ,\[eq:RDM1\] where we have defined \_[1]{}a\_[1]{} - ,\_[2]{}-b\_[1]{}+ , . Thus, of the original six (real) parameters on which the total system depends, three are fixed by specifying data for the subsystem. Our strategy is then to minimize the complexity over the remaining three parameters. That is, we purify the subsystem with a second oscillator, so that the total wavefunction can be written in the form , and the parameters of this purification are those which effect the minimum complexity.
To rephrase this in the language of covariance matrices employed in the main text: the two-oscillator system is described by a four-by-four covariance matrix. In the basis $\xi^a=\{x^1,p^1,x^2,p^2\}$ (cf. ), this is naturally organized into diagonal two-by-two blocks that describe the subsystems (i.e., the individual oscillators), and off-diagonal blocks which describe the entanglement between them. It is straightforward to show that specifying the three parameters $\alpha_1,\alpha_2,\beta$ in the reduced density matrix completely fixes the first diagonal block corresponding to the $x_1$ oscillator; explicitly, G\_=
&\
&
. \[eq:G1\] Note that this corresponds to fixing the expectation values $\langle x_{1}^2 \rangle$, $\langle p_1^2\rangle$, and $\langle x_1p_1+p_1x_1\rangle$; see the discussion below Fig. \[fig:purification\].
We then apply our formula for the complexity with $N\!=\!2$, =, \[eq:C2osc\] where $\chi_i\!=\!\frac{1}{2}\mr{tr}\,\tilde G_i$, and $\tilde G_i\!=\!SG_iS^T$ is the squeezed covariance matrix, cf. . This expression depends on six real parameters, three of which are fixed in terms of $\alpha_1,\alpha_2,\beta$. To simplify the numerical minimization procedure over the remaining three parameters, it is convenient to perform a change of variables to spherical coordinates, so that two of them become compact; the range of the remaining non-compact (radial) parameter can then be constrained by experimentation.
After repeating this process for the second oscillator (that is, fixing the parameters in the reduced density matrix $\rho_2$, which are *a priori* independent), we can compute the subsystem complexities $\CC_A$ and $\CC_{\bar A}$ used in verifying the superadditivity property . It would of course be interesting to test this conjecture for varying subsystem sizes, rather than the fixed equipartition forced upon us by the two-oscillator case. Unfortunately, it is computationally challenging to extend this procedure to $N\!>\!2$, but we hope to return to this question in future work.
Let us briefly contrast the complexity of purification with the information accessed by entanglement. Consider the case in which we know the reduced density matrices for both subsystems, which for simplicity we may fix symmetrically, i.e., $\langle x_{1}^2 \rangle = \langle x_{2}^2 \rangle$, $\langle p_{1}^2 \rangle = \langle p_{2}^2 \rangle$ and $\langle x_{1} \, p_{1} \rangle = \langle x_{2} \, p_{2} \rangle$. This leaves a one-parameter freedom in the full covariance matrix, which we can choose to be $\langle x_{1} \, p_{2} \rangle = \langle x_{2} \, p_{1} \rangle$. Obviously, the entanglement entropy for either subsystem will be insensitive to the information this correlator contains about the total state. In contrast, the complexity varies as a function of this parameter, and therefore provides a complementary probe to entanglement for a fixed bipartition. We stress however that this dependence is not necessarily monotonic, which prevents us from directly identifying complexity as an order parameter for the wavefunction. It would certainly be interesting to investigate the relative merits of entanglement and complexity in this regard, and the reader is referred to the upcomming work [@notyet] for a more thorough treatment of entanglement in this context.\
D. Scalings and choice of norm {#appx:normscale}
==============================
Here we comment briefly on how the scaling behaviours of the critical complexity emerge in the field theory in different norms. As shown in Fig. \[fig:QC-scalings\], the zero-mode complexity exhibits a linear growth with $\delta t$ which smoothly transitions to the $\log\!\delta t$ KZ behaviour at $\delta t^{-1}\!\sim\!\omega_0$. As we move towards the UV, both the exact solution as well as the KZ approximation for non-zero modes (see Fig. \[fig:QC-mode\]) saturate to smaller values of complexity at lower values of $\delta t$. Thus when adding the modes using the $L_p$-norm (with $p\!\geq\!2$), the field theory critical complexity is dominated by the zero mode and hence inherits its scaling behaviour. If we instead use the $L_1$-norm, the mode-by-mode saturation values are no longer sufficiently suppressed for the field theory to simply inherit the scaling behaviour of the zero mode, and thus it is unclear whether universal scalings can be extracted in this case. However, previous studies [@Jefferson:2017sdb; @Chapman:2017rqy] found that $L_1$ exhibits better agreement with holography than $L_2$. Hence it would be very interesting to explore critical quenches holographically (non-critical quenches have been studied in, e.g., [@Moosa:2017yvt; @Chapman:2018dem; @Chapman:2018lsv; @Ageev:2018nye]).
Meanwhile in the case of entanglement entropy studied in [@Caputa:2017ixa], the authors observed only a narrow range in which the fast scaling is applicable with quadratic dependence on the quench rate $\delta t$. We believe that the range of fast scalings in [@Caputa:2017ixa] depends on the subsystem size. The analogue of this additional dimensionful parameter in the case of complexity is the reference state scale, which indeed controls the range over which we observe fast scalings. Additionally, we note that in this work we consider field theories on a circle; in the decompactification limit, we expect the zero mode to give a subdominant contribution to the dynamics, as observed recently in [@TFD].
|
---
abstract: 'In [@Sarria1], we derived representation formulae for spatially periodic solutions to the generalized, inviscid Proudman-Johnson equation and studied their regularity for several classes of initial data. The purpose of this paper is to extend these results to larger classes of functions including those having arbitrary local curvature near particular points in the domain.'
address:
- |
Department of Mathematics\
University of New Orleans\
New Orleans, LA, 70148, USA
- |
Department of Mathematics\
University of New Orleans\
New Orleans, LA, 70148, USA
author:
- Alejandro Sarria
- Ralph Saxton
title: 'The Role of Initial Curvature in Solutions to the Generalized Inviscid Proudman-Johnson Equation'
---
Introduction
============
\[sec:intro\]
In this article, we extend the analysis initiated in [@Sarria1] concerning blow-up, and blow-up properties, in solutions to the initial boundary value problem for the generalized, inviscid Proudman-Johnson equation ([@Proudman1], [@Childress], [@Okamoto1]) $$\label{eq:nonhomo}
\begin{cases}
u_{xt}+uu_{xx}-\lambda u_x^2=I(t),\,\,\,\,t>0,
\\
u(x,0)=u_0(x),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\in[0,1],
\\
I(t)=-(\lambda+1)\int_0^1{u_x^2\,dx},
\end{cases}$$ where $\lambda\in\mathbb{R}$ and solutions are subject to periodic boundary conditions $$\label{eq:pbc}
\begin{split}
u(0,t)=u(1,t),\,\,\,\,\,\,u_x(0,t)=u_x(1,t).
\end{split}$$
We note that the equation arises in several important applications, in the presence or absence of the nonlocal term $I(t)$. For $\lambda=-1,$ (\[eq:nonhomo\])i), iii) reduces to the inviscid Burgers’ equation of gas dynamics differentiated once in space. If $\lambda=-1/2,$ the Hunter Saxton equation (HS) describes the orientation of waves in a massive director field of a nematic liquid crystal ([@Hunter1], [@Bressan1], [@Dafermos1], [@Yin1]). For periodic functions, the HS equation also has a deep geometric meaning as it describes geodesics on a group of orientation preserving diffeomorphisms on the unit circle modulo rigid rotations with respect to a right-invariant metric ([@Khesin1], [@Bressan1], [@Tiglay1], [@Lenells1]). If $\lambda=\frac{1}{n-1},\,n\geq2,$ (\[eq:nonhomo\]) i), iii) can be obtained directly from the $n-$dimensional incompressible Euler equations
$$\boldsymbol{u}_t+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}=-\nabla p,\,\,\,\,\,\,\,\,\nabla\cdot \boldsymbol{u}=0$$
using stagnation point-form velocities $\boldsymbol{u}(x,\boldsymbol{x}^\prime,t)=(u(x,t),-\lambda\boldsymbol{x}^\prime u_x(x,t))$, $\boldsymbol{x}^\prime=\{x_2,...,x_n\},$ or through the cylindrical coordinate representation $u^r=-\lambda ru_x(x,t),$ $u^{\theta}=0$ and $u^x=u(x,t)$, where $r=\left|\boldsymbol{x}^\prime\right|,$ ([@Childress], [@Weyl1], [@Saxton1], [@Okamoto1], [@Escher1]). Finally, in the local case $I(t)=0$, the equation appears as a special case of Calogero’s equation $$u_{xt}+uu_{xx}-\Phi(u_x)=0$$ for arbitrary functions $\Phi(\cdot)$ ([@Calogero1]).
In [@Sarria1] we derived representation formulae for periodic solutions to (\[eq:nonhomo\])-(\[eq:pbc\]) and, for several classes of mean-zero initial data, examined their $L^p$ regularity for $p\in[1,+\infty]$. For convenience of the reader, the main results established in [@Sarria1] are summarized in Theorems \[thm:sarria1\]-\[thm:sarria2\] below.
\[thm:sarria1\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]). There exist smooth, mean zero initial data such that:
1. \[it:1\] For $\lambda\in(-\infty,-2]\cup(1,+\infty)$, there is a finite $t_*>0$ such that $\lim_{t\uparrow t_*}\left|u_x(x,t)\right|=+\infty$ for every $x\in[0,1]$. Additionally, the blow-up is two-sided (two-sided, everywhere blow-up).
2. \[it:2\] For $\lambda\in(-2,0),$ there is a finite time $t_*>0$ and a finite number of $\underline x_j\in[0,1]$, $j\in\mathbb{N}$, such that $\lim_{t\uparrow t_*}u_x(\underline x_j,t)=-\infty$ (one-sided, discrete blow-up).
3. \[it:3\] For $\lambda\in[0,1],$ solutions persist globally in time. More particularly, these vanish as $t\uparrow t_*=+\infty$ for $\lambda\in(0,1)$ but converge to a nontrivial steady state for $\lambda=1.$
For $t_*>0$ as in Theorem \[thm:sarria1\] above, Theorem \[thm:lpintro\] below examines $L^p(0,1)$ regularity of $u_x$ for $t\in[0,t_*)$ and $p\in[1,+\infty)$.
\[thm:lpintro\] Let $u$ in Theorem \[thm:sarria1\] be a solution to the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) defined for $t\in[0,t_*)$. Then
1. \[it:lp2\] For $p\geq1$ and $\frac{2}{1-2p}<\lambda\leq1,\, \lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty.$
2. \[it:lp1\] For $p\in(1,+\infty)$ and $\lambda\in(-\infty,-2/p]\cup(1,+\infty)$, $\lim_{t\uparrow t_*}\left\|u_x\right\|_p=+\infty$.
3. \[it:ener\] The energy $E(t)=\left\|u_x\right\|_2^2$ diverges if $\lambda\in\mathbb{R}\backslash(-2/3,1]$ as $t\uparrow t_*$ but remains finite for $t\in[0,t_*]$ otherwise. Moreover, $\dot E(t)$ blows up to $+\infty$ as $t\uparrow t_*$ when $\lambda\in\mathbb{R}\backslash[-1/2,1]$ and $\dot E(t)\equiv0$ for $\lambda=-1/2$; whereas, $\lim_{t\uparrow t_*}\dot E(t)=-\infty$ if $\lambda\in(-1/2,-2/5]$ but remains bounded when $\lambda\in(-2/5,1]$ for all $t\in[0,t_*]$.
See §\[sec:preliminaries\] for details on the class of initial data used to establish Theorems \[thm:sarria1\] and \[thm:lpintro\]. Lastly, let $PC_{\mathbb{R}}(0,1)$ denote the family of piecewise constant functions with zero mean in $[0,1]$. Then, in [@Sarria1] we proved the following:
\[thm:sarria2\] For the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]),
1. \[it:4\] Suppose $u_0''(x)\in PC_{\mathbb{R}}(0,1)$ and $\lambda>1/2$. Then, there exist solutions and a finite $t_*>0$ for which $u_x$ undergoes a two-sided, everywhere blow-up as $t\uparrow t_*$. If $\lambda<0,$ a one-sided discrete blow-up may occur instead. In contrast, for $\lambda\in[0,1/2]$, solutions may persist globally in time. More particularly, these either vanish as $t\uparrow t_*=+\infty$ if $\lambda\in(0,1/2)$, or converge to a nontrivial steady-state for $\lambda=1/2$.
2. \[it:5\] Suppose $u_0'(x)\in PC_{\mathbb{R}}(0,1)$ and assume solutions are defined for all $t\in[0,T],\, T>0.$ Then no $W_{\mathbb{R}}^{1,\infty}(0,1)$ solution may exist for $T\geq t_*$, where $0<t_*<+\infty$ if $\lambda<0$, and $t_*=+\infty$ for $\lambda\geq0$. Further, $\lim_{t\uparrow t_*}\left\|u_x\right\|_1=+\infty$ when $\lambda<-1$, while $$\lim_{t\uparrow t_*}\left\|u_x\right\|_p=
\begin{cases}
C,\,\,\,\,\,\,\,\,&-\frac{1}{p}\leq\lambda<0,\,\,\,\,\,\,\,\,\,p\geq1,
\\
+\infty,\,\,\,\,\,\,\,\,&-1\leq\lambda<-\frac{1}{p},\,\,\,\,p>1,
\end{cases}$$ where the constants $C\in\mathbb{R}^+$ depend on the choice of $\lambda$ and $p.$
The reader may refer to [@Sarria1] for details, and the works [@Okamoto2], [@Aconstantin1], [@Hunter2], [@Wunsch2], [@Wunsch3], [@Wunsch1] for additional background. The purpose of this work is to extend the above results to initial data which belongs to classes of functions with varying concavity profile near certain points in the domain. More particularly, we suppose throughout that $u_0'(x)$ is bounded and, at least, $C^0(0,1)$ $a.e.$ Then, for $\lambda>0$, we will assume there are constants $q, M_0\in\mathbb{R}^+$ and $C_1\in\mathbb{R}^-$, and a finite number of points $\overline\alpha_i\in[0,1]$ such that, near $\overline\alpha_i$, $$\label{eq:expnew02}
u_0^\prime(\alpha)\sim M_0+C_1\left|\alpha-\overline\alpha_i\right|^q.$$ Analogously, for $\lambda<0$, we suppose there are constants $C_2\in\mathbb{R}^+$, $m_0\in\mathbb{R}^-$, and a finite number of locations $\underline\alpha_j\neq\overline\alpha_i$ in $[0,1]$ such that, in a neighbourhood of $\underline\alpha_j$, $$\label{eq:expnew002}
u_0^\prime(\alpha)\sim m_0+C_2\left|\alpha-\underline\alpha_j\right|^q.$$ We refer to §\[subsec:dataclass\] for specifics of the above. It is worth mentioning that, for $q\in(0,1)$, the above local estimates may lead to cusps in the graph of $u_0'$, therefore possible jump discontinuities in $u_0''$ of *infinite* magnitude across $\overline\alpha_i$ and/or $\underline\alpha_j$. In contrast, a jump discontinuity of *finite* magnitude in $u_0''$ may occur if $q=1$. As we will see in the coming sections, the finite or infinite character in the size of this jump plays a decisive role, particularly in the formation of spontaneous singularities for the special case of stagnation point-form solutions to the three dimensional incompressible Euler equations.
The remaining of the paper is organized as follows. In §\[sec:preliminaries\], we provide an outline for the derivation of the representation formulae established in [@Sarria1] and provide further details on the class of initial data to be considered in this article. Then, new blow-up results are stated and proved in §\[sec:blowup\], while specific examples are to be found in §\[sec:examples\].
Preliminaries {#sec:preliminaries}
=============
The General Solution
--------------------
\[subsec:sol\]
In [@Sarria1], we used the method of characteristics to derive a representation formula for periodic solutions to (\[eq:nonhomo\]). For convenience of the reader, below we give a brief outline of the derivation.
Define the characteristics, $\gamma,$ as the solution to the initial value problem $$\label{eq:cha}
\dot\gamma(\alpha,t)=u(\gamma(\alpha,t),t),\,\,\,\,\,\,\,\,\,\,\,\,\gamma(\alpha,0)=\alpha\in[0,1],$$ so that $$\label{eq:jacid}
\begin{split}
\dot\gamma_{\alpha}(\alpha,t)=u_x(\gamma(\alpha,t),t)\cdot\gamma_{\alpha}(\alpha,t).
\end{split}$$ Then, using (\[eq:nonhomo\])i), iii) and the above, we obtain $$\label{eq:chain}
\begin{split}
\ddot\gamma_\alpha&=(u_{xt}+uu_{xx})\circ\gamma\cdot\gamma_\alpha+(u_x\circ\gamma)\cdot\dot\gamma_\alpha
\\
&=(u_{xt}+uu_{xx})\circ\gamma\cdot\gamma_\alpha+u_x^2\circ\gamma\cdot\gamma_\alpha
\\
&=(\lambda+1)\left(u_x^2\circ\gamma-\int_0^1{u_x^2dx}\right)\cdot\gamma_\alpha
\\
&=(\lambda+1)\left((\gamma^{-1}_{\alpha}\cdot\dot\gamma_{\alpha})^2-\int_0^1{u_x^2dx}\right)\cdot\gamma_\alpha\,,
\end{split}$$ which for $\lambda\neq0$, $I(t)=-(\lambda+1)\int_0^1{u_x^2dx}$, and $\omega(\alpha,t)=\gamma_{\alpha}(\alpha,t)^{-\lambda}$, can be written as $$\label{eq:nonhomo2}
\begin{split}
\ddot\omega(\alpha,t)+\lambda I(t)\omega(\alpha,t)=0.
\end{split}$$ Assume we have two linearly independent solutions $\phi_1(t)$ and $\phi_2(t)$ to (\[eq:nonhomo2\]) satisfying $\phi_1(0)=\dot{\phi}_2(0)=1$ and $\dot{\phi}_1(0)=\phi_2(0)=0$. Then, since $\dot{\omega}=-\lambda\gamma_\alpha^{-(\lambda+1)}\dot{\gamma_\alpha}$ and $\gamma_\alpha(\alpha,0)=1$, we deduce that $$\label{eq:compat}
\begin{split}
\omega(\alpha,t)=\phi_1(t)\left(1-\lambda \eta(t)u_0'(\alpha)\right),\,\,\,\,\,\,\,\,\,\,\,\,\eta(t)=\int_0^t\frac{ds}{\phi_1^2(s)}.
\end{split}$$ Now, uniqueness of solution to (\[eq:cha\]) and periodicity implies that $$\label{eq:chaperio}
\begin{split}
\gamma(\alpha+1,t)-\gamma(\alpha,t)=1
\end{split}$$ for as long as $u$ is defined. Consequently, simplifying and integrating (\[eq:compat\])i) with respect to $\alpha$ gives $$\label{eq:sum}
\gamma_\alpha={\mathcal K}_0/{\bar{\mathcal K}}_0$$ where we define $$\label{eq:def}
\begin{split}
\mathcal{K}_i(\alpha, t)=\frac{1}{\mathcal{J}(\alpha,t)^{i+{\frac{1}{\lambda}}}},\,\,\,\,\,\,\,\,\,\,\,\,\bar{\mathcal{K}}_i(t)=\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{i+\frac{1}{\lambda}}}},
\end{split}$$ for $i\in\mathbb{N}\cup\{0\}$, and $$\label{eq:J}
\begin{split}
\mathcal{J}(\alpha,t)=1-\lambda\eta(t)u_0^\prime(\alpha),\,\,\,\,\,\,\,\,\,\,\,\,\mathcal{J}(\alpha,0)=1.
\end{split}$$ As a result, (\[eq:jacid\]) and (\[eq:J\])i) yield, after further simplification, $$\label{eq:mainsolu}
\begin{split}
u_x(\gamma(\alpha,t),t)=\frac{1}{\lambda\eta(t){\bar{\mathcal K}_0(t)}^{2\lambda}}
\left(\frac{1}{\mathcal{J}(\alpha, t)}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal K}_0(t)}
\right).
\end{split}$$ The strictly increasing function $\eta(t)$ satisfies the initial value problem $$\label{eq:etaivp}
\begin{split}
\dot{\eta}(t)=\bar{\mathcal{K}}_0(t)^{^{-2\lambda}},\,\,\,\,\,\,\,\,\,\,\,\eta(0)=0,
\end{split}$$ from which the existence of an eventual finite blow-up time $t_*>0$ for (\[eq:mainsolu\]) will depend, in turn, upon the existence of a finite, positive limit $$\label{eq:assympt}
\begin{split}
t_*\equiv\lim_{\eta\uparrow\eta_*}\int_0^{\eta}{\left(\int_0^1{\frac{d\alpha}{(1-\lambda\mu u_0^\prime(\alpha))^{\frac{1}{\lambda}}}}\right)^{2\lambda}\,d\mu}
\end{split}$$ for $\eta_*>0$ to be defined. Moreover, assuming sufficient smoothness, (\[eq:sum\]) and (\[eq:mainsolu\]) imply that $$\label{eq:preserv1}
\begin{split}
u_{xx}(\gamma(\alpha,t),t)=\frac{u_0^{\prime\prime}(\alpha)}{\mathcal{J}(\alpha,t)^{2-\frac{1}{\lambda}}}\bar{\mathcal{K}}_0(t)^{1-2\lambda},
\end{split}$$ so that, for as long as it exists, $u$ maintains its initial concavity profile.
The Data Classes
----------------
\[subsec:dataclass\]
Suppose solutions exist for $t\in[0,t_*)$, $0<t_*\leq+\infty$. Define $$\label{eq:max}
\begin{split}
M(t)\equiv\sup_{\alpha\in[0,1]}\{u_x(\gamma(\alpha,t),t)\},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,M(0)=M_0
\end{split}$$ and $$\label{eq:min22}
\begin{split}
m(t)\equiv\inf_{\alpha\in[0,1]}\{u_x(\gamma(\alpha,t),t)\},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m(0)=m_0,
\end{split}$$ where $\overline\alpha_i$, $i=1,2,...,m$, and $\underline\alpha_j$, $j=1,2,...,n$, denote the finite number of locations in $[0,1]$ where $u_0'(\alpha)$ attains its greatest and least values $M_0>0>m_0$, respectively. Then, it follows from (\[eq:mainsolu\]) ([@Sarria1]) that $$\label{eq:maxmin}
\begin{split}
M(t)=u_x(\gamma(\overline\alpha_i,t),t),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m(t)=u_x(\gamma(\underline\alpha_j,t),t)
\end{split}$$ for $0\leq t<t_*$. Now, the results of Theorems \[thm:sarria1\]-\[thm:sarria2\] suggest that the curvature of $u_0'$ near $\overline\alpha_i$ and/or $\underline\alpha_j$ plays a decisive role in the regularity of solutions to (\[eq:nonhomo\]). Therefore, in the following sections, we further examine this interaction by considering a large class of functions in which $u_0'(x)$ is assumed to be bounded, at least $C^0(0,1)\, a.e.$, and has arbitrary curvature near the location(s) in question. More particularly, for $\lambda>0$, we will assume there are constants $q\in\mathbb{R}^+$ and $C_1\in\mathbb{R}^-$ such that $$\label{eq:expnew0}
u_0^\prime(\alpha)\sim M_0+C_1\left|\alpha-\overline\alpha_i\right|^q$$ for $0\leq\left|\alpha-\overline\alpha_i\right|\leq r$, and small enough $0<r\leq1$, $r\equiv\min_{1\leq i\leq m}\{r_i\}$. Similarly, for $\lambda<0$, we suppose there is $C_2\in\mathbb{R}^+$ such that $$\label{eq:expnew00}
u_0^\prime(\alpha)\sim m_0+C_2\left|\alpha-\underline\alpha_j\right|^q$$ for $0\leq\left|\alpha-\underline\alpha_j\right|\leq s$ and $0<s\leq1$, $s\equiv\min_{1\leq j\leq n}\{s_j\}$. See Figure \[fig:data0\] below. Now, for $r$ and $s$ as above, define $$\mathcal{D}_i\equiv[\overline\alpha_i-r,\overline\alpha_i+r],\,\,\,\,\,\,\,\,\,\,\,\,\,\mathcal{D}_j\equiv[\underline\alpha_j-s,\underline\alpha_j+s].$$ Then, below we list some of the data classes that admit the asymptotic behaviour (\[eq:expnew0\]) and/or (\[eq:expnew00\]) for particular values of $q>0$.
- $u_0(x)\in C^{\infty}(0,1)$ for $q=2k$ and $k\in\mathbb{Z}^+$ (see definition \[def:order\]).
- If $q=1,$ $u_0''(x)\in PC(\mathcal{D}_i)$ for $\lambda>0$, or $u_0''(x)\in PC(\mathcal{D}_j)$ if $\lambda<0$.
- In the limit as $q\to+\infty$, $u_0'(x)\in PC(\mathcal{D}_i)$ for $\lambda>0$, or $u_0'(x)\in PC(\mathcal{D}_j)$ if $\lambda<0$.
- From (\[eq:expnew0\]), we see that the quantity
$$\label{eq:holder2}
[u_0']_{_{q;\overline\alpha_i}}=\sup_{\alpha\in\mathcal{D}_i}\frac{\lvert u_0'(\alpha)-u_0'(\overline\alpha_i)\rvert}{\lvert\alpha-\overline\alpha_i\rvert^q}$$
is finite. As a result, for $0<q\leq1$ and $\lambda>0$, $u_0'$ is H$\ddot{\text{o}}$lder continuous at $\overline\alpha_i$. Analogously for $\lambda<0$, since
$$\label{eq:holder3}
[u_0']_{_{q;\underline\alpha_j}}=\sup_{\alpha\in\mathcal{D}_j}\frac{\lvert u_0'(\alpha)-u_0'(\underline\alpha_j)\rvert}{\lvert\alpha-\underline\alpha_j\rvert^q}$$
is defined by (\[eq:expnew00\]).
- For $\lambda>0$ and either $N<q<N+1$, $N\in\mathbb{N}$, or $q>0$ odd, $u_0'(\alpha)\in C^{^{N+1}}(\mathcal{D}_i)$. Similarly for $\lambda<0$.
![Local behaviour of $u_0'(\alpha)$ satisfying (\[eq:expnew0\]) for several values of $q>0$, $\overline\alpha=1/2$, $M_0=1$ and $C_1=-1$.[]{data-label="fig:data0"}](data000.png)
Blow-up
=======
\[sec:blowup\]
In this section, we study regularity properties in solutions to (\[eq:nonhomo\])-(\[eq:pbc\]) which, according to the sign of $\lambda$, arise from initial data satisfying (\[eq:expnew0\]) and/or (\[eq:expnew00\]). More particularly, finite-time blow-up and global existence in time are examined using $L^p(0,1)$ Banach spaces for $p\in[1,+\infty]$. Set $$\label{eq:defeta*}
\begin{split}
\eta_*=
\begin{cases}
\frac{1}{\lambda M_0},\,\,\,\,\,\,\,&\lambda>0,
\\
\frac{1}{\lambda m_0},\,\,\,\,\,\,\,&\lambda<0.
\end{cases}
\end{split}$$ Then, as $\eta\uparrow \eta_*,$ the space-dependent term in (\[eq:mainsolu\]) will diverge for certain choices of $\alpha$ and not at all for others. Specifically, for $\lambda>0,$ $\mathcal{J}(\alpha,t)^{-1}$ blows up earliest as $\eta\uparrow \eta_*$ at $\alpha=\overline\alpha_i,$ since $$\begin{split}
\mathcal{J}(\overline\alpha_i,t)^{-1}=(1-\lambda\eta(t)M_0)^{-1}\to+\infty\,\,\,\,\,\,\,\text{as}\,\,\,\,\,\,\,\eta\uparrow\eta_*=\frac{1}{\lambda M_0}.
\end{split}$$ Similarly for $\lambda<0,\, \mathcal{J}(\alpha,t)^{-1}$ diverges first at $\alpha=\underline\alpha_j$ and $$\begin{split}
\mathcal{J}(\underline\alpha_j,t)^{-1}=(1-\lambda\eta(t)m_0)^{-1}\to+\infty\,\,\,\,\,\,\,\text{as}\,\,\,\,\,\,\,\eta\uparrow\eta_*=\frac{1}{\lambda m_0}.
\end{split}$$ However, blow-up of (\[eq:mainsolu\]) does not necessarily follow from this; we will need to estimate the behaviour of the time-dependent integrals $$\begin{split}
\bar{\mathcal{K}}_0(t)=
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda}}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\bar{\mathcal{K}}_1(t)=\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{1+\frac{1}{\lambda}}}}
\end{split}$$ as $\eta\uparrow\eta_*.$ To this end, in some of the proofs we find convenient the use of the Gauss hypergeometric series ([@Barnes1], [@Magnus1], [@Gasper1]) $$\label{eq:2f1}
\begin{split}
{}_2F_1\left[a,b;c;z\right]\equiv\sum_{k=0}^{\infty}\frac{\left(a\right)_k(b)_k}{\left(c\right)_k\,k!}z^k,\,\,\,\,\,\,\,\,\,\,\,\lvert z\rvert< 1,
\end{split}$$ for $c\notin\mathbb{Z}^-\cup\{0\}$ and $(x)_k,\, k\in\mathbb{N}\cup\{0\}$, the Pochhammer symbol $(x)_0=1$, $(x)_k=x(x+1)...(x+k-1).$ Also, we will make use of the following results:
\[lem:analcont\] Suppose $\lvert\text{arg}\left(-z\right)\rvert<\pi$ and $a,b,c,a-b\notin\mathbb{Z},$ then the analytic continuation for $\lvert z\rvert>1$ of the series (\[eq:2f1\]) is given by $$\label{eq:analform}
\begin{split}
{}_2F_1[a,b;c;z]=&\frac{\Gamma(c)\Gamma(a-b)(-z)^{-b}{}_2F_1[b,1+b-c;1+b-a;z^{-1}]}{\Gamma(a)\Gamma(c-b)}
\\
&+\frac{\Gamma(c)\Gamma(b-a)(-z)^{-a}{}_2F_1[a,1+a-c;1+a-b;z^{-1}]}{\Gamma(b)\Gamma(c-a)}
\end{split}$$ where $\Gamma(\cdot)$ denotes the standard gamma function.
See for instance [@Magnus1], [@Gasper1].
\[lem:diff\] Suppose $b<2$, $0\leq\left|\beta-\beta_0\right|\leq1$ and $\epsilon\geq C_0$ for some $C_0>0.$ Then $$\label{eq:derseries}
\begin{split}
\frac{1}{\epsilon^b}\,\frac{d}{d\beta}\left((\beta-\beta_0)\,{}_2F_1\left[\frac{1}{q},b;1+\frac{1}{q};-\frac{C_0\left|\beta-\beta_0\right|^q}{\epsilon}\right]\right)=(\epsilon+C_0\left|\beta-\beta_0\right|^q)^{-b}
\end{split}$$ for all $q\in\mathbb{R}^+$ and $b\neq1/q.$
Lemma \[lem:diff\] above is a generalization of Lemma 4.5 in [@Sarria1]. Its proof follows similar reasoning. Finally, the next Lemma provides us with additional tools for estimating the behaviour, as $\eta\uparrow\eta_*$, of time-dependent integrals of the type $\bar{\mathcal{K}}_i(t)$. Its proof is deferred to §\[subsec:generalcase\].
\[lem:general\] For some $q\in\mathbb{R}^+$, suppose $u_0'(\alpha)$ satisfies (\[eq:expnew0\]) when $\lambda\in\mathbb{R}^+$, or (\[eq:expnew00\]) if $\lambda\in\mathbb{R}^-$. It holds:
1\. If $\lambda\in\mathbb{R}^+$ and $b>\frac{1}{q}$, $$\label{eq:generalestimate}
\begin{split}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^b}}\sim C\mathcal{J}(\overline\alpha_i,t)^{\frac{1}{q}-b}
\end{split}$$ for $\eta_*-\eta>0$ small and positive constants $C$ given by $$\label{eq:generalconstant}
\begin{split}
C=\frac{2m\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(b-\frac{1}{q}\right)}{\Gamma\left(b\right)}\left(\frac{M_0}{\left|C_1\right|}\right)^{\frac{1}{q}}.
\end{split}$$ Here, $m\in\mathbb{N}$ denotes the finite number of locations $\overline\alpha_i$ in $[0,1]$.
2\. If $\lambda\in\mathbb{R}^-$ and $b>\frac{1}{q}$, $$\label{eq:generalestimate2}
\begin{split}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^b}}\sim C\mathcal{J}(\underline\alpha_j,t)^{\frac{1}{q}-b}
\end{split}$$ for $\eta_*-\eta>0$ small and positive constants $C$ determined by $$\label{eq:generalconstant2}
\begin{split}
C=\frac{2n\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(b-\frac{1}{q}\right)}{\Gamma\left(b\right)}\left(\frac{\left|m_0\right|}{C_2}\right)^{\frac{1}{q}}.
\end{split}$$ Above, $n\in\mathbb{N}$ represents the finite number of points $\underline\alpha_j$ in $[0,1]$.
3\. Suppose $q>1/2$ and $b\in(0,1/q)$, or $q\in(0,1/2)$ and $b\in(0,2)$, satisfy $\frac{1}{q}$, $b$, $b-\frac{1}{q}\notin\mathbb{Z}$. Then for $\lambda\neq0$ and $\eta_*$ as defined in (\[eq:defeta\*\]), $$\label{eq:generalestimate3}
\begin{split}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^b}}\sim C
\end{split}$$ for $\eta_*-\eta>0$ small and positive constants $C$ that depend on the choice of $\lambda$, $b$ and $q$. Similarly, the integral remains bounded, and positive, for all $\eta\in[0,\eta_*]$ and $\lambda\neq0$ when $b\leq0$ and $q\in\mathbb{R}^+$.
The outline of this section is as follows. In §\[subsec:lin\], we examine $L^p$, $p\in[1,+\infty]$ regularity of solutions arising from initial data satisfying (\[eq:expnew0\]) and/or (\[eq:expnew00\]) for $q=1$. Then, in §\[subsec:generalcase\] the case of arbitrary $q\in\mathbb{R}^+$ is studied. Also, regularity results concerning a class of smooth initial data larger than the one studied in [@Sarria1] are discussed. We remark that the case $q=1$ is considered separately from the more general argument in §\[subsec:generalcase\], due to the assumptions in Lemma \[lem:general\].
Global Estimates and Blow-up for $q=1$
--------------------------------------
\[subsec:lin\]
In [@Sarria1], we showed that for a particular choice of piecewise linear $u_0'(\alpha)$, a special class of solutions to the 2D Euler equations ($\lambda=1$) could develop a singularity in finite-time, whereas, for the corresponding 3D problem $(\lambda=1/2)$, solutions may converge to a nontrivial steady state as $t\to+\infty$. Therefore, it is of particular interest to determine how these results generalize to initial data satisfying (\[eq:expnew0\]) for $q=1$. In fact, in this section we will examine $L^p$ regularity in $u_x$ for $\lambda\in\mathbb{R}$ and $p\in[1,+\infty]$.
### $L^{\infty}$ Regularity for $q=1$ {#subsubsec:linlinfty}
\[thm:p=1\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) with $u_0'(\alpha)$ satisfying, for $q=1$, either (\[eq:expnew0\]) when $\lambda>0$, or (\[eq:expnew00\]) if $\lambda<0$. It holds,
1. \[it:two\] For $\lambda>1/2$, there exists a finite $t_*>0$ such that both the maximum $M(t)$ and the minimum $m(t)$ diverge to $+\infty$ and respectively to $-\infty$ as $t\uparrow t_*$. Moreover, for every $\alpha\notin\bigcup_{i,j}\{\overline\alpha_i\}\cup\{\underline\alpha_j\},$ $\lim_{t\uparrow t_*}u_x(\gamma(\alpha,t),t)=-\infty$ (two-sided, everywhere blow-up).
2. \[it:one\] For $\lambda\in[0,1/2]$, solutions exist globally in time. More particularly, these vanish as $t\uparrow t_*=+\infty$ for $\lambda\in(0,1/2)$ but converge to a nontrivial steady-state if $\lambda=1/2.$
3. \[it:three\] For $\lambda<0$, there is a finite $t_*>0$ such that only the minimum diverges, $m(t)\to-\infty,$ as $t\uparrow t_*$ (one-sided, discrete blow-up).
Let $C$ denote a positive constant which may depend on $\lambda\neq0$.
**Proofs of Statements** (\[it:two\]) **and** (\[it:one\])
For simplicity, we prove (\[it:two\]) and (\[it:one\]) for the case where $M_0$ occurs at a single location $\overline\alpha\in(0,1)$. By (\[eq:expnew0\]), there is $0<r\leq1$ small enough such that $\epsilon+M_0-u_0^\prime(\alpha)\sim\epsilon-C_1\left|\alpha-\overline\alpha\right|$ for $0\leq\left|\alpha-\overline\alpha\right|\leq r$, $C_1<0$ and $\epsilon>0$. Then $$\label{eq:app}
\begin{split}
\int_{\overline\alpha-r}^{\overline\alpha+r}{\frac{d\alpha}{(\epsilon+M_0-u_0'(\alpha))^{\frac{1}{\lambda}}}}&\sim\int_{\overline\alpha-r}^{\overline\alpha+r}{\frac{d\alpha}{(\epsilon-C_1\left|\alpha-\overline\alpha\right|)^{\frac{1}{\lambda}}}}
\\
&=\frac{2\lambda}{\left|C_1\right|(1-\lambda)}\left(\epsilon^{1-\frac{1}{\lambda}}-(\epsilon+\left|C_1\right| r)^{1-\frac{1}{\lambda}}\right)
\end{split}$$ for $\lambda\in(0,+\infty)\backslash\{1\}$. Consequently, setting $\epsilon=\frac{1}{\lambda\eta}-M_0$ in (\[eq:app\]) gives $$\label{eq:phiest1}
\begin{split}
\bar{\mathcal{K}}_0(t)\sim
\begin{cases}
C,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&\lambda>1,
\\
\frac{2\lambda M_0}{\left|C_1\right|(1-\lambda)}\mathcal{J}(\overline\alpha,t)^{1-\frac{1}{\lambda}},\,\,\,\,\,\,\,\,\,\,\,&\lambda\in(0,1)
\end{cases}
\end{split}$$ for $\eta_*-\eta>0$ small, $\eta_*=\frac{1}{\lambda M_0}$ and $\mathcal{J}(\overline\alpha,t)=1-\lambda\eta(t)M_0.$ Following a similar argument, or using Lemma \[lem:general\](1) with $b=1+\frac{1}{\lambda}$ and $q=1$, we estimate $$\label{eq:phi12}
\begin{split}
\bar{\mathcal{K}}_1(t)\sim\frac{2\lambda M_0}{\left|C_1\right|}\mathcal{J}(\overline\alpha,t)^{-\frac{1}{\lambda}}
\end{split}$$ for any $\lambda>0$. Suppose $\lambda>1.$ Then, (\[eq:mainsolu\]), (\[eq:phiest1\])i) and (\[eq:phi12\]) give $$\label{eq:est21}
\begin{split}
u_x(\gamma(\alpha,t),t)\sim C\left(\frac{1}{\mathcal{J}(\alpha,t)}-\frac{C}{\mathcal{J}(\overline\alpha,t)^{\frac{1}{\lambda}}}\right)
\end{split}$$ for $\eta_*-\eta>0$ small. Setting $\alpha=\overline\alpha$ into (\[eq:est21\]) and using (\[eq:maxmin\])i) implies that $$\begin{split}
M(t)\sim\frac{C}{\mathcal{J}(\overline\alpha,t)}\rightarrow+\infty
\end{split}$$ as $\eta\uparrow\eta_*$. However, if $\alpha\neq\overline\alpha,$ the second term in (\[eq:est21\]) dominates and $$\begin{split}
u_x(\gamma(\alpha,t),t)\sim-\frac{C}{\mathcal{J}(\overline\alpha,t)^{\frac{1}{\lambda}}}\rightarrow-\infty.
\end{split}$$ The existence of a finite $t_*>0$ for all $\lambda>1$ follows from (\[eq:etaivp\]) and (\[eq:phiest1\])i), which imply $$\begin{split}
t_*-t\sim C(\eta_*-\eta).
\end{split}$$ Now let $\lambda\in(0,1).$ Using (\[eq:phiest1\])ii) and (\[eq:phi12\]) on (\[eq:mainsolu\]), yields $$\label{eq:u1}
\begin{split}
u_x(\gamma(\alpha,t),t)\sim C\left(\frac{1}{\mathcal{J}(\alpha,t)}-\frac{1-\lambda}{\mathcal{J}(\overline\alpha,t)}\right)\mathcal{J}(\overline\alpha,t)^{2(1-\lambda)}
\end{split}$$ for $\eta_*-\eta>0$ small. Setting $\alpha=\overline\alpha$ in (\[eq:u1\]) implies $$M(t)\sim C\mathcal{J}(\overline\alpha,t)^{1-2\lambda}\to
\begin{cases}
0,\,\,\,\,\,\,\,\,\,\,&\lambda\in(0,1/2),
\\
+\infty,\,\,\,\,\,\,\,\,\,&\lambda\in(1/2,1)
\end{cases}$$ as $\eta\uparrow \eta_*.$ If instead $\alpha\neq\overline\alpha,$ $$u_x(\gamma(\alpha,t),t)\sim -C\mathcal{J}(\overline\alpha,t)^{1-2\lambda}\to
\begin{cases}
0,\,\,\,\,\,\,\,\,\,\,&\lambda\in(0,1/2),
\\
-\infty,\,\,\,\,\,\,\,\,\,&\lambda\in(1/2,1)
\end{cases}$$ as $\eta\uparrow\eta_*.$ For the threshold parameter $\lambda=1/2,$ we keep track of the constants and find that, as $\eta\uparrow\eta_*,$ $$u_x(\gamma(\alpha,t),t)\to
\begin{cases}
\,\,\,\,\,\frac{\left|C_1\right|}{4},\,\,\,\,\,\,\,\,\,\,\,\,\,\,&\alpha=\overline\alpha
\\
-\frac{\left|C_1\right|}{4},\,\,\,\,\,\,\,\,\,\,\,\,\,&\alpha\neq\overline\alpha.
\end{cases}$$ Finally, (\[eq:etaivp\]) and (\[eq:phiest1\])ii) imply that $dt\sim C\mathcal{J}(\overline\alpha,t)^{2(\lambda-1)}d\eta$ so that $$t_*=\lim_{\eta\uparrow\eta_*}t(\eta)\sim
\begin{cases}
\frac{C}{2\lambda-1}\left(C-\lim_{\eta\uparrow\eta_*}(\eta_*-\eta)^{2\lambda-1}\right),\,\,\,\,\,\,\,\,\,&\lambda\in(0,1)\backslash\{1/2\},
\\
-C\lim_{\eta\uparrow\eta_*}\log(\eta_*-\eta),\,\,\,\,\,\,\,\,\,\,\,&\lambda=1/2.
\end{cases}$$ As a result, $t_*=+\infty$ if $\lambda\in(0,1/2]$ but $0<t_*<+\infty$ for $\lambda\in(1/2,1)$. Lastly, $$\begin{split}
\label{eq:l1}
\bar{\mathcal{K}}_0(t)\sim-\frac{2M_0}{\left|C_1\right|}\,\text{log}(\eta_*-\eta)
\end{split}$$ for $0<\eta_*-\eta<<1$ small and $\lambda=1$. Then, two-sided, everywhere blow-up in finite-time follows just as above from (\[eq:mainsolu\]), (\[eq:etaivp\]), (\[eq:phi12\]) and (\[eq:l1\]). Finally, the case $\lambda=0$ follows from the results in [@Sarria1].
**Proof of Statement** (\[it:three\])
For $\lambda<0$, set $\eta_*=\frac{1}{\lambda m_0}$. Then $\bar{\mathcal{K}}_0(t)$ remains finite, and positive, for all $\eta\in[0,\eta_*]$. In fact, one can easily show that $$\label{eq:quick3p}
\begin{split}
1\leq\bar{\mathcal{K}}_0(t)\leq\left(1+\frac{M_0}{\left|m_0\right|}\right)^{\frac{1}{\left|\lambda\right|}}
\end{split}$$ if $\lambda\in[-1,0)$, while $$\label{eq:newestq}
\begin{split}
0<\int_0^1{\left(1+\frac{u_0'(\alpha)}{\left|m_0\right|}\right)^{\frac{1}{\left|\lambda\right|}}d\alpha}\leq\bar{\mathcal{K}}_0(t)\leq1
\end{split}$$ for $\lambda<-1$. Similarly, when $\lambda\in[-1,0)$ and $\eta\in[0,\eta_*]$, $$\label{eq:quick4p}
\begin{split}
1\leq\bar{\mathcal{K}}_1(t)\leq\left(\frac{\left|m_0\right|}{M_0+\left|m_0\right|}\right)^{1+\frac{1}{\lambda}}.
\end{split}$$ However, if $\lambda<-1,$ we need to estimate $\bar{\mathcal{K}}_1(t)$ for $\eta_*-\eta>0$ small. To do so, we proceed analogously to the derivation of (\[eq:phiest1\]). For simplicity, assume $u_0'(\alpha)$ achieves its least value $m_0<0$ at a single point $\underline\alpha\in(0,1)$. Then (\[eq:expnew00\]) with $q=1$ implies that $u_0'(\alpha)\sim m_0+C_2\left|\alpha-\underline\alpha\right|$ for $0\leq\left|\alpha-\underline\alpha\right|\leq s$, $C_2>0$ and $0<s\leq1$. It follows that $$\begin{split}
\label{eq:l-}
\int_{\underline\alpha-s}^{\underline\alpha+s}{\frac{d\alpha}{(\epsilon+u_0'(\alpha)-m_0)^{1+\frac{1}{\lambda}}}}&\sim\int_{\underline\alpha-s}^{\underline\alpha+s}{\frac{d\alpha}{(\epsilon+C_2\left|\alpha-\underline\alpha\right|)^{1+\frac{1}{\lambda}}}}
\\
&= \frac{2\left|\lambda\right|}{C_2}\left((\epsilon+C_2s)^{\frac{1}{\left|\lambda\right|}}-\epsilon^{\frac{1}{\left|\lambda\right|}}\right)
\end{split}$$ for $\epsilon>0$. By substituting $\epsilon=m_0-\frac{1}{\lambda\eta}$ into (\[eq:l-\]), we find that $\bar{\mathcal{K}}_1(t)$ has a finite, positive limit as $\eta\uparrow\eta_*$ for $\lambda<-1$. This implies that for $\lambda<0$, both time-dependent integrals in (\[eq:mainsolu\]) remain bounded and positive for all $\eta\in[0,\eta_*]$. Consequently, blow-up of (\[eq:mainsolu\]), as $\eta\uparrow\eta_*$, will follow from the space-dependent term, $\mathcal{J}(\alpha,t)^{-1}$, evaluated at $\alpha=\underline\alpha$. In this way, we set $\alpha=\underline\alpha$ into (\[eq:mainsolu\]) and use (\[eq:maxmin\])ii) to obtain $$m(t)\sim\frac{Cm_0}{\mathcal{J}(\overline\alpha,t)}\to-\infty$$ as $\eta\uparrow\eta_*$. In contrast, for $\alpha\neq\underline\alpha,$ the definition of $m_0$ implies that the space-dependent term now remains bounded for $\eta\in[0,\eta_*]$. Finally, the existence of a finite blow-up time $t_*>0$ for the minimum follows from (\[eq:etaivp\]) and the estimates on $\bar{\mathcal{K}}_0(t)$. In fact, by (\[eq:etaivp\]), $t_*=\eta_*$ for $\lambda=-1$, while ([@Sarria1]) $$\begin{cases}
\label{eq:timebounds}
\eta_*\leq t_*<+\infty,\,\,\,&\lambda<-1,
\\
\eta_*\left(1-\frac{M_0}{m_0}\right)^{-2}\leq t_*\leq\eta_*,\,\,\,\,\,&\lambda\in(-1,0).
\end{cases}$$ See §\[sec:examples\] for examples.
In preparation for the next section, we recall some formulas, as well as upper and lower bounds, derived in [@Sarria1] for the $L^p$ norm of $u_x$. For as long as a solution exists, (\[eq:sum\]) and (\[eq:mainsolu\]) imply that $$\begin{split}
\left\|u_x(\cdot,t)\right\|_p^p=\frac{1}{\left|\lambda\eta(t)\right|^p\bar{\mathcal{K}}_0(t)^{^{1+2\lambda p}}}\int_0^1{\left|\frac{1}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}\right|^pd\alpha}
\end{split}$$ for $\lambda\neq 0$ and $p\in[1,+\infty)$. Using the above and some standard inequalities yields $$\label{eq:upper}
\begin{split}
\left\|u_x(\cdot,t)\right\|_p^p\leq\frac{2^{p-1}}{\left|\lambda\eta(t)\right|^p\bar{\mathcal{K}}_0(t)^{^{1+2\lambda p}}}\left(\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{p+\frac{1}{\lambda }}}}}+\frac{\bar{\mathcal{K}}_1(t)^{^p}}{\bar{\mathcal{K}}_0(t)^{^{p-1}}}\right)
\end{split}$$ and $$\label{eq:lower}
\begin{split}
\left\|u_x(\cdot,t)\right\|_p\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|.
\end{split}$$ Moreover, the energy function $E(t)\equiv\left\|u_x(\cdot,t)\right\|_2^2$ is explicitly given by $$\label{eq:energy}
\begin{split}
E(t)=\left(\lambda\eta(t)\bar{\mathcal{K}}_0(t)^{1+2\lambda}\right)^{-2}\left(\bar{\mathcal{K}}_0(t)\bar{\mathcal{K}}_2(t)-\bar{\mathcal{K}}_1(t)^2\right).
\end{split}$$ Lastly, multiplying (\[eq:nonhomo\])i) by $u_x$, integrating by parts, and using (\[eq:pbc\]), (\[eq:sum\]) and (\[eq:mainsolu\]), gives $$\label{eq:derenergy}
\begin{split}
\dot E(t)&=(1+2\lambda)\int_0^1{u_x(x,t)^3dx}=(1+2\lambda)\int_0^1{u_x(\gamma(\alpha,t),t)^3\gamma_\alpha(\alpha,t)\,d\alpha}
\\
&=\frac{1+2\lambda}{(\lambda\eta(t))^3}
\left[\frac{\bar{\mathcal{K}}_3(t)}{\bar{\mathcal{K}}_1(t)}-\frac{3\bar{\mathcal{K}}_2(t)}{\bar{\mathcal{K}}_0(t)}+2\left(\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\right)^2\right]\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)^{1+6\lambda}}.
\end{split}$$ The reader may refer to [@Sarria1] for details on the above.
### Further $L^p$ Regularity for $\lambda\neq0$, $q=1$ and $p\in[1,+\infty)$
\[subsubsec:linlp\]
In the previous section, we established the existence of a finite $t_*>0$ such that $\left\|u_x\right\|_{\infty}$ diverges as $t\uparrow t_*$ for all $\lambda\in\mathbb{R}\backslash[0,1/2]$ and initial data satisfying (\[eq:expnew0\]) and/or (\[eq:expnew00\]) for $q=1$ relative to the sign of $\lambda$. If instead, $\lambda\in[0,1/2]$, we proved that solutions remain in $L^{\infty}$ for all time. In this section, we examine further $L^p$ regularity of $u_x$, as $t\uparrow t_*$, for $\lambda\in\mathbb{R}\backslash[0,1/2]$ and $p\in[1,+\infty)$.
\[thm:q=1,p>=1\] For the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]), let $t_*>0$ denote the finite $L^{\infty}$ blow-up time for $u_x$ in Theorem \[thm:p=1\]. Further, for $q=1$, suppose $u_0'(\alpha)$ satisfies (\[eq:expnew0\]) when $\lambda>0$, or (\[eq:expnew00\]) if $\lambda<0$.
1. \[it:onep\] For $\lambda>1/2$ and $p>1$, $\lim_{t\uparrow t_*}\left\|u_x\right\|_p=+\infty$.
2. \[it:twop\] For $\lambda<0$ and $t\in[0,t_*]$, $u_x$ remains integrable; moreover, if $\frac{1}{1-p}<\lambda<0$ and $p>1$, then $u_x\in L^p$ for all $t\in[0,t_*]$.
3. \[it:threep\] The energy $E(t)=\left\|u_x\right\|_2^2$ diverges if $\lambda\in(-\infty,-1]\cup(1/2,+\infty)$ as $t\uparrow t_*$ but remains finite for $t\in[0,t_*]$ if $\lambda\in(-1,0)$. Also, $\lim_{t\uparrow t_*}\dot E(t)=+\infty$ when $\lambda\in(-\infty,-1/2)\cup(1/2,+\infty)$, whereas, $\dot E(t)\equiv0$ if $\lambda=-1/2$ while $\dot E(t)$ stays bounded for $t\in[0,t_*]$ if $\lambda\in(-1/2,0)$.
Let $C$ denote a positive constant that may depend on the choice of $\lambda$ and $p\in[1,+\infty)$.
**Proof of Statement** (\[it:onep\])
First, suppose $\lambda>0$ and set $\eta_*=\frac{1}{\lambda M_0}$. For simplicity, we prove part (\[it:onep\]) under the assumption that $M_0>0$ occurs at a single point $\overline\alpha\in(0,1)$. Using Lemma \[lem:general\](1) with $b=1+\frac{1}{\lambda p}$, $q=1$ and $p\geq1$, yields
$$\label{eq:p13}
\begin{split}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{1+\frac{1}{\lambda p}}}}\sim\frac{2\lambda pM_0}{\left|C_1\right|}\mathcal{J}(\overline\alpha,t)^{-\frac{1}{\lambda p}}
\end{split}$$
for $\eta_*-\eta>0$ small. Similarly, taking $b=p+\frac{1}{\lambda}$ we find that $$\label{eq:p141}
\begin{split}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{p+\frac{1}{\lambda}}}}\sim \frac{2\lambda M_0}{\left|C_1\right|(\lambda(p-1)+1)}\mathcal{J}(\overline\alpha,t)^{1-p-\frac{1}{\lambda}}.
\end{split}$$ Moreover, following the argument that led to estimate (\[eq:phiest1\]), with $\frac{1}{\lambda p}$ instead of $\frac{1}{\lambda}$, gives $$\label{eq:p14}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\sim
\begin{cases}
\frac{2\lambda pM_0}{\left|C_1\right|(1-\lambda p)}\mathcal{J}(\overline\alpha,t)^{1-\frac{1}{\lambda p}},\,\,&\lambda\in(0,1/p),
\\
C,\,\,\,\,&\lambda>1/p
\end{cases}$$ for $p\geq1$ and $\eta_*-\eta>0$ small. Suppose $\lambda, p>1$ so that $\lambda>1/p$. Then, using (\[eq:phiest1\])i), (\[eq:phi12\]), (\[eq:p13\]) and (\[eq:p14\])ii) in (\[eq:lower\]), implies that $$\begin{split}
\left\|u_x(\cdot,t)\right\|_p&\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|
\\
&\sim C\left|C\mathcal{J}(\overline\alpha,t)^{-\frac{1}{\lambda p}}-\mathcal{J}(\overline\alpha,t)^{-\frac{1}{\lambda}}\right|
\\
&\sim C\mathcal{J}(\overline\alpha,t)^{-\frac{1}{\lambda}}\to+\infty
\end{split}$$ as $\eta\uparrow\eta_*$. Next, let $p\in(1,2)$ and $\lambda\in(1/2,1/p)\subset(1/2,1)$. Then, using (\[eq:phiest1\])ii), (\[eq:phi12\]), (\[eq:p13\]) and (\[eq:p14\])i) in (\[eq:lower\]), gives $$\begin{split}
\left\|u_x(\cdot,t)\right\|_p&\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|
\\
&\sim C\left|1-\frac{1-\lambda}{1-\lambda p}\right|\mathcal{J}(\overline\alpha,t)^{\rho(\lambda,p)}
\\
&=C\mathcal{J}(\overline\alpha,t)^{\rho(\lambda,p)}
\end{split}$$ for $\eta_*-\eta>0$ small and $\rho(\lambda,p)=2(1-\lambda)-\frac{1}{p}$. However, for $\lambda$ and $p$ as prescribed, we see that $\rho(\lambda,p)<0$ for $1-\frac{1}{2p}<\lambda<\frac{1}{p}$ and $p\in(1,3/2)$. Therefore, for any $\lambda\in(1/2,1)$ there is $1-p>0$ arbitrarily small such that $\left\|u_x\right\|_p\to+\infty$ as $\eta\uparrow\eta_*$. Finally, if $\lambda=1$ we have $\lambda>1/p$ for $p>1$, as a result, (\[eq:phi12\]), (\[eq:l1\]), (\[eq:p13\]) and (\[eq:p14\])iii) imply that $$\begin{split}
\left\|u_x(\cdot,t)\right\|_p&\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|
\\
&\sim C\mathcal{J}(\overline\alpha,t)^{-1}(-\log(\eta_*-\eta))^{-3-\frac{1}{p}}
\end{split}$$ for $0<\eta_*-\eta<<1$ small, and so, $\left\|u_x\right\|_p\to+\infty$ as $\eta\uparrow\eta_*$. The existence of a finite blow-up time $t_*>0$ follows from Theorem \[thm:p=1\].
**Proof of Statement** (\[it:twop\])
Suppose $\lambda<0$ and set $\eta_*=\frac{1}{\lambda m_0}$. First, recall from the proof of Theorem \[thm:p=1\] that $\bar{\mathcal{K}}_i(t),\, i=0,1$ remain finite and positive for all $\eta\in[0,\eta_*]$. Furthermore, in Theorem \[thm:p=1\] we established the existence of a finite blow-up time $t_*>0$ for the minimum $m(t)$. Consequently, the upper bound (\[eq:upper\]) implies that $$\label{eq:ref2}
\begin{split}
\lim_{t\uparrow t_*}\left\|u_x(\cdot,t)\right\|_p<+\infty\,\,\,\,\,\,\,\Leftrightarrow\,\,\,\,\,\,\,\lim_{t\uparrow t_*}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{p+\frac{1}{\lambda}}}}<+\infty
\end{split}$$ for $\lambda<0$ and $p\geq1$. However, if $p=1$, (\[eq:ref2\])ii) is just $\bar{\mathcal{K}}_1(t)$, which remains finite as $t\uparrow t_*$. As a result, $u_x\in L^1$ for all $t\in[0,t_*]$ and $\lambda<0$. If $p>1$, we recreate the argument in (\[eq:l-\]), with $p+\frac{1}{\lambda}$ instead of $1+\frac{1}{\lambda}$, and find that for $\frac{1}{1-p}<\lambda<0$ and $p>1$, the integral remains finite and positive as $\eta\uparrow\eta_*$. Consequently, (\[eq:ref2\]) implies that $$\lim_{t\uparrow t_*}\left\|u_x(\cdot,t)\right\|_p<+\infty$$ for all $\frac{1}{1-p}<\lambda<0$ and $p>1$. We remark that the lower bound (\[eq:lower\]) yields no information regarding $L^p$ blow-up of $u_x$, as $t\uparrow t_*$, for parameter values $-\infty<\lambda<\frac{1}{1-p}$, $p>1$. Nonetheless, we can use (\[eq:energy\]) and (\[eq:derenergy\]) to obtain additional blow-up information on energy-related quantities.
**Proof of Statement** (\[it:threep\])
For $\lambda>1/2$, blow-up of $E(t)$ and $\dot E(t)$ to $+\infty$ as $t\uparrow t_*$ is a consequence of part (\[it:onep\]) above. Further, setting $p=2$ in part (\[it:twop\]) implies that $E(t)$ remains bounded for all $\lambda\in(-1,0)$ and $t\in[0,t_*]$. Now, (\[eq:derenergy\])i) yields $$\label{eq:l3}
\left|\dot E(t)\right|\leq\left|1+2\lambda\right|\left\|u_x(\cdot,t)\right\|_3^3,$$ and so setting $p=3$ in part (\[it:twop\]) implies that $\dot E(t)$ remains finite for $\lambda\in[-1/2,0)$ and $t\in[0,t_*]$. According to these results, we have yet to determine the behaviour of $E(t)$ as $t\uparrow t_*$ for $\lambda\leq-1$ and $\dot E(t)$ when $\lambda<-1/2$. To do so, we will use formulas (\[eq:energy\]) and (\[eq:derenergy\]). From Lemma \[lem:general\](2) with $b=3+\frac{1}{\lambda}$, $q=1$ and $\lambda<-1/2$, we find that $$\label{eq:k3}
\bar{\mathcal{K}}_3(t)\sim \frac{2\lambda\left|m_0\right|}{C_2(1+2\lambda)}\mathcal{J}(\underline\alpha,t)^{-2-\frac{1}{\lambda}}
$$ for $\eta_*-\eta>0$ small. Also, following the argument in (\[eq:l-\]), with $2+\frac{1}{\lambda}$ instead of $1+\frac{1}{\lambda}$, we derive $$\label{eq:k2}
\bar{\mathcal{K}}_2(t)\sim
\begin{cases}
\frac{2\lambda\left|m_0\right|}{C_2(1+\lambda)}\mathcal{J}(\underline\alpha,t)^{-1-\frac{1}{\lambda}},\,\,\,\,&\lambda<-1,
\\
-C\log(\eta_*-\eta),\,\,\,\,&\lambda=-1,
\\
C,\,\,\,\,\,\,\,&\lambda\in(-1,0).
\end{cases}$$ Since both $\bar{\mathcal{K}}_i(t)$, $i=0,1$ stay finite and positive for all $\eta\in[0,\eta_*]$ and $\lambda<0$, (\[eq:energy\]) tells us that blow-up in $\bar{\mathcal{K}}_2(t)$ leads to a diverging $E(t)$. Then, (\[eq:k2\])i) implies that for $\lambda<-1$, $$E(t)\sim C\mathcal{J}(\underline\alpha,t)^{-1-\frac{1}{\lambda}}\to+\infty$$ as $\eta\uparrow\eta_*$. Similarly for $\lambda=-1$ by using (\[eq:k2\])ii) instead. Clearly, this also implies blow-up of $\dot E(t)$ to $+\infty$ as $t\uparrow t_*$ for all $\lambda\leq-1$. Finally, from (\[eq:derenergy\])ii), (\[eq:k3\]) and (\[eq:k2\])iii), $$\dot E(t)\sim\frac{Cm_0^3(1+2\lambda)}{\mathcal{J}(\underline\alpha,t)^{2+\frac{1}{\lambda}}}\to+\infty$$ as $\eta\uparrow\eta_*$ for all $\lambda\in(-1,-1/2)$. The existence of a finite $t_*>0$ follows from Theorem \[thm:p=1\](\[it:three\]).
From the results established thus far, we are able to obtain a complete description of the $L^3$ regularity for $u_x$: if $\lambda\in[0,1/2]$, $\lim_{t\to +\infty}\left\|u_x\right\|_3=C$ where $C\in\mathbb{R}^+$ for $\lambda=1/2$ but $C=0$ if $\lambda\in(0,1/2)$, while, for $t_*>0$ the finite $L^{\infty}$ blow-up time for $u_x$ in Theorem \[thm:p=1\], $$\label{eq:l3q=1}
\lim_{t\uparrow t_*}\left\|u_x(\cdot,t)\right\|_3=
\begin{cases}
+\infty,\,\,\,\,\,\,\,&\lambda\in(-\infty,-1/2]\cup(1/2,+\infty),
\\
C\in\mathbb{R}^+,\,\,\,\,&\lambda\in(-1/2,0).
\end{cases}$$
For $t_*>0$ the finite $L^{\infty}$ blow-up time for $u_x$ in Theorem \[thm:p=1\], we may use (\[eq:derenergy\]), (\[eq:k3\]) and (\[eq:k2\]), as well as Theorem \[thm:q=1,p>=1\], to establish a global bound on $\int_0^1{u_x^3dx}$ if $\lambda\in[0,1/2]$, or for $t\in[0,t_*]$ when $\lambda\in(-1/2,0)$, whereas $$\label{eq:3integral}
\lim_{t\uparrow t_*}\int_0^1{u_x(x,t)^3dx}=
\begin{cases}
+\infty,\,\,\,\,\,\,\,&\lambda>1/2,
\\
-\infty,\,\,\,\,&\lambda\leq-1/2.
\end{cases}$$ We also note that, unlike the result in Theorem \[thm:lpintro\](\[it:ener\]) of §\[sec:intro\], (\[eq:3integral\]) and the change in sign through $\lambda=-1/2$ of the term $1+2\lambda$ in (\[eq:derenergy\]), prevent the possibility of blow-up of $\dot E(t)$ towards $-\infty$, which might otherwise have played a role in the study of weak solutions from the point of view of energy dissipation.
Notice that the two-sided, everywhere blow-up found in Theorem \[thm:p=1\] for $\lambda>1/2$ corresponds, in Theorem \[thm:q=1,p>=1\], to $L^p$ blow-up of $u_x$ for any $p>1$. On the other hand, $u_x$ remains integrable for all $\lambda<0$ and $t\in[0,t_*]$ but, as $t\uparrow t_*$, undergoes an $L^{\infty}$ blow-up of the one-sided, discrete type for $\lambda<0$. Then, as the magnitude of $\lambda<0$ decreases, $u_x$ is guaranteed to remain, for $t\in[0,t_*]$, in smaller $L^p$ spaces with $p\in(1,+\infty)$. In the coming sections, we will find that a similar correspondence between the “strengths” of the $L^{\infty}$ and $L^p$, $p\in[1,+\infty)$, blow-up in $u_x$, as $t\uparrow t_*$, also holds for other $q>0$.
Global Estimates and Blow-up for $\lambda\in\mathbb{R}$ and $q>0$
-----------------------------------------------------------------
\[subsec:generalcase\]
In this section, we study the case of arbitrary $q>0$. As in the previous sections, $L^{p}$ regularity of $u_x$ for $\lambda\in\mathbb{R}$ and $p\in[1,+\infty]$ is examined. In addition, the behaviour of the jacobian (\[eq:sum\]) is considered. Particularly, we will show that if $q\geq1$, no blow-up occurs in stagnation point-form solutions to the 3D incompressible Euler equations, whereas, for the corresponding 2D case, no spontaneous singularity forms when $q\geq2$. Finally, a class of smooth, periodic initial data larger than the one considered in [@Sarria1] is studied. Before stating and proving our results, we first establish Lemma \[lem:general\] and obtain estimates on $\bar{\mathcal{K}}_0(t)$ and $\bar{\mathcal{K}}_1(t)$.
**Proof of Lemma** \[lem:general\](1)
For simplicity, we prove statement (1) for functions $u_0'$ that attain their greatest value $M_0>0$ at a single location $\overline\alpha\in(0,1)$. The case of several $\overline\alpha_i\in[0,1]$ follows similarly. From (\[eq:expnew0\]), there is $0<r\leq1$ such that $\epsilon+M_0-u_0'(\alpha)\sim\epsilon-C_1\left|\alpha-\overline\alpha\right|^q$ for $q\in\mathbb{R}^+$, $\epsilon>0$ and $0\leq\left|\alpha-\overline\alpha\right|\leq r$. Therefore $$\begin{split}
&\int_{\overline\alpha-r}^{\overline\alpha+r}{\frac{d\alpha}{(\epsilon+M_0-u_0'(\alpha))^b}}\sim\int_{\overline\alpha-r}^{\overline\alpha+r}{\frac{d\alpha}{(\epsilon-C_1\left|\alpha-\overline\alpha\right|^q)^b}}
\\
&=\epsilon^{-b}\left[\int_{\overline\alpha-r}^{\overline\alpha}{\left(1+\frac{\left|C_1\right|}{\epsilon}\left(\overline\alpha-\alpha\right)^q\right)^{-b}d\alpha}+\int_{\overline\alpha}^{\overline\alpha+r}{\left(1+\frac{\left|C_1\right|}{\epsilon}\left(\alpha-\overline\alpha\right)^q\right)^{-b}d\alpha}\right]
\end{split}$$ for $b\in\mathbb{R}$. Making the change of variables $$\sqrt{\frac{\left|C_1\right|}{\epsilon}}(\overline\alpha-\alpha)^{\frac{q}{2}}=\tan\theta,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt{\frac{\left|C_1\right|}{\epsilon}}(\alpha-\overline\alpha)^{\frac{q}{2}}=\tan\theta$$ in the first and second integrals inside the bracket, respectively, we find that $$\label{eq:general2}
\begin{split}
&\int_{\overline\alpha-r}^{\overline\alpha+r}{\frac{d\alpha}{(\epsilon+M_0-u_0'(\alpha))^b}}\sim\frac{4}{q\left|C_1\right|^{\frac{1}{q}}\epsilon^{b-\frac{1}{q}}}\int_0^{\frac{\pi}{2}}{\frac{(\cos\theta)^{^{2b-\frac{2}{q}-1}}}{(\sin\theta)^{^{1-\frac{2}{q}}}}d\theta}
\end{split}$$ for small $\epsilon>0$. Suppose $b>\frac{1}{q}$, then setting $\epsilon=\frac{1}{\lambda\eta}-M_0$ in (\[eq:general2\]) implies $$\label{eq:general3}
\begin{split}
\int_{0}^{1}{\frac{d\alpha}{\mathcal{J}(\alpha,t)^b}}\sim\frac{C}{\mathcal{J}(\overline\alpha,t)^{^{b-\frac{1}{q}}}}
\end{split}$$ for $\eta_*-\eta>0$ small, $\eta_*=\frac{1}{\lambda M_0}$ and $$\label{eq:generalcst}
\begin{split}
C=\frac{4}{q}\left(\frac{M_0}{\left|C_1\right|}\right)^{\frac{1}{q}}\int_0^{\frac{\pi}{2}}{\frac{(\cos\theta)^{^{2b-\frac{2}{q}-1}}}{(\sin\theta)^{^{1-\frac{2}{q}}}}d\theta}.
\end{split}$$ Now, since the beta function satisfies (see for instance [@Gamelin1]): $$\label{eq:gammarel}
\begin{split}
B(p,s)=\int_0^1{t^{p-1}(1-t)^{s-1}dt}=\frac{\Gamma(p)\Gamma(s)}{\Gamma(p+s)},\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma(1+y)=y\Gamma(y)
\end{split}$$ for $p,s,y>0$, then, letting $t=\sin^2\theta$, $p=\frac{1}{q}$ and $s=b-\frac{1}{q}$ into (\[eq:gammarel\])i), and using (\[eq:gammarel\])ii), one has $$\label{eq:gammarel2}
\begin{split}
2\int_0^{\frac{\pi}{2}}{\frac{(\cos\theta)^{^{2b-\frac{2}{q}-1}}}{(\sin\theta)^{^{1-\frac{2}{q}}}}d\theta}=\frac{q\,\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(b-\frac{1}{q}\right)}{\Gamma(b)},\,\,\,\,\,\,\,\,\,\,\,\,b>\frac{1}{q}.
\end{split}$$ The result follows from (\[eq:general3\]), (\[eq:generalcst\]) and (\[eq:gammarel2\]).
**Proof of Lemma** \[lem:general\](2)
Follows from an analogous argument using (\[eq:expnew00\]) and $\eta_*=\frac{1}{\lambda m_0}$ instead.
**Proof of Lemma** \[lem:general\](3)
The last claim in (3) follows trivially if $b\leq0$ and $q\in\mathbb{R}^+$ due to the “almost everywhere” continuity and boundedness of $u_0'$. To establish the remaining claims, we make use of Lemmas \[lem:analcont\] and \[lem:diff\]. However, in order to use the latter, we require that $b\in(0,2)$ and $b\neq1/q$. Since the case $b>1/q$ was established in parts (1) and (2) above, suppose that $b\in(0,1/q)$ and $b\in(0,2)$, or equivalently $q>1/2$ and $b\in(0,1/q)$, or $q\in(0,1/2)$ and $b\in(0,2)$. First, for $q$ and $b$ as prescribed, consider $\lambda>0$ and, for simplicity, assume $M_0$ occurs at a single point $\overline\alpha\in(0,1)$. Then, (\[eq:expnew0\]) and Lemma \[lem:diff\] imply that $$\label{eq:lastgen1}
\begin{split}
\int_{\overline\alpha-r}^{\overline\alpha+r}{\frac{d\alpha}{(\epsilon+M_0-u_0'(\alpha))^b}}&\sim\int_{\overline\alpha-r}^{\overline\alpha+r}{\frac{d\alpha}{(\epsilon-C_1\left|\alpha-\overline\alpha\right|^q)^b}}
\\
&=2r\epsilon^{-b}\,{}_2F_1\left[\frac{1}{q},b,1+\frac{1}{q},\frac{C_1r^q}{\epsilon}\right]
\end{split}$$ for $\epsilon\geq\left|C_1\right|\geq\left|C_1\right|r^q>0$ and $0\leq\left|\alpha-\overline\alpha\right|\leq r$. Now, the restriction on $\epsilon$ implies that $-1\leq\frac{C_1r^q}{\epsilon}<0$. However, our ultimate goal is to let $\epsilon$ vanish, so that, eventually, the argument $\frac{C_1r^q}{\epsilon}$ of the series in (\[eq:lastgen1\])ii) will leave the unit circle, particularly $\frac{C_1r^q}{\epsilon}<-1$. At that point, definition \[eq:2f1\] for the series no longer holds and we turn to its analytic continuation in Lemma \[lem:analcont\]. Accordingly, taking $\epsilon>0$ small enough such that $\left|C_1\right|r^q>\epsilon>0$, we apply Lemma \[lem:analcont\] to (\[eq:lastgen1\]) and obtain $$\label{eq:lastgen2}
\begin{split}
\frac{2r}{\epsilon^{b}}\,{}_2F_1\left[\frac{1}{q},b,1+\frac{1}{q},\frac{C_1r^q}{\epsilon}\right]=\frac{2r^{1-qb}}{(1-bq)\left|C_1\right|^b}+\frac{2\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(b-\frac{1}{q}\right)}{\Gamma(b)\left|C_1\right|^{\frac{1}{q}}\epsilon^{b-\frac{1}{q}}}+\psi(\epsilon)
\end{split}$$ for $\psi(\epsilon)=o(1)$ as $\epsilon\to0$, and either $q>1/2$ and $b\in(0,1/q)$, or $q\in(0,1/2)$ and $b\in(0,2)$. In addition, due to the assumptions in Lemma \[lem:analcont\] we require that $\frac{1}{q}$, $b$, $b-\frac{1}{q}\notin\mathbb{Z}$. Finally, since $b-\frac{1}{q}<0$, substituting $\epsilon=\frac{1}{\lambda\eta}-M_0$ into (\[eq:lastgen1\]) and (\[eq:lastgen2\]), implies that $$\label{eq:lastgen3}
\begin{split}
\int_{0}^{1}{\frac{d\alpha}{\mathcal{J}(\alpha,t)^b}}\sim C
\end{split}$$ for $\eta_*-\eta>0$ small, $\eta_*=\frac{1}{\lambda M_0}$, and positive constants $C$ that depend on $\lambda>0$, $b$ and $q$. An analogous argument follows for $\lambda<0$ by using (\[eq:expnew00\]) instead of (\[eq:expnew0\]).$\square$
Using Lemma \[lem:general\], we now derive estimates for $\bar{\mathcal{K}}_i(t)$, $i=0,1$, which will be used in subsequent regularity Theorems.
### Estimates for $\bar{\mathcal{K}}_0(t)$ and $\bar{\mathcal{K}}_1(t)$
\[subsubsec:integralestimates\]
**For parameters $\lambda>0.$**
For $\lambda>0$, we set $b=\frac{1}{\lambda}$ into Lemma \[lem:general\](1)-(3) to obtain $$\label{eq:firstint1}
\bar{\mathcal{K}}_0(t)\sim
\begin{cases}
C,\,\,\,\,\,\,\,\,\,\,\,\,\,&\,\lambda>q>\frac{1}{2}\,\,\,\,\,\,\text{or}\,\,\,\,\,q\in(0,1/2),\,\,\,\,\lambda>\frac{1}{2},
\\
C_3\mathcal{J}(\overline\alpha_i,t)^{\frac{1}{q}-\frac{1}{\lambda}},\,\,\,\,\,\,&q>0,\,\,\,\,\lambda\in(0,q)
\end{cases}$$ for $\eta_*-\eta>0$ small and positive constants $C_3$ given by $$\label{eq:cst1}
C_3=\frac{2m\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(\frac{1}{\lambda}-\frac{1}{q}\right)}{\Gamma\left(\frac{1}{\lambda}\right)}\left(\frac{M_0}{\left|C_1\right|}\right)^{\frac{1}{q}}.$$ Also, in (\[eq:firstint1\])i) we assume that $\lambda$ and $q$ satisfy, whenever applicable, $$\label{eq:lemmaass1}
\lambda\neq\frac{q}{1-nq},\,\,\,\,\,\,\,\,q\neq\frac{1}{n}\,\,\,\,\,\,\,\forall\,\,\,\,\,\,\,n\in\mathbb{N}.$$ We note that corresponding estimates for the missing values may be obtained via a simple continuity argument.
Similarly, taking $b=1+\frac{1}{\lambda}$ we find $$\label{eq:secint1}
\bar{\mathcal{K}}_1(t)\sim
\begin{cases}
C,\,\,&q\in(1/2,1),\,\,\lambda>\frac{q}{1-q}\,\,\,\,\,\,\text{or}\,\,\,\,\,\,q\in(0,1/2),\,\,\lambda>1,
\\
C_4\mathcal{J}(\overline\alpha_i,t)^{\frac{1}{q}-\frac{1}{\lambda}-1},&q\in(0,1),\,\,0<\lambda<\frac{q}{1-q}\,\,\,\,\,\text{or}\,\,\,\,\,q\geq1,\,\,\lambda>0
\end{cases}$$ with positive constants $C_4$ determined by $$\label{eq:const2}
C_4=\frac{2m\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(1+\frac{1}{\lambda}-\frac{1}{q}\right)}{\Gamma\left(1+\frac{1}{\lambda}\right)}\left(\frac{M_0}{\left|C_1\right|}\right)^{\frac{1}{q}}.$$ Additionally, for (\[eq:secint1\])i) we assume that $\lambda$ and $q$ satisfy (\[eq:lemmaass1\]).
**For parameters $\lambda<0.$**
For $\lambda<0$ and $b=\frac{1}{\lambda}$, Lemma \[lem:general\](3) implies that $$\label{eq:thirdint1}
\bar{\mathcal{K}}_0(t)\sim C$$ for $\eta_*-\eta>0$ small. Similarly, parts (2) and (3), now with $b=1+\frac{1}{\lambda}$, yield $$\label{eq:lastest}
\bar{\mathcal{K}}_1(t)\sim C$$ for either $$\label{eq:condthirdint1}
\begin{cases}
q>0,\,\,\,\,&\lambda\in[-1,0),
\\
q\in(0,1),\,\,\,\,&\lambda<-1\,\,\,\,\,\,\,\,\,\,\,\,\text{satisfying}\,\,\,\,(\ref{eq:lemmaass1}),
\\
q>1,\,\,\,\,&\frac{q}{1-q}<\lambda<-1,
\end{cases}$$ whereas $$\label{eq:lastest1}
\bar{\mathcal{K}}_1(t)\sim C_5\mathcal{J}(\underline\alpha_j,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}$$ for $q>1$, $\lambda<\frac{q}{1-q}$ and positive constants $C_5$ determined by $$\label{eq:const3}
C_5=\frac{2n\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(1+\frac{1}{\lambda}-\frac{1}{q}\right)}{\Gamma\left(1+\frac{1}{\lambda}\right)}\left(\frac{\left|m_0\right|}{C_2}\right)^{\frac{1}{q}}.$$
### $L^{\infty}$ Regularity for $\lambda\in\mathbb{R}^+\cup\{0\}, q\in\mathbb{R}^+$
\[subsubsec:generalcaselambdapospinf\]
In this section, we use the estimates in §\[subsubsec:integralestimates\] to examine the $L^{\infty}$ regularity of $u_x$ for $\lambda\in\mathbb{R}^+\cup\{0\}$ and $u_0'$ satisfying (\[eq:expnew0\]) for some $q\in\mathbb{R}^+$. Furthermore, the behaviour of the jacobian (\[eq:sum\]) is also studied.
\[thm:lambdapos\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) for $u_0'(\alpha)$ satisfying (\[eq:expnew0\]).
1. \[it:global\] If $q\in\mathbb{R}^+$ and $\lambda\in[0,q/2],$ solutions exist globally in time. More particularly, these vanish as $t\uparrow t_*=+\infty$ for $\lambda\in(0,q/2)$ but converge to a nontrivial steady state if $\lambda=q/2$.
2. \[it:blow1\] If $q\in\mathbb{R}^+$ and $\lambda\in(q/2,q)$, there exists a finite $t_*>0$ such that both the maximum $M(t)$ and the minimum $m(t)$ diverge to $+\infty$ and respectively to $-\infty$ as $t\uparrow t_*$. Moreover, $\lim_{t\uparrow t_*}u_x(\gamma(\alpha,t),t)=-\infty$ for $\alpha\notin\bigcup_{i,j}\{\overline\alpha_i\}\cup\{\underline\alpha_j\}$ (two-sided, everywhere blow-up).
3. \[it:blow2\] For $q\in(0,1/2)$ and $\lambda>1$ such that $q\neq\frac{1}{n}$ and $\lambda\neq\frac{q}{1-nq}$ for all $n\in\mathbb{N},$ there is a finite $t_*>0$ such that only the maximum blows up, $M(t)\to+\infty,$ as $t\uparrow t_*$ (one-sided, discrete blow-up). Further, if $\frac{1}{2}<\lambda<\frac{q}{1-q}$ for $q\in(1/3,1/2)$, a two-sided, everywhere blow-up (as described in (\[it:blow1\]) above) occurs at a finite $t_*>0$.
4. \[it:blow3\] Suppose $q\in(1/2,1)$. Then for $q<\lambda<\frac{q}{1-q}$, there exists a finite $t_*>0$ such that, as $t\uparrow t_*$, two-sided, everywhere blow-up develops. If instead $\lambda>\frac{q}{1-q}$, only the maximum diverges, $M(t)\to+\infty$, as $t\uparrow t_*<+\infty$.
5. \[it:blow4\] For $\lambda>q>1$, there is a finite $t_*>0$ such that $u_x$ undergoes a two-sided, everywhere blow-up as $t\uparrow t_*$.
Suppose $\lambda,q>0$, let $C$ denote a positive constant which may depend on $\lambda$ and $q$, and set $\eta_*=\frac{1}{\lambda M_0}.$
**Proof of Statements** (\[it:global\]) **and** (\[it:blow1\])
Suppose $\lambda\in(0,q)$ for some $q>0$. Then, for $\eta_*-\eta>0$ small $\bar{\mathcal{K}}_0(t)$ satisfies (\[eq:firstint1\])ii) while $\bar{\mathcal{K}}_1(t)$ obeys (\[eq:secint1\])ii). Consequently, (\[eq:mainsolu\]) implies that $$\label{eq:ok3}
u_x(\gamma(\alpha,t),t)\sim \frac{M_0}{C_3^{^{2\lambda}}}\left(\frac{\mathcal{J}(\overline\alpha_i,t)}{\mathcal{J}(\alpha,t)}-\frac{C_4}{C_3}\right)\mathcal{J}(\overline\alpha_i,t)^{1-\frac{2\lambda}{q}}$$ for positive constants $C_3$ and $C_4$ given by (\[eq:cst1\]) and (\[eq:const2\]). But for $y_1=\frac{1}{\lambda}-\frac{1}{q}$ and $y_2=\frac{1}{\lambda}$, (\[eq:gammarel\])ii), (\[eq:cst1\]) and (\[eq:const2\]) yield $$\label{eq:gammaid}
\frac{C_4}{C_3}=\frac{\Gamma(y_1+1)\,\Gamma(y_2)}{\Gamma(y_1)\,\Gamma(y_2+1)}=\frac{y_1}{y_2}=1-\frac{\lambda}{q}\in(0,1),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lambda\in(0,q).$$ As a result, setting $\alpha=\overline\alpha_i$ in (\[eq:ok3\]) and using (\[eq:maxmin\])i) implies that $$\label{eq:ux5}
M(t)\sim\frac{M_0}{C_3^{^{2\lambda}}}\left(\frac{\lambda}{q}\right)\mathcal{J}(\overline\alpha_i,t)^{1-\frac{2\lambda}{q}}$$ for $\eta_*-\eta>0$ small, whereas, if $\alpha\neq\overline\alpha_i$, $$\label{eq:ux6}
u_x(\gamma(\alpha,t),t)\sim-\left(1-\frac{\lambda}{q}\right)\frac{M_0}{C_3^{^{2\lambda}}}\mathcal{J}(\overline\alpha_i,t)^{1-\frac{2\lambda}{q}}.$$ Clearly, when $\lambda=q/2$, $$M(t)\to\frac{M_0}{2C_3^{\,q}}>0$$ as $\eta\uparrow\eta_*$, while, for $\alpha\neq\overline\alpha_i$, $$u_x(\gamma(\alpha,t),t)\to-\frac{M_0}{2C_3^{\,q}}<0.$$ If $\lambda\in(0,q/2)$, (\[eq:ux5\]) now implies that $$M(t)\to0^+$$ as $\eta\uparrow\eta_*$, whereas, using (\[eq:ux6\]) for $\alpha\neq\overline\alpha_i$, $$u_x(\gamma(\alpha,t),t)\to0^-.$$ In contrast, if $\lambda\in(q/2,q)$, $1-\frac{2\lambda}{q}<0$. Then (\[eq:ux5\]) and (\[eq:ux6\]) yield $$\label{eq:ux52}
M(t)\to+\infty$$ as $\eta\uparrow\eta_*$, but $$\label{eq:ux62}
u_x(\gamma(\alpha,t),t)\to-\infty$$ for $\alpha\neq\overline\alpha_i$. Lastly, rewriting (\[eq:etaivp\]) as $$\label{eq:time}
dt=\bar{\mathcal{K}}_0(t)^{2\lambda}d\eta$$ and using (\[eq:firstint1\])ii), we obtain $$\label{eq:ux7}
t_*-t\sim C\int_{\eta}^{\eta_*}{(1-\lambda\mu M_0)^{\frac{2\lambda}{q}-2}d\mu}$$ or equivalently $$\label{eq:timegen}
t_*-t\sim
\begin{cases}
\frac{C}{2\lambda-q}\left(C(\eta_*-\eta)^{\frac{2\lambda}{q}-1}-\lim_{\mu\uparrow\eta_*}(\eta_*-\mu)^{\frac{2\lambda}{q}-1}\right),\,\,\,\,&\lambda\in(0,q)\backslash\{q/2\},
\\
C\left(\log(\eta_*-\eta)-\lim_{\mu\uparrow\eta_*}\log(\eta_*-\mu)\right),\,\,\,&\lambda=q/2.
\end{cases}$$ Consequently, $t_*=+\infty$ for $\lambda\in(0,q/2]$, while $0<t_*<+\infty$ if $\lambda\in(q/2,q)$. Lastly, the case $\lambda=0$ follows from the results in [@Sarria1].
**Proof of Statement** (\[it:blow2\])
First, suppose $q\in(0,1/2)$ and $\lambda>1$ satisfy (\[eq:lemmaass1\]). Then $\bar{\mathcal{K}}_0(t)$ and $\bar{\mathcal{K}}_1(t)$ satisfy (\[eq:firstint1\])i) and (\[eq:secint1\])i), respectively. Therefore, (\[eq:mainsolu\]) implies that $$\label{eq:blow3case}
u_x(\gamma(\alpha,t),t)\sim C\left(\frac{1}{\mathcal{J}(\alpha,t)}-C\right)$$ for $\eta_*-\eta>0$ small. Setting $\alpha=\overline\alpha_i$ into (\[eq:blow3case\]) and using (\[eq:maxmin\])i) gives $$M(t)\sim\frac{C}{\mathcal{J}(\overline\alpha_i,t)}\to+\infty$$ as $\eta\uparrow\eta_*$, while, if $\alpha\neq\overline\alpha_i$, $u_x(\gamma(\alpha,t),t)$ remains finite for all $\eta\in[0,\eta_*]$ due to the definition of $M_0$. The existence of a finite blow-up time $t_*>0$ for the maximum is guaranteed by (\[eq:firstint1\])i) and (\[eq:time\]), which lead to $$\label{eq:timeblow3case}
t_*-t\sim C(\eta_*-\eta).$$ Next, suppose $\frac{1}{2}<\lambda<\frac{q}{1-q}$ for $q\in(1/3,1/2)$, so that $\frac{q}{1-q}\in(1/2,1)$. Then, using (\[eq:firstint1\])i) and (\[eq:secint1\])ii) in (\[eq:mainsolu\]), we find that $$\label{eq:blow4case}
u_x(\gamma(\alpha,t),t)\sim C\left(\frac{C}{\mathcal{J}(\alpha,t)}-\mathcal{J}(\overline\alpha_i,t)^{^{\frac{1}{q}-\frac{1}{\lambda}-1}}\right)$$ for $\eta_*-\eta>0$ small. Set $\alpha=\overline\alpha_i$ into the above and use $\lambda>q$ to obtain $$\label{eq:blow4case1}
M(t)\sim\frac{C}{\mathcal{J}(\overline\alpha_i,t)}\to+\infty$$ as $\eta\uparrow\eta_*$. On the other hand, for $\alpha\neq\overline\alpha_i$, the space-dependent in (\[eq:blow4case\]) now remains finite for all $\eta\in[0,\eta_*]$. As a result, the second term dominates and $$\label{eq:blow4case2}
u_x(\gamma(\alpha,t),t)\sim-C\mathcal{J}(\overline\alpha_i,t)^{^{\frac{1}{q}-\frac{1}{\lambda}-1}}\to-\infty$$ as $\eta\uparrow\eta_*$. The existence of a finite blow-up time $t_*>0$, follows, as in the previous case, from (\[eq:time\]) and (\[eq:firstint1\])i).
**Proof of Statement** (\[it:blow3\])
Part (\[it:blow3\]) follows from an argument analogous to the one above. Briefly, if $q<\lambda<\frac{q}{1-q}$ for $q\in(1/2,1)$, we use estimates (\[eq:firstint1\])i) and (\[eq:secint1\])ii) on (\[eq:mainsolu\]) to get (\[eq:blow4case\]), with different positive constants $C$. Two-sided, everywhere blow-up in finite-time then follows just as above. If instead $\lambda>\frac{q}{1-q}$ for $q\in(1/2,1)$, then (\[eq:firstint1\])i) still holds but $\bar{\mathcal{K}}_1(t)$ now remains bounded for all $\eta\in[0,\eta_*]$; it satisfies (\[eq:secint1\])i). Therefore, up to different positive constants $C$, (\[eq:mainsolu\]) leads to (\[eq:blow3case\]), and so only the maximum diverges, $M(t)\to+\infty$, as $t$ approaches some finite $t_*>0$ whose existence is guaranteed by (\[eq:timeblow3case\]).
**Proof of Statement** (\[it:blow4\])
For $\lambda>q>1$, (\[eq:firstint1\])i), (\[eq:secint1\])ii) and (\[eq:mainsolu\]) imply (\[eq:blow4case\]). Then, we follow the argument used to establish the second part of (\[it:blow2\]) to show that two-sided, everywhere finite-time blow-up occurs. See §\[sec:examples\] for examples.
Theorems \[thm:p=1\] and \[thm:lambdapos\] allow us to predict the regularity of stagnation point-form (SPF) solutions to the two ($\lambda=1$) and three ($\lambda=1/2$) dimensional incompressible Euler equations assuming we know something about the curvature of the initial data $u_0$ near $\overline\alpha_i$. Setting $\lambda=1$ into Theorem \[thm:lambdapos\](\[it:global\]) implies that SPF solutions in the 2D setting persist for all time if $u_0'$ satisfies (\[eq:expnew0\]) for arbitrary $q\geq2$. On the contrary, Theorems \[thm:p=1\] and \[thm:lambdapos\](\[it:blow1\])-(\[it:blow3\]), tell us that if $q\in(1/2,2)$, two-sided, everywhere finite-time blow-up occurs. Analogously, solutions to the corresponding 3D problem exist globally in time for $q\geq1$, whereas, two-sided, everywhere blow-up develops when $q\in(1/2,1)$. See Table \[table:thecase\] below. Finally, we remark that finite-time blow-up in $u_x$ is expected for both the two and three dimensional equations if $q\in(0,1/2]$. See for instance §\[sec:examples\] for a blow-up example in the 3D case with $q=1/3$.
[ | m[2.0cm]{} | m[3.4cm]{} | m[3.4cm]{} | ]{} & & $\text{\,\,\,\,\,\,\,\,\,\,\,\,\,\,3D Euler}$\
&$\text{Finite time blow up}$ & $\text{Finite time blow up}$\
&$\text{Finite time blow up}$ & $\text{\,\,\,\,\,\,Global in time}$\
&$\text{\,\,\,\,\,\,\,\,Global in time}$ &$\text{\,\,\,\,\,\,Global in time}$\
\[table:thecase\]
Corollary \[coro:lambda>1/2\] below briefly examines the behaviour, as $t\uparrow t_*$, of the jacobian (\[eq:sum\]) for $t_*>0$ as in Theorem \[thm:lambdapos\].
\[coro:lambda>1/2\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) with $u_0'(\alpha)$ satisfying (\[eq:expnew0\]) for $q\in\mathbb{R}^+$, and let $t_*>0$ be as in Theorem \[thm:lambdapos\]. It follows,
1. \[it:item1\] For $q>0$ and $\lambda\in(0,q)$, $$\label{eq:jacone}
\lim_{t\uparrow t_*}\gamma_{\alpha}(\alpha,t)=
\begin{cases}
+\infty,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&\alpha=\overline\alpha_i,
\\
0,\,\,\,\,\,\,\,\,&\alpha\neq\overline\alpha_i
\end{cases}$$ where $t_*=+\infty$ for $\lambda\in(0,q/2]$, while $0<t_*<+\infty$ if $\lambda\in(q/2,q)$.
2. \[it:item1.5\] Suppose $\lambda>q>1/2$, or $q\in(0,1/2)$ and $\lambda>1/2$, satisfy (\[eq:lemmaass1\]). Then, there exists a finite $t_*>0$ such that $$\label{eq:jacone22}
\lim_{t\uparrow t_*}\gamma_{\alpha}(\alpha,t)=
\begin{cases}
+\infty,\,\,\,\,\,\,\,\,\,\,&\alpha=\overline\alpha_i,
\\
C(\alpha),\,\,\,\,\,\,\,\,&\alpha\neq\overline\alpha_i
\end{cases}$$ where $C(\alpha)\in\mathbb{R}^+$ depends on the choice of $\lambda$, $q$ and $\alpha\neq\overline\alpha_i$.
The limits (\[eq:jacone\]) and (\[eq:jacone22\]) follow straightforwardly from (\[eq:sum\]) and estimates (\[eq:firstint1\])ii) and (\[eq:firstint1\])i), respectively; whereas, the finite or infinite character of $t_*>0$ is a consequence of Theorem \[thm:lambdapos\].
### Further $L^p$ Regularity for $\lambda\in[0,+\infty), q\in\mathbb{R}^+$ and $p\in[1,+\infty)$
\[subsubsec:generalcaselambdapospfin\]
From Theorem \[thm:lambdapos\], if $\lambda\in[0,q/2]$ for $q\in\mathbb{R}^+$, solutions remain in $L^{\infty}$ for all time; otherwise, $\left\|u_x\right\|_{\infty}$ diverges as $t$ approaches some finite $t_*>0$. In this section, we study further properties of $L^p$ regularity in $u_x$, as $t\uparrow t_*$, for $\lambda>q/2$, $p\in[1,+\infty)$ and initial data $u_0'(\alpha)$ satisfying (\[eq:expnew0\]). To do so, we will use the upper and lower bounds (\[eq:upper\]) and (\[eq:lower\]). Consequently, for $\eta_*-\eta>0$ small and $\eta_*=\frac{1}{\lambda M_0}$, estimates on the behaviour of the time-dependent integrals $$\label{eq:remaining}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{1+\frac{1}{\lambda p}}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{p+\frac{1}{\lambda}}}}$$ are required. Since these may be obtained directly from Lemma \[lem:general\](1)-(3), we omit the details and state our findings below. For $p\in[1,+\infty)$, $$\label{eq:lpos3}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\sim
\begin{cases}
C,\,\,&q\in(0,1/2),\, \lambda>\frac{1}{2p}\,\,\,\,\text{or}\,\,\,q>\frac{1}{2},\,\,\, \lambda>\frac{q}{p},
\\
C_6\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda p}}, &q>0,\,\,\, \lambda\in(0,q/p)
\end{cases}$$ with positive constants $$\label{eq:c6}
C_6=\frac{2\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(\frac{1}{\lambda p}-\frac{1}{q}\right)}{\Gamma\left(\frac{1}{\lambda p}\right)}\left(\frac{M_0}{\left|C_1\right|}\right)^{\frac{1}{q}}.$$ Also $$\label{eq:lpos4}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{1+\frac{1}{\lambda p}}}}\sim C$$ for either $$\label{eq:lpos5}
\begin{cases}
&q\in(0,1/2),\,\,\,\,\,\, \lambda>\frac{1}{p},
\\
&q\in(1/2,1),\,\,\,\,\,\, \lambda>\frac{q}{p(1-q)},
\end{cases}$$ whereas $$\label{eq:lpos6}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{1+\frac{1}{\lambda p}}}}\sim C_7\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda p}-1}$$ for $$\label{eq:lpos7}
\begin{cases}
&q\in(0,1),\,\,\,\,\, 0<\lambda<\frac{q}{p(1-q)},
\\
&q\geq1,\,\,\,\,\, \lambda>0.
\end{cases}$$ The positive constants $C_7$ in (\[eq:lpos6\]) are obtained by replacing every $\frac{1}{\lambda p}$ term in (\[eq:c6\]) by $1+\frac{1}{\lambda p}$. Also, due to Lemma \[lem:general\], (\[eq:lpos3\])i) and (\[eq:lpos4\]) are valid for $$\label{eq:req1lambdapos1}
\lambda\neq\frac{q}{p(1-nq)},\,\,\,\,\,\,\,\,\,q\neq\frac{1}{n}\,\,\,\forall\,\,\,\,n\in\mathbb{N}\cup\{0\},$$ where a simple continuity argument may again be used (see (\[eq:lemmaass1\])) to obtain estimates for the missing values. Finally $$\label{eq:lpos8}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{p+\frac{1}{\lambda}}}}\sim C$$ for either $$\label{eq:lpos9}
\begin{cases}
&q\in(0,1/2),\,\,\,\,p\in[1,2),\,\,\,\,\,\,\,\,\,\, \lambda>\frac{1}{2-p},
\\
&q\in(1/2,1),\,\,\,\, p\in[1,1/q),\,\,\,\, \lambda>\frac{q}{1-pq},
\end{cases}$$ while $$\label{eq:lpos10}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{p+\frac{1}{\lambda}}}}\sim C\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda}-p}$$ if $$\label{eq:lpos11}
\begin{cases}
&q\in(0,1],\,\,\,\,\,\, p\in[1,1/q),\,\,\,\, 0<\lambda<\frac{q}{1-pq},
\\
&q\in(0,1],\,\,\,\,\,\, p\geq\frac{1}{q},\,\,\,\,\,\, \lambda>0,
\\
&q>1,\,\,\,\,\,\,\,\,\,\,\,\,\,\, p\geq1,\,\,\,\,\,\,\, \lambda>0.
\end{cases}$$ Estimate (\[eq:lpos8\]) is in turn valid for $$\label{eq:req1lambdapos2}
\lambda\neq\frac{q}{1+q(n-p)},\,\,\,\,\,\,\,\,\,q\neq\frac{1}{n}\,\,\,\,\forall\,\,\,n\in\mathbb{N}.$$ In what follows, $t_*>0$ will denote the $L^{\infty}$ blow-up time for $u_x$ in Theorem \[thm:lambdapos\]. Also, we will assume that (\[eq:lemmaass1\]), (\[eq:req1lambdapos1\]) and (\[eq:req1lambdapos2\]) hold whenever their corresponding estimates are used. We begin by considering the lower bound (\[eq:lower\]). In particular, we will show that two-sided, everywhere blow-up in Theorem \[thm:lambdapos\] corresponds to a diverging $\left\|u_x\right\|_p$ for all $p>1$. Then, by studying the upper bound (\[eq:upper\]), we will find that if $q\in\mathbb{R}^+$ and $\lambda>q$ are such that only the maximum diverges at a finite $t_*>0$, then $u_x$ remains integrable for all $t\in[0,t_*]$, whereas, its regularity in smaller $L^p$ spaces for $t\in[0,t_*]$ will vary according to the value of the parameter $\lambda$ as a function of either $p$, $q$, or both.
Suppose $q/2<\lambda<q/p$ for $q\in\mathbb{R}^+$ and $p\in(1,2)$. Then (\[eq:lpos3\])ii) holds as well as (\[eq:firstint1\])ii), since $(q/2,q/p)\subset(0,q)$. Now, if $q\in(0,1)$ then $0<\frac{q}{2}<\lambda<\frac{q}{p}<q<\frac{q}{1-q},$ and so (\[eq:secint1\])ii) applies, otherwise, (\[eq:secint1\])ii) also holds for $q\geq1$ and $\lambda>0$. Similarly for $q\in(0,1)$, we have that $0<\frac{q}{2}<\lambda<\frac{q}{p}<\frac{q}{p(1-q)}$ so that (\[eq:lpos6\]) is valid. Alternatively, this last estimate also holds if $q\geq1$ for $\lambda>0$. Accordingly, using these estimates in (\[eq:lower\]) yields, after simplification, $$\begin{split}
\left\|u_x(\cdot,t)\right\|_p&\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|
\\
&\sim C(p-1)\mathcal{J}(\overline\alpha,t)^{\sigma(p,q,\lambda)}
\end{split}$$ for $\eta_*-\eta>0$ small and $\sigma(p,q,\lambda)=1+\frac{1}{q}\left(1-\frac{1}{p}-2\lambda\right).$ Consequently, $\left\|u_x\right\|_p$ will diverge as $\eta\uparrow\eta_*$ if $\sigma(p,q,\lambda)<0$, or equivalently for $p(1+q-2\lambda)-1<0$. Since $q/2<\lambda<q/p$ for $q>0$ and $p\in(1,2)$, we find this to be the case as long as $$q\in\mathbb{R}^+,\,\,\,\,\,\,\,\,\,1<p<1+\frac{q}{1+q},\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2}\left(q+1-\frac{1}{p}\right)<\lambda<\frac{q}{p}.$$ Therefore, by taking $p-1>0$ arbitrarily small, we find that $$\lim_{t\uparrow t_*}\left\|u_x(\cdot,t)\right\|_p=+\infty$$ for $\lambda\in(q/2,q)$ and $q>0$. The existence of a finite blow-up time $t_*>0$ follows from Theorem \[thm:lambdapos\](\[it:blow1\]), while the embedding $$\label{eq:embedding}
\begin{split}
L^s\hookrightarrow L^p,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,s\geq p,
\end{split}$$ yields $L^p$ blow-up for any $p>1$. Next, for $q\in(1/3,1/2)$ we consider values of $\lambda$ lying between stagnation point-form solutions to the 2D ($\lambda=1$) and 3D ($\lambda=1/2$) incompressible Euler equations. Suppose $\frac{1}{2}<\lambda<\frac{q}{p(1-q)}$ for $1<p<\frac{2q}{1-q}$ and $q\in(1/3,1/2)$. The condition on $p$ simply guarantees that $\frac{q}{p(1-q)}>\frac{1}{2}$ for $q$ as specified. Furthermore, we have that $$0<\frac{1}{2p}<\frac{1}{2}<\lambda<\frac{q}{p(1-q)}<\frac{q}{1-q}\in(1/2,1),$$ so that relative to our choice of $\lambda$ and $q$, $\lambda\in(1/2,1).$ Using the above, we find that (\[eq:firstint1\])i), (\[eq:secint1\])ii), (\[eq:lpos3\])i) and (\[eq:lpos6\]) hold, and so (\[eq:lower\]) leads to $$\label{eq:touse1}
\begin{split}
\left\|u_x(\cdot,t)\right\|_p&\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|
\\
&\sim C\left|C\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda p}-1}-\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}\right|
\\
&\sim C\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}
\end{split}$$ for $\eta_*-\eta>0$ small. Therefore, as $\eta\uparrow\eta_*$, $\left\|u_x\right\|_p$ will diverge for all $\frac{1}{2}<\lambda<\frac{q}{p(1-q)}$, $q\in(1/3,1/2)$ and $1<p<\frac{2q}{1-q}$. Here, we can take $p-1>0$ arbitrarily small and use (\[eq:embedding\]) to conclude the finite-time blow-up, as $t\uparrow t_*$, of $\left\|u_x\right\|_p$ for all $\frac{1}{2}<\lambda<\frac{q}{1-q}$, $q\in(1/3,1/2)$ and $p>1$. The existence of a finite blow-up time $t_*>0$ is guaranteed by the second part of Theorem \[thm:lambdapos\](\[it:blow2\]). Now suppose $q\in(1/2,1)$ and $q<\lambda<\frac{q}{p(1-q)}$ for $1<p<\frac{1}{1-q}$. This means that $\lambda>q>1/2$ and $$0<\frac{q}{p}<q<\lambda<\frac{q}{p(1-q)}<\frac{q}{1-q}.$$ Consequently, using (\[eq:firstint1\])i), (\[eq:secint1\])ii), (\[eq:lpos3\])i) and (\[eq:lpos6\]) in (\[eq:lower\]), implies (\[eq:touse1\]), possibly with distinct positive constants $C$. Then, as $\eta\uparrow\eta_*$, $$\left\|u_x\right\|_p\to+\infty$$ for all $q<\lambda<\frac{q}{p(1-q)}$, $q\in(1/2,1)$ and $1<p<\frac{1}{1-q}$. Similarly, if $q$ and $p$ are as above, but $\frac{q}{p(1-q)}<\lambda<\frac{q}{1-q}$, (\[eq:firstint1\])i), (\[eq:secint1\])ii), (\[eq:lpos3\])i) and (\[eq:lpos4\]) imply $$\begin{split}
\left\|u_x(\cdot,t)\right\|_p&\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|
\\
&\sim C\left|C-\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}\right|
\\
&\sim C\mathcal{J}(\overline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}\to+\infty
\end{split}$$ as $\eta\uparrow\eta_*$. From these last two results and (\[eq:embedding\]), we see that, as $\eta\uparrow\eta_*$, $\left\|u_x\right\|_p\to+\infty$ for all $q<\lambda<\frac{q}{1-q}$, $q\in(1/2,1)$ and $p>1$. The existence of a finite $t_*>0$ follows from Theorem \[thm:lambdapos\](\[it:blow3\]). Lastly, suppose $\lambda>q>1$ and $p>1$. Then, estimates (\[eq:firstint1\])i), (\[eq:secint1\])ii), (\[eq:lpos3\])i) and (\[eq:lpos6\]) hold for $\eta_*-\eta>0$ small. As a result, (\[eq:lower\]) implies (\[eq:touse1\]), which in turn leads to $L^p$ blow-up of $u_x$ for any $\lambda>q>1$ and $p>1$, as $\eta\uparrow\eta_*$. The existence of a finite $t_*>0$ is due to Theorem \[thm:lambdapos\](\[it:blow4\]).
Notice from the results established so far, that some values of $\lambda>q/2$ for $q>0$ are missing. These are precisely the cases for which the lower bound (\[eq:lower\]) yields inconclusive information about the $L^p$ regularity of $u_x$ for $p\in(1,+\infty)$. To examine some aspects of the $L^p$ regularity of $u_x$ for $t\in[0,t_*]$ and $p\in[1,+\infty)$ in these particular cases, we consider the upper bound (\[eq:upper\]). First, suppose $q\in(0,1/2)$ and $\lambda>\frac{1}{2-p}$ for $p\in[1,2)$. Then $\lambda>\frac{1}{2-p}>1>\frac{q}{1-q}>q,$ so that (\[eq:firstint1\])i), (\[eq:secint1\])i) and (\[eq:lpos8\]), imply that the integral terms in (\[eq:upper\]) remain bounded, and nonzero, for $\eta\in[0,\eta_*]$. We conclude that $$\label{eq:upp1}
\begin{split}
\lim_{t\uparrow t_*}\left\|u_x(\cdot,t)\right\|_p<+\infty
\end{split}$$ for all $\lambda>\frac{1}{2-p}$, $q\in(0,1/2)$ and $p\in[1,2)$. Here, $t_*>0$ denotes the finite $L^{\infty}$ blow-up time for $u_x$ established in the first part of Theorem \[thm:lambdapos\](\[it:blow2\]). Particularly, this result implies that even though $\lim_{t\uparrow t_*}\left\|u_x\right\|_{\infty}=+\infty$ for all $\lambda>1$ when $q\in(0,1/2)$, $u_x$ remains integrable for $t\in[0,t_*]$. Finally, suppose $q\in(1/2,1)$ and $\lambda>\frac{q}{1-pq}$ for $p\in[1,1/q)$. Then $\lambda>\frac{q}{1-pq}\geq\frac{q}{1-q}>1>q>\frac{1}{2}$, and so (\[eq:firstint1\])i), (\[eq:secint1\])i) and (\[eq:lpos8\]) hold. Consequently, (\[eq:upper\]) implies that $\lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty$ for all $\lambda>\frac{q}{1-pq}$, $q\in(1/2,1)$ and $p\in[1,1/q)$. This time, $t_*>0$ stands as the finite $L^{\infty}$ blow-up time for $u_x$ established in the second part of Theorem \[thm:lambdapos\](\[it:blow3\]). Furthermore, this result tells us that even though $\lim_{t\uparrow t_*}\left\|u_x\right\|_{\infty}=+\infty$ for $\lambda>\frac{q}{1-q}$ and $q\in(1/2,1)$, $u_x$ stays integrable for all $t\in[0,t_*]$. These last two results on the integrability of $u_x$, for $t\in[0,t_*]$, become more apparent if we set $p=1$ in (\[eq:upper\]) to obtain $$\left\|u_x(\cdot,t)\right\|_1\leq\frac{2\bar{\mathcal{K}}_1(t)}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{1+2\lambda}}.$$ The result then follows from the above inequality and estimates (\[eq:firstint1\])i) and (\[eq:secint1\])i). Theorem \[thm:lplambdapos\] below summarizes the above results.
\[thm:lplambdapos\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) for $u_0'(\alpha)$ satisfying (\[eq:expnew0\]), and let $t_*>0$ be as in Theorem \[thm:lambdapos\].
1. \[it:lplambdapos1\] For $q>0$ and $\lambda\in[0,q/2]$, $\lim_{t\to+\infty}\left\|u_x\right\|_p<+\infty$ for all $p\geq1$. More particularly, $\lim_{t\to+\infty}\left\|u_x\right\|_p=0$ for $\lambda\in(0,q/2)$, while, as $t\to+\infty$, $u_x$ converges to a nontrivial, $L^{\infty}$ function when $\lambda=q/2$.
2. \[it:lplambdapos2\] Let $p>1$. Then, there exists a finite $t_*>0$ such that for all $q>0$ and $\lambda\in(q/2,q)$, $\lim_{t\uparrow t_*}\left\|u_x\right\|_p=+\infty$. Similarly for $\lambda>q>1$, or $\frac{1}{2}<\lambda<\frac{q}{1-q}$, $q\in(1/3,1/2)$.
3. \[it:lplambdapos3\] For all $q\in(0,1/2)$, $\lambda>\frac{1}{2-p}$ and $p\in[1,2)$, there exists a finite $t_*>0$ such that $\lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty$ (see Theorem \[thm:lambdapos\](\[it:blow2\])).
4. \[it:lplambdapos4\] Suppose $q\in(1/2,1)$. Then, there exists a finite $t_*>0$ such that $\lim_{t\uparrow t_*}\left\|u_x\right\|_p=+\infty$ for $q<\lambda<\frac{q}{1-q}$ and $p>1$, whereas, if $\lambda>\frac{q}{1-pq}$ and $p\in[1,1/q)$, $\lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty$ (see Theorem \[thm:lambdapos\](\[it:blow3\])).
### $L^{\infty}$ regularity for $\lambda<0$ and $q\in\mathbb{R}^+$
\[subsubsec:generalcaselambdanegpinf\]
We now examine the $L^{\infty}$ regularity of $u_x$ for parameters $\lambda<0$ and initial data satisfying (\[eq:expnew00\]) for arbitrary $q\in\mathbb{R}^+$. We prove Theorem \[thm:lambdaneg\] below.
\[thm:lambdaneg\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) for $u_0'(\alpha)$ satisfying (\[eq:expnew00\]). Furthermore,
1. \[it:lambdaneg1\] Suppose $\lambda\in[-1,0)$ and $q>0$. Then, there exists a finite $t_*>0$ such that only the minimum diverges, $m(t)\to-\infty,$ as $t\uparrow t_*$ (one-sided, discrete blow-up).
2. \[it:lambdaneg2\] Suppose $\lambda<-1$ and $q\in(0,1)$ satisfy $\lambda\neq\frac{q}{1-nq}$ and $q\neq\frac{1}{n}$ $\forall$ $n\in\mathbb{N}$. Then, a one-sided discrete blow-up, as described in (\[it:lambdaneg1\]), occurs in finite-time. Similarly for $\frac{q}{1-q}<\lambda<-1$ and $q>1$.
3. \[it:lambdaneg3\] Suppose $\lambda<\frac{q}{1-q}$ and $q>1$. Then, there is a finite $t_*>0$ such that both the maximum $M(t)$ and the minimum $m(t)$ diverge to $+\infty$ and respectively to $-\infty$ as $t\uparrow t_*$. Moreover, $\lim_{t\uparrow t_*}u_x(\gamma(\alpha,t),t)=+\infty$ for $\alpha\notin\bigcup_{i,j}\{\overline\alpha_i\}\cup\{\underline\alpha_j\}$ (two-sided, everywhere blow-up).
Finally, for $\lambda<0$, $q>0$ and $t_*>0$ as above, the jacobian (\[eq:sum\]) satisfies $$\label{eq:lastrealjac}
\lim_{t\uparrow t_*}\gamma_{\alpha}(\alpha,t)=
\begin{cases}
0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&\alpha=\underline\alpha_j,
\\
C(\alpha),\,\,\,\,\,\,\,\,\,\,\,&\alpha\neq\underline\alpha_j
\end{cases}$$ where $C(\alpha)\in\mathbb{R}^+$ depends on the choice of $\lambda$, $q$ and $\alpha\neq\underline\alpha_j$.
Throughout, let $C$ denote a positive constant that may depend on $\lambda<0$, $q>0$ and recall that $\eta_*=\frac{1}{\lambda m_0}.$
**Proof of Statement** (\[it:lambdaneg1\])
Suppose $\lambda\in[-1,0)$ and assume $u_0'(\alpha)$ satisfies (\[eq:expnew00\]) for some $q>0$. Then (\[eq:thirdint1\]) and (\[eq:lastest\]) imply that both integral terms in (\[eq:mainsolu\]) remain finite and nonzero as $\eta\uparrow\eta_*$. More particularly, one can show that (\[eq:quick3p\]) and (\[eq:quick4p\]) hold for all $\eta\in[0,\eta_*].$ Therefore, blow-up of (\[eq:mainsolu\]) depends, solely, on the behaviour of the space-dependent term $\mathcal{J}(\alpha,t)^{-1}$. Accordingly, we set $\alpha=\underline\alpha_j$ into (\[eq:mainsolu\]) and use (\[eq:maxmin\])ii) to find that the minimum diverges, $m(t)\to-\infty$, as $\eta\uparrow\eta_*$. However, if $\alpha\neq\underline\alpha_j$, the definition of $m_0$ implies that the space-dependent term now remains bounded, and positive, for $\eta\in[0,\eta_*]$. The existence of a finite blow-up time $t_*>0$ for the minimum follows from (\[eq:etaivp\]) and (\[eq:thirdint1\]). In fact, we may use (\[eq:etaivp\]) and (\[eq:quick3p\]) to obtain the estimate $$\label{eq:t1}
\begin{split}
\frac{\left|m_0\right|}{\left|\lambda\right|(m_0-M_0)^2}\leq t_*\leq\eta_*.
\end{split}$$
**Proof of Statements** (\[it:lambdaneg2\]) and (\[it:lambdaneg3\])
Now suppose $\lambda<-1$. As in the previous case, the term $\bar{\mathcal{K}}_0(t)$ remains finite, and positive, for all $\eta\in[0,\eta_*]$. Particularly, $\bar{\mathcal{K}}_0(t)$ satisfies (\[eq:newestq\]) for all $\eta\in[0,\eta_*]$. On the other hand, $\bar{\mathcal{K}}_1(t)$ now either converges or diverges, as $\eta\uparrow\eta_*$, according to (\[eq:lastest\]) or (\[eq:lastest1\]), respectively. If $\lambda<-1$ and $q>0$ are such that (\[eq:lastest\]) holds, then part (\[it:lambdaneg2\]) follows just as part (\[it:lambdaneg1\]). However, if $q>1$ and $\lambda<\frac{q}{1-q}$, we use (\[eq:thirdint1\]) and (\[eq:lastest1\]) on (\[eq:mainsolu\]), to obtain $$u_x(\gamma(\alpha,t),t)\sim Cm_0\left(\frac{1}{\mathcal{J}(\alpha,t)}-C\mathcal{J}(\underline\alpha_j,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}\right)$$ for $\eta_*-\eta>0$ small. Setting $\alpha=\underline\alpha_j$ into the above implies that $$m(t)\sim\frac{Cm_0}{\mathcal{J}(\underline\alpha,t)}\to-\infty$$ as $\eta\uparrow\eta_*$, whereas, for $\alpha\neq\underline\alpha_i$, the space-dependent term now remains bounded, as a result, the second term dominates and $$u_x(\gamma(\alpha,t),t)\sim C\left|m_0\right|\mathcal{J}(\underline\alpha_j,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}\to+\infty$$ as $\eta\uparrow\eta_*$. The existence of a finite blow-up time $t_*>0$ follows as in the case $\lambda\in[-1,0)$. In fact, (\[eq:etaivp\]) and (\[eq:newestq\]) yield the lower bound $\eta_*\leq t_*$. Finally, (\[eq:lastrealjac\]) is derived straightforwardly from (\[eq:sum\]) and (\[eq:thirdint1\]). See §\[sec:examples\] for examples.
### Further $L^p$ regularity for $\lambda\in\mathbb{R}^-$, $q\in\mathbb{R}^+$ and $p\in[1,+\infty)$
\[subsubsec:generalcaselambdanegpfin\]
Let $t_*>0$ denote the finite $L^{\infty}$ blow-up time for $u_x$ in Theorem \[thm:lambdaneg\] above. In this last section, we briefly examine the $L^p$ regularity of $u_x$, as $t\uparrow t_*$, for $\lambda\in\mathbb{R}^-$, $p\in[1,+\infty)$ and $u_0'$ satisfying (\[eq:expnew00\]) for some $q\in\mathbb{R}^+$. As in §\[subsubsec:generalcaselambdapospfin\], we will make use of (\[eq:upper\]) and (\[eq:lower\]). First of all, by the last part of Lemma \[lem:general\](3), we have that for $q>0$ and $p\geq1$, $$\label{eq:reallast}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\sim C$$ for $\eta_*-\eta>0$ small, $\eta_*=\frac{1}{\lambda m_0}$ and $\lambda<0$. Similarly $$\label{eq:fin1}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{p+\frac{1}{\lambda}}}}\sim C$$ for $-\frac{1}{p}\leq\lambda<0$. Moreover, due to the first part of $(3)$ in the Lemma, estimate (\[eq:fin1\]) is also seen to hold, with different positive constants $C$, for $\lambda<-\frac{1}{p}$, $p\geq1$ and $q>0$ satisfying either of the following $$\label{eq:laste0}
\begin{cases}
q\in(0,1/2),\,\,\, &p\in[1,2],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \lambda<-\frac{1}{p},
\\
q\in(0,1/2),\,\,\, &p>2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \frac{1}{2-p}<\lambda<-\frac{1}{p},
\\
q\in(1/2,1),\,\,\, &p\in[1,1/q],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \lambda<-\frac{1}{p},
\\
q\in(1/2,1),\,\,\, &p>\frac{1}{q},\,\,\,\,\,\,\,\,\,\,\,\,\, \frac{q}{1-pq}<\lambda<-\frac{1}{p},
\\
q>1,\, &p\geq1,\,\,\,\,\,\,\,\,\,\,\,\,\, \frac{q}{1-pq}<\lambda<-\frac{1}{p},
\end{cases}$$ as well as $$\label{eq:req2lambdaneg1}
\lambda\notin\left\{\frac{q}{1-q(p+n)},\frac{1}{1-p}\right\},\,\,\,\,\,\,\,\,\,\,\,q\neq\frac{1}{n}\,\,\,\,\forall\,\,\,\,\,n\in\mathbb{N}.$$ We remark that in the cases where (\[eq:fin1\]) diverges, it dominates the other terms in (\[eq:upper\]), regardless of whether these converge or diverge, and so no information on the behaviour of $\left\|u_x\right\|_p$ is obtained. Consequently, we will omit those instances. Finally, using Lemma \[lem:general\](2), one finds that $$\label{eq:fin2}
\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{1+\frac{1}{\lambda p}}}}\sim C\mathcal{J}(\underline\alpha_j,t)^{\frac{1}{q}-\frac{1}{\lambda p}-1}$$ for $q>1$, $p\geq1$ and $\lambda<\frac{q}{p(1-q)}$. Analogously, if (\[eq:fin2\]) converges, the lower bound (\[eq:lower\]) yields no information on the $L^p$ regularity of $u_x$. For the remaining of this section, we will assume that (\[eq:lemmaass1\]) holds whenever (\[eq:lastest\]) is used for $\lambda<-1$ and $q\in(0,1)$. Also, (\[eq:req2lambdaneg1\]) will be valid in those cases where estimate (\[eq:fin1\]) is considered for $\lambda$, $p$ and $q$ as in (\[eq:laste0\]). Suppose $\frac{q}{1-q}<\lambda<\frac{q}{p(1-q)}$ for $q>1$ and $p>1$. Then, using (\[eq:thirdint1\]), (\[eq:lastest\]), (\[eq:reallast\]) and (\[eq:fin2\]), in the lower bound (\[eq:lower\]), implies that $$\lim_{t\uparrow t_*}\left\|u_x(\cdot,t)\right\|_p=+\infty.$$ If instead, $\lambda<\frac{q}{1-q}$ for $q>1$ and $p>1$, then (\[eq:thirdint1\]), (\[eq:lastest1\]), (\[eq:reallast\]) and (\[eq:fin2\]) give $$\begin{split}
\left\|u_x(\cdot,t)\right\|_p&\geq\frac{1}{\left|\lambda\eta(t)\right|\bar{\mathcal{K}}_0(t)^{^{2\lambda+\frac{1}{p}}}}\left|\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{^{1+\frac{1}{\lambda p }}}}}-\frac{\bar{\mathcal{K}}_1(t)}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda p}}}}\right|
\\
&\sim C\left|C\mathcal{J}(\underline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda p}-1}-\mathcal{J}(\underline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda}-1}\right|
\\
&\sim C\mathcal{J}(\underline\alpha,t)^{\frac{1}{q}-\frac{1}{\lambda p}-1}\to+\infty
\end{split}$$ as $\eta\uparrow\eta_*$. For the upper bound (\[eq:upper\]), we simply mention that estimates (\[eq:thirdint1\]), (\[eq:lastest\]) and (\[eq:fin1\]) lead to several instances where $\left\|u_x\right\|_p$ remains finite for all $t\in[0,t_*]$. This can be shown, just as above, by using the appropriate estimates. For simplicity, we omit the details and summarize the results in Theorem \[thm:genlplambdaneg\] below.
\[thm:genlplambdaneg\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) for $u_0'(\alpha)$ satisfying (\[eq:expnew00\]), and let $t_*>0$ denote the finite $L^{\infty}$ blow-up time for $u_x$ as described in Theorem \[thm:lambdaneg\].
1. \[it:plambdaneg1\] Let $q\in(0,1/2)$. Then, $\lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty$ for either $\lambda<0$ and $p\in[1,2]$, or $\frac{1}{2-p}<\lambda<0$ and $p>2$.
2. \[it:plambdaneg2\] Let $q\in(1/2,1)$. Then, $\lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty$ for either $\lambda<0$ and $p\in[1,1/q]$, or $\frac{q}{1-pq}<\lambda<0$ and $p>1/q$.
3. \[it:plambdaneg3\] Let $q>1$. Then $\lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty$ for $\frac{q}{1-pq}<\lambda<0$ and $p\geq1$, whereas $\lim_{t\uparrow t_*}\left\|u_x\right\|_p=+\infty$ for $\lambda<\frac{q}{p(1-q)}$ and $p>1$.
When applicable, (\[eq:lemmaass1\]) and (\[eq:req2lambdaneg1\]) apply to (\[it:plambdaneg1\]) and (\[it:plambdaneg2\]) above.
### Further regularity results for smooth initial data {#subsubsec:arbitrarysmooth}
\[def:order\] Suppose a smooth function $f(x)$ satisfies $f(x_0)=0$ but $f$ is not identically zero. We say $f$ has a zero of order $k\in\mathbb{N}$ at $x=x_0$ if $$f(x_0)=f'(x_0)=...=f^{(k-1)}(x_0)=0,\,\,\,\,\,\,\,\,\,\,\,\,f^{(k)}(x_0)\neq0.$$
In [@Sarria1], we examined a class of smooth, mean-zero, periodic initial data characterized by $u_0''(\alpha)$ having zeroes of order $k=1$ at the finite number of locations $\overline\alpha_i$ for $\lambda>0$, or at $\underline\alpha_j$ if $\lambda<0$, that is, $u_0'''(\overline\alpha_i)<0$ or $u_0'''(\underline\alpha_j)>0$. Consequently, in each case, we were able to use an appropriate Taylor expansion up to quadratic order to account for the local behaviour of $u_0'$ near these points. This approach, in turn, led to the results summarized in Theorems \[thm:sarria1\] and \[thm:lpintro\] of §\[sec:intro\]. Assuming the order $k$ of these particular zeroes, $\overline\alpha_i$ or $\overline\alpha_j$, of $u_0''$ is the same regardless of location, and noticing that $k\geq1$ must be odd due to $u_0'$ being even in a small neighbourhood of these points, we may use definition \[def:order\] to generalize the results in [@Sarria1] to a larger class of smooth, mean-zero, periodic initial data characterized by $u_0''$ having zeroes of higher orders, $k=1,3,5,...$, at every $\overline\alpha_i$ if $\lambda>0$, or $\underline\alpha_j$ for $\lambda<0$. Since this corresponds to replacing $q$ in (\[eq:expnew0\]) or (\[eq:expnew00\]) by $k+1$, we obtain our results simply by substituting $q$ in Theorems \[thm:lambdapos\], \[thm:lplambdapos\], \[thm:lambdaneg\] and \[thm:genlplambdaneg\], by $1+k$ in those cases where $q\geq2$. The results are summarized in Corollary \[coro:gensmooth\] below.
\[coro:gensmooth\] Consider the initial boundary value problem (\[eq:nonhomo\])-(\[eq:pbc\]) for smooth, mean-zero, periodic initial data. Furthermore,
1. Suppose $u_0''(\alpha)$ has a zero of order $k\geq1$ at every $\overline\alpha_i$, $i=1,2,...,m$. Then
- \[it:globalgen\] For $0\leq\lambda\leq\frac{1+k}{2}$, solutions exist globally in time. More particularly, these vanish as $t\uparrow t_*=+\infty$ for $0<\lambda<\frac{1+k}{2}$ but converge to a nontrivial steady state if $\lambda=\frac{1+k}{2}$.
- \[it:blow1gen\] For $\frac{1+k}{2}<\lambda<+\infty$, there exists a finite $t_*>0$ such that both the maximum $M(t)$ and the minimum $m(t)$ diverge to $+\infty$ and respectively to $-\infty$ as $t\uparrow t_*$. Furthermore, $\lim_{t\uparrow t_*}u_x(\gamma(\alpha,t),t)=-\infty$ if $\alpha\notin\bigcup_{i,j}\{\overline\alpha_i\}\cup\{\underline\alpha_j\}$ and $\lim_{t\uparrow t_*}\left\|u_x\right\|_{p}=+\infty$ for all $p>1$.
2. Suppose $u_0''(\alpha)$ has a zero of order $k\geq1$ at each $\underline\alpha_j$, $j=1,2,...,n$. Then
- \[it:lambdaneg1gen\] For $-\frac{1+k}{k}<\lambda<0$, there exists a finite $t_*>0$ such that only the minimum diverges, $m(t)\to-\infty,$ as $t\uparrow t_*$, whereas, for $\frac{1+k}{1-p(1+k)}<\lambda<0$ and $p\geq1$, $\lim_{t\uparrow t_*}\left\|u_x\right\|_p<+\infty$.
- \[it:lambdaneg3gen\] For $\lambda<-\frac{1+k}{k}$, there is a finite $t_*>0$ such that both $M(t)$ and $m(t)$ diverge to $+\infty$ and respectively to $-\infty$ as $t\uparrow t_*$. Additionally, $\lim_{t\uparrow t_*}u_x(\gamma(\alpha,t),t)=+\infty$ for $\alpha\notin\bigcup_{i,j}\{\overline\alpha_i\}\cup\{\underline\alpha_j\}$ and $\lim_{t\uparrow t_*}\left\|u_x\right\|_p=+\infty$ if $\lambda<-\frac{1+k}{pk}$ and $p>1$.
\[rem:exception\] It turns out that, unless the initial data is smooth, the results established in this paper for periodic boundary conditions extend to a Dirichlet setting. For smooth initial data, if there are $\overline\alpha_i\in\{0,1\}$ when $\lambda>0$, or $\underline\alpha_j\in\{0,1\}$ for $\lambda<0$, then the results in the periodic setting will extend to Dirichlet boundary conditions as long as $u_0''$ vanishes at those end-points. This last condition prevents a Lipschitz-type behaviour of $u_0'$ at the boundary, which could otherwise lead to finite-time blow-up from smooth initial data under Dirichlet boundary conditions. Details on this will be presented in an upcoming paper. Also, notice that letting $q\to+\infty$ in either (\[eq:expnew0\]) or (\[eq:expnew00\]) implies that $u_0'\sim M_0$ near $\overline\alpha_i$, or $u_0'\sim m_0$ for $\alpha\sim\underline\alpha_j$, respectively. Then, letting $k\to+\infty$ in Corollary \[coro:gensmooth\](\[it:globalgen\]) implies that, for this particular class of locally constant $u_0'$, a solution that exists locally in time for any $\lambda\in\mathbb{R}$, will persist for all time.
Examples {#sec:examples}
========
Examples for Theorems \[thm:p=1\], \[thm:lambdapos\] and \[thm:lambdaneg\] are now presented. For simplicity, we consider initial data satisfying Dirichlet boundary conditions $$u(0,x)=u(1,t)=0,$$ and we note that (\[eq:mainsolu\]) is equivalent to the representation formula (see [@Sarria1]) $$\label{eq:finalsolu}
\begin{split}
u_x(\gamma(\alpha,t),t)=\frac{1}{\bar{\mathcal{K}}_0(t)^{^{2\lambda}}}\left(\frac{u_0'(\alpha)}{\mathcal{J}(\alpha,t)}-\frac{1}{\bar{\mathcal{K}}_0(t)}\int_0^1{\frac{u_0'(\alpha)d\alpha}{\mathcal{J}(\alpha,t)^{1+\frac{1}{\lambda}}}}\right).
\end{split}$$ For several choices of $\lambda\in\mathbb{R}$, the time-dependent integrals in (\[eq:finalsolu\]) are evaluated and pointwise plots are generated using <span style="font-variant:small-caps;">Mathematica</span>. Whenever possible, plots in the Eulerian variable $x,$ instead of the Lagrangian coordinate $\alpha,$ are provided. For practical reasons, details of the computations in most examples are omitted. Also, due to the difficulty in solving for the time variable $t$ through the IVP (\[eq:etaivp\]) for $\eta(t)$, most plots for $u_x(\gamma(\alpha,t),t)$ are against the variable $\eta$ rather than $t$.
Example 1 below applies to stagnation point-form (SPF) solutions to the incompressible 3D Euler equations ($\lambda=1/2$). We consider two types of data, one satisfying (\[eq:expnew0\]) for $q\in(0,1)$, and other having $q>1$. Recall from Table \[table:thecase\] that if $q\geq1$, global existence in time follows, while, for $q\in(1/2,1)$, finite-time blow-up occurs. Below, we see that a spontaneous singularity may also form if $q=1/3$.
**Example 1. Regularity of SPF solutions to 3D Euler for $q=1/3$ and $q=6/5$**
First, for $\lambda=1/2$ and $\alpha\in[0,1]$, let $$\label{eq:data3d}
\begin{split}
u_0(\alpha)=\alpha(1-\alpha^{\frac{1}{3}}).
\end{split}$$ Then $u_0'(\alpha)=1-\frac{4}{3}\alpha^{\frac{1}{3}}$ achieves its maximum $M_0=1$ at $\overline\alpha=0$. Also, $q=1/3$, $\eta_*=2$, and $u_0'(\alpha)\notin C^1(0,1)$, i.e. $\lim_{\alpha\downarrow0}u_0''(\alpha)=-\infty$; a jump discontinuity of infinite magnitude in $u_0''$. Evaluating the integrals in (\[eq:finalsolu\]), we obtain $$\label{eq:3d1}
\begin{split}
\bar{\mathcal{K}}_0(t)=-\frac{54(\eta(t)-6)\eta(t)-81(2-\eta(t))(6+\eta(t))\,\text{arctanh}\left(\frac{2\eta(t)}{\eta(t)-6}\right)}{4(6+\eta(t))\eta(t)^3}
\end{split}$$ and $$\label{eq:3d2}
\begin{split}
&\int_0^1{\frac{u_0'(\alpha)\,d\alpha}{\mathcal{J}(\alpha,t)^{3}}}=-\frac{27\left(9(2-\eta(t))(6+\eta(t))^2\,\log\left(\frac{24}{\eta(t)+6}-3\right)\right)}{8(6+\eta(t))^2\eta(t)^4}
\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{27\left(8\eta(t)(54-(\eta(t)-9)\eta(t))+6\eta(t)(6+\eta(t))^2\,\text{arctanh}\left(\frac{2\eta(t)}{\eta(t)-6}\right)\right)}{8(6+\eta(t))^2\eta(t)^4}
\end{split}$$ for $0\leq\eta<2$. Furthermore, in the limit as $\eta\uparrow\eta_*=2$, $\bar{\mathcal{K}}_0(t_*)=27/16$ whereas $\int_0^1{\frac{u_0'(\alpha)\,d\alpha}{\mathcal{J}(\alpha,t)^{3}}}\to+\infty.$ Also, (\[eq:etaivp\]) and (\[eq:3d1\]) yield $$t(\eta)=-\frac{9\left(2\eta(6-5\eta)+9(\eta-2)^2\text{arctanh}\left(\frac{2\eta}{\eta-6}\right)\right)}{16\eta^2},$$ so that $t_*=\lim_{\eta\uparrow2}t(\eta)=9/4.$ Using (\[eq:3d1\]) and (\[eq:3d2\]) on (\[eq:finalsolu\]), we find that $u_x(\gamma(\alpha,t),t)$ undergoes a two-sided, everywhere blow-up as $t\uparrow 9/4$.
Next, replace $q=1/3$ in (\[eq:data3d\]) by $q=6/5$. Then, $u_0'(\alpha)=1-\frac{11}{5}\alpha^{\frac{6}{5}}$ so that $u_0''$ is now defined as $\alpha\downarrow0$. Also, for this data, both integrals now diverge to $+\infty$ as $\eta\uparrow2$. Particularly, this causes a balancing effect amongst the terms in (\[eq:finalsolu\]) that was previously absent when $q=1/3$. Ultimately, we find that as $t\to t_*=+\infty$, $u_x(\gamma(\alpha,t),t)\to0$ for every $\alpha\in[0,1]$. See Figure \[fig:ex7\] below.
![Example 1 for $\lambda=1/2$ and $q\in\{1/3,6/5\}$. Figure $A$ depicts two-sided, everywhere blow-up of $u_x(\gamma(\alpha,t),t)$ for $q=1/3$ as $\eta\uparrow 2$ ($t\uparrow9/4$), whereas, for $q=6/5$, Figure $B$ represents its vanishing as $\eta\uparrow 2$ ($t\to+\infty$).[]{data-label="fig:ex7"}](ex712p13-mod2.pdf "fig:") ![Example 1 for $\lambda=1/2$ and $q\in\{1/3,6/5\}$. Figure $A$ depicts two-sided, everywhere blow-up of $u_x(\gamma(\alpha,t),t)$ for $q=1/3$ as $\eta\uparrow 2$ ($t\uparrow9/4$), whereas, for $q=6/5$, Figure $B$ represents its vanishing as $\eta\uparrow 2$ ($t\to+\infty$).[]{data-label="fig:ex7"}](ex712p65-mod2.pdf "fig:")\
In [@Sarria1] (see Theorem \[thm:sarria1\] in §\[sec:intro\]), we showed that for a class of smooth, periodic initial data $(q=2)$, finite-time blow-up occurs for all $\lambda>1$. Example 2 below is an instance of Theorem \[thm:lambdapos\](\[it:global\]). For $\lambda\in\{2,5/4\}$, we consider initial data satisfying (\[eq:expnew0\]) for $q\in\{5,5/2\}$, respectively, and find that solutions persist globally in time. Also, the example illustrates the two possible global behaviours: convergence of solutions, as $t\to+\infty$, to nontrivial or trivial steady states.
**Example 2. Global existence for $\lambda=2$, $q=5$ and $\lambda=5/4$, $q=5/2$**
First, let $\lambda=2$ and $$\label{eq:dat00}
\begin{split}
u_0(\alpha)=\alpha(1-\alpha^5).
\end{split}$$ Then $u_0^\prime(\alpha)=1-6\alpha^5$ achieves its greatest value $M_0=1$ at $\overline\alpha=0$ and $\eta_*=1/2$. Since $\lambda=2\in[0,5/2)=[0,q/2)$, Theorem \[thm:lambdapos\](\[it:global\]) implies global existence in time. Particularly, $u_x(\gamma(\alpha,t),t)\to0$ as $t\to+\infty.$ See Figure \[fig:ex34\]$(A)$. Now, suppose $\lambda=5/4$ and replace $q=5$ in (\[eq:dat00\]) by $q=5/2$. Then, $u_0'(\alpha)=1-\frac{7}{2}\alpha^{5/2}$ attains $M_0=1$ at $\overline\alpha=0$ and $\eta_*=4/5$. Because $\lambda=5/4=q/2$, Theorem \[thm:lambdapos\](\[it:global\]) implies that $u_x$ converges to a nontrivial steady-state as $t\to+\infty$. See Figure \[fig:ex34\]$(B)$.
![For example 2, Figure $A$ represents the vanishing of $u_x(\gamma(\alpha,t),t)$ as $\eta\uparrow1/2$ ($t\to+\infty$) for $\lambda=2$ and $q=5$, whereas, Figure $B$ illustrates its convergence to a nontrivial steady state as $\eta\uparrow4/5$ ($t\to+\infty$) if $q=5/2$ and $\lambda=5/4=q/2$.[]{data-label="fig:ex34"}](uxgloballambda2p5-mod.pdf "fig:") ![For example 2, Figure $A$ represents the vanishing of $u_x(\gamma(\alpha,t),t)$ as $\eta\uparrow1/2$ ($t\to+\infty$) for $\lambda=2$ and $q=5$, whereas, Figure $B$ illustrates its convergence to a nontrivial steady state as $\eta\uparrow4/5$ ($t\to+\infty$) if $q=5/2$ and $\lambda=5/4=q/2$.[]{data-label="fig:ex34"}](steady-mod.pdf "fig:")
**Example 3. Two-sided, everywhere blow-up for $\lambda=\frac{11}{2}$ and $q=6$.**
Suppose $\lambda=11/2$ and $u_0(\alpha)=\frac{\alpha}{11}(1-\alpha^6)$. Then, $u_0'(\alpha)=\frac{1}{11}(1-7\alpha^6)$ attains its greatest value $M_0=1/11$ at $\overline\alpha=0.$ Also, $\eta_*=2$ and $\lambda=11/2\in(q/2,q)$. According to Theorem \[thm:lambdapos\](\[it:blow1\]), two-sided, everywhere finite-time blow-up occurs. The estimated blow-up time is $t_*\sim 22.5.$ See Figure \[fig:ex340\]$(A)$.
**Example 4. One-sided, discrete blow-up for $\lambda=-5/2$ and $q=3/2$**
Let $\lambda=-5/2$ and $u_0(\alpha)=\alpha(\alpha^{\frac{3}{2}}-1)$. Then $u_0'$ attains its minimum $m_0=-1$ at $\underline\alpha=0$ and $\eta_*=2/5.$ Since $\frac{q}{1-q}<\lambda<-1,$ Theorem \[thm:lambdaneg\](2) implies one-sided, discrete finite-time blow-up and $t_*\sim0.46$. See Figure \[fig:ex340\]$(B)$. We remark that in [@Sarria1], the same value for $\lambda$ with smooth, periodic initial data, and $q=2$ led to two-sided, everywhere blow-up instead.
![Figure $A$ for example 3 depicts two-sided, everywhere blow-up of $u_x(\gamma(\alpha,t),t)$ as $\eta\uparrow2$ $(t\uparrow22.5)$ for $\lambda=11/2$ and $q=6$, while, Figure $B$ for example 4 illustrates one-sided, discrete blow-up, $m(t)=u_x(0,t)\to-\infty$, as $\eta\uparrow2/5$ ($t\uparrow t_*\sim0.46$) for $\lambda=-5/2$ and $q=3/2$.[]{data-label="fig:ex340"}](ex5112p6-mod.pdf "fig:") ![Figure $A$ for example 3 depicts two-sided, everywhere blow-up of $u_x(\gamma(\alpha,t),t)$ as $\eta\uparrow2$ $(t\uparrow22.5)$ for $\lambda=11/2$ and $q=6$, while, Figure $B$ for example 4 illustrates one-sided, discrete blow-up, $m(t)=u_x(0,t)\to-\infty$, as $\eta\uparrow2/5$ ($t\uparrow t_*\sim0.46$) for $\lambda=-5/2$ and $q=3/2$.[]{data-label="fig:ex340"}](ex5-52p32-mod.pdf "fig:")
In these last two examples, we consider smooth data with either mixed local behaviour near two distinct locations $\underline\alpha_j$ for $\lambda=-1/3$, or $M_0$ occurring at both endpoints for $\lambda=1$.
**Example 5. One-sided, discrete blow-up for $\lambda=-1/3$ and $q=1,2$.**
For $\lambda=-1/3$, let $$u_0(\alpha)=\alpha(1-\alpha)(\alpha-\frac{3}{4})\left(\alpha-\frac{1+4\sqrt{22}}{36}\right).$$ Then $m_0\sim-0.113$ occurs at both $\underline\alpha_1=1$ and $\underline\alpha_2=\frac{4+\sqrt{22}}{24}\sim0.36.$ Now, near $\underline\alpha_2$, $u_0'$ behaves quadratically ($q=2$), whereas, for $1-\alpha>0$ small, it behaves linearly ($q=1$). The quadratic behaviour is due to $u_0''$ having zero of order one at $\underline\alpha_2\sim0.36$, thus, Corollary \[coro:gensmooth\] implies a discrete, one-sided blow-up. Similarly in the case of linear behaviour according to Theorem \[thm:p=1\]. After evaluating the integrals, we find that $m(t)\to-\infty$ as $t\uparrow t_*\sim17.93$. Due to the Dirichlet boundary conditions, we have that $\gamma(0,t)\equiv0$ and $\gamma(1,t)\equiv1$ for a s long a s $u$ is defined. Then, one blow-up location is given by the boundary $\underline{x}_1=1$, while the interior blow-up location, $\underline x_2$, is obtained by integrating (\[eq:sum\]). This yields the characteristics: $$\gamma(\alpha,t)=\int_0^{\alpha}{\frac{dy}{\mathcal{J}(y,t)^{\frac{1}{\lambda}}}}\left(\int_0^{1}{\frac{d\alpha}{\mathcal{J}(\alpha,t)^{\frac{1}{\lambda}}}}\right)^{-1}.$$ Setting $\alpha=\alpha_2$ and letting $\eta\uparrow\eta_*=\frac{3}{\left|m_0\right|}$, we find that $\underline{x}_2\sim0.885$. See Figure \[fig:weird\]$(A)$.
**Example 6. Two-sided, everywhere blow-up of SPF solutions to 2D Euler ($\lambda=1$) for $q=1$.**
For $\lambda=1$, let $u_0(\alpha)=\alpha(\alpha-1)(\alpha-1/2).$ Then, $M_0=1/2$ occurs at both endpoints $\overline\alpha_i=\{0,1\}$. Also $\eta_*=2$ and since $$u_0'(\alpha)=M_0-3\alpha+3\alpha^2=M_0-3\left|\alpha-1\right|+3(\alpha-1)^2,$$ the local behaviour of $u_0'$ near both endpoints is linear ($q=1$). The integrals in (\[eq:finalsolu\]) evaluate to $$\bar{\mathcal{K}}_0(t)=\frac{2\,\text{arctanh}(y(t))}{\sqrt{3\eta(t)(4+\eta(t))}},\,\,\,\,\,\,\,\,\,\,\int_0^1{\frac{u_0'(\alpha)\,d\alpha}{\mathcal{J}(\alpha,t)^2}}=\frac{\,\,d\bar{\mathcal{K}}_0(t)}{d\eta}$$ for $0\leq\eta<2$ and $y(t)=\frac{\sqrt{3\eta(t)(4+\eta(t))}}{2(1+\eta(t))}$. Using the above on (\[eq:finalsolu\]), we find that $M(t)=u_x(0,t)=u_x(1,t)\to+\infty$ as $\eta\uparrow2$, while $u_x(x,t)\to-\infty$ for all $x\in(0,1)$. The blow-up time is estimated from (\[eq:etaivp\]) and $\bar{\mathcal{K}}_0(t)$ above as $t_*\sim2.8$. See Figure \[fig:weird\]$(B)$.
![Figure $A$ for example 5 with $\lambda=-1/3$ and $q=1,2$, depicts one-sided, discrete blow-up, $m(t)=\to-\infty$, as $t\uparrow17.93$. The blow-up locations are $\underline{x}_1=1$ and $\underline{x}_2\sim0.885$. Then, Figure $B$ for example 6 with $\lambda=1$ and $q=1$, represents two-sided, everywhere blow-up of $u_x(x,t)$, as $t\uparrow2.8$.[]{data-label="fig:weird"}](uxblowalpha-1-3intbound-mod.pdf "fig:") ![Figure $A$ for example 5 with $\lambda=-1/3$ and $q=1,2$, depicts one-sided, discrete blow-up, $m(t)=\to-\infty$, as $t\uparrow17.93$. The blow-up locations are $\underline{x}_1=1$ and $\underline{x}_2\sim0.885$. Then, Figure $B$ for example 6 with $\lambda=1$ and $q=1$, represents two-sided, everywhere blow-up of $u_x(x,t)$, as $t\uparrow2.8$.[]{data-label="fig:weird"}](uxblowalpha2-mod.pdf "fig:")
[99]{}
E.W. Barnes, “*A New Development of the Theory of Hypergeometric Functions*”, Proc. London Math. Soc. (2) **6** (1908), 141-177. A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal 37(3) (2005), 996-1026. F. Calogero, A solvable nonlinear wave equation, Stud. Appl. Math., 70, 3, (1984), 189-199. S. Childress, G.R. Ierley, E.A. Spiegel and W.R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes equations having stagnation-point form, J. Fluid Mech. 203 (1989), 1-22. C. H. Cho and M. Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation, J. Diff. Eqns. 249, (2010), 392-413. C.H. Cho and M. Wunsch, Global weak solutions to the generalized Proudman-Johnson equation, Commun. Pure Appl. Ana. Vol 11, No. 4, (2012), 1387-1396. A. Constantin and M. Wunsch, On the inviscid Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci., 85, 7, (2009), 81-83. C.M. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbol. Differ. Eq., Vol. 8, No. 1 (2011), 159-168. A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, “*Higher transcendental functions, Vol. I*”, McGraw-Hill, (1981), 56-119. J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, pre-print, arXiv:1009.1029, 2010. T.W. Gamelin, “*Complex Analysis*”, Undergraduate Texts in Mathematics, Springer (2000), 361-365. G. Gasper and M. Rahman, “*Basic Hypergeometric Series*”, Encyclopedia of Mathematics and Its Applications, Vol. 96, Second ed., Cambridge University Press (2004), 113-119. J.K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51(6) (1991), 1498-1521. J.K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation: I. Global existence of weak solutions, Arch. Rat. Mech. Anal. 129 (1995) 305-353. B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003) no. 1, 116-144. J. Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation, Discrete Contin. Dyn. Syst. 18 (4) (2007), 643-656. H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Taiwanese J. Math. 4 (2000), 65-103. H. Okamoto, Well-posedness of the generalized Proudman-Johnson equation without viscosity, J. Math. Fluid Mech. 11 (2009), 46-59. I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech. 12 (1962), 161-168. A. Sarria and R. Saxton, Blow-up of solutions to the generalized inviscid Proudman-Johnson equation, J. Math. Fluid Mech., online first, (2012), DOI 10.1007/s00021-012-0126-x. R. Saxton and F. Tiglay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal. 4 (2008), 1499-1515. F. Tiglay, The periodic Cauchy problem of the modified Hunter-Saxton equation, J. Evol. Eq. 5 (4) (2005), 509-527. H. Weyl, On the differential equations of the simplest boundary-layer problems, Ann. Math. 43 (1942), 381-407. M. Wunsch, The generalized Proudman-Johnson equation revisited, J. Math. Fluid Mech. 13 (1) (2009), 147-154. Z. Yin, On the Structure of Solutions to the Periodic Hunter-Saxton Equation, SIAM J. Math. Anal., Vol. 36, No. 1, (2004), 272-283.
|
---
author:
- 'Rishi Khatri,'
- 'Rashid A. Sunyaev'
bibliography:
- 'cmbBB.bib'
title: 'Creation of the CMB spectrum: precise analytic solutions for the blackbody photosphere'
---
Introduction
============
There are several important events in the history of the Universe which provide the foundations for the standard cosmological model. The first event is the *big bang*, which created the present expanding Universe filled with matter, radiation and dark energy. One of the goals of cosmology is to reconstruct these initial conditions and learn about the high energy physics at early times. We should emphasize here that the blackbody spectrum of cosmic microwave background (CMB) is *not* an initial condition. The blackbody spectrum is created and maintained dynamically throughout the early history of the Universe ($z\gtrsim 2\times 10^6$) by standard physics processes and how this happens is the main topic of the present paper. The CMB spectrum thus provides information about the physics of the Universe after the big bang.
The other important events in standard cosmology are: the formation of light elements in primordial nucleosynthesis [@wfh67], recombination of electrons and ions to neutral atoms [@zks68; @peebles68], and reionization of the Universe [@gp1965] by the radiation emitted by first stars and galaxies. We should also mention electron-positron annihilation at $10^{10} \gtrsim z\gtrsim 10^8$ which more than doubles the entropy of CMB and raises its temperature by $\sim 40\%$. The important events in the history of the Universe are sketched in Fig. \[sketch\]. Big bang nucleosynthesis (BBN) theory together with measurement of light elements abundances constrains the photon to baryon number density at $z\sim 10^8$ [see @iocco2009 for a review] and is the earliest direct evidence and measurement of electromagnetic radiation in the early Universe. The fact that the photon energy density inferred from BBN and Cosmic microwave background (CMB) at $z\sim 10^3$ [@wmap7] are close to each other (consistent within $\sim 2-\sigma$) also implies that we do not have arbitrary freedom in adding energy to CMB between these two epochs. [There is, however, no direct way to constrain the energy density in photons before BBN and the epoch of electron-positron annihilation.]{}
{width="15cm"}
[The CMB (anisotropy and polarization)]{} is at present the most precise cosmological probe. The CMB spectrum was created at $z\gtrsim 2\times 10^{6}$, and this critical redshift defines the *blackbody surface* for our Universe. Spatial fluctuations in the temperature of the CMB were imprinted much later, when the electron and protons recombined to form hydrogen atoms at $z\approx 1100$. This second boundary defines the well known *last scattering surface*, the structure of which is encoded in the photon visibility function [first studied by @sz1970c]. The anisotropies and physics at the last scattering surface have been very well studied and accurate analytic [@hu1995] and numerical solutions [@mabert95; @cmbfast] have been available for some time, motivated by the precise experiments such as WMAP [@wmap7] and Planck [@planck].
[The blackbody spectrum, once created at high redshifts (for example before the time of electron-positron annihilation), is preserved by the adiabatic expansion of the Universe at all subsequent times. However, if there is energy release at lower redshifts , for example by particle decay and annihilation or Silk damping, it will distort the CMB spectrum away from the Planck form. In this case, Zeldovich and Sunyaev [@zs1969] first demonstrated that bremsstrahlung alone cannot recreate blackbody spectrum until very high redshifts, almost up to the time of electron-positron annihilation. The problem of evolution of the CMB spectrum through the blackbody surface, in the presence of heating, was solved analytically by [@sz1970] including the processes of comptonization and emission and absorption of photons with special emphasis on bremsstrahlung. Comptonization is the process of redistribution of photons over frequency, resulting from the Doppler and recoil effects of Compton scattering of photons on thermal electrons [@k1956]. Since the double Compton cross section [@lightman; @thorne81] has a dependence on frequency similar to bremsstrahlung, the solution of Sunyaev and Zeldovich [@sz1970] also allowed inclusion of double Compton emission and absorption, [which is dominant over bremsstrahlung in a low baryon density Universe such as ours and was first considered by [@dd1982].]{} Double Compton emission or absorption is just the first radiative correction to the process of Compton scattering just as bremsstrahlung is the first radiative correction to the scattering of electrons on nuclei [see discussion in @pss1983].]{} The approximations used in these analytic solutions result in accuracy of $\sim 5-10\%$ at redshifts ($z\lesssim 3\times 10^6$) and worse than $10\%$ at higher redshifts. Much better accuracy is, of course, achievable in numerical solutions [@is1975; @bdd91; @hs1993; @pb2009; @cs2011].
The type of spectrum we get for the CMB is determined by the Compton $y$ parameter, defined in Eq. . Bose-Einstein spectrum can be created only at high redshifts $z\gtrsim 10^5, y>1$ [@sz1970c], when comptonization of CMB is very efficient. Any addition of energy and photons to CMB at $z\lesssim 10^5, y<<1$ inevitably distorts the CMB spectrum with a $y$-type distortion [@zs1969]. The $y$-type distortions, in contrast to $\mu$-type distortions, can be created at low redshifts up to $z=0$. Reionization at $z\sim 10$ in particular is expected to create $y$-type distortions of magnitude $y_{\rm e}\sim \tau_{ri}{{{k_{\rm B}}}}{{{T_{\rm e}}}}/({{{m_{\rm e}}}}c^2)\sim 10^{-7}$ where $\tau_{ri}\approx 0.1$ is the optical depth due to reionization, ${{{T_{\rm e}}}}\approx 10^4$ is the average temperature during reionization, ${{{k_{\rm B}}}}$ is the Boltzmann’s constant, $c$ is the speed of light and ${{{m_{\rm e}}}}$ is the mass of electron. Due to uncertainties in reionization physics, it will be extremely difficult to separate these low redshift distortions from the $y$-type distortions created before recombination. COBE FIRAS experiment [@cobe] measured the CMB spectrum with high precision and placed a constraint on the chemical potential of Bose-Einstein spectrum for CMB of $\mu\lesssim 9\times 10^{-5}$, thus confirming that the CMB has a Planck spectrum at very high accuracy.
In standard model of cosmology we can get constraints on the energy density of radiation from two distinct and very precise observables. The first is the deuterium abundance, which gives the baryon number to photon number ratio $\eta=(5.7\pm 0.3)\times 10^{-10}$ during primordial nucleosynthesis at $4\times 10^8 \gtrsim z\gtrsim 4\times 10^7$ [@fields2001; @iocco2009]. The second is the measurement of CMB anisotropies, which constrains the baryon to photon ratio $\eta=(6.18\pm 0.15)\times 10^{-10}$ [@wmap7] during recombination at $z\approx 1100$. The fact that these two values of baryon to photon ratio are almost identical with small error bars means that we do not have arbitrary freedom in adding energy to CMB, for example, with the introduction of new physics. [In fact any addition of energy/entropy to CMB between primordial nucleosynthesis and recombination cannot be more than a small percentage ($\lesssim 10\%$) of the photon energy density during BBN.]{}
In standard cosmology, the chemical potential of CMB is expected to have a magnitude of $\mu\sim 10^{-9}-10^{-8}$ resulting from the heating of CMB by Silk damping. A similar magnitude but opposite sign distortion, $\mu\approx -2.7\times 10^9$, is expected from cooling of photons due to energy losses to baryons and electrons, which have a different adiabatic index (5/3) than radiation (4/3) and as a result cool faster than the radiation as the Universe expands [@cs2011; @ksc2011]. A detection of a chemical potential of magnitude greater than $10^{-8}$ will therefore mean existence of non-standard physics [(or a small scale power spectrum that is bigger than what is expected from extrapolating the large scale power spectrum as measured by CMB and large scale structure)]{} at these redshifts. Proposed experiment Pixie [@pixie] has exactly this level of sensitivity. Constraining high energy physics using $\mu$-type distortions requires precise calculation of evolution of the CMB spectrum through the blackbody surface at $z\approx 2\times
10^6$. Analytic solutions provide valuable physical insight in addition to being much easier to compute. Motivated by these factors we try to improve the method of [@sz1970] to achieve better than $1\%$ precision in analytic solutions. These solutions should prove useful in predicting the signal from models of high energy physics which can provide energy injection to CMB at these high redshifts.
In the last part of the paper we apply our analytic solution to examples from the standard model of cosmology including electron-positron annihilation and dark matter annihilation. These simple examples in particular demonstrate the efficiency of processes responsible for maintaining thermal equilibrium in the early Universe. They also demonstrate the difficulty of creating spectral distortion in CMB at high redshifts. There remains thus only a narrow window at redshifts $10^5<z<2\times 10^6$ when a Bose-Einstein spectrum can be created.
We use WMAP [@wmap7] best fit parameters for ${\rm \Lambda CDM}$ cosmology for numerical calculations.
Thermalization of CMB
=====================
Compton scattering is responsible for creating a Bose-Einstein spectrum of photons if the rate of comptonization [(i.e. the redistribution of photons over the entire spectrum by Compton scattering)]{} is greater than the expansion (Hubble) rate of the Universe [@zs1969]. This condition is satisfied at redshifts $z\gtrsim 10^5$. In the non-relativistic regime comptonization is described by Kompaneets equation [@k1956]. Compton scattering, however, conserves photon number and therefore the spectrum obtained as a result of comptonization will in general have a non-zero chemical potential. At redshifts $z\gtrsim 2\times 10^6$ double Compton scattering, and to a lesser extent bremsstrahlung, can emit/absorb photons at low frequencies very efficiently because [the optical depth and the absorption coefficient]{} of these two processes increase with decreasing frequency as $\propto \nu^{-2}$. Compton scattering is then able to redistribute these photons over the entire spectrum. The net result is that the chemical potential is exponentially suppressed and a Planck spectrum is established.
The equilibrium electron temperature in a radiation field with occupation number $n(x)$, where $x\equiv h\nu/kT_{\gamma}$ and $T_{\gamma}$ is the temperature of reference blackbody, is given by [@zl1970; @ls1971] $$\begin{aligned}
\frac{{{{T_{\rm e}}}}}{T_{\gamma}}=\frac{\int(n+n^2)x^4{{{\rm d}}}x}{4\int n x^3 {{{\rm d}}}x}\label{te}\end{aligned}$$ The rate ($8 {{{\sigma_{\rm T}}}}E_{\gamma}/(3 {{{m_{\rm e}}}}c)\times {{{n_{\rm e}}}}/({{{n_{\rm e}}}}+n_i)$) at which electron/baryon plasma achieves equilibrium temperature ${{{T_{\rm e}}}}$ given by Eq. is shown in Fig. \[rates\] in the topmost curve. Rates of bremsstrahlung absorption ${{{{K_{\rm br}}}}}(e^{{{{{x_{\rm e}}}}}}-1)/{{{{x_{\rm e}}}}}^3$, double Compton absorption ${{{{K_{\rm dC}}}}}(e^{{{{{x_{\rm e}}}}}}-1)/{{{{x_{\rm e}}}}}^3$ and Compton scattering ${{{{K_{\rm C}}}}}$ are also compared with the Hubble rate in Fig. \[rates\]. ${{{n_{\rm e}}}}$ is the electron number density, ${{{\sigma_{\rm T}}}}$ is the Thomson cross section, ${{{T_{\rm e}}}}$ is the electron temperature, $n_i$ is the number density of ions, ${{{{x_{\rm e}}}}}=h\nu/{{{k_{\rm B}}}}{{{T_{\rm e}}}}$ is the dimensionless frequency corresponding to frequency $\nu$, $h$ is the Planck’s constant, $E_{\gamma}$ is the energy density of photons. [${{{T_{\rm e}}}}=T_{\gamma}={{{T_{\rm CMB}}}}(1+z)$, ${{{T_{\rm CMB}}}}=2.725{\rm K}$ and blackbody spectrum is assumed for this figure.]{} The rate coefficients ${{{{K_{\rm C}}}}},{{{{K_{\rm dC}}}}},{{{{K_{\rm br}}}}}$ are given by: $$\begin{aligned}
{{{{K_{\rm C}}}}}&={{{n_{\rm e}}}}{{{\sigma_{\rm T}}}}c\frac{{{{k_{\rm B}}}}{{{T_{\rm e}}}}}{{{{m_{\rm e}}}}c^2}\nonumber\\
&\equiv {{{{a_{\rm C}}}}}(1+z)^4=2.045\times 10^{-30}(1+z)^4\left(\frac{{{{{\Omega_{\rm b}}}}}h_0^2}{0.0226}\right)\left(\frac{1-Y_{He}/2}{0.88}\right){\rm s}^{-1}\\
{{{{K_{\rm dC}}}}}&={{{n_{\rm e}}}}{{{\sigma_{\rm T}}}}c \frac{4{{{{\alpha_{\rm fs}}}}}}{3\pi} \left(\frac{{{{k_{\rm B}}}}{{{T_{\rm e}}}}}{{{{m_{\rm e}}}}c^2}\right)^2 {{{{g_{\rm dC}}}}}({{{{x_{\rm e}}}}})I_{dC}\nonumber\\
&\equiv {{{{a_{\rm dC}}}}}(1+z)^5=7.561\times 10^{-41}(1+z)^5{{{{g_{\rm dC}}}}}({{{{x_{\rm e}}}}})
\left(\frac{{{{{\Omega_{\rm b}}}}}h_0^2}{0.0226}\right)\left(\frac{1-Y_{He}/2}{0.88}\right){\rm s}^{-1}\\
{{{{K_{\rm br}}}}}&={{{n_{\rm e}}}}{{{\sigma_{\rm T}}}}c\frac{{{{{\alpha_{\rm fs}}}}}{{{n_{\rm B}}}}}{(24\pi^3)^{1/2}}\left(\frac{{{{k_{\rm B}}}}{{{T_{\rm e}}}}}{{{{m_{\rm e}}}}c^2}\right)^{-7/2}\left(\frac{h}{{{{m_{\rm e}}}}c}\right)^3{{{{g_{\rm br}}}}}({{{{x_{\rm e}}}}},{{{T_{\rm e}}}})\nonumber\\
&\equiv {{{{a_{\rm br}}}}}(1+z)^{5/2}=2.074\times 10^{-27}(1+z)^{5/2}{{{{g_{\rm br}}}}}({{{{x_{\rm e}}}}},{{{T_{\rm e}}}})\left(\frac{{{{{\Omega_{\rm b}}}}}h_0^2}{0.0226}\right)^2\left(\frac{1-Y_{He}/2}{0.88}\right){\rm s^{-1}}.\end{aligned}$$ ${{{{\alpha_{\rm fs}}}}}$ is the fine structure constant and ${{{n_{\rm B}}}}$ is the baryon number density. $I_{dC}=\int {{{\rm d}}}{{{{{x_{\rm e}}}}}}~ {{{{x_{\rm e}}}}}^4n(1+n)\approx 25.976$ for a blackbody spectrum at temperature ${{{T_{\rm e}}}}$, ${{{{\Omega_{\rm b}}}}}$ is the baryon density parameter, $h_0$ is the Hubble parameter and ${{{{Y_{\rm He}}}}}$ is the primordial helium mass fraction. ${{{{g_{\rm br}}}}}=\sum_i Z_i^2n_i{{{{g_{\rm ff}}}}}(Z_i,{{{{x_{\rm e}}}}},{{{T_{\rm e}}}})/{{{n_{\rm B}}}}$ is the average gaunt factor for bremsstrahlung, $n_i$ is the number density of ion species $i$ and $Z_i$ the charge of ion. The sum is over all ionic species, which for the primordial plasma at high redshifts consists of protons and helium nuclei. Accurate fitting formulas for ${{{{g_{\rm ff}}}}}$ have been provided by [@itoh2000]. ${{{{g_{\rm dC}}}}}$ is the gaunt factor for double Compton scattering and accurate fitting formula for it has been calculated recently by [@cs2011]. We use these fitting formulae for numerical solution. For reference at ${{{{x_{\rm e}}}}}=0.01, z=2\times 10^6$, ${{{{g_{\rm dC}}}}}=1.005$ and ${{{{g_{\rm br}}}}}=2.99$ and we use these values for our analytic solutions. These gaunt factors are slowly varying functions of time and frequency and can be assumed to be constant in the redshift range of interest for analytic calculations. We justify this assumption below.
The division into $y$-type and Bose-Einstein regions depends on the Compton $y$ parameter, $$\begin{aligned}
y(z)=\int_{0}^{z}{{{\rm d}}}z\frac{{{{k_{\rm B}}}}{{{\sigma_{\rm T}}}}}{{{{m_{\rm e}}}}c}\frac{{{{n_{\rm e}}}}T_{\gamma}}{H(1+z)}{.}\label{yz}\end{aligned}$$ During radiation domination ($z\gg z_{\rm eq}=3.2\times 10^4$) the integral can be carried out analytically, giving $y(z)\approx 4.9\times 10^{-11}(1+z)^2$, where $z_{\rm eq}$ is the redshift when matter energy density equals radiation energy density and we have assumed $3.046$ effective species of massless neutrinos [@mm2005]. We have, for WMAP $\Lambda CDM$ cosmological parameters [@wmap7], $y(1.5\times 10^4)\approx 0.01$, $y(4.7\times 10^4)\approx 0.1$ and $y(1.5\times 10^5)\approx 1$. For $y\lesssim 0.01<<1$ we have a $y$-type distortion and for $y\gtrsim 1$ a Bose-Einstein spectrum. The spectrum attained for $0.01\lesssim y\lesssim 1$ is in between a pure $y$-type and Bose-Einstein. We choose $z=5\times 10^4$ as an approximate division between the two types of distortions for our estimates. This division is accurate if the energy injection between $1.5\times
10^4\lesssim z\lesssim 1.5\times 10^5$ is small compared to the total energy injection at $z\lesssim 2\times 10^6$. If most of the energy injection happens between $1.5\times
10^4\lesssim z\lesssim 1.5\times 10^5$, then numerical calculations must be performed to get accurate final spectrum. We note that for small distortions, we can calculate distortions arising from different physical processes separately and add them linearly.
![\[rates\]Rates of bremsstrahlung, comptonization and double Compton scattering. Bremsstrahlung and double Compton absorption have a dependence on frequency $\sim x^{-2}$. Rates above are for ${{{{x_{\rm e}}}}}=0.01$ which is the typical value of critical frequency at which double Compton/ bremsstrahlung absorption becomes equal to comptonization rate (Eq. and Fig. \[xrates\]). Also shown is the rate at which electron/baryon plasma achieves equilibrium temperature ${{{T_{\rm e}}}}$ given by Eq. . ${{{T_{\rm e}}}}=T_{\gamma}$ and blackbody spectrum is assumed for this figure. Compton scattering is able to create Bose-Einstein spectrum at $z\gtrsim 10^5$ while for redshifts $z\lesssim 15000$ any energy injection gives rise to a $y$-type distortion [@zs1969]. The distortion has a shape in-between $y$-type and $\mu$-type for redshifts $1.5\times 10^4\lesssim
z\lesssim 10^5$ and we choose $z=5\times 10^4$ to divide our estimates between $\mu$-type and $y$-type.](rates.eps){width="14cm"}
Kinetic equation
----------------
We will follow [@sz1970] in solving the kinetic equation for photon distribution using stationarity assumption and then do an iteration to relax this assumption to arrive at a more accurate solution. The kinetic equation for the evolution of photon occupation number $n({{{{x_{\rm e}}}}})$ in the presence of Compton scattering, double Compton scattering and bremsstrahlung is given by [see also @bdd91; @hs1993] $$\begin{aligned}
\frac{\partial n({{{{x_{\rm e}}}}},t)}{\partial t}&={{{{K_{\rm C}}}}}\frac{1}{{{{{x_{\rm e}}}}}^2}\frac{\partial
}{\partial {{{{x_{\rm e}}}}}}{{{{x_{\rm e}}}}}^4\left[\frac{\partial n}{\partial
{{{{x_{\rm e}}}}}}+n+n^2\right]+\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)\frac{e^{-{{{{x_{\rm e}}}}}}}{{{{{x_{\rm e}}}}}^3}\left[1-n(e^{{{{{x_{\rm e}}}}}}-1)\right]+{{{{x_{\rm e}}}}}\frac{\partial n}{\partial {{{{x_{\rm e}}}}}}\frac{\partial} {\partial t}\left[\ln \left(\frac{{{{T_{\rm e}}}}}{{{{T_{\rm CMB}}}}(1+z)}\right)\right]\label{kineticeq}\end{aligned}$$ The first term with coefficient ${{{{K_{\rm C}}}}}$ is the Kompaneets term describing Compton scattering. The three terms in the square brackets in Kompaneets term describe photon diffusion in frequency due to the Doppler effect, electron recoil and induced recoil effects respectively. The second term represents emission and absorption of photons due to double Compton (${{{{K_{\rm dC}}}}}$) and bremsstrahlung (${{{{K_{\rm br}}}}}$). The last term in Eq. arises because we are evaluating the time derivative at constant ${{{{x_{\rm e}}}}}=h \nu/{{{k_{\rm B}}}}{{{T_{\rm e}}}}$ instead of constant frequency $\nu$, and electron temperature changes with time[^1] because the photon distribution is evolving (see Eq. ).
We are interested in the regime where comptonization is efficient and deviation from a Planck spectrum is small. In this regime the photon spectrum is described by a Bose-Einstein distribution with a chemical potential much smaller than unity in magnitude. $$\begin{aligned}
n({{{{x_{\rm e}}}}})&=\frac{1}{e^{{{{{x_{\rm e}}}}}+\mu}-1}\approx \frac{1}{e^{{{{{x_{\rm e}}}}}}-1}-\mu\frac{e^{{{{{x_{\rm e}}}}}}}{(e^{{{{{x_{\rm e}}}}}}-1)^2}\end{aligned}$$ The total energy density and number density of photons is then given by [^2] $$\begin{aligned}
E&=\frac{{{{a_{\rm R}}}}{{{T_{\rm e}}}}^4}{I_3}\int {{{\rm d}}}{{{{x_{\rm e}}}}}{{{{x_{\rm e}}}}}^3 n({{{{x_{\rm e}}}}})\approx{{{a_{\rm R}}}}{{{T_{\rm e}}}}^4\left(1-\mu\frac{6\zeta (3)}{I_3}\right)\label{energy}\\
N&=\frac{{{{b_{\rm R}}}}{{{T_{\rm e}}}}^3}{I_2}\int {{{\rm d}}}{{{{x_{\rm e}}}}}{{{{x_{\rm e}}}}}^2
n({{{{x_{\rm e}}}}})\approx{{{b_{\rm R}}}}{{{T_{\rm e}}}}^3\left(1-\mu\frac{\pi^2}{3I_2}\right),\label{number}\end{aligned}$$ where $a_R=\frac{8 \pi^5{{{k_{\rm B}}}}^4}{15 c^3 h^3}$ is the radiation constant, ${{{b_{\rm R}}}}=\frac{16 \pi{{{k_{\rm B}}}}^3\zeta(3)}{ c^3 h^3}$, $\zeta$ is the Riemann zeta function with $\zeta(3)\approx 1.20206$, $I_3=\int x^3 n_{\rm pl}(x) {{{\rm d}}}x=\pi^4/15$,$I_2=\int x^2 n_{\rm
pl}(x) {{{\rm d}}}x=2\zeta(3)$, and $n_{\rm pl}(x)=1/(e^x-1)$.
[In order to cancel the effect of the expansion of the Universe we will use the blackbody spectrum with temperature $T_{\gamma}={{{T_{\rm CMB}}}}(1+z)$, $E_{\gamma}$ is its energy density and $N_{\gamma}$ its number density. If we have a source injecting energy density at a rate ${{{\dot{{\mathcal{E}}}}}}$ and photon number density at rate ${{\dot{\mathcal{N}}}}$, where ${{{{{\mathcal{E}}}}}}=E/E_{\gamma}$, ${{{\mathcal{N}}}}=N/N_{\gamma}$,]{} we get using Eqs. and and with $|\mu|<<1$ $$\begin{aligned}
\frac{{{{\rm d}}}}{{{{\rm d}}}t}\ln\left(\frac{E}{E_{\gamma}}\right)&\equiv\frac{{{{\dot{{\mathcal{E}}}}}}}{{{{{{\mathcal{E}}}}}}}=4\frac{{{{\rm d}}}}{{{{\rm d}}}t}\ln\left(\frac{{{{T_{\rm e}}}}}{T_{\gamma}}\right)-\frac{6\zeta
(3)}{I_3}\frac{{{{\rm d}}}\mu}{{{{\rm d}}}t}\label{edt}\\
\frac{{{{\rm d}}}}{{{{\rm d}}}t}\ln\left(\frac{N}{N_{\gamma}}\right)&\equiv\frac{{{\dot{\mathcal{N}}}}}{{{{\mathcal{N}}}}}=3\frac{{{{\rm d}}}}{{{{\rm d}}}t}\ln\left(\frac{{{{T_{\rm e}}}}}{T_{\gamma}}\right)-\frac{\pi^2}{3I_2}\frac{{{{\rm d}}}\mu}{{{{\rm d}}}t}\label{ndt}.\end{aligned}$$ [ For small distortions $E\approx E_{\gamma}, N\approx N_{\gamma}$ and we have at lowest order, ${{{\dot{{\mathcal{E}}}}}}/{{{{{\mathcal{E}}}}}}\approx {{{\dot{{\mathcal{E}}}}}}$ and ${{\dot{\mathcal{N}}}}/{{{\mathcal{N}}}}\approx {{\dot{\mathcal{N}}}}$. The photon production due to double Compton and bremsstrahlung can be calculated by taking the time derivative of Eq. . Multiplying the kinetic equation Eq. by ${{{{x_{\rm e}}}}}^2$ and integrating over ${{{{x_{\rm e}}}}}$ and using it in the the time derivative of Eq. , the terms involving $\partial {{{T_{\rm e}}}}/\partial t$ cancel out and only the bremsstrahlung and double Compton terms contribute, giving]{} $$\begin{aligned}
\frac{{{{\rm d}}}}{{{{\rm d}}}t}\ln\left(\frac{N}{N_{\gamma}}\right)&=\frac{1}{I_2}\int_0^{\infty}{{{\rm d}}}{{{{x_{\rm e}}}}}\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)\frac{e^{-{{{{x_{\rm e}}}}}}}{{{{{x_{\rm e}}}}}}\left[1-n(e^{{{{{x_{\rm e}}}}}}-1)\right]\nonumber\\
&\approx\frac{\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)}{I_2}\int{{{\rm d}}}{{{{x_{\rm e}}}}}\frac{\mu}{{{{{x_{\rm e}}}}}\left(e^{{{{{x_{\rm e}}}}}}-1\right)}.
\label{ndtint}\end{aligned}$$ In general there may be additional sources of photons, for example, if energy is injected as an electromagnetic shower resulting from decay of a heavy particle, the resulting cascade may produce non-negligible amount of photons. However in most cases of interest and at high redshifts this is much smaller than the photon production from bremsstrahlung and double Compton.
Numerical solution
------------------
In order to calculate the precision of the analytic formulae derived below, we compare them with the numerical solution of the coupled system of Eqs. and . The initial spectrum for the numerical solution is a $\mu$ type distortion of magnitude $10^{-5}$ at $z=5\times 10^6$ with the chemical potential decaying exponentially with decreasing frequencies at $x<<1$ (see Eq. below). The results are not sensitive to the exact form of initial spectrum if the starting redshift is $\gg 10^6$, what matters is the total energy input into an initial blackbody. We solve equation iteratively in small time steps (using Compton $y$ parameter as the time variable) of $\delta y = 0.1$. In the first iteration we use the analytic solution for the evolution of electron temperature. In the second iteration we use the solution of first iteration to calculate the electron temperature using Eq. . As a frequency variable we use a $x=h\nu/{{{k_{\rm B}}}}T$ where $T$ is the reference temperature which evolves just by redshifting due to cosmological expansion and is equal to the electron temperature at the start of each iteration step. This gets rid of the last term in Eq. and introduces factors of ${{{T_{\rm e}}}}/T$ in the Kompaneets and bremsstrahlung/double Compton terms. We also write the total spectrum as a sum of blackbody part at reference temperature $T$ and a distortion part, keep only the terms linear in distortions and solve the linearized Eq. for distortions, as the zeroth order blackbody part vanishes. Implicit backward differentiation method is used to solve the PDE. We use logarithm of $x$ as the second independent variable with the variable step size in the $x$ direction.
The main source of error is the deviation of electron temperature used in solving the PDE from the correct temperature given by Eq. , resulting in violation of energy conservation. In our solution, the maximum error in energy conservation as a fraction of energy in the distortion occurs at high redshifts, when there is strong evolution of $\mu$, but even this is $\sim 10^{-5}$ in each iterative step. The error with respect to the total energy density in photons is, therefore, $\sim 10^{-10}$, since we start with an initial distortion of $\sim 10^{-5}$, and is much smaller in the later steps as the distortion decays exponentially. An important point to note here is that since we change the reference temperature to the current electron temperature at the beginning of each iterative step, and solve and track only the distortion part, the error in $\mu$ in individual iterative steps does not accumulate but is in fact suppressed in the subsequent evolution of the spectrum by the visibility factor (defined below). The solution obtained can thus be considered almost exact for the purpose of the present paper.
The high frequency spectrum is forced to be Wien at $120\ge x>100$ with the chemical potential and temperature given by analytic solution. The low frequency boundary is at $x=10^{-5}$ and the spectrum at $x<2\times 10^{-5}$ is forced to be blackbody with the temperature equal to the analytic electron temperature. Since our boundaries are far away in the distant Wien/Rayleigh-Jeans tails, where there are negligible amount of photons/energy , the solution is not sensitive to the exact boundary conditions. Since our boundary conditions are approximately equal to the true solution, we are able to use large steps in the $x$ direction resulting in considerable speedup in the numerical calculation compared to a calculation with arbitrary (but reasonable) boundary conditions. The initial spectrum is evolved until recombination ($z=1100$) although the $\mu$ distortion at $x\gtrsim
0.1$ is effectively frozen-in after $z\sim 10^4$. Further details on numerical issues can be found in [@bdd91; @pb2009; @cs2011].
Solution using stationarity approximation
=========================================
In order to calculate the integral in Eq. we need to know the occupation number $n({{{{x_{\rm e}}}}})$. Note that we cannot use Bose-Einstein distribution as an approximation since the integral in this case diverges at small ${{{{x_{\rm e}}}}}$. The reason is that double Compton and bremsstrahlung rates diverge at small frequencies and establish Planck spectrum. We can get an approximate solution from the kinetic equation Eq. [@k1956] by assuming that the instantaneous spectrum is stationary. Thus, neglecting time derivatives, making chemical potential function of frequency,[^3] $\mu({{{{x_{\rm e}}}}})$, and also assuming, ${{{{x_{\rm e}}}}}<<1,\mu({{{{x_{\rm e}}}}})<<1$ Eq. simplifies considerably,
$$\begin{aligned}
0&=-{{{{K_{\rm C}}}}}\frac{1}{{{{{x_{\rm e}}}}}^2}\frac{{{{\rm d}}}}{{{{\rm d}}}{{{{x_{\rm e}}}}}}{{{{x_{\rm e}}}}}^2\frac{{{{\rm d}}}\mu}{{{{\rm d}}}{{{{x_{\rm e}}}}}}+\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)\frac{\mu}{{{{{x_{\rm e}}}}}^4}\label{stateq}\end{aligned}$$
The solution of this ordinary differential equation with the boundary condition $\mu(0)=0$ is given by [@sz1970] $$\begin{aligned}
\mu({{{{x_{\rm e}}}}})&={{{{\mu_{\rm c}}}}}e^{-{{{{x_{\rm c}}}}}/{{{{x_{\rm e}}}}}}\nonumber\\
{{{{x_{\rm c}}}}}&\approx \left(\frac{{{{{K_{\rm dC}}}}}({{{{x_{\rm c}}}}})+{{{{K_{\rm br}}}}}({{{{x_{\rm c}}}}})}{{{{{K_{\rm C}}}}}}\right)^{1/2}\nonumber\\
&=\left(\frac{{{{{a_{\rm dC}}}}}(1+z)+{{{{a_{\rm br}}}}}(1+z)^{-3/2}}{{{{{a_{\rm C}}}}}}\right)^{1/2}\nonumber\\
&\approx
\left(7.43\times 10^{-5}\left(\frac{1+z}{2\times 10^6}\right)+1.07\times
10^{-6}\left(\frac{1+z}{2\times 10^6}\right)^{-3/2}\right)^{1/2},\label{xc}\end{aligned}$$ where ${{{{\mu_{\rm c}}}}}$ is normalization specified by chemical potential at large ${{{{x_{\rm e}}}}}$ . Thus $\mu({{{{x_{\rm e}}}}})$ decays exponentially at small frequencies and goes to constant at large frequencies. ${{{{x_{\rm c}}}}}$ is also the frequency at which comptonization rate is equal to photon absorption rate due to double Compton and bremsstrahlung. Similarly we can also define a frequency at which the photon absorption rate is equal to the Hubble rate $H(z)$ [[@hs1993],]{} $$\begin{aligned}
{{{{x_{\rm H}}}}}&\approx \left(\frac{{{{{K_{\rm dC}}}}}({{{{x_{\rm H}}}}})+{{{{K_{\rm br}}}}}({{{{x_{\rm H}}}}})}{H}\right)^{1/2}\nonumber\\
&= \left(\frac{{{{{a_{\rm dC}}}}}(1+z)^3+{{{{a_{\rm br}}}}}(1+z)^{1/2}}{H(0){{{{\Omega_{\rm r}}}}}^{1/2}}\right)^{1/2}\nonumber\\
&\approx \left(0.029\left(\frac{1+z}{2\times 10^6}\right)^3+4.18\times
10^{-4}\left(\frac{1+z}{2\times 10^6}\right)^{1/2}\right)^{1/2}
,\label{xH}\end{aligned}$$ ${{{{x_{\rm c}}}}}$ and ${{{{x_{\rm H}}}}}$ are plotted in Fig. \[xrates\]. We note that ${{{{x_{\rm c}}}}}<<1$ in the redshift range of interest. This is consistent with assumptions made in the derivation. Also, since ${{{{x_{\rm c}}}}}\sim 0.01$ in the redshift range of interest and gaunt factors are slowly varying functions of frequency and temperature, we can assume ${{{{g_{\rm dC}}}}}\approx
{{{{g_{\rm dC}}}}}(0.01)=1.005$ and ${{{{g_{\rm br}}}}}\approx 2.99$.
[We should emphasize that the sole purpose of finding an accurate solution for the spectrum at low frequencies is to calculate precisely the photon emission/absorption due to bremsstrahlung and double Compton. In particular, only the spectrum at ${{{{x_{\rm e}}}}}\gtrsim 0.1$ is frozen-in at $x\ll 10^5$. The low frequency spectrum continues to be affected by bremsstrahlung (at smaller redshifts, during recombination and after), which tries to bring the spectrum in equilibrium with the electrons, which are cooler than the radiation due to adiabatic expansion [@cs2011; @ksc2011]. Furthermore, most of the photons are created around the critical frequency ${{{{x_{\rm c}}}}}$. The double Compton gaunt factor at the critical frequency ${{{{g_{\rm dC}}}}}({{{{x_{\rm c}}}}})$ deviates from the value $1.005$ by less than $0.5\%$ in the interesting redshift range of $10^5<z<10^7$. The bremsstrahlung gaunt factor has a maximum deviation of $\sim 7\%$ from $2.99$ in the same redshift range, but since it is only a small correction to the dominant double Compton process, the error in the final solution for $\mu$ evolution is small. Thus the assumption of constant gaunt factor is an excellent one for the present problem and is further justified by a comparison of the analytic and numerical solutions.]{}
![\[xrates\]The frequency ${{{{x_{\rm c}}}}}$ and ${{{{x_{\rm H}}}}}$ at which the Compton and Hubble rates respectively are equal to the sum of the bremsstrahlung and double Compton rate. If Compton rate is also greater than the Hubble rate, then bremsstrahlung and double Compton can establish complete thermodynamic equilibrium (blackbody spectrum) below ${{{{x_{\rm c}}}}}$ otherwise complete thermodynamic equilibrium is established below ${{{{x_{\rm H}}}}}$. [Above ${{{{x_{\rm c}}}}}$, at redshifts $z>z_{\rm c}\approx 10^5$, a frequency dependent chemical potential $\mu$ is established. At $x>>x_c$, the chemical potential has an almost constant (frequency independent) value ${{{{\mu_{\rm c}}}}}$ and we have thus a Bose-Einstein spectrum. The chemical potential decreases with time due to the photons created by bremsstrahlung and double Compton at low frequencies and redistributed by Compton scattering over the entire spectrum.]{}](xrates.eps){width="14cm"}
We can now use the solution Eq. to evaluate the integral Eq. (ignoring the ${{{{x_{\rm e}}}}}$ dependence of gaunt factors), $$\begin{aligned}
\frac{{{{\rm d}}}}{{{{\rm d}}}t}\ln\left(\frac{N}{N_{\gamma}}\right)&\approx\frac{\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)}{I_2}\int{{{\rm d}}}{{{{x_{\rm e}}}}}\frac{{{{{\mu_{\rm c}}}}}e^{-{{{{x_{\rm c}}}}}/{{{{x_{\rm e}}}}}}}{{{{{x_{\rm e}}}}}^2}=\frac{{{{{\mu_{\rm c}}}}}}{I_2}\frac{{{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}}{{{{{x_{\rm c}}}}}}=\frac{{{{{\mu_{\rm c}}}}}}{I_2}\left[\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right){{{{K_{\rm C}}}}}\right]^{1/2}.
\label{ndtintsol}\end{aligned}$$ Equations and along with the above solution give the following equation for the evolution of chemical potential $\mu={{{{\mu_{\rm c}}}}}$ at $x>1$, $$\begin{aligned}
\frac{{{{\rm d}}}\mu}{{{{\rm d}}}z}=\frac{C}{(1+z)H}\left[\mu\left[\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right){{{{K_{\rm C}}}}}\right]^{1/2}-B\frac{{{{\dot{{\mathcal{E}}}}}}}{{{{{{\mathcal{E}}}}}}}+\frac{4 B}{3}\left.\frac{{{\dot{\mathcal{N}}}}}{{{{\mathcal{N}}}}}\right|_{\rm Extra}\right],\label{mueq}\end{aligned}$$ where $C=0.7768$, $B=1.803$. $\left.\frac{{{\dot{\mathcal{N}}}}}{{{{\mathcal{N}}}}}\right|_{\rm Extra}$ are the extra photons injected from processes other than the low frequency bremsstrahlung and double Compton photons calculated above, for example, from the same source which injects energy. We assume this extra term to be negligible in the rest of the paper. The solution of Eq. at $z=0$ is given by $$\begin{aligned}
\mu(0)=\mu(z_i)e^{-{{{{\mathcal{T}}}}}(z_i)}+CB\int_{z_{\rm min}}^{z_i}\frac{{{{\rm d}}}z}{(1+z)H}\left( \frac{{{{\dot{{\mathcal{E}}}}}}}{{{{{{\mathcal{E}}}}}}}-\frac{4 }{3}\left.\frac{{{\dot{\mathcal{N}}}}}{{{{\mathcal{N}}}}}\right|_{\rm Extra}\right)e^{-{{{{\mathcal{T}}}}}(z)}\label{musol}\end{aligned}$$ where $z_{\rm min}\approx 5\times 10^4$, $z_i$ is the initial/maximum energy injection redshift and we have defined the effective *blackbody optical depth* (which is frequency independent at $x\gtrsim {{{{x_{\rm c}}}}}$)
$$\begin{aligned}
{{{{\mathcal{T}}}}}(z)&=\int_0^{z}{{{\rm d}}}z' \frac{C
\left[\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right){{{{K_{\rm C}}}}}\right]^{1/2}}{(1+z')H}\nonumber\\
&\approx
\left[\left(\frac{1+z}{1+{{{{z_{\rm dC}}}}}}\right)^5+\left(\frac{1+z}{1+{{{{z_{\rm br}}}}}}\right)^{5/2}\right]^{1/2}+\epsilon
\ln\left[\left(\frac{1+z}{1+z_{\epsilon}}\right)^{5/4}+\sqrt{1+\left(\frac{1+z}{1+z_{\epsilon}}\right)^{5/2}}\right],\label{optold}\end{aligned}$$
where $$\begin{aligned}
{{{{z_{\rm dC}}}}}&=\left[\frac{25 {{{{\Omega_{\rm r}}}}}H(0)^2}{4 C^2{{{{a_{\rm C}}}}}{{{{a_{\rm dC}}}}}}\right]^{1/5}=1.96\times
10^6\nonumber\\
{{{{z_{\rm br}}}}}&=\left[\frac{25 {{{{\Omega_{\rm r}}}}}H(0)^2}{4 C^2{{{{a_{\rm C}}}}}{{{{a_{\rm br}}}}}}\right]^{2/5}=1.05\times
10^7\nonumber\\
z_{\epsilon}&=\left[\frac{{{{{a_{\rm br}}}}}}{{{{{a_{\rm dC}}}}}}\right]^{2/5}=3.67\times
10^5\nonumber\\
\epsilon&=\left[\frac{4 C^2{{{{a_{\rm br}}}}}^2{{{{a_{\rm C}}}}}}{25 {{{{a_{\rm dC}}}}}{{{{\Omega_{\rm r}}}}}H(0)^2}\right]^{1/2}=0.0151,\end{aligned}$$ ${{{{\Omega_{\rm r}}}}}$ is the total radiation density parameter, and $H(0)$ is the Hubble constant today. It is interesting to note that in the absence of double Compton scattering we would have ${{{{z_{\rm br}}}}}\approx 6\times 10^6$. The presence of double Compton increases the critical frequency ${{{{x_{\rm c}}}}}$ from its bremsstrahlung only value, and thus reducing the bremsstrahlung emission.
It is also straightforward to calculate, if needed, the chemical potential at any intermediate redshift $z'>z_{\rm min}$ using Eq. by replacing $\mathcal{T}(z)$ (and similarly for $\mathcal{T}(z_i)$) with $\mathcal{T}(z)-\mathcal{T}(z')$ and also replacing the lower limit $z_{\rm min}$ in the integral with $z'$, $$\begin{aligned}
\mu(z')=\mu(z_i)e^{-\left[{{{{\mathcal{T}}}}}(z_i)-{{{{\mathcal{T}}}}}(z')\right]}+CB\int_{z_{\rm '}}^{z_i}\frac{{{{\rm d}}}z}{(1+z)H}\left( \frac{{{{\dot{{\mathcal{E}}}}}}}{{{{{{\mathcal{E}}}}}}}-\frac{4 }{3}\left.\frac{{{\dot{\mathcal{N}}}}}{{{{\mathcal{N}}}}}\right|_{\rm Extra}\right)e^{-\left[{{{{\mathcal{T}}}}}(z)-{{{{\mathcal{T}}}}}(z')\right]}\label{musolz}\end{aligned}$$
The solution given in Eq. corresponds to Eq. (15) in [@sz1970] but including the double Compton process. Similarly, the solution in Eq. generalizes Eq. (20) of [@sz1970]. The dominant term (${{{{z_{\rm dC}}}}}$)in Eq. is due to the double Compton process with the bremsstrahlung term (${{{{z_{\rm br}}}}}$) providing a small but important correction.
Improved solution by approximating non-stationarity using previous solution
===========================================================================
We will see below that the solution arrived at in the previous section underestimates the photon production. It turns out that the stationary solution, which is normalized at high frequencies, underestimates the chemical potential at small frequencies where most of the photons are being produced/absorbed. [We find below the correction for the normalization of chemical potential, which enables us to improve the formula for blackbody optical depth Eq. in the solution Eq. . The result of the computations, using the new analytic formula Eq. for the blackbody optical depth, deviates from numerical solution by less than $1\%$.]{}
A very simple correction to the normalization can be arrived at as follows. An immediate improvement over the solution of [@sz1970] is possible by approximating the non-stationarity in Eq. using the solution Eq. (ignoring the energy injection term)[^4] $$\begin{aligned}
\frac{\partial n}{\partial t}\approx \frac{-1}{{{{{x_{\rm e}}}}}^2}\frac{\partial
\mu}{\partial t}\approx \frac{C\mu\left[\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right){{{{K_{\rm C}}}}}\right]^{1/2}}{{{{{x_{\rm e}}}}}^2}\end{aligned}$$ Note that the time derivative of temperature in Eq. can be neglected at small frequencies as its effect is suppressed by a factor of ${{{{x_{\rm e}}}}}$ with respect to the term with the time derivative of $\mu$. Equation with the above approximation for the non-stationary term gives Bessel’s equation $$\begin{aligned}
\frac{{{{\rm d}}}}{{{{\rm d}}}{{{{x_{\rm e}}}}}}{{{{x_{\rm e}}}}}^2\frac{{{{\rm d}}}\mu}{{{{\rm d}}}{{{{x_{\rm e}}}}}}-\left(\frac{{{{{x_{\rm c}}}}}^2}{{{{{x_{\rm e}}}}}^2}-C{{{{x_{\rm c}}}}}\right)\mu&=0\label{stateq2}\end{aligned}$$ The solution is given in terms of modified Bessel function of second kind $K_{\nu}(x)$, $$\begin{aligned}
\mu({{{{x_{\rm e}}}}})&=A\mu_c\sqrt{\frac{2}{\pi}}\sqrt{\frac{{{{{x_{\rm c}}}}}}{{{{{x_{\rm e}}}}}}}K_{0.5\sqrt{1-4C{{{{x_{\rm c}}}}}}}({{{{x_{\rm c}}}}}/{{{{x_{\rm e}}}}}).\label{mux2}\end{aligned}$$ [This result provides a more precise dependence of $\mu$ on frequency compared to Eq. .]{} Choosing normalization to give $\mu(0.5)\approx {{{{\mu_{\rm c}}}}}$ at ${{{{x_{\rm e}}}}}=0.5$ gives $A=1.007 + 3.5{{{{x_{\rm c}}}}}$, where this fit is accurate for $5\times
10^{-3}<{{{{x_{\rm c}}}}}<2\times 10^{-2}$. This fit thus covers all the interesting range for critical frequency ${{{{x_{\rm c}}}}}$ (see Fig. \[xrates\]). This choice of normalization frequency (${{{{x_{\rm e}}}}}=0.5$) provides a good fit to the numerical solution.
[ The normalization frequency is chosen so that (i) it is $<1$, since this is the assumption made in deriving the analytic solution and (ii) it is also large enough so that $\mu\approx {\rm constant}$. Since we only want to use this solution to calculate the total photon emission, it need only be accurate at ${{{{x_{\rm e}}}}}\lesssim 1$. The only requirement at ${{{{x_{\rm e}}}}}>1$ is that its contribution to the photon emission/absorption should be negligible at ${{{{x_{\rm e}}}}}\gg 1$ and that it should be approximately constant around ${{{{x_{\rm e}}}}}=1$, as expected from a correct solution. We should also point out that $\mu({{{{x_{\rm e}}}}})$ decreases with increasing ${{{{x_{\rm e}}}}}$ at ${{{{x_{\rm e}}}}}\gg 1$ for the solution in Eq. and $\mu({{{{x_{\rm e}}}}}\rightarrow\infty)=0$. Thus our solution satisfies the requirements outlined above. Obviously it cannot be normalized at ${{{{x_{\rm e}}}}}=\infty$, as was done with the original solution Eq. . The normalization must be done by comparison with the numerical solution, taking into account the assumptions made in arriving at this solution, resulting in our choice of ${{{{x_{\rm e}}}}}=0.5$.[^5] We show a snapshot of the numerical solution (chosen at random) at $z=3.48\times 10^6$, original solution Eq. and improved solution Eq. in Fig \[muxfig\]. The critical frequency at this redshift is ${{{{x_{\rm c}}}}}=0.0114$. Needless to say that the shape of the spectrum is well described by our solution at all redshifts and we have chosen a random snapshot in Fig. \[muxfig\] for illustration. The final justification for all our assumptions and approximations is of course given by a comparison of the final numerical and analytic solutions for the evolution of the chemical potential with redshift as described below.]{}
![\[muxfig\]Snapshot of the numerical solution, original solution Eq. marked SZ1970 and improved solution Eq. at $z =3.48\times 10^6$ (chosen randomly) with the initial $\mu=10^{-5}$ at $z=5\times 10^6$ including both double Compton and bremsstrahlung photon production and no additional energy injection. The critical frequency at $z=3.48\times 10^6$ is ${{{{x_{\rm c}}}}}=0.0114$. Both the analytical solutions are plotted with the high frequency distortion ${{{{\mu_{\rm c}}}}}=1.39\times 10^{-9}$. The numerical and improved solutions are indistinguishable in the figure while the original solution underestimates $\mu({{{{x_{\rm e}}}}})$ at low frequencies. ](mux.eps){width="14cm"}
We can now use our improved solution to calculate the photon production rate $$\begin{aligned}
\frac{{{{\rm d}}}}{{{{\rm d}}}t}\ln\left(\frac{N}{N_{\gamma}}\right)&\approx\frac{\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)}{I_2}\int_0^{\infty}{{{\rm d}}}{{{{x_{\rm e}}}}}\frac{\mu({{{{x_{\rm e}}}}}) }{{{{{x_{\rm e}}}}}^2}\nonumber\\
&\approx\frac{{{{{\mu_{\rm c}}}}}}{I_2}({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}})(1.007+3.5
{{{{x_{\rm c}}}}})\left(\frac{1}{{{{{x_{\rm c}}}}}}-C\ln(2)\right)\nonumber\\
&\approx \frac{{{{{\mu_{\rm c}}}}}}{I_2}\left[1.007\left[\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right){{{{K_{\rm C}}}}}\right]^{1/2}+2.958\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)\right].
\label{ndtintsolnew}\end{aligned}$$ Proceeding as before we get improved formula for blackbody optical depth, $$\begin{aligned}
{{{{\mathcal{T}}}}}(z)&=\int_0^{z}{{{\rm d}}}z' \frac{1.007C
\left[\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right){{{{K_{\rm C}}}}}\right]^{1/2}+2.958C\left({{{{K_{\rm dC}}}}}+{{{{K_{\rm br}}}}}\right)}{(1+z')H}\nonumber\\
&\approx
1.007\left[\left(\frac{1+z}{1+{{{{z_{\rm dC}}}}}}\right)^5+\left(\frac{1+z}{1+{{{{z_{\rm br}}}}}}\right)^{5/2}\right]^{1/2}+1.007
\epsilon
\ln\left[\left(\frac{1+z}{1+z_{\epsilon}}\right)^{5/4}+\sqrt{1+\left(\frac{1+z}{1+z_{\epsilon}}\right)^{5/2}}\right]\nonumber\\
&+\left[\left(\frac{1+z}{1+{{{{z_{\rm dC}}}}}'}\right)^3+\left(\frac{1+z}{1+{{{{z_{\rm br}}}}}'}\right)^{1/2}\right],\label{optnew}\end{aligned}$$ where we have defined $$\begin{aligned}
{{{{z_{\rm dC}}}}}'&=\left[\frac{3 {{{{\Omega_{\rm r}}}}}^{1/2} H(0)}{2.958 C{{{{a_{\rm dC}}}}}}\right]^{1/3}=7.11\times
10^6\nonumber\\
{{{{z_{\rm br}}}}}'&=\left[\frac{ {{{{\Omega_{\rm r}}}}}^{1/2} H(0)}{5.916 C{{{{a_{\rm br}}}}}}\right]^{2}=5.41\times
10^{11}\nonumber\\\end{aligned}$$ The improved solution for evolution of $\mu$ is still given by the original equation on substituting the improved optical depth given by Eq. .
[The improved solution has a broad region of validity and covers the entire redshift range of interest. The physics used to derive Eq. is applicable for redshifts $z\lesssim 8 \times 10^7$. Thus we can use the blackbody optical depth, Eq. , for redshift interval $10^5 < z <
8\times 10^7$, the upper limit is well behind blackbody surface at $z\sim 2 \times 10^6$. At $z\approx 8\times
10^7$ the number density of positrons becomes comparable to the number density of electrons/baryons due to pair production. At higher redshifts, the number density of electrons and positrons, and thus the rates of Compton scattering, double Compton and electron-electron and electron-positron bremsstrahlung, increase exponentially with increasing redshift. Thus the blackbody optical depth, $\mathcal{T}$ also starts increasing exponentially instead of a power law as in our solution and Eq. is no longer applicable. However, we already have $\mathcal{T}\sim 10^4$ at $z\sim 8\times 10^7$, and creation of a distortion in photon spectrum is thus impossible at higher redshifts.]{}
We plot the optical depth for the numerical solution, the total improved solution as well as the individual terms in Eq. in Fig. \[optfig\]. At high redshifts, double Compton terms dominate with the new double Compton term ${{{{z_{\rm dC}}}}}'$ also contributing. Bremsstrahlung term becomes important at low redshifts. At high redshifts, our improved solution is indistinguishable from the numerical solution. As blackbody optical depth becomes small the quasi-static assumptions made in arriving at the analytic solution also breakdown and the error grows. The new solution is however an excellent approximation to the numerical result, and definitely better than the double Compton only result, over the entire redshift range of interest, where the optical depth is greater than a few $\%$. The deviations from the numerical result for the double Compton only formula and our new result are plotted in Fig. \[opterror\]. The analytic solution overestimates the photon production at low redshifts. The reason becomes clear by looking at ${{{{x_{\rm H}}}}}$ in Fig. \[xrates\]. Photon production at ${{{{x_{\rm e}}}}}>{{{{x_{\rm H}}}}}$ would be suppressed since the photon production rate is smaller than the expansion rate. At high redshifts, ${{{{x_{\rm H}}}}}\gtrsim 1$ and the error introduced by including ${{{{x_{\rm e}}}}}>{{{{x_{\rm H}}}}}$ is negligible since photon production is negligible at these frequencies anyway. At low redshifts, ${{{{x_{\rm H}}}}}$ becomes less than unity and starts approaching ${{{{x_{\rm c}}}}}$ and Eqs. and overestimate photon production. We show the *blackbody visibility* factors[^6] $\mathcal{G}\equiv e^{-\mathcal{T}}$ in Figs. \[visfigzoom\] and \[visfig\] for analytic solution Eq. , for the double Compton only term ${{{{z_{\rm dC}}}}}$ and the numerical result. The accuracy of blackbody visibility is better than $1\%$ with the new solution Eq. .
![\[optfig\]Blackbody optical depth as a function of redshift (independent of frequency) calculated using numerical solution, and improved solution Eq. . Individual terms in Eq. are also shown. At high redshifts double Compton terms dominate with the new double Compton term ${{{{z_{\rm dC}}}}}'$ also contributing. Bremsstrahlung term becomes important at low redshifts. At high redshifts our improved solution is indistinguishable from the numerical solution. The new solution is an excellent approximation to the numerical result over the entire redshift of interest where the optical depth is greater than a few $\%$. ](bbtau.eps){width="14cm"}
![\[opterror\] Deviation ($\%$) from the numerical solution in blackbody optical depth with respect to the numerical solution for the standard double Compton only analytic solution, Eq. , which includes both bremsstrahlung and double Compton using the method of [@sz1970] and our new solution. The error in $\mathcal{T}$ at low redshifts does not have a significant effect on the final spectrum. The error in visibility for our improved solution is better than $1\%$ at all redshifts and shown in Fig. \[visfig\].](bbtauerror.eps){width="14cm"}
![\[visfigzoom\][The characteristics of the blackbody photosphere as given by the blackbody visibility]{} $\mathcal{G}\equiv e^{-\mathcal
{T}}$ for analytic solution Eq. , for the double Compton only solution ${{{{z_{\rm dC}}}}}$ and the numerical result. We have introduced *blackbody surface* as the boundary where the blackbody optical depth $\mathcal{T}$=1.](bbviszoom.eps){width="14cm"}
![\[visfig\] Same as Fig. \[visfigzoom\] but going to higher redshifts and also showing the errors, relative to the numerical solution, for the different analytic solutions. Deviations from the numerical solution is also shown for Eq. based on the method of [@sz1970] but including both the double Compton and bremsstrahlung. The errors are negative at high redshifts and positive at low redshifts with a spike where they change sign.](bbvis.eps){width="14cm"}
Examples from standard cosmology
================================
Upper limit to energy release after BBN and before recombination
----------------------------------------------------------------
[In standard model of cosmology, we can get constraints on energy density in radiation from two distinct and very precise observables. The first is the deuterium abundance, which gives the baryon number to photon number ratio $\eta=(5.7\pm 0.3)\times 10^{-10}$ during primordial nucleosynthesis at $4\times 10^8 \gtrsim z\gtrsim 4\times 10^7$ [@fields2001; @iocco2009]. The second is the measurement of CMB anisotropies, which constrains the baryon to photon ratio $\eta=(6.18\pm 0.15)\times 10^{-10}$ [@wmap7] during recombination at $z\approx 1100$. The fact that these two values of baryon to photon ratio are almost identical, with small error bars means that we do not have arbitrary freedom in adding energy to CMB, for example, with the introduction of new physics. In fact any addition of energy/entropy to CMB between primordial nucleosynthesis and recombination cannot be more than a small percentage ($\sim 7\%$ for CMB and BBN to be consistent within $2-\sigma$) of the already existing radiation energy density. COBE limit [@cobe] of $\mu\lesssim
9\times 10^{-5}$ implies that a $\sim 7\%$ energy can be added only at redshifts $z\gtrsim 4.1\times 10^6$. These limits also justify our assumption of small distortions in the analytic calculations. Any energy injection into photons from non-standard processes before electron-positron annihilation, however, is unconstrained.]{}
Electron-positron annihilation
------------------------------
It is, of course, possible to add of order unity energy to radiation before primordial nucleosynthesis. [This happens in standard cosmology during electron-positron annihilation [@alpher53; @peebles66; @zeld66], which more than doubles the energy density of photons and increases their temperature by $\sim 40\%$.]{} In the early stages of electron-positron annihilation, the electrons/positrons far outnumber the photons. In this era, therefore, electron-positron annihilation and electron-electron/electron-positron bremsstrahlung dominate the thermalization process. [In the very late stages, when most of the electron-positrons have annihilated, Compton and double Compton scattering are dominant, electron number is conserved and their density evolves according to the non-relativistic adiabatic law, and our analytic formulae become applicable.]{} [We, of course, do not expect any observable $\mu$ distortion from electron positron annihilation [@sz1970].]{} It is still interesting to calculate the magnitude of the distortion to demonstrate the effectiveness of double Compton scattering and comptonization in restoring the equilibrium between matter and radiation.
At redshifts $z\lesssim 10^8$, most of the positrons have annihilated and their number density falls below that of electrons. The number density of electrons is high enough (as a result of $\sim 10^{-9}$ asymmetry in matter anti-matter) to maintain the annihilation rate much faster than the expansion rate. We can, thus, use Saha equation to follow the positron number density during the last stages of positron annihilation. Using the fact that the positron number density is much smaller than the electron number density and that the electron number density is unaffected by annihilation, we get for the positron number density $n_{+}$, $$\begin{aligned}
n_{+}&\approx\frac{{{{n_{\rm eq}}}}^2}{{{{n_{\rm e}}}}},\end{aligned}$$ where the equilibrium (zero chemical potential) number density of electrons/positrons is $$\begin{aligned}
{{{n_{\rm eq}}}}&=\frac{2}{h^3}\left(2\pi {{{m_{\rm e}}}}{{{k_{\rm B}}}}T\right)^{3/2}e^{-\frac{{{{m_{\rm e}}}}c^2}{{{{k_{\rm B}}}}T}}\end{aligned}$$ The rate of energy injection is given by, $$\begin{aligned}
{{{\dot{{\mathcal{E}}}}}}=H(1+z)\frac{2{{{m_{\rm e}}}}c^2}{{{{a_{\rm R}}}}T^4}\frac{{{{\rm d}}}n_{+}}{{{{\rm d}}}z}\end{aligned}$$
We have plotted the resulting $\mu$ injection rate multiplied by redshift, $BC (1+z)\frac{2{{{m_{\rm e}}}}c^2}{{{{a_{\rm R}}}}T^4} \frac{{{{\rm d}}}n_{+}}{{{{\rm d}}}z}e^{-\mathcal{T}}$ in Fig. \[muex\]. Visibility function suppresses the high redshift contribution, while the exponentially decreasing positron number density suppresses the low redshift contribution, giving the peak at $z\sim
1.3\times 10^7$. The chemical potential from electron-positron annihilation is suppressed by an astronomical factor of $10^{178}$! Thus it is impossible to create a deviation from blackbody spectrum at high redshifts.
![\[muex\] Chemical potential $\mu$ from electron positron annihilation. The CMB blackbody spectrum is maintained at extraordinary precision of $10^{-178}$! Also shown, in the inset, is the actual rate of energy injection multiplied by $(1+z)$. At high redshifts ($z\gtrsim 10^8$) we have used entropy conservation to calculate the rate of heating. ](examples.eps){width="14cm"}
Primordial nucleosynthesis
--------------------------
Big bang nucleosynthesis (BBN) at $z\sim 3\times 10^8$ results in binding of almost all neutrons into helium (${{{n_{\rm ^4He}}}}/{{{n_{\rm B}}}}\approx 6\times 10^{-2}$) along with the production of small amount of deuterium (${{{n_{\rm { }^2H}}}}/{{{n_{\rm B}}}}\approx 2\times 10^{-5}$), helium-3 ${{{n_{\rm ^3He}}}}/{{{n_{\rm B}}}}\approx 8\times 10^{-6}$, tritium ${{{n_{\rm { }^3H}}}}/{{{n_{\rm B}}}}\approx 7\times
10^{-8}$, beryllium-7/lithium-7 (${{{n_{\rm { }^7Be}}}}/{{{n_{\rm B}}}}\approx 3\times 10^{-10}$), lithium-6 (${{{n_{\rm { }^6Li}}}}/{{{n_{\rm B}}}}\approx
10^{-14}$) and trace amounts of heavier elements [see @serpico for a recent calculation]. We can get a rough estimate of the energy released during this main part of nucleosynthesis by calculating the total binding energy of helium-4 produced.[^7] Thus we have an energy release of $\Delta E/E\sim E_{bind}{{{n_{\rm ^4He}}}}/E_{\gamma} \sim
6\times 10^{-9}$. The blackbody optical depth at $z\sim 10^8$ is $\sim 10^5$ and we have the final $\mu\sim 0$. However, tritium and beryllium-7 survive for a long time before decaying into helium-3 and lithium-7 respectively. [Although, the energy released in the decay of beryllium-7 and lithium-7 is much smaller than that released during helium production in BBN, the distortions are much larger because these decays happen in front of the blackbody surface, when the blackbody visibility is almost unity. Also, the energy density released is proportional to the number density of beryllium-7 and lithium-7, which has decreased as $(1+z)^3$ compared to the $(1+z)^4$ decrease of the radiation energy density, thus giving a larger $\Delta E/E$ than if the decay had happened at the same time as the main BBN.]{} Tritium has a half life of $12.32$ years. It, therefore, decays at $z\sim
2.5\times 10^5$ to helium-3 releasing an electron with average energy $5.7$KeV. Most of this energy release happens at $z\gtrsim 10^5$ and causes a $\mu$-distortion with $\mu=2\times 10^{-15}$.
Neutral beryllium atom decays by electron capture with a half life of 53.2 days. Fully ionized beryllium in the low density plasma in the early Universe is however stable. It has to wait for $\approx 800$ years until $z\approx 3\times 10^4$ when it recombines to hydrogen like beryllium. The recombined beryllium can now capture the orbital electron and decay to lithium-7 with a half-life of 106.4 days, which is twice the half-life of a fully recombined beryllium [@ks2011]. $89.6\%$ of the decays go to the ground state of lithium-7 and most of the energy released is carried away by neutrinos, which would appear today as a narrow line in the cosmic neutrino spectrum. $10.4\%$ of beryllium decays into an excited state of lithium. About half of the total decay energy in this case also is lost to neutrinos forming a second lower energy line in the cosmic neutrino spectrum. The excited lithium nucleus then de-excites, almost immediately, to the ground state, emitting a $Q=477.6$ KeV photon, which delivers most of its energy to plasma by Compton scattering on electrons (recoil effect). The Compton $y$ parameter at $z=30000$ is $0.04$, which lies intermediate between pure $y$ and $\mu$ type eras. The heating, therefore, results in a distortion intermediate between the $y$ and the $\mu$ type distortions of magnitude (using formula for $y$-type distortion) $\approx
(1/4)\Delta E/E \approx (1/4)0.104 Q {{{n_{\rm { }^7Be}}}}/E_{\gamma}\sim 10^{-16}$.
Dark matter annihilation
------------------------
A natural and favored candidate for dark matter is a weakly interacting massive particle (WIMP) with several candidates in high energy theories beyond the standard model [@dm]. A very attractive feature of WIMP is that if they have weak scale interactions then the correct amount of dark matter (which is close to the critical density) observed today can be thermally produced in the early Universe. This is remarkable since a priori there is no reason to suspect any relation between the weak scale interactions and the present critical density of the Universe and this coincidence is sometimes referred to as the *WIMP miracle*. For the thermally produced WIMPs, the dark matter density is related to the velocity averaged cross section as $$\begin{aligned}
\left<\sigma v\right>&\approx \frac{3\times 10^{-27}}{{{{{\Omega_{\rm dm}}}}}h_0^2}{\rm cm^3s^{-1}},\end{aligned}$$ where $h_0\approx 0.702$ is the Hubble parameter and ${{{{\Omega_{\rm dm}}}}}$ is the dark matter density as a fraction of critical density today. The above values for annihilation cross sections are of similar order of magnitude as the current upper limits from Fermi-LAT experiment probing dark matter annihilation in the local Universe [@fermi2; @fermi1; @hutsi]. [The actual annihilation cross section can, of course, be much smaller than the Fermi upper limits.]{} [Energy released from dark matter annihilation also changes recombination history [@ss1985] and it can thus be constrained through its effect on the CMB power spectrum [@ck2004; @pf2005] and recombination spectrum [@c2010]. Effect of dark matter annihilation also changes the abundance of elements produced during BBN [@bbn1; @bbn2]; these constraints are complementary but less stringent compared to the current CMB constraints.]{}
Initially, dark matter is in thermal equilibrium with other constituents of the Universe, and is being continuously created and annihilated. As the Universe cools and expands, these interactions freeze out and thereafter the dark matter number is conserved. However, a small number of residual annihilations keep happening throughout the history of the Universe. The rate of energy released into CMB from these residual annihilations[^8] is given by, $$\begin{aligned}
{{{\dot{{\mathcal{E}}}}}}&=f_{\gamma}\frac{{{{{m_{\rm WIMP}}}}}c^2n_{dm}^2\left<\sigma
v\right>}{a T^4}\nonumber\\
&\approx 1.4\times 10^{-29}(1+z)^2f_{\gamma}\left(\frac{10{\rm
GeV}}{{{{{m_{\rm WIMP}}}}}}\right)\left(\frac{{{{{\Omega_{\rm dm}}}}}h_0^2}{0.105}\right),\end{aligned}$$ where ${{{{m_{\rm WIMP}}}}}$ is the mass of the dark matter particle and $f_{\gamma}$ is the fraction of energy that goes into particles with electromagnetic interactions, and is deposited in the plasma.
We plot the rate of energy release into the CMB $$\begin{aligned}
(1+z)\mathcal{G}\frac{{{{\rm d}}}{{{{{\mathcal{E}}}}}}}{{{{\rm d}}}z}&= {{{\dot{{\mathcal{E}}}}}}\frac{e^{-\mathcal{T}}}{H}\end{aligned}$$ in Fig. \[dmfig\] for the fraction of energy going into the plasma[^9] $f_{\gamma}=1$. Hubble rate is proportional to $(1+z)^2$ during radiation domination, which gives us the flat portion of the curve. Energy release rate decreases faster than the expansion rate $\propto (1+z)^{3/2}$ during matter domination, giving the low redshift declining tail in the plot. The total $\mu$ distortion (using energy release from $z> 5\times 10^4$) is $\mu\approx 3\times 10^{-9}$. The $y$- type distortion from energy release $z\lesssim 5\times 10^4$ is $y\approx 5\times
10^{-10}$, these distortions were also calculated by [@cs2011]. These numbers are of similar order of magnitude as the distortions from Silk damping and Bose-Einstein condensation of CMB discussed in the next two sections. COBE constraint of $\mu<9\times 10^{-5}$ [@cobe] constrains WIMP mass to be ${{{{m_{\rm WIMP}}}}}>0.3f_{\gamma}{\rm MeV}$ while PIXIE [@pixie] would be able to constrain up to ${{{{m_{\rm WIMP}}}}}\sim 3f_{\gamma}{\rm GeV}$.
![\[dmfig\] Energy injection from dark matter annihilation for a $10~{\rm GeV}$ WIMP with $f_{\gamma}=1$ (solid line). The exponential suppression at high redshifts is because of the decrease in visibility, $e^{-\mathcal{T}}$. Dashed line shows the energy injection $(1+z)\frac{{{{\rm d}}}{{{{{\mathcal{E}}}}}}}{{{{\rm d}}}z}$ without the visibility factor. ](dmann.eps){width="14cm"}
Silk damping
------------
Sound waves are excited in the primordial baryon-electron-photon plasma by primordial perturbations. They decay on small scales because of shear viscosity, with thermal conduction also becoming important near the time of recombination. This damping of primordial perturbations was first calculated by Silk [@silk] including only thermal conduction. The full calculation, including both shear viscosity and thermal conduction and also including the effects of photon polarization was done by Kaiser [@kaiser]. Silk damping transfers energy from sound waves to the average CMB spectrum, resulting in effective energy injection into CMB [@sz1970b; @daly1991; @hss94; @cs2011; @cks2012]. Microscopically, shear viscosity and thermal conduction arise due to the diffusion of photons which are repeatedly scattered by the electrons. This diffusion of photon results in mixing of blackbodies from different phases of the sound waves on diffusion scales. $2/3$ of the dissipated energy in sound waves just increases the average temperature of CMB while $1/3$ results in the spectral distortions of $\mu$ and $y$ type. Depending on the primordial perturbation power spectrum at these very small scales of comoving wavenumbers $50\lesssim k\lesssim 10^4$, the $\mu$ distortion can be in the range $10^{-8}-10^{-10}$ [@cks2012] for the parameter space allowed in the standard cosmological model [@wmap7]. We refer to [@cks2012] for a detailed discussion, including fitting formulae for spectral distortions from adiabatic initial conditions, and constraints from the future experiments on initial power spectrum spectral index and its running.
Bose-Einstein condensation of CMB
---------------------------------
After the epoch of electron-positron annihilation, electrons and baryons are non-relativistic and cool adiabatically (with adiabatic index $5/3$) as a result of the expansion of the Universe, ${{{T_{\rm e}}}}\propto (1+z)^2$. Radiation (photons) has adiabatic index $4/3$ and cools slower than baryons, $T_{\gamma}\propto (1+z)$ [@zks68]. Comptonization however is very efficient before recombination and efficiently transfer energy from photons to electrons/baryons, keeping them at same temperature as photons. This cooling of CMB [@cs2011], along with thermalization from comptonization, results in Bose-Einstein condensation of CMB [@ksc2011]. The photons thus move from high to low frequencies where they are efficiently destroyed by bremsstrahlung (and at high redshifts also by double Compton scattering). Since the amount of cooling is small, linear theory for small distortions applies. The resulting distortions have the same shape as that caused by heating of CMB in previous examples, but with opposite sign. Thus we have negative $\mu$ and negative $y$ distortions which partially cancel the distortions due to dark matter annihilation and Silk damping. Surprisingly, the $\mu$ (and $y$) distortions have a magnitude which is similar to those from dark matter annihilation and Silk damping. A comparison of $\mu$ distortions from Bose-Einstein condensation as well as all previous examples is presented in Table \[tbl\]. [We also show comparison of $y$-type distortions in Table \[tbl2\]. $y$-type distortions are dominated by the low redshift contributions, during and after reionization, from the intergalactic medium and clusters. Early universe physics is therefore difficult to constrain using the $y$-type distortions.]{}
----------------------------------------------------------------------------------------------
Process $ \mu$
--------------------------------- ------------------------------------------------------------
electron-positron annihilation $ 10^{-178}$
BBN tritium decay $ 2\times 10^{-15}$
BBN $^7{\rm Be}$ decay $ 10^{-16}$
WIMP dark matter annihilation $ 3\times
10^{-9}f_{\gamma}\frac{10{\rm GeV}}{{{{{m_{\rm WIMP}}}}}}$
Silk damping $ 10^{-8} - 10^{-9}$
Adiabatic cooling of matter and
Bose-Einstein condensation $ -2.7\times 10^{-9}$
----------------------------------------------------------------------------------------------
: \[tbl\]Census of energy release and $\mu$ distortions in standard cosmological model. The negative distortion from adiabatic cooling of matter is shown in red.
-----------------------------------------------------------------------------------------------------------------
Process $y$
--------------------------------------------------- -------------------------------------------------------------
WIMP dark matter annihilation $ 6\times
10^{-10}f_{\gamma}\frac{10{\rm GeV}}{{{{{m_{\rm WIMP}}}}}}$
Silk damping $ 10^{-8} - 10^{-9}$
Adiabatic cooling of matter and
Bose-Einstein condensation $ -6\times 10^{-10}$
Reionization $ 10^{-7}$
Mixing of blackbodies: CMB $\ell\ge 2$ multipoles $ 8\times 10^{-10}$
-----------------------------------------------------------------------------------------------------------------
: \[tbl2\]Census of energy release and $y$ distortions in standard cosmological model. We also give the value of $y$-type distortion expected from the mixing of blackbodies when averaging our CMB sky [@cs2004]. The negative distortion from adiabatic cooling of matter is shown in red. $y$ type distortion is clearly dominated by the contributions, during and after reionization, from the intergalactic medium and clusters of galaxies, and the early Universe contributions are difficult to constrain.
Energy released from recombination of plasma
--------------------------------------------
We should also mention that recombination lines also create a distortion of amplitude $\Delta T/T \sim 10^{-8}-10^{-9}$ [see @rcs2008 for a detailed calculation]. The distorted spectrum would heat the electrons, adding a $y$-distortion at the time of HeIII$\rightarrow$HeII recombination of $\sim y(6000)\times 10^{-9}\sim
10^{-12}$. Additionally, the Ly-$\alpha$ and 2s-1s (2-photon decay) photons from recombination with energy $\sim 40$eV, $x\sim 30$, escape as they redshift out of resonance, (Compton) scatter on electrons and heat the plasma through recoil effect. The heating can be estimated using the analytic solution of Kompaneets equation with only the recoil effect [@arons; @is72]. In the limit of small Compton-$y$ parameter ($xy\ll 1$), the fraction of energy lost by photons at frequency $x$ is $\sim y\times
x\sim 1/30$, giving an additional $y$-type distortion of $\sim
(1/4)(40{\rm eV}){{{n_{\rm ^4He}}}}x y/E_{\gamma}\sim 10^{-12}$. [The distortions from HeI and HI recombination are much smaller since they happen later, when $y$ is much smaller, although the energy released is comparable to HeII recombination.]{}
Conclusions
===========
Future experiments, such as PIXIE [@pixie], would be able to constrain/measure spectral distortions in the CMB at high accuracy. There are several sources of spectral distortions possible from standard and new physics. Using the results of experiments like PIXIE to constrain new physics would require precision calculations of evolution of the CMB spectrum, especially around the blackbody surface at $z\sim 2\times 10^6$. So far, precise calculations have only been possible numerically, although analytic solutions with $5-10\%$ precision around the blackbody surface have been available for a long time. We have presented new analytic solutions, which take into account both double Compton scattering (important at high redshifts) and bremsstrahlung (important at low redshifts). We also take into account the non-stationarity of the problem which is important to achieve high precision. The new solutions are presented in Eq. (ignoring non-stationarity) and in Eq. (including the non-stationarity of the problem). Equation gives accuracy of better than $1\%$ in blackbody visibility at all redshifts. We also present examples from the standard $\Lambda CDM$ model of cosmology, [which do not require new physics,]{} illustrating the structure of blackbody surface. In particular, electron-positron annihilation and BBN demonstrate the effectiveness of Compton and double Compton scattering in maintaining equilibrium at high redshifts. We also point out the coincidence/degeneracy among the three significant sources of distortions in standard cosmology, Bose-Einstein condensation of CMB, Silk damping and dark matter annihilation. All of these create distortions which have roughly the same order of magnitude, especially for low dark matter particle masses. This is remarkable considering that the three sources of distortions have completely different physical origins. Bose-Einstein condensation is fixed by standard cosmological parameters, which are now known with high precision. However, the degeneracy between Silk damping and dark matter annihilation must be taken into account when using spectral distortions to constrain the primordial power spectrum or dark matter parameters.
[^1]: This is in addition to the usual $(1+z)$ dependence due to the expansion of the Universe. The variable ${{{{x_{\rm e}}}}}$ is invariant w.r.t. the expansion of the Universe.
[^2]: [The chemical potential is not constant but a function of frequency at low frequencies but this dependence is important only for calculating the photon production rate. We can ignore the frequency dependence in calculating the total energy and number density in the spectrum since the contribution from low frequencies ($x\ll 1$) to these quantities is small and the constant $\mu$ assumption introduces negligible error.]{}
[^3]: We note that any spectrum can be described by a frequency dependent chemical potential.
[^4]: The energy injection term will add a inhomogeneous term to the homogeneous equation for $\mu(x)$. The effect of this term is to change the overall normalization ${{{{\mu_{\rm c}}}}}$ without significantly affecting the shape of the spectrum. This term can therefore be neglected for the purpose of calculating the photon creation/absorption.
[^5]: [Since $\mu({{{{x_{\rm e}}}}})$ is approximately constant around ${{{{x_{\rm e}}}}}=0.5$ (variation in the analytic solution is less than $1\%$ for $0.4<{{{{x_{\rm e}}}}}<1$), the exact value of normalization frequency is not important, and we get similar precision if we choose to normalize at a slightly different frequency around ${{{{x_{\rm e}}}}}=0.5$, for example at ${{{{x_{\rm e}}}}}=0.6$. We should also mention that the numerical solution also shows a tiny decrease ($\sim 1\%$ from ${{{{x_{\rm e}}}}}=1$ to ${{{{x_{\rm e}}}}}=100$) in the chemical potential at ${{{{x_{\rm e}}}}}>1$ because of the increasing efficiency of the recoil effect at high frequencies.]{}
[^6]: [This is really the visibility of distortions. When the visibility is small, the distortions are not visible, and when the visibility is unity, distortions survive and are visible today.]{}
[^7]: We ignore the fact that some of the energy will be lost to neutrinos.
[^8]: We assume self annihilating Majorana particles. For Dirac particles the energy release would be smaller by a factor of 2.
[^9]: In general we expect some energy to be lost to neutrinos and other dark particles and therefore $f_{\gamma}$ would be less than unity.
|
---
abstract: 'We present a study of the spectral properties like the energy spectrum, the eigenmodes and density of states of a classical finite system of two-dimensional (2D) charged particles which are confined by a quadratic potential. Using the method of Newton optimization we obtain the ground state and the metastable states. For a given configuration the eigenvectors and eigenfrequencies for the normal modes are obtained using the Householder diagonalization technique for the dynamical matrix whose elements are the second derivative of the potential energy. For small clusters the lowest excitation corresponds to an intershell rotation. The energy barrier for such rotations is calculated. For large clusters the lowest excitation consists of a vortex/anti-vortex pair. The Lindeman melting criterion is used to calculate the order-disorder transition temperature for intershell rotation and intershell diffusion. The value of the transition temperature at which intershell rotation becomes possible depends very much on the configuration of the cluster, i.e. the distribution of the particles between the different shells. Magic numbers are associated to clusters which are most stable against intershell rotation. The specific heat of the cluster is also calculated using the Monte-Carlo technique which we compare with an analytical calculation where effects due to anharmonicity are incorporated.'
address: |
*Departement Natuurkunde, Universiteit Antwerpen (UIA),\
Universiteitsplein 1, B-2610 Antwerpen, Belgium*
author:
- 'Vitaly A. Schweigert [@*:gnu] and François M. Peeters [@f:gnu]'
title: 'Spectral properties of classical two-dimensional clusters.'
---
cond-mat/9409054
————————————————————————-
Introduction
=============
During the last few years considerable attention has been paid to the study of the properties of mesoscopic systems consisting of a finite number of neutral or charged particles. The particles are confined by an artificial external confining field. Behavior of either ions in a radio-frequency trap (Paul trap) or a Penning trap [@11; @42] and heavy-ion ring storage [@12] can serve as an illustration of three-dimensional (3D) Coulomb clusters. Very large Coulomb clusters have been created recently in strongly coupled rf dusty plasmas [@40]. Examples of two-dimensional (2D) Coulomb clusters are electrons on the surface of liquid He [@13] and electrons in quantum dots [@14]. The vortex clusters in an isotropic superfluid [@15] and in superconducting grains [@ree] have many common features with those of 2D charged particles [@16]. Refs. [@17; @18] have been devoted to the investigation of the ground state of 3D clusters of charged particles. Below we give a short overview of previous theoretical work on 2D clusters of charged particles.
Clusters of particles in 2D with Coulomb repulsion were investigated by Lozovik and co-workers [@2] in the case of parabolic confinement. They found that for low temperature and in the case of a small number of particles the cluster has a shell structure. A two step order-disorder transition was found. With increasing temperature, first intershell rotation starts, and intershell diffusion may be possible at high temperature. When the size of the cluster is sufficiently large, the simple shell structure gradually disappears in the center and features of a Wigner lattice appear. Then cluster melting occurs around the 2D Wigner lattice melting temperature.
Bolton and Rössler [@6] considered the case of parabolic confinement for a small number of particles: $1-40$. They investigated the ground state as well as some metastable states. For clusters consisting of 6 particles they determined the barrier height for transition from the configuration (1,5) (these are the number of electrons in each shell) to the configuration (6).
Systematic and detailed investigation of the structure of 2D clusters was carried out by Bedanov and Peeters [@5]. They considered both parabolic and hard wall confinement. A table of Mendeleev was constructed for clusters with: $2-52$, $82$, $151$, $230$ number of particles. Using the Lindeman melting criterion these authors determined the temperature for the order-disorder transition for radial and angular displacement.
In all of the above works on 2D systems with a finite number of charged particles the Monte-Carlo simulation technique was used. We found that in some cases this method is rather slow in finding the ground state of the cluster. The reason is that the Monte-Carlo technique spends too much time in the vicinity of metastable states such that for a finite simulation time not necessarily the correct ground state is found. This becomes more of a problem for clusters with larger number of particles which have many more metastable states. In Ref.[@5] this drawback was partially avoided by heating up the system and cooling it down repeatedly. In the present work we will present an alternative approach. To find the ground state we choose the Newton method with initial configurations determined randomly. In this way we are able to obtain not only the ground state but also the metastable states. The latter are relevant in the calculation of thermodynamic properties and the barrier height for intershell rotation.
In previous work [@2; @6; @5] the ground state properties and melting temperatures were obtained. Here we will investigate the spectral properties of the system. This paper is organized as follows. In Sec. II we describe the model and introduce the dimensionless units. In Sec. III our numerical technique to obtain the ground and metastable states is outlined and compared to the Monte-Carlo technique. Sec. IV is devoted to the stable configurations and the spectrum of normal modes is determined. The barrier height for intershell rotation is obtained in Sec. V. Intershell rotation is the lowest excitation for small clusters. We correlate the strong dependence of the height of the barrier for intershell rotation to the number of particles placed in various shells. In Sect. VI we discuss large clusters for which we calculate the density of states and discuss their lowest excitation which consists of a vortex/anti-vortex pair. In Sec. VII the zero temperature results for the excitation spectrum are used in order to calculate the melting temperatures using the Lindeman melting criterion. These results are compared with earlier results [@5] which were based on the Monte Carlo simulation technique. As an example of the use of metastable states in the calculation of thermodynamic property we calculate the heat capacity in Sec. VIII. We compare the Monte-Carlo results with an analytical approach in which we include anharmonicity effects in an approximate way. Our conclusions are presented in Sec. IX.
Model system
=============
The model system was defined in Ref. [@5]. But for completeness we recall the main features. Our system is described by the Hamiltonian $$\label{eq1}
H=\frac {q^2}{\epsilon }
\sum_{i>j} \frac {1}{\mid \vec {r}_i-\vec {r}_j\mid}
+\sum_{i}V(\vec {r}_i),$$ where $q$ is the particle charge, $\epsilon$ is the dielectric constant of the medium the particles are moving in and the confinement potential $V(\vec {r})=\frac{1}{2}m\omega_{0}^{2}r^2$ is taken parabolic. Particle motion is described by classical mechanics in the plane $\vec{r} = (x,y)$. To exhibit the scaling of the system we introduce the characteristic scales in the problem: $r_0=(2q^2/m\epsilon \omega _{0}^2)^{1/3}$ for the length, $E_0=(m\omega _{0}^2q^4/2\epsilon^2)^{1/3}$ for the energy, and $T_0=(m\omega _{0}^2q^4/2\epsilon^2)^{1/3}k_B^{-1}$ for the temperature. These scales will be used as our new units and all our results will be given in these units. In so doing the Hamiltonian can be written as $$\label{eq5}
H=\sum_{i>j} \frac {1}{\mid \vec {r}_i-\vec {r}_j\mid}
+\sum_{i}V(\vec {r}_i) ,$$ with $V(\vec {r})=x^2+y^2$. The numerical values for the parameters $\omega _0$, $r_0$, $E_0$, $T_0$ for some typical experimental systems were given in [@5].
In the present paper we will consider only classical systems. Although a classical approach for the description of the behavior of electrons in quantum dots is not applicable, nevertheless it is possible that certain features of the classical system may survive in a quantum system. For example in the quantum study of the transition from a crystal to a liquid in the absence of a magnetic field[@7], we know that the parameter for formation of a Wigner crystal is $r_s=l_{0}/a_0=37\pm5$, where $l_0$ is the mean distance between the particles. If the number of particles is small, the interparticle distance in the case of parabolic confinement is close to $r_0$. Thus for typical parameters for a quantum dot in GaAs with $m=0.067$, $\epsilon =13$, $ \hbar \omega_0
= 1\: meV$ we obtain $r_s=7.8$. Reducing the confinement $\omega_0$ or applying a magnetic field [@102] will give us a possibility to investigate the existence of a Wigner crystal or another ordered state for a finite number of particles. In Ref. [@5] it was found that a classical 2D cluster with a finite number of charged particles can be more or less stable than a 2D crystal for the same parameter $\Gamma=q^2/\epsilon l_0 k_BT$. We expect that a similar quantity will be relevant in the quantum case and therefore it is expected that also a Wigner crystal like state can exist in quantum dots.
Numerical approach
==================
The Monte-Carlo simulation technique [@8] is relatively simple and provides relatively rapid convergence and a reliable estimate of the total energy of the system in cases that a relative small number of Metropolis steps is sufficient. However, the accuracy of this method in calculating the explicit states may be poor in certain cases. We can understand this as follows: for the present system of axial symmetric confinement some configurations have very small frequencies for intershell rotation $\omega_{min}=10^{-3}\div 10^{-4}$ which may lock the simulation in an unstable state. Using the Monte-Carlo method with an acceptable number of steps $10^4\div 10^5$, in order to limit the computer time, we may obtain the energy $E$ up to an error $\delta $, but the error in the coordinates will be proportional to $\delta ^{1/2}/\omega_{min}$ which in such a case can be large.
To circumvent this problem we used a different numerical approach which is mainly based on our experience from which we learned that with different modifications to the gradient method and the method of molecular dynamics using artificial viscosity we were able to obtain more reliable results than with the Monte-Carlo technique. To be more explicit, to find the state with the minimal energy we used the modified Newton technique. Since this method is practically not applied in the present field we will give a short outline. Let us suppose that the coordinates of the particles in a cluster are given by {$r_{\alpha ,i}^n$; $\alpha =x,y$, $i=1,\ldots N$} after $n$-steps in the simulation. Then the potential energy in the vicinity of this configuration can be written in the following quadratic form $$\label{eq6}
H=H(r_{\alpha,i}^n)
-\sum_i \sum_{\alpha} H_{\alpha,i}(r_{\alpha,i}-r_{\alpha,i}^n)
+\frac {1}{2} \sum_{i,j} \sum_{\alpha ,\beta} H_{\alpha \beta, \: ij}
(r_{\alpha,i}-r_{\alpha,i}^n)(r_{\beta,j}-r_{\beta,j}^n) ,$$ where $H_{\alpha,i}=-\partial H/\partial r_{\alpha,i}$ is the force and $H_{\alpha \beta \:,ij}$ is the dynamical matrix $$\label{eq7}
H_{\alpha \beta , \:ij}=
\frac {\partial ^2H}{\partial r_{\alpha,i}\partial r_{\beta,j}} .$$ The next step in our simulation is based on the condition of minimal total energy $$\label {eq8}
\sum_j \sum_{\beta} (\eta \delta_{\alpha \beta ,\: ij}
+H_{\alpha \beta \:,ij})
(r_{\beta,j}-r_{\beta,j}^n)=H_{\alpha,i} ,$$ where $\delta _{\alpha \beta ,\: ij}$ is the unit matrix and the coefficient $\eta $ is added to assure the stability of the algorithm. It is easy to show that the iteration procedure converges if $\eta > -\lambda _{min}$, where $\lambda_{min}$ is the minimal eigenvalue of the dynamical matrix. The system of linear equations (\[eq8\]) is solved using Gaussian elimination. The calculation of the matrix and solving the system of linear equations takes about $N^2$ numerical operations. This is equivalent to a Monte-Carlo step where also about $N^2$ operations are needed to find the energy, but the coefficient in front of $N^2$ is less for the latter. The reason is that to obtain the spectrum of the matrix is more laborious. The usual approach guarantees only convergence in the vicinity of the minimum. Therefore we introduced an empirical dumping coefficient $\eta $. In the first few iterations the value for $\eta $ is set to be large: $\eta =10\div 100$. If in the next step the total energy of the system decreases the dumping coefficient is reduced, while in the opposite case the value $\eta$ is increased. From our experience we know that such an algorithm for choosing the dumping parameter guarantees convergency of the iteration process. Furthermore, near the last steps, the dumping parameter becomes less than the minimal value of the eigenvalue of the dynamical matrix and the rate of convergency becomes square ( $\delta_{n+1}\sim \delta_n^2$). The accuracy of the calculated energy $\delta $ is now only limited by rounding errors. For systems with axial symmetry there exists an eigenvalue with value zero which corresponds to turning the system as a whole around the axis of symmetry. In such a case the second eigenvalue $\lambda_2$ has to be taken as the minimal eigenvalue. We found that in order to obtain the configuration with minimal energy with an accuracy of $\mid H_{\alpha,i}\mid =10^{-9}\div 10^{-10}$ takes about $10 \div 100$ steps, the exact number of steps depends on the number of particles.
After finding the state with the minimal energy we obtain the eigenvalues and eigenvectors of the dynamical matrix (\[eq7\]). The eigenfrequencies of it are the eigenvalues squared. The condition that the minimal eigenvalue is positive guarantees that the obtained configuration is stable. Of course also the present method does not guarantee that all stable and metastable configurations and the configuration with the lowest energy are found. To overcome this difficulty partially we consider a large number (typically $200$) of initial configurations which are generated randomly. From these initial configurations a few stable configurations remain, the number of which increases fast when $N>30\div 40$. Among these stable configurations the state with the lowest energy is taken to be the ground state of the system. The fact, that usually the state with the minimal energy is achieved already after a small number of steps, gives us confidence that this is likely the actual ground state of the cluster. Usually, the radius of convergence of the ground state is sufficiently large. We confirmed that the present approach for $N<80$ leads to the ground state configurations of Ref. [@5] which were obtained using the Monte-Carlo method with about $10^5 \div 10^6$ simulation steps.
The efficiency of the present method is illustrated in Fig.1 where we plot the precision of the energy, which is defined as the difference from the exact energy value, as function of the number of simulation steps for a cluster of $13$ particles. It is apparent that the present technique converges much faster, about an increase with a factor of $200$ is found. Furthemore, we discovered that even if within the Monte-Carlo approach the error in the energy is only of order $10^{-11}$, the obtained cluster configuration was unstable. This was found by calculating the minimal eigenvalue of the matrix which consists of the second derivative of the potential energy with respect to the position coordinates which for the obtained configuration was negative. The present Newton optimization approach did not exhibit such a deficiency. In contrast to the Monte-Carlo approach of Bolton and Rössler [@6] who found more than one stable configuration for the case of $N=13$ particles, the present approach in which $200$ initial configurations were considered, demonstrates that there exist only one stable configuration which is (4,9). But for this configuration the minimal excitation frequency $\omega _{min}\approx 6\cdot 10^{-4}$ is very small which may be the reason for the error in Ref. [@6].
Eigenvalues and eigenvectors
============================
A detailed description of the features of the lattice structure, the interparticle distance scale in the various shells, and the Mendeleev table for the configurations with $N=2-52$, $82$, $151$, $230$ particles was given in Ref. [@5]. Here, we will discuss the excitation spectrum corresponding to the ground state configuration of the system. This spectrum is shown in Fig. 2 as function of the number of particles for $N$ ranging from 2 to 50. The eigenfrequency in this figure is in units of $\omega_o/\sqrt{2}$. Notice that there are three eigenfrequencies which are independent of N: i) for any axial symmetric system the system as a whole can rotate which gives an eigenfrequency $\omega=0$. This is illustrated in Fig. 3 (figure indicated by $k=1$; $k$ counts the eigenvalues in increasing order) where the arrows indicate the direction of movement of the different particles (i.e. the eigenvectors of the excitation) for a cluster with $N=9$; ii) there is a twofold degenerate vibration of the center of mass with frequency $\omega =\sqrt {2}=1.4142$ (see Fig. 3, $k=7$ ); and iii) the third eigenfrequency corresponds to a vibration of the mean square radius $N^{-1}\sum _{i}(x_i^2+y_i^2)$ with frequency $\omega=\sqrt {6}=2.4495$ (see Fig. 3, $k=15$). The value of this breathing mode can easily be obtained analytically.
For clusters of sufficient large size (i.e. $N>8$) a typical feature of its spectrum is the occurrence of a very low eigenfrequency. Because of the scale in Fig. 2 this frequency is not discernable from the $\omega=0$ frequency and therefore we have listed it in Table I as $\omega_{min}$. For a number of clusters the eigenvectors corresponding to these minimal eigenvalues are shown in Figs. 3($k=2$ ), 4 and 5($k=2$ ). For $N=19$ and $N=20$ the central particle does not move for this specific excitation and consequently its displacement vector has length zero and is therefore not visible in Fig. 4. For the clusters with $N=9, 19$ and $20$ particles, the motion with the minimal frequency $\omega_{min}$ corresponds to intershell rotation. The necessary condition for the existence of intershell rotation is the presence of at least two particles on the inner shell in order to conserve total angular momentum. With changing configuration, the minimal eigenfrequency can vary by several orders of magnitude (see Table I). For instance, for $N=19$ with the ground state configuration ($1,6,12$), the minimal eigenfrequency is $\omega _{min}\approx0.67$, and for $N=20$ and configuration ($1,7,12$), $\omega _{min}\approx1.0\times10^{-4}$. In both cases the minimal eigenfrequencies correspond to intershell rotation (Fig. 4). This large change in the size of the minimal eigenfrequency is connected with the shell configuration, and not with the total number of particles. For example, if for $N=19$ we take the metastable configuration ($1,7,11$) whose energy is an amount $1.66\times10^{-2}$ larger than the ground state energy, we obtain $\omega _{min}\approx1.1\times10^{-4}$ which coincides practically with $\omega _{min}$ for the cluster with $20$ particles.
From the data given in Table I we infer the following law: [*a high frequency value for intershell rotation is obtained for configurations such that the number of particles on the outer shell is an integral number times the number of particles on the inner shell*]{}. For example: $N=12$ ($3,9$), $N=15$ ($5,10$), $N=16$ ($1,5,10$), and $N=19$ ($1,6,12$). For clusters with more than two shells (i.e. $N>21$) a large $\omega_{min}$ for intershell rotation is found for ground state configurations in which the number of particles in the different shells are multiples of an integer number. The latter is usually the number of particles in the inner shell. For example: $N=22$ ($2,8,12$), $N=30$ ($5,10,15$), $N=45$ ($3,9,15,18$) and to a lesser extend also $N=34$ ($1,6,12,15$). These cluster numbers can be considered as the [*magic numbers*]{}, because they represent the clusters which are most stable against intershell rotations. In previous work by others on 3D clusters magic numbers were determined on the basis of energy calculations of the cluster configuration. We found [@5] that for 2D clusters no clear steps are found in the cluster energy versus the number of particles in the cluster and therefore the stability argument is more appropriate in the present case. On the other hand, a configuration with small $\omega_{min}$ for intershell rotation is realized when the number of particles in the different shells have no common denominator. For example: $N=13$ ($4,9$) and $N=20$ ($1,7,12$).
The above rules can be understood from the Hamiltonian by analyzing the intershell interactions using cylindrical coordinates. Let us consider the most simple configuration with two shells. From the outset we notice that the occurrence of an particle in the center of the system, for example, for a cluster with $20$ particles, does not disturb the intershell rotation. Therefore, we do not have to consider such a case separately. Let us discuss the rotation between two outer shells. The interparticle distance and the distance between the shells changes only by a few percent when we alter the number of particles and/or the configuration. Therefore, in an initial approximation we can describe each shell by an ideal polygon and thus the interaction Hamiltonian between two shells can be reduced to the form $$\label {eq9}
H=\frac {1}{2} \sum _{i=1}^{N_1}\sum _{j=1}^{N_2}
(R_1^2+R_2^2+2R_1R_2\cos {(i\theta _1-j\theta _2-\theta)})^{-1/2} ,$$ where $R_1$, $R_2$, $\theta_1=2\pi/N_1$, $\theta_2=2\pi/N_2$ are the radii and angles between particles of the first and second shell which have $N_1$ and $N_2$ particles respectively, and $\theta $ is the intershell angular distance. The sum (\[eq9\]) over the two indexes can be reduced to the sum over one index only $$\label {eq10}
H=\frac {N_1N_2}{2K} \sum_{i=1}^{I}
(R_1^2+R_2^2+2R_1R_2\cos {(i\theta _{\star }-\theta )})^{-1/2},$$ where $$\label {eq11}
\theta_{\star}=\frac {2\pi} {I},$$ and $I$ is an integer which is the minimal divider of the number of particles ($N_1$, $N_2$, ...) in the different shells. From expression (\[eq10\]) it follows that the Hamiltonian for intershell interaction is periodic in $\theta$. Moreover the period $\theta _{\star}$, as a rule is less than the angular interparticle distance within a shell. To evaluate the strength of the intershell interaction we deduct from the Hamiltonian the following value $$\label {eq112}
\frac {N_1N_2}{4\pi } \int _{0}^{2\pi} dx
(R_1^2+R_2^2+2R_1R_2\cos { (x-\theta) })^{-1/2} ,$$ which is independent of $\theta$. This result was obtained from Eq. (\[eq10\]) by replacing the summation over $k$ by an integration. We proved that the error we make in doing so is proportional to $\theta_{\star }^2$. Numerical summation of ($\ref{eq10}$) gives even a more weaker dependence of the interaction energy on $N$ for two ideal polygons. When we compare the computed results for the barrier height for intershell rotation with those found from Eq.(9) we found that Eq.(9) gives a good qualitative description but quantitatively the results are not satisfactory. Therefore we may conclude that for small eigenvalues, the exact value of the barrier height is strongly influenced by the [*non-ideality*]{} of the polygons. Indeed in order to obtain Eq.(6) we assumed that the particles were placed at the edges of an ideal polygon. Because intershell rotation is a collective phenomena, one can easily understand that the actual barrier height is less than that given by Eq. (\[eq10\]) due to the deformation of the polygons during the motion. Indeed, during the rotational motion not only the intershell distance changes but also the interparticle distance within a shell is altered. This is illustrated in Figs. 3 and 4. From these figures we notice that the eigenvectors for the particles in the inner shell have practically the same length and are orthogonal to the radius-vector of the particle. For the outer shell the situation is different and the eigenvectors have also components in the radial direction and futhermore, the length of the eigenvectors are different for the different particles. Therefore the vibrations in the radial and axial directions of the outer shell follow the intershell rotational motion of the particles. Only for clusters in which the number of particle on the inner shell is a multiplicative integer factor of those of the outer shell, i.e. when a large intershell rotation frequency is found, are the polygons almost ideal which can be understood from symmetry reasons and from our numericl results. The characteristics and modelling of the intershell rotation will be given in next section.
When we increase the number of particles, we found that for $N=39$ the motion with the minimal eigenfrequency no longer corresponds to intershell rotation, but rather consists of rotation of different individual regions of the cluster. For $N\geq 60$ (see Fig. 4) the rotation of an inner shell is followed by the rotation of individual polygons at the periphery of the cluster. For $N\geq115$ we found that the minimal frequency $\omega_{min}$ no longer corresponds to intershell rotation but corresponds to the excitation of a vortex/anti-vortex pair (see Fig. 5 for N=151). Higher excitations (see Fig. 5 with $k=4$ and $k=6$) may consist of multiples of such pairs. In case of a cluster of $N=151$ particles the $7^{th}$ lowest excitation corresponds now to an intershell rotation. A more detailed discussion on the nature of low energy excitations of large clusters is postponed to Sect. VI.
Barrier for intershell rotation
================================
In the present section we give a more detailed discussion of the lowest non-trivial excitation in case of small clusters, which is the intershell rotation. The barrier for intershell rotation was obtained using the following procedure. Let us assume that after n-steps in our simulation the coordinates of the particles are given by $\{\vec{r}_i^n; i=1,...N\}$. After diagonalizing the dynamical matrix $H_{\alpha \beta \:,ij}$ we obtain the eigenvectors $\vec{A}_i(k)$ and the eigenvalues $\lambda_k
=\omega_k$. The particle coordinates for a next time step is then given by $$\vec{r}_i^{n+1}=\vec{r}_i^n+\sum_{k=2}^{2N}\tau_k\vec{A}_i(k)\ .$$ Denote $\tau=\tau_{k^*}$ as being the lowest frequency for intershell rotation which is taken to be constant and which sets the size of the time step. The values of all other coefficients $\tau_k$ are found from the condition of minimal potential energy. This is done as follows: substitute the above expression in Eq.(3) which gives us the total energy for the next step $$H_{n+1}=H_{n}
+\sum_{i=1}^{N}\sum_{k=2}^{2N} \tau_k
\frac{\partial H}{\partial\vec{r}_i} \cdot \vec{A}_i(k)
+\frac{1}{2} \sum_{k=2}^{2N} \lambda_k\tau_k^2 \ ,$$ from which we readily find the coefficients $$\tau_k=-\lambda_k^{-1} \sum_{i=1}^{N} \frac{\partial
H}{\partial \vec{r}_i}\cdot \vec{A}_i(k)\ .$$
Trajectories of the particles in four different clusters are depicted in Fig. 6. As was mentioned above, intershell rotation takes place in conjuction with radial oscillations. The latter are more noticeable for clusters with high symmetry, which have a relative large frequency and consequently large barrier heights for intershell rotation. In clusters with only two shells, particles in different shells rotate in opposite direction in order to conserve total angular momentum. Such a motion is defined completely by the angle of rotation $\varphi$ of one shell relative to the second. When there are three shells or more it is convenient to introduce the angle of rotation $\varphi $ of the shell with the maximum angular velocity as an independent parameter. Fig. 7 illustrates the dependence of the potential energy on this parameter $\varphi$ for two different clusters with $N=9$ and $N=40$ particles. In general this function is well approximated by the simple relation $$\label {eq15}
U=\frac {U_{\star}}{2}\left[1-\cos{(\frac{2\pi\varphi}
{\varphi_{\star }})}\right] ,$$ where $U_{\star }$, and $\varphi _{\star }$ are the barrier height and the period, respectively. The values for $U_{\star}$ and $\varphi_{\star}$ are given in Table I. The above procedure is not able to give the value of the barrier height for the clusters $N=39,42,47,51$ and $70$. The reason is that for those clusters the minimal eigenvalue does not correspond to intershell rotation. Notice that the cluster with $N=40$ has two minima in the potential energy. One of this minima corresponds to a metastable state. Of course $U_{\star}$ and $\varphi_{\star}$ in Table I are determined by the global minimum and maximum. In a few cases, the maxima in potential energy are sharper than that given by expression (\[eq15\]). The reason is that the energy for clusters with three and more shells are not only a function of the angular position of the shell.
For clusters with two shells, the parameter $\varphi $ characterizes the motion of the inner shell. The angle of rotation of the outer shell relatively to the inner one can be obtained from the condition of zero total momentum $$\label {eq16}
\theta =(1+\frac {N_1R_1^2}{N_2R_2^2})\varphi\ ,$$ where $N_i, R_i$ are the number of particles and the radius of shell $i$, respectively. For clusters with two shells, the value $\varphi _{\star}$ presented in Table I is correctly approximated by the simple analytical formulas (\[eq11\]) and (\[eq16\]). The Hamiltonian for intershell rotation taking into account kinetic energy can be written in the form $$\label {eq17}
H=\frac{1}{2} N_1R_1^2{\dot{\varphi }}^2 (1+\frac {N_1R_1^2}{N_2R_2^2})
+\frac{1}{2}U_{\star}(1-\cos{(\frac{2\pi\varphi}{\varphi_{\star }})}).$$ For clusters with more than two shells, we can only propose phenomenologic generalization to expression (\[eq17\]). Let us label {$\vec {A}_i; i=1,...,N$} the set of eigenvectors corresponding to intershell rotation. Then to first approximation the Hamiltonian for intershell rotation becomes $$\label {eq18}
H=\frac{1}{2} R_{\star}^2{\dot{\varphi}}^2
+\frac{1}{2} U_{\star}(1-cos(\frac{2\pi \varphi}{\varphi _{\star}})),$$ with $$\label {eq59}
R_{\star}^{-1} = \frac{1}{M} \sum_{i=1}^{M} [\vec{r_i}
\times \vec {A}_i]/r_i^2 \ ,$$ where the summation is carried out over the particles of the shell with the maximum angular velocity. The value of the parameter $R_{\star}$ is also given in Table I. Once we have the Hamiltonian it is not difficult to find the connection between the barrier angular value $\varphi_{\star}$ and the characteristic frequency for intershell rotation $$\label {eq19}
U_{\star}=2(\omega_{min}\frac{R_{\star} \varphi_{\star}}{2\pi})^2
=2(\omega_{min} \delta)^2 \ .$$ The parameter $\delta$ has a clear physical meaning: it is the length which a particle travels within a shell when it moves over the angle $\varphi_{\star}$. The approximate expression (\[eq19\]) is shown in Fig. 8 by the solid curve together with the results of our simulation which are given by the symbols. Notice that Eq.(\[eq19\]) describes our numerical results very well over an energy barrier height variation of more than 8 orders of magnitude.
Density of states and vortex excitations
========================================
From Fig. 2 we notice that the maximum frequency in the excitation spectrum, on the average, slowly increases with increasing number of particles. We can easily explain this with the aid of the theory of an infinite system. As it follows from our calculations, and has been mentioned in previous work [@5], the minimal interparticle distance decreases slowly with the growth of the cluster size due to the compression of the inner shell by particles placed at the periphery of the cluster. As a consequence, the maximum value of the wave vector $k\approx \pi/l_0$ ($l_0$ is the mean distance between the particles) and also the wave frequency will increase weakly with the cluster size.
For large clusters, it is more interesting to consider the density of states (DOS) of excitations (phonons) which can be obtained by a summation of the energy levels, displayed in Fig. 2, over a frequency interval which we took $\delta \omega =\omega _{max}/40$, where $\omega _{max}$ is the maximal eigenfrequency. The results for $N=80$ and $N=300$ particles is shown in Fig. 9. A characteristic feature in the DOS for all clusters is the occurrence of two broad maxima. From earlier investigations [@9] of classical infinite 2D systems we know that there are two types of waves in a 2D Wigner crystal: the lateral sound waves with dispersion relation $\omega \sim k$ and the longitudinal plasma wave with $\omega \sim \sqrt{k}$, in the long-wavelength limit. Using an analytical approximation for the frequency of sound $\omega _1 ^2 \approx 0.00363\omega _p^2 k^2l_0^2$ and the plasma frequency $\omega _2^2 \approx \omega _p^2 kl_0(1-0.181483kl_0)$, taken from [@9], it is possible to show that the positions of the broad maxima in Fig. 9 are in qualitative agreement with the ones for an infinite crystal. In our dimensionless units $\omega _p=2\pi/\rho l_0$. To obtain the value of $\omega _p$ we used the average particle density $\rho =N/\pi R_o^2$, where $R_o$ is the radius of the most outer shell. The maximum frequency of plasma like waves $\omega _{2,max}\approx 1.17\omega _p$ for the cluster with $N=80$ equals $4.67$ and for $N=300$ is about $5.77$. Let us assume that the maximum frequency for sound waves is achieved at $kl_0=\pi $. Then for $N=80$ we obtain $\omega _{1,max}\approx 2.38$ and for $N=300$ we find $\omega _{1,max}\approx 2.94$ which are slightly larger than the position of the first maximum appearing in Fig. 9.
From continuum elastic theory, a 2D electron crystal can be considered as incompressible at low frequencies [@10]. In order to check if this is still the case for the present finite system we consider the z-component of the rotor $\Psi_r(k) = \vec{e}_z \cdot
rot \Psi(k)$ and the divergence $\Psi_d(k) = div \Psi(k)$ of the field of eigenvectors of mode $k$
\[eq:all\] $$\Psi_d(k)= \frac{1}{N} \sum_{i=1}^{N} \psi_{d,i}^2(k) ,
\label{eq:a}$$ $$\Psi_r(k)= \frac{1}{N} \sum_{i=1}^{N} \psi_{r,i}^2(k) \ .
\label{eq:b}$$
The values $\psi_{d,i}(k)$, and $\psi_{r,i}(k)$ for the $i^{th}$ particle are given by
\[eq2:all\] $$\psi_{d,i}(k)=\sum_{m=1}^M \left[(\vec{r}_i - \vec{r})\cdot
\vec{A}_i(k) +
(\vec{r}_m - \vec{r})\cdot \vec{A}_m(k) \right]/
\mid \vec{r}_i - \vec{r}_m \mid \ , \\
\label{eq2:a}$$ $$\psi_{r,i}(k)= \sum_{m=1}^M \left[(\vec{r}_i - \vec{r})\times
\vec{A}_i(k) +
(\vec{r}_m - \vec{r})\times \vec{A}_m(k) \right]/
\mid \vec{r}_i - \vec{r}_m \mid \ ,
\label{eq2:b}$$
where $\vec{r}_m$ are the coordinates of the neighbor particles and $\vec{A}_i(k)$ is the eigenvector of particle $i$ for mode $k$. The rotor and divergence of the eigenvector field are shown in Fig. 10 as function of the excitation frequency for clusters of size $N=80$ and $N=300$. Notice that for small values of the frequency the rotor of the field of eigenvectors is larger than the divergence. As a consequence, in a finite system but with $N$ sufficiently large, the system is incompressible and one can expect that the low frequency excitation consists of vortex motion in which the particle density is not disturbed. From our computer calculations we found that for $N=151$ the minimum eigenfrequency corresponds indeed to the formation of a vortex/anti-vortex structure (Fig. 5). Since the total angular moment has to be equal to zero, those vortexes always come into pairs. With higher eigenvalues, the number of vortexes rises, although this function is not necessarily monotonic (see Fig. 5). Thus when $N$ is sufficiently large the cluster of charged particles can be described as a viscious non-compressible fluid in case of small wave vectors. Vortex motion is only expected for sufficiently large $N$ because the velocity of dissipation of the vortex energy is inversely proportional to $R^2$, where $R$ is the characteristic radius, which increases with increasing $N$.
Melting temperatures
=====================
In Ref. [@5] it was shown that the $T=0$ ordered state of the cluster is destroyed with increasing temperature ($T$). The melting temperature for this order-disorder transition was obtained by investigating the radial displacement, the relative intrashell and intershell angular displacements as function of temperature. Here we will start from the excitation spectrum of the zero temperature ordered state in order to calculate the melting temperature using the Lindeman melting criterion [@46]. In the harmonic approximation the mean square displacement is given by the following expression $$\label {eq12}
<u^2>=\frac {T}{N} \sum_{k=2}^{2N} \omega_k^{-2} \ ,$$ from which we find the melting temperature $T$ using the relation $<u^2>=\gamma l_0^2$, where $\gamma=0.10$ for a 2D Wigner crystal [@22; @23], and $l_0$ is the mean interparticle distance. As discussed in previous section, there exists a number of configurations with $\omega _{min}$ very small which will give a very large contribution to the sum (\[eq12\]). In order to see what the effect is of these very low frequency excitations on the melting temperature we also considered the sum without the first term. Then we will find the temperature for intershell diffusion. Because for large clusters, the value of the interparticle distance around the center and near the periphery can be considerably different, therefore we will use the mean value of relative displacement in order to define the melting temperature $$\label {eq13}
T=\gamma N \left [\sum _{i=1}^{N} l_i^{-2}
\sum_{k=2}^{2N} \vec {A}_i^2(k)/ \omega_k^2 \right]^{-1} \ ,$$ where $\vec{A}_i(k)$ is the displacement vector for the $i$-particle in mode $k$, and $l_i$ is the mean interparticle distance for the $i$th particle.
The numerical results are shown in Fig. 11. As we expect there is a significant difference in the transition temperature whether intershell rotation is taken into account or not. These results agree qualitatively with the results of Ref. [@5] where it was found that: i) for clusters with a small number of particles the angular order is destroyed at much lower temperatures than the radial order which agrees with the large difference in melting temperatures shown in Fig. 11; ii) for larger clusters both temperatures are practically equal as is also apparent in Fig. 11. Orientational order and radial order disappear practically at the same temperature for $N>40$; and iii) the melting temperature at which intershell diffusion sets in, is a decreasing function of the number of particles in the cluster up to about $N \sim 20 \div 40$ beyond which it starts to increase, which agrees qualitatively with Fig. 11. The magnitude of the transition temperature found in the present approach is slightly higher than found in the Monte Carlo study of Ref. [@5]. This is a consequence of the present harmonic approximation which has a limited validity near the melting temperature. The melting temperature for intershell rotation (top part of Fig. 11) is strongly influenced by the value of $\omega_{min}$ which is proportional to the rigidity of the cluster agains intershell rotations. In fact it is a measure of the stability of the cluster against intershell rotations. As was mentioned before the value of $\omega_{min}$, and also $U_{\star}$, is determined by the configuration of the cluster. Clusters with a magic number of particles have a large melting temperature for intershell rotation. These fine details were not present in Ref. [@5].
It is known that for an infinite crystal the sum $(\ref {eq13})$ diverges logarithmically in the low-wavelength which is due to the presence of lateral sound waves. Therefore one uses the average square displacement of interparticle distance in Lindeman’s melting criterion. In our case, such criterion gives the relation $$\label {eq14}
T=\gamma N\left[ \sum_{i=1}^{N} l_i^{-2} \sum_{k=2}^{2N} \omega_k^{-2}
\frac{1}{M} \sum_{m=1}^{M} (\vec{A}_i(k)-\vec{A}_m(k))^2 \right]^{-1}$$ where the sum over $m$ runs over the M neighbor particles. The numerical results obtained using Eq.(\[eq14\]) is shown in Fig. 12. These results are very close to those found in Ref. [@5] with the exception that here near $N \sim 150$ a maximum is found while the Monte Carlo results slowly increases towards the $N \rightarrow \infty$ value. We want to emphasize that if the number of particles is not too large, the transition temperature obtained with the second criterion (\[eq14\]) is lower than the one from Eq. (\[eq13\]). This indicates that the particles mainly move towards each other, and only for $N\geq200$, the effect of small wave vectors begin to appear. In the latter case the neighbor particles move with the same velocity and the difference in the value of critical temperatures obtained using the spectrum of the eigen vibrations (Fig. 11) and the Monte-Carlo technique [@5] is very small.
Specific heat
=============
Before we already mentioned, that in order to obtain the ground state we generated many initial configurations and in so doing not only stable states but also metastable states were obtained. Thus, if we also calculate the spectrum of normal modes $\omega_{k}$ for each local minimum we can easily obtain the partition function within the harmonic approximation and consequently all the thermodynamic quantities like the free energy, the specific heat,... Such an approach was followed in Refs. [@19] and [@20] for 3D clusters where also the influence of anharmonicity and saddle points on the partition function was studied. In Refs. [@19; @20] only, the main characteristics of the spectrum of normal modes was used. Here we know the complete spectrum of our finite 2D system and are therefore able to calculate the partition function more correctly.
In the quasiclassical approximation the partition function for a cluster with N particles is given by [@21] $$Z(T)=(2\pi \hbar )^{-2N} \int
d\vec {q}d\vec{p} \exp{(-H(\vec{q},\vec{p})/k_bT)}\ ,$$ where $\vec{q}= (\vec{r}_1,...,\vec{r}_N)$, $\vec{p} = (\vec{p}_1,...,\vec{p}_N)$ are $2N$ dimensional vectors. The partition function can be written as $$\label{eq20}
Z(T)=\sum _{m=1}^{M}\exp {(-U_m/T)}Z_m(T) ,$$ where $Z_m$ is the partition function of the $m$th metastable state whose energy differs with the ground state by an amount $U_m$. The dimensionless units for temperature and specific heat are used here and below. In the vicinity of this $m$th metastable state, the Hamiltonian is quadratic in the normal coordinates. Because the energy barrier for intershell rotation is small, the effect of anharmonicity will already appear at low temperatures. Therefore we will integrate only over a small region of particle motion $\mid q_i\mid \leq \sqrt {2U_m(k)/T\omega_{k,m}}$ which results in $$\label {eq21}
Z_m=g_mZ_{rot}
\prod_{k=2}^{2N} \frac {T} {\hbar \omega_{k,m}}
erf(\sqrt{U_m(k)/T} ),$$ where $Z_{rot}\propto \sqrt {T}$ is the part of the partition function resulting from the rotational degrees of freedom, $g_m=2\pi/\theta_{\star}$ is the degeneracy of the $m$th state, which is determined by the number of particles occupying a shell, $U_m(k)$ is the barrier height for normal mode $k$, and $erf(x)$ is the error function. These parameters are given in Table II for a number of metastable states.
For convenience let us consider only one normal mode. At low temperature $T\ll U_m(k)$ expression (\[eq21\]) results in the usual value for the specific heat for a harmonic oscillator $C=1$. For high temperature $T\ll U_m(k)$ the specific heat equals $1/2$ as for free motion. For the intermediate temperature region $T\sim U_m(k)$ expression (\[eq21\]) gives an interpolation between these two limiting cases. Unfortunately, we know only the value of the barrier height for intershell rotation. For the remaining normal modes, we will use the analogy with the Lindeman criterion to write the phenomenological relation $$U_m(k)=\gamma_u N \frac {\omega_{k,m}^2l_0^2}{2}\ ,$$ where $l_0$ is the mean interparticle distance, and $\gamma _u
=0.2\div 0.3 $. The above expression is then used in the numerical evaluation of the partition function (\[eq21\]) and (\[eq20\]). Below we will mainly deal with the specific heat $$C=\frac {\partial}{\partial T} T^2 \frac {\partial \log
{Z}}{\partial T} \ ,$$ which is shown by the solid curve in Fig. 13.
General features of the behavior of the specific heat as function of $T$ and $N$ can be predicted without detailed information regarding the metastable states and the spectrum of the normal modes. At low temperatures such that $T\ll (U_m(k),\: U_m)$ the specific heat is only determined by the ground state, and the effect of anharmonicity is not essential. Consequently $C=2N-1/2$ as is apparent in Fig. 13.. Usually the barrier for intershell rotation is the smallest energy, which is also smaller than the difference in energy between the ground state and the metastable states. In the temperature range $U_1(k=2) \ll T \ll (U_1(k\ne 2),\: U_{m}) $ the specific heat will be constant and having the value $C=2N-1$. Such a small reduction in $C$ is visible in Fig. 13 near $T \sim 10^{-3}$. With further increase of the temperature, the behavior of the specific heat is determined by the competition of two processes. On the one hand, transitions to metastable states which lead to an increase of the specific heat, and on the other hand, the effect of anharmonicity which will reduce $C$. This interplay will lead to peaks in the specific heat as is apparent in Fig. 13. Note that the position of the peak does not equal the melting temperature.
In the order-disorder transition region the applicability of the above approach is questionable. Therefore, we also calculated the specific heat using the standard Monte-Carlo technique. As the initial state we took the ground state of our system. Then we fix the temperature and execute $10^5$ steps of the Metropolis algorithm to allow the system to achieve equilibrium. Next about $(4\div 10)\cdot 10^6$ steps of the Metropolis algorithm are made in order to reduce the statistical error. The specific heat is then found using the following formula $$C=N+(<E^2>-<E>^2)/T^2\ ,$$ where $E$ is the potential energy for the system with $N$ particles. In Fig. 13 we compare the results from the Monte-Carlo simulation (full dots) with the above results (full curve) which are based on the excitation spectrum of the $T=0$ stable and metastable states. Note that for the small cluster with $N=9$ very good agreement is obtained. For the other two clusters good quantitative agreement is found at low temperature while at intermediate and high temperatures only the qualitative behavior is correctly described. Thus for large clusters the approximate model is not able to give a satisfactory description of the effect of anharmonicity. Nevertheless there is qualitative agreement in the position of the maxima. We have tried to vary the parameter $\gamma _u$ and to change the integration interval for allowed particle motion in Eq.(26) but we were not able to obtain any better agreement.
Conclusion
==========
We have presented the results of a numerical simulation of the ground state and the spectrum of normal modes of classical 2D clusters with quadratic confinement. The barriers for intershell rotation and the specific heat are also obtained. The Lindeman melting criterion in conjunction with the $T=0$ excitation spectrum of the ground state configuration was used to obtain the order-disorder transition temperatures for angular and radial melting.
For systems with axial symmetry, and an intermediate number of particles the normal mode with the lowest frequency corresponds to intershell rotation if there are at least two shells. A low excitation energy for intershell rotation is found for clusters which have a shell configuration such that the number of particles on each shell have no common multiple. If the number of particles in the outer shell is an integer multiple of the number of particles in the inner shell, the cluster will be most stable against intershell rotation which define the clusters with magic numbers. Such clusters also have a large melting temperature for intershell rotation. Distortion of the axial symmetry of the external potential, will lead to a rise in the eigenfrequency and in the barrier height for intershell rotation. For large clusters, i.e. $N>100$, the normal mode with the lowest frequency corresponds to a vortex/anti-vortex excitation.
Acknowledgments
================
We wish to thank our colleague V.M. Bedanov for fruitful discussions. Part of this work is supported by INTAS-93-1495, the Human Capital and Mobility network programme No. ERBCHRXT 930374 and the Belgian National Science Foundation.
[BM]{} Permanent address: Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk 630090, Russia. To whom correspondence should be addressed. Electronic mail: peeters@ nats.uia.ac.be P.E. Toschek, [*New trends in atomic physics*]{}, edited by G. Grynberg and R. Stora, Vol. I (North-Holland, Amsterdam, 1984), p. 383. B.G. Levi, Phys. Today [**41**]{}, 17 (September 1988); G. Birkl, S. Kassner, and H. Walther, Europhys. News [**23**]{}, 143 (1992). A. Rahman and J.P. Schiffer, in [*Condensed Matter Theories*]{} Vol.2, edited by P. Vashishta, R.K. Kalia, and R.F. Bishop, (Plenum, New York, 1987), p. 33. J.H. Chu and Lin I, Phys. Rev. Lett. [**72**]{}, 4009 (1994); H. Thomas, G.E. Morfill, V. Demmel, J. Goree, B. Feuerbacher, and D. Möhlmann, Phys. Rev. Lett. [**73**]{}, 652 (1994). P. Leiderer, W. Ebner, and V.B. Shikin, Surf. Sci. [**113**]{}, 405 (1987). , edited by M.A. Reed and W.P. Kirk (Academic, Boston, 1989) Y. Kondo, J.S. Korhonen, M. Krusius, V.V. Dmitriev, E.V. Thuneberg, and G.E. Volovik, Phys. Rev. Lett. [**68**]{}, 3331 (1992). D. Reefman and H.B. Brom, Physica [**C183**]{}, 212 (1991). G.E. Volovik and U. Parts, Pis’ma Zh. Eksp. Teor. Fiz. [**58**]{}, 826 (1993) \[JETP Lett. [**58**]{}, 774 (1993)\]. R. Rafac, J.P. Schiffer, J.S. Hangst, D.H.E. Dubin, and D.J. Wales, Proc. Natl. Acad. Sci. U.S.A. [**88**]{}, 483 (1991). K. Tsuruta and S. Ichimaru, Phys. Rev. [**A 48**]{}, 1339 (1993). Yu. E. Lozovik and L.M. Pomirchy, Phys. Stat. Sol.(b) [**161**]{}, K11 (1990); Yu. E. Lozovik and V.A. Mandelshtam, Phys. Lett. [**A145**]{}, 269 (1990); Yu.E. Lozovik and V.A. Mandelshtam, Phys. Lett. [**A165**]{}, 469 (1992). F. Bolton and U. Rössler, Superl. and Microstr. [**13**]{}, 139 (1993) . V.M. Bedanov and F.M. Peeters, Phys. Rev. [**B49**]{}, 2667 (1994) . B. Tanatur and D.M. Ceperley, Phys. Rev. [**B39**]{}, 5005 (1989). K. Jauregui, W. Häusler, and B. Kramer, Europhys. Lett. [**24**]{}, 581 (1993). N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.M. Teller, and E. Teller, J. Chem. Phys. [**21**]{}, 1087 (1953). L. Bonsall and A.A. Maradudin, Phys. Rev. [**B15**]{}, 1959 (1977). D.S. Fisher, B.I. Halperin, and R. Morf, Phys. Rev. [**B20**]{}, 4692 (1979). F. Lindeman, Z. Phys. [**11**]{}, 609 (1910). Yu.E. Lozovik and V.M. Fartzdinov, Solid Stat. Commun. [**54**]{}, 725 (1985). V.M. Bedanov, G.V. Gadiyak, and Yu.E. Lozovik, Phys. Lett. [**A109**]{}, 289 (1985). M. Bixon and J. Jortner, J. Chem. Phys. [**91**]{}, 1631 (1989). S.F. Chekmarev and I.N. Umirzakov, Z. Phys. [**D26**]{}, 373 (1993). , L.D. Landau and E.M. Lifshitz (Pergamon Press, New York, 1980), p.91.
[ TABLE I]{}
Table I. Shell configuration ($N_1,N_2,\ldots $) for clusters with N-particles with parabolic confinement. The minimal excitation frequency ($\omega _{min}$ in units of $\omega_o /\sqrt{2}$), the period ($\varphi _{\star }$) in degree units and the barrier height ($U_{\star }$) for intershell rotation are given together with the parameter $R_{\star }$ for the ground state of the cluster.
$N$ $N_1,N_2,\ldots$ $\omega _{min}$ $\varphi _{\star }$ $U_{\star }$ $R_{\star }$
------ ------------------- ---------------------- --------------------- ---------------------- --------------
$9$ $2,7$ $1.268\times10^{-1}$ $24.9$ $8.44\times10^{-5}$ $1.349 $
$10$ $2,8$ $8.910\times10^{-2}$ $43.9$ $1.20\times10^{-4}$ $1.427 $
$11$ $3,8$ $2.451\times10^{-2}$ $14.2$ $2.42\times10^{-6}$ $0.882$
$12$ $3,9$ $5.308\times10^{-1}$ $38.3$ $7.33\times10^{-3}$ $0.912$
$13$ $4,9$ $6.002\times10^{-4}$ $ 9.7$ $1.06\times10^{-9}$ $0.672$
$14$ $4,10$ $4.940\times10^{-2}$ $16.8$ $2.60\times10^{-5}$ $0.644$
$15$ $5,10$ $4.599\times10^{-1}$ $32.8$ $1.11\times10^{-2}$ $0.564$
$16$ $1,5,10$ $4.924\times10^{-1}$ $31.9$ $2.03\times10^{-2}$ $0.449$
$17$ $1,6,10$ $5.416\times10^{-2}$ $10.4$ $3.44\times10^{-5}$ $0.374$
$18$ $1,6,11$ $6.141\times10^{-3}$ $ 4.8$ $1.03\times10^{-7}$ $0.360$
$19$ $1,6,12$ $6.676\times10^{-1}$ $26.6$ $3.14\times10^{-2}$ $0.396$
$20$ $1,7,12$ $1.031\times10^{-4}$ $ 4.0$ $2.01\times10^{-11}$ $0.334$
$21$ $1,7,13$ $3.174\times10^{-3}$ $ 3.5$ $2.18\times10^{-8}$ $0.294$
$22$ $2,8,12$ $2.934\times10^{-1}$ $12.5$ $5.44\times10^{-3}$ $0.257$
$23$ $2,8,13$ $1.287\times10^{-1}$ $11.6$ $4.74\times10^{-4}$ $0.258$
$24$ $3,8,13$ $2.762\times10^{-2}$ $ 2.7$ $5.86\times10^{-6}$ $0.118$
$25$ $3,9,13$ $1.138\times10^{-1}$ $ 7.4$ $2.47\times10^{-4}$ $0.210$
$26$ $3,9,14$ $1.041\times10^{-1}$ $ 7.0$ $1.81\times10^{-4}$ $0.212$
$27$ $4,9,14$ $1.311\times10^{-2}$ $12.0$ $1.04\times10^{-6}$ $0.611$
$28$ $4,10,14$ $5.682\times10^{-2}$ $ 5.6$ $8.31\times10^{-5}$ $0.179$
$29$ $4,10,15$ $3.911\times10^{-2}$ $ 9.5$ $1.58\times10^{-4}$ $0.183$
$30$ $5,10,15$ $2.974\times10^{-1}$ $18.6$ $1.47\times10^{-2}$ $0.172$
$31$ $5,11,15$ $2.351\times10^{-2}$ $ 5.1$ $1.52\times10^{-5}$ $0.167$
$32$ $1,5,11,15$ $2.971\times10^{-2}$ $ 5.1$ $2.57\times10^{-5}$ $0.137$
$33$ $1,6,11,15$ $6.805\times10^{-2}$ $10.0$ $1.15\times10^{-4}$ $0.257$
$34$ $1,6,12,15$ $2.379\times10^{-1}$ $ 8.6$ $3.68\times10^{-3}$ $0.131$
$35$ $1,6,12,16$ $6.585\times10^{-2}$ $ 5.5$ $1.12\times10^{-4}$ $0.134$
$36$ $1,6,12,17$ $8.963\times10^{-3}$ $ 2.6$ $5.04\times10^{-7}$ $0.131$
$37$ $1,7,12,17$ $3.214\times10^{-3}$ $ 5.3$ $4.66\times10^{-8}$ $0.317$
$38$ $1,7,13,17$ $6.134\times10^{-3}$ $ 4.8$ $3.28\times10^{-6}$ $0.113$
$39$ $2,8,12,17$ $2.231\times10^{-1}$ - - $0.091$
$40$ $4,6,13,17$ $1.242\times10^{-1}$ $14.8$ $2.21\times10^{-3}$ $0.118$
$41$ $4,6,14,17$ $1.237\times10^{-1}$ $ 4.8$ $5.90\times10^{-4}$ $0.092$
$42$ $3,8,14,17$ $3.340\times10^{-2}$ - - $0.096$
$43$ $3,9,14,17$ $5.010\times10^{-2}$ $ 7.0$ $2.97\times10^{-4}$ $0.200$
$44$ $3,9,14,18$ $1.552\times10^{-1}$ $ 7.5$ $3.40\times10^{-3}$ $0.092$
$45$ $3,9,15,18$ $1.962\times10^{-1}$ $13.4$ $2.66\times10^{-2}$ $0.094$
$46$ $3,9,15,19$ $8.425\times10^{-1}$ $ 4.3$ $2.40\times10^{-4}$ $0.093$
$47$ $4,10,15,18$ $1.850\times10^{-1}$ - - $0.034$
$48$ $4,10,15,19$ $1.242\times10^{-1}$ $ 9.1$ $2.37\times10^{-3}$ $0.130$
$49$ $4,10,16,19$ $1.511\times10^{-1}$ $ 6.2$ $2.05\times10^{-3}$ $0.079$
$50$ $4,10,16,20$ $7.535\times10^{-2}$ $12.0$ $3.32\times10^{-3}$ $0.088$
$51$ $5,11,16,19$ $7.530\times10^{-2}$ - - $0.189$
$60$ $1,7,13,18,21$ $7.420\times10^{-2}$ $ 7.0$ $7.11\times10^{-4}$ $0.059$
$70$ $6,6,15,20,23$ $1.220\times10^{-2}$ - - $0.097$
$80$ $1,6,12,17,22,22$ $1.840\times10^{-2}$ $ 4.9$ $1.19\times10^{-5}$ $0.117$
[ TABLE II]{}
Table II. Shell configuration ($N_1,N_2,\ldots $) for some metastable states for a number of different clusters. $U_m$ is the energy difference of the metastable configuration with the ground state energy and $W_m=\prod {\omega_{k,m=1}}/\prod {\omega _{k,m}}$ is the relative statistical weight.
$N$ $N_1,N_2,\ldots$ $U_m$ $W_m$
------ ------------------ ---------------------- ---------
$9$ $1,8 $ $5.526\times10^{-2}$ $3.14$
$25$ $3,8,14 $ $5.308\times10^{-2}$ $0.81$
- $4,8,13 $ $1.013\times10^{-1}$ $22.70$
$34$ $1,6,11,16 $ $9.114\times10^{-3}$ $15.91$
- $1,7,11,15 $ $1.003\times10^{-2}$ $79.70$
- $1,5,11,17 $ $1.581\times10^{-2}$ $ 7.22$
- $6,12,16 $ $1.847\times10^{-2}$ $60.60$
|
---
abstract: 'Light thermalised at room temperature in an optically pumped, dye-filled microcavity resembles a model system of non-interacting Bose-Einstein condensation in the presence of dissipation. We have experimentally investigated some of the steady-state properties of this unusual state of light and found features which do not match the available theoretical descriptions. We have seen that the critical pump power for condensation depends on the pump beam geometry, being lower for smaller pump beams. [Far below threshold, both intracavity photon number and thermalised photon cloud size]{} depend on pump beam size, with optimal coupling when pump beam matches the thermalised cloud size. We also note that the critical pump power for condensation depends on the cavity cutoff wavelength and longitudinal mode number, which suggests that [energy-dependent thermalisation and loss mechanisms are important]{}.'
author:
- 'J. Marelic'
- 'R. A. Nyman'
bibliography:
- 'photon\_bec\_refs.bib'
title: '[Experimental Evidence for Inhomogeneous-Pumping and Energy-Dependent Effects in Photon Bose-Einstein Condensation]{}'
---
[The decision to categorise an experimentally observed phenomenon as Bose-Einstein condensation (BEC) goes hand-in-hand with the consensus microscopic description. ]{}For example, the popular definition of BEC by Penrose and Onsager [@Penrose56] of extensive or macroscopic occupancy by identical bosons of a single quantum state was chosen to extend the original idea of Bose and Einstein to interacting particles, implicitly assuming homogeneity, in their case superfluid Helium.
In general, BEC at thermal equilibrium arises [because the Bose-Einstein distribution]{} diverges when the chemical potential is at least equal to the energy of ground state. In dissipative, non-equilibrium condensation of [exciton-polaritons]{} in semiconductors (e.g. [@Kasprzak06; @Deng06; @Balili07]) or of polaritons in organic molecules [@Daskalakis14; @Plumhof14], the system may be effectively homogeneous, so the Penrose and Onsager definition of BEC applies, but thermal equilibrium is not always strongly established. [In these cases, BEC is widely accepted when thermal equilibrium is experimentally demonstrated to be a good description, and a macroscopic population is observed in the lowest energy state, despite the strong interactions.]{}
[Photons thermalised in a dye-filled microcavity probably have the weakest interactions of any system to have exhibited BEC. In this intrinsically inhomogeneous system, thermal equilibrium and macroscopic occupancy of the ground state are the usual criteria for BEC, and both have been observed despite the dissipation [@Klaers10a; @Klaers10b], so BEC is uncontroversially assigned. Interactions are so weak, that questions have been asked about the mechanism by which the condensate forms [@Snoke13]. There has been considerable recent activity developing microscopic models of this physical system, but most of the models, e.g. by Kruchkov [@Kruchkov14], assume that near-thermal-equilibrium conditions hold.]{}
[ Using principals of detailed balance [@Klaers12] and hierarchical maximum entrance [@Sobyanin12; @Sobyanin13], fluctuations of the condensate population about the thermal equilibrium have been predicted and subsequently observed [@Schmitt14]. Likewise, low-energy excitations about the condensate mean-field such as the Bogoliubov dispersion [@Zhang12; @Nyman14] have also been calculated. Phase fluctuations can only be predicted by fully-quantised models including dissipation [@Leeuw13; @Chiocchetta14].]{}
One published model [@Kirton13; @Kirton14private] looks at the limits of the thermalisation process itself, and hence can state when BEC is and is not a good description. When thermalisation is slower than loss, threshold may still be reached, but the macroscopically occupied mode may no longer be the ground state. In other words, BEC breaks down when thermalisation breaks down. Validation of this and all the other models requires new experimental evidence.
We report here our own observations of dye-filled microcavity photon condensation in the steady state. We have seen that the critical pump power varies strongly with the pump beam geometry, in stark contrast to the predictions of a simple, equilibrium model [@Klaers10b; @Kruchkov14]. We demonstrate that even below threshold the model is incorrect: the thermalised cloud size and photon number are also pump-geometry dependent. We also measure critical pump power as ground-state energy and overall cavity length vary, and we explain our observations through energy-dependent losses. These steady-state features should be explained described by any successful model of photon BEC.
We note that very recent experiments have looked at aspects of the time-resolved behaviour of photon thermalisation [@Schmitt15], and the crossover to lasing when thermalisation fails.
[Our experimental method is similar to that of ]{}Klaers *et al* [@Klaers10a; @Klaers10b]. A fluorescent dye, Rhodamine 6G dissolved in Ethylene Glycol at 2 mM concentration, is held by surface tension between two dielectric mirrors placed apart, as shown in [Fig. \[fig:making bec\]]{}(top). One of the two mirrors is spherical with a radius of curvature $R=0.25$ m and the other is planar, cut down to 1 mm diameter. We pump the dye [incoherently using $\lambda_{pump}=532$ nm light]{}, passing through the dielectric mirror at a transmission maximum angle around 37$^\circ$ to the normal. To prevent shelving of the dye in the triplet state, we pulse the pump on for 500 ns at a repetition rate of 500 Hz.
![Schematic diagram of the apparatus (top) and some example data demonstrating BEC of photons in a dye-filled microcavity. Bottom left: spectra at varying intracavity pump powers, showing the saturation of the excited-state population and then condensation into the lowest-energy mode available. Bottom right: an image of a condensate just above threshold, in real colours, albeit with the intensity adjusted for visibility when printed. []{data-label="fig:making bec"}](fig1){width="48.00000%"}
We collect the light leaking through the cavity mirrors, the photoluminescence, using a 50 mm focal length, 25 mm diameter, achromatic doublet as an objective lens in an afocal setup: an image is effectively formed at infinity. The light is split and imaged onto a camera and a commercial spectrometer, whose entrance slit is easily replaceable. The length of the cavity is controlled using a piezoelectric actuator, and stabilised by [reference to a collimated helium-neon laser]{} (at 633 nm) as observed by a secondary camera. Pump and stabilisation wavelengths are separated from photoluminescence using a combination of dichroic, notch and short-pass filters.
We show an example image of BEC of photons in [Fig. \[fig:making bec\]]{}(bottom right). The colours correspond nearly to those observed by the camera. The thermal component is the broad Gaussian cloud around the condensate, which shows up as a bright central spot. In the spectrum, [Fig. \[fig:making bec\]]{}(bottom left), the condensation shows up as an increase in the population of the lowest energy state, at the cavity cutoff. The thermal component is compatible with a room-temperature thermal distribution, although here the 50 [$\mu{\rm m}$]{} spectrometer entrance slit cuts off some of the higher energy components.
We determine the threshold power using the [deviation of the spatial variation of photoluminescence from a Gaussian near the centre]{}. This measure has proven to be precise and robust and is as well performed by eye as by any quantitative measure we have tried, e.g. output power as a function of pump power, or fitting of the spectra.
The simplest theory of thermalised BEC of dye-filled microcavity photons[, as used in Ref. [@Klaers10b],]{} assumes that a number of photons are trapped in the cavity at thermal equilibrium, and that above the critical number a condensate will form. The critical particle number per spin state for equilibrium Bose-Einstein condensation in a symmetric two-dimensional (2D) harmonic oscillator is $N_C = g \frac{\pi^2}{6}\left( \frac{k_B T}{\hbar \Omega}\right)^2$ where $T$ is the temperature, $\Omega = (c/n_L)\sqrt{1/LR}$ is the angular trapping frequency for [photons a cavity of length $L$, filled with a medium of refractive index]{} $n_L$, and $g$ is the spin degeneracy.
The photon number stored in a dye-filled microcavity is equal to the light power absorbed from the pump, $P_{abs} = P_{pump} n_{mol} \sigma_{abs} \frac{\lambda_0}{2n_L}(q-q_0)$ [(assuming that the pump couples well to the cavity)]{}, times the time the light circulates for, [$\tau_{cav} = \frac{F\lambda_0}{c}q$]{}, divided by the [typical energy per photon absorbed, $h c / \lambda_{pump}$]{}. Here, $P_{pump}$ is the pump light power inside the cavity, $n_{mol}$ is the dye-molecule volumetric number density, $\sigma_{abs}$ is the absorption cross-section for light at the pump wavelength. The lowest-energy mode of the cavity at the cutoff wavelength $\lambda_0$ is in the $q^{\rm th}$ longitudinal mode, the parameter $q_0$ indicates how far the light penetrates the surface of the mirrors, and $F$ is the typical number of round trips light will make in the cavity before decay [(i.e. equal to the finesse divided by $\pi$)]{}. [The intracavity photon number is then ]{} In such a cavity, a thermal cloud, below threshold, is expected to have a root-mean-square size of .
The critical pump power is then predicted to be: [ ]{} For a given longitudinal mode, the critical pump power is expected to decrease with increasing cutoff wavelength, albeit only slightly. The critical power also decreases as the longitudinal mode number is increased. [We note that the predictions of this model are independent of the spatial distribution of pump light. It is assumed that all the pump light absorbed is fully thermalised, and a single lifetime is assigned to all cavity modes at all energies.]{}
The pump light is focussed into the cavity through planar mirror. To measure the pump spot size, we lengthen the cavity to around 30 [$\mu{\rm m}$]{}, and observe only the photoluminescence at 625 nm and more, to eliminate the possibility of observing thermalised light. We fit a 2D Gaussian and infer two width parameters (root-mean-square size, which is half the beam waist parameter sometimes used in optics), since our spots are often elliptical or even more distorted. We give uncertainties in spot size which are half the difference between the two width parameters.
![Critical intracavity pump power variation with pump spot size for the 8^th^ longitudinal mode and cavity cutoff wavelength 586 nm. The line is a guide to the eye, proportional to . Error bars are from the differences in inferred size of the two principal axes of a two-dimensional Gaussian fitted to the pump spot images.[]{data-label="fig:Pcrit vs spot size"}](fig2){width="35.00000%"}
The theory we have presented assumes that how photons are pumped into the cavity is irrelevant, so the critical pump power should be independent of pump geometry. In [Fig. \[fig:Pcrit vs spot size\]]{} we present experimental observations of critical pump power as a function of pump spot size which directly contradict the prediction [of our simple model, which gives around 1 mW for these parameters. Clearly the pump is not well coupled to the cavity photons.]{} The data can be fitted well to a power law, $P_C \propto ({\rm spot\, size})^{1.5 \pm 0.1}$, although there is no reason to believe that the power law extends beyond the range of our measurements. The critical pump intensity decreases with increasing pump spot size.
There are two main saturation mechanisms in dye fluorescence, stimulated emission and pumping into a dark state, neither of which can explain our observations. The stimulated-emission saturation intensity for absorption at our pump wavelength is approximately $I_{sat} = h c /{\lambda \sigma_{abs} \tau_{sp}} = 4\times10^9$ W m$^{-2}$, taking [the spontaneous emission lifetime as]{} [@Magde99; @Schaefer]. The highest intensity we find at threshold is [$6\times 10^6$ W m$^{-2}$]{}; stimulated emission is a negligible effect. The rate of non-radiative events, mostly intersystem crossing into the triplet state, is found from the fluorescence quantum yield, $\Phi$, about 95% for rhodamine-6G in polar organic solvents[@Magde02]. The typical timescale for these events to occur with our weak pump intensities is [$\tau_{ST}=\tau_{sp}\frac{I_{sat}}{I(1-\Phi)} > 50~\mu$s]{}, much longer than our pulses, so singlet depopulation during a single pulse is negligible. Recovery from the triplet state is known to be no slower than $400~\mu$s [@Zondervan03], implying that all molecules return to the singlet state in the between pulses.
[Since saturation mechanisms are not responsible, we suppose that spatial redistribution of photons from the pump to the thermalised distribution is not fully efficient. It is worth noting that our smallest pump spots, 15 [$\mu{\rm m}$]{}, are not much larger than the smallest cavity mode, whose typical size is 6 [$\mu{\rm m}$]{}.]{} Where threshold behaviour is seen, independent of the pump geometry, it is always the lowest transverse mode which is macroscopically occupied. Along with the thermal excitations seen in both the images and the spectra, this feature points to BEC being a good description of the system[@Klaers10b; @Kirton13].
Having established that pump spot size affects the critical power, we now look for indications that below threshold the thermalisation behaviour also shows pump geometry dependence. We have observed both the number of thermalised photons and their spatial distribution as pump spot size varies, for pump power well below threshold.
We infer the intracavity photon number $N_{cav}$ by measuring the output light power and accounting for the mirror transmission and cavity round-trip time: .
![Cavity photon number normalised to intracavity CW pump power well below threshold, as pump spot size varies. The thermalised cloud is observed in the 8^th^ longitudinal mode with [cavity cutoff]{} 590 nm. Horizontal error bars are from the differences in inferred size of the two principal axes of a two-dimensional Gaussian fitted to the images. Vertical error bars from variation among three series of measurements spanning a factor 15 in pump power.[]{data-label="fig: Ncav vs spot size"}](fig3){width="35.00000%"}
The transmission $T_M(\lambda)$ is a slowly-varying function of the wavelength. We have calibrated the normal-incidence transmission of our mirrors at two wavelengths (532 and 568 nm), and gathered wavelength variation information using fluorescence spectroscopy for each of our mirrors. When compared to the reflection specification given by the manufacturer (Ultrafast Innovations), there is a shift in the wavelengths, probably due to an unwanted tilt during coating, and an overall scaling.
There are no predictions for how the inhomogeneous pump beam couples to thermalised intracavity photons. Our observations in [Fig. \[fig: Ncav vs spot size\]]{} indicate variations in coupling efficiency up to a factor about 2[, although always much lower efficiency than our simple model, which predicts about 20000 photons per mW of pump. One explanation for this poor pumping efficiency may be the large angle of incidence of the pump, giving pump photons far greater in-plane momenta than thermally accessible.]{} The observations are for continuous-wave (CW) pumping, always less than 20 % of critical power. We have performed experiments over a span of a factor 15 in pump power, and observed no systematic power-dependent effects, ruling out saturation phenomena. The standard deviations over the multiple experiments are incorporated into the data shown in [Fig. \[fig: Ncav vs spot size\]]{} as standard error bars. The optimal coupling between pump and thermalised light is found when the pump spot approximately matches the expected thermal spatial distribution of photons. [It seems that the pump couples to spatially matched modes, and that re-distribution is not fully efficient, supporting the explanation we have for the threshold behaviour, as in [Fig. \[fig:Pcrit vs spot size\]]{}. Larger pump spots couple more weakly, and it is possible that some of the pump light strikes damaged regions of our mirrors. Smaller pump spot sizes couple very much more weakly.]{} Combined with the results of [Fig. \[fig:Pcrit vs spot size\]]{}, we conclude that the threshold photon number decreases rapidly for decreasing pump spot size below the typical thermal cloud size, faster than linearly.
![Thermalised photon cloud size variation with pump spot size for CW pump well below threshold. For comparison, the expected thermal cloud size for room temperature is plotted, alongside a line equal to the pump spot size. Error bars are from the differences in inferred size of the two principal axes of a two-dimensional Gaussian fitted to the images. The thermalised cloud is observed in the 10^th^ longitudinal mode with [cavity cutoff ]{}590 nm.[]{data-label="fig:thermal spot size"}](fig4){width="35.00000%"}
A second surrogate measure of thermalisation, cloud size, gives further information about the coupling between pump and thermalised photons. We measure the thermalised cloud size by fitting a 2D Gaussian to the photoluminescence from the cavity. Uncertainties in the data are from differences between the characteristic sizes fitted in the two dimensions. In [Fig. \[fig:thermal spot size\]]{} we see that for large pump spots the cloud is always somewhere between the pump spot size and the expected thermal size, with the thermalisation being more dominant for larger pump spots. The implication is that [light from ]{}larger pump spots thermalises better than small spots, but that pumping with a spot the same size as the thermal cloud gives optimised coupling.
Thermalisation depends on the scattering rate being faster than the cavity loss rate, both of which are wavelength dependent. Photon scattering from the dye decreases exponentially with increasing wavelength, and our cavity mirrors have maximum reflectivity at about 550 nm. Thermalisation is then less effective at longer wavelengths. We have made below-threshold, CW, cloud-size measurements for [cavity cutoff wavelengths]{} from 575–610 nm at concentrations from 0.02–2 mM. Under those circumstances, the rate of scattering, hence the thermalisation rate, varies by a factor of 20000. For small pump spots we see no systematic variation in cloud size with scattering rate. For large pump spots there is some evidence that higher scattering rates (high concentration, short cutoff wavelength) are associated with cloud sizes that better match the expected thermal cloud size.
Threshold observations, as in [Fig. \[fig:Pcrit vs lam0\]]{}, reveal that for each longitudinal mode number $q$ the critical pump power for BEC shows a minimum as a function of the cavity cutoff wavelength. Cavity cutoff is determined by the peak emission wavelength above threshold, i.e. the BEC wavelength. There are predictions that the critical pump power decreases exponentially with increasing wavelength, either with an offset [@Kruchkov14] or without[@Kirton13], because, close to the molecular resonance (short wavelengths), excitations are attached to dye molecules and it is only the free photons that are involved in the BEC. Going to long wavelengths, when the cavity loss rate becomes comparable to the thermalisation rate, the thermalisation breaks down and the BEC threshold pump power increases[@Kirton14private]. Non-radiative scattering, which occurs about once for every radiative, thermalising scattering events, becomes an important loss mechanism at short wavelengths. Since the scattering rate varies by that factor every 9 nm or so, the critical pump power varies on this scale.
![Critical power variation with cavity cutoff wavelength (the wavelength at which BEC appears), for various longitudinal mode numbers, $q$, indicative of energy-dependent loss mechanisms. The pump spot size was 170 [$\mu{\rm m}$]{}. Power is estimated intracavity pump power. Error bars come from uncertainty in determining threshold and from the observed wavelength jitter during experiments.[]{data-label="fig:Pcrit vs lam0"}](fig5){width="35.00000%"}
At longer cavity lengths, larger $q$ values, the lowest critical pump power generally shifts to longer wavelengths. For longer cavities, the photons meet the mirrors less frequently, and hence the cavity loss rate is lower. In this way the balance between loss mechanisms shifts to longer wavelengths, where scattering events are more infrequent, although the magnitude of the shift is larger than would be expected.
[ In conclusion, we have observed dye-filled microcavity photon BEC, and seen that the macroscopic occupation of the lowest-energy state is a robust phenomenon. We have noted behaviour]{} which was dependent on both the pump beam geometry and the cutoff wavelength of the cavity. The critical pump power increases faster than linearly with pump spot size over the range that we have measured. The efficiency of coupling from pump light to intracavity photon number also increases with spot size, for spots smaller than the typical thermal size, implying that critical photon number increases dramatically. The size of the intracavity photon cloud also depends on pump spot size. [This evidence suggests that the pump beam couples poorly to cavity photons, but better to spatially well-matched modes, and that spatial redistribution of light is not complete.]{} We have also observed that critical pump power depends on ground-state-energy, with an optimum dictated by a balance between loss mechanisms: cavity photon loss and non-radiative photon scattering by the dye. We believe a model of dye-filled microcavity photon BEC that included these effects would be able to fully explain our results, and that the inclusion of these effects would render predictions of phenomena such as coherence and fluctuations more accurate.
We thank Peter Kirton, Jonathan Keeling and Jan Klaers for many informative discussions, and Lydia Zajiczek for experimental assistance. This work was funded by EPRSC fellowship EP/J017027/1.
|
---
abstract: 'Dust growth is often neglected when building models of protoplanetary disks due to its complexity and computational expense. However, it does play a major role in shaping the evolution of protoplanetary dust and planet formation. In this paper, we present a numerical model coupling 2-D hydrodynamic evolution of a protoplanetary disk, including a Jupiter-mass planet, and dust coagulation. This is obtained by including multiple dust fluids in a single grid-based hydrodynamic simulation and solving the Smoluchowski equation for dust coagulation on top of solving for the hydrodynamic evolution. We find that fragmentation of dust aggregates trapped in a pressure bump outside of the planetary gap leads to an enhancement in density of small grains. We compare the results obtained from the full coagulation treatment to the commonly used, fixed dust size approach and to previously applied, less computationally intensive methods for including dust coagulation. We find that the full coagulation results cannot be reproduced using the fixed-size treatment, but some can be mimicked using a relatively simple method for estimating the characteristic dust size in every grid cell.'
author:
- Joanna Drażkowska
- Shengtai Li
- Til Birnstiel
- 'Sebastian M. Stammler'
- Hui Li
bibliography:
- 'paper.bib'
title: |
Including Dust Coagulation in Hydrodynamic Models of Protoplanetary Disks:\
Dust Evolution in the Vicinity of a Jupiter-mass Planet
---
= 10000
Introduction {#sec:intro}
============
Due to the expense of including dust coagulation in already expensive hydrodynamic models of protoplanetary disks, (magneto-)hydrodynamic codes usually adopt either fixed size or fixed Stokes number approach, and the size distribution is taken into account either by stacking results of series of single-sized models or including multiple dust fluids representing different dust sizes in one simulation (but without the possibility to exchange mass between the different size fluids as would happen during coagulation, see, e.g. ).
Dust coagulation is usually studied in azimuthally and vertically averaged setups . The dust component is typically treated as a fluid and the Smoluchowski equation is used to solve for dust coagulation. The alternative is to treat dust as (super-)particles and use the Monte Carlo approach to model collisions . A limited number of hybrid algorithms, which connect the two approaches have been developed as well [@2012ApJ...753..119C; @2018ApJ...864...78K].
The connection between hydrodynamic simulations was previously done by taking azimuthally averaged profile of gas obtained in hydrodynamic simulations and performing the dust coagulation calculation in a post-processing step .
@2018ApJ...863...97T included a simplified prescription for dust growth in hydrodynamic code RoSSBi , where dust is represented by a single fluid but dust size may be different in every cell and is set in a sub-grid algorithm based on the work of , and demonstrated that this approach yields significantly different outcome than fixed-size treatment. implemented a similar method, with two dust populations, where dust growth is limited by barriers as proposed by . However, the method proposed by was developed for azimuthally averaged, smooth disk and was previously not tested in a full disk setup. @2017MNRAS.467.1984G included dust coagulation in a 3-D smoothed-particle hydrodynamics code (SPH), however the particle size distribution is not taken into account in the coagulation calculation (i.e. single-sized growth is modeled inside of every SPH dust super-particle).
In this paper, for the first time, we test different approaches to include dust coagulation in 2-D hydrodynamic simulations of protoplanetary disks against the fully self-consistent dust coagulation prescription included in the hydrodynamic grid code `LA-COMPASS` [@2005ApJ...624.1003L; @2009ApJ...690L..52L; @2014ApJ...788L..41F; @2014ApJ...795L..39F; @2019ApJ...878...39L].
We focus on the problem of dust growth in the vicinity of a Jupiter-mass planet, which interacts with the protoplanetary disk and modifies the evolution of gas and dust. The problem of dust evolution in the neighbourhood of a gap-opening planet is particularly important for further growth of the planet by pebble accretion . Early accretion of Jupiter in the Solar System is thought to impact the subsequent accretion of the other planets . The growing Jupiter is also considered as a barrier for mixing of different reservoirs in the early solar nebula [@2017PNAS..114.6712K; @2018NatAs...2..873A].
This paper is organized as follows. We describe our setup and numerical methods in section \[sect:methods\]. We describe the results in section \[sect:results\]. In section \[sect:discussion\], we discuss the limitations and implications of the results. Finally, we summarize our work in section \[sect:summary\].
Methods {#sect:methods}
=======
[ll]{}\[tbp!\] Gas surface density at 1 AU & 1700 g cm$^{-2}$\
Gas surface density exponent & -1.5\
Dust-to-gas ratio & 0.01\
Temperature at 1 AU & 195 K\
Temperature exponent & -0.5\
Turbulence strength parameter $\alpha$ & 10$^{-3}$\
Planet mass & 1 M$_{\rm J}$\
Planet semi-major axis & 10 AU\
Initial / minimum dust size & $10^{-4}$ cm\
Internal density of dust grains & 1.2 g cm$^{-3}$\
Dust fragmentation threshold & 10 m s$^{-1}$\
We study the evolution of a protoplanetary disk around a solar mass star and follow the Minimum Mass Solar Nebula model (MMSN, ) characterized by the gas surface density profile $$\Sigma_{\rm g,t=0} = 1700 \cdot \left(\frac{r}{1~\textrm{AU}}\right)^{-3/2} \mathrm{g~cm}^{-2},$$ where $r$ is the radial distance to the central star. The initial distribution of dust follows the gas profile with radially constant dust-to-gas ratio of 1%. In models including dust coagulation, the initial size of grains is set to $a_0=1~\mu$m.
The isothermal temperature structure of the disk is set by the sound speed in the gas $c_\mathrm{s}$ and $$\label{eq:cs}
c_{\mathrm s} = 83745.82 \cdot \left(\frac{r}{1~\textrm{AU}}\right)^{-1/4} \mathrm{cm~s}^{-1},$$ which translates into $T = 195~\mathrm{K} \cdot (r/{1~\mathrm{AU}})^{-1/2}$ (we adopt the mean molecular weight of $\mu=2.3$ proton mass). The temperature profile is fixed during the simulation. We assume that the scale-height of the disk is $H_{\mathrm g}=c_{\mathrm s}/\Omega_{\mathrm K}$, where $\Omega_{\mathrm K}$ is Keplerian frequency. With these assumptions, $c_{\mathrm s}/v_{\mathrm{K}} = H_{\mathrm g}/r \propto r^{1/4}$, so the disk is slightly flaring. We assume a gas kinematic viscosity $\nu = \alpha c_{\mathrm s} H_{\mathrm g}$ and we set $\alpha=10^{-3}$. We focus on the region between 4 AU and 34 AU and place a Jupiter mass planet at a fixed, circular orbit with a semi-major axis of 10 AU. With the adopted disk parameters, $H_{\mathrm g}/r=0.05$ at the planet location. The planet gradually opens a gap in the disk, modifying the radial distribution of both gas and dust. Table \[tab:input\] summarizes the input values used in all the models. A very similar setup was used in @2018ApJ...863...97T, however they assumed an inviscid disk.
In this paper, we use four different methods to model the evolution of this system of gas, planet, and dust:
- [**Fixed size**]{}: 2-D hydrodynamic simulations with one dust fluid representing dust of a fixed size.
- [**1-D coagulation**]{}: 1-D dust coagulation simulation using the azimuthally averaged gas evolution obtained from the 2-D hydrodynamic models as an input to simulate multiple 1D dust fluids to resolve dust coagulation and radial transport.
- [**Simple coagulation**]{}: 2-D hydrodynamic simulation with one dust fluid and a sub-grid method that sets the dust size in each cell according to an expected coagulation outcome (similar to the method proposed by @2018ApJ...863...97T).
- [**Full coagulation**]{}: 2-D hydrodynamic simulation with multiple dust fluids representing the full size distribution, and dust coagulation algorithm which redistributes mass between the fluids.
We describe each of these methods in detail below.
2-D models {#sub:2Dmodels}
----------
All the 2-D ($r+\phi$) hydrodynamic models are performed using code which is a part of `LA-COMPASS` (which stays for Los Alamos CoMPutational AStrophysics Suite) and was described by @2005ApJ...624.1003L [@2009ApJ...690L..52L]. The protoplanetary disk is assumed to be geometrically thin so that the hydrodynamical equations can be reduced to two-dimensional Navier-Stokes equations by considering vertically integrated quantities. We adopt a locally isothermal equation of state $$P_{\rm g}(r) = \Sigma_{\rm g} c_{\rm s}^2,$$ where $P_{\rm g}$ is the vertically integrated pressure, $\Sigma_{\rm g}$ is the gas surface density and $c_{\rm s}$ is the sound speed, which only depends on distance $r$ here (see ).
Dust is treated as a pressureless fluid in a bi-fluid model that is governed by conservation laws . The gas and dust equations are coupled together through source terms that model the drag between the two fluids, i.e. including the backreaction from dust to gas. We include both Epstein and Stokes drag regimes. We have implemented a Godunov Riemann solver for dust equations. We also implement dust diffusion due to the turbulence and consistently combine it with the dynamic model of bi-fluid [see @2014ASPC..488...96L; @2014ApJ...795L..39F]. To deal with multiple timescales of coupling different dust species with gas dynamics, we develop an efficient and robust $L$-stable method to solve the coupled gas and dust equations [@2017AIPC.1863X0004L].
The planet motion is governed by Newton’s laws, whose equations are solved with an adaptive high-order Runge-Kutta method. Planet’s gravitational potential is smoothed over 0.7 disk scale-height $H_{\mathrm g}$ at the planet location. The disk self-gravity is not included in the models presented in this paper.
We used linear polar grid with uniform spacing between the cells. The grid resolution is $N_{\mathrm r}\times N_\phi$ = $1024\times1024$. With this resolution, the disk scale-height $H_{\mathrm g}$ is resolved with 17 cells and the Hill radius of the planet is resolved with 23 cells at the planet location.
The computational domain is set from 4 AU to 34 AU. We keep the gas density constant at the inner and outer boundaries. This is justified because no significant viscous evolution is expected for the duration of the simulations ($\sim10^5$ years). For dust, we have open inner boundary to allow outflow. We keep the dust density at the outer boundary constant for the first 1000 planet orbits, which is equivalent to a steady inflow, and then we close the boundary, mimicking a decreasing flux of dust from the outer disk to the planet region. In the initial condition, velocities of gas and dust are set according to their equilibrium values derived by @1986Icar...67..375N. At the inner and outer boundaries, the velocities (both radial and azimuthal) are kept constant during the simulation.
### Fixed size dust
In the default version of the code, dust is treated as a single fluid with a fixed particle size. We run a series of models covering dust sizes between 1 micron ($10^{-4}$ cm) and 10 centimeter.
### Full dust coagulation treatment
The default code has been modified to include multiple dust fluids representing different dust sizes. Collisional evolution of dust is solved using explicit integration of the Smoluchowski equation. We include the Brownian motion, turbulence, differential radial and azimuthal drift, and vertical settling as sources of the collision velocities. The values of radial and azimuthal velocities for each dust species are obtained directly from the hydrodynamic solver and the other three sources are calculated in the same way as in .
When calculating the collision probabilities, we take into account midplane density of dust, which we calculate for each dust species $i$ from the surface density $\Sigma_{\rm d}^i$ used in the 2-D version of `LA-COMPASS`: $$\label{eq:rhod}
\rho_{\rm d}^i = \frac{\Sigma_{\rm d}^i}{\sqrt{2\pi}H_{\rm d}^i},$$ where we assume Gaussian distribution of the grains around the midplane with scale-height following @1995Icar..114..237D $$\label{eq:hd}
H_{\rm d}^i = H_{\rm g} \cdot\sqrt{\frac{{\alpha}}{\alpha+{St^i}}},$$ where $H_{\rm g}$ is the gas scale-height. For small grains, this equation is consistent with the work of @2007Icar..192..588Y. In this approach, we assume that turbulent mixing is fast enough to always keep the vertical structure in the settling-mixing equilibrium. This assumption might break in a low turbulence case, when the interplay between settling and dust growth leads to the so-called sedimentation-driven coagulation .
We assume that grains are compact spheres with internal density of $\rho_{\rm s}=1.2$ g cm$^{-3}$. Collisional outcomes include sticking for collisions with the impact speed below $v_{\rm f}=10$ m s$^{-1}$, fragmentation for collisions speeds above $v_{\rm f}$, and erosion for collisions speeds above $v_{\rm f}$ when the mass ratio of colliding particles is greater than 10. Numerical implementation of the collisional evolution is the same as described in .
The dust size distribution is resolved with 151 dust fluids covering sizes between 1 $\mu$m and 100 cm, which corresponds to 8.4 grid points per mass decade, a typical resolution used in dust coagulation models. In the initial condition, all the dust has a radius of $\SI{1}{\mu m}$.
Due to the computational expense of solving dust coagulation, we call the coagulation solver every 50 time-steps of the hydrodynamic solver. We tested that this sub-stepping routine does not impact the results significantly by running an analogical simulation where coagulation was solved at every time step (but with a shorter duration).
A more detailed description of the code will be given in a corresponding paper by Li et al. (in prep).
### Simple dust coagulation approach {#sect:simple}
We implemented a simple, sub-grid method for dust growth in the `LA-COMPASS` code, which is an updated version of the method adopted by @2018ApJ...863...97T. In this method, dust is treated as a single fluid but its size is not fixed. The dust size is calculated at every time-step and in every cell based on local conditions. In the initial condition, the size in all cells is set to $a_0 = 1~\mu$m. The initial Stokes number of grains is calculated as $$\label{eq:st0}
St_0 = \frac{\pi}{2}\frac{a_0 \rho_{\rm s}}{\Sigma_{\rm g}},$$ as all the micron-sized grains in our computational domain are in the Epstein drag regime. Dust growth is modeled as $$\label{eq:stini}
a_i = a_{i-1} + \dot{a} \cdot \Delta t$$ where $a_{i-1}$ is dust size obtained in the given cell in the previous time-step, $\Delta t$ is the length of the time-step, and the growth speed $\dot{a}$ is calculated as $$\label{eq:dotSt}
\dot{a} = \frac{\rho_{\rm d} \Delta v}{\rho_{\rm s}},$$ where $\rho_{\rm d}$ is the midplane density of dust (see equation \[eq:rhod\]), $\Delta v$ is impact velocity between grains of Stokes number equivalent to $St_{i-1}$ and $0.5\cdot St_{i-1}$ (where $St_{i-1}$ corresponds to the Stokes number of grains with size $a_{i-1}$), and $\rho_{\rm s}$ is the internal density of grains. When calculating the impact speed $\Delta v$, we take into account turbulence, radial drift and azimuthal drift. The impact speeds are calculated from the radial and azimuthal velocities returned by the hydrodynamic solver assuming that the radial speed depends on the Stokes number as $v_{\rm r}\propto{St}$ (correct for ${St}<1$) and the azimuthal speed as $v_{\phi}\propto1/(1+{St}^2)$.
Dust growth can be halted either by fragmentation or radial drift. To take this into account, we calculate the maximum dust size that could grow using the semi-analytic expressions derived by . The maximum Stokes number with turbulence-driven fragmentation is $$\label{eq:stfrag}
{St}_{\rm frag} = {\rm f_f} \cdot \frac{v_{\rm f}^2}{3\alpha c_{\rm s}^2},$$ where the fudge factor ${\rm f_f}=0.37$, and $v_{\rm f}$ is the fragmentation threshold velocity which we set to 10 m s$^{-1}$ in this paper. This equation was derived assuming that the turbulence driven impact velocities scale as $\Delta v_{\rm t}\propto\sqrt{St}$, which applies for grains in so-called fully intermediate regime of . In fact, grains which hit the fragmentation barrier are typically in this regime as shown by . Fragmentation can also be caused by the differential drift and the maximum Stokes number for the drift-induced fragmentation is $$\label{eq:stdf}
{St}_{\rm df} = {\rm f_f} \cdot \frac{v_{\rm f}}{|\eta| v_{\rm K}},$$ where $\eta v_{\rm K}$ is the maximum drift speed calculated using the midplane radial pressure gradient $$\label{eq:eta}
\eta v_{\rm K} = \frac{1}{2 \rho_{\rm g} \Omega_{\rm K}}\cdot\frac{dP_{\rm g,z=0}}{dr},$$ where $\rho_{\rm g}$ is the midplane gas density and $P_{\rm g, z=0}$ is midplane gas pressure.
Fragmentation is the dominant factor in setting dust size when the coagulation timescale is shorter than the drift timescale. However, in a realistic disk, this is not always true. Particularly, in the outer part of the disk, radial drift may be faster than coagulation. This sets a limit on how far the growth can proceed before the grains are removed faster that they can grow. This effect is naturally recovered in the full coagulation models, in which each dust fluid can be advected at its own speed. Although the radial drift is still accurately modeled by the hydrodynamic solver, in the simple coagulation approach the advection of dust does not have a direct effect on the representative size. Therefore we must include the drift limit explicitly in the size calculation. The maximum Stokes number which can remain at given location taking into account radial drift is $$\label{eq:stdrift}
{St}_{\rm drift} = {\rm f_d} \cdot \frac{1}{2|\eta|}\frac{\Sigma_{\rm d}}{\Sigma_{g}},$$ where the fudge factor ${\rm f_d}=0.55$.
The values of the fudge factors ${\rm f_f}$ and ${\rm f_d}$ that we adopted were derived by by comparing simple coagulation results to 1-D coagulation in a framework of a global, smooth disk.
The new Stokes number is decided by choosing the minimum of the values calculated when taking into account growth (${St}_i$, corresponding to the size $a_i$ obtained in equation \[eq:stini\]), and the possible barriers (equations \[eq:stfrag\], \[eq:stdf\], and \[eq:stdrift\]): $$\label{eq:st}
{St} = \min{\left({{St}_i},{St}_{\rm frag},{St}_{\rm df},{St}_{\rm drift} \right)}.$$ We found that, particularly in case when pressure gradient is briefly enhanced by the spiral wakes (see figure \[fig:eta\]), the Stokes number recovered from this treatment can be much lower than the one given by full coagulation results. This is because the radial advection of size and the timescale needed to fragment all particles in a given cell are not taken into account. To minimize this effect, we limit the impact of fragmentation by comparing the Stokes number obtained in equation \[eq:st\] to the Stokes number from the previous time-step ${St}_{i-1}$, and if ${St}<{St}_{i-1}$ we set $$\label{eq:fraglim}
{St} = \min\left(1,{\rm f_n}\right)\cdot {St} + \max\left(0,(1-{\rm f_n})\right)\cdot{St}_{i-1},$$ where the fudge-factor ${\rm f_n}={\Delta t}/{t_{\rm{coag}}}$ is a ratio of the simulation time-step $\Delta t$ and the coagulation timescale calculated as $t_{\rm{coag}}={a}/\dot{a}$ (see equation \[eq:dotSt\]). This way we avoid any sudden, local drops of the Stokes number but let it decrease gradually. Finally we limit the minimum value of the Stokes number to the Stokes number of the smallest, micron-sized grains: $${St} = \max{\left({St},{St_0} \right)}.$$
The main difference between our implementation of the simple coagulation and the algorithm presented by @2018ApJ...863...97T is in the treatment of the initial dust growth phase. We changed the dust growth prescription from an exponential function implemented by @2018ApJ...863...97T to calculating the growth rate based on the local conditions (see equation \[eq:dotSt\]). We have also introduced the limit on how much can the size decrease between two consecutive time steps (see equation \[eq:fraglim\]).
1-D coagulation
---------------
To run the azimuthally averaged models, we used the `DustPy` code. The code, developed by S. M. Stammler and T. Birnstiel, is `Python`-based version of the commonly used dust coagulation code described by . It solves dust coagulation and radial surface density evolution in azimuthally and vertically averaged framework, performing implicit integration of Smoluchowski equation and advection-diffusion equation.
To test the impact of solving dust coagulation in azimuthally averaged framework, and reproduce the approach previously used to study dust coagulation in the presence of gap opening planet, we set up a model where the `DustPy` code uses azimuthally averaged gas evolution obtained in the `LA-COMPASS` simulation as an input. This is done in the following way: the azimuthally averaged output of the 2-D model is stored at every 10 planet orbits. An interpolation routine is used to generate input at time instances needed by the 1-D model. Otherwise, we use the coagulation setup with parameters given in Table \[tab:input\].
Results {#sect:results}
=======
Full coagulation
----------------
![Surface density of gas and dust (left and right panel, respectively) obtained in the full coagulation run after 1000 planet orbits (corresponding to 31622.8 years). The inserts zoom in on the planet region.[]{data-label="fig:full_coag_maps"}](full_coag_maps_zoom.pdf){width="\linewidth"}
![Azimuthally averaged time evolution of the full coagulation run. The snapshots were taken every 1000 planets orbits. [*a)*]{} surface density of gas, [*b)*]{} surface density of dust, [*c)*]{} vertically integrated dust-to-gas ratio, [*d)*]{} density averaged grain size.[]{data-label="fig:full_coag_evo"}](full_coag_evo.pdf){width="0.9\linewidth"}
As expected, the massive planet placed in a viscous disk quickly clears a gap, however some accretion flow through the gap is retained [@2006ApJ...641..526L], which is visible in the Figure \[fig:full\_coag\_maps\]. The initial power-law density profiles are modified not only by planetary gap opening, but also by the planet induced spiral density waves and by dust drift. Dust evolution depends strongly on grain sizes, which influence their aerodynamic interaction with gas.
We run the simulation for 4000 planet orbits (corresponding to $t=1.26\cdot10^5$ years). Figure \[fig:full\_coag\_evo\] presents time evolution of the gas and dust surface density as well as characteristic dust size. As can be seen, the gap profile is practically saturated at the end of the simulation. As dust growth timescale is on the order of 100 local orbits (or, to be more precise, the growth timescale is orbital timescale divided by the dust-to-gas ratio, see, e.g., ), the dust sizes quickly reach a steady-state and therefore they do not change significantly after the gap is fully open.
![Radial pressure gradient parameter $\eta$ at 4000 planet orbits. The grey lines correspond to 20 azimuthal disk sectors. The red line is calculated based on azimuthally averaged gas density. The negative values of $\eta$ correspond to inward and positive values to outward dust drift (see the red arrows). The dotted vertical line marks position of the planet.[]{data-label="fig:eta"}](eta_drift_2D.pdf){width="\linewidth"}
Interaction between the planet and the disk causes formation of a pressure bump at the outer edge of the planet gap. Figure \[fig:eta\] shows the $\eta$ parameter (see equation \[eq:eta\]), which defines the maximum possible drift speed of dust and its direction. The red line corresponds to azimuthally averaged value of $\eta$, which determines the overall evolution of dust. Negative values of $\eta$ translate into inward drift and positive values mean outward drift of dust. As $\eta>0$ between 11 AU and 15 AU, the inward drift is reversed and thus we can expect that the radial dust drift is halted around 15 AU. Indeed, as visible in the middle panel of figure \[fig:full\_coag\_evo\], dust is radially concentrated around 15 AU. However, trapping is not 100% efficient and the inner region of the domain is not completely cleared, but some population of grains is retained throughout the simulation. This is because the effect of trapping is compromised by viscosity . Since the radial drift speed in the pressure bump is directed towards $\eta=0$ and it increases with size, there is a critical size for which the particle will always drift back to the pressure bump after being displaced by random turbulent movements. Thus, small grains are expected to pass through the gap, while large grains are expected to stay outside of the gap. Gas flows quickly through the gap and small grains, which are well-coupled, are carried along [@2006MNRAS.373.1619R; @2012ApJ...755....6Z]. This is confirmed in the lower panel of figure \[fig:full\_coag\_evo\], where the density averaged size at each location is plotted. The typical size of grains in the gap is much smaller than outside of the gap. We will discuss this effect in more details in the subsequent section, where we focus on comparing the full coagulation run to models employing the fixed size approach.
It is worth noting that the radial profiles of the $\eta$ parameter plotted for different disk sectors (the grey lines in figure \[fig:eta\]) display significant variations, driven mostly by the spiral wakes. These wakes sweep the disk, causing temporary, small-scale pressure bumps. However, in the full coagulation approach, these do not seem to modify dust evolution considerably.
Fixed dust size versus full coagulation
---------------------------------------
{width="0.75\linewidth"}
![[*Upper panel:*]{} Azimuthally averaged gas (dashed blue line) and dust (solid lines) surface density profiles obtained in runs with fixed dust size after 1000 planet orbits. The red dashed line is the total dust density assuming the MRN size distribution. [*Lower panel:*]{} Split of the total dust density obtained in the full coagulation run (red dashed line) into contributions from different size bins (solid lines). The blue dashed line shows surface density of gas. The dotted vertical line marks the position of the planet.[]{data-label="fig:filtering"}](filtering_split_MRN.pdf){width="0.9\linewidth"}
Figure \[fig:2D\_maps\_coag\] compares dust distribution in the protoplanetary disk obtained in the series of models assuming fixed dust size and in the model with full coagulation. In agreement with previously published results , we find that large grains are trapped outside of the gap opened by the planet and cannot pass it. The larger the grains, the larger and deeper gap they open in dust density distribution. On the other hand, small grains that are coupled to the gas, pass through the gap. The critical size of grains that can pass trough the gap is about 1 millimeter in our setup. The millimeter sized grains open a clear gap but at the same time do not form a distinct peak outside of the planetary gap, which is characteristic for simulations including larger dust sizes.
Figure \[fig:filtering\] compares the azimuthally averaged dust density profiles obtained in the series of fixed-size models (upper panel) to the results of the full coagulation model (lower panel). We find that the results obtained when applying the full dust coagulation treatment cannot be adequately fitted using a single fixed-size model. Dust distribution resulting from the interplay between multi-size dust advection and coagulation shows both confined peak outside of the planetary gap, characteristic for centimeter-sized and larger grains, and the partially filled planetary gap, characteristic for sub-millimeter grains. Overall, the slope of dust density through the outer edge of the planet gap is much shallower in the full coagulation model than in most of the fixed-size models.
This can be understood when considering what are the contributions to the total dust density from grains of different sizes (see the lower panel of figure \[fig:filtering\]). While the maximum dust sizes that can be obtained in the pressure bump outside of the planet orbit are on the order of few centimeters, the gap is filled exclusively with grains smaller than 300 micron.
![Comparison of the MRN size distribution with maximum size of 10 cm (dashed line) and the global dust size distribution obtained in the full coagulation model at different times (integrated over the whole disk, solid lines).[]{data-label="fig:sizedistr_MRN"}](sizedistr_MRN.pdf){width="0.9\linewidth"}
In models that do not solve dust coagulation but need input on size distribution (mostly for comparison to observations), it is often assumed that the dust size distribution follows the so called MRN distribution with $n(a)\propto a^{-3.5}$ [@1977ApJ...217..425M]. In the upper panel of figure \[fig:filtering\], we summed up contributions from each single-size model assuming the MRN distribution. The resulting total density profile (red dashed line) is relatively similar to the density profile obtained in the full coagulation model presented in the bottom panel. In figure \[fig:sizedistr\_MRN\] we compare the global size distribution obtained in the full models to the assumed MRN profile. At 1000 orbits of the planet, it does indeed match the power-law distribution reasonably well, although the slope of the distribution is generally shallower and there is significantly less of the largest grains. The MRN profile may be a reasonable assumption for the overall size distribution, although a good estimate of the maximum dust size is necessary, as most of the mass is contained in the largest grains. The size distribution is significantly different at the beginning of the simulation, when the grains have not reached their maximum sizes yet. Also, the maximum size of grains decreases toward the end of the simulation, and falls from 10 cm to about 3 cm at 4000 planet orbits. This is caused by the decreasing dust influx at the outer boundary (see Section \[sub:2Dmodels\]). With a lower dust-to-gas ratio, the drift barrier affects smaller grains . Thus, the MRN size distribution with a fixed maximum dust size may not be very useful to model disk evolution over a long period of time.
![Dust size distributions obtained at 10 AU ([*lower panel*]{}) and at 16 AU ([*upper panel*]{}) in the full coagulation model. The red line corresponds to azimuthally averaged profile while the gray lines represent sample distributions across 20 homogeneously distributed angles.[]{data-label="fig:sizedistr"}](sizedistr.pdf){width="0.9\linewidth"}
A significant difference between the full coagulation and fixed-size models is that in the full models fragmentation constantly replenishes small grains that can pass through the planetary gap. The impact of fragmentation is noticeable when comparing the upper and lower panels of figure \[fig:filtering\]. In the full coagulation simulation (lower panel), the small grains follow the density profile of larger grains outside of the planet orbit, different than expected from the fixed-size simulations (upper panel). This is because the small grains are constantly produced in collisions between larger aggregates. Our results suggest that, if fragmentation is efficient, density of small aggregates should be enhanced in dust traps, despite they are not trapped themselves. Indeed, a look into the size distribution of grains presented in figure \[fig:sizedistr\] reveals that outside of the planet orbit, the size distribution is close to coagulation-fragmentation equilibrium .
Another effect that distinguishes the full-coagulation simulation from the fixed-size models is the growth of small particles after they passed the gap. Dust density in the gap is too low to allow for efficient coagulation. But in the inner region of the simulation domain, where dust density increases, grains larger than 3 centimeters are present, which would not be expected from the fixed-size models.
It is worth noting that the dust distribution obtained outside of the planet gap (the upper panel of figure \[fig:sizedistr\]) is remarkably symmetric, with little deviation when considering different azimuthal angles. This is not true in the planet co-orbital region (lower panel of figure \[fig:sizedistr\]). Some of the small grains that pass through the gap are trapped either in direct vicinity of the planet (we do not consider accretion onto the planet) or in the Lagrange points (this is visible in figure \[fig:full\_coag\_maps\]). Due to this asymmetric nature of the co-orbital region, the size distributions sampled around the planet orbit exhibit different profiles.
Since the density and size distribution profiles are generally symmetric (outside of the planetary gap region) and the size distribution generally follows the expected coagulation-fragmentation equilibrium profile, we can expect that it would be possible to recover results of the full coagulation model using less computationally expensive methods. In the subsequent section, we compare results of three methods of including dust coagulation.
Comparison of different treatments for dust coagulation
-------------------------------------------------------
We aimed to reproduce the results of the full coagulation run using two less computationally intensive methods. The first method relies on semi-analytical estimate of the coagulation outcome and is based on the work of . The second method relies on extracting the gas evolution from the full, 2-D simulation and running 1-D, azimuthally averaged dust evolution model in a post-processing step. The estimated computational expense of the three methods, per 1000 planet orbits, is as follows:
- [Full coagulation:]{} 27 684 CPU hours (12 hours on 2304 CPUs).
- [Simple coagulation:]{} 192 CPU hours (20 minutes on 576 CPUs, very similar to an analogical fixed-size simulation).
- [1-D coagulation:]{} 78 CPU hours.
![[*Upper panel:*]{} comparison of azimuthally averaged dust surface density profiles obtained in runs with full coagulation, simple coagulation, and 1-D coagulation after 4000 planet orbits. [*Lower panel:*]{} azimuthally averaged profiles of dust size obtained in the three simulations.[]{data-label="fig:coag_lineplots"}](400compare_coag_lineplots.pdf){width="0.9\linewidth"}
Figure \[fig:coag\_lineplots\] compares azimuthally averaged dust densities and dust sizes obtained using the three methods. As can be inferred from this plot, the simple coagulation method is generally better in reproducing the full results than the azimuthally averaged approach.
The main problem of 1-D coagulation is that azimuthal averaging of gas density “kills” the effects of spiral wakes (see figure \[fig:eta\]): they induce additional impact speeds, limiting the maximum size possible to obtain, but they also induce extra mixing. The 1-D coagulation predicts almost one order of magnitude higher peak density in the trap outside of the planet orbit. The peak is also narrower than in full coagulation results. This effect has multiple reasons: first of all, due to averaging out of the spiral wakes, the 1-D coagulation predicts larger particles, that are trapped more efficiently. On top of that, in the full coagulation results, the exact position of the trap may change with the azimuthal angle because of the asymmetric nature of planet-disk interaction. Thus the trap is radially “smeared” in the full coagulation results. Additionally, as we mentioned before, the spiral wakes induce additional mixing which is not taken into account in the 1-D model. The full coagulation model includes the effect of backreaction of dust to gas, which additionally increase the width of dust ring (although this is not a significant contribution in our setup, see section \[sect:br\]). It is worth noting that similar results of widening the peak in 2-D versus 1-D models was found by @2018ApJ...854..153W for fixed size grains.
The negligence of the 2-D effects of planet-disk interaction has the most significant outcome in the planetary gap region. The 1-D coagulation predicts significantly more dust inside of the gap than the full and simple coagulation models. In 2-D simulations, dust can only flow through the gap if it enters the streamline around the plane (see figure \[fig:full\_coag\_maps\]). The 1-D model cannot take this subtlety into account. In our case, using the azimuthally averaged gas density information to calculate pressure gradient and drift speed leads to an increased dust flux through the gap. This is an opposite conclusion than presented by @2018ApJ...854..153W who also compared a 2-D and 1-D approach to dust dynamics in the vicinity of Jupiter-mass planet, using a fixed-size model. However, they did not extract the density information from 2-D simulation but run a self-consistent 1-D model with planet, which led to a different density profile in 1-D and 2-D runs.
The simple coagulation results do not reproduce the full coagulation perfectly either. The main problem of the simple model is that the size is calculated locally, without an input from neighbouring cells, thus the effect of dust mixing is not taken into account in all aspects. As shown by , the simple method reproduces results of full coagulation very well in the case a smooth, axisymmetric disk. We find that including a massive planet that induces spiral wakes, locally enhancing the pressure gradient and thus impact speeds, leads to violent fragmentation events. In the current simple coagulation approach, we only track one “representative” particle size per cell. If this size suddenly drops due to the spiral wake induced fragmentation, dust will take relatively long time to re-grow at this position. In the full coagulation method, this effect is significantly reduced as the fluids representing different sizes mix, leading to a similar size distribution in neighbouring cells. This is why we had to limit the effect of fragmentation in the simple coagulation model (see section \[sect:simple\]).
Despite these difficulties, the simple model reproduces the full coagulation results reasonably well. Outside of the region where the spiral wakes have the strongest effect ($\sim$10 AU to 25 AU), the dust size calculated in the sub-grid method fits the density averaged size obtained in the full coagulation model almost perfectly (see the lower panel of figure \[fig:coag\_lineplots\]). It is worth noting that the implementation of the simple coagulation practically does not increase the computational cost of the 2-D hydrodynamic model, so this calculation is as fast as a fixed-size run.
Discussion {#sect:discussion}
==========
Limitations
-----------
We presented results of computational models utilizing state-of-the-art methods for modeling dust evolution in protoplanetary disk. However, our models are not free from limitations, which we discuss in this section.
We performed 2-D models, solving for radial and azimuthal structure of the protoplanetary disk, assuming that the vertical density distribution is Gaussian, and depends on dust size in a simple way (see equations \[eq:rhod\] and \[eq:hd\]). Thus we neglect potential effect that sedimentation-driven coagulation could have on dust growth . It is known that in some cases, the 3-D effects may change the conclusions of 2-D hydrodynamic models [see, e.g., @2018RNAAS...2d.195L].
We adopted a simple, isothermal protoplanetary disk model with a fixed temperature structure. Thus we do not take into account the effects of planet heating the protoplanetary disk [@2015MNRAS.453.1768P; @2017ApJ...842..103S], which could potentially change the outcome of dust coagulation as the collisional speeds are dependent on the sound speed (see equation \[eq:stfrag\]).
Our computational domain covers a patch of the protoplanetary disk ranging from 4 AU and 34 AU. While this domain allows us to cover most of the physics connected to dust drift, it is still a relatively small fraction of the global disk, which could extend to several hundreds AU. One of the problems associated with not including the outer parts of the protoplanetary disk directly is that, without a proper boundary condition, we would quickly “run out” of dust. In the models presented in this paper, we adopted an outer boundary condition which allows inflow of gas and dust, thus preventing the density at the outer edge from dropping significantly. However, particularly for a long runtime of the simulation, this condition cannot adequately account for an evolving pebble flux that is expected from global disk simulations .
In the full coagulation and 1-D coagulation models, we adopted a relatively simple collision model with only two possible collision outcomes: sticking and fragmentation. We have assumed a single fragmentation threshold value in the whole domain ($v_{\rm f}=10$ m s$^{-1}$). While more complex collision models can be developed based on results of laboratory experiments [see, e.g., @2018SSRv..214...52B], these are much harder to implement in the Smoluchowski equation solver . We have also neglected the evolution of porosity of dust aggregates, which can potentially lead to a different coagulation pattern .
Dust filtering and pebble isolation mass
----------------------------------------
Despite these limitations, our results may have implications for the theory of pebble isolation mass. This concepts assumes that delivery of solids to a growing gap-opening planets is halted if grains are large enough to be trapped in the pressure maximum outside of the planet orbit . However, our results suggest that those large grains will fragment and constantly replenish the population of small grains, which are able to pass through the gap and potentially re-grow in the planet co-orbital region (although the resolution of our models does not allow us to resolve the potential circumplanetary disk, in which the growth would be most efficient, see @2017ApJ...846...81S [@2018ApJ...866..142D]). Dust coagulation and fragmentation could thus increase the pebble isolation mass.
Similarly, our results cast doubt on the efficiency of dust filtration by growing Jupiter which is postulated to explain some features of the Solar System. Efficient isolation of different reservoirs by a gap-opening planet, as postulated by, e.g., @2017PNAS..114.6712K [@2018NatAs...2..873A; @2019arXiv190312274H] would only be possible if particles do not fragment during collisions (but at the same time are large enough to undergo efficient trapping). Because of the efficient fragmentation outside of the planet, small grains passing the gap around growing Jupiter could still transport water into the inner Solar System, in contrast to the idea proposed by @2016Icar..267..368M, where the proto-Jupiter blocks the delivery of water when it reaches mass of about 20 M$_{\oplus}$.
Importantce of backreaction {#sect:br}
---------------------------
![Azimuthally averaged surface densities of gas and dust obtained after 4000 planet orbits in the run with (solid lines) and without (dashed lines) the effect of backreaction of dust on gas included.[]{data-label="fig:br"}](br_nobr.pdf){width="0.9\linewidth"}
The 2-D hydrodynamic models include the effect of backreaction of dust on gas. We tested the importance of this effect by running a setup analogical to the full coagulation run but with backreaction switched off. The comparison of these two runs is presented in figure \[fig:br\]. The gas density is not modified significantly, but the effect of including backreaction is visible in the dust distribution. The dust ring formed outside of the planetary gap is placed a little bit further away and it is slightly wider in the run including backreaction. This is because the large grains in overdense region push on the gas, leading to a slight modification of the pressure gradient. The outward drift of dust in the pressure bump is sped up, leading to the wider ring profile. This effect was also observed by @2018ApJ...868...48K. @2015MNRAS.454L..36G suggested that backreaction may lead to formation of a second pressure maximum and, in consequence, second dust ring caused by a single planet. We do not observe such an effect in our results, but this may be due to difference in setup: our planet is significantly less massive than 5 M$_{\rm J}$ implemented by @2015MNRAS.454L..36G.
The limited effect of backreaction we observe is a consequence of assuming a viscous disk with $\alpha=10^{-3}$. Viscosity prevents the dust-to-gas ratio from becoming high: in the full coagulation model, the maximum vertically integrated dust-to-gas ratio stays below 10% (see figure \[fig:full\_coag\_evo\]). In disks with lower viscosity, planet-disk interactions lead to development of a vortex outside of the planet orbit [@2009ApJ...690L..52L; @2014ApJ...788L..41F; @2017MNRAS.466.3533H]. The vortices are able to significantly concentrate dust and the effects of backreaction are more pronounced, including destruction of the vortex [@2014ApJ...795L..39F; @2015ApJ...804...35R; @2015MNRAS.450.4285C; @2016ApJ...831...82S], although this effect is mitigated in 3-D models [@2018RNAAS...2d.195L]. We plan to study effects of dust coagulation inside of a vortex in a next paper.
Summary {#sect:summary}
=======
Dust coagulation is the first step toward forming planetesimals and planets. In this paper, we presented results of coupling dust coagulation to hydrodynamics in simulation of a protoplanetary disk including a massive planet. We compared our model to the usual, fixed-size approach and showed that the results differ considerably. We have also compared the full coagulation results to previously used, azimuthally averaged approach and to a simple, sub-grid growth prescription. The main findings of this work may be summarized in the following points:
- Stacking fixed size simulations cannot reproduce the full coagulation results as it does not take into account the exchange of mass between dust populations of different sizes. Particularly, the fragmentation of large grains leads to enhanced density of small grains in the trap region, while the fixed size simulation does not predict trapping of small grains.
- Fragmentation of large grains limits the effect of trapping and increases the permeability of planet-induced gap.
- None of the cheaper methods of solving dust coagulation that we tested is able to recover the full coagulation result perfectly. However, both methods give a reasonable estimate of dust sizes. Any of the two methods is better in reproducing the dust density evolution than a single fixed-size approach.
We thank the referee for their valuable comments. J.D., T.B., and S.M.S. acknowledge funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No. 714769 and the support from the DFG Research Unit “Transition Disks” (FOR 2634/1, ER 685/8-1). S.L. and H.L. gratefully acknowledge the support by LANL/CSES and NASA/ATP. Part of this work was performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. This research used resources provided by the Los Alamos National Laboratory Institutional Computing Program, which is supported by the U.S. Department of Energy National Nuclear Security Administration under Contract No. 89233218CNA000001.
|
---
abstract: 'Using the Optimal Filter Technique applied to Sloan Digital Sky Survey photometry, we have found extended tails stretching about $1^\circ$ (or several tens of half-light radii) from either side of the ultra-faint globular cluster Palomar 1. The tails contain roughly as many stars as does the cluster itself. Using deeper Hubble Space Telescope data, we see that the isophotes twist in a chacteristic S-shape on moving outwards from the cluster centre to the tails. We argue that the main mechanism forming the tails may be relaxation driven evaporation and that Pal 1 may have been accreted from a now disrupted dwarf galaxy $\sim 500$ Myr ago.'
author:
- |
M.Niederste-Ostholt$^1$[^1], V. Belokurov$^1$, N.W. Evans$^1$, S. Koposov$^{1,2}$, M. Gieles$^1$, M.J. Irwin$^1$\
$^1$ Institute of Astronomy, Madingley Rd, Cambridge, CB3 0HA\
$^2$ Sternberg Astronomical Institute, Universitetskiy pr. 13, 119992 Moscow, Russia
date: May 2010
title: 'The Tidal Tails of the Ultra-Faint Globular Cluster Palomar 1'
---
-.6in
\[firstpage\]
globular clusters: tidal disruption – globular clusters: individual (Palomar 1)
Introduction
============
Fig. \[fig:pal1\_image\] shows a Sloan Digital Sky Survey (SDSS) image of the sparsely populated, young halo globular cluster Palomar 1. It was originally discovered by @Ab55, and lies $3.7$ kpc above the Galactic disk and $17.3$ kpc from the Galactic Centre [@Ro98a; @Ro98b; @Sa07]. Its size and low luminosity are very similar to recently discovered Milky Way globular clusters, such as Segue 3, Koposov 1 and 2 [@Be10; @Ko07], as well as Whiting 1, AM 4 and E 3 [@Wh02; @Ca05; @vdB80; @Ca09]. The Sloan Digital Sky Survey color-magnitude diagram (CMD) of Pal 1 shows a red clump, a main sequence turn-off and a well defined main-sequence down approximately two magnitudes from the turn-off. The giant branch on the other hand is very sparsely populated [@BS95; @Sa07].
Owing to its unusually flat mass function, @Ro98a suggested that Pal 1 has either experienced strong dynamical evolution (either tidal shocks or evaporation) or that its initial mass function is significantly different from other halo clusters. @vdB00 notes that Pal 1 is part of a group of young halo globular clusters (including Palomar 12, Ruprecht 106, IC 4499, Arp 2, Terzan 7, Palomar 3, Palomar 4, Eridanus, Fornax 4, NGC 4590). It has often been suggested that these young globular clusters were formed in dwarf companions of the Milky Way, which have since been accreted and destroyed, leaving their clusters behind. For example, @Cr03 proposed that Pal 1 is part of the Monoceros ring (along with NGC 2808, NGC 5286, NGC 2298), whilst @Be07 suggested it may be associated with the accretion event that formed the Orphan Stream (along with Ruprecht 106 and possibly Terzan 7).
Pal 1, together with Segue 3, Koposov 1 and 2, Whiting 1, E 3, and AM 4, comprise [*an ultra-faint population*]{} of globular clusters. Pal 1 is the nearest of the six contained within the SDSS footprint (though E3 is still closer to the Sun and Galactic Centre). Compared to the other young globular clusters, Pal 1 is the faintest and smallest. Its tiny size and low luminosity (and hence presumably low mass) make it an attractive target to look for the effect of tides or dynamical evolution, which is the purpose of this [*Letter*]{}.
Tail Detection
==============
Pal 1 is located at $(l,b)=(130.1^{\circ},19.0^{\circ})$. It has an absolute magnitude of $M_V=-2.5\pm0.5$ and a half-light radius of $R_{\rm h}\approx2.2$ pc and its heliocentric radial velocity is $-82.3\pm3.3$ kms$^{-1}$ [@Ro98a; @Ro98b; @OR85]. Pal 1 lies within an SDSS data release 7 stripe [@Ab09], covering $2.5^{\circ}$ in galactic longitude and $11.3^{\circ}$ in galactic latitude. Henceforth, all SDSS magnitudes are corrected for extinction using the maps of @Sc98.
The panels of Fig. \[fig:pal1\_overview\] show $82\times 18$ pixel maps of the the density of stars around Pal 1, an estimate of the possible smooth underlying distribution of stars, the distribution of galaxies in the area, as well as the mean $g$-band extinction. Pal 1 is clearly visible as an overdensity in the topmost panel. To search for stars torn from Pal 1, we employ the optimal filter technique which works by calculating conditional probabilities of cluster and foreground membership from densities in colour-magnitude space, known as Hess diagrams. From this weighted distribution, a smooth distribution of field stars is then subtracted [see e.g. @Od03; @Ni09]. The field star density is estimated by removing a $0.3^{\circ}\times0.3^{\circ}$ box around the cluster, replacing it by a patch at $(\ell,b)=(131.1^{\circ},22.0^{\circ})$, and smoothing the resultant distribution with a Gaussian kernel of FWHM $15$ pixels and a box-car smoothing over $2$ pixels.
The left-hand panel of Fig. \[fig:cmfollowup\_pal1\] shows the Hess difference between stars within $0.16^{\circ}$ and those further than $0.5^{\circ}$ from the centre of Pal 1. The mask shown encloses both the main sequence and the red clump, though the inclusion of the red clump makes little difference to our results as the giant-branch is so poorly populated. The mask closely follows the main sequence at the red edge but allows more leeway on the blue side. This is done since the contamination from disk and halo stars is more severe at the red side of the distribution. The ratio of the Hess diagrams is used as the weights in the optimal filter analysis.
Fig. \[fig:pal1\_optfilter\] shows the results of the optimal filter analysis. The distribution is smoothed with a Gaussian kernel over $3$ pixels. Pal 1 stands out as the most significant overdensity in the field but there are significant overdensities extending from it towards higher and lower galactic latitudes. There is also a high signal region at the lower edge of the plot, but follow-up analysis shows this to be unrelated to the cluster and likely to be an artefact caused by the high extinction in this region, together with edge effects. The lower panel of Fig. \[fig:pal1\_optfilter\] shows the results with a higher spatial resolution ($123\times27$ pixels), smoothed over $3$ pixels. Again, the tails are noticeable in an otherwise nearly noise-free area. Color-magnitude analyses of the tails are shown in Fig. \[fig:cmfollowup\_pal1\]. There are suggestive similarities between the main body of Pal 1 and this debris. The stars in both extensions are concentrated around Pal 1’s main sequence turn-off with a faint continuation down the main sequence.
Pal 1 does not appear to be embedded in a noisy field. It is not possible to conclude from the optimal filter analysis whether or not it was once part of a dwarf companion. Of course, such a stream may be hidden by the small field-of-view available, or it may already have dispersed.
There exist deep [*Hubble Space Telescope*]{} (HST) ACS data covering Pal 1 [@Sa07], which allow us to explore the central regions more closely. By applying our source extraction to the HST images, we select cluster stars based on their position in color-magnitude space, as shown in the inset in Fig. \[fig:opt\_hst\]. We then overplot the density contours derived from the HST data on a higher resolution ($574$ by $126$ pixels, smoothed over $1$ pixel) version of the optimal filter analysis. This high resolution plot of the central $0.2^{\circ}\times0.2^{\circ}$ does not allow us to identify any debris. However, we can see that the central region of the cluster is elongated, pointing towards the Galactic Centre, and that the cluster stars are spread over several half-light radii ($R_{\rm h} \approx
0.01^{\circ}$). The outer isodensity contours, given by the optimal filter technique, twist and align with the direction of the debris. Such an S-shaped misalignment between tidal debris and the elongation of the central regions is observed in simulations when the object is near the apocentre of its orbit [see for example the bottom row of Fig. 4 in @Jo02].
To estimate the number of stars in the tails, we use a coordinate system ($\ell',b'$), in which the extensions lie along a great circle (see lower panel of Fig. \[fig:pal1\_optfilter\]). In this system, Pal 1’s possible debris lie at roughly constant latitude ($b'\approx
0^{\circ}$). The top panel of Fig. \[fig:starcounts\] shows the density of stars in a $2^\circ$ area around Pal 1 in this new coordinate system. There are $9$ bins along, and 21 bins perpendicular to, the stream direction. By using larger bins in the direction along the stream, we can assess the significance of the enhancement around Pal 1. The influence of the cluster itself has been removed from this plot by excluding stars that are within $4$ half-light radii ($\sim
0.04^{\circ}$) of the cluster centre. To amplify the signal of Pal 1 stars, this plot is restricted to stars lying in the rectangular box in color-magnitude space shown in Fig. \[fig:cmfollowup\_pal1\] left panel, which picks up the Pal 1 main sequence turn-off and the blue edge of the main sequence. This avoids contamination from the field stars, which lie slightly to the red of the main sequence. To account for possible contamination in this plot, we subtract a background estimate, derived by applying a linear fit to each column of constant $\ell'$, to arrive at the upper panel of Fig. \[fig:starcounts\]. The stream stands out as a continuous overdensity of pixels along $b' \approx 0$. It is now straightforward to count the number of stream stars present by summing the weighted, background subtracted counts in bins of constant latitude $b'$ as shown in the lower panel of Fig. \[fig:starcounts\]. There is a significant spike at $b'\approx
0^{\circ}$, which we identify as the tails of Pal 1. There seem to be on order the same number of stars in the tails (approximately $70$) as in the cluster (approximately $85$), suggesting that the mass in the tails is comparable to the mass in the cluster.
Conclusions
===========
We have discovered probable tails around the globular cluster Pal 1 in Sloan Digital Sky Survey data. The tails cover at least $2^{\circ}$ on the sky and extend northward and southward from the cluster centre, possibly up to $\approx 80$ half-light radii. The tails contain roughly as many stars as does the cluster itself. [*Hubble Space Telescope*]{} data reveal that the central parts of the cluster are clearly elongated, with cluster stars spread over several half-light radii. The tails constrain the direction of Pal 1’s projected proper motion, but this, together with its measured radial velocity, is not sufficient to establish meaningful constraints on the orbital parameters of Pal 1.
Pal 1 is a prototype of the population of ultrafaint globular clusters with $M_v$ fainter than $-3$. This also includes Segue 3, Koposov 1 and 2, as well as Whiting 1, E3 and AM 4. If the ultrafaint globular clusters are born in their present state, then they would evaporate within 1 Gyr [@Ro98a; @Ko07]. Hence, it would be unlikely that they survived long enough for us to see them, unless a large number of such clusters are born like this continuously. It is much more likely that globular clusters evolve into the ultrafaint régime, either through tides or through evaporation causing catastrophic loss of stars. Mass loss due to evaporation is accelerated when the cluster orbits in a tidal field, as for example is the case if an accreted dwarf galaxy originally hosted the cluster.
Could the tails of Pal 1 be caused primarily by evaporation? First, assuming a mass-to-light ratio of $\sim 2$ gives a present cluster mass of $\sim 1400$ $M_{\odot}$. Pal 1 has a half-mass relaxation time [@Sp71] of $\trelax\sim 0.16$ Gyr. Based on the results of dynamical models, a lower-limit to the mass-loss rate due to relaxation driven evaporation is 1.4 $M_\odot$ Myr$^{-1}$ for an assumed circular orbit [@Ba03], which scales like $(1-e)^{-1}$ for an orbit with eccentricity $e$. Assuming that stars escape and drift with velocities $\sim 1$ kms$^{-1}$ as suggested by eq. (18) of [@Ku10], then the mass within a length $L_{\rm Tail} \sim 400$ pc of the tidal tails ($\sim 2^{\circ}$ on the sky) is $M_{\rm
Tail}\sim\dot{M}\times L_{\rm Tail}/\sigma\sim 560 M_{\odot}$ for a circular orbit. This rises to $\sim 1100 M_\odot$ for an eccentricity $e=0.5$, which is comparable to the mass of the cluster. Secondly, the Jacobi radius at Galactocentric radius of $\sim 17$ kpc for an object of the mass of Pal 1 is $r_{\rm J} \sim 26$ pc, so that $r_{\rm
h}/r_{\rm J} \gtrsim 0.1$. This is surprisingly low for a dynamically evolved system. @Ba10 find that the expected $r_{\rm h}/r_{\rm J} \sim 0.3$ for extended and low-mass Galactic globular clusters. This suggests that the cluster is underfilling its Roche-surface. Both arguments favour evaporation rather than disk shocking as the main formation mechanism.
Pal 1 is therefore a rather different object to Pal 5, which is well-known to have extended tails [@Od03]. Pal 5 is overfilling its Roche surface and its tails are believed to have formed primarily by disk shocking [@De04]. Certainly, given its present state of disruption, Pal 1 must have passed within $8$ kpc of the Galactic Centre in order for disk shocking to account for the tidal tails, as judged by Figure 14 of @VH97.
If Pal 1 has been evolving in a satellite galaxy with a stronger tidal field, we expect it to have it a high density with respect to its current Galactocentric radius. This may explain why it is currently underfilling its Roche surface. However, the accretion must then have happened very recently, because the relaxation time is so low that adjustment to the new tidal field would happen quickly. For a cluster that is not too severely affected by the tidal field its (half-mass) relaxation time, $\trelax$, increases linearly with its age [@He65; @Ba02] because of expansion driven by binaries and 2-body relaxation. This expansion is ultimately stopped by the tidal field and then $\trelax$ decreases with age [@He61]. We assume that the cluster was in this Roche-lobe filling evolution in the original host galaxy. Let us denote by sub-script 0 the time of accretion, and assume that Pal 1 after accretion in the Milky Way started expanding again due to the weaker tidal field. @Gi10 showed that for globular clusters $ \trelax=0.3t $, where t is the age of the cluster. This means that $$\trelax \approx \trelaxzero + 0.3(t-t0)
\Rightarrow t-t0 \approx 3.3(\trelax-\trelaxzero)$$ Putting $\trelaxzero\approx0$ gives a rough upper limit on the accretion time of $\sim 500$ Myr in the past.
Acknowledgments {#acknowledgments .unnumbered}
===============
MNO is funded by the Gates Cambridge Trust, the Isaac Newton Studentship fund and the Science and Technology Facilities Council (STFC), whilst VB acknowledges financial support from the Royal Society. We thank M.G. Walker for helpful advice and support and the referee, Laszlo Kiss, for constructive feedback.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The paper is partly based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Institute. STScI is operated by the association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555.
[99]{} Abazaijan, K.N. et al. 2009, ApJS, 182, 543
Abell, G.O. 1995, PASP, 67, 258
Baumgardt H., Hut P., Heggie D. C., 2002, MNRAS, 336, 1069
Baumgardt, H., & Makino, J. 2003, MNRAS, 340, 227
Baumgardt, H., Parmentier, G., Gieles, M., & Vesperini, E. 2010, MNRAS, 401, 1832
Belokurov, V., et al. 2007, ApJ, 658, 337
Belokurov, V., et al. 2010, ApJL, 712, 103
Borissova, J., & Spassova, N. 1995, A&AS, 110, 1
Carraro, G. 2005, ApJ, 621, 61L
Carraro, G. 2009, AJ, 137, 3809
Clem, J.L., 2005, PhD Thesis, Univ. Victoria
Crane, J.D., Majewski, S.R., Rocha-Pinto, H.J., Frinchaboy, P.M., Skrutskie, M.F., Law, D.R. 2003, ApJ, 594, L119
Dehnen, W., Odenkirchen, M., Grebel, E. K., & Rix, H.-W. 2004, AJ, 127, 2753
Gieles, M., Baumgardt, H., Heggie, D., & Lamers, H. 2010, MNRAS, in press (arXiv:1007.2333)
Hénon M., 1961, Annales dÕAstrophysique, 24, 369
Hénon M., 1965, Annales dÕAstrophysique, 28, 62
Inman, R. T., & Carney, B. W. 1987, AJ, 93, 1166
Johnston, K. V., Choi, P. I., Guhathakurta, P. 2002, AJ, 124,127
Kinman, T.D., & Rosino, L. 1962, PASP, 74, 499
Koposov, S., et al. 2007, ApJ, 669, 337
K[ü]{}pper, A. H. W., Kroupa, P., Baumgardt, H., & Heggie, D. C. 2010, MNRAS, 401, 105
Niederste-Ostholt, M., Belokurov, V., Evnas, N.W., Gilmore, G., Wyse, R.F.G., Norris, J.E. 2009, MNRAS, 398, 1771
Odenkirchen, M. et al. 2003, AJ, 126, 2385
Ortolani, S., & Rosino, L. 1985, Mem. Soc. Astron. Italiana, 56, 105
Rosenberg, A., Saviane, I., Piotto, G., Aparicio, A., Zaggia, S.R. 1985, AJ, 115, 648
Rosenberg, A., Piotto, G., Saviane, I., Aparicio, A., Gratton R. 1985, AJ, 115, 658
Sarajedini, A., et al. 2007, AJ, 133, 1658
Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525
Spitzer, L., Jr., & Hart, M. H. 1971, ApJ, 164, 399
van den Bergh, S., Demers, S., & Kunkel, W. E. 1980, ApJ, 239, 112
van den Bergh, S. 2000, ApJ, 530, 777
Vesperini, E., & Heggie, D. C. 1997, MNRAS, 289, 898
Whiting, A. B., Hau, G. K. T., & Irwin, M. 2002, ApJS, 141, 123
\[lastpage\]
[^1]: E-mail:[email protected]
|
---
abstract: 'HH 900 is a peculiar protostellar outflow emerging from a small, tadpole-shaped globule in the Carina nebula. Previous H$\alpha$ imaging with *HST*/ACS showed an ionized outflow with a wide opening angle that is distinct from the highly collimated structures typically seen in protostellar jets. We present new narrowband near-IR \[Fe [ii]{}\] images taken with the Wide Field Camera 3 on the *Hubble Space Telescope* that reveal a remarkably different structure than H$\alpha$. In contrast to the unusual broad H$\alpha$ outflow, the \[Fe [ii]{}\] emission traces a symmetric, collimated bipolar jet with the morphology and kinematics that are more typical of protostellar jets. In addition, new Gemini adaptive optics images reveal near-IR H$_2$ emission coincident with the H$\alpha$ emission, but not the \[Fe [ii]{}\]. Spectra of these three components trace three separate and distinct velocity components: (1) H$_2$ from the slow, entrained molecular gas, (2) H$\alpha$ from the ionized skin of the accelerating outflow sheath, and (3) \[Fe [ii]{}\] from the fast, dense, and collimated protostellar jet itself. Together, these data require a driving source inside the dark globule that remains undetected behind a large column density of material. In contrast, H$\alpha$ and H$_2$ emission trace the broad outflow of material entrained by the jet, which is irradiated outside the globule. As it get dissociated and ionized, it remains visible for only a short time after it is dragged into the H [ii]{} region.'
author:
- |
Megan Reiter$^{1}$[^1], Nathan Smith$^{1}$, Megan M. Kiminki$^{1}$, John Bally$^{2}$, and Jay Anderson$^{3}$\
$^{1}$ Steward Observatory, University of Arizona, Tucson, AZ 85721, USA\
$^{2}$ Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309, USA\
$^{3}$ Space Telescope Science Institute, Baltimore, MD 21218, USA
date: 'Accepted 2014 Oct ??. Received 2014 Oct ??; in original form 2014 October ??'
title: Disentangling the outflow and protostars in HH 900 in the Carina Nebula
---
\[firstpage\]
stars: formation – jets – outflows
Introduction {#s:intro}
============
Protostellar outflows are a beacon of star formation, signaling active disc accretion even in deeply embedded regions where the disc, and sometimes protostars themselves, cannot be detected directly. We discuss one such example in this paper. The detailed physics of jet launch and collimation is not yet understood [see, e.g. @fer06], but disc accretion ultimately must fuel the jet. Observations of relatively unobscured young stars measure this jet directly, but trace an epoch long after the most active accretion. CO observations trace outflows from young sources and those in more embedded regions. The observed emission may be dominated by ambient molecular material entrained by the jet [e.g. @arc05] and/or from a slower disc wind [e.g. @kla13].
Spatially resolved outflows can be a powerful tool to study star formation at higher stellar mass where evidence for discs remains elusive. There have been only a few direct detections of disks around intermediate-mass protostars [e.g. @kra10; @pre11; @car12]. Structure in the form of clumps and knots along the jet axis points to the variable nature of accretion and outflow, and provides one of the few ways to infer a given protostar’s accretion history. The longest jets extend parsecs on the sky, sampling a significant fraction of the accretion age [e.g. @mar93; @dev97; @smi04]. Many outflows from moderate- to high-mass young stars appear to be scaled-up versions of low-mass systems [e.g. @guz11; @rei13; @rei14], although this is not always the case [see, e.g. @she03]. Other outflows, such as the Becklin-Neugebauer / Kleinmann-Low (BN/KL) outflow in Orion OMC1, show evidence for violent processes related to the dynamical interaction and ejection of high-velocity stars that creates an explosive morphology [@bal11; @god11]. However, similar outflow behavior from a large number of protostars over a wide mass range provides compelling evidence that massive star formation can be understood as a scaled-up version of low-mass star formation. It is then essential to study intermediate-mass stars ($\sim 2-8$ M$_{\odot}$) where any changes in the dominant physics of formation between low- and high-masses are expected to occur.
Outflows from young, intermediate-mass protostars are typically observed at millimeter wavelengths where emission from entrained molecules penetrates the high column densities that characterize massive star-forming regions [e.g. @tak07; @beu08; @bel08]. However, in an H [ii]{} region, much of the obscuring gas and dust may have been cleared. In such environments, ultraviolet (UV) radiation from nearby massive stars will illuminate protostellar jets propagating into the H [ii]{} region cavity. Unlike outflows in quiescent regions where emission at visual wavelengths traces only shock-heated material, external irradiation in H [ii]{} regions lights up the body of the entire jet, revealing outflow material that would otherwise remain invisible. This allows the physical properties of irradiated jets to be measured using the diagnostics of photoionized gas [e.g. @bal01; @yus05] with the high angular resolution observations that are available at shorter wavelengths. HH 900 is one of the many HH jets discovered by @smi10 as part of their H$\alpha$ survey of the Carina nebula using the Advanced Camera for Surveys (ACS) onboard the *Hubble Space Telescope (HST)*. HH 900 is an unusual bipolar outflow emerging from a small tadpole-shaped globule located $\sim 3$ pc (in projection) away from $\eta$ Carinae (see Figure \[fig:hst\_ims\]). In earlier ground-based images, @smi03 identified it as a candidate proplyd, although neither of the two ‘tails’ emerging from the globule point away from $\eta$ Car. Higher resolution *HST* images show that these two ‘tails’ appear to be a wide bipolar outflow, and reveal a number of other peculiarities. Along the western limb of the broad bipolar outflow lies a strong H$\alpha$ filament and a point source nearly at its center, raising the possibility that the filament is a microjet driven by the star [@smi10]. Without kinematic information, it remains unclear whether the point source actually drives the putative H$\alpha$ microjet, and if so, whether it is physically related to the larger bipolar outflow. Estimated mass-loss rates (derived using the H$\alpha$ emission measure) from the wide inner jet in the eastern limb of the outflow and the microjet along the western limb are the highest of all the outflows found by @smi10.
@shi13 present ground-based observations of \[Fe [ii]{}\] 1.64 emission in the Carina nebula. Based on the morphology of the \[Fe [ii]{}\] emission from HH 900, they propose that the HH 900 driving source is one of the young stellar objects (YSOs) modeled in the Pan-Carina YSO Catalog [PCYC, @pov11]. A second *Spitzer*-identified YSO lies along the western limb of the flow, although with angular resolution $\gtrsim 1.5$, it is unclear how this source relates to the putative microjet. @ohl12 conclude that *Herschel* emission near HH 900 is from the externally heated globule and is unlikely to come from young stellar objects in the region.
In this paper, we present new optical and IR observations of HH 900 that allow us to investigate its morphology and kinematics in detail. New narrowband IR images obtained with *HST* probe the \[Fe [ii]{}\] 1.26 and 1.64 lines that are often assumed to be shock-excited in protostellar jets, although this is not necessarily the case in regions with significant FUV radiation. Indeed, @rei13 showed that these two \[Fe [ii]{}\] lines probe dense, low-ionization jet material not traced by H$\alpha$. Near-IR \[Fe [ii]{}\] lines can also penetrate the extinction inside the dusty birthplaces of these jets, allowing us to connect the larger H$\alpha$ outflows to the *Spitzer*-identified protostars that drive them [@rei13]. Both \[Fe [ii]{}\] 1.26 and 1.64 originate from the same $a^4D$ level, so their flux ratio is insensitive to excitation conditions. Variations in the flux ratio along the jet therefore provide one way to measure variations in the reddening of the immediate jet environment. This may be particularly interesting in HH 900 where extended H$_2$ emission (Hartigan et al. 2015, in preparation) and dark filaments in H$\alpha$ images suggest the presence of molecules and dust in the broad bipolar flow.
We also present new, second-epoch H$\alpha$ images from *HST* that allow us to measure jet motions in the plane of the sky. The outward motion of jet knots provides a direct kinematic identification of candidate jet driving sources. Together with ground-based optical and IR spectra, our H$\alpha$ proper motions and \[Fe [ii]{}\] images from *HST* offer a new view of HH 900 that simultaneously help to unravel but also deepen the mystery of this unusual outflow.
Observations {#s:obs}
============
Instrument Filter / Position Date Int. time Comment
----------------- ------------------------- ------------- ----------- ------------------------------
ACS/*HST* F658N 2005 Jul 18 1000s H$\alpha$ $+$ \[N [ii]{}\]
ACS/*HST* F658N 2014 Aug 04 1000s H$\alpha$ $+$ \[N [ii]{}\]
GSAOI/*Gemini* K$_\mathrm{s}$ N cont. 2013 Mar 23 1080s
GSAOI/*Gemini* H$_2$ 2013 Mar 23 1080s
WFC3-IR/*HST* F126N 2013 Dec 28 2397 s \[Fe [ii]{}\] $\lambda12567$
WFC3-IR/*HST* F128N 2013 Dec 28 2397 s continuum
WFC3-IR/*HST* F164N 2013 Dec 28 2397 s \[Fe [ii]{}\] $\lambda16435$
WFC3-IR/*HST* F170N 2013 Dec 28 2397 s continuum
EMMI/*NTT* H$\alpha$, \[S [ii]{}\] 2003 Mar 09 900 s P.A. $=63^{\circ}$
FIRE/*Magellan* West 2014 Jan 15 600 s P.A. $=77^{\circ}$
FIRE/*Magellan* East 2014 Jan 16 600 s P.A. $=60^{\circ}$
\[t:foobar\]
High-resolution images
----------------------
**\[Fe [ii]{}\]:** Table \[t:obs\] lists the details of the images and spectroscopy presented in this paper. Near-IR, narrowband \[Fe [ii]{}\] images and corresponding off-line narrowband continuum images were obtained with WFC3-IR on board the *Hubble Space Telescope (HST)* under program GO-13391 (PI: N. Smith) in Cycle 21. For HH 900, we duplicated the observing strategy used previously to obtain \[Fe [ii]{}\] images of four other jets in the Carina nebula, HH 666, HH 901, HH 902, and HH 1066 [@rei13]. We employed a box-dither pattern to avoid dead-pixel artifacts and to provide modest resolution enhancement. Using the same integration time for all filters provided a similar signal-to-noise ratio for the \[Fe [ii]{}\] and adjacent continuum images.
**H$_2$:** We obtained near-IR, narrowband H$_2$ 2.12 images and complementary off-line narrowband K(short) continuum images (central wavelength of 2.093 ) during early science with the Gemini South Adaptive Optics Imager [GSAOI, @mcg04; @gem_car12]. GSAOI is a near-IR imager, used in combination with Gemini Multi-Conjugate Adaptive Optics System [GeMS, @rig14; @nei14] with natural guide stars to provide near-diffraction limited images over a field of view of 85 $\times$ 85 in the 0.9-2.5 range. HH 900 was observed 22 March 2013 in queue observing mode with a 9-point dither pattern and individual integrations of 120 s. We resampled the aligned, flat-fielded images to the same pixel scale as the WFC3-IR images. Simultaneous K(short) continuum images allow us to subtract bright IR continuum emission from the edge of the globule and therefore search for extended H$_2$ emission from the jet.
**H$\alpha$:** H$\alpha$ images of HH 900 were obtained with *HST*/ACS for the first time on 18 July 2005 [@smi10]. We obtained a second epoch on 04 August 2014 using the the same instrument and filter, duplicating the first-epoch observational setup in order to measure proper motions (program GO-13390, PI: N. Smith). Together, this provides a $\sim 9$ yr time baseline between observations, allowing us to measure the motion of faint jet features to $\sim 25$ km s$^{-1}$. Using the same orientation and coordinates for both epochs minimizes position-dependent errors when determining proper motions.
We follow a method similar to that of @and08a [@and08b], @and10, and @soh12 to align and then measure precise proper motions in the ACS images. This method is based on PSF photometry of the bias-subtracted, flat-fielded, and CTE-corrected [flc]{} images produced by the *HST* pipeline. Unlike the drizzled [drc]{} images, which have been calibrated, flat-fielded, CTE- and geometrically-corrected, and dither-combined (via AstroDrizzle), the [flc]{} images have not been resampled to correct for geometric distortion and thus allow for more accurate PSF fitting.
In brief, we first measure centroid positions for bright, relatively isolated stars in the [drc]{} images, since these images have an astrometric solution that allows us to construct an initial reference frame. As in @and08a, our reference frame has a 50 mas pixel$^{-1}$ scale and a north-aligned *y* axis. Next, we perform PSF photometry on the individual flc exposures using the program [img2xym\_WFC.09x10]{} [@and06], which uses a library of models of the spatially-variable effective ACS PSF. Once measured, stellar positions are corrected for distortion using the @and05 corrections. Stars in common between the distortion-corrected frame of each [flc]{} exposure and the reference frame are identified, allowing us to determine six-parameter linear transformations from the former to the latter. We iterate on this process once, replacing the centroid positions that initially defined the reference frame with the average transformed reference-frame positions from the [flc]{} PSF photometry, then recomputing the linear transformation from each [flc]{} image to the reference frame.
We relate both epochs to a single reference frame, so the measured positions in each epoch can be directly compared. Relative proper motions for point sources (including the YSOs adjacent to HH 900) are simply the difference in average reference frame position between 2005 and 2014. The positional errors in each dimension for each epoch are calculated by dividing the rms scatter of the individual measurements from each exposure, divided by the square root of the number of exposures (N=2-4) used to compute the average position.
Our *HST* observations are such that the western bow shock of HH 900 was observed in a different orbit than the globule, inner jet, and eastern bow shock. Each orbit consists of three overlapping pointings, with two dithered exposures per pointing; however, the overlap in area between orbits is small. This requires that we construct a separate reference frame for each orbit. These reference frames are defined by the positions of the stars and are not tied to an absolute proper-motion zero point. Therefore, all motions are measured relative to the average motion of the stars in the image. Because the HH 900 western bow shock was observed in a separate frame, there may be a systematic offset between the proper motion of the western bow shock and the rest of HH 900. We expect this offset to be $\lesssim1$ km s$^{-1}$, but certainly no larger than the stellar velocity dispersion of the Tr 16 cluster, or $\lesssim5$ km s$^{-1}$.
To measure the proper motions of extended features in HH 900, we create stacked images by resampling the [flc]{} exposures into our reference frame [see @and08a for details of the stacking algorithm used]. We then select bright jet features that do not change significantly in morphology between the two epochs and measure their motion using the modified cross-correlation technique described by @cur96 [@har01; @mor01]. After subtracting a median-filtered image to account for background emission, we select a small box around each jet feature. This subimage is shifted relative to the reference image, and for each shift, we compute the total of the square of the difference over the box region between the two images. The minimum of this array corresponds to the pixel offset between the two images. For the typical signal-to-noise in our images, the uncertainty of this procedure is $\approx 0.03$ pixel, corresponding to $\approx 10$ km s$^{-1}$.
We also measure the proper motions of point sources in the image by fitting a spatially variable PSF to the stars. This allows the motion of stars to be measured to greater precision than nebulous jet features, typically a few km s$^{-1}$ (see Table 2).
$\begin{array}{c}
\includegraphics[trim=10mm 0mm 0mm 0mm,angle=0,scale=0.45]{hh900_halpha_annotated.eps} \\
\includegraphics[trim=10mm 0mm 0mm 0mm,angle=0,scale=0.45]{hh900_fe_annotated.eps} \\
\includegraphics[trim=10mm 0mm 0mm 0mm,angle=0,scale=0.45]{hh900_f126_m_f130.eps} \\
\includegraphics[trim=10mm 0mm 0mm 0mm,angle=0,scale=0.45]{hh900_f164_m_f167.eps} \\
\end{array}$
Spectroscopy
------------
**FIRE:** We obtained near-IR spectra of HH 900 with the Folded-Port InfraRed Echellette (FIRE) near-IR spectrograph [@sim08; @sim10; @sim13] on the Magellan Baade 6.5-m telescope on 17 January 2014 (see Table \[t:obs\]). FIRE’s $0.8 - 2.5$ wavelength coverage includes multiple emission lines from the jet. We positioned the 7 slit along bright \[Fe [ii]{}\] emission in the eastern and western limbs of the outflow and used a 1 slit width to accommodate $\gtrsim 1\farcs1$ seeing. Figure \[fig:feii\_pvs\] shows the slit positions. The YSO and putative microjet that lie along the western limb of the larger bipolar outflow fall within the slit used to observe the western limb of the \[Fe [ii]{}\] jet. To account for sky emission, we employed a nodding strategy, pointing on- and off-source in an ABBA sequence with an average sky offset located $\sim2$ below the jet. Wavelength calibration was done using the internal ThAr lamp and data reduction was performed with the [Firehose IDL]{} pipeline. After correcting for the motion of the earth, we report Doppler velocities in the heliocentric velocity frame.
**EMMI:** We obtained visual-wavelength spectra of HH 900 with the ESO Multi-Mode Instrument (EMMI) on the New Technology Telescope (NTT) on 09 March 2003. We observed HH 900 in echelle mode, using the cross-dispersing grism to obtain H$\alpha$ and \[S [ii]{}\] simultaneously. The echelle slit length is 25 and the jet was observed with a 1 slit width. We aligned the slit through the middle of the globule and the star in the western limb of the flow. We estimate sky emission from separate frames, with the slit offset $\sim 4$ perpendicular to the jet. We compute the wavelength solution from the sky frames, using the nebular \[N [ii]{}\] lines for the H$\alpha$ dispersion solution and the \[S [ii]{}\] lines themselves. Thus our derived radial velocities are relative to the Doppler shift of ambient ionized gas in the Carina H [ii]{} region.
Results {#s:results}
=======
$\begin{array}{c}
\includegraphics[trim=10mm 0mm 0mm 0mm,angle=0,scale=0.55]{hh900_f657_w_fe_contours.eps} \\
\includegraphics[trim=10mm 0mm 0mm 0mm,angle=0,scale=0.55]{hh900_h2_w_fe_contours.eps} \\
\end{array}$
$\begin{array}{c}
\includegraphics[trim=-5mm 0mm 0mm 0mm,angle=0,scale=0.375]{feii_ratio_tracing_location.eps} \\
\includegraphics[angle=90,scale=0.325]{feii_ratio_tracing.eps} \\
\includegraphics[angle=90,scale=0.325]{both_sides_feii_ratio_tracing.eps} \\
\end{array}$
Morphology in the IR images {#ss:morph}
---------------------------
Figure \[fig:hst\_ims\] shows the new \[Fe [ii]{}\] 1.26 and 1.64 images of HH 900 obtained with *HST*/WFC3-IR (Figures \[fig:hst\_ims\]b, c, and d) and the ACS H$\alpha$ image (Figure \[fig:hst\_ims\]a). Bright, collimated \[Fe [ii]{}\] emission that extends to the east and west of the dark tadpole globule stands out in the continuum-subtracted images (Figures \[fig:hst\_ims\]c and d). Near-IR \[Fe [ii]{}\] emission traces a symmetric, collimated bipolar jet, unlike H$\alpha$, which emerges into the H [ii]{} region with almost the same geometric width as the globule (see Figure \[fig:fe\_contours\]). Figure \[fig:fe\_contours\] shows a more detailed comparison of the H$\alpha$ and \[Fe [ii]{}\]. Intensity tracings through the eastern and western limbs of the \[Fe [ii]{}\] jet show that they are remarkably similar (see Figure \[fig:feii\_tracings\] and \[fig:feii\_ratio\_tracings\]).
Both sides of the jet show a $\gtrsim 1.5$ gap between the edge of the globule and bright \[Fe [ii]{}\] emission from the jet. Unlike the images of jets in our previous study using archival data [@rei13], our observations of HH 900 include simultaneous line-free continuum images that allow us to subtract the continuum and therefore separate photodissociation region (PDR) emission along the globule edge from fainter \[Fe [ii]{}\] emission emitted by the inner jet. Subtracting the line-free continuum from HH 900 demonstrates that the offset of the \[Fe [ii]{}\] jet from the edge of the globule is real (see Figure \[fig:fe\_contours\]). No tenuous \[Fe [ii]{}\] emission from the inner jet extends back to the globule edge in the continuum-subtracted image, demonstrating that the offset is not just confusion with continuum emission that grows increasingly bright close to the irradiated edge of the dusty globule.
$\begin{array}{c}
\includegraphics[trim=0mm 0mm 0mm 0mm,angle=0,scale=0.40]{feii_ratio_y_tracing_location.eps} \\
\begin{array}{cc}
\includegraphics[trim=30mm 0mm 10mm 0mm,angle=0,scale=0.25]{feii_ratio_y_tracing_east.eps} &
\includegraphics[trim=40mm 0mm 0mm 0mm,angle=0,scale=0.25]{feii_ratio_y_tracing_west.eps} \\
\end{array}
\end{array}$
@smi10 identify two bow shocks associated with HH 900 that are $\sim 10$ beyond the end of the inner jet in H$\alpha$ images. Both bow shocks are also bright in \[Fe [ii]{}\] emission (see Figure \[fig:hst\_ims\]).
Three dark, filamentary streaks in the western limb of HH 900 are seen in absorption in the H$\alpha$ image (see Figure \[fig:hst\_ims\]a). @smi10 interpret these dark filaments as extinction due to dust entrained by the jet along the walls of the outflow cavity, similar to the features that @wal04 observed in HH 280 (another jet that appears to burst out of a globule). Bright emission extending off the western edge of the globule in the \[Fe [ii]{}\] image in Figure \[fig:hst\_ims\]a disappears in the continuum-subtracted image, indicating that it is continuum emission. This may be starlight scattered by the dusty filaments.
Narrowband H$_2$ images from *GSAOI* show H$_2$ emission from the surface of the globule and outside it that, like the H$\alpha$ emission, appears broad with nearly the same projected width as the globule itself (see Figure \[fig:fe\_contours\]). The intensity of the H$_2$ emission appears to be slightly ($\sim 1.5\times$) greater along the edges of the outflow than in the center (i.e. limb brightened). H$_2$ emission from the outflow fades below our detection limit $\sim 1\farcs5$ away from the globule edge; this is the same place where the width of the H$\alpha$ emission appears to decrease and the collimated \[Fe [ii]{}\] jet begins (see Figure \[fig:fe\_contours\]).
\[Fe [ii]{}\] ratio {#ss:ratio}
-------------------
Both the \[Fe [ii]{}\] 1.26 and 1.64 lines originate from the same upper level, so the observed flux ratio therefore provides a measure of the reddening. In unobscured environments, the 1.26 line will be brighter, with a derived flux ratio, $\mathcal{R} = \lambda12567$/$\lambda16435 = 1.49$ [@sh06]. Smaller values of $\mathcal{R}$ indicate more reddening and extinction. Along the length of the \[Fe [ii]{}\] jet, the ratio remains consistent, with a value of $\sim 0.95$ (see Figure \[fig:feii\_tracings\]). This corresponds to an E(B$-$V) $\approx 0.9$ [@rei13]. Using the ratio of total to selective extinction measured toward Carina of $R=4.8$ [@smi87; @smi02], this corresponds to $A_V \approx 4.5$ mag along the length of the jet. We also measure $\mathcal{R}$ through the width of the jet (perpendicular to the direction of propagation, see Figure \[fig:feii\_y\_tracing\]). Reddening through the width of the jet increases further away from $\eta$ Car and Tr 16, corresponding to a decrease in $\mathcal{R}$ of $\sim 0.2$ across the width of the jet. The measured decrease in $\mathcal{R}$ corresponds to $\sim 1.5$ mag more extinction on the side of the jet further away from the ionizing source.
$\begin{array}{c}
\includegraphics[angle=0,scale=0.375]{all_tracing.eps} \\
\includegraphics[angle=0,scale=0.375]{both_sides_all_tracing.eps} \\
\end{array}$
Candidate driving sources {#ss:ysos}
-------------------------
@rei13 showed that in HH 666 and HH 1066, the driving protostar lies along the projected jet axis and that \[Fe [ii]{}\] connects the jet to the *Spitzer*-identified driving sources. However, \[Fe [ii]{}\] emission in HH 900 does not reach back to the edge of the tadpole-shaped globule, and no \[Fe [ii]{}\] emission from the jet is detected inside the globule.
Two IR-bright YSOs have been found in the vicinity of the HH 900 globule [@shi13]. The jet axis traced by \[Fe [ii]{}\] emission lies $\sim 1.5$ north of the YSO that lies at the bottom of the tadpole globule [PCYC 842; @pov11 see Figure \[fig:hst\_ims\]], clearly demonstrating that it cannot be the driving source of the bipolar flow. The second *Spitzer*-identified YSO (PCYC 838) lies along the western limb of the flow, in the middle of the putative microjet. While the morphology raises the possibility that this protostar drives a separate jet that happens to be aligned with the larger H$\alpha$ outflow lobe, images alone do not indicate whether there is a separate microjet or if PCYC 838 is its driving source. As we discuss in Section \[ss:pm\], however, radial velocities and proper motions lead us to reject this hypothesis.
Evidence for an additional protostellar source along the jet axis embedded in the tadpole globule remains elusive due to the small size of the globule ($\sim 2$) and the coarser resolution of available mid- and far-IR observations. We do not detect any evidence of an IR-bright point source inside the globule in any of the WFC3-IR images. The upper limit in all four bands is $\geq 21$ mag. High frequency, high spatial resolution observations with ALMA are required to determine whether there is an additional protostar embedded in the globule.
Velocity structure {#ss:velocity}
------------------
Among the peculiarities of HH 900 revealed with H$\alpha$ images from *HST* is the morphology of the dark globule. A small, wiggly, dark tail extends from the southeastern edge of the globule, creating a tadpole-like morphology (see Figure \[fig:hst\_ims\]). The tail of the dark tadpole globule appears to lie in front of the jet, leading @smi10 to suggest that the eastern limb of the jet is redshifted, and the western limb of the jet is blueshifted. Both optical and IR spectra now confirm this conjecture (see Figures \[fig:feii\_pvs\] and \[fig:vis\_pvs\]). H$\alpha$ emission from the western limb of the jet is blueshifted and shows a Hubble-like velocity structure, with Doppler velocities extending up to roughly $-40$ km s$^{-1}$. A similar velocity structure is barely discernible in the \[S [ii]{}\] spectrum (the average of the $\lambda 6717$ and $\lambda 6731$ emission) due to the lower signal-to-noise.
In contrast to the Hubble-like flow in the optical emission, near-IR \[Fe [ii]{}\] lines from both sides of the jet show a relatively constant velocity with redshifted velocities measured to be $\sim 30-60$ km s$^{-1}$ in the eastern limb of the jet and blueshifted velocities of $\sim 0 - 20$ km s$^{-1}$ in the western limb (suggesting $v_{sys} \approx 19$ km s$^{-1}$, see Figure \[fig:feii\_pvs\]). The velocity difference between the eastern and western limbs of the jet is symmetric about the globule, indicating that the \[Fe [ii]{}\] emission is dominated by a single bipolar jet launched from within the globule. The H$_2$ emission from the eastern and western limbs of the jet is also redshifted and blueshifted, respectively. However, H$_2$ emission appears to be $\sim 20$ km s$^{-1}$ bluer than \[Fe [ii]{}\] velocities in both limbs. Position-velocity diagrams illustrate that the locations of H$_2$ and \[Fe [ii]{}\] emission from the jet appear to be mutually exclusive (see Figure \[fig:feii\_pvs\]). In fact, H$\alpha$, \[Fe [ii]{}\], and H$_2$ emission all trace different velocity components (see Figures \[fig:feii\_pvs\] and \[fig:vis\_pvs\]). Blueshifted velocities in the H$\alpha$ position-velocity diagram increase from $\sim0$ km s$^{-1}$ to the higher velocity traced by the \[Fe [ii]{}\] jet ($\sim -40$ km s$^{-1}$) while H$_2$ emission appears to be $\sim 20$ km s$^{-1}$ bluer than $v_{sys}$ inferred from the \[Fe [ii]{}\] emission from the innermost part of the jet.
$\begin{array}{ccc}
\includegraphics[trim=100mm 0mm 0mm 10mm,angle=90,scale=0.60]{fire_slit_pos.eps} &
\begin{array}{c}
\includegraphics[trim=0mm 10mm 10mm 0mm,angle=0,scale=0.325]{hh900_east_fe126_quicklook_pv.eps} \\
\includegraphics[trim=0mm 0mm 10mm 10mm,angle=0,scale=0.325]{hh900_west_fe126_quicklook_pv.eps} \\
\end{array}
\begin{array}{c}
\includegraphics[trim=20mm 10mm 10mm 0mm,angle=0,scale=0.325]{hh900_east_fe164_quicklook_pv.eps} \\
\includegraphics[trim=20mm 0mm 10mm 10mm,angle=0,scale=0.325]{hh900_west_fe164_quicklook_pv.eps} \\
\end{array}
\begin{array}{c}
\includegraphics[trim=20mm 10mm 15mm 0mm,angle=0,scale=0.325]{hh900_east_h2_quicklook_pv.eps} \\
\includegraphics[trim=20mm 0mm 15mm 10mm,angle=0,scale=0.325]{hh900_west_h2_quicklook_pv.eps} \\
\end{array}
\begin{array}{c}
\includegraphics[trim=15mm 10mm 0mm 0mm,angle=0,scale=0.325]{hh900_east_all_points.eps} \\
\includegraphics[trim=15mm 0mm 0mm 10mm,angle=0,scale=0.325]{hh900_west_all_points.eps} \\
\end{array}
\end{array}$
$\begin{array}{ccc}
\includegraphics[trim=15mm 0mm 0mm 0mm,angle=0,scale=0.495]{hh900_halpha.eps} &
\includegraphics[trim=0mm -26mm 15mm 0mm,angle=0,scale=0.275]{hh900_emmi_slit_pos.eps} &
\includegraphics[trim=10mm 0mm 0mm 0mm,angle=0,scale=0.495]{hh900_sii.eps} \\
\end{array}$
H$\alpha$ proper motions and 3-D velocities of jet features {#ss:pm}
-----------------------------------------------------------
With a $\sim 9$ yr time baseline between the two *HST*/ACS epochs, we are sensitive to transverse velocities of $\sim 25$ km s$^{-1}$ in nebulous jet features seen in H$\alpha$, limited by the signal-to-noise of the jet knot compared to the bright background of the H [ii]{} region. We report proper motions and radial velocities in Table \[t:pm\]. Despite the relatively long time baseline, the morphology of the outflow has remained remarkably constant. Only the bow shocks and knots capping either side of the broad bipolar outflow show noticeable motion between the two epochs. Both knots move away from the globule along the jet axis with transverse velocities $\sim 100$ km s$^{-1}$ (see Figure \[fig:pm\]). The H$\alpha$-bright filament along the western limb of the jet does not appear to move at all ($<25$ km s$^{-1}$).
The eastern and western bow shocks move in opposite directions, with proper motion velocities near the tip of the shock of $\sim 60$ km s$^{-1}$. For jets that lie near the plane of the sky (as appears to be the case for HH 900, see Table \[t:pm\]), the difference between the inclination-corrected proper motions of adjacent knots can be used to estimate the shock velocity. Both bow shocks have velocities $\sim 40$ km s$^{-1}$ slower than the inner jet, suggesting shock velocities $\sim 40$ km s$^{-1}$. Fast, dissociative J-shocks ($v_{shock} > 30-40$ km s$^{-1}$) are expected to be bright in near-IR \[Fe [ii]{}\] lines [@nis02; @pod06] and indeed, both shocks have bright \[Fe [ii]{}\] emission (see Section \[ss:morph\] and Figure \[fig:hst\_ims\]).
Combining the transverse velocity of jet knots with the radial velocity from spectra, we can calculate the tilt angle, $\alpha = \mathrm{tan}^{-1} (v_r / v_T)$ and 3-D space velocity. With the low radial velocities in the H$\alpha$ spectrum, we find that the jet lies close to the plane of the sky (although we note the poor signal-to-noise of the spectrum). We derive a tilt angle of the jet away from the plane of the sky of $\lesssim 10^{\circ}$ (see Table \[t:pm\]).
---------- ----------- ------------- ---------------------- ---------------------- ------------------------- ------------- --------------------
Object $\delta$x $\delta$y v$_T$$^{\mathrm{a}}$ v$_R$$^{\mathrm{b}}$ velocity$^{\mathrm{c}}$ $\alpha$ age$^{\mathrm{d}}$
mas mas \[km s$^{-1}$\] \[km s$^{-1}$\] \[km s$^{-1}$\] \[degrees\] yr
HH 900 A 43 (1) -24 (0.4) 60 (1) ... 60 (1) ... 3523 (115)
HH 900 B 31 (1) -2 (2) 37 (2) ... 37 (2) ... 5364 (307)
HH 900 C 55 (0.1) -11 (1) 68 (1) ... 68 (2) ... 2803 (87)
HH 900 D 74 (1) -31 (4) 97 (3) ... 97 (3) ... 859 (34)
HH 900 E 76 (2) -25 (1) 97 (3) ... 97 (3) ... 682 (25)
HH 900 F -60 (1) 42 (2) 89 (2) -16 (4) 90 (5) -10 (0.3) 691 (24)
HH 900 G -77 (1) 39 (1) 104 (2) -7 (10) 105 (10) -4 (0.1) 746 (24)
HH 900 H -40 (3) 14 (1) 51 (3) ... 51 (3) ... 5379 (365)
HH 900 I -42 (1) 28 (1) 61 (2) ... 61 (2) ... 4664 (174)
PCYC 838 -3 (1) 2 (1) 4 (1) ... 4 (1) ... 11201 (3126)
PCYC 842 5 (3) -11.8 (0.4) 15 (2) ... 15 (2) ... 1393 (140)
---------- ----------- ------------- ---------------------- ---------------------- ------------------------- ------------- --------------------
-------------------------------------------------------------------------------------------------------------
Proper motions measured for the YSOs and the knots marked in Figure \[fig:pm\].
Uncertainties are listed in parentheses alongside the best-fit value.
$^a$ The transverse velocity, assuming a distance of 2.3 kpc.
$^b$ The radial velocity measured from spectra.
$^c$ The total velocity, assuming the average inclination when a radial velocity is not measured directly.
$^d$ Time for the object to reach its current location at the measured velocity, assuming ballistic motion.
-------------------------------------------------------------------------------------------------------------
\[t:pm\]
Protostar kinematics {#ss:yso_kin}
--------------------
We can also constrain the motion of the two protostars near HH 900. The YSO that lies along the western limb of the outflow (PCYC 838, see Figure \[fig:hst\_ims\]) falls within the slit we use to observe the western limb of the jet. We extract the spectrum of this source separately. After subtracting off extended emission, we find that the central velocity of the hydrogen recombination lines in the spectrum of the YSO constrain the heliocentric Doppler velocity of the protostar to be $\lesssim |5|$ km s$^{-1}$ [compared to the heliocentric systemic velocity of $-8.1\pm 1$ km s$^{-1}$ found for $\eta$ Car by @ns04].
We also measure the proper motions of both protostars. PCYC 838 moves in roughly the same direction as the jet with a transverse velocity of $4.3 \pm 1.3$ km s$^{-1}$ (see Table 2). Together, the radial and transverse velocities constrain the motion of PCYC 838 to $\lesssim 7$ km s$^{-1}$. We do not have spectra of the protostar near the bottom of the dark globule, PCYC 842 (see Figure \[fig:hst\_ims\]), but we can measure its proper motion in the plane of the sky. The projected motion of PCYC 842 is *toward* the globule with a transverse velocity of $15.4 \pm 3.0$ km s$^{-1}$ (see Figure \[fig:pm\]).
![New *HST*/ACS H$\alpha$ image of HH 900 with boxes used to measure proper motions overplotted. []{data-label="fig:pm"}](hh900_boxes.ps)
Discussion {#s:discussion}
==========
Spatial offset of \[Fe [ii]{}\] emission {#ss:feii_offset}
----------------------------------------
Bright \[Fe [ii]{}\] emission on either side of the dusty tadpole globule in HH 900 traces a symmetric, narrow jet. Like HH 901 and HH 902, two externally irradiated jets that are also in the Carina nebula, the brightest \[Fe [ii]{}\] emission turns on more than 1 away from the edge of the globule, indicating a gap between the globule edge and the beginning of bright \[Fe [ii]{}\] emission from the jet. The offset of the \[Fe [ii]{}\] emission may be interpreted as evidence that the mass-loss rate in the jet has recently decreased, since a lower density jet may not maintain a column of material sufficient to shield Fe$^+$ from further ionization [@rei13]. However, HH 900, HH 901, and HH 902 all have gaps of similar size, and a nearly synchronized end of an outflow phase from three physically unrelated jets would seem highly suspicious. The greatest similarity between the three sources is their proximity ($\lesssim 3$ pc in projection) to a young massive cluster with log(Q$_{\mathrm{H}}$)$> 50$ photons s$^{-1}$ (Tr 14 for HH 901 and HH 902, Tr 16 for HH 900). This argues that the gap is produced by something in the environment, rather than due to a recent change in the mass-loss rate.
The three dark, narrow streaks in the western limb of HH 900 that @smi10 interpret as dust filaments in the walls of the outflow cavity suggest another explanation for the offset \[Fe [ii]{}\] emission. These dusty streaks hint at a potentially large column of cold, dusty material that was blasted out of the globule by the jet. Close to the cloud edge, recently ejected dust and molecules may prevent UV emission from illuminating the inner jet.
We can measure reddening in the HH 900 jet using the ratio $\mathcal{R} = \lambda12567$/$\lambda16435$. $\mathcal{R}$ increases through the width of the jet (perpendicular to the direction of propagation), supporting the idea that external irradiation from massive stars in Tr 16 dominate the ionization and dust survival in the jet, not the nearby protostars or shocks. The increase in $\mathcal{R}$ also suggests that some entrained dust survives even in the outer regions of the jet.
Dust entrained in the outflow may obscure the inner jet, explaining the offset between \[Fe [ii]{}\] emission and the edge of the globule. We propose that \[Fe [ii]{}\] emission in HH 900 traces the collimated jet while broad H$\alpha$ and H$_2$ emission near the globule trace the wider body of an externally illuminated outflow cavity (see Figure \[fig:cartoon\]). Narrow \[Fe [ii]{}\] emission tracing the body of the jet has been seen in other HH jets in Carina [e.g. HH 666 and HH 1066, see @rei13]. Broad H$_2$ emission in HH 900 extends beyond the edge of the globule along the outflow direction, but only to the point where \[Fe [ii]{}\] emission begins, indicating that molecules persist in this intermediate zone. Extinction in the H$\alpha$ image also hints at the presence of entrained dust that survives in the H [ii]{} region close to the globule. Continuum-subtracted \[Fe [ii]{}\] and H$_2$ images are required to determine whether entrained dust and molecules also obscure the inner regions of HH 901 and HH 902.
HH 900 has two bow shocks separated from continuous emission in the inner jet, demonstrating that there has been a previous breakout of the jet into the H [ii]{} region. Despite this, new near-IR images show evidence for a significant column of dust and molecules outside the globule entrained by the jet. This may be expected from the youngest sources that are still surrounded by a substantial molecular envelope with only a small cavity cleared by the jet [@arc06]. Significant entrainment of molecular and dusty material by jets driven by more evolved protostars, or after many jet bursts, may be a direct consequence of the feedback-dominated environment (see Section \[ss:enviro\]). Compression of the molecular globule by massive star feedback may refill the cavity along the jet axis, increasing the amount of material entrained by a subsequent pulse of the jet.
Both HH 901 and HH 902 also show evidence for previous jet activity. If entrained material obscures the inner jet in all three cases, then this argues for a large column of material to be dragged out of the globule repeatedly by jet bursts. @rei13 estimate $n_H \gtrsim 10^5$ cm$^{-3}$ for the pillar housing HH 901, and we expect similarly high densities for the HH 902 and HH 900 clouds (also see Section \[ss:ysos\_globs\]). All three jets are embedded in the hot stellar wind bubble created by the many O-type stars in the nearby Tr 14/Tr 16 star cluster, so they are subject to similar amounts of feedback. Jets emerging from more diffuse globules [e.g. HH 666, see @smi04; @rei13 Reiter et al. in preparation] do not show this same offset from the globule edge.
The putative H$\alpha$ microjet {#ss:microjet}
-------------------------------
Based on the morphology in the H$\alpha$ image, @smi10 identify a possible microjet along the western limb of the broad HH 900 outflow. A bright point source with colours consistent with being a YSO lies in the middle of this bright H$\alpha$ filament and was identified as the possible driving source for the putative microjet. After examining the new imaging and spectroscopy presented here, we find it unlikely that this H$\alpha$-bright feature is a separate microjet. The key arguments against the microjet hypothesis are (1) the morphology in the \[Fe [ii]{}\] images, (2) radial velocities in spectra, and (3) proper motions as detailed below.
\(1) The \[Fe [ii]{}\] emission indicates that the main body of the HH 900 jet is symmetric about the center of the globule and shows no significant deviation at the position of the putative microjet. \[Fe [ii]{}\] intensity tracings through both sides of the jet are similar, with no significant difference in the brightness of the western limb as compared to the east (see Figure \[fig:feii\_tracings\] and Section \[ss:morph\]). It is possible that the microjet, driven by an unobscured star, is completely ionized, and therefore is only visible in H$\alpha$. However, if this were the case, the jet would have to be projected in front of the blueshifted limb of HH 900.
\(2) Linewidths of the \[Fe [ii]{}\] and H$_2$ emission from the western limb are similar to the eastern side of the jet, and appear to trace a single jet (see Figure \[fig:feii\_pvs\] and Section \[ss:velocity\]). Moreover, velocity changes on either side of the jet are monotonic. There is no evidence for two jet velocity components in the H$\alpha$ spectrum (although the signal-to-noise and velocities are low, see Figure \[fig:vis\_pvs\]).
\(3) The high degree of symmetry in the H$\alpha$ filament on either side of the protostar combined with the lack of any radial velocity changes suggests that if this feature is indeed a jet, it must lie almost exactly in the plane of the sky. However, our proper motion measurements constrain any oppositely directed, outward motions in the plane of the sky to $\lesssim 25$ km s$^{-1}$. Together, proper motions and spectra require that any outflowing gas must move slower than $\sim 30$ km s$^{-1}$, unusually slow for a jet driven by an unobscured protostar [see, e.g. @rei01; @rei14].
Without evidence for any bipolar motion in the H$\alpha$ filament away from PCYC 838, it seems unlikely that this putative “microjet” is a separate jet at all. Hypothetically, the YSO that appears to lie at its center may have been ejected from the globule in a dynamical encounter, with the motion of the ejected star through the larger bipolar outflow creating a bright H$\alpha$ tidal tail that looks like an H$\alpha$ microjet. However, hydrogen emission lines in the spectrum of the YSO (seen intersecting the jet in Figure \[fig:feii\_pvs\]) constrain the Doppler velocity of the protostar to be $v_{sys} \lesssim - 5$ km s$^{-1}$ whereas the H$\alpha$ emission at the same position is $\sim -12$ km s$^{-1}$. Moreover, we can constrain the proper motion of the YSO with the two epochs of *HST*/ACS data. We find that the star moves almost imperceptibly in the H$\alpha$ images, with a measured velocity in the plane of the sky of $4.3 \pm 1.3$ km s$^{-1}$. The YSO is not moving faster than the random motion of stars in the Carina nebula, making the ejection scenario unlikely. At these low velocities, the dynamical time required for the star to reach its current position is $\sim 11,000$ yr, longer than the estimated dynamical age of the two outer bow shocks in HH 900 ($\sim 2200$ yr, see Section \[ss:morph\]), and much longer than the age of the inner jet ($\sim 1000$ yr, see Section \[ss:ysos\_globs\]). Given the discrepancy in the timescales for the star and the jet, any interaction between the star and HH 900 is likely due to the jet recently moving past the position of the YSO. The simplest explanation is a chance alignment of the YSO with the apparent center of the H$\alpha$-bright filament. Indeed, @wol11 find a relatively high spatially averaged source density in the Tr 16 subcluster near HH 900 (45 src pc$^{-2}$ in sub-cluster 12).
The H$\alpha$-bright filament may result from local irradiation due to the small separation between the YSO and the HH 900 outflow. Local irradiation from the driving protostar has been invoked to explain bright optical and IR emission from the inner regions of protostellar jets in more quiescent regions [e.g. @rei00]. The IR-bright YSO that appears to lie in the middle of this filament must lie extremely close to the broad, bipolar outflow ($\lesssim 4.5 \times 10^{-4}$ pc or $\lesssim92$ au in the plane of the sky). Uncertainties in the three-dimensional geometry of the system may hide a slightly greater separation, although the presence of especially bright H$\alpha$ emission on either side of the YSO suggests that the protostar may affect the larger, apparently separate outflow. Thus, while our kinematic data rule out the possibility that this bright H$\alpha$ feature is a separate microjet, its origin remains elusive.
Mass-loss rate in the neutral jet {#ss:mdot}
---------------------------------
@smi10 estimated mass-loss rates, $\dot{M}_{jet}$, of the jets in Carina using the H$\alpha$ emission measure in ACS images. For HH 900, the wide inner jet in the east and the H$\alpha$-bright filament (the now refuted microjet) in the west represent the two highest $\dot{M}_{jet}$ in the sample. However, mass-loss rates calculated from the H$\alpha$ emission measure reflect the total $\dot{M}_{jet}$ only for fully ionized jets, and otherwise provide a lower-limit. @rei13 argue that the HH jets in Carina are not fully ionized, and that \[Fe [ii]{}\] emission traces neutral material in jets that are sufficiently dense to self-shield (i.e. to prevent Fe$^+ \rightarrow$ Fe$^{++}$). Relaxing the assumption that the jets are fully ionized and calculating the density required to self-shield generally increases the estimated mass-loss rate by at least an order of magnitude [@rei13]. For HH 900, the situation is unfortunately even more complex. H$\alpha$ and \[Fe [ii]{}\] have different spatial distributions and appear to trace different velocity components, suggesting that the mass-loss rate estimated from H$\alpha$ only samples the mass contained in a thin ionized skin on the outside of the outflow sheath entrained by the \[Fe [ii]{}\] jet. This is remarkable since the mass-loss rate derived from the H$\alpha$ emission measure was already the highest of all the jets in Carina [@smi10].
We can estimate the mass-loss rate of the HH 900 \[Fe [ii]{}\] jet by requiring that a neutral, cylindrical jet survives out to a distance $L_1$ before it is completely photoablated [@bal06; @rei13]. Here, we take $L_1$ to be the length of the continuous \[Fe [ii]{}\] jet (not including the bow shocks). For a cylindrical jet, $$\dot{M}_{jet} \approx \frac{L_1 f \mu m_H c_{II}}{2D} \left[ \frac{\alpha_B}{\pi r_{jet} Q_H sin(\beta)} \right]^{-1/2}$$ where $f \approx 1$ is the filling factor, $\mu \approx 1.35$ is the mean molecular weight, $m_H$ is the mass of hydrogen, $c_{II} \approx 11$ km s$^{-1}$ is the sound speed in ionized gas, and $\alpha_B \approx 2.6 \times 10^{-13}$ cm$^3$ s$^{-1}$ is the Case B recombination coefficient. CPD-59 2641 is an O5 V star located $\sim1$ from HH 900, or a projected distance $D\sim 0.7$ pc (which may underestimate the true separation by $\sim \sqrt{2}$). Assuming that CPD-59 2641 dominates the ionization, the ionizing photon luminosity is log$(Q_H)=49.22$ s$^{-1}$ [@smi06]. We measure the jet radius, $r_{jet} \approx 0.007$ pc, from the \[Fe [ii]{}\] images and assume that the angle $\beta$ between the jet axis and the direction of the ionizing radiation, is $\beta \approx 90^{\circ}$. From this, we find $\dot{M}_{jet} \gtrsim 7 \times 10^{-6}$ M$_{\odot}$ yr$^{-1}$ and $\dot{M}_{jet} \gtrsim 5 \times 10^{-6}$ M$_{\odot}$ yr$^{-1}$, respectively, for the eastern and western limbs of the jet [compared to $5.68 \times 10^{-7}$ M$_{\odot}$ yr$^{-1}$ and $6.20 \times 10^{-7}$ M$_{\odot}$ yr$^{-1}$, respectively, estimated from the H$\alpha$ emission measure, see @smi10].
As @rei13 found for other HH jets in Carina, using \[Fe [ii]{}\] as a diagnostic and without assuming the jet is fully ionized, $\dot{M}_{jet}$ is an order of magnitude larger than @smi10 estimate from the H$\alpha$ emission measure. Assuming that the mass *accretion* rate, $\dot{M}_{acc}$, onto the protostar is $\sim 10-100$ times larger than the mass-loss rate in the jet [e.g. @har95; @cal98], we find $\dot{M}_{acc} \approx 10^{-4}$ M$_{\odot}$ yr$^{-1}$. This points to a high luminosity from the driving protostar, either due to its relatively high mass or a recent accretion burst.
Protostars and globule survival {#ss:ysos_globs}
-------------------------------
The jet axis defined by the collimated \[Fe [ii]{}\] emission rules out the hypothesis that the driving source of the jet is the protostar at the bottom of the HH 900 globule [identified by @shi13]. Although the jet axis defined by the \[Fe [ii]{}\] emission in the WFC3-IR images from *HST* indicates a clear mis-match of the star and the jet (see Figure \[fig:fe\_contours\]), is it possible that the star began in the globule and was ejected, perhaps in a small-N stellar interaction [e.g. @rei10; @rei12]? If this is the case, the star must have been in the globule at the time that the innermost jet emission we detect in \[Fe [ii]{}\] was ejected. Assuming a jet velocity of 100 km s$^{-1}$ for the \[Fe [ii]{}\] emission located $\sim 1\farcs5$ from the globule edge (thus $\sim 2\farcs5$ from the presumed position of the driving source), this means the star would need to have been ejected within the last $\sim 275$ yr. To travel to its current location at the bottom of the globule in that time, the star would have to be moving at $\gtrsim 60$ km s$^{-1}$. However, PCYC 842 has an observed proper motion of $15.4$ km s$^{-1}$ *toward* the globule (see Section \[ss:yso\_kin\] and Table \[t:pm\]), clearly inconsistent with the ejection hypothesis.
Without compelling evidence for the recent ejection of the jet-driving source, we must conclude that an additional protostar remains hidden inside the dark globule. Symmetric \[Fe [ii]{}\] emission from the eastern and western sides of the jet, including a similar offset from the globule edge (see Figure \[fig:feii\_tracings\]), argues that the driving source is located inside the globule. Dark lanes in the blueshifted western limb of the jet apparent in the H$\alpha$ image connect the jet directly with the globule and require that it is driven by a protostar embedded within it. However, we do not detect IR emission associated with a protostar in the globule.
The high mass-loss rate in HH 900 makes the non-detection of a protostar particularly puzzling. @smi10 and @rei13 have argued that the HH jets in Carina are driven by intermediate-mass ($\sim 2-8$ M$_{\odot}$) protostars based on their high mass-loss rates, and the luminosity of candidate driving sources found for some of the jets support this interpretation [@smi04; @ohl12; @rei13; @rei14]. Even if the HH 900 driving source were a low-mass protostar, it would need to be undergoing an FU-Orionis-type outburst to create a jet of this strength [@cro87; @cal93], and would therefore likely have a high accretion luminosity. However, we note that the dynamical age of the inner jet is $\sim 1000$ years (for a jet velocity of $100$ km s$^{-1}$), a factor of $\sim 10$ longer than the typical decay time of an FU Orionis outburst [@hk96]. Small velocity differences between the redshifted and blueshifted sides of the jet suggest that the outflow axis lies close to the plane of the sky, and we find a small inclination angle for the jet($\alpha \lesssim 10^{\circ}$). Thus the circumstellar disc of the driving source will be seen almost edge-on. A YSO viewed through the midplane of an optically thick disc will be heavily extincted by a large column of circumstellar material. If this YSO is embedded in a dense and sufficiently opaque globule, no scattered light will escape the cloud and the protostar will remain unseen.
To estimate the amount of circumstellar material required to completely obscure the driving source, we use the YSO models from @rob06. Since we have only upper limits on the protostar postulated to be in the globule (see Section \[ss:ysos\]), we instead do a parameter search of the @rob06 models. We make the conservative estimate that the protostellar mass is $1.95 - 2.05$ M$_{\odot}$ and the opening angle of the envelope cavity angle is $5-10^{\circ}$. We estimate a small cavity opening angle based on the fact that H$\alpha$ and H$_2$ emission are not wider than the globule. Within this search criteria, we sample disc accretion rates as high as $\sim 10^{-5}$ M$_{\odot}$ yr$^{-1}$, on the lower end of the accretion rates we expect if $\dot{M}_{jet} \approx 0.01 - 0.1 \times \dot{M}_{acc}$ (see Section \[ss:mdot\]). The disc accretion rate will be even higher (leading to a higher accretion luminosity) if the driving source is more massive (closer to 8 M$_{\odot}$), or undergoing an FU Orionis outburst [with disc accretion rates $\gtrsim 10^{-4}$ M$_{\odot}$ yr$^{-1}$, e.g. @hc95; @cal98]. For the 12 models that satisfy this search criteria, the estimated line-of-sight extinction to the stellar surface ranges from $A_v \approx 5 \times 10^3 - 2 \times 10^7$. Allowing for larger cavity opening angles of $40-45^{\circ}$ corresponding to more evolved driving sources, $A_v$ may be as low as $\approx 3 \times 10^2$. Taking the lower bound, $A_v = 10^2$, we can make a crude estimate of the globule mass. Using $N_H \approx 1.8 \times 10^{21} \times A_v$ cm$^{-2}$ [@guv09], the column density along the line of sight must be at least $\sim 2 \times 10^{23}$ cm$^{-2}$. Assuming the globule is a uniform sphere, we can estimate the volume density. The diameter of the globule is $\sim 2$, corresponding to a linear size of $\approx 0.02$ pc at the distance of Carina. For a globule radius of $0.01$ pc or $3\times10^{16}$ cm, this rough estimate yields an average volume density $n \sim 6 \times 10^6$ cm$^{-3}$, among the higher volume densities inferred from dust emission of low-mass star forming cores ($\sim 10^4 - 10^6$ cm$^{-3}$, @eno07 [@eno08], although higher density cores have also been observed, e.g. @tok14). Taking the globule to be a uniform sphere, this density suggests a globule mass of $\gtrsim 1$ M$_{\odot}$.
@gre14 use the H$\alpha$ images of @smi10 to estimate the extinction in the HH 900 globule and calculate a mass of $\sim 14$ M$_{\mathrm{Jupiter}}$, two orders of magnitude smaller than our estimate. This single-wavelength assessment of the extinction provides only a lower limit on the true column of material in the globule and is compromised by emission from the surface of the limb-brightened globule. Indeed, requiring that the globule has survived photoevaporation from the many O-type stars in Tr 16 for $\gtrsim 3.5$ Myr [@smi06] argues for a much larger mass reservoir. The photoevaporation mass-loss rate of the globule is $$\label{eq:mdotphot}
\dot{M}_{phot} \approx 4 \pi r^2 n_H \mu m_H c_{II}$$ where $r$ is the radius of the globule, and $n_H$ is the density of neutral hydrogen. We can estimate the density at the ionization front from the requirement that ionizations are balanced by recombinations. This gives $$n_{IF} \approx \sqrt{ \frac{Q_H}{8 \pi D^2 r \alpha_B} }$$ where, as in Section \[ss:mdot\], we assume that the nearby O-type star CPD-59 2641 dominates the ionization. This yields $n_{IF} \sim 4000$ cm$^{-3}$. We can connect this to the density in the globule using the continuity equation, $n_{IF}c_{II} = n_H c_I$ where $c_I \approx 3$ km s$^{-1}$ is the sound speed in the molecular globule. For the estimated $n_H \sim 1.5 \times 10^4$ cm$^{-3}$, we find $\dot{M}_{phot} \sim 7 \times 10^{-6}$ M$_{\odot}$ yr$^{-1}$. This is within a factor of two of the photoevaporation rate @smi04b calculate for the nearby finger globule in Carina ($\sim 2 \times 10^{-5}$ M$_{\odot}$ yr$^{-1}$) as the higher flux of the ionizing source (log$(Q_H)=50.0$ s$^{-1}$) is roughly canceled by its larger separation ($\sim 6$). With this high photoevaporation rate, a 14 M$_{Jup}$ globule will rapidly be ablated, and completely destroyed in $\sim 2000$ years. In contrast, the remaining lifetime of a $\gtrsim 1$ M$_{\odot}$ globule will be $\gtrsim 10^5$ years.
$\begin{array}{|c|}
\hline
\includegraphics[trim=25mm 20mm 25mm 17.5mm,angle=90,scale=0.325]{hh900_cartoon_t1.ps} \\
\hline
\includegraphics[trim=25mm 20mm 25mm 17.5mm,angle=90,scale=0.325]{hh900_cartoon_t2.ps} \\
\hline
\includegraphics[trim=25mm 20mm 25mm 17.5mm,angle=90,scale=0.325]{hh900_cartoon_t3.ps} \\
\hline
\end{array}$
Comparison to molecular outflows {#ss:mol_outflows}
--------------------------------
@rei13 postulated that the irradiated jets in Carina are similar to the jets that drive molecular outflows, but are laid bare by the harsh UV radiation permeating Carina. In this picture, molecules entrained in the outflow that are swept into the H [ii]{} region will be quickly dissociated, and should only be observed within a few arcsec of the edge of the natal globule. Indeed, H$_2$ emission in HH 900 survives only $\sim 1.5$ away from the edge of the molecular globule, and must have been introduced into the H [ii]{} region recently. Other authors have invoked a two-component outflow model to explain the coexistence of a high velocity jet and a slower, broader molecular outflow created by a wide-angle wind (e.g. HH 111, @nag97 [@lee00]; HH 315, @arc02; HH 46/47, @arc13).
In HH 900, there is the further complication that the H$\alpha$ position-velocity diagram traces a smooth increase in velocity with increasing distance from the dark globule. In the case of HH 111, higher velocity emission far from the driving source has been seen in maps of the CO emission and interpreted as evidence of jet-bow-shock entrainment [e.g. @lee00; @lef07]. As a bow shock propagates through its parent cloud, it will sweep up material, creating a sheath that envelopes the jet traveling in its wake. The initial impulse of the jet turning on (or up) creates the Hubble-like velocity structure. A corresponding shell of material swept up in HH 900 will be externally irradiated by Tr 16 upon exiting the globule, creating an ionized outer layer that will emit H$\alpha$. Within $\sim 1.5$ of the globule, this ionized sheath still exhibits H$_2$ emission. The overall blueshift of the H$_2$ velocities ($\sim 20$ km s$^{-1}$) compared to \[Fe [ii]{}\] suggests that H$_2$ emission is dominated by the expansion of the outflow cavity. The abrupt end of H$_2$ emission where \[Fe [ii]{}\] emission begins may signal a transition in the column of obscuring, entrained material from a mix of atoms and molecules to primarily atomic gas. Beyond this point, H$\alpha$ emission narrows until it converges with the \[Fe [ii]{}\] morphology at the head of the inner jet, as expected for prompt entrainment. Indeed, the Hubble-like velocities in the H$\alpha$ position-velocity diagram suggest that HH 900 is the irradiated analog of an entrained molecular outflow.
Environmental interaction {#ss:enviro}
-------------------------
Theoretical work on small globules in H [ii]{} regions has explored their photoevaporation and the possibility that radiative compression may trigger star formation [e.g. @ber89; @ber90; @ber96; @gor02; @dal07a; @dal07b]. The small size of HH 900 and its survival despite its proximity to $\eta$ Car hint that the protostar forming within it may have been triggered by radiation-driven implosion. There remains an ambiguity between the triggering of new star formation and the uncovering, and/or possible acceleration, of star formation that would have happened anyway [@dal07b]. However, in feedback dominated environments, back-pressure from the photoevaporative flow off the globule will continue to compress the globule even as the protostar evolves. From detailed study of the ‘defiant finger,’ another small globule in the Carina nebula, @smi04b find that back-pressure from the photoevaporative flow is a factor of $\sim 5$ greater than thermal support in the globule. If similar conditions apply to HH 900, then pressure from the ionization front may alter the physical conditions of the local star forming environment. By increasing the local pressure, feedback from massive stars may affect the envelope accretion rate, the amount of molecular and dusty material entrained by the jet (as well as the degree that it can clear an outflow cavity in the molecular envelope), and the rate at which the developing protostar destroys the globule from within. Clearly, hydrodynamic models of the evolution of protostars in irradiated globules would be interesting.
Conclusions {#s:conclusion}
===========
We present new imaging and spectroscopy of HH 900, a peculiar protostellar outflow in the Carina nebula. H$\alpha$ images reveal an unusually wide-body bipolar outflow while new narrowband \[Fe [ii]{}\] 1.26 and 1.64 images of HH 900 obtained with WFC3-IR trace a collimated jet. \[Fe [ii]{}\] emission remains remarkably symmetric on either side of the globule, suggesting that the unseen driving source lies roughly at the center of the globule. New narrowband H$_2$ images from GSAOI reveal extended molecular emission from the outflow just outside the globule that disappears at the same separation where the \[Fe [ii]{}\] emission from the jet grows bright ($\sim 1.5$). The H$_2$ $\rightarrow$ \[Fe [ii]{}\] transition zone is seen on both sides of the outflow and is roughly symmetric. Both H$\alpha$ and H$_2$ emission are as wide as the globule when they emerge into the H [ii]{} region. Their width and kinematics suggest that these lines trace dusty and molecular material entrained by the jet that was recently dragged into the H [ii]{} region.
Optical and near-IR spectra reveal separate and distinct velocity components traced by \[Fe [ii]{}\], H$_2$, and H$\alpha$ emission. Both \[Fe [ii]{}\] lines trace the steady jet emission from the red- and blueshifted (eastern and western, respectively) limbs of the jet. On the other hand, velocities in the H$\alpha$ and H$_2$ spectra increase further away from the globule, similar to the Hubble-like velocity structure observed in molecular outflows.
Unlike other HH jets with a *Spitzer*-identified candidate driving source near the jet axis, \[Fe [ii]{}\] emission from HH 900 does not connect the jet to either of the two YSOs near the globule. The jet axis defined by \[Fe [ii]{}\] emission clearly bisects the globule, and dust lanes near the globule surface reveal the impact of the jet breaking out of the globule. Together, this argues for the existence of a third protostar that remains deeply embedded in the globule and is obscured by high column density, perhaps by an edge-on circumstellar disc. The invisibility of this protostar may be interpreted as evidence that the source is extremely young, that the globule has been compressed to high densities by radiation from nearby massive stars, or both.
Our data allow us to reject the hypothesis that the YSO (PCYC 838) along the western limb of broader bipolar outflow drives a separate microjet. The \[Fe [ii]{}\] jet structure is symmetric and center on the globule (not the star) and there is no evidence for multiple velocity components in either H$\alpha$ or \[Fe [ii]{}\] spectra. Proper motions measured in H$\alpha$ images taken with *HST* $\sim 9$ yr apart are consistent with zero velocity, calling into question the nature of this unusual feature. Low velocities rule out a dynamical origin. Nevertheless, the small projected separation between HH 900 and this YSO suggests that the YSO may illuminate the nearby bipolar outflow, creating an H$\alpha$-bright filament. However, the simplest interpretation is that this YSO is simply a chance alignment with the HH 900 outflow.
The bright, collimated \[Fe [ii]{}\] jet, together with wider-angle H$\alpha$ and H$_2$ components point to a powerful jet breaking out of a small globule that is sufficiently massive to have survived the harsh radiative environment of the Carina nebula. Different morphologies and velocity structures from the optical and IR emission lines in HH 900 reveal the protostellar jet itself and offer a rare glimpse of the material it entrains before it is destroyed in the H [ii]{} region. This supports the interpretation that the irradiated outflows in the Carina reflect the same underlying phenomena driving molecular outflows in embedded regions, but that they are laid bare and illuminated by the harsh radiative environment.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Rob Simcoe for his assistance with reduction of the FIRE data and Jayadev Rajagopal for assistance with the Gemini Observations. MR would like to thank Kaitlin Kratter for helpful discussions. Support for this work was provided by NASA through grants AR-12155, GO-13390, and GO-13391 from the Space Telescope Science Institute. This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These *HST* observations are associated with programs GO 10241, 10475, 13390, and 13391. Gemini observations are from the GS-2013A-Q-12 science program.
Anderson J., 2006, “The 2005 HST calibration workshop: Hubble after the transition to two-gyro mode”, Proceedings of a workshop held at the Space Telescope Science Institute, Baltimore, Maryland, October 26-28, 2005. Eds Koekemoer A.M., Goudfrooij P., Dressel L.L. Greenbelt, MD, 11 Anderson J., King I.R., 2006, “PSFs, Photometry, and Astronomy for the ACS/WFC”, 2006, 1 Anderson J., et al., 2008a, AJ, 135, 2055 Anderson J., et al., 2008b, AJ, 135,2114 Anderson J., van der Marel R.P., 2010, ApJ, 710,1032 Arce H.G., Goodman A.A., 2002, ApJ, 575, 928 Arce H.G., Sargent A.I., 2005, ApJ, 624, 232 Arce H.G., Sargent A.I., 2006, ApJ, 646, 1070 Arce H.G., Mardones D., Corder S.A., Garay G., Noriega-Crespo A., Raga A.C., 2013, ApJ, 774, 39 Bally J., Reipurth B., 2001, ApJ, 546, 299 Bally J., Licht D., Smith N., Walawender J., 2006, AJ, 131, 473 Bally J., Cunningham N.J., Moeckel N., Burton M.G., Smith N., Frank A., Nordlund A., 2011, ApJ, 727, 113 Bertoldi F., 1989, ApJ, 346, 735 Bertoldi F., McKee C.M., 1990, ApJ, 354, 529 Bertoldi F., Draine B.T., 1996, ApJ, 458, 222 Beltrán M.T., Estalella R., Girart J.M., Ho P.T.P., Anglada G., 2008, A&A, 481, 93 Beuther H. et al., 2008, A&A, 481, 169 Calvet N., 1998, AIPC, 431, 495 Calvet N., Hartmann L., Kenyon S.J., 1993, ApJ, 402, 623 Carrasco E.R., et al., 2012, Proc. SPIE 8447, Adaptive Optics System III, 84470N Carrasco-González C., et al., 2012, ApJL, 752, 29 Croswell K., Hartmann L., Avrett E.H., 1987, ApJ, 312, 227 Currie D.G. et al., 1996, AJ, 112, 1115 Dale J.E., Bonnell I.A., Whitworth A.P., 2007a, MNRAS, 375, 1219 Dale J.E., Clark P.C., Bonnell I.A., 2007b, MNRAS, 377, 535 Devine D., Bally J., Reipurth B. Heathcote S., 1997, AJ, 114, 2095 Enoch M.L., Glenn J., Evans N.J., II, Sargent A.I., Young K.E., Huard T.L., 2007, ApJ, 666, 982 Enoch M.L., Evans N.J., II, Sargent A.I., Glenn J., Rosolowsky E., Myers P., 2008, ApJ, 684, 1240 Ferreira J., Dougados C., Cabrit S., 2006, A&A, 453, 785 Goddi C., Humphreys E.M.L., Greenhill L.J., Chandler C.J., Matthews L.D., 2011, ApJ, 728, 15 Gorti U., Hollenbach D., 2002, ApJ, 573, 215 Grenman T., Gahm G., 2014, A&A, 565, 107 Güver T., Özel F., 2009, MNRAS, 400, 2050 Guzmán A.E., Garay G., Brooks K.J., Voronkov M.A., 2011, ApJ, 736, 150 Hartigan P., Edwards S. Ghandour L.J. 1995, ApJ, 452, 736 Hartigan P., Morse J.A., Reipurth B., Heathcote S., Bally J., 2001, ApJL, 559, 157 Hartigan P., Reiter M., Smith N., Bally J., 2015, AJ, submitted Hartmann L., Calvet N., 1995, AJ, 109, 1846 Hartmann L., Kenyon, S.J., 1996, ARA&A, 34, 207 Klaassen P.D., et al. 2013, A&A, 555, 73 Kraus S., et al. 2010, *Nature*, 466, 339 Lee C.-F., Mundy L.G., Reipurth B., Ostriker E.C., Stone J.M., 2000, ApJ, 542, 925 Lefloch B., Cernicharo J., Reipurth B., Pardo J.R., Neri R., 2007, ApJ, 658, 498 Martí J., Rodriguez L.F., Reipurth B., 1993, ApJ, 416, 208 McGregor P., et al., 2004, Proc. SPIE, 5492, Ground-based Instrumentation for Astronomy, 1033 Morse J.A., Kellogg J.R., Bally J., Davidson K., Balick B., Ebbets D., 2001, ApJL, 548, 207 Nagar N.M., Vogel S.N., Stone J.M., Ostriker E.C., 1997, ApJ, 482, L195 Neichel B., et al. 2014, MNRAS, 440, 1002 Nisini S., Caratti o Garatti A., Giannini T., Lorenzetti D., 2002, A&A, 393, 1035 Ohlendorf H., Preibisch T., Gaczkowski B., Ratzka T., Grellmann R., McLeod A.F., 2012, A&A, 540, 810 Podio L., Bacciotti F., Nisini B., Eislöffel J., Massi F., Giannini T., Ray T.P., 2006, A&A, 456, 189 Povich M. et al. 2011, ApJS, 194, 14 Preibisch T., Ratzka T., Gehring T., Ohlendorf H., Zinnecker H., King R.R., McCaughrean M.J., Lewis J.R., 2011, A&A, 530, 40 Reipurth B., Yu K.C., Heathcote S., Bally J. Rodríguez L.F., 2000, AJ, 120, 1449 Reipurth B., Bally J., 2001, ARA&A, 39, 403 Reipurth B., Mikkola S., Connelley M., Valtonen M., 2010, ApJ, 725, 56 Reipurth B., Mikkola S., 2012, *Nature*, 492, 221 Reiter M., Smith N., 2013, MNRAS, 433, 2226 Reiter M., Smith N., 2014, MNRAS, accepted Rigaut F., et al. 2014, MNRAS, 437, 2361 Robitaille T.P., Whitney B.A., Indebetouw R., Wood K., Denzmore P., 2006, ApJS, 167, 256 Shepherd D.S., Testi L. Stark D.P., 2003, ApJ, 584, 882 Shinn J.-H. et al., 2013, ApJ, 777, 45 Simcoe R.A. et al., 2008, SPIE, 7014, 27 Simcoe R.A. et al., 2010, SPIE, 773, 38 Simcoe R.A. et al., 2013, PASP, 125, 270 Smith R.G., 1987, MNRAS, 227, 943 Smith N., 2002, MNRAS, 331, 7 Smith N., Bally J., Morse J.A., 2003, ApJ, 587, L105 Smith N., 2004, MNRAS, 351, L15 Smith N., Bally J., Brooks K.J., 2004a, AJ, 127, 2793 Smith N., Barbá R.H., Walborn N.R., 2004b, MNRAS, 351, 1457 Smith N., 2006, MNRAS, 367, 763 Smith N., Hartigan P., 2006, ApJ, 638, 1045 Smith N., Bally J., Walborn N., 2010, MNRAS, 405, 1153 Smith N. et al., 2010, MNRAS, 406, 952 Sohn S.T., Anderson J., van der Marel R.P., 2012, ApJ, 753, 7 Takahashi S., Shimajiri Y., Takakuwa S., Saito M., Kawabe R., 2007, IAUS, 237, 479 Tokuda K., et al., 2014, ApJL, 789, 4 Walawender J., Bally J., Reipurth B., Aspin C. 2004, AJ, 124, 2809 Wolk S.J., et al., 2011, ApJS, 194, 12 Yusef-Zadeh F., Biretta J., Wardle M., 2005, ApJ, 624, 246
\[lastpage\]
[^1]: E-mail: [email protected]
|
addtoreset[equation]{}[section]{}
[Perfect Lattice Perturbation Theory:\
\
A Study of the Anharmonic Oscillator]{}
W. Bietenholz$^{\rm a}$ and T. Struckmann$^{\rm b}$
$^{\rm a}$ HLRZ c/o Forschungszentrum Jülich\
52425 Jülich, Germany\
$^{\rm b}$ Physics Department\
University of Wuppertal\
D-42097 Wuppertal, Germany
Preprint HLRZ 1997-67, WUB 97-35
As an application of perfect lattice perturbation theory, we construct an $O(\lambda)$ perfect lattice action for the anharmonic oscillator analytically in momentum space. In coordinate space we obtain a set of 2-spin and 4-spin couplings $\propto \lambda$, which we evaluate for various masses. These couplings never involve variables separated by more than two lattice spacings.
The $O(\lambda)$ perfect action is simulated and compared to the standard action. We discuss the improvement for the first two energy gaps $\Delta E_{1}$, $\Delta E_{2}$ and for the scaling quantity $\Delta E_{2}/\Delta E_{1}$ in different regimes of the interaction parameter, and of the correlation length.
Introduction
============
The only non-perturbative access to complicated 4d quantum field theories, such as QCD, which proved successful, are Monte Carlo simulations on the lattice. They necessarily take place at a finite lattice spacing $a$ and in a finite size $L$. In order to reveal information about continuum physics in an infinite volume, we have to require $a << \xi << L$, where $\xi$ is the correlation length. In particular the finiteness of $\xi /a$ causes serious systematic errors in practical simulations. It is now very fashionable to fight such artifacts by using “improved lattice actions” [@LAT97]. These are discretizations of the continuum action, which are supposed to display the correct scaling behavior down to a much shorter correlation length in lattice units, than it is the case for the standard lattice action.
In the literature, there are mainly two strategies to construct improved lattice actions, in particular for QCD. The first one is called Symanzik’s program [@Sym]. One tries to eliminate the lattice spacing artifacts order by order in $a$ – similar to the Runge and Kutta procedure for the numerical solution of ordinary differential equations. This is achieved by adding irrelevant operators. On the classical level, and in the framework of on-shell improvement [@OnShell], the standard Wilson action for QCD could be improved to $O(a)$ analytically by adding the so-called clover term [@SW]. On the quantum level, the coefficient of this term gets renormalized, and the quantum correction was first estimated numerically by a mean field approach [@Tad]. The complete $O(a)$ improvement was finally determined by the ALPHA collaboration based on extensive simulations [@alpha]. However, it seems hardly feasible to carry on this program beyond $O(a)$.
The alternative method uses renormalization group concepts to construct quasi-perfect actions. These are approximations to perfect actions, i.e. to actions which are completely free of cutoff artifacts [@WilKog]. As a fundamental difference from Symanzik’s program, this method is non-perturbative with respect to $a$. As a first step, this program can be realized perturbatively (in the interaction), which yields analytic expressions for the perfect quark-gluon and 3-gluon vertex functions [@QuaGlu; @StL; @KODiss]. [^1] Thus one eliminates all artifacts of $O(a^{n})$ and $O(ga^{n})$, such that the remaining artifacts are of $O(g^{2}a)$ and beyond ($g$ is the gauge coupling). This is opposed to the action of Ref. [@alpha], which is free of artifacts in $O(g^{n}a)$, but plagued for instance by systematic errors in $O(a^{2})$.
An extension of this program is the construction of “classically perfect actions” [@Has]. This approach, which is designed particularly for asymptoti-cally free models, is non-perturbative also with respect to the coupling $g$. Using a multigrid procedure, one identifies the fixed point action of an renormalization group transformation. This can be done solely by minimization – the functional integral reduces to a classical field theory problem – and the fixed point action then serves as an approximatively perfect (“classically perfect”) action at finite correlation length too. In a sequence of toy models, it turned out that classically perfect actions are excellent approximations to (quantum) perfect actions, in the sense that they drastically suppress lattice spacing artifacts. The improvement achieved in this way goes far beyond first order Symanzik improvement. This has been observed for the 2d O(3) and CP(3) model [@Has; @Ruedi], the Schwinger model [@Lang] and the 1d XY model [@rotor]. In principle that program can be extended also beyond classical perfection, if one performs e.g. one real space MCRG step at finite correlation length, starting from a classically perfect action.
The construction, which is non-perturbative in $a$ and in $g$, is presumably the climax of the improvement program. However, in perfect and also in classically perfect actions the couplings tend to involve infinite distances, and we can at best achieve locality in the sense of their exponential decay. For practical purposes a truncation is needed, which does some harm to the quality of the improvement. This is the main reason why the second, more sophisticated, improvement program could not be applied yet in a satisfactory way to QCD.\
Here we focus on the perturbatively perfect action. It has potential applications with two respects: it can either be used directly, or as a starting point of the non-perturbative multigrid improvement [@StL]. A direct application of a truncated perfect quark-gluon vertex function – together with truncated perfect free quarks – to heavy quarks is presently under investigation. Preliminary results for the charmonium spectrum are given in Ref. [@vert].
The purpose of this paper is to test specifically such a direct application in a very simple situation. Our model is the 1d $\lambda \phi^{4}$ model, or anharmonic oscillator.
As a toy model, the anharmonic oscillator has a number of virtues: we can achieve an excellent locality, such that our $O(\lambda )$ perfect action does not need any truncation of the couplings. Thus the perturbative improvement can be tested separately, without admixture of truncation effects. Moreover, our construction is based on continuum perturbation theory, and there we do not encounter any divergent loop integrals, in contrast to field theory ($d>1$). Finally, the reduction to quantum mechanics has the advantage that the couplings we identify do not get renormalized in the full theory.
On the other hand, it is exceptionally difficult in our model to demonstrate an improvement compared to the standard action, because the latter is also very good in this case. For the harmonic oscillator it is even perfect itself, for small interaction – the regime of interest here – it is still very good, and even for moderate interactions it performs amazingly well. As a further problem we note that the performance of continuum perturbation theory, which our improvement is based on, is rather poor in this model.
The advantages and disadvantages listed above are specific for the one dimensional case.
The model in the continuum
==========================
The observables we are going to consider can be evaluated directly in the continuum to a fantastic accuracy. Our interest is of course not in their values, but solely in the comparison of lattice artifacts in different discretizations. We want to test the success of a specific improvement program for the lattice action.
Nevertheless we have to start by recalling some properties of the conti-nuum system. To fix the (field theoretic) notation, we denote the Euclidean action as $$\label{contact}
s[{\varphi}] = \int dt \ \Big[ \frac{1}{2} \dot {\varphi}(t)^{2}
+ \frac{m^{2}}{2} {\varphi}(t)^{2} + \lambda {\varphi}(t)^{4} \Big] .$$ Throughout this paper we assume $m,\ \lambda \geq 0$, hence we only study the “symmetric phase” (as opposed to the double well). We consider the energy eigenvalues $E_{n}$, more precisely we are going to measure directly the energy gaps $\Delta E_{n} \doteq E_{n}-E_{0}$. An additive constant all over the spectrum is out of control, and not much of interest either. The simplest scaling quantity is $$\label{scal}
\frac{\Delta E_{2}}{\Delta E_{1}} \equiv \Delta E_{2} \cdot \xi \ ,$$ where $\xi$ is the correlation length. Moreover, there is the simple relation $$\label{rescale}
\frac{\Delta E_{n}(\mu^{2}m^{2}, \mu^{3} \lambda )}
{\Delta E_{n}(m^{2},\lambda )} = \mu \ , \quad {\rm any~}\mu >0 \ .$$ This just follows from rescaling $t \to t /\mu$ and momentum $k \to \mu k$ in the Hamiltonian ([*not*]{} rescaling all dimensional quantities). [^2]
The quantity (\[scal\]) is a [*scaling quantity*]{} in the strict sense, i.e. a dimensionless ratio of physical observables. On the other hand, quantities like (\[rescale\]) may also involve unphysical normalization factors. In a field theoretic language they correspond to the [*asymptotic scaling*]{}. Actually, improved actions are designed for an improvement of scaling, but the influence on asymptotic scaling is of interest too. It has been observed before for the Gross Neveu model [@GN] and for pure SU(3) gauge theory [@TdG] that “accidentally” the latter is also improved for (quasi-)perfect actions.
For $m>0$, the strength of the interaction depends on the dimensionless parameter $$\tilde \lambda \doteq \frac{\lambda}{m^{3}} \ ,$$ which is obvious from eq. (\[rescale\]). The energy eigenvalues can be expanded in $\tilde \lambda$, but these expansions diverge at large orders (the coefficients oscillate and their absolute values grow faster than any polynomial). [^3] However, a truncated series is still useful at ${\tilde \lambda}<<1$ (this situation is familiar from QED). [^4] The coefficients of these expansions have been derived many times in the literature, for instance in Ref. [@expa], $$\begin{aligned}
\frac{\Delta E_{1}}{m}({\tilde \lambda}) & \simeq &
1 + 3 \tilde \lambda - 18 {\tilde \lambda}^{2} + \frac{1791}{8} {\tilde \lambda}^{3}
- 3825 {\tilde \lambda}^{4} , \nonumber \\
\frac{\Delta E_{2}}{m}({\tilde \lambda}) & \simeq &
2 + 9 \tilde \lambda - \frac{297}{4} {\tilde \lambda}^{2} +
\frac{9873}{8} {\tilde \lambda}^{3} - \frac{1772685}{64} {\tilde \lambda}^{4} ,
\nonumber \\ \label{expand}
\frac{\Delta E_{2}}{\Delta E_{1}} ({\tilde \lambda}) & \simeq &
2 + 3 \tilde \lambda - \frac{189}{4} {\tilde \lambda}^{2} +
\frac{7857}{8} {\tilde \lambda}^{3} - \frac{1569069}{64} {\tilde \lambda}^{4} \ .
$$ Tables of explicit values at finite ${\tilde \lambda}$ are given for example in Refs. [@Ban; @Bis; @BBCK]. However, in particular for small ${\tilde \lambda}$ they can easily be reproduced from an eigenvalue problem, as described for instance in Ref. [@Bis].
Since our construction in the following sections is also perturbative in ${\tilde \lambda}$, it is important to know how the perturbation series for the above quantities behave. Figs. \[figE1\] and \[figE21\] compare the exact function to the truncated expansion in first, second, third and fourth order.
We see that the applicability of truncated expansions is restricted to very small values of ${\tilde \lambda}$. The exact range depends on the quantity considered; it agrees with the relative magnitude of the coefficients in the expansions (\[expand\]). This range gradually expands as we proceed to higher orders.
The $O(\lambda )$ perfect action in momentum space
==================================================
If a system is given by some lattice action, then its physical properties remain unaltered under a block variable renormalization group transformation (RGT) [@WilKog]. For suitable RGT parameters and infinite correlation length, an infinite number of iterations may lead to a finite fixed point action (FPA). A FPA is an example for a perfect action, since it is invariant under an RGT, and hence insensitive to the lattice spacing. Perfect actions also exist at any finite correlation length [@WilKog]. They reveal exact continuum scaling at any lattice spacing. For free or perturbatively interacting fields, they can be computed analytically in momentum space. This calculation simplifies if we send the blocking factor to infinity, which amounts to a technique that we call “blocking from the continuum”. Recently this method has been applied to the Schwinger model [@Schwing] and to QCD [@QuaGlu]. Here we want to apply it to construct a lattice action for the anharmonic oscillator, which is perfect to $O({\tilde \lambda})$. Our blocking uses the standard, piece-wise constant weight distribution for the original variables, and a Gaussian transformation term. For an alternative ansatz, closer to the spirit of a decimation RGT, see Ref. [@NKF].
Our perfect lattice action $S[\phi ]$ is determined by the functional integral $$\begin{aligned}
e^{-S[\phi ]} &=& \int D{\varphi}\exp \{ -s[\varphi ]-R[\phi ,\varphi ] \} \ , \nonumber \\
R[\phi ,\varphi ] &=& \frac{1}{2 \alpha} \sum_{x \in {Z \!\!\! Z}} \Big(
\phi_{x} - \int_{x-1/2}^{x+1/2} dt \ \varphi (t) \Big)^{2} \ .\end{aligned}$$ Here $\phi$ is the lattice field, $x$ are the lattice sites and the continuum action $s[\varphi ]$ is given in eq. (\[contact\]). The RGT parameter $\alpha > 0$ is arbitrary; for any value of $\alpha$ the RGT keeps the partition function invariant, $$Z = \int D {\varphi}\ e^{-s[{\varphi}]} \propto \int D \phi \ e^{-S[\phi ]},
\quad (D\phi \doteq \prod_{x\in {Z \!\!\! Z}} \int d\phi_{x}),$$ and with it all expectation values. The limit $\alpha \to 0$ corresponds to the well known “$\delta$ function RGT”.
In momentum space, this expression can be written as $$\begin{aligned}
e^{-S[\phi ]} &=& \int D{\varphi}D \sigma \nonumber
\exp \Big\{ -\frac{1}{2\pi} \int_{-\pi}^{\pi} dk \ \times \\
&& \Big[ \frac{1}{2} \sum_{l\in {Z \!\!\! Z}} {\varphi}(-k-2\pi l)
[(k+2\pi l)^{2}+m^{2}] {\varphi}(k+2\pi l) \nonumber \\
&& + i \sigma (-k)[\phi (k) - \sum_{l\in {Z \!\!\! Z}}{\varphi}(k+2\pi l)
\Pi (k+2\pi l)] \nonumber \\
&& + \frac{1}{2} \alpha \ \sigma (-k) \sigma (k) \Big] \Big\}
\times \nonumber \\
&& \Big\{ 1 - \frac{\lambda}{(2\pi )^{3}} \int d^{4}p \
{\varphi}(p_{1}) {\varphi}(p_{2}) {\varphi}(p_{3}) {\varphi}(p_{4}) \
\delta (\sum_{i=1}^{4}p_{i}) \nonumber \\
&& + O(\lambda^{2}) \Big\} , \nonumber \\
&& \Pi (k) \doteq \frac{\hat k}{k} \ , \quad
\hat k \doteq 2 \sin \frac{k}{2} \ ,\end{aligned}$$ where we have introduced an auxiliary lattice field $\sigma$.
We denote the free continuum propagator as $$\Delta (k) = \frac{1}{k^{2}+m^{2}} ,$$ and $$G(k) = \sum_{l\in {Z \!\!\! Z}} \Delta (k+2\pi l) \Pi(k+2\pi l)^{2}+\alpha$$ is the perfect free lattice propagator, as we will see. At $m=0$ this is the fixed point propagator, which has been calculated for free scalar theories first by Bell and Wilson, and it characterizes the FPA for the O($N$) model in the large $N$ limit as well [@HS].
We now choose the special value $\alpha =
(\sinh m - m)/m^{3}$, which renders the free lattice action “ultralocal” [@Anton], i.e. it only couples nearest neighbor lattice variables, [^5] $$\label{nn}
G(k) = \frac{\sinh m \cdot \hat m^{2}}{m^{3}}
\frac{1}{\hat k^{2} + \hat m^{2}} \ , \quad
\hat m \doteq 2 \sinh \frac{m}{2} .$$
Our first step is the substitution $$\tilde {\varphi}(k+2\pi l) \doteq {\varphi}(k+2\pi l) - i \sigma (k)
\Delta (k + 2\pi l) \Pi (k+2\pi l),$$ which allows us to integrate out the continuum variable $\tilde {\varphi}$. We omit the constant factor in the Gaussian integral [^6] and obtain $$\begin{aligned}
e^{-S[\phi ]} &=& \int D \sigma \exp \Big\{ - \frac{1}{2\pi }
\int_{-\pi}^{\pi} dk \ [i\sigma (-k) \phi (k) +
\frac{1}{2} \sigma (-k) G(k) \sigma (k)] \Big\}
\times \nonumber \\
&& \Big\{ 1 - 3 \lambda \Big( \frac{1}{2m} \Big)^{2} \nonumber \\
&& + 6 \lambda \frac{1}{2m} \Big[ \prod_{i=1}^{2}
\sum_{n_{i}\in {Z \!\!\! Z}} \frac{1}{2\pi} \int_{-\pi}^{\pi}
dp_{i} \ \sigma (p_{i}) \Delta (p_{i}+ 2\pi n_{i})
\Pi (p_{i}+2\pi n_{i}) \Big] \nonumber \\
&& \qquad \times 2 \pi \delta (p_{1}+p_{2}) \delta_{n_{1},-n_{2}}
\nonumber \\
&& - \lambda \Big[ \prod_{i=1}^{4} \sum_{n_{i}} \frac{1}{2\pi}
\int_{-\pi}^{\pi} dp_{i} \ \sigma (p_{i}) \Delta (p_{i}
+ 2\pi n_{i}) \Pi (p_{i}+2\pi n_{i}) \Big] \nonumber \\
&& \qquad \times 2 \pi \delta ( \sum_{i=1}^{4}[p_{i}+2\pi n_{i}])
+ O(\lambda^{2})\Big\} .\end{aligned}$$ In a sense, this computation goes beyond the perfect QCD vertex function of Ref. [@QuaGlu], because it includes – for the first time in the construction of a perfect action – a loop calculation, i.e. a [*quantum correction*]{}. The continuum loop integral reads $$\Delta (x)\vert_{x=0} = \frac{1}{2\pi} \int dk \
\Delta (k) = \frac{1}{2m} \qquad (m>0),$$ which has been inserted above. In field theory we would encounter divergences at this point, which could be regularized by some standard technique in the continuum. Here the expression is finite from the beginning, and we will see that even the limit $m\to 0$ can safely been taken at the end, when we identify the couplings in the $O({\tilde \lambda})$ perfect lattice action.
After performing a second substitution, $$\tilde \sigma (k) \doteq \sigma (k) + iG(k)^{-1} \phi (k),$$ we can integrate $\tilde \sigma$, and we arrive at a lattice action of the form $$\begin{aligned}
S[\phi ] &=& \frac{1}{2\pi} \int_{-\pi}^{\pi} dk \ \frac{1}{2}
\phi (-k) G(k)^{-1} \phi (k) \nonumber \\
&& + \lambda \Big[ A + \frac{1}{2\pi} \int_{-\pi}^{\pi}
dk \ \phi (-k) B(k) \phi (k) \nonumber \\
&& + \frac{1}{(2\pi )^{3}} \int_{-\pi}^{\pi} d^{4}p \
C(p) \phi (p_{1}) \phi (p_{2}) \phi (p_{3}) \phi (p_{4})
\Big] + O(\lambda^{2}) . \label{form}\end{aligned}$$ This confirms that $G$ is the perfect free lattice propagator for the RGT chosen here. Since it differs from the standard propagator only by a constant factor and a transformation of the mass, we can also confirm the statement that for the [*harmonic*]{} oscillator the standard action is perfect already. [^7] This is very specific for the case $d=1$ considered here.
The functions $B(k)$ and $C(p)=C(p_{1},p_{2},p_{3},p_{4})$ represent additional 2-variable and 4-variable couplings, while $A$ is a constant, which is not really of interest to $O(\lambda )$.
The Wick contraction of two isolated $\tilde \sigma$ variables yields the lattice loop integral $$\begin{aligned}
\gamma (m) &\doteq & \frac{1}{2\pi} \sum_{l \in {Z \!\!\! Z}}
\int_{-\pi}^{\pi} G(k)^{-1} \Delta (k + 2\pi l)^{2}
\Pi (k+2\pi l)^{2} \nonumber \\ \nonumber
&=& \frac{1}{m\cdot \sinh m \cdot \hat m^{2}}
\Big[ 2 \cosh m + e^{-m} (1+\sinh m)- \frac{3}{m} \sinh m \Big] \\
&=& \frac{1}{2m} - \frac{7}{30} + \frac{11}{630}m^{2} + O(m^{4}) ,\end{aligned}$$ which obeys $\gamma (m) - \gamma (-m) = 1/m$.
If we just insert this everywhere, we obtain the following $O(\lambda)$ terms, $$\begin{aligned}
A_{0} &=& 3 \Big[ \frac{1}{2m} - \gamma (m) \Big]^{2} \nonumber \\
B_{0}(k) &=& 3 \Big[ \frac{1}{m} - 2\gamma (m)\Big]
G(k)^{-2} \sum_{l\in {Z \!\!\! Z}} \Delta (k+2\pi l)^{2}
\Pi (k+2\pi l)^{2} \nonumber \\
&=& \Big[ \frac{1}{m} - 2 \gamma (m) \Big] \frac{3m^{2}}
{2 (\sinh m \cdot \hat m^{2})^{2}} \times \nonumber \\
&& \Big\{ \hat k^{4}(\frac{1}{2} \hat m^{2} + \tilde m )
+ \hat k^{2} \hat m^{2} \tilde m + 2 \hat m^{4} \Big\}
\nonumber \\
{\rm where} && \tilde m \doteq 3 \Big( 1 - \frac{\sinh m}{m}
\Big) \nonumber \\
C(p) &=& \Big[ \prod_{i=1}^{4}
G(p_{i})^{-1} \sum_{n_{i}\in {Z \!\!\! Z}} \Delta (p_{i}+2\pi n_{i})
\Pi (p_{i}+2\pi n_{i})\Big] \times \nonumber \\
&& \delta ( \sum_{i=1}^{4} [p_{i} + 2\pi n_{i}]) \ . \label{naco}\end{aligned}$$
Note, however, that in the contractions in the $\sigma^{4}$ term, an expectation value $\langle \tilde \sigma (p_{i}) \tilde \sigma (p_{j}) \rangle$ only enforces $p_{i}=-p_{j}$ (‘pairing’ of the lattice momenta), while the related summation integers $n_{i},\ n_{j}$ remain independent (no ‘pairing’ of the continuum momenta). Therefore, such contractions yield an additional contribution $S(k), ~ (k \in ]-\pi ,\pi ]) $, which does not factorize, $$\begin{aligned}
S(k) & \doteq & \frac{1}{2\pi} \int_{-\pi}^{\pi} dq \
G(q)^{-1} \Big[ \sum_{n_{1},n_{2},n_{3},n_{4}} \Delta (q+2\pi n_{1})
\Pi (q + 2\pi n_{2}) \times \nonumber \\
&& \Delta (-q+2\pi n_{2}) \Pi (-q + 2\pi n_{2})
\Delta (k+2\pi n_{3}) \Pi (k + 2\pi n_{3}) \nonumber \\
&& \Delta (-k+2\pi n_{4}) \Pi (-k + 2\pi n_{4}) \Big] \
\delta_{\sum_{i}n_{i},0} \ (1-\delta_{n_{1},-n_{2}}) . \label{sdef}\end{aligned}$$ (The case $n_{1}+n_{2}=0=n_{3}+n_{4}$ is excluded here, because it has been included before in eq. (\[naco\]).) This modifies the constant and the bilinear term to $$\begin{aligned}
\nonumber
A &=& A_{0}+ \frac{3}{2\pi} \int_{-\pi}^{\pi} dk \ G(k)^{-1}S(k), \\
B(k) &=& B_{0}(k) - 6 G(k)^{-2}S(k),\end{aligned}$$ while the vertex function $C(p)$, given in eq. (\[naco\]), is not affected. It does not pick up any loop contributions, and its structure can easily be understood in the language of “building blocks” as introduced for the quark-gluon vertex [@StL].
The lattice spacing artifacts in the standard action can be classified in magnitudes as $\lambda a^{2},\ \lambda a^{4},
\ \lambda a^{6} ,\dots ,\ \lambda^{2} a^{2}, \lambda^{2} a^{4},
\dots , \lambda^{3} a^{2} \dots $. In the action we have constructed now, all artifacts $\propto \lambda $ are erased. The remaining artifacts can still be $\propto a^{2}$, but they are multiplied at least by $\lambda^{2}$. This is analogous to QCD with the perfect vertex function, where the gauge coupling $g$ plays the rôle of $\lambda$. There the artifacts of the Wilson action start even in $O(ga)$, and the perturbative perfection pushes the leading artifact to $O(g^{2}a)$.
The perfect couplings in coordinate space
=========================================
The interaction terms involving $B$ and $C$, which we derived in momentum space, turn into convolutions in coordinate space. Hence $B(r)$ describes additional 2-variable couplings (“2-spin couplings” in a solid state language) over a distance $r\in {Z \!\!\! Z}$, and $C(r_{1},r_{2},r_{3},0)
\doteq C(\vec r )$ introduces 4-spin couplings, $$\lambda \sum_{x,r \in {Z \!\!\! Z}} B(r) \phi_{x} \phi_{x+r} +
\lambda \sum_{x \in {Z \!\!\! Z},\vec r \in {Z \!\!\! Z}^{3}}
C(\vec r) \phi_{x+r_{1}} \phi_{x+r_{2}} \phi_{x+r_{3}} \phi_{x}.$$ We exploit lattice translational invariance, in the latter case by setting $r_{4}$ to the arbitrary value 0. $C(r_{1},r_{2}r_{3},0)$ is invariant under permutation of its components, and $C(\vec r )= C(-\vec r)$ (but there is [*no*]{} invariance under sign flip of just one or two components of $\vec r$).
The 2-spin couplings
--------------------
The function $B(k)$ is even, hence $$B(r) \doteq B_{0}(r)+B_{1}(r) =
\frac{1}{2\pi} \int_{-\pi}^{\pi} dk \ [B_{0}(k)
- 6 G(k)^{-2}S(k)] \cos (kr) .$$ The first term, $B_{0}(r)$, can be computed analytically. It turns out to be the dominating contribution to $B(r)$, $$\begin{aligned}
B_{0}(r) &=& 3 \ \Big[ \frac{1}{2m}-\gamma (m) \Big] \
\frac{m^{2}}{(\sinh m \cdot \hat m^{2})^{2}} \times \nonumber \\
&& \Big\{ \beta_{0} \delta_{r,0} + \beta_{1} [\delta _{r,1}+
\delta_{r,-1}] + \beta_{2} [\delta_{r,2}+\delta_{r,-2}] \Big\} ,
\nonumber \\
\beta_{0} &=& 2 \ (3+\hat m^{2})\tilde m + 3\hat m^{2} + 2 \hat m^{4}
, \nonumber \\
\beta_{1} &=& -(4+\hat m^{2})\tilde m - 2 \hat m^{2} ,
\nonumber \\
\beta_{2} &=& \tilde m + \frac{1}{2} \hat m^{2}.\end{aligned}$$ (The quantities $\hat m$ and $\tilde m$ are defined in eqs. (\[nn\]) and (\[naco\]).)
Expanding in small $m$, we recognize the finiteness of this expression in the limit $m\to 0$, [^8] $$\begin{aligned}
B_{0}(r)&=& b_{0}^{(0)} \delta_{r,0} + b_{1}^{(0)}
[ \delta_{r,1}+\delta_{r,-1} ] + b_{2}^{(0)}
[ \delta_{r,2}+\delta_{r,-2} ] , \nonumber \\
b_{0}^{(0)} &=& \frac{77}{100} - \frac{419}{1400} m^{2}
+O(m^{4}) , \nonumber \\
b_{1}^{(0)} &=& \frac{91}{300} - \frac{1637}{12600} m^{2}
+O(m^{4}) , \nonumber \\
b_{2}^{(0)} &=& \frac{7}{600} - \frac{31}{5040} m^{2}
+O(m^{4}).\end{aligned}$$
The additional term, $B_{1}(r)$, has to be evaluated numerically. It is significantly suppressed, essentially because the case $n_{i}=0, \ i=1 \dots 4$ is excluded from the summation in eq. (\[sdef\]). It typically affects the bilinear couplings only in third digit.
It turns out that also in $B_{1}(r)$ the couplings are restricted to distances $\leq 2$. Therefore, the entire bilinear term $B(r)$ is given by $b_{i}=b_{i}^{(0)}+b_{i}^{(1)}$, $i=0,1,2$. These couplings are shown as functions of the mass in Fig. \[figc2\], and some precise values are given Table \[tabcop\]. For completeness we also include the constant $A$.
$m=0$ $m=0.2$ $m=0.3$
----------- -------------- -------------- --------------
$A$ 0.1634448589 0.1627118789 0.1615030089
$b_{0}$ 0.7648784176 0.7560248574 0.7415398190
$b_{1}$ 0.3034599619 0.2995928547 0.2932745430
$b_{2}$ 0.0112505782 0.0110757933 0.0107911914
$C_{0}$ 0.2242402815 0.2202292735 0.2137198561
$C_{1}$ 0.0579948140 0.0568473063 0.0549896682
$C_{2}$ 0.0002458836 0.0002396038 0.0002295148
$C_{11}$ 0.0386197986 0.0378246484 0.0365388182
$C_{12}$ 0.0008364070 0.0008162208 0.0007837264
$C_{22}$ 0.0000983088 0.0000956674 0.0000914309
$C_{112}$ 0.0042341197 0.0041403776 0.0039890713
: *The $O(\lambda )$ perfect couplings for masses $0 \dots 1$.*[]{data-label="tabcop"}
$m=0.4$ $m=0.5$ $m=1$
----------- -------------- -------------- ---------------
$A$ 0.1598369028 0.1577387270 0.14198250413
$b_{0}$ 0.7218143338 0.6973665904 0.52780256487
$b_{1}$ 0.2846878032 0.2740741694 0.20145092650
$b_{2}$ 0.0104063833 0.0099339708 0.00680856517
$C_{0}$ 0.2049621122 0.1942810514 0.12580113799
$C_{1}$ 0.0524997264 0.0494780115 0.03056403944
$C_{2}$ 0.0002161447 0.0002001630 0.00010708640
$C_{11}$ 0.0348181119 0.0327344171 0.01982579170
$C_{12}$ 0.0007405382 0.0006887131 0.00038135880
$C_{22}$ 0.0000858306 0.0000791587 0.00004091627
$C_{112}$ 0.0037871600 0.0035435707 0.00206185706
: *The $O(\lambda )$ perfect couplings for masses $0 \dots 1$.*[]{data-label="tabcop"}
In addition we have of course the bilinear couplings of the free theory (resp. harmonic oscillator), given in eqs. (\[nn\]) and (\[form\]). In coordinate space the free action reads $$S[\phi ]_{\lambda =0} = \frac{m^{3}}{2 \sinh m \cdot \hat m^{2}}
\sum_{x,y \in {Z \!\!\! Z}} \phi_{x} \Big[ (2+\hat m^{2})\delta_{x,y}
- \delta_{x,y+1} - \delta_{x,y-1} \Big] \phi_{y} \ .$$
The 4-spin couplings
--------------------
We recall that we describe the 4-spin couplings by $C(\vec r )$, $\vec r = (r_{1},r_{2},r_{3}) \in {Z \!\!\! Z}^{3}$, $r_{4}=0$, that we have permutation invariance among $r_{1},\dots ,r_{4}$ and invariance under $\vec r \to -\vec r$, hence $$C(\vec r ) = \frac{1}{(2\pi )^{3}} \int_{-\pi}^{\pi} d^{4}p\
C(p) \ \cos (\vec p \cdot \vec r ) \ .$$
Note that all the singularities at $p_{i}=0$ are removable, both, for finite and for vanishing mass.
It turns out that again the couplings never involve any two spins separated by a distance larger than 2. A general argument for that is given in the following subsection. This means that there are just 7 independent 4-spin couplings. We denote them as $$\begin{aligned}
&& \hspace{-7mm} C_{0} \doteq C(\vec 0 ); \
C_{1} \doteq C(1,0,0) ; \
C_{2} \doteq C(2,0,0) \\
&& \hspace{-7mm} C_{11} \doteq C(1,1,0) ; \
C_{12} \doteq C(1,2,0) ; \
C_{22} \doteq C(2,2,0) ; \
C_{112} \doteq C(1,1,2) .\end{aligned}$$ They all represent equivalence classes, which contain a total of 65 nontrivial couplings (keeping $r_{4}=0$ fixed). Some exact values are given in Table \[tabcop\], and their mass dependence is illustrated in Fig. \[figc4-1\].
\[figc4-1\]
Locality
--------
Assume that we calculate the perfect action to $O(\lambda^{n})
\ (n \geq 1)$. This involves a number of perturbative correction terms, which arise from the expectation values of the continuum field ${\varphi}$ to some power. The highest power is $\langle {\varphi}^{4n}\rangle $. There, Wick contractions lead to several kinds of terms, which we classify by the power of the inverse free propagator $G^{-1}$. The maximal power is $G^{-2n}$.
For instance, in the bilinear term, which depends only on one momentum, the maximal factor is $G(k)^{-2n}$. Therefore the maximal power of $\hat k$ is $\hat k^{4n} \propto (1-\cos k)^{2n}$, which can be decomposed into terms $\propto \cos k, \dots ,\cos 2nk$. In coordinate space this yields couplings $\propto
\delta_{x,1}+\delta_{x,-1}, \dots ,
[\delta_{x,2n}+\delta_{x,-2n}]$.
In the terms, which couple more than two lattice variables, an analogous consideration leads to a ‘maximal’ factor $\prod_{i=1}^{2n} \cos p_{i}$, where the momenta $p_{i}$ may be all or partially different. The observation that variables can not be coupled over distances $>2n$ still holds.
We have seen this restriction explicitly for $n=1$. Since the couplings are confined to such a short range, we can easily include all of them. Unlike field theory, no truncation – which does harm to the improved properties of the action – is needed. This allows us to study the quality of a perturbatively improved action separately, whereas in field theory one can only study a superposition of the improvement and the truncation scheme.
We also note that – within the limited set of couplings we deal with – locality becomes even better if the mass increases. For instance, a larger mass suppresses the couplings over distance 2 even more, relative to the leading coupling constants (see Table \[tabcop\]).
The restriction of the couplings to a finite range is in qualitative agreement with the effectively 1d quark-gluon vertex function (quark fields constant in all but one direction) [@QuaGlu]. Moreover, also the increase of locality for rising mass agrees with fermionic models and with scalar fields in higher dimensions [@WB].
Numerical results for the energy gaps
=====================================
As a warming up exercise we computed the lattice partition function from direct integration. If we do so at ${\tilde \lambda}=0$ and some ${\tilde \lambda}>0$, we can extract an estimation for $E_{0}({\tilde \lambda})-E_{0}(0)$. In fact, this works much better for the perturbatively perfect action than for the standard action; for instance at $m=\lambda =1$ the continuum ground state energy $E_{0,cont}=0.8038$ is approximated well using the improved action, 0.8021, whereas the standard action yields 0.7111. However, this involves additive constants, which may depend on ${\tilde \lambda}$, so we focus on the energy [*gaps*]{} now.
The simulation
--------------
To compare the performance of the standard discretization and the perturbatively perfect action, we simulated both actions at correlation lengths ranging from $\xi\approx2 \dots 5$, on a $L=30$ lattice with periodic boundary conditions, using a standard Metropolis multi-hit algorithm. The first two energy gaps were extracted from the correlation functions $\langle 0|\phi (x) \phi (0)|0 \rangle $ and $\langle 0|\phi (x)^{2} \phi(0)^{2}|0 \rangle$, and the statistical errors were estimated by jackknife analysis. The decay of the $\phi^{\ell}$ correlation function (in infinite volume) is given by [^9] $$\langle 0|\phi (x)^{\ell}\phi (0)^{\ell}|0 \rangle =
\sum_{n=0}^{\infty}
|\langle 0|\phi ^{\ell}|n \rangle |^{2}\exp(-\Delta E_{n}x),$$ ($\Delta E_{n} \doteq E_{n}-E_{0}$). For $\ell =1,2$ it reduces to [^10] $$\begin{aligned}
\langle 0|\phi (x) \phi (0)|0 \rangle & = & \sum_{n=0}^{\infty}
|\langle 0|\phi |2n+1 \rangle |^{2}\exp(-\Delta E_{2n+1}x ), \nonumber \\
\langle 0|\phi (x)^{2}\phi (0)^{2}|0 \rangle & = & \sum_{n=0}^{\infty}
| \langle 0|\phi^{2}|2n \rangle |^{2}\exp(-\Delta E_{2n}x) ,\end{aligned}$$ and for the harmonic oscillator we are left with $$\begin{aligned}
\langle 0|\phi (x)\phi (0)|0\rangle & = &
|\langle 0|\phi |1 \rangle |^{2}\exp(-\Delta E_{1}x), \nonumber \\
\langle 0|\phi (x)^{2}\phi (0)^{2}|0 \rangle &=&
|\langle 0|\phi^{2}|0\rangle |^{2} +
|\langle 0|\phi^{2}|2 \rangle |^{2} \exp(-\Delta E_{2}x). \label{decay}\end{aligned}$$ In our simulations we study small interaction parameters ${\tilde \lambda}\leq 0.2$. In this regime, varying the fitting ranges reveals that the $\ell = 1,2$ correlation functions do not pick up significant contributions from energy gaps higher than the leading ones given in eq. (\[decay\]).
Numerical results
-----------------
Simulations were done for masses and anharmonic couplings in the range $m = 0.2 \dots 0.5$ and ${\tilde \lambda}= 0.001 \dots 0.2$. We compared the first two energy gaps as “asymptotic scaling quantities”, as well as $\Delta E_{2}/\Delta E_{1}$ as a scaling quantity. For a direct evaluation, we divide the lattice results by the corresponding continuum values.
Figs. \[figp1\] and \[figp2\] show the gaps $\Delta E_{1} ({\tilde \lambda})$ and $\Delta E_{2}({\tilde \lambda})$, measured at $m=0.5$ for the standard and perturbatively perfect action (and normalized by their corresponding continuum gaps $\Delta E_{n,cont}$). These plots show that for anharmonic couplings up to ${\tilde \lambda}\approx 0.05$ the perturbatively perfect action reproduces the continuum gaps much better than the standard action. Comparing Figs. \[figp1\], \[figp2\] with Fig. \[figE1\], one recognizes roughly the same reliability range for first order perturbation theory in the continuum. This qualitative behavior of the improvement holds for a variety of masses.
For the harmonic oscillator in infinite volume, the perfect action reproduces the continuum gaps exactly, whereas the standard action yields $$\Delta E_{n,stan}(m) = n \cdot {\rm arccosh} (1 + m^{2}/2) .$$ (The value $\Delta E_{1,stan}(m=0.5) \simeq 0.495$ agrees with Fig. \[figp1\]). The fact that even the standard action is perfect at ${\tilde \lambda}=0$ is reflected by the exact values of the gap ratios.
The scaling ratio $\frac{\Delta E_{2}}{\Delta E_{1}}({\tilde \lambda})$ – again measured at $m=0.5$ and normalized by the continuum value – is shown in Fig. \[figp4\]. Unfortunately there is hardly any conclusive difference between the two types of action for that quantity.
Nevertheless, in Fig. \[figp5\], which shows the gap ratios versus $1/\xi$ at fixed ${\tilde \lambda}=0.005$, a better performance of the perturbatively perfect action is visible at intermediate correlation length. This is a scaling plot, based on $10^{9}$ Monte Carlo sweeps.
As we know from Fig. \[figE21\], the applicability of first order perturbation theory for $\Delta E_{2}/\Delta E_{1}$ is restricted to really tiny values of ${\tilde \lambda}$. The hope that the perturbatively perfect action could still perform well beyond that range can not be confirmed. [^11] In that range itself, the improvement is extremely difficult to demonstrate, because the standard action – being perfect at ${\tilde \lambda}= 0$ – is also excellent there. The available accuracy is mainly limited by the fitting precision of the exponential decays.
Conclusions and outlook
=======================
We have constructed a lattice action for the anharmonic oscillator, which is perfect to first order in perturbation theory. This is the first manifestly one loop perfect lattice action [^12]. Comparing this action to the standard lattice formulation, we observe a clear improvement for the energy gaps $\Delta E_{1}$, $\Delta E_{2}$ up to ${\tilde \lambda}\doteq
\lambda /m^{3} \sim 0.2$, over a variety of correlation lengths. As a consequence, (pseudo-)scaling laws of the type of eq. (\[rescale\]) are improved at small interactions. In field theory, this corresponds to an improved asymptotic scaling. Unfortunately, for the scaling quantity $\Delta E_{2}/
\Delta E_{1}$ an improvement could only be demonstrated laboriously. It seems to be restricted to very small values of ${\tilde \lambda}$, in agreement with the performance of continuum perturbation theory. There the linear approximation is useful only for ${\tilde \lambda}\leq O(10^{-2})$. One could have hoped that the improved action is successful also beyond this regime, but it turned out that this is not the case. This can be viewed as a negative sign for the direct application of perturbatively perfect actions, but the outcome might of course depend on the model. As a further test one could simulate the perturbatively classically perfect action for the 2d O(3) model, which has been presented – but not tested – in Ref. [@Has] (Table 2).
Moreover, in the regime of tiny ${\tilde \lambda}$, the standard action is also exceptionally successful in this toy model. Therefore, an accuracy of 4 or 5 digits is required to distinguish the results of the two actions in that regime. The fit of the exponential decay does not allow for such a high precision. The behavior of other scaling quantities, like $\Delta E_{3}/
\Delta E_{1}$ etc., is even worse, i.e. the regime of successful first order perturbation theory is even smaller. [^13]
Actually this construction could be carried on to $O({\tilde \lambda}^{2})$, but this includes couplings over distances 4, involving up to 8 lattice variables. Moreover, continuum perturbation theory suggests that the $O({\tilde \lambda}^{2})$ perfect action would only help to proceed to slightly larger interactions, see Fig. \[figE21\].\
Still the anharmonic oscillator – and in particular the ratio of its mass gaps – may serve as a good testing ground for quasi-perfect actions, if we proceed to a [*non-perturbative improvement*]{} scheme [@prep]. Then one expects a progress also for moderate and large interactions. Thus one overcomes the disadvantages of this model listed at the end of Section 1, although some truncation of the couplings will be needed.
In particular, one may introduce an inverse temperature $\beta$ in the expression for the partition function plus RGT transformation term, and send $\beta \to \infty$. Then minimization is sufficient for a multigrid inverse blocking, and in this way one can identify a [*classically perfect action*]{}. Using the standard action on the fine lattice and a fixed coarse configuration, we compared the minima for different blocking factors, and we typically observed a good convergence around blocking factor 10. [^14] As a test, we run the minimizer at small ${\tilde \lambda}$ and reproduced in this way the 4-spin couplings of Section 4. [^15] The 2-spin couplings $b_{i}$ arise from loop corrections, hence they are quantum effects, which are not present in the classically perfect action at small ${\tilde \lambda}$.
We can identify the classically perfect couplings over a wide range of interaction parameters. As an example, Fig. \[figC0m1\] compares $C_{0}({\tilde \lambda})$ for the perturbatively perfect and for the classically perfect action. This figure does not imply that the perturbatively perfect approximation should be rather poor already at ${\tilde \lambda}\sim 0.1$, as we have to conclude from the simulation results.
Of course, the model is not asymptotically free, and the parameter $\lambda$ is relevant (not just “weakly relevant”, i.e. in leading order marginal), so it does not belong to the class of models the classically perfect action is designed for. In fact, from the Table \[tabcop\] we see that the bilinear contributions – that the classical approximation misses – are important. However, it might have a chance to perform well if we simulate at $\beta >1$, where the quantum corrections to the perfect action are suppressed. The possibility that classically perfect actions work to some extent also for models, which are not asymptotically free, is conceivable and deserves being tested. [^16]
Finally, thanks to the simplicity of the model, one can also perform the full path integral (numerically) instead, which yields a (quantum) perfect action. As a test, we also reproduced roughly some bilinear couplings $b_{i}$ from Section 4 in this way. There one is restricted to small blocking factors and lattices, such that a good convergence of the action requires iteration. Then it is interesting to compare the couplings and their performance for the classically perfect action, and the action, which is – up to numerical errors and truncation – quantum perfect. The latter is more promising, but the classically perfect action is much of interest, because in complicated field theoretic models, this is what one – maximally – has at hand.
An other important question is the convergence velocity in the multigrid procedure. In non-Abelian gauge theories, only very few iteration steps, with a small blocking factor (typically 2) are possible, hence a fast convergence to the FPA is crucial. We hope that starting from a perturbatively perfect action helps to accelerate the convergence. Also this can also be tested in the toy model discussed here.
Furthermore one can use this model to test the “cycling” procedure of shifted forward and backward blocking, proposed by the Boulder group [@bold]. It is tractable in higher dimensions, but not strictly based on the renormalization group. Hence a toy model analysis of the errors emerging in the “cycling” process is of interest.
At last – as we mentioned in the introduction – one may speculate that in complicated models the improvement could be pushed beyond classical perfection, by performing, say, one full block factor 2 RGT step at finite $\beta$, starting from a classically perfect action. This method can also be tested for the anharmonic oscillator.\
[*Acknowledgment*]{} We are indebted to R. Brower, who first suggested this project. In addition we thank S. Chandrasekharan, S. Güsken, H. Hoeber, Th. Lippert, A. Okopinskaya, G. Ritzenhöfer, A. Seyfried and U.-J. Wiese for useful comments.
[50]{}
Proceedings of LATTICE’97, to appear in Nucl. Phys. B (Proc. Suppl.).
K. Symanzik, Nucl. Phys. B226 (1983) 187; 205.
M. Lüscher and P. Weisz, Comm. Math. Phys. 97 (1985) 59.
B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259 (1985) 572.
G.P. Lepage and P. Mackenzie, Phys. Rev. D48 (1993) 2250.\
M. Alford, W. Dimm, G.P. Lepage, G. Hockney, P. Mackenzie, Phys. Lett. B361 (1995) 87.\
G.P. Lepage, Nucl. Phys. B (Proc. Suppl.) 47 (1996) 3.
M. Lüscher, S. Sint, R. Sommer and P. Weisz, Nucl. Phys. B478 (1996) 365.\
M. Lüscher, S. Sint, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B491 (1997) 323.
K. Wilson and J. Kogut, Phys. Rep. C12 (1974) 75.\
K. Wilson, Rev. Mod. Phys. 47 (1975) 773.
W. Bietenholz and U.-J. Wiese, Nucl. Phys. B464 (1996) 319.
W. Bietenholz, R. Brower, S. Chandrasekharan and U.-J. Wiese, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 921.
K. Orginos, Ph.D. Thesis, Brown University (1997).
W. Bietenholz and U.-J. Wiese, Phys. Lett. B378 (1996) 222.
F. Farchioni and V. Laliena, hep-lat/9709040.
P. Hasenfratz and F. Niedermayer, Nucl. Phys. B414 (1994) 785.
R. Burkhalter, Phys. Rev. D54 (1996) 4121.
C. Lang and T. Pany, hep-lat/9707024.
W. Bietenholz, R. Brower, S. Chandrasekharan and U.-J. Wiese, Phys. Lett. B407 (1997) 283.
K. Orginos, W. Bietenholz, R. Brower, S. Chandrasekharan and U.-J. Wiese, hep-lat/9709100, to appear in Ref. [@LAT97].
K. Banerjee, Proc. R. Soc. Lond. A364 (1978) 265.
C. Bender and T.T. Wu, Phys. Rev. D7 (1973) 1620.
W. Bietenholz, E. Focht and U.-J. Wiese, Nucl. Phys. B436 (1995) 385.
T. DeGrand, A. Hasenfratz, P. Hasenfratz and F. Niedermayer, Phys. Lett. B365 (1996) 233.
W. Caswell, Ann. Phys. (N.Y.) 123 (1979) 153.\
Appendix A gives a general prescription how to compute the coefficients. An alternative derivation is given in\
A. Okopinskaya, Ann. Phys. (N.Y.) 249 (1996) 367.
S. Biswas, K. Datta, R. Saxena, P. Srivastava and V. Varma, J. Math. Phys. 14 (1973) 1190.\
F. Hioe, D. MacMillen and E. Montroll, Phys. Rep. 43, No. 7 (1978) 305.
K. Banerjee, S. Bhatnagar, V. Choudhry and S. Kanwal, Proc. R. Soc. Lond. A360 (1978) 575.\
B. Bacus, Y. Meurice and A. Soemadi, J. Phys. A28 (1995) L381.
M. Nauenberg, F. Kutter and M. Furman, Phys. Rev. A13 (1976) 1185.
T. Bell and K. Wilson, Phys. Rev. B11 (1975) 3431.
J. Hirsch and S. Shenker, Phys. Rev. B27 (1983) 1736.
A. Tsapalis and U.-J. Wiese, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 948.
W. Bietenholz, hep-lat/9709117, to appear in Ref. [@LAT97].
P. Hasenfratz and F. Niedermayer, hep-lat/9706002.
W. Bietenholz and T. Struckmann, in preparation.
T. DeGrand, A. Hasenfratz and T. Kovács, hep-lat/9705009.
[^1]: Perturbatively perfect actions have also be studied for the Schwinger model, [@Schwing; @Farc].
[^2]: This argument was made rigorous first by K. Symanzik (unpublished). The point is that the rescaling can be implemented unitarily [@Ban].
[^3]: Note that the point $\tilde \lambda =0$ is non analytic. For a discussion of large orders, see e.g. Ref. [@BeWu].
[^4]: We expect the same behavior also for the the couplings in the perturbatively perfect action.
[^5]: This choice for the RGT parameter $\alpha$ also provides optimal locality in $d=4$ [@WB].
[^6]: This is an example of an uncontrolled additive constant in $S[\phi ]$, which motivates the consideration of the energy [*gaps*]{}, rather than the single eigenvalues. The same holds for the subsequent integration over $\tilde \sigma$, see below.
[^7]: In fact, any lattice action for the harmonic oscillator is perfect [@HN].
[^8]: The finiteness at $m=0$ (“quartic oscillator”) is a very sensitive consistency test. We did not study that case – where ${\tilde \lambda}= \infty$ and $E_{n} = c_{n} \lambda^{1/3}$ – extensively, since our improved action is designed for small ${\tilde \lambda}$. However, we observed that $E_{0} \propto
\lambda^{1/3}$ can be fitted better for the perturbatively perfect action than for the standard action.
[^9]: At $L<\infty$ we actually obtain cosh functions, but we can easily measure the decay in a region, where this difference is negligible.
[^10]: Due to the mirror symmetry of the potential, eigenfunctions for $E_n$ have parity $(-)^n$.
[^11]: We also looked at stronger interactions, such as ${\tilde \lambda}= 0.5$, for masses $m=0.3$ and $m=0.5$. In that regime, the standard action scales even better than the perturbatively perfect action.
[^12]: For some time it was claimed that for asymptotically free theories, FPAs are not only classically perfect, but automatically also one loop quantum perfect. However, this claim has recently been disproved [@HN].
[^13]: As a general trend, the perturbation series gets worse if higher gaps are involved.
[^14]: For a finite blocking factor $n$ one has to use the modified RGT parameter $\alpha_{n} = \alpha (1-1/n^{2})$.
[^15]: Similarly, for the Schwinger model, the $O(g)$ (truncated) perfect plaquette couplings [@StL] could be reproduce to percent level from the classically perfect action [@Lang] (even though a slightly different RGT was used).
[^16]: One may also think about an extension to “semi-classical perfection”.
|
---
abstract: |
Let $(X_n)$ be a sequence of integrable real random variables, adapted to a filtration $(\mathcal{G}_n)$. Define $$C_n=\sqrt{n}\,\bigl\{\frac{1}{n}
\sum_{k=1}^nX_k-E(X_{n+1}\mid\mathcal{G}_n)\bigr\}\quad\text{and}\quad
D_n=\sqrt{n}\,\bigl\{E(X_{n+1}\mid\mathcal{G}_n)-Z\bigr\}$$ where $Z$ is the a.s. limit of $E(X_{n+1}\mid\mathcal{G}_n)$ (assumed to exist). Conditions for $(C_n,D_n)\longrightarrow\mathcal{N}(0,U)\times\mathcal{N}(0,V)$ stably are given, where $U,\,V$ are certain random variables. In particular, under such conditions, one obtains $$\sqrt{n}\,\bigl\{\frac{1}{n}
\sum_{k=1}^nX_k-Z\bigr\}=C_n+D_n\longrightarrow\mathcal{N}(0,U+V)\quad\text{stably}.$$This CLT has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced generalized Polya urns.
address:
- 'Patrizia Berti, Dipartimento di Matematica Pura ed Applicata ”G. Vitali”, Universita’ di Modena e Reggio-Emilia, via Campi 213/B, 41100 Modena, Italy'
- 'Irene Crimaldi, Dipartimento di Matematica, Universita’ di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy'
- 'Luca Pratelli, Accademia Navale, viale Italia 72, 57100 Livorno, Italy'
- 'Pietro Rigo (corresponding author), Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100 Pavia, Italy'
author:
- Patrizia Berti
- Irene Crimaldi
- Luca Pratelli
- Pietro Rigo
date: 'April 24, 2009. First version: April 6, 2009. http://arxiv.org/abs/0904.0932v1'
title: |
A central limit theorem and its applications to\
multicolor randomly reinforced urns
---
Introduction and motivations {#intro}
============================
As regards asymptotics in urn models, there is not a unique reference framework. Rather, there are many (ingenious) disjoint ideas, one for each class of problems. Well known examples are martingale methods, exchangeability, branching processes, stochastic approximation, dynamical systems and so on; see [@P].
Those limit theorems which unify various urn problems, thus, look of some interest.
In this paper, we focus on the CLT. While thought for urn problems, our CLT is stated for an arbitrary sequence $(X_n)$ of real random variables. Accordingly, it potentially applies to every urn situation, but it has generally a broader scope. Suppose $E{\lvertX_n\rvert}<\infty$ and define $Z_n=E\bigl(X_{n+1}\mid\mathcal{G}_n\bigr)$ where $(\mathcal{G}_n)$ is some filtration which makes $(X_n)$ adapted. Under various assumptions, one obtains $Z_n\overset{a.s.,L_1}\longrightarrow Z$ for some random variable $Z$. Define further $\overline{X}_n=\frac{1}{n} \sum_{k=1}^nX_k$ and $$\begin{gathered}
C_n=\sqrt{n}\,\bigl(\overline{X}_n-Z_n),\quad
D_n=\sqrt{n}\,\bigl(Z_n-Z),\quad
W_n=\sqrt{n}\,\bigl(\overline{X}_n-Z).\end{gathered}$$
The limit distribution of $C_n$, $D_n$ or $W_n$ is a main goal in various fields, including Bayesian statistics, discrete time filtering, gambling and urn problems. See [@AMS], [@BCL], [@BPR], [@BCPR], [@CLP], [@C], [@GR] and references therein. In fact, suppose the next observation $X_{n+1}$ is to be predicted basing on the available information $\mathcal{G}_n$. If the predictor $Z_n$ cannot be evaluated in closed form, one needs some estimate $\widehat{Z}_n$ and $C_n$ reduces to the scaled error when $\widehat{Z}_n=\overline{X}_n$. And $\overline{X}_n$ is a sound estimate of $Z_n$ under some distributional assumptions on $(X_n)$, for instance when $(X_n)$ is exchangeable, as it is usual in Bayesian statistics. Similarly, $D_n$ and $W_n$ are of interest provided $Z$ is regarded as a random parameter. In this case, $Z_n$ is the Bayesian estimate (of $Z$) under quadratic loss and $\overline{X}_n$ can be often viewed as the the maximum likelihood estimate. Note also that, in the trivial case where $(X_n)$ is i.i.d. and $\mathcal{G}_n=\sigma(X_1,\ldots,X_n)$, one obtains $C_n=W_n=\sqrt{n}\,\bigl(\overline{X}_n-EX_1)$ and $D_n=0$. As to urn problems, $X_n$ could be the indicator of $\{$black ball at time $n\}$ in a multicolor urn. Then, $Z_n$ becomes the proportion of black balls in the urn at time $n$ and $\overline{X}_n$ the observed frequency of black balls at time $n$.
Our main result (Theorem \[main\]) provides conditions for $$\label{piot} (C_n,D_n)\longrightarrow
\mathcal{N}(0,U)\times \mathcal{N}(0,V)\quad\text{stably}$$ where $U,\,V$ are certain random variables and $\mathcal{N}(0,L)$ is the Gaussian kernel with mean $0$ and variance $L$. A nice consequence is that $$W_n=C_n+D_n\longrightarrow
\mathcal{N}(0,U+V)\quad\text{stably}.$$ Stable convergence, in the sense of Aldous and Renyi, is a strong form of convergence in distribution; the definition is recalled in Section \[prel\].
To check the conditions for , it is fundamental to know something about the convergence rate of $$\begin{gathered}
Z_{n+1}-Z_n=E\bigl(X_{n+2}\mid\mathcal{G}_{n+1}\bigr)-E\bigl(X_{n+1}\mid\mathcal{G}_n\bigr),
\\E\bigl(Z_{n+1}-Z_n\mid\mathcal{G}_n\bigr)=E\bigl(X_{n+2}-X_{n+1}\mid\mathcal{G}_n\bigr).\end{gathered}$$ If $(X_n)$ is conditionally identically distributed with respect to $(\mathcal{G}_n)$, in the sense of [@BPR], then $(Z_n)$ is a $(\mathcal{G}_n)$-martingale and thus only $Z_{n+1}-Z_n$ plays a role. This happens in particular if $(X_n)$ is exchangeable and $\mathcal{G}_n=\sigma(X_1,\ldots,X_n)$.
To illustrate how the CLT works, three applications are given: $r$-step predictions, Poisson-Dirichlet sequences, and [*randomly reinforced generalized Polya urns*]{}. We next describe the latter, the main of such applications, and we refer to Subsections \[kstep\] and \[poidir\] for the remaining two.
An urn contains black and red balls. At each time $n\geq 1$, a ball is drawn and then replaced together with a random number of balls of the same color. Say that $B_n$ black balls or $R_n$ red balls are added to the urn according to whether $X_n=1$ or $X_n=0$, where $X_n$ is the indicator of $\{$black ball at time $n\}$. Suppose $$\begin{gathered}
B_n\geq 0,\quad R_n\geq 0,\quad EB_n=ER_n\quad\text{for all }n,
\\\sup_nE\bigl\{(B_n+R_n)^u\bigr\}<\infty\quad\text{for some }u>2,
\\m:=\lim_nEB_n>0,\quad q:=\lim_nEB_n^2,\quad \quad s:=\lim_nER_n^2.\end{gathered}$$ Letting $\mathcal{G}_n=\sigma(X_1,B_1,R_1,\ldots,X_n,B_n,R_n)$, suppose also that $(B_{n+1},R_{n+1})$ is independent of $\mathcal{G}_n\vee\sigma(X_{n+1})$. Then, as shown in Corollary \[poi\], the conditions for are satisfied with $$U=Z(1-Z)\,\bigl(\frac{(1-Z)q+Zs}{m^2}-1\bigr) \quad\text{and}\quad
V=Z(1-Z)\,\frac{(1-Z)q+Zs}{m^2}.$$
Corollary \[poi\] improves the existing result on this type of urns, obtained in [@AMS], under two respects. First, Corollary \[poi\] implies convergence of the pairs $(C_n,D_n)$ and not only of $D_n$. Hence, one also gets $W_n\longrightarrow\mathcal{N}(0,U+V)$ stably. Second, unlike [@AMS], neither the sequence $((B_n,R_n))$ is identically distributed nor the random variables $B_n+R_n$ have compact support.
By just the same argument used for two color urns, multicolor versions of Corollary \[poi\] are easily manufactured. To our knowledge, results of this type were not available so far. Briefly, for a $d$-color urn, let $X_{n,j}$ be the indicator of $\{$ball of color $j$ at time $n\}$ where $n\geq 1$ and $1\leq j\leq d$. Suppose $A_{n,j}$ balls of color $j$ are added in case $X_{n,j}=1$. The random variables $A_{n,j}$ are requested exactly the same conditions asked above to $B_n$ and $R_n$. Then, $$\begin{gathered}
\bigl({\bf C_n},\,{\bf D_n}\bigr)\longrightarrow\mathcal{N}_d(0,{\bf
U})\times\mathcal{N}_d(0,{\bf V})\quad\text{stably,}\end{gathered}$$ where ${\bf C_n}$ and ${\bf D_n}$ are the vectorial versions of $C_n$ and $D_n$ while ${\bf U},\,{\bf V}$ are certain random covariance matrices; see Corollary \[prio98poi\].
A last note is the following. In the previous urn, the $n$-th reinforce matrix is $${\bf A_n}=\text{diag}\bigl(A_{n,1},\ldots,A_{n,d}\bigr).$$ Since $EA_{n,1}=\ldots=EA_{n,d}$, the leading eigenvalue of the mean matrix $E{\bf A_n}$ has multiplicity greater than 1. Even if significant for applications, this particular case (the leading eigenvalue of $E{\bf A_n}$ is not simple) is typically neglected; see [@BH], [@J04], [@J05], and page 20 of [@P]. Our result, and indeed the result in [@AMS], contribute to fill this gap.
Stable convergence {#prel}
==================
Stable convergence has been introduced by Renyi in [@REN] and subsequently investigated by various authors. In a sense, it is intermediate between convergence in distribution and convergence in probability. We recall here basic definitions. For more information, we refer to [@AE], [@CLP], [@HH] and references therein.
Let $(\Omega,\mathcal{A},P)$ be a probability space and $S$ a metric space. A [*kernel*]{} on $S$ (or a [*random probability measure*]{} on $S$) is a measurable collection $N=\{N(\omega):\omega\in\Omega\}$ of probability measures on the Borel $\sigma$-field on $S$. Measurability means that $$N(\omega)(f)=\int f(x)\,N(\omega)(dx)$$ is $\mathcal{A}$-measurable, as a function of $\omega\in\Omega$, for each bounded Borel map $f:S\rightarrow\mathbb{R}$.
Let $(Y_n)$ be a sequence of $S$-valued random variables and $N$ a kernel on $S$. Both $(Y_n)$ and $N$ are defined on $(\Omega,\mathcal{A},P)$. Say that $Y_n$ converges [*stably*]{} to $N$ in case $$\begin{gathered}
P\bigl(Y_n\in\cdot\mid H\bigr)\rightarrow E\bigl(N(\cdot)\mid
H\bigr)\quad\text{weakly}
\\\text{for all }H\in\mathcal{A}\text{ such that }P(H)>0.\end{gathered}$$ Clearly, if $Y_n\rightarrow N$ stably, then $Y_n$ converges in distribution to the probability law $E\bigl(N(\cdot)\bigr)$ (just let $H=\Omega$). Moreover, when $S$ is separable, it is not hard to see that $Y_n\overset{P}\rightarrow Y$ if and only if $Y_n$ converges stably to the kernel $N=\delta_Y$.
We next mention a strong form of stable convergence, introduced in [@CLP], to be used later on. Let $\mathcal{F}_n\subset\mathcal{A}$ be a sub-$\sigma$-field, $n\geq
1$. Say that $Y_n$ converges to $N$ [*stably in strong sense*]{}, with respect to the sequence $(\mathcal{F}_n)$, in case $$E\bigl(f(Y_n)\mid\mathcal{F}_n\bigr)\overset{P}\longrightarrow
N(f)\quad\text{for each }f\in C_b(S)$$ where $C_b(S)$ denotes the set of real bounded continuous functions on $S$.
Finally, we state a simple but useful fact as a lemma.
\[plm\] Suppose that $S$ is a separable metric space and
$C_n$ and $D_n$ are $S$-valued random variables on $(\Omega,\mathcal{A},P)$, $n\geq 1$;
$M$ and $N$ are kernels on $S$ defined on $(\Omega,\mathcal{A},P)$;
$(\mathcal{G}_n:n\geq 1)$ is an (increasing) filtration satisfying $$\sigma(C_n)\subset\mathcal{G}_n\quad\text{and}\quad\sigma(D_n)\subset\mathcal{G}_\infty\quad\text{for
all }n,\text{ where }\mathcal{G}_\infty=\sigma(\cup_n\mathcal{G}_n).$$ If $C_n\rightarrow M$ stably and $D_n\rightarrow N$ stably in strong sense, with respect to $(\mathcal{G}_n)$, then $$(C_n,D_n)\longrightarrow M\times N\quad\text{stably}.$$ (Here, $M\times N$ is the kernel on $S\times S$ such that $\bigl(M\times N\bigr)(\omega)=M(\omega)\times N(\omega)$ for all $\omega$).
By standard arguments, since $S$ is separable and $\sigma(C_n,D_n)\subset\mathcal{G}_\infty$, it suffices to prove that $E\bigl\{I_H\, f_1(C_n)\,f_2(D_n)\}\rightarrow E\bigl\{I_H\,
M(f_1)\,N(f_2)\}$ whenever $H\in\cup_n\mathcal{G}_n$ and $f_1,\,f_2\in C_b(S)$. Let $L_n=
E\bigl(f_2(D_n)\mid\mathcal{G}_n\bigr)-N(f_2)$. Since $H\in\cup_n\mathcal{G}_n$, there is $k$ such that $H\in\mathcal{G}_n$ for $n\geq k$. Thus, $$\begin{gathered}
E\bigl\{I_H\, f_1(C_n)\,f_2(D_n)\}=E\bigl\{I_H\,
f_1(C_n)\,E\bigl(f_2(D_n)\mid\mathcal{G}_n\bigr)\}
\\=E\bigl\{I_H\,f_1(C_n)\,N(f_2)\}+E\bigl\{I_H\, f_1(C_n)\,L_n\}\quad\text{for all }n\geq k.\end{gathered}$$ Finally, ${\lvertE\bigl\{I_H\,
f_1(C_n)\,L_n\}\,\rvert}\leq\sup{{\lvertf_1\rvert}}\,E{\lvertL_n\rvert}\rightarrow 0$, since $D_n\rightarrow N$ stably in strong sense, and $E\bigl\{I_H\,f_1(C_n)\,N(f_2)\}\rightarrow
E\bigl\{I_H\,M(f_1)\,N(f_2)\}$ as $C_n\rightarrow M$ stably.
Main result {#main}
===========
In the sequel, $(X_n:n\geq 1)$ is a sequence of real random variables on the probability space $(\Omega,\mathcal{A},P)$ and $(\mathcal{G}_n:n\geq 0)$ an (increasing) filtration. We assume $E{\lvertX_n\rvert}<\infty$ and we let $$Z_n=E(X_{n+1}\mid\mathcal{G}_n)\quad\text{and}\quad
\overline{X}_n=\frac{1}{n} \sum_{k=1}^nX_k.$$
In case $\sup_nEX_n^2<\infty$ and $$\label{basic}
E\bigr\{\bigl(E(Z_{n+1}\mid\mathcal{G}_n)-Z_n\bigr)^2\bigr\}=o(n^{-3}),$$ the sequence $(Z_n)$ is an uniformly integrable quasi-martingale; see e.g. page 532 of [@K]. Accordingly, $$Z_n\overset{a.s.,L_1}\longrightarrow Z$$ for some real random variable $Z$. Define $$\begin{gathered}
C_n=\sqrt{n}\,\bigl(\overline{X}_n-Z_n\bigr),\quad
D_n=\sqrt{n}\,\bigl(Z_n-Z\bigr).\end{gathered}$$
Let $\mathcal{N}(a,b)$ denote the one-dimensional Gaussian law with mean $a$ and variance $b\geq 0$ (where $\mathcal{N}(a,0)=\delta_a$). Note that $\mathcal{N}(0,L)$ is a kernel on $\mathbb{R}$ for each real non negative random variable $L$. We are now in a position to state our main result.
\[main\] Suppose $\sigma(X_n)\subset\mathcal{G}_n$ for each $n\geq 1$, $(X_n^2)$ is uniformly integrable and condition holds. Let us consider the following conditions
- $\frac{1}{\sqrt{n}}\,E\bigl\{\max_{1\leq k\leq
n}k\,{\lvertZ_{k-1}-Z_k\rvert}\bigr\}\longrightarrow 0$,
- $\frac{1}{n}\sum_{k=1}^n\bigl\{X_k-Z_{k-1}+k(Z_{k-1}-Z_k)\bigr\}^2\overset{P}\longrightarrow
U$,
- $\sqrt{n}\,E\bigl\{\sup_{k\geq n}{\lvertZ_{k-1}-Z_k\rvert}\,\bigr\}\longrightarrow
0$,
- $n\sum_{k\geq n}(Z_{k-1}-Z_k)^2\overset{P}\longrightarrow
V$,
where $U$ and $V$ are real non negative random variables. Then, $C_n\rightarrow\mathcal{N}(0,U)$ stably under (a)-(b), and $D_n\rightarrow\mathcal{N}(0,V)$ stably in strong sense, with respect to $(\mathcal{G}_n)$, under (c)-(d). In particular,$$(C_n,D_n)\longrightarrow
\mathcal{N}(0,U)\times\mathcal{N}(0,V)\quad\text{stably under
(a)-(b)-(c)-(d)}.$$
Since $\sigma(C_n)\subset\mathcal{G}_n$ and $Z$ can be taken $\mathcal{G}_\infty$-measurable, Lemma \[plm\] applies. Thus, it suffices to prove that $C_n\rightarrow\mathcal{N}(0,U)$ stably and $D_n\rightarrow\mathcal{N}(0,V)$ stably in strong sense.
[**“$C_n\rightarrow\mathcal{N}(0,U)$ stably”.**]{} Suppose conditions (a)-(b) hold. First note that $$\begin{gathered}
\sqrt{n}\,C_n=n\,\overline{X}_n-n\,Z_n=\sum_{k=1}^nX_k+\sum_{k=1}^n\bigl((k-1)Z_{k-1}-kZ_k\bigr)
\\=\sum_{k=1}^n\bigl\{X_k-Z_{k-1}+k(Z_{k-1}-Z_k)\bigr\}.\end{gathered}$$ Letting $$Y_{n,k}=\frac{X_k-Z_{k-1}+k\bigl(E(Z_k\mid\mathcal{G}_{k-1})-Z_k\bigr)}{\sqrt{n}}\quad\text{and}\quad
Q_n=\frac{1}{\sqrt{n}}\,\sum_{k=1}^nk\bigl(Z_{k-1}-E(Z_k\mid\mathcal{G}_{k-1})\bigr),$$ it follows that $C_n=\sum_{k=1}^nY_{n,k}\,+\,Q_n$. By , $$\begin{gathered}
E{\lvertQ_n\rvert}\leq\frac{1}{\sqrt{n}}\,\sum_{k=1}^nk\,\sqrt{E\bigr\{\bigl(Z_{k-1}-E(Z_k\mid\mathcal{G}_{k-1})\bigr)^2\bigr\}}
=\frac{1}{\sqrt{n}}\,\sum_{k=1}^n\text{o}(k^{-1/2})\longrightarrow
0.\end{gathered}$$ Hence, it suffices to prove that $\sum_{k=1}^nY_{n,k}\rightarrow\mathcal{N}(0,U)$ stably. Letting $\mathcal{F}_{n,k}=\mathcal{G}_k$, $k=1,\ldots,n$, one obtains $E\bigl(Y_{n,k}\mid\mathcal{F}_{n,k-1}\bigr)=0$ a.s.. Thus, by Corollary 7 of [@CLP], $\sum_{k=1}^nY_{n,k}\rightarrow\mathcal{N}(0,U)$ stably whenever $$\text{(i)}\,\,\,E\bigl\{\max_{1\leq k\leq
n}{\lvertY_{n,k}\rvert}\bigr\}\longrightarrow 0;\quad
\text{(ii)}\,\,\,\sum_{k=1}^nY_{n,k}^2\overset{P}\longrightarrow U.$$
As to (i), first note that $$\sqrt{n}\max_{1\leq k\leq n}{\lvertY_{n,k}\rvert}\leq\max_{1\leq k\leq
n}{\lvertX_k-Z_{k-1}\rvert}\,+\,\sum_{k=1}^nk\,{\lvertE(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\rvert}\,+\,
\max_{1\leq k\leq n}k\,{\lvertZ_{k-1}-Z_k\rvert}.$$ Since $(X_n^2)$ is uniformly integrable, $((X_n-Z_{n-1})^2)$ is uniformly integrable as well, and this implies $\frac{1}{n}\,E\bigl\{\max_{1\leq k\leq
n}(X_k-Z_{k-1})^2\bigr\}\longrightarrow 0$. By condition , $$\frac{1}{\sqrt{n}}\sum_{k=1}^nk\,E{\Bigl\lvertE(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\Bigr\rvert}=
\frac{1}{\sqrt{n}}\,\sum_{k=1}^n\text{o}(k^{-1/2})\longrightarrow 0.$$ Thus, (i) follows from condition (a).
As to (ii), write $$\begin{gathered}
\sum_{k=1}^nY_{n,k}^2=\frac{1}{n}\sum_{k=1}^n\bigl(X_k-Z_{k-1}+k(Z_{k-1}-Z_k)\bigr)^2+\,
\frac{1}{n}\sum_{k=1}^nk^2\bigl(E(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\bigr)^2+
\\+\,\frac{2}{n}\sum_{k=1}^n\bigl(X_k-Z_{k-1}+k(Z_{k-1}-Z_k)\bigr)\,k\,\bigl(E(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\bigr)
\\=R_n+S_n+T_n\quad\text{say}.\end{gathered}$$ Then, $R_n\overset{P}\rightarrow U$ by (b) and $E{\lvertS_n\rvert}=ES_n\rightarrow 0$ by . Further $T_n\overset{P}\longrightarrow 0$, since $$\begin{gathered}
\frac{T_n^2}{4}\leq
\frac{1}{n}\sum_{k=1}^n\bigl(X_k-Z_{k-1}+k(Z_{k-1}-Z_k)\bigr)^2\,\cdot\,\frac{1}{n}\sum_{k=1}^nk^2\bigl(E(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\bigr)^2=R_n\,S_n.\end{gathered}$$ Hence, (ii) holds, and this concludes the proof of $C_n\rightarrow\mathcal{N}(0,U)$ stably.
[**“$D_n\rightarrow\mathcal{N}(0,V)$ stably in strong sense”.**]{} Suppose conditions (c)-(d) hold. We first recall a known result; see Example 6 of [@CLP]. Let $(L_n)$ be a $(\mathcal{G}_n)$-martingale such that $L_n\overset{a.s.,L_1}\longrightarrow L$ for some real random variable $L$. Then, $$\begin{gathered}
\sqrt{n}\,\bigl(L_n-L\bigr)\longrightarrow\mathcal{N}(0,V)\quad\text{stably
in strong sense with respect to }(\mathcal{G}_n),\end{gathered}$$ provided $$\text{(c*)}\,\,\,\sqrt{n}\,E\bigl\{\sup_{k\geq
n}{\lvertL_{k-1}-L_k\rvert}\,\bigr\}\longrightarrow 0;\quad\text{(d*)}\,\,\,
n\sum_{k\geq n}(L_{k-1}-L_k)^2\overset{P}\longrightarrow V.$$ Next, define $L_0=Z_0$ and $$\begin{gathered}
L_n=Z_n-\sum_{k=0}^{n-1}\bigl(E(Z_{k+1}\mid\mathcal{G}_k)-Z_k\bigr).\end{gathered}$$ Then, $(L_n)$ is a $(\mathcal{G}_n)$-martingale. Also, $L_n\overset{a.s.,L_1}\longrightarrow L$ for some $L$, as $(Z_n)$ is an uniformly integrable quasi martingale. In particular, $L_n-L$ can be written as $L_n-L=\sum_{k\geq n}(L_k-L_{k+1})$ a.s.. Similarly, $Z_n-Z=\sum_{k\geq n}(Z_k-Z_{k+1})$ a.s.. It follows that $$\begin{gathered}
E{\Bigl\lvertD_n-\sqrt{n}(L_n-L)\Bigr\rvert}=\sqrt{n}\,E{\Bigl\lvert(Z_n-Z)-(L_n-L)\Bigr\rvert}
\\=\sqrt{n}\,E{\Bigl\lvert\,\sum_{k\geq
n}\bigl\{(Z_k-L_k)-(Z_{k+1}-L_{k+1})\bigr\}\Bigr\rvert}
\\\leq\sqrt{n}\,\sum_{k\geq
n}E{\Bigl\lvertZ_k-E(Z_{k+1}\mid\mathcal{G}_k)\Bigr\rvert}=\sqrt{n}\,\sum_{k\geq
n}\text{o}(k^{-3/2})\longrightarrow 0.\end{gathered}$$ Thus, $D_n\rightarrow\mathcal{N}(0,V)$ stably in strong sense if and only if $\sqrt{n}(L_n-L)\rightarrow\mathcal{N}(0,V)$ stably in strong sense, and to conclude the proof it suffices to check conditions (c\*)-(d\*). In turn, (c\*)-(d\*) are a straightforward consequence of conditions , (c), (d) and $$L_{k-1}-L_k=\bigl(Z_{k-1}-Z_k\bigr)+\bigl(E(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\bigr).$$
Some remarks on Theorem \[main\] are in order.
In real problems, one of the quantities of main interest is $$W_n=\sqrt{n}\,\bigl(\overline{X}_n-Z).$$ And, under the assumptions of Theorem \[main\], one obtains $$W_n=C_n+D_n\longrightarrow\mathcal{N}(0,U+V)\quad\text{stably}.$$
Condition trivially holds when $(X_n)$ is conditionally identically distributed, in the sense of [@BPR], with respect to the filtration $(\mathcal{G}_n)$. In this case, in fact, $(Z_n)$ is even a $(\mathcal{G}_n)$-martingale. In particular, holds if $(X_n)$ is exchangeable and $\mathcal{G}_n=\sigma(X_1,\ldots,X_n)$.
Under (c), condition (a) can be replaced by
- $\sup_n\,\frac{1}{n}\sum_{k=1}^nk^2E\bigl\{(Z_{k-1}-Z_k)^2\bigr\}<\infty$.
Indeed, (a\*) and (c) imply (a) (we omit calculations). Note that, for proving $C_n\rightarrow\mathcal{N}(0,U)$ stably under (a\*)-(b)-(c), one can rely on more classical versions of the martingale CLT, such as Theorem 3.2 of [@HH].
To check conditions (b) and (d), the following simple lemma can help.
\[hftuimn\] Let $(Y_n)$ be a $(\mathcal{G}_n)$-adapted sequence of real random variables. If $\sum_{n=1}^\infty\frac{EY_n^2}{n^2}<\infty$ and $E\bigl(Y_{n+1}\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow
Y$, for some random variable $Y$, then $$n\sum_{k\geq n}\frac{Y_k}{k^2}\,\overset{a.s.}\longrightarrow
Y\quad\text{and}\quad\frac{1}{n}\sum_{k=1}^nY_k\overset{a.s.}\longrightarrow
Y.$$
Let $L_n=\sum_{k=1}^n\frac{Y_k-E\bigl(Y_k\mid\mathcal{G}_{k-1}\bigr)}{k}$. Then, $L_n$ is a $(\mathcal{G}_n)$-martingale such that $$\sup_nEL_n^2\leq 4\,\sum_k\frac{EY_k^2}{k^2}<\infty.$$ Thus, $L_n$ converges a.s. and Abel summation formula yields $$n\sum_{k\geq
n}\,\frac{Y_k-E\bigl(Y_k\mid\mathcal{G}_{k-1}\bigr)}{k^2}\,\overset{a.s.}\longrightarrow
0.$$ Since $E\bigl(Y_{n+1}\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow
Y$ and $n\sum_{k\geq n}\,\frac{1}{k^2}\longrightarrow 1$, it follows that $$\begin{gathered}
n\sum_{k\geq n}\,\frac{Y_k}{k^2}=n\sum_{k\geq
n}\,\frac{Y_k-E\bigl(Y_k\mid\mathcal{G}_{k-1}\bigr)}{k^2}\,+\,n\sum_{k\geq
n}\,\frac{E\bigl(Y_k\mid\mathcal{G}_{k-1}\bigr)}{k^2}\overset{a.s.}\longrightarrow
Y.\end{gathered}$$ Similarly, Kroneker lemma and $E\bigl(Y_{n+1}\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow
Y$ yield $$\begin{gathered}
\frac{1}{n}\sum_{k=1}^nY_k=\frac{1}{n}\sum_{k=1}^nE(Y_k\mid\mathcal{G}_{k-1})\,+\,\frac{1}{n}\sum_{k=1}^nk\,\frac{Y_k-E\bigl(Y_k\mid\mathcal{G}_{k-1}\bigr)}{k}\overset{a.s.}\longrightarrow
Y.\end{gathered}$$
Our last comment needs a formal remark.
\[stabqc\] As regards $D_n$, a natural question is whether $$\label{pevb}
E\bigl(f(D_n)\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow
\mathcal{N}(0,V)(f)\quad\text{for each }f\in C_b(\mathbb{R}).$$ This is a strengthening of $D_n\rightarrow\mathcal{N}(0,V)$ stably in strong sense, as $E\bigl(f(D_n)\mid\mathcal{G}_n\bigr)$ is requested to converge a.s. and not only in probability. Let $(X_n)$ be a (non necessarily $(\mathcal{G}_n)$-adapted) sequence of integrable random variables. Then, for to be true, it is enough that $(Z_n)$ is uniformly integrable and $$\begin{gathered}
\sum_{k\geq
1}\sqrt{k}\,E{\Bigl\lvert\,E(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\Bigr\rvert}<\infty,
\\E\bigl\{\sup_{k\geq
1}\sqrt{k}\,{\lvertZ_{k-1}-Z_k\rvert}\bigr\}<\infty,\quad n\sum_{k\geq
n}(Z_{k-1}-Z_k)^2\overset{a.s.}\longrightarrow V.\end{gathered}$$ The proof is essentially the same as that of Theorem \[main\], up to using Theorem 2.2 of [@C] instead of Example 6 of [@CLP].
Applications {#appl}
============
This section is split into four subsections, arranged in increasing order of length.
$r$-step predictions {#kstep}
--------------------
Suppose we are requested to make conditional forecasts on a sequence of events $A_n\in\mathcal{G}_n$. To fix ideas, for each $n$, we aim to predict $$A_n^*=\bigl(\cap_{j\in J}A_{n+j}\bigr)\cap\bigl(\cap_{j\in J^c}A_{n+j}^c\bigr)$$ conditionally on $\mathcal{G}_n$, where $J$ is a given subset of $\{1,\ldots,r\}$ and $J^c=\{1,\ldots,r\}\setminus J$. Letting $X_n=I_{A_n}$, the predictor can be written as $$\begin{gathered}
Z_n^*=E\bigl\{\,\prod_{j\in J}X_{n+j}\,\prod_{j\in J^c}(1-X_{n+j})\mid\mathcal{G}_n\bigr\}.\end{gathered}$$
In the spirit of Section \[intro\], when $Z_n^*$ cannot be evaluated in closed form, one needs to estimate it. Under some assumptions, in particular when $(X_n)$ is exchangeable and $\mathcal{G}_n=\sigma(X_1,\ldots,X_n)$, a reasonable estimate of $Z_n^*$ is $\overline{X}_n^h(1-\overline{X}_n)^{r-h}$ where $h=\,$card$(J)$. Usually, under such assumptions, one also has $Z_n\overset{a.s.}\longrightarrow Z$ and $Z_n^*\overset{a.s.}\longrightarrow Z^h(1-Z)^{r-h}$ for some random variable $Z$. So, it makes sense to define $$\begin{gathered}
C_n^*=\sqrt{n}\,\bigl\{\overline{X}_n^h(1-\overline{X}_n)^{r-h}-Z_n^*\bigr\},\quad
D_n^*=\sqrt{n}\,\bigl\{Z_n^*-Z^h(1-Z)^{r-h}\bigr\}.\end{gathered}$$
Next result is a straightforward consequence of Theorem \[main\].
Let $(X_n)$ be a $(\mathcal{G}_n)$-adapted sequence of indicators satisfying . If conditions (a)-(b)-(c)-(d) of Theorem \[main\] hold, then $$\begin{gathered}
(C_n^*,D_n^*)\longrightarrow
\mathcal{N}(0,\sigma^2U)\times\mathcal{N}(0,\sigma^2V)\quad\text{stably,
where}
\\\sigma^2=\bigl\{h\,Z^{h-1}(1-Z)^{r-h}-(r-h)\,Z^h(1-Z)^{r-h-1}\bigr\}^2.\end{gathered}$$
We just give a sketch of the proof. Let $f(x)=x^h(1-x)^{r-h}$. Basing on (c), it can be shown that $\sqrt{n}\,E{\Bigl\lvert\,Z_n^*-f(Z_n)\Bigr\rvert}\longrightarrow 0$. Thus, $C_n^*$ can be replaced by $\sqrt{n}\,\bigl\{f(\overline{X}_n)-f(Z_n)\bigr\}$ and $D_n^*$ by $\sqrt{n}\,\bigl\{f(Z_n)-f(Z)\bigr\}$. By the mean value theorem, $$\sqrt{n}\,\bigl\{f(\overline{X}_n)-f(Z_n)\bigr\}=\sqrt{n}\,f'(M_n)\,(\overline{X}_n-Z_n)=f'(M_n)\,C_n$$ where $M_n$ is between $\overline{X}_n$ and $Z_n$. By , $Z_n\overset{a.s}\longrightarrow Z$ and $\overline{X}_n\overset{a.s}\longrightarrow Z$. Hence, $f'(M_n)\overset{a.s}\longrightarrow f'(Z)$ as $f'$ is continuous. By Theorem \[main\], $C_n\rightarrow \mathcal{N}(0,U)$ stably. Thus, $$\sqrt{n}\,\bigl\{f(\overline{X}_n)-f(Z_n)\bigr\}\longrightarrow
f'(Z)\,\mathcal{N}(0,U)=\mathcal{N}(0,\sigma^2U)\quad\text{stably}.$$ By a similar argument, it can be seen that $\sqrt{n}\,\bigl\{f(Z_n)-f(Z)\bigr\}\longrightarrow\mathcal{N}(0,\sigma^2V)$ stably in strong sense. An application of Lemma \[plm\] concludes the proof.
Poisson-Dirichlet sequences {#poidir}
---------------------------
Let $\mathcal{Y}$ be a finite set and $(Y_n)$ a sequence of $\mathcal{Y}$-valued random variables satisfying $$P\bigl(Y_{n+1}\in A\mid Y_1,\ldots,Y_n\bigr)=\frac{\sum_{y\in
A}(S_{n,y}-\alpha)\,I_{\{S_{n,y}\neq
0\}}+\bigl(\theta+\alpha\sum_{y\in\mathcal{Y}}I_{\{S_{n,y}\neq
0\}}\bigr)\,\nu(A)}{\theta+n}$$ a.s. for all $A\subset\mathcal{Y}$ and $n\geq 1$. Here, $0\leq
\alpha<1$ and $\theta>-\alpha$ are constants, $\nu$ is the probability distribution of $Y_1$ and $S_{n,y}=\sum_{k=1}^nI_{\{Y_k=y\}}$.
Sequences $(Y_n)$ of this type play a role in various frameworks, mainly in population-genetics. They can be regarded as a generalization of those exchangeable sequences directed by a two parameter Poisson-Dirichlet process; see [@PY]. For $\alpha=0$, $(Y_n)$ reduces to a classical Dirichlet sequence (i.e., an exchangeable sequence directed by a Dirichlet process). But, for $\alpha\neq 0$, $(Y_n)$ may even fail to be exchangeable.
From the point of view of Theorem \[main\], however, the only important thing is that $P\bigl(Y_{n+1}\in\cdot\mid Y_1,\ldots,Y_n\bigr)$ can be written down explicitly. Indeed, the following result is available.
Let $\mathcal{G}_n=\sigma(Y_1,\ldots,Y_n)$ and $X_n=I_A(Y_n)$, where $A\subset\mathcal{Y}$. Then, condition holds (so that $Z_n\overset{a.s.}\longrightarrow Z$) and $$(C_n,D_n)\longrightarrow\delta_0\times\mathcal{N}\bigl(0,Z(1-Z)\bigr)\quad\text{stably}.$$
Let $Q_n=-\alpha\,\sum_{y\in A}I_{\{S_{n,y}\neq 0\}}+\bigl(\theta+\alpha\sum_{y\in\mathcal{Y}}I_{\{S_{n,y}\neq 0\}}\bigr)\,\nu(A)$. Since $$Z_n=P\bigl(Y_{n+1}\in A\mid Y_1,\ldots,Y_n\bigr)=\frac{n\,\overline{X}_n\,+\,Q_n}{\theta+n}\quad\text{and}\quad {\lvertQ_n\rvert}\leq c$$ for some constant $c$, then $C_n\overset{a.s.}\longrightarrow 0$. By Lemma \[plm\] and Theorem \[main\], thus, it suffices to check conditions , (c) and (d) with $V=Z(1-Z)$. On noting that $$Z_{n+1}-Z_n=\frac{X_{n+1}-Z_n}{\theta+n+1}\,+\,\frac{Q_{n+1}-Q_n}{\theta+n+1},$$ condition (c) trivially holds. Since $S_{n+1,y}=S_{n,y}+I_{\{Y_{n+1}=y\}}$, one obtains $$\begin{gathered}
Q_{n+1}-Q_n=-\alpha\,\nu(A^c)\,\sum_{y\in A}I_{\{S_{n,y}=0\}}I_{\{Y_{n+1}=y\}}\,+\,\alpha\,\nu(A)\,\sum_{y\in A^c}I_{\{S_{n,y}=0\}}I_{\{Y_{n+1}=y\}}.\end{gathered}$$ It follows that $$E\bigl\{{\lvertQ_{n+1}-Q_n\rvert}\mid\mathcal{G}_n\bigr\}\leq
2\,\sum_{y\in\mathcal{Y}}I_{\{S_{n,y}=0\}}\,P\bigl(Y_{n+1}=y\mid\mathcal{G}_n\bigr)\leq\frac{d}{\theta+n}\quad\text{a.s.}$$ for some constant $d$, and this implies $${\Bigl\lvertE\bigl(Z_{n+1}\mid\mathcal{G}_n\bigr)-Z_n\Bigr\rvert}=\frac{{\Bigl\lvertE\bigl(Q_{n+1}-Q_n\mid\mathcal{G}_n\bigr)\Bigr\rvert}}{\theta+n+1}\leq\frac{d}{(\theta+n)^2}\quad\text{a.s.}.$$ Hence, condition holds. To check (d), note that $\sum_kk^2E\bigl\{(Z_{k-1}-Z_k)^4\bigr\}<\infty$. Since $Z_k\overset{a.s.}\longrightarrow Z$ (by ) one also obtains $$\begin{gathered}
E\bigl\{(X_k-Z_{k-1})^2\mid\mathcal{G}_{k-1}\bigr\}=Z_{k-1}-Z_{k-1}^2\overset{a.s.}\longrightarrow
Z(1-Z),
\\E\bigl\{(Q_k-Q_{k-1})^2\mid\mathcal{G}_{k-1}\bigr\}\,+\,
2\,E\bigl\{(X_k-Z_{k-1})\,(Q_k-Q_{k-1})\mid\mathcal{G}_{k-1}\bigr\}\overset{a.s.}\longrightarrow
0.\end{gathered}$$ Thus, $k^2E\bigl\{(Z_{k-1}-Z_k)^2\mid\mathcal{G}_{k-1}\bigr\}\overset{a.s.}\longrightarrow
Z(1-Z)$. Letting $Y_k=k^2(Z_{k-1}-Z_k)^2$ and $Y=Z(1-Z)$, Lemma \[hftuimn\] implies $$n\sum_{k\geq
n}(Z_{k-1}-Z_k)^2=n\sum_{k\geq
n}\frac{Y_k}{k^2}\overset{a.s.}\longrightarrow Z(1-Z).$$
As it is clear from the previous proof, all conditions of Remark \[stabqc\] are satisfied. Therefore, $D_n$ meets condition with $V=Z(1-Z)$.
Two color randomly reinforced generalized Polya urns {#2col}
----------------------------------------------------
An urn contains $b>0$ black balls and $r>0$ red balls. At each time $n\geq 1$, a ball is drawn and then replaced together with a random number of balls of the same color. Say that $B_n$ black balls or $R_n$ red balls are added to the urn according to whether $X_n=1$ or $X_n=0$, where $X_n$ is the indicator of $\{$black ball at time $n\}$.
Urns of this type have some history: see [@AMS], [@BCL], [@BPR], [@C], [@MF], [@P] and references therein.
To model such urns, we assume $X_n,\,B_n,\,R_n$ random variables on the probability space $(\Omega,\mathcal{A},P)$ such that
- $X_n\in\{0,1\}$, $\,\,\,\,B_n\geq 0$, $\,\,\,\,R_n\geq 0$, $$\begin{gathered}
(B_n,R_n)\text{ independent of }\,
\bigl(X_1,B_1,R_1,\ldots,X_{n-1},B_{n-1},R_{n-1},X_n\bigr),
\\Z_n=P\bigl(X_{n+1}=1\mid\mathcal{G}_n\bigr)=\frac{b+\sum_{k=1}^nB_kX_k}{b+r+\sum_{k=1}^n\bigl(B_kX_k+R_k(1-X_k)\bigr)}\,\,\text{
a.s.},\end{gathered}$$ for each $n\geq 1$, where $$\mathcal{G}_0=\{\emptyset,\Omega\},\quad\mathcal{G}_n=\sigma\bigl(X_1,B_1,R_1,\ldots,X_n,B_n,R_n\bigr).$$
In the particular case $B_n=R_n$, in Example 3.5 of [@BPR], it is shown that $C_n$ converges stably to a Gaussian kernel whenever $EB_1^2<\infty$ and $B_n\sim B_1$ for all $n$. Further, in Corollary 4.1 of [@C], $D_n$ is shown to satisfy condition . The latter result on $D_n$ is extended to $B_n\neq R_n$ in [@AMS], under the assumptions that $B_1+R_1$ has compact support, $EB_1=ER_1$, and $(B_n,R_n)\sim (B_1,R_1)$ for all $n$.
Basing on Theorem \[main\], condition can be shown to hold more generally. Indeed, it is fundamental that $EB_n=ER_n$ for all $n$ and the three sequences $(EB_n)$, $(EB_n^2)$, $(ER_n^2)$ approach a limit. But identity in distribution of $(B_n,R_n)$ can be dropped and compact support of $B_n+R_n$ can be replaced by a moment condition such as $$\label{momcon}
\sup_n E\bigl\{(B_n+R_n)^u\bigr\}<\infty\quad\text{for some }u>2.$$ Under these conditions, not only $D_n$ meets , but the pairs $(C_n,D_n)$ converge stably as well. In particular, one obtains stable convergence of $W_n=C_n+D_n$ which is of potential interest in urn problems.
\[poi\] In addition to ($*$) and , suppose $EB_n=ER_n$ for all $n$ and $$\begin{gathered}
m:=\lim_nEB_n>0,\quad q:=\lim_nEB_n^2,\quad s:=\lim_nER_n^2.\end{gathered}$$ Then, condition holds (so that $Z_n\overset{a.s.}\longrightarrow Z$) and $$\begin{gathered}
(C_n,D_n)\longrightarrow
\mathcal{N}(0,U)\times\mathcal{N}(0,V)\quad\text{stably, where}
\\U=Z(1-Z)\,\bigl(\frac{(1-Z)q+Zs}{m^2}-1\bigr)\quad\text{and}\quad
V=Z(1-Z)\,\frac{(1-Z)q+Zs}{m^2}.\end{gathered}$$ In particular, $W_n=C_n+D_n\longrightarrow\mathcal{N}(0,U+V)$ stably. Moreover, $D_n$ meets condition , that is, $E\bigl(f(D_n)\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow
\mathcal{N}(0,V)(f)$ for each $f\in C_b(\mathbb{R})$.
It is worth noting that, arguing as in [@AMS] and [@MF], one obtains $P(Z=z)=0$ for all $z$. Thus, $\mathcal{N}(0,V)$ is a non degenerate kernel. In turn, $\mathcal{N}(0,U)$ is non degenerate unless $q=s=m^2$, and this happens if and only if both $B_n$ and $R_n$ converge in probability (necessarily to $m$). In the latter case ($q=s=m^2$), $C_n\overset{P}\longrightarrow 0$ and condition holds with $V=Z(1-Z)$. Thus, in a sense, randomly reinforced urns behave as classical Polya urns (i.e., those urns with $B_n=R_n=m$) whenever the reinforcements converge in probability.
The proof of Corollary \[poi\] is deferred to the Appendix as it needs some work. Here, to point out the underlying argument, we sketch such a proof under the superfluous but simplifying assumption that $B_n\vee R_n\leq c$ for all $n$ and some constant $c$. Let $$S_n=b+r+\sum_{k=1}^n\bigl(B_kX_k+R_k(1-X_k)\bigr).$$
After some algebra, $Z_{n+1}-Z_n$ can be written as $$\begin{gathered}
Z_{n+1}-Z_n=\frac{(1-Z_n)\,X_{n+1}\,B_{n+1}\,-\,Z_n\,(1-X_{n+1})\,R_{n+1}}{S_{n+1}}
\\=\frac{(1-Z_n)\,X_{n+1}\,B_{n+1}}{S_n+B_{n+1}}\,-\,\frac{Z_n\,(1-X_{n+1})\,R_{n+1}}{S_n+R_{n+1}}.\end{gathered}$$ By ($*$) and $EB_{n+1}=ER_{n+1}$, $$\begin{gathered}
E\bigl(Z_{n+1}-Z_n\mid\mathcal{G}_n\bigr)=Z_n(1-Z_n)\,E\bigl\{\,\frac{B_{n+1}}{S_n+B_{n+1}}-\frac{R_{n+1}}{S_n+R_{n+1}}
\mid\mathcal{G}_n\bigr\}
\\=Z_n(1-Z_n)\,E\bigl\{\,\frac{B_{n+1}}{S_n+B_{n+1}}-\frac{B_{n+1}}{S_n}-\frac{R_{n+1}}{S_n+R_{n+1}}+\frac{R_{n+1}}{S_n}
\mid\mathcal{G}_n\bigr\}
\\=Z_n(1-Z_n)\,E\bigl\{\,-\frac{B_{n+1}^2}{S_n(S_n+B_{n+1})}\,+\,\frac{R_{n+1}^2}{S_n(S_n+R_{n+1})}\,\mid\mathcal{G}_n\bigr\}\quad\text{a.s.}.\end{gathered}$$ Thus, ${\Bigl\lvert\,E\bigl(Z_{n+1}\mid\mathcal{G}_n\bigr)-Z_n\Bigr\rvert}\leq\frac{EB_{n+1}^2+ER_{n+1}^2}{S_n^2}\,$ a.s.. Since $\sup_n\bigl(EB_n^2+ER_n^2\bigr)<\infty$ and $E(S_n^{-p})=\,$O$(n^{-p})$ for all $p>0$ (as shown in Lemma \[jrblk\]) then $$E\bigl\{{\lvertE(Z_{n+1}\mid\mathcal{G}_n)-Z_n\rvert}^p\bigr\}=\,\text{O}(n^{-2p})\quad\text{for
all }p>0.$$ In particular, condition holds and $\sum_k\sqrt{k}\,E{\Bigl\lvert\,E(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\Bigr\rvert}<\infty$.
To conclude the proof, in view of Lemma \[plm\], Theorem \[main\] and Remark \[stabqc\], it suffices to check conditions (a), (b) and $$\text{(i)}\,\,\,E\bigl\{\sup_{k\geq
1}\sqrt{k}\,{\lvertZ_{k-1}-Z_k\rvert}\bigr\}<\infty;\quad\text{(ii)}\,\,\,n\sum_{k\geq
n}(Z_{k-1}-Z_k)^2\overset{a.s.}\longrightarrow V.$$ Conditions (a) and (i) are straightforward consequences of ${\lvertZ_{n+1}-Z_n\rvert}\leq\frac{c}{S_n}$ and $E(S_n^{-p})=\,$O$(n^{-p})$ for all $p>0$. Condition (b) follows from the same argument as (ii). And to prove (ii), it suffices to show that $E(Y_{n+1}\mid\mathcal{G}_n)\overset{a.s.}\longrightarrow V$ where $Y_n=n^2(Z_{n-1}-Z_n)^2$; see Lemma \[hftuimn\]. Write $(n+1)^{-2}E(Y_{n+1}\mid\mathcal{G}_n)$ as $$\begin{gathered}
Z_n(1-Z_n)^2E\bigl\{\frac{B_{n+1}^2}{(S_n+B_{n+1})^2}\mid\mathcal{G}_n\bigr\}\,+\,Z_n^2(1-Z_n)E\bigl\{\frac{R_{n+1}^2}{(S_n+R_{n+1})^2}\mid\mathcal{G}_n\bigr\}.\end{gathered}$$ Since $\frac{S_n}{n} \overset{a.s.}\longrightarrow m$ (by Lemma \[jrblk\]) and $B_{n+1}\leq c$, then $$\begin{gathered}
n^2E\bigl\{\frac{B_{n+1}^2}{(S_n+B_{n+1})^2}\mid\mathcal{G}_n\bigr\}\leq n^2E\bigl\{\frac{B_{n+1}^2}{S_n^2}\mid\mathcal{G}_n\bigr\}=n^2\frac{EB_{n+1}^2}{S_n^2}\overset{a.s.}\longrightarrow\frac{q}{m^2}\,\text{ and}
\\n^2E\bigl\{\frac{B_{n+1}^2}{(S_n+B_{n+1})^2}\mid\mathcal{G}_n\bigr\}\geq n^2E\bigl\{\frac{B_{n+1}^2}{(S_n+c)^2}\mid\mathcal{G}_n\bigr\}=n^2\frac{EB_{n+1}^2}{(S_n+c)^2}\overset{a.s.}\longrightarrow\frac{q}{m^2}.\end{gathered}$$ Similarly, $n^2E\bigl\{\frac{R_{n+1}^2}{(S_n+R_{n+1})^2}\mid\mathcal{G}_n\bigr\}\overset{a.s.}\longrightarrow\frac{s}{m^2}$. Since $Z_n\overset{a.s.}\longrightarrow Z$, it follows that $$\begin{gathered}
E(Y_{n+1}\mid\mathcal{G}_n)\overset{a.s.}\longrightarrow \,\,Z(1-Z)^2\frac{q}{m^2}+Z^2(1-Z)\frac{s}{m^2}=V.\end{gathered}$$ This concludes the (sketch of the) proof.
In order to $(C_n,D_n)\longrightarrow
\mathcal{N}(0,U)\times\mathcal{N}(0,V)$ stably, some of the assumptions of Corollary \[poi\] can be stated in a different form. We mention two (independent) facts.
First, condition can be weakened into uniform integrability of $(B_n+R_n)^2$.
Second, $(B_n,R_n)$ independent of $\mathcal{G}_{n-1}\vee\sigma(X_n)$ can be replaced by the following four conditions:
- $(B_n,R_n)$ conditionally independent of $X_n$ given $\mathcal{G}_{n-1}$;
- Condition holds for some $u>4$;
- There are an integer $n_0$ and a constant $l>0$ such that $$\begin{gathered}
E\bigl(B_n\wedge n^{1/4}\mid\mathcal{G}_{n-1}\bigr)\geq
l\,\,\text{and}\,\,E\bigl(R_n\wedge
n^{1/4}\mid\mathcal{G}_{n-1}\bigr)\geq l\,\text{ a.s. whenever
}n\geq n_0; \end{gathered}$$
- There are random variables $m,\,q,\,s$ such that $$\begin{gathered}
E\bigl(B_n\mid\mathcal{G}_{n-1}\bigr)=E\bigl(R_n\mid\mathcal{G}_{n-1}\bigr)\overset{P}\longrightarrow
m,\quad
E\bigl(B_n^2\mid\mathcal{G}_{n-1}\bigr)\overset{P}\longrightarrow
q,\quad
E\bigl(R_n^2\mid\mathcal{G}_{n-1}\bigr)\overset{P}\longrightarrow s.\end{gathered}$$
Even if in a different framework, conditions similar to (i)-(iv) are in [@BH].
The multicolor case {#multi}
-------------------
To avoid technicalities, we firstly investigated two color urns, but the results in Subsection \[2col\] extend to the multicolor case.
An urn contains $a_j>0$ balls of color $j\in\{1,\ldots,d\}$ where $d\geq 2$. Let $X_{n,j}$ denote the indicator of $\{$ball of color $j$ at time $n\}$. In case $X_{n,j}=1$, the ball which has been drawn is replaced together with $A_{n,j}$ more balls of color $j$. Formally, we assume $\bigl\{X_{n,j},\,A_{n,j}:n\geq 1,\,1\leq j\leq
d\bigr\}$ random variables on the probability space $(\Omega,\mathcal{A},P)$ satisfying
- $\,\,\,\,X_{n,j}\in\{0,1\}$, $\,\,\,\,\sum_{j=1}^dX_{n,j}=1$, $\,\,\,\,A_{n,j}\geq 0$, $$\begin{gathered}
(A_{n,1},\ldots,A_{n,d})\text{ independent of }\,
\bigl(A_{k,j},\,X_{k,j},\,X_{n,j}:1\leq k<n,\,1\leq j\leq d\bigr),
\\Z_{n,j}=P\bigl(X_{n+1,j}=1\mid\mathcal{G}_n\bigr)=\frac{a_j+\sum_{k=1}^nA_{k,j}X_{k,j}}{\sum_{i=1}^da_i+\sum_{k=1}^n\sum_{i=1}^dA_{k,i}X_{k,i}}\,\,\text{
a.s.},
\\\text{where}\quad\mathcal{G}_0=\{\emptyset,\Omega\},\quad\mathcal{G}_n=\sigma\bigl(A_{k,j},\,X_{k,j}:1\leq
k\leq n,\,1\leq j\leq d\bigr).\end{gathered}$$
Note that $$\begin{gathered}
Z_{n+1,j}-Z_{n,j}=(1-Z_{n,j})\,\frac{A_{n+1,j}\,X_{n+1,j}}{S_n+A_{n+1,j}}\,-\,Z_{n,j}\sum_{i\neq
j}\frac{A_{n+1,i}\,X_{n+1,i}}{S_n+A_{n+1,i}}
\\\text{where
}\,S_n=\sum_{i=1}^da_i+\sum_{k=1}^n\sum_{i=1}^dA_{k,i}X_{k,i}.\end{gathered}$$
In addition to ($**$), as in Subsection \[2col\], we ask the moment condition $$\label{hg5dr}
\sup_n
E\bigl\{\bigl(\,\sum_{j=1}^dA_{n,j}\bigr)^u\bigr\}<\infty\quad\text{for
some }u>2.$$ Further, it is fundamental that $$\begin{gathered}
\label{pawbn}
EA_{n,j}=EA_{n,1}\quad\text{for each }n\geq 1\text{ and }1\leq j\leq
d,\,\text{ and}
\\m:=\lim_nEA_{n,1}>0,\quad q_j:=\lim_nEA_{n,j}^2\quad\text{for each }1\leq j\leq d.\notag\end{gathered}$$
Fix $1\leq j\leq d$. Since $EA_{n,i}=EA_{n,1}$ for all $n$ and $i$, the same calculation as in Subsection \[2col\] yields $$\begin{gathered}
{\Bigl\lvert\,E\bigl(Z_{n+1,j}\mid\mathcal{G}_n\bigr)-Z_{n,j}\Bigr\rvert}\leq\frac{\sum_{i=1}^dEA_{n+1,i}^2}{S_n^2}\quad\text{a.s.}.\end{gathered}$$ Also, $E(S_n^{-p})=\,$O$(n^{-p})$ for all $p>0$; see Remark \[chissa1\]. Thus, $$\label{aw38j}
E\bigl\{{\lvertE\bigl(Z_{n+1,j}\mid\mathcal{G}_n\bigr)-Z_{n,j}\rvert}^p\bigr\}=\,\text{O}(n^{-2p})\quad\text{for
all }p>0.$$ In particular, $Z_{n,j}$ meets condition so that $Z_{n,j}\overset{a.s.}\longrightarrow Z_{(j)}$ for some random variable $Z_{(j)}$. Define $$\begin{gathered}
C_{n,j}=\sqrt{n}\,\bigl(\,\frac{1}{n}\sum_{k=1}^nX_{k,j}\,-\,Z_{n,j}\bigr)\quad\text{and}\quad
D_{n,j}=\sqrt{n}\,\bigl(Z_{n,j}-Z_{(j)}\bigr).\end{gathered}$$ Next result is quite expected at this point.
\[priophg\] Suppose conditions ($**$), , hold and fix $1\leq j\leq d$. Then, $$\begin{gathered}
\bigl(C_{n,j},\,D_{n,j}\bigr)\longrightarrow\mathcal{N}(0,U_j)\times\mathcal{N}(0,V_j)\quad\text{stably,
where}
\\U_j=V_j-Z_{(j)}(1-Z_{(j)})\quad\text{and}\quad
V_j=\frac{Z_{(j)}}{m^2}\,\bigl\{\,q_j\,(1-Z_{(j)})^2\,+\,Z_{(j)}\sum_{i\neq
j}q_i\,Z_{(i)}\,\bigr\}.\end{gathered}$$ Moreover, $E\bigl(f(D_{n,j})\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow\mathcal{N}(0,V_j)(f)$ for each $f\in C_b(\mathbb{R})$, that is, $D_{n,j}$ meets condition .
Just repeat the proof of Corollary \[poi\] with $X_{n,j}$ in the place of $X_n$.
A vectorial version of Corollary \[priophg\] can be obtained with slight effort. Let $\mathcal{N}_d(0,\Sigma)$ denote the $d$-dimensional Gaussian law with mean vector 0 and covariance matrix $\Sigma$ and $${\bf C_n}=\bigl(C_{n,1},\ldots,C_{n,d}\bigr),\quad {\bf
D_n}=\bigl(D_{n,1},\ldots,D_{n,d}\bigr).$$
\[prio98poi\] Suppose conditions ($**$), , hold. Then, $$\begin{gathered}
\bigl({\bf C_n},\,{\bf D_n}\bigr)\longrightarrow\mathcal{N}_d(0,{\bf
U})\times\mathcal{N}_d(0,{\bf V})\quad\text{stably,}\end{gathered}$$ where ${\bf U},\,{\bf V}$ are the $d\times d$ matrices with entries $U_{j,j}=U_j$, $V_{j,j}=V_j$, and $$\begin{gathered}
U_{i,j}=V_{i,j}+Z_{(i)}Z_{(j)},\quad
V_{i,j}=\frac{Z_{(i)}Z_{(j)}}{m^2}\,\bigl\{\sum_{h=1}^dq_hZ_{(h)}-q_i-q_j\bigr\}\quad\text{
for }i\neq j.\end{gathered}$$ Moreover, $E\bigl(f({\bf
D_n})\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow\mathcal{N}_d(0,{\bf
V})(f)$ for each $f\in C_b(\mathbb{R}^d)$.
Given a linear functional $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$, it suffices to see that $$\begin{gathered}
\phi({\bf C_n})\longrightarrow\mathcal{N}_d(0,{\bf
U})\circ\phi^{-1}\quad\text{stably, and} \\E\bigl(g\circ \phi({\bf
D_n})\mid\mathcal{G}_n\bigr)\overset{a.s.}\longrightarrow\mathcal{N}_d(0,{\bf
V})(g\circ \phi)\quad\text{for each }g\in C_b(\mathbb{R}).\end{gathered}$$ To this purpose, note that $$\begin{gathered}
\phi({\bf
C_n})=\sqrt{n}\,\bigl\{\,\frac{1}{n}\sum_{k=1}^n\phi(X_{k,1},\ldots,X_{k,d})\,-\,E\bigl(\phi(X_{n+1,1},\ldots,X_{n+1,d})\mid\mathcal{G}_n\bigr)\,\bigr\},
\\\phi({\bf
D_n})=\sqrt{n}\,\bigl\{\,E\bigl(\phi(X_{n+1,1},\ldots,X_{n+1,d})\mid\mathcal{G}_n\bigr)\,-\,\phi(Z_{(1)},\ldots,Z_{(d)})\,\bigr\},\end{gathered}$$ and repeat again the proof of Corollary \[poi\] with $\phi(X_{n,1},\ldots,X_{n,d})$ in the place of $X_n$.
A nice consequence of Corollary \[prio98poi\] is that $${\bf W_n}={\bf C_n+D_n}\longrightarrow\mathcal{N}_d(0,{\bf
U+V})\quad\text{stably}$$ provided conditions ($**$)-- hold, where ${\bf W_n}=\bigl(W_{n,1},\ldots,W_{n,d}\bigr)$ and $W_{n,j}=\sqrt{n}\,\bigl(\,\frac{1}{n}\sum_{k=1}^nX_{k,j}\,-\,Z_{(j)}\bigr)$.
Finally, we briefly mention a possible development of the above material. Suppose condition is turned into $$\begin{gathered}
EA_{n,j}=EA_{n,1}\quad\text{whenever }n\geq 1\text{ and }1\leq j\leq
d_0,
\\\liminf_n\,\bigl(EA_{n,1}-EA_{n,j}\bigr)>0\quad\text{whenever }j>d_0,\notag
\\m:=\lim_nEA_{n,1}>0,\quad q_j:=\lim_nEA_{n,j}^2\quad\text{whenever }1\leq j\leq d_0,\notag\end{gathered}$$ for some integer $1\leq d_0\leq d$. Roughly speaking, this means that some colors (those labelled from $d_0+1$ to $d$) are dominated by the others. So far, we dealt with $d_0=d$ but the case $d_0<d$ is not unusual in applications. The main trouble is that condition may fail when $d_0<d$. It is still possible to get a CLT but one should decide how to handle dominated colors. There are essentially two options.
One is to make assumptions on dominated colors. A classical assumption is $$\limsup_n\frac{EA_{n,j}}{EA_{n,1}}<\frac{1}{2}\quad\text{for each
}j>d_0.$$ Under this condition, using some ideas from [@MF], an analogous of Corollary \[priophg\] can be proved for $(C_{n,j},\,D_{n,j})$ with $j=1,\ldots,d_0$.
The other option is to neglect dominated colors, that is, to replace $Z_{n,j}$ and $\frac{1}{n}\sum_{k=1}^nX_{k,j}$ by $$\begin{gathered}
Z_{n,j}^*=\frac{a_j+\sum_{k=1}^nA_{k,j}X_{k,j}}{\sum_{i=1}^{d_0}a_i+\sum_{k=1}^n\sum_{i=1}^{d_0}A_{k,i}X_{k,i}}\quad\text{and}\quad
M_{n,j}^*=\frac{\sum_{k=1}^nX_{k,j}}{1+\sum_{k=1}^n\sum_{i=1}^{d_0}X_{k,i}}.\end{gathered}$$ Again, an analogous of Corollary \[priophg\] can be shown for $$\begin{gathered}
C_{n,j}^*=\sqrt{n}\,\bigl(M_{n,j}^*-Z_{n,j}^*\bigr)\quad\text{and}\quad
D_{n,j}^*=\sqrt{n}\,\bigl(Z_{n,j}^*-Z_{(j)}\bigr),\,\,\,\,j=1,\ldots,d_0.\end{gathered}$$
The case $d_0<d$ will be deepened in a forthcoming paper.
[**APPENDIX**]{}
In the notation of Subsection \[2col\], let $S_n=b+r+\sum_{k=1}^n\bigl(B_kX_k+R_k(1-X_k)\bigr)$.
\[jrblk\] Under the assumptions of Corollary \[poi\], $$\frac{n}{S_n}\longrightarrow\frac{1}{m}\quad\text{a.s. and in
}L_p\text{ for all }p>0.$$
Let $Y_n=B_nX_n+R_n(1-X_n)$. By ($*$) and $EB_{n+1}=ER_{n+1}$, $$\begin{gathered}
E\bigl(Y_{n+1}\mid\mathcal{G}_n\bigr)=EB_{n+1}\,E\bigl(X_{n+1}\mid\mathcal{G}_n\bigr)\,+\,ER_{n+1}\,E\bigl(1-X_{n+1}\mid\mathcal{G}_n\bigr)
\\=Z_n\,EB_{n+1}\,+\,(1-Z_n)\,EB_{n+1}=EB_{n+1}\overset{a.s.}\longrightarrow
m.\end{gathered}$$ Since $m>0$, Lemma \[hftuimn\] implies $\frac{n}{S_n}=\frac{1}{S_n/n}\overset{a.s.}\longrightarrow\frac{1}{m}$. To conclude the proof, it suffices to see that $E(S_n^{-p})=\,$O$(n^{-p})$ for all $p>0$. Given $c>0$, define $$S_n^{(c)}=\sum_{k=1}^n\bigl\{X_k\bigl(B_k\wedge c-E(B_k\wedge c)\bigr)+(1-X_k)\bigl(R_k\wedge c-E(R_k\wedge c)\bigr)\bigr\}.$$ By a classical martingale inequality (see e.g. Lemma 1.5 of [@LT]) $$P\bigl({\lvertS_n^{(c)}\rvert}>x\bigr)\leq 2\,\exp{\bigl(-x^2/2\,\,c^2\,n\bigr)}\quad\text{for all }x>0.$$ Since $EB_n=ER_n\longrightarrow m$ and both $(B_n)$, $(R_n)$ are uniformly integrable (as $\sup_n\,\bigl(EB_n^2+ER_n^2\bigr)<\infty$), there are $c>0$ and an integer $n_0$ such that $$m_n:=\sum_{k=1}^n\min\bigl\{E(B_k\wedge c),\,E(R_k\wedge c)\bigr\}>n\frac{m}{2}\quad\text{for all }n\geq n_0.$$ Fix one such $c>0$ and let $l=m/4>0$. For every $p>0$, one can write $$\begin{gathered}
E(S_n^{-p})=p\,\int_{b+r}^\infty t^{-p-1}P(S_n<t)\,dt
\\\leq\frac{p}{(b+r)^{p+1}}\,\int_{b+r}^{b+r+n\,l}P(S_n<t)\,dt\,+\,p\,\int_{b+r+n\,l}^\infty t^{-p-1}\,dt.\end{gathered}$$ Clearly, $p\,\int_{b+r+n\,l}^\infty t^{-p-1}\,dt=(b+r+n\,l)^{-p}=\,$O$(n^{-p})$. Further, for each $n\geq n_0$ and $t<b+r+n\,l$, since $m_n>n\,2\,l$ one obtains $$\begin{gathered}
P(S_n<t)\leq P\bigl(S_n^{(c)}<t-b-r-m_n\bigr)\leq P\bigl(S_n^{(c)}<t-b-r-n\,2\,l\bigr)
\\\leq P\bigl({\lvertS_n^{(c)}\rvert}>b+r+n\,2\,l-t\bigr)\leq 2\,\exp{\bigl(-(b+r+n\,2\,l-t)^2/2\,\,c^2\,n\bigr)}.\end{gathered}$$ Hence, $\int_{b+r}^{b+r+n\,l}P(S_n<t)\,dt\leq n\,2\,l\,\exp{\bigl(-n\,\frac{l^2}{2\,c^2}\bigr)}$ for every $n\geq n_0$, so that $E(S_n^{-p})=\,$O$(n^{-p})$.
\[chissa1\] As in Subsection \[multi\], let $S_n=\sum_{i=1}^da_i+\sum_{k=1}^n\sum_{i=1}^dA_{k,i}X_{k,i}$. Under conditions ($**$)--, the previous proof still applies to such $S_n$. Thus, $\frac{n}{S_n}\longrightarrow\frac{1}{m}$ a.s. and in $L_p$ for all $p>0$.
\[[**Proof of Corollary \[poi\]**]{}\] By Lemma \[plm\], it is enough to prove $C_n\rightarrow\mathcal{N}(0,U)$ stably and $D_n$ meets condition . Recall from Subsection \[2col\] that $$\begin{gathered}
Z_{n+1}-Z_n=\frac{(1-Z_n)\,X_{n+1}\,B_{n+1}\,-\,Z_n\,(1-X_{n+1})\,R_{n+1}}{S_{n+1}}
\\\text{and }\,\,E\bigl\{{\lvertE(Z_{n+1}\mid\mathcal{G}_n)-Z_n\rvert}^p\bigr\}=\,\text{O}(n^{-2p})\quad\text{for all }p>0.\end{gathered}$$ In particular, condition holds and $\sum_k\sqrt{k}\,E{\Bigl\lvert\,E(Z_k\mid\mathcal{G}_{k-1})-Z_{k-1}\Bigr\rvert}<\infty$.
[**“$D_n$ meets condition ”.**]{} By and Lemma \[jrblk\], $$\begin{gathered}
E\bigl\{{\lvertZ_{k-1}-Z_k\rvert}^u\bigr\} \leq
E\bigl\{\frac{(B_k+R_k)^u}{S_{k-1}^u}\bigr\}=E\bigl\{(B_k+R_k)^u\bigr\}\,E(S_{k-1}^{-u})=\text{O}(k^{-u}).\end{gathered}$$ Thus, $E\bigl\{\sup_k\sqrt{k}\,{\lvertZ_{k-1}-Z_k\rvert}\bigr\}^u\leq\sum_k\,k^{\frac{u}{2}}E\bigl\{{\lvertZ_{k-1}-Z_k\rvert}^u\bigr\}<\infty$ as $u>2$. In view of Remark \[stabqc\], it remains only to prove that $$\begin{gathered}
n\sum_{k\geq n}(Z_{k-1}-Z_k)^2=n\sum_{k\geq
n}\bigl(\frac{(1-Z_{k-1})X_kB_k}{S_k}-\frac{Z_{k-1}(1-X_k)R_k}{S_k}\bigr)^2
\\=n\sum_{k\geq n}\frac{(1-Z_{k-1})^2X_kB_k^2}{(S_{k-1}+B_k)^2}\,+\,n\sum_{k\geq
n}\frac{Z_{k-1}^2(1-X_k)R_k^2}{(S_{k-1}+R_k)^2}\end{gathered}$$ converges a.s. to $V=Z(1-Z)\,\frac{(1-Z)q+Zs}{m^2}$. It is enough to show that $$n\sum_{k\geq
n}\frac{(1-Z_{k-1})^2X_kB_k^2}{(S_{k-1}+B_k)^2}\overset{a.s.}\longrightarrow
Z(1-Z)^2\frac{q}{m^2}\,\text{ and }\, n\sum_{k\geq
n}\frac{Z_{k-1}^2(1-X_k)R_k^2}{(S_{k-1}+R_k)^2}\overset{a.s.}\longrightarrow
Z^2(1-Z)\frac{s}{m^2}\,.$$ These two limit relations can be proved by exactly the same argument, and thus we just prove the first one. Let $U_n=B_nI_{\{B_n\leq \sqrt{n}\}}$. Since $P(B_n>\sqrt{n})\leq
n^{\frac{-u}{2}}EB_n^u$, condition yields $P(B_n\neq
U_n,$ i.o.$)=0$. Hence, it suffices to show that $$\label{vhqosd}
n\sum_{k\geq
n}\frac{(1-Z_{k-1})^2X_kU_k^2}{(S_{k-1}+U_k)^2}\overset{a.s.}\longrightarrow
Z(1-Z)^2\frac{q}{m^2}.$$ Let $Y_n=n^2\frac{(1-Z_{n-1})^2X_nU_n^2}{(S_{n-1}+U_n)^2}$. Since $(B_n^2)$ is uniformly integrable, $EU_n^2\longrightarrow q$. Furthermore, $\frac{S_n}{n}\overset{a.s.}\longrightarrow m$ and $Z_n\overset{a.s.}\longrightarrow Z$. Thus, $$\begin{gathered}
E\bigl(Y_{n+1}\mid\mathcal{G}_n\bigr)\leq
(1-Z_n)^2(n+1)^2E\bigl(\frac{X_{n+1}U_{n+1}^2}{S_n^2}\mid\mathcal{G}_n\bigr)
\\=Z_n(1-Z_n)^2\frac{(n+1)^2}{S_n^2}\,EU_{n+1}^2\overset{a.s.}\longrightarrow
Z(1-Z)^2\frac{q}{m^2}\quad\text{ and}
\\E\bigl(Y_{n+1}\mid\mathcal{G}_n\bigr)\geq
(1-Z_n)^2(n+1)^2E\bigl(\frac{X_{n+1}U_{n+1}^2}{(S_n+\sqrt{n+1})^2}\mid\mathcal{G}_n\bigr)
\\=Z_n(1-Z_n)^2\frac{(n+1)^2}{(S_n+\sqrt{n+1})^2}\,EU_{n+1}^2\overset{a.s.}\longrightarrow
Z(1-Z)^2\frac{q}{m^2}.\end{gathered}$$ By Lemma \[hftuimn\], for getting relation , it suffices that $\sum_n\frac{EY_n^2}{n^2}<\infty$. Since $$\begin{gathered}
\frac{EU_n^4}{n^2}\leq\frac{E\bigl\{B_n^2I_{\{B_n^2\leq\sqrt{n}\}}\bigr\}}{n^{\frac{3}{2}}}+\frac{E\bigl\{B_n^2I_{\{B_n^2>\sqrt{n}\}}\bigr\}}{n}
\leq\frac{EB_n^2}{n^{\frac{3}{2}}}+\frac{EB_n^u}{n^{1+\frac{u-2}{4}}}\,,\end{gathered}$$ condition implies $\sum_n\frac{EU_n^4}{n^2}<\infty$. By Lemma \[jrblk\], $E(S_{n-1}^{-4})=\,$O$(n^{-4})$. Then, $$\begin{gathered}
\sum_n\frac{EY_n^2}{n^2}\leq
\sum_nn^2E\bigl\{\frac{U_n^4}{S_{n-1}^4}\bigr\}=\sum_nn^2E(S_{n-1}^{-4})\,EU_n^4\leq
c\,\sum_n\frac{EU_n^4}{n^2}<\infty\end{gathered}$$ for some constant $c$. Hence, condition holds.
[**“$C_n\rightarrow\mathcal{N}(0,U)$ stably”.**]{} By Theorem \[main\], it suffices to check conditions (a) and (b) with $U=Z(1-Z)\,\bigl(\frac{(1-Z)q+Zs}{m^2}-1\bigr)$. As to (a), since $E\bigl\{{\lvertZ_{k-1}-Z_k\rvert}^u\bigr\}=\,$O$(k^{-u})$, $$\begin{gathered}
\bigl(\,n^{-\frac{1}{2}}\,E\bigl\{\max_{1\leq k\leq
n}k\,{\lvertZ_{k-1}-Z_k\rvert}\bigr\}\,\bigr)^u\leq
n^{-\frac{u}{2}}\,\sum_{k=1}^nk^uE\bigl\{{\lvertZ_{k-1}-Z_k\rvert}^u\bigr\}\longrightarrow
0.\end{gathered}$$ We next prove condition (b). After some algebra, one obtains $$\begin{gathered}
E\bigl\{(X_n-Z_{n-1})(Z_{n-1}-Z_n)\mid\mathcal{G}_{n-1}\bigr\}
=-Z_{n-1}(1-Z_{n-1})\,E\bigl\{\frac{B_n}{S_{n-1}+B_n}\mid\mathcal{G}_{n-1}\bigr\}\,+\\+\,Z_{n-1}^2(1-Z_{n-1})\,E\bigl\{\frac{B_n}{S_{n-1}+B_n}-\frac{R_n}{S_{n-1}+R_n}\mid\mathcal{G}_{n-1}\bigr\}\quad\text{a.s.}.\end{gathered}$$ Arguing as in the first part of this proof (“$D_n$ meets condition ”), $$n\,E\bigl\{\frac{B_n}{S_{n-1}+B_n}\mid\mathcal{G}_{n-1}\bigr\}\overset{a.s.}\longrightarrow
1\quad\text{and}\quad
n\,E\bigl\{\frac{R_n}{S_{n-1}+R_n}\mid\mathcal{G}_{n-1}\bigr\}\overset{a.s.}\longrightarrow
1.$$ Thus, $n\,E\bigl\{(X_n-Z_{n-1})(Z_{n-1}-Z_n)\mid\mathcal{G}_{n-1}\bigr\}\overset{a.s.}\longrightarrow
-Z(1-Z)$. Further, $$E\bigl\{\bigl(X_n-Z_{n-1})^2\mid\mathcal{G}_{n-1}\bigr\}=Z_{n-1}-Z_{n-1}^2\overset{a.s.}\longrightarrow
Z(1-Z).$$ Thus, Lemma \[hftuimn\] implies $$\begin{gathered}
\frac{1}{n}\sum_{k=1}^n(X_k-Z_{k-1})^2\,+\,\frac{2}{n}\sum_{k=1}^nk\,(X_k-Z_{k-1})\,(Z_{k-1}-Z_k)\overset{a.s.}\longrightarrow
-Z(1-Z).\end{gathered}$$ Finally, write $\frac{1}{n}\sum_{k=1}^nk^2(Z_{k-1}-Z_k)^2=\frac{1}{n}\sum_{k=1}^nk^2\bigl\{\,\frac{(1-Z_{k-1})^2X_kB_k^2}{(S_{k-1}+B_k)^2}+\frac{Z_{k-1}^2(1-X_k)R_k^2}{(S_{k-1}+R_k)^2}\,\bigr\}$. By Lemma \[hftuimn\] and the same truncation technique used in the first part of this proof, $\frac{1}{n}\sum_{k=1}^nk^2(Z_{k-1}-Z_k)^2\overset{a.s.}\longrightarrow
V$. Squaring, $$\begin{gathered}
\frac{1}{n}\sum_{k=1}^n\bigr\{X_k-Z_{k-1}+k(Z_{k-1}-Z_k)\bigl\}^2\,\overset{a.s.}\longrightarrow
V-Z(1-Z)=U,\end{gathered}$$ that is, condition (b) holds. This concludes the proof.
[99]{}
Aldous D.J. and Eagleson G.K. (1978) On mixing and stability of limit theorems, [*Ann. Probab.*]{}, 6, 325-331.
Aletti G., May C. and Secchi P. (2008) A central limit theorem, and related results, for a two-color randomly reinforced urn, [*Preprint*]{}, currently available at: `ArXiv:math.PR/0811.2097v1`
Bassetti F., Crimaldi I. and Leisen F. (2008) Conditionally identically distributed species sampling sequences, [*Preprint*]{}, currently available at: `http://amsacta.cib.unibo.it/archive/00002479/`
Bay Z.-D. and Hu F. (2005) Asymptotics in randomized urn models, [*Ann. Appl. Probab.*]{}, 15, 914-940.
Berti P., Pratelli L. and Rigo P. (2004) Limit theorems for a class of identically distributed random variables, [*Ann. Probab.*]{}, 32, 2029-2052.
Berti P., Crimaldi I., Pratelli L. and Rigo P. (2009) Rate of convergence of predictive distributions for dependent data, [*Bernoulli*]{}, to appear, currently available at: `http://amsacta.cib.unibo.it/archive/00002538/`
Crimaldi I., Letta G. and Pratelli L. (2007) A strong form of stable convergence, [*Sem. de Probab. XL*]{}, LNM, 1899, 203-225.
Crimaldi I. (2007) An almost sure conditional convergence result and an application to a generalized Polya urn, [*Internat. Math. Forum*]{}, to appear, currently available at: `http://www.m-hikari.com/forth/crimaldiIMF21-24-2009.pdf`
Ghosh J.K. and Ramamoorthi R.V. (2003) [*Bayesian nonparametrics*]{}, Springer.
Hall P. and Heyde C.C. (1980) [*Martingale limit theory and its applications*]{}, Academic Press.
Janson S. (2004) Functional limit theorems for multitype branching processes and generalized Polya urns, [*Stoch. Proc. Appl.*]{}, 110, 177-245.
Janson S. (2005) Limit theorems for triangular urn schemes, [*Probab. Theo. Rel. Fields*]{}, 134, 417-452.
Kallenberg O. (2002) [*Foundations of modern probability*]{}, Springer.
Ledoux M. and Talagrand M. (1991) [*Probability in Banach Spaces*]{}, Springer-Verlag.
May C. and Flournoy N. (2009) Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn, [*Ann. Statist.*]{}, 37, 1058-1078.
Pemantle R. (2007) A survey of random processes with reinforcement, [*Probab. Surveys*]{}, 4, 1-79.
Pitman J. and Yor M. (1997) The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, [*Ann. Probab.*]{}, 25, 855-900.
Renyi A. (1963) On stable sequences of events, [*Sankhya*]{} A, 25, 293-302.
|
---
abstract: 'The long-time relaxation of the average conductance in a 2D mesoscopic sample is studied within the method recently suggested by Muzykantskii and Khmelnitskii and based on a saddle-point approximation to the supermatrix $\sigma$–model. The obtained far asymptotics is in perfect agreement with the result of renormalization group treatment by Altshuler, Kravtsov and Lerner.'
address:
- 'Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany'
- ' and Petersburg Nuclear Physics Institute, 188350 Gatchina, St.Petersburg, Russia.'
author:
- 'Alexander D. Mirlin'
title: 'Long–time relaxation of current in a 2D weakly disordered conductor.'
---
In the recent paper [@MK], Muzykantskii and Khmelnitskii (MK) considered the relaxation phenomena in disordered conductors in the framework of the supersymmetric $\sigma$–model approach. They suggested a nice idea that the long-time asymptotics of the conductance $G(t)$ is governed by a non-trivial saddle point of the $\sigma$–model. Their original goal was to reproduce in a more direct way the result of Altshuler, Kravtsov and Lerner (AKL) [@AKL], who found the logarithmically normal (LN) “tail” in the time dispersion in two and $(2+\epsilon)$ dimensions. However, MK found in 2D a different, power-law decay for moderately large times. They put forward a hypothesis that the LN asymptotics could hold for longer times. Here I will show that this is indeed the case, and that this result can be obtained via the method developed by MK.
Following MK, I consider a 2D disk-shaped sample of a radius $R$. I will consider the unitary symmetry (broken time reversal invariance) in course of the calculations. For systems of the orthogonal and symplectic symmetries, the treatment is completely analogous, and I simply present the corresponding results in the end of the paper. The problem can be described by the $\sigma$–model with the action [@efe] $$S=-{\pi\nu\over 4}\int d^2 r\,\mbox{Str}[D(\nabla Q)^2+2i\omega\Lambda Q]
\label{1a}$$ Here $Q(\bbox{r})$ is $4\times 4$ supermatrix field, $D$ is the diffusion constant, $\nu$ the density of states, $\omega$ the frequency, $\mbox{Str}$ denotes the supertrace, and $\Lambda=\mbox{diag}(1,1,-1,-1)$. The saddle point equation of MK reads: $$\Delta_L\theta+\gamma^2\sinh\theta=0\ ,
\label{2a}$$ where $\theta(\bbox{r})$ is the “non-compact angle” parametrizing the $\sigma$-model field $Q(\bbox{r})$, $\Delta_L$ is the Laplace operator and $\gamma^2=i\omega/D$. It should be supplemented by the boundary conditions on the boundary with leads $$\theta|_{\mbox{leads}}=0
\label{3a}$$ and on insulating boundary $$\nabla_{\bbox{n}} \theta|_{\mbox{insulator}}=0\ ,
\label{4a}$$ where $\nabla_{\bbox{n}}$ denotes the normal derivative.
We can consider the two leads attached to the disk boundary to be of almost semicircular shape, with relatively narrow insulating intervals between them. Then we can approximate the boundary conditions by using eq.(\[3a\]) for all the boundary, as it was done by MK. In fact, in view of the logarithmic dependence of the saddle point action on $R$ (see below), the result should not depend to the leading aproximation on the specific shape of the sample and the leads attached. With the rotationally invariant form of the boundary condition, the minimal action corresponds to the function $\theta$ depending on the radius $r$ only. We get therefore the radial equation $$\theta'' + \theta'/r+\gamma^2\sinh\theta=0\ ;\qquad 0\le r\le R
\label{1}$$ (the prime denotes the derivative $d/dr$) with the boundary conditions: $$\begin{aligned}
&&\theta(R)=0 \label{6a}\ ,\\
&& \theta'(0)=0 \label{7a}\end{aligned}$$ The condition (\[7a\]) follows from the requirment of analyticity of the field in the disk center.
Assuming that characteristic values of $\theta$ satisfy the condition $\theta\gg 1$ (we will find below the corresponding restriction on the time $t$), one can replace $\sinh\theta$ by $e^\theta/2$. Eq.(\[1\]) can be then easily integrated, and its general solution reads: $$e^{\theta(r)}={4C_1^2\over\gamma^2}
{C_2r^{C_1-2}\over (C_2 r^{C_1}+1)^2}\ ,
\label{g}$$ with two integration constants $C_1$ and $C_2$. To satisfy the boundary condition (\[7a\]), we have to choose $C_1=2$. Furthermore, the above assumption $\theta(0)\gg1$ implies that $4C_2/\gamma^2\gg1$. Therefore, the second boundary condition (\[6a\]) is satisfied if $C_2\simeq (4/\gamma R^2)^2$, and the solution can be written in the form $$e^{\theta(r)}\simeq[(r/R)^2 + (\gamma R/4)^2]^{-2}
\label{3}$$ Using now the self-consistency equation of MK, $$2\pi\int_0^R dr\,r(\cosh\theta-1)=t/\pi\nu\ ,
\label{4}$$ one finds $\gamma^2=8\pi^2\nu/t$. Finally, the action $$S\simeq\pi^2\nu D\int dr\,r(\theta^{'2}-\gamma^2 e^\theta)
\label{5}$$ is equal on the saddle point (\[3\]) to $$S\simeq 8\pi^2\nu D\ln(t\Delta)\ ,
\label{5b}$$ where $\Delta=1/(\nu\pi R^2)$ is the mean level spacing. Eq.(\[5\]) coincides exactly with the result of MK. This consideration is valid provided $\theta'(r)<l^{-1}$ on the saddle point solution, which is the condition of the applicability of the diffusion approximation (here $l$ is the mean free path). In combination with the assumption $\theta(0)\gg 1$ this means that $1\ll
t\Delta\ll (R/l)^2$.
Now I consider the ultra-long-time region, $t\gg \Delta^{-1}
(R/l)^2$. In order to support the applicability of the diffusion approximation, we should search for a function $\theta(r)$ minimizing the action with an additional restriction $\theta'\le l^{-1}$. Since the derivative has a tendency to increase in the vicinity of $r=0$, the restriction can be implemented via replacing the boundary conditions (\[7a\]) by $\theta'(r_*)=0$, where the parameter $r_*$ will be specified below. The solution reads now: $$e^{\theta(r)}=\frac { (r/R)^{C-2}} {[(r/R)^C + {C+2\over C-2}
(r_*/R)^C]^2}\ ;\ \ r_*\le r\le R
\label{6}$$ The function $\theta(r)$ is ment as being constant within the vicinity $|r|\le r_*$ of the disk center. The condition $\theta'\le l^{-1}$ yields $r_*\sim lC$. It is important to note that the result does not depend on details of the cut-off procedure. For example one gets the same results if one chooses the boundary condition in the form $\theta'(r_*)=1/l$. The crucial point is that the maximal derivative $\theta'$ should not exceed $1/l$. The constant $C$ is to be found from the self-consistency equation (\[4\]) which can be reduced to the following form: $$\left({R\over r_*}\right)^C= {2t\over\pi^2\nu R^2} {C^2\over C-2}
\label{8a}$$ Neglecting corrections of the $\ln(\ln\cdot)$ form, we find $$C\simeq {\ln (t\Delta) \over \ln (R/r_*)}\simeq
{\ln (t\Delta) \over \ln (R/l)}
\label{7}$$ The action (\[5\]) is then equal to $$S\simeq\pi^2\nu D (C+2)^2\ln (R/r_*)
\simeq\pi^2\nu D \frac {\ln^2[t\Delta(R/l)^2]} {\ln (R/l)}
\label{8}$$
For the orthogonal and symplectic ensembles, the saddle-point equation (\[1\]) has the same form, with the only difference that the action (\[5\]) is multiplied by the factor $\beta/2$, where $\beta=1,2,4$ for the orthogonal, unitary and symplectic symmetries respectively. Combining eqs.(\[5\]) and (\[8\]), we get thus for the long-time asymptotics of the average conductance $G(t)\sim e^{-S}$ in all three symmetry cases: $$\begin{aligned}
& G(t)\sim
(t\Delta)^{-2\pi\beta g}\ , & \quad 1\ll t\Delta\ll (R/l)^2 \label{9d}\\
& G(t)\sim \exp\left\{-{\pi\beta g\over 4} {\ln^2(t/g\tau)\over
\ln(R/l)}\right\}\ , &\quad
t\Delta\gg (R/l)^2
\label{9}\end{aligned}$$ where $g=2\pi\nu D$ is the dimensionless conductance per square in 2D and $\tau$ is the mean free time.
The far asymptotical behavior (eq.(\[9\])) is of the LN form and very similar to that found by AKL (see eq.(7.8) in Ref.[@AKL]). It differs only by the factor $1/g$ in the argument of $\ln^2$. It is easy to see however that this difference disappears if one does the last step of the AKL calculation with a better accuracy. Let us consider for this purpose the intermediate expression of AKL (Ref.[@AKL], eq.(7.11)): $$G(t)\propto -{\sigma\over\tau}\int_0^\infty e^{-t/t_\phi}\exp\left[
-{1\over 4u}\ln^2{t_\phi\over\tau}\right]{dt_\phi\over t_\phi}
\label{10}$$ where $u\simeq{1\over 2\pi^2\nu D}\ln{R\over l}$ in the weak localization region in 2D, which we are considering. Evaluating the integral (\[10\]) by the saddle point method, we find $$\begin{aligned}
G(t)&\sim& \exp\left\{-{1\over 4u}\ln^2{2ut\over\tau}\right\}
\nonumber\\
&\sim& \exp\left\{-{\pi g\over 4} {\ln^2(t/g\tau)\over
\ln(R/l)}\right\} \ ,
\label{11}\end{aligned}$$ where we have kept only the leading term in the exponent. Eq.(\[11\]) is in [*exact*]{} agreement with eq.(\[9\]) for $\beta=1$ (AKL assumed the orthogonal symmetry of the ensemble). Therefore, the supersymmetric treatment confirms the AKL result and also establishes the region of its validity. It is instructive to represent the obtained results in terms of the superposition of simple relaxation processes with mesoscopically distributed relaxation times $t_\phi$: $$G(t)\sim\int {dt_\phi\over t_\phi} e^{-t/t_\phi} P(t_\phi)
\label{11s}$$ Then we have from eqs.(\[9d\]), (\[9\]) for the distribution function $P(t_\phi)$: $$P(t_\phi)\sim\left\{
\begin{array}{ll}
(t_\phi/t_D)^{-2\pi\beta g}\ , &\ \
t_D\ll t_\phi\ll t_D \left({R\over l}\right)^2 \\
\exp\left\{-{\pi\beta g\over 4} {\ln^2(t_\phi/\tau)\over \ln
(R/l)}\right\} \ ,&\ \ t_\phi\gg t_D \left({R\over l}\right)^2\ ,
\end{array}
\right.
\label{11t}$$ where $t_D\simeq R^2/D$ is the time of diffusion through the sample.
For completness, we list also the results for quasi-1D and 3D systems. For a quasi-1D sample (wire) of the length $L$ (which is assumed to be much shorter than the localization length $\xi=2\beta\pi\nu D$) the asymptotics read $$G(t)\sim\exp\left\{-{\beta\pi\nu D\over L}\ln^2(t\Delta)\right\}\ ,\ \
t\Delta\gg 1
\label{12}$$ (for $\beta=2$ this is just eq.(16) of MK). It is interesting to note that eq.(\[12\]) has essentially the same form as the asymptotical formula for $G(t)$ found by Altshuler and Prigodin [@AP] for the [*strictly*]{} 1D sample with a length much [*exceeding*]{} the localization length: $$G(t)\sim\exp\left\{-{l\over L}\ln^2(t/\tau)\right\}
\label{13}$$ If we replace in eq.(\[13\]) the 1D localization length $\xi=2l$ by the quasi-1D localization length $\xi=2\beta\pi\nu D$, we reproduce the asymptotics (\[14\]) (up to a normalization of $t$ in the argument of $\ln^2$, which does not affect the leading term in the exponent for $t\to\infty$). This leads us to make the following two conclusions. Firstly, this confirms once more the general conjecture [@FM] that the statistical properties of smooth envelopes of the wave functions in 1D and quasi-1D samples are identical. Secondly, this shows that the asymptotical “tail” (\[12\]) in the metallic sample is indeed due to “quasi-localized” eigenstates, as has been conjectured [@MK; @AP; @AP1; @Kravtsov].
In 3D, the analysis proceeds along the same line as for the ultra-long-time region in 2D. This is essentially what has been done by MK in their consideration of the 3D case. The result at $t\gg(k_fl)^2t_D$ (where $k_f$ is the Fermi momentum) reads: $$G(t)\sim\exp\{-S(t)\}\ ,\qquad S(t)\sim (k_f l)^2\ln^3\left[{t\over
\tau(k_f l)^2}\right]
\label{14}$$ In contrast to the 2D case, the exact numerical coefficient in the exponent in eq.(\[14\]) cannot be found within the diffusion approximation.
We note in conclusion, that the obtained long-time asymptotics of the average conductance have a very similar form to the asymptotical behavior of the distribution function $P(\rho)$ of local density of states (LDOS) [@LDOS]. In both cases, the result is of the LN form in quasi-1D and 2D, and of a somewhat different (though very similar) $\exp\{-(k_f l)^2\ln^3(\cdot)\}$ form in 3D. As in the case of LDOS distribution [@LDOS], we have found a perfect agreement with the result of renormalization group (RG) treatment [@AKL] in 2D. I believe this agreement between the RG and supersymmetric treatments of $G(t)$ and $P(\rho)$ to be of considerable conceptual importance. To make this point clear, I remind the reader that one of the first achievments of the supersymmetry method as applied to disordered electronic systems was the detailed study of the Anderson metal–insulator transition on the (effectively infinite-dimensional) Bethe lattice [@BL]. The found non-power-law critical behavior seemed at first sight to be in contradiction with the scaling hypothesis and with the results of RG treatment. Since the solution of the Bethe lattice problem was exact, this apparent contradiction questioned the validity of the scaling and RG approaches. These doubts were supported by the fact the solution in [@BL] heavily relied on the non-compact structure of the supersymmetric $\sigma$-model manifold and was dominated by the large values $\theta\gg 1$ of the “non-compact angle” $\theta$. On the other hand, the RG consideration is just a resummation of the perturbative expansion and does not distinguish between the compact and non-compact versions of the $\sigma$-model.
However, we have been able to show recently [@BL-LDOS] that the exotic critical behavior found in [@BL] is the property of infinite-dimensional models only and transits to a power-law one for a finite value $d<\infty$ of the space dimension, in qualitative agreement with predictions of the scaling and RG approaches. Results of [@LDOS] and of the present paper show a perfect quantitative agreement of supersymmetry and RG methods when applied to the problem of asymptotical behavior of various distributions in the ensemble of mesoscopic metallic samples in the weak localization region. This provides strong support to other results obtained within the RG approach in the weak localization region and in the vicinity of the Anderson transition [@AKL]. On the other hand, we see that the supersymmetry method is in many cases able to reproduce results of RG treatment in a more elegant way. Furthermore, it is not restricted like RG to the spatial dimension $d=2$ and can be successfully applied to quasi-1D and 3D systems as well. Besides the study of conductivity relaxation $G(t)$ and LDOS distribution $P(\rho)$ discussed above, I would like to mention in this context the recent progress in understanding of the statistical properties of eigenfunctions [@FM; @FM1; @FE]. Seeing that the two approaches are in amazingly good agreement, we can (depending on the problem considered) use any of them or even combine them to complete our understanding of the properties of mesoscopic disordered systems.
I am grateful to V.E.Kravtsov and D.E.Khmelnitskii for valuable discussions and comments. This work was supported by SFB 195 der Deutschen Forschungsgemeinschaft.
B.A.Muzykantskii, D.E.Khmelnitskii, Phys.Rev. [**B 51**]{}, 5480 (1995). B.L.Altshuler, V.E.Kravtsov, I.V.Lerner, in [*Mesoscopic Phenomena in Solids*]{}, ed. by B.L.Altshuler, P.A.Lee and R.A.Webb (North Holland, Amsterdam , 1991). K.B.Efetov, Adv.Phys. [**32**]{}, 53 (1983). B.L.Altshuler and V.N.Prigodin, Pis’ma Zh. Eksp. Teor. Fiz. [**47**]{}, 36 (1988) \[JETP Lett. [**47**]{}, 43 (1988)\]. B.L.Altshuler, V.N.Prigodin, Zh. Eksp. Teor. Fiz. [**95**]{}, 348 (1989) \[Sov. Phys. JETP [**68**]{}, 198 (1989)\]. V.E.Kravtsov, Habilitationsschrift, Heidelberg, 1992. Y.V.Fyodorov, A.D.Mirlin, Int. Journ. Mod. Phys. B [**8**]{}, 3795 (1994). A.D.Mirlin, preprint cond-mat/9507082, submitted to Phys.Rev.B. K.B.Efetov, [*Zh.Eksp.Theor.Fiz.*]{}, [**92**]{}, 638 (1987). \[[*Sov.Phys.JETP*]{}, [**65**]{}, 360 (1987)\]; M.R.Zirnbauer, [*Phys.Rev.*]{} [**B34**]{}, 6394 (1986); [*Nucl.Phys.*]{} [**B265**]{}, 375 (1986); A.D.Mirlin, Y.V.Fyodorov, [*Nucl.Phys.*]{} [**B366**]{} \[FS\], 507 (1991). A.D.Mirlin, Y.V.Fyodorov, Phys.Rev.Lett. [**72**]{}, 526 (1994); A.D.Mirlin, Y.V.Fyodorov, J.Phys. France I, [**4**]{}, 665 (1994). Y.V.Fyodorov, A.D.Mirlin, Phys.Rev. [**B 51**]{}, 13403 (1995). V.I.Fal’ko, K.B.Efetov, preprints cond-mat/9503096, cond-mat/9507091.
|
---
abstract: 'Let $U_\varepsilon(\g)$ be the standard simply–connected version of the Drinfeld–Jumbo quantum group at an odd m-th root of unity $\varepsilon$. De Concini, Kac and Proicesi observed that isomorphism classes of irreducible representations of $U_\varepsilon(\g)$ are parameterized by the conjugacy classes in the connected simply connected algebraic group $G$ corresponding to the simple complex Lie algebra $\g$. They also conjectured that the dimension of a representation corresponding to a conjugacy class $\mathcal{O}$ is divisible by $m^{\frac{1}{2}{\rm dim}~\mathcal{O}}$. We show that if $\mathcal{O}$ intersects one of special transversal slices $\Sigma_s$ to the set of conjugacy classes in $G$ then the dimension of every finite–dimensional irreducible representation of $U_\varepsilon(\g)$ corresponding to $\mathcal{O}$ is divisible by $m^{\frac{1}{2}{\rm codim}~\Sigma_s}$. This reduces the De Concini–Kac–Proicesi conjecture to constructing appropriate transversal slices $\Sigma_s$ such that ${\rm dim}~\mathcal{O}={\rm codim}~\Sigma_s$ for conjugacy classes $\mathcal{O}$ of exceptional elements in $G$ intersecting $\Sigma_s$. Our result also implies an equivalence between a category of finite–dimensional $U_\varepsilon(\g)$–modules and a category of finite–dimensional representations of a q-W algebra which can be regarded as a truncation of the quantized algebra of regular functions on $\Sigma_s$.'
address: |
Institute of Mathematics, University of Aberdeen\
Aberdeen AB24 3UE, United Kingdom\
e-mail: [email protected]
author:
- 'A. Sevostyanov'
title: |
A proof of De Concini–Kac–Procesi conjecture I.\
Representations of quantum groups at roots of unity and q-W algebras
---
[[*Proof.* ]{}]{}
[g]{}
[n]{}
[h]{}
[U()]{}
[U()]{}
[U()]{}
[U()]{}
[U([b]{})]{}
[p]{}
[z]{}
[m]{}
[[**k**]{}\_]{}
[[Hk]{}\^(A,A\_0,)]{}
[A-[mod]{}]{}
[[D]{}\^-(A)]{}
[[Hom]{}\_A]{}
\[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Conjecture]{} \[theorem\][Proposition]{}
Introduction
============
It is very well known that the number of simple modules for a finite–dimensional algebra over an algebraically closed field is finite. However, often it is very difficult to classify such representations. In some important particular examples even dimensions of simple modules over finite–dimensional algebras are not known.
One of the important examples of that kind is representation theory of semisimple Lie algebras over algebraically closed fields of prime characteristic. Let $\frak g'$ be the Lie algebra of a semisimple algebraic group $G'$ over an algebraically closed field $\bf k$ of characteristic $p>0$. Let $x\mapsto x^{[p]}$ be the $p$-th power map of $\frak g'$ into itself. The structure of the enveloping algebra of $\frak g'$ is quite different from the zero characteristic case. Namely, the elements $x^{p}-x^{[p]},~x\in \frak g'$ are central. For any linear form $\theta$ on $\frak g'$, let $U_{\theta}$ be the quotient of the enveloping algebra of $\frak g'$ by the ideal generated by the central elements $x^{p}-x^{[p]}-\theta (x)^{p}$ with $x\in \frak g'$. Then $U_{\theta }$ is a finite–dimensional algebra. Kac and Weisfeiler proved that any simple $\frak g'$-module can be regarded as a module over $U_{\theta }$ for a unique $\theta$ as above (this explains why all simple $\frak g'$–modules are finite–dimensional). The Kac–Weisfeiler conjecture formulated in [@KW] and proved in [@Pr1] says that if the $G'$–coadjoint orbit of $\theta$ has dimension $d$ then $p^{\frac d2}$ divides the dimension of every finite–dimensional $U_{\theta }$–module.
One can identify $\theta $ with an element of $\frak g'$ via the Killing form and reduce the proof of the Kac–Weisfeiler conjecture to the case of nilpotent $\theta $. In that case Premet defines a subalgebra $U_\theta(\m_\theta)\subset U_{\theta }$ generated by a Lie subalgebra $\m_\theta\subset \g'$ such that $U_\theta(\m_\theta)$ has dimension $p^{\frac d2}$ and every finite–dimensional $U_{\theta }$–module is $U_\theta(\m_\theta)$–free. Verification of the latter fact uses the theory of support varieties (see [@FP1; @FP2; @FP; @Pr3]). Namely, according to the theory of support varieties, in order to prove that a $U_{\theta }$–module is $U_\theta(\m_\theta)$–free one should check that it is free over every subalgebra $U_\theta(x)$ generated in $U_\theta(\m_\theta)$ by a single element $x\in \m_\theta$.
There is a more elementary and straightforward proof of the Kac–Weisfeiler conjecture given in [@Pr2]. A proof of the conjecture for $p>h$, where $h$ is the Coxeter number of the corresponding root system, using localization of $\mathcal{D}$–modules is presented in [@BMR].
Another important example of finite–dimensional algebras is related to the theory of quantum groups at roots of unity. Let $\mathfrak g$ be a complex finite–dimensional semisimple Lie algebra. A remarkable property of the standard Drinfeld-Jimbo quantum group $U_q(\mathfrak g)$ associated to $\mathfrak g$, where $q$ is a primitive $m$-th root of unity, is that its center contains a huge commutative subalgebra isomorphic to the algebra $Z_G$ of regular functions on (a finite covering of a big cell in) a complex algebraic group $G$ with Lie algebra $\mathfrak g$. In this paper we consider the simply–connected version of $U_q(\mathfrak g)$ and the case when $m$ is odd. In that case $G$ is the connected, simply connected algebraic group corresponding to $\g$.
Consider finite–dimensional representations of $U_q(\mathfrak g)$, on which $Z_G$ acts according to nontrivial characters $\eta_g$ given by evaluation of regular functions at various points $g\in G$. Note that all irreducible representations of $U_q(\mathfrak g)$ are of that kind, and every such representation is a representation of the algebra $U_{\eta_g}=U_q(\mathfrak g)/U_q(\mathfrak g){\rm Ker}~\eta_g$ for some $\eta_g$. In [@DKP1] De Concini, Kac and Procesi showed that if $g_1$ and $g_2$ are two conjugate elements of $G$ then the algebras $U_{\eta_{g_1}}$ and $U_{\eta_{g_2}}$ are isomorphic. Moreover in [@DKP1] De Concini, Kac and Procesi formulated the following conjecture.
0.3cm
[**De Concini–Kac–Procesi conjecture.**]{} [*The dimension of any finite–dimensional representation of the algebra $U_{\eta_g}$ is divisible by $m^{\frac{1}{2}{\rm dim}\mathcal{O}_g}$, where $\mathcal{O}_g$ is the conjugacy class of $g$.*]{}
0.3cm
This conjecture is the quantum group counterpart of the Kac–Weisfeiler conjecture for semisimple Lie algebras over fields of prime characteristic.
As it is shown in [@DK1] it suffices to verify the De Concini–Kac–Procesi conjecture in case of exceptional elements $g\in G$ (an element $g\in G$ is called exceptional if its centralizer in $G$ has a finite center). However, the De Concini–Kac–Procesi conjecture is related to the geometry of the group $G$ which is much more complicated than the geometry of the linear space $\g'$ in case of the Kac–Weisfeiler conjecture.
The De Concini–Kac–Procesi conjecture is known to be true for the conjugacy classes of regular elements ([@DKP2]), for the subregular unipotent conjugacy classes in type $A_n$ when $m$ is a power of a prime number ([@10]), for all conjugacy classes in $A_n$ when $m$ is a prime number ([@8]), for the conjugacy classes $\mathcal{O}_g$ of $g\in SL_n$ when the conjugacy class of the unipotent part of $g$ is spherical ([@9]), and for spherical conjugacy classes ([@CCC]). In [@KR] a proof of the De Concini–Kac–Procesi using localization of quantum $\mathcal{D}$–modules is outlined in case of unipotent conjugacy classes.
In this paper following Premet’s philosophy we construct certain subalgebras $U_{\eta_g}(\m_-)$ in $U_{\eta_g}$ over which $U_{\eta_g}$–modules are free, at least for some $g\in G$. Since the De Concini–Kac–Procesi conjecture is related to the structure of conjugacy classes in $G$ it is natural to look at transversal slices to the set of conjugacy classes. It turns out that the definition of the subalgebras $U_{\eta_g}(\m_-)$ is related to the existence of some special transversal slices $\Sigma_s$ to the set of conjugacy classes in $G$. These slices $\Sigma_s$ associated to (conjugacy classes of) elements $s$ in the Weyl group of $\g$ were introduced by the author in [@S6]. The slices $\Sigma_s$ play the role of Slodowy slices in algebraic group theory. In the particular case of elliptic Weyl group elements these slices were also introduced later by He and Lusztig in paper [@Lus] within a different framework.
A remarkable property of a slice $\Sigma_s$ is that if $g$ is conjugate to an element in $\Sigma_s$ then $U_{\eta_g}$ has a subalgebra of dimension $m^{\frac{1}{2}{\rm codim}~\Sigma_s}$ with a nontrivial character. If $g\in \Sigma_s$ (in fact $g$ may belong to a larger variety) then the corresponding subalgebra $U_{\eta_g}(\m_-)$ can be explicitly described in terms of quantum group analogues of root vectors. There are also analogues of subalgebras $U_{\eta_g}(\m_-)$ in $U_q(\mathfrak g)$ in case of generic $q$ (see [@S9]).
In Section \[FREE\] we prove, in particular, that if $g\in \Sigma_s$ then every finite–dimensional $U_{\eta_g}$–module is free over a subalgebra $\widetilde{U}_{\eta_g}(\m_-)$ isomorphic to $U_{\eta_g}(\m_-)$. Thus the dimension of every such module is divisible by $m^{\frac{1}{2}{\rm codim}~\Sigma_s}$, and if the conjugacy class of $g$ intersects $\Sigma_s$ strictly transversally in the sense that ${\rm codim}~\Sigma_s={\dim}~\mathcal{O}_g$, this proves the De Concini–Kac–Procesi conjecture. Thus the De Concini–Kac–Proicesi conjecture is reduced to constructing appropriate transversal slices $\Sigma_s$ such that ${\rm dim}~\mathcal{O}_g={\rm codim}~\Sigma_s$ for conjugacy classes $\mathcal{O}_g$ of exceptional elements in $G$. This will be proved in a subsequent paper. In Section \[FREE\] it is also shown that the rank of every finite–dimensional $U_{\eta_g}$–module $V$ over $\widetilde{U}_{\eta_g}(\m_-)$ is equal to the dimension of the space $V_\chi$ of the so-called Whittaker vectors in $V$, which consists of elements $v\in V$ such that $xv=\chi(x)v$, $x\in \widetilde{U}_{\eta_g}(\m_-)$, and $\chi$ is a nontrivial character of $\widetilde{U}_{\eta_g}(\m_-)$. Whittaker vectors are studied in detail in Section \[WHITT\].
The proof of the main statement of Section \[FREE\] is reduced to the fact that for certain $g$ every finite–dimensional $U_{\eta_g}$–module $V$ is free over every subalgebra ${U}_{\eta_g}(f)$ in ${U}_{\eta_g}(\m_-)$ generated by a quantum analogue $f$ of a root vector in a Lie subalgebra $\m_-\subset \g$. The support variety technique can not be transferred to the case of quantum groups straightforwardly. The notion of the support variety is still available in case of quantum groups (see [@Dr; @GK; @O]). But in practical application it is much less efficient since in case of quantum groups there is no underlying linear space. However, one can show that $V$ is free over ${U}_{\eta_g}(\m_-)$ using a complicated induction over appropriately ordered set of root vectors in $\m_+$. In case of restricted representations of a small quantum group this was done in [@Dr]. The situation in [@Dr] is rather similar to the case of the trivial character $\eta_1$ corresponding to the identity element $1\in G$. In the case considered in this paper the induction is even more complicated because the algebra ${U}_{\eta_g}(\m_-)$ has the Jacobson radical $\mathcal{J}$ (see Section \[WHITT\]), and the quotient ${U}_{\eta_g}(\m_-)/\mathcal{J}$ is a nontrivial semisimple algebra. This shows a major difference between Lie algebras and quantum groups: in case of Lie algebras $\g'$ over fields of prime characteristic the algebras $U_\theta(\m_\theta)$ are local while in the quantum group case the algebras ${U}_{\eta_g}(\m_-)$, which play the role of $U_\theta(\m_\theta)$, are not local.
Slices $\Sigma_s$ also appear in Section \[SKR\] in a different incarnation. Namely, we show that for $g$ conjugate to an element in $\Sigma_s$ the category of finite–dimensional $U_{\eta_g}$–modules is equivalent to a category of finite–dimensional modules over an algebra $W_q^s(G)$ which can be regarded as a noncommutative deformation of a truncated version of the algebra of regular functions on $\Sigma_s$. In case of generic $q$ such algebras, called q-W algebras, were introduced and studied in [@S9]. In fact $U_{\eta_g}$ is the algebra of matrices of size $m^{\frac{1}{2}{\rm codim}~\Sigma_s}$ over the algebra $W_q^s(G)$ which has dimension $m^{\frac{1}{2}{\rm dim}~\Sigma_s}$. In case of Lie algebras over fields of prime characteristic similar results were obtained in [@Pr].
The proofs of statements in Sections \[WHITT\], \[FREE\] and \[SKR\] require some preliminary results which are presented in Sections \[notation\]–\[crossect\].
Notation
========
Fix the notation used throughout of the text. Let $G$ be a connected simply connected finite–dimensional complex simple Lie group, $
{\frak g}$ its Lie algebra. Fix a Cartan subalgebra ${\frak h}\subset {\frak
g}\ $and let $\Delta $ be the set of roots of $\left( {\frak g},{\frak h}
\right)$. Let $\alpha_i,~i=1,\ldots l,~~l=rank({\frak g})$ be a system of simple roots, $\Delta_+=\{ \beta_1, \ldots ,\beta_N \}$ the set of positive roots. Let $H_1,\ldots ,H_l$ be the set of simple root generators of $\frak h$.
Let $a_{ij}$ be the corresponding Cartan matrix, and let $d_1,\ldots , d_l$ be coprime positive integers such that the matrix $b_{ij}=d_ia_{ij}$ is symmetric. There exists a unique non–degenerate invariant symmetric bilinear form $\left( ,\right) $ on ${\frak g}$ such that $(H_i , H_j)=d_j^{-1}a_{ij}$. It induces an isomorphism of vector spaces ${\frak h}\simeq {\frak h}^*$ under which $\alpha_i \in {\frak h}^*$ corresponds to $d_iH_i \in {\frak h}$. We denote by $\alpha^\vee$ the element of $\frak h$ that corresponds to $\alpha \in {\frak h}^*$ under this isomorphism. The induced bilinear form on ${\frak h}^*$ is given by $(\alpha_i , \alpha_j)=b_{ij}$.
Let $W$ be the Weyl group of the root system $\Delta$. $W$ is the subgroup of $GL({\frak h})$ generated by the fundamental reflections $s_1,\ldots ,s_l$, $$s_i(h)=h-\alpha_i(h)H_i,~~h\in{\frak h}.$$ The action of $W$ preserves the bilinear form $(,)$ on $\frak h$. We denote a representative of $w\in W$ in $G$ by the same letter. For $w\in W, g\in G$ we write $w(g)=wgw^{-1}$. For any root $\alpha\in \Delta$ we also denote by $s_\alpha$ the corresponding reflection.
Let ${{\frak b}_+}$ be the positive Borel subalgebra and ${\frak b}_-$ the opposite Borel subalgebra; let ${\frak n}_+=[{{\frak b}_+},{{\frak b}_+}]$ and ${\frak n}_-=[{\frak b}_-,{\frak b}_-]$ be their nilradicals. Let $H=\exp {\frak h},N_+=\exp {{\frak n}_+},
N_-=\exp {\frak n}_-,B_+=HN_+,B_-=HN_-$ be the Cartan subgroup, the maximal unipotent subgroups and the Borel subgroups of $G$ which correspond to the Lie subalgebras ${\frak h},{{\frak n}_+},{\frak n}_-,{\frak b}_+$ and ${\frak b}_-,$ respectively.
We identify $\frak g$ and its dual by means of the canonical invariant bilinear form. Then the coadjoint action of $G$ on ${\frak g}^*$ is naturally identified with the adjoint one. We also identify ${{\frak n}_+}^*\cong {\frak n}_-,~{{\frak b}_+}^*\cong {\frak b}_-$.
Let ${\frak g}_\beta$ be the root subspace corresponding to a root $\beta \in \Delta$, ${\frak g}_\beta=\{ x\in {\frak g}| [h,x]=\beta(h)x \mbox{ for every }h\in {\frak h}\}$. ${\frak g}_\beta\subset {\frak g}$ is a one–dimensional subspace. It is well–known that for $\alpha\neq -\beta$ the root subspaces ${\frak g}_\alpha$ and ${\frak g}_\beta$ are orthogonal with respect to the canonical invariant bilinear form. Moreover ${\frak g}_\alpha$ and ${\frak g}_{-\alpha}$ are non–degenerately paired by this form.
Root vectors $X_{\alpha}\in {\frak g}_\alpha$ satisfy the following relations: $$[X_\alpha,X_{-\alpha}]=(X_\alpha,X_{-\alpha})\alpha^\vee.$$
Note also that in this paper we denote by $\mathbb{N}$ the set of nonnegative integer numbers, $\mathbb{N}=\{0,1,\ldots \}$.
Quantum groups
==============
The standard simply connected quantum group $U_q({\frak g})$ associated to a complex finite–dimensional simple Lie algebra $\frak g$ is the algebra over $\mathbb{C}(q)$ generated by elements $L_i,L_i^{-1},~X_i^+,~X_i^-,~i=1,\ldots ,l$, and with the following defining relations: $$\label{qgr}
\begin{array}{l}
[L_i,L_j]=0,~~L_iL_i^{-1}=L_i^{-1}L_i=1,~~ L_iX_j^\pm L_i^{-1}=q_i^{\pm \delta_{ij}}X_j^\pm, \\
\\
X_i^+X_j^- -X_j^-X_i^+ = \delta _{i,j}{K_i -K_i^{-1} \over q_i -q_i^{-1}} , \\
\\
\mbox{where }K_i=\prod_{j=1}^lL_j^{a_{ji}},~~q_i=q^{d_i},
\end{array}$$ and the quantum Serre relations: $$\label{qserre}
\begin{array}{l}
\sum_{r=0}^{1-a_{ij}}(-1)^r
\left[ \begin{array}{c} 1-a_{ij} \\ r \end{array} \right]_{q_i}
(X_i^\pm )^{1-a_{ij}-r}X_j^\pm(X_i^\pm)^r =0 ,~ i \neq j ,\\ \\
\mbox{ where }\\
\\
\left[ \begin{array}{c} m \\ n \end{array} \right]_q={[m]_q! \over [n]_q![n-m]_q!} ,~
[n]_q!=[n]_q\ldots [1]_q ,~ [n]_q={q^n - q^{-n} \over q-q^{-1} }.
\end{array}$$ $U_q({\frak g})$ is a Hopf algebra with comultiplication defined by $$\begin{array}{l}
\Delta(L_i^{\pm 1})=L_i^{\pm 1}\otimes L_i^{\pm 1},\\
\\
\Delta(X_i^+)=X_i^+\otimes K_i+1\otimes X_i^+,
\end{array}$$ $$\Delta(X_i^-)=X_i^-\otimes 1 +K_i^{-1}\otimes X_i^-,$$ antipode defined by $$S(L_i^{\pm 1})=L_i^{\mp 1},~~S(X_i^+)=-X_i^+K_i^{-1},~~S(X_i^-)=-K_iX_i^-,$$ and counit defined by $$\varepsilon(L_i^{\pm 1})=1,~~\varepsilon(X_i^\pm)=0.$$
Now we shall explicitly describe a basis for $U_q({\frak g})$. First following [@ChP] we recall the construction of root vectors of $U_q({\frak g})$ in terms of a braid group action on $U_q({\frak g})$. Let $m_{ij}$, $i\neq j$ be equal to $2,3,4,6$ if $a_{ij}a_{ji}$ is equal to $0,1,2,3$. The braid group $\mathcal{B}_\g$ associated to $\g$ has generators $T_i$, $i=1,\ldots, l$, and defining relations $$T_iT_jT_iT_j\ldots=T_jT_iT_jT_i\ldots$$ for all $i\neq j$, where there are $m_{ij}$ $T$’s on each side of the equation.
There is an action of the braid group $\mathcal{B}_\g$ by algebra automorphisms of $U_q({\frak g})$ defined on the standard generators as follows: $$\begin{aligned}
T_i(X_i^+)=-X_i^-K_i,~T_i(X_i^-)=-K_i^{-1}X_i^+,~T_i(L_j)=L_jK_i^{-1}, \\
\\
T_i(X_j^+)=\sum_{r=0}^{-a_{ij}}(-1)^{r-a_{ij}}q_i^{-r}
(X_i^+ )^{(-a_{ij}-r)}X_j^+(X_i^+)^{(r)},~i\neq j,\\
\\
T_i(X_j^-)=\sum_{r=0}^{-a_{ij}}(-1)^{r-a_{ij}}q_i^{r}
(X_i^-)^{(r)}X_j^-(X_i^-)^{(-a_{ij}-r)},~i\neq j,\end{aligned}$$ where $$(X_i^+)^{(r)}=\frac{(X_i^+)^{r}}{[r]_{q_i}!},~(X_i^-)^{(r)}=\frac{(X_i^-)^{r}}{[r]_{q_i}!},~r\geq 0,~i=1,\ldots,l.$$
Recall that an ordering of a set of positive roots $\Delta_+$ is called normal if all simple roots are written in an arbitrary order, and for any three roots $\alpha,~\beta,~\gamma$ such that $\gamma=\alpha+\beta$ we have either $\alpha<\gamma<\beta$ or $\beta<\gamma<\alpha$.
Any two normal orderings in $\Delta_+$ can be reduced to each other by the so–called elementary transpositions (see [@Z], Theorem 1). The elementary transpositions for rank 2 root systems are inversions of the following normal orderings (or the inverse normal orderings): $$\label{rank2}
\begin{array}{lr}
\alpha,~\beta & A_1+A_1 \\
\\
\alpha,~\alpha+\beta,~\beta & A_2 \\
\\
\alpha,~\alpha+\beta,~\alpha+2\beta,~\beta & B_2 \\
\\
\alpha,~\alpha+\beta,~2\alpha+3\beta,~\alpha+2\beta,~\alpha+3\beta,~\beta & G_2
\end{array}$$ where it is assumed that $(\alpha,\alpha)\geq (\beta,\beta)$. Moreover, any normal ordering in a rank 2 root system is one of orderings (\[rank2\]) or one of the inverse orderings.
In general an elementary inversion of a normal ordering in a set of positive roots $\Delta_+$ is the inversion of an ordered segment of form (\[rank2\]) (or of a segment with the inverse ordering) in the ordered set $\Delta_+$, where $\alpha-\beta\not\in \Delta$.
For any reduced decomposition $w_0=s_{i_1}\ldots s_{i_D}$ of the longest element $w_0$ of the Weyl group $W$ of $\g$ the set $$\beta_1=\alpha_{i_1},\beta_2=s_{i_1}\alpha_{i_2},\ldots,\beta_D=s_{i_1}\ldots s_{i_{D-1}}\alpha_{i_D}$$ is a normal ordering in $\Delta_+$, and there is one to one correspondence between normal orderings of $\Delta_+$ and reduced decompositions of $w_0$ (see [@Z1]).
Now fix a reduced decomposition $w_0=s_{i_1}\ldots s_{i_D}$ of the longest element $w_0$ of the Weyl group $W$ of $\g$ and define the corresponding root vectors in $U_q({\frak g})$ by $$\label{rootvect}
X_{\beta_k}^\pm=T_{i_1}\ldots T_{i_{k-1}}X_{i_k}^\pm.$$
\[rootprop\] For $\beta =\sum_{i=1}^lm_i\alpha_i,~m_i\in {\Bbb N}$ $X_{\beta}^\pm $ is a polynomial in the noncommutative variables $X_i^\pm$ homogeneous in each $X_i^\pm$ of degree $m_i$.
Note that one can construct root vectors in the Lie algebra $\g$ in a similar way. Namely, if $X_{\pm \alpha_i}$ are simple root vectors of $\g$ then one can define an action of the braid group $\mathcal{B}_\g$ by algebra automorphisms of ${\frak g}$ defined on the standard generators as follows: $$\begin{aligned}
T_i(X_{\pm \alpha_i})=-X_{\mp \alpha_i},~T_i(H_j)=H_j-a_{ji}H_i, \\
\\
T_i(X_{\alpha_j})=\frac{1}{(-a_{ij})!}
{\rm ad}_{X_{\alpha_i} }^{-a_{ij}}X_{\alpha_j},~i\neq j,\\
\\
T_i(X_{-\alpha_j})=\frac{(-1)^{a_{ij}}}{(-a_{ij})!}
{\rm ad}_{X_{-\alpha_i} }^{-a_{ij}}X_{-\alpha_j},~i\neq j.\end{aligned}$$ Now the root vectors $X_{\pm \beta_k}\in \g_{\pm \beta_k}$ of $\g$ can be defined by $$\label{rootvectg}
X_{\pm \beta_k}=T_{i_1}\ldots T_{i_{k-1}}X_{\pm \alpha_{i_k}}.$$
The root vectors $X_{\beta}^-$ satisfy the following relations: $$\label{qcom}
X_{\alpha}^-X_{\beta}^- - q^{(\alpha,\beta)}X_{\beta}^-X_{\alpha}^-= \sum_{\alpha<\delta_1<\ldots<\delta_n<\beta}C(k_1,\ldots,k_n)
{(X_{\delta_n}^-)}^{(k_n)}{(X_{\delta_{n-1}}^-)}^{(k_{n-1})}\ldots {(X_{\delta_1}^-)}^{(k_1)},~~\alpha<\beta,$$ where for $\alpha \in \Delta_+$ we put ${(X_{\alpha}^\pm)}^{(k)}=\frac{(X_\alpha^\pm)^{k}}{[k]_{q_\alpha}!}$, $k\geq 0$, $q_\alpha =q^{d_i}$ if the positive root $\alpha$ is Weyl group conjugate to the simple root $\alpha_i$, $C(k_1,\ldots,k_n)\in {\Bbb C}[q,q^{-1}]$.
Let $U_q({\frak n}_+),U_q({\frak n}_-)$ and $U_q({\mathfrak h})$ be the subalgebras of $U_q({\frak g})$ generated by the $X_i^+$, by the $X_i^-$ and by the $L_i^{\pm 1}$, respectively.
Now using the root vectors $X_{\beta}^\pm$ we can construct a basis of $U_q({\frak g})$. Define for ${\bf r}=(r_1,\ldots ,r_D)\in {\Bbb N}^D$, $$(X^+)^{(\bf r)}=(X_{\beta_1}^+)^{(r_1)}\ldots (X_{\beta_D}^+)^{(r_D)},$$ $$(X^-)^{(\bf r)}=(X_{\beta_D}^-)^{(r_D)}\ldots (X_{\beta_1}^-)^{(r_1)},$$ and for ${\bf s}=(s_1,\ldots s_l)\in {\Bbb Z}^{l}$, $$L^{\bf s}=L_1^{s_1}\ldots L_l^{s_l}.$$
[**([@kh-t], Proposition 3.3)**]{}\[PBW\] The elements $(X^+)^{(\bf r)}$, $(X^-)^{(\bf t)}$ and $L^{\bf s}$, for ${\bf r},~{\bf t}\in {\Bbb N}^N$, ${\bf s}\in {\Bbb Z}^l$, form topological bases of $U_q({\frak n}_+),U_q({\frak n}_-)$ and $U_q({\frak h})$, respectively, and the products $(X^+)^{(\bf r)}L^{\bf s}(X^-)^{(\bf t)}$ form a basis of $U_q({\frak g})$. In particular, multiplication defines an isomorphism of ${\Bbb C}(q)$–modules: $$U_q({\frak n}_-)\otimes U_q({\frak h}) \otimes U_q({\frak n}_+)\rightarrow U_q({\frak g}).$$
Let $U_\mathcal{A}(\g)$ be the subalgebra in $U_q(\g)$ over the ring $\mathcal{A}=\mathbb{C}[q,q^{-1}]$ generated over $\mathcal{A}$ by the elements $L_i^{\pm 1},~{K_i -K_i^{-1} \over q_i -q_i^{-1}},~X_i^\pm,~i=1,\ldots ,l$. The most important for us is the specialization $U_\varepsilon(\g)$ of $U_\mathcal{A}(\g)$, $U_\varepsilon(\g)=U_\mathcal{A}(\g)/(q-\varepsilon)U_\mathcal{A}(\g)$, $\varepsilon\in \mathbb{C}^*$. $U_\mathcal{A}(\g)$ and $U_\varepsilon(\g)$ are Hopf algebras with the comultiplication induced from $U_q(\g)$. If in addition $\varepsilon^{2d_i}\neq 1$ for $i=1,\ldots ,l$ then $U_\varepsilon(\g)$ is generated over $\mathbb{C}$ by $L_i^{\pm 1},~X_i^\pm,~i=1,\ldots ,l$ subject to relations (\[qgr\]) and (\[qserre\]) where $q=\varepsilon$. We also have the following obvious consequence of Proposition \[PBW\].
\[PBW1\] Let $U_\varepsilon({\frak n}_+),U_\varepsilon({\frak n}_-)$ and $U_\varepsilon({\mathfrak h})$ be the subalgebras of $U_\varepsilon({\frak g})$ generated by the $X_i^+$, by the $X_i^-$ and by the $L_i^{\pm 1}$, respectively. The elements $(X^+)^{\bf r}=(X_{\beta_1}^+)^{r_1}\ldots (X_{\beta_D}^+)^{r_D}$, $(X^-)^{\bf t}=(X_{\beta_D}^-)^{t_D}\ldots (X_{\beta_1}^-)^{t_1}$ and $L^{\bf s}$, for ${\bf r},~{\bf t}\in {\Bbb N}^N$, ${\bf s}\in {\Bbb Z}^l$, form bases of $U_\varepsilon({\frak n}_+),U_\varepsilon({\frak n}_-)$ and $U_\varepsilon({\frak h})$, respectively, and the products $(X^+)^{\bf r}L^{\bf s}(X^-)^{\bf t}$ form a basis of $U_\varepsilon({\frak g})$. In particular, multiplication defines an isomorphism of vector spaces: $$U_\varepsilon({\frak n}_-)\otimes U_\varepsilon({\frak h}) \otimes U_\varepsilon({\frak n}_+)\rightarrow U_\varepsilon({\frak g}).$$ The root vectors $X_{\beta}^+$ satisfy the following relations in $U_\varepsilon({\frak g})$: $$\label{qcom1}
X_{\alpha}^-X_{\beta}^- - \varepsilon^{(\alpha,\beta)}X_{\beta}^-X_{\alpha}^-= \sum_{\alpha<\delta_1<\ldots<\delta_n<\beta}C(k_1,\ldots,k_n)
{(X_{\delta_n}^-)}^{(k_n)}{(X_{\delta_{n-1}}^-)}^{(k_{n-1})}\ldots {(X_{\delta_1}^-)}^{(k_1)},~~\alpha<\beta,$$ where $C(k_1,\ldots,k_n)\in {\Bbb C}$.
Quantum groups at roots of unity {#1root}
================================
Let $m$ be a an odd positive integer number, and $m>d_i$ for all $i$, $\varepsilon$ a primitive $m$-th root of unity. In this section, following [@ChP], Section 9.2, we recall some results on the structure of the algebra $U_\varepsilon({\frak g})$. We keep the notation introduced in Section \[notation\].
Let $Z_\varepsilon$ be the center of $U_\varepsilon({\frak g})$.
[**([@DK], Corollary 3.3, [@DKP1], Theorems 3.5, 7.6 and Proposition 4.5)**]{}\[ue\] Fix the normal ordering in the positive root system $\Delta_+$ corresponding a reduced decomposition $w_0=s_{i_1}\ldots s_{i_D}$ of the longest element $w_0$ of the Weyl group $W$ of $\g$ and let $X_{\alpha}^\pm$ be the corresponding root vectors in $U_\varepsilon({\frak g})$, and $X_{\pm \beta_k}$ the corresponding root vectors in $\g$. Let $x_{\alpha}^-=(\varepsilon_{\alpha}-\varepsilon_{\alpha}^{-1})^m(X_{\alpha}^-)^m$, $x_{\alpha}^+=(\varepsilon_{\alpha}-\varepsilon_{\alpha}^{-1})^mT_0(X_{\alpha}^-)^m$, where $T_0=T_{i_1}\ldots T_{i_D}$, $\alpha\in \Delta_+$ and $l_i=L_i^m$, $i=1,\ldots ,l$ be elements of $U_\varepsilon({\frak g})$.
Then the following statements are true.
\(i) The elements $x_{\alpha}^\pm$, $\alpha\in \Delta_+$, $l_i$, $i=1,\ldots ,l$ lie in $Z_\varepsilon$.
\(ii) Let $Z_0$ ($Z_0^\pm$ and $Z_0^0$) be the subalgebras of $Z_\varepsilon$ generated by the $x_{\alpha}^\pm$ and the $l_i^{\pm 1}$ (respectively by the $x_{\alpha}^\pm$ and by the $l_i^{\pm 1}$). Then $Z_0^\pm\subset U_\varepsilon({\frak n}_\pm)$, $Z_0^0\subset U_\varepsilon({\frak h})$, $Z_0^\pm$ is the polynomial algebra with generators $x_{\alpha}^\pm$, $Z_0^0$ is the algebra of Laurent polynomials in the $l_i$, $Z_0^\pm=U_\varepsilon({\frak n}_\pm)\bigcap Z_0$, and multiplication defines an isomorphism of algebras $$Z_0^-\otimes Z_0^0 \otimes Z_0^+ \rightarrow Z_0.$$
\(iii) $U_\varepsilon({\frak g})$ is a free $Z_0$–module with basis the set of monomials $(X^+)^{\bf r}L^{\bf s}(X^-)^{\bf t}$ in the statement of Proposition \[PBW1\] for which $0\leq r_k,t_k,s_i<m$ for $i=1,\ldots ,l$, $k=1,\ldots ,D$.
\(iv) ${\rm Spec}(Z_0)=\mathbb{C}^{2D}\times(\mathbb{C}^*)^l$ is a complex affine space of dimension equal to ${\rm dim}~\g$, ${\rm Spec}(Z_\varepsilon)$ is a normal affine variety and the map $$\tau: {\rm Spec}(Z_\varepsilon) \rightarrow {\rm Spec}(Z_0)$$ induced by the inclusion $Z_0\hookrightarrow Z_\varepsilon$ is a finite map of degree $m^l$.
\(v) The subalgebra $Z_0$ is preserved by the action of the braid group automorphisms $T_i$.
\(vi) Let $G$ be the connected simply connected Lie group corresponding to the Lie algebra $\g$ and $G^*_0$ the solvable algebraic subgroup in $G\times G$ which consists of elements of the form $(L_+',L_-')\in G\times G$, $$(L_+',L_-')=(t,t^{-1})(n_+',n_-'),~n_\pm' \in N_\pm,~t\in H.$$
Then ${\rm Spec}(Z_0^0)$ can be naturally identified with the maximal torus $H$ in $G$, and the map $$\widetilde{\pi}: {\rm Spec}(Z_0)={\rm Spec}(Z_0^+)\times {\rm Spec}(Z_0^0) \times {\rm Spec}(Z_0^-)\rightarrow G^*_0,$$ $$\widetilde{\pi}(u_+,t,u_-)=(t{\bf X}^+(u_+),t^{-1}{\bf X}^-(u_-)^{-1}),~u_\pm\in {\rm Spec}(Z_0^\pm),~t\in {\rm Spec}(Z_0^0),$$ $${\bf X}^\pm: {\rm Spec}(Z_0^\pm) \rightarrow N_\pm$$ $${\bf X}^-=\exp(x_{\beta_D}^-X_{-\beta_D})\exp(x_{\beta_{D-1}}^-X_{-\beta_{D-1}})\ldots \exp(x_{\beta_1}^-X_{-\beta_1}),$$ $${\bf X}^+=\exp(x_{\beta_D}^+T_0(X_{-\beta_D}))\exp(x_{\beta_{D-1}}^+T_0(X_{-\beta_{D-1}}))\ldots \exp(x_{\beta_1}^+T_0(X_{-\beta_1})),$$ where $x_{\beta_i}^\pm$ should be regarded as complex-valued functions on ${\rm Spec}(Z_0)$, is an isomorphism of varieties independent of the choice of reduced decomposition of $w_0$.
In fact ${\rm Spec}(Z_0)$ carries a natural structure of a Poisson–Lie group, and the map $\widetilde{\pi}$ is an isomorphism of algebraic Poisson–Lie groups if $G^*_0$ is regarded as the dual Poisson–Lie group to the Poisson–Lie group $G$ equipped with the standard Sklyanin bracket (see [@DKP1], Theorem 7.6). We shall not need this fact in this paper.
Let ${\bf K}:{\rm Spec}(Z_0^0)\rightarrow H$ be the map defined by ${\bf K}(h)=h^2$, $h\in H$.
[**([@DKP1], Corollary 4.7)**]{} Let $G^0=N_-HN_+$ be the big cell in $G$. Then the map $$\pi={\bf X}^-{\bf K}{\bf X}^+:{\rm Spec}(Z_0)\rightarrow G^0$$ is independent of the choice of reduced decomposition of $w_0$, and is an unramified covering of degree $2^l$.
Define derivations $\underline{x}_i^\pm$ of $U_{\mathcal{A}}(\g)$ by $$\label{qder}
\underline{x}_i^+(u)=\left[ \frac{(X_i^+)^m}{[m]_{q_i}!},u \right],~\underline{x}_i^-(u)=T_0\underline{x}_i^+T_0^{-1}(u),~i=1,\ldots ,l.$$
Let $\widehat{Z}_0$ be the algebra of formal power series in the $x_{\alpha}^\pm$, $\alpha\in \Delta_+$, and the $l_i^{\pm1}$, $i=1,\ldots ,l$, which define holomorphic functions on ${\rm Spec}(Z_0)=\mathbb{C}^{2D}\times(\mathbb{C}^*)^l$. Let $$\widehat{U}_\varepsilon(\g)=U_\varepsilon(\g)\otimes_{Z_0}\widehat{Z}_0,~~
\widehat{Z}_\varepsilon=Z_\varepsilon\otimes_{Z_0}\widehat{Z}_0.$$
[**([@DK], Propositions 3.4, 3.5, [@DKP1], Proposition 6.1)**]{}\[qcoadj\]
(i)On specializing to $q=\varepsilon$, (\[qder\]) induces a well–defined derivation $\underline{x}_i^\pm$ of ${U}_\varepsilon(\g)$.
(ii)The series $$\exp(t\underline{x}_i^\pm)=\sum_{k=0}^\infty \frac{t^k}{k!}(\underline{x}_i^\pm)^k$$ converge for all $t\in \mathbb{C}$ to a well–defined automorphisms of the algebra $\widehat{U}_\varepsilon(\g)$.
(iii)Let $\mathcal{G}$ be the group of automorphisms generated by the one–parameter groups $\exp(t\underline{x}_i^\pm)$, $i=1,\ldots, l$. The action of $\mathcal{G}$ on $\widehat{U}_\varepsilon(\g)$ preserves the subalgebras $\widehat{Z}_\varepsilon$ and $\widehat{Z}_0$, and hence acts by holomorphic automorphisms on the complex algebraic varieties ${\rm Spec}(Z_\varepsilon)$ and ${\rm Spec}(Z_0)$.
(iv)Let $\mathcal{O}$ be a conjugacy class in $G$. The intersection $\mathcal{O}^0=\mathcal{O}\bigcap G^0$ is a smooth connected variety, and the connected components of the variety $\pi^{-1}(\mathcal{O}^0)$ are $\mathcal{G}$–orbits in ${\rm Spec}(Z_0)$.
(v)If $\mathcal{P}$ is a $\mathcal{G}$–orbit in ${\rm Spec}(Z_0)$ then the connected components of $\tau^{-1}(\mathcal{P})$ are $\mathcal{G}$–orbits in ${\rm Spec}(Z_\varepsilon)$.
Given a homomorphism $\eta:{Z}_0 \rightarrow \mathbb{C}$, let $$U_\eta(\g)={U}_\varepsilon(\g)/I_\eta,$$ where $I_\eta$ is the ideal in ${U}_\varepsilon(\g)$ generated by elements $z-\eta(z)$, $z\in Z_0$. By part (iii) of Proposition \[ue\] $U_\eta(\g)$ is an algebra of dimension $m^{{\rm dim}~\g}$ with basis the set of monomials $(X^+)^{\bf r}L^{\bf s}(X^-)^{\bf t}$ in the statement of Proposition \[PBW1\] for which $0\leq r_k,t_k,s_i<m$ for $i=1,\ldots ,l$, $k=1,\ldots ,D$.
If $V$ is an irreducible finite–dimensional representation of ${U}_\varepsilon(\g)$ then by the Schur lemma $zv=\theta(z)v$ for all $v\in V$ and $z\in Z_\varepsilon$ and some character $\theta:{Z}_\varepsilon \rightarrow \mathbb{C}$. Therefore we get a natural map $$X:{\rm Rep}({U}_\varepsilon(\g)) \rightarrow {\rm Spec}(Z_\varepsilon),$$ where ${\rm Rep}({U}_\varepsilon(\g))$ is the set of equivalence classes of irreducible finite-dimensional representations of ${U}_\varepsilon(\g)$, and $V$ is in fact a representation of the algebra $U_\eta(\g)$ for $\eta=X(\theta)$. We shall identify this representation with $V$. Observe that every finite–dimensional irreducible representation in ${\rm Rep}({U}_\varepsilon(\g))$ is a representation of $U_\eta(\g)$ for some $\eta\in {\rm Spec}(Z_0)$.
If $\widetilde{g}\in \mathcal{G}$ then for any $\eta\in {\rm Spec}(Z_0)$ we have $\widetilde{g}\eta\in {\rm Spec}(Z_0)$ by part (iii) of Proposition \[qcoadj\], and by part (ii) of the same proposition $\widetilde{g}$ induces an isomorphism of algebras, $$\widetilde{g}:U_\eta(\g) \rightarrow U_{\widetilde{g}\eta}(\g).$$ This establishes a bijection between the sets ${\rm Rep}(U_\eta(\g))$ and ${\rm Rep}(U_{\widetilde{g}\eta}(\g))$ of equivalence classes of irreducible finite-dimensional representations of $U_\eta(\g)$ and $U_{\widetilde{g}\eta}(\g)$, $$\label{repiso}
\widetilde{g}:{\rm Rep}(U_\eta(\g)) \rightarrow {\rm Rep}(U_{\widetilde{g}\eta}(\g)).$$
For every finite-dimensional representation $V$ of $U_\eta(\g)$, and $\widetilde{g}\in \mathcal{G}$ we denote by $V^{\widetilde{g}}$ the corresponding representation of $U_{\widetilde{g}\eta}(\g)$.
For any element $g\in G$ let $g_s,g_u\in G$ be the semisimple and the unipotent part of $g$ so that $g=g_sg_u$. Recall that $g$ is called exceptional if the centralizer of $g_s$ in $G$ has a finite center.
Let $\varphi=\pi \tau X: {\rm Rep}({U}_\varepsilon(\g)) \rightarrow G^0$ be the composition of the three maps $\pi$, $\tau$ and $X$ defined above. A finite–dimensional irreducible representation $V$ of ${U}_\varepsilon(\g)$ is called exceptional if $\varphi(V)\in G^0\subset G$ is an exceptional element.
Observe that the conjugacy class of every non–exceptional element contains an element $g\in G$ such that $$\label{can1}
g_s\in H,~~g_u\in N_-,$$ $$\label{can2}
{\rm the~Lie~algebra}~{\mathfrak h}_g~{\rm of~the~center~of~the~centralizer~of}~g_s~{\rm in}~G~{\rm is~nontrivial},$$ and $$\label{can3}
\Delta'=\{\alpha\in \Delta:\alpha|_{{\mathfrak h}_g}=0\}=\mathbb{Z}\Gamma'\cap \Delta,$$ where $\Gamma'\subset \Gamma$ is a proper subset of the set of simple positive roots $\Gamma$.
Therefore if $V$ is a non–exceptional irreducible finite–dimensional representation of ${U}_\varepsilon(\g)$ then $V$ can be regarded as a representation of the algebra $U_\eta(\g)$ for $\eta=\tau X(V)$, and by part (iv) of Proposition \[qcoadj\] there exists an element $\widetilde{g}\in \mathcal{G}$ such that $\pi(\widetilde{g}\eta)$ satisfies properties (\[can1\])-(\[can3\]), and by (\[repiso\]) $\widetilde{g}V$ can be regarded as a representation of the algebra $U_{\widetilde{g}\eta}(\g)$.
Replacing $V$ with $\widetilde{g}V$ we may assume that $V$ is an irreducible representation of the algebra $U_\eta(\g)$ such that $g=\pi(\eta)$ satisfies (\[can1\])-(\[can3\]).
Let ${U}'_\varepsilon(\g)$ be the subalgebra of ${U}_\varepsilon(\g)$ generated by ${U}_\varepsilon({\mathfrak h})$ and all the elements $X_i^\pm$ such that $\alpha_i\in \Gamma'$. Denote by $U'_\eta(\g)$ the quotient of ${U}'_\varepsilon(\g)$ by the ideal generated by elements $z-\eta(z)$, $z\in Z_0\cap {U}'_\varepsilon(\g)$. Now let ${U}_\varepsilon^g(\g)={U}'_\varepsilon(\g)U_\varepsilon(\n_+)$ and ${U}_\eta^g(\g)$ be the quotient of ${U}_\varepsilon^g(\g)$ by the ideal generated by elements $z-\eta(z)$, $z\in Z_0\cap {U}^g_\varepsilon(\g)$. The algebras ${U}_\varepsilon^g(\g)$ and ${U}_\eta^g(\g)$ can be regarded as quantum analogues of the parabolic subalgebras associated to the subset $\Gamma'$ of simple roots. Let also $U''_\eta(\g)$ be the subalgebra of $U'_\eta(\g)$ generated by all the elements $X_i^\pm$ and $L_i^{\pm1}$ such that $\alpha_i\in \Gamma'$. $U''_\eta(\g)$ can be regarded as the semisimple part of the Levi factor $U'_\eta(\g)$.
The following fundamental proposition states that $V$ is in fact induced from a representation of the algebra ${U}_\eta^g(\g)$.
[**([@DKP1], Theorem 6.8, [@DK1], §8, Theorem)**]{}\[except\]
(i)The ${U}_\eta(\g)$–module $V$ contains a unique irreducible ${U}_\eta^g(\g)$–submodule $V'$ which remains irreducible when restricted to $U''_\eta(\g)$.
(ii)The ${U}_\eta(\g)$–module $V$ is induced from the ${U}_\eta^g(\g)$–module $V'$, $$V={U}_\eta(\g)\otimes_{{U}_\eta^g(\g)}V',$$ with the left action defined by left multiplication on ${U}_\eta^g(\g)$. In particular, ${\rm dim}~V=m^{t/2}{\rm dim}~V'$, where $t=|\Delta\setminus \Delta'|$.
(iii)The map $V\mapsto V'$ establishes a bijection ${\rm Rep}(U_\eta(\g)) \rightarrow {\rm Rep}(U''_\eta(\g))$, and $V'$ can be regarded as an exceptional representation of the algebra $U_\varepsilon(\g')$, where $\g'$ is the Lie subalgebra of $\g$ generated by the Chevalley generators corresponding to $\alpha_i\in \Gamma'$.
Realizations of quantum groups associated to Weyl group elements {#wqreal}
================================================================
Some important ingredients that will be used in the proof of the main statement in Section \[FREE\] are certain subalgebras of the quantum group. These subalgebras are defined in terms of realizations of the algebra $U_\varepsilon(\g)$ associated to Weyl group elements. We introduce these realizations in this section. A similar construction in case of quantum groups $U_q(\g)$ with generic $q$ was introduced in [@S9].
Let $s$ be an element of the Weyl group $W$ of the pair $(\g,{\mathfrak h})$, and ${\mathfrak h}'$ the orthogonal complement, with respect to the Killing form, to the subspace of ${\mathfrak h}$ fixed by the natural action of $s$ on ${\mathfrak h}$. The restriction of the natural action of $s$ on ${\mathfrak h}^*$ to the subspace ${\mathfrak h}'^*$ has no fixed points. Therefore one can define the Cayley transform ${1+s \over 1-s }P_{{{\mathfrak h}'}^*}$ of the restriction of $s$ to ${{\mathfrak h}'}^*$, where $P_{{{\mathfrak h}'}^*}$ is the orthogonal projection operator onto ${{{\mathfrak h}'}^*}$ in ${\mathfrak h}^*$, with respect to the Killing form.
Recall also that in the classification theory of conjugacy classes in the Weyl group $W$ of the complex simple Lie algebra $\g$ the so-called primitive (or semi–Coxeter in another terminology) elements play a primary role. The primitive elements $w\in W$ are characterized by the property ${\rm det}(1-w)={\rm det}~a$, where $a$ is the Cartan matrix of $\g$. According to the results of [@C] the element $s$ of the Weyl group of the pair $(\g,{\mathfrak h})$ is primitive in the Weyl group $W'$ of a regular semisimple Lie subalgebra $\g'\subset \g$ of the form $$\g'={\mathfrak h}'+\sum_{\alpha\in \Delta'}\g_\alpha,$$ where $\Delta'$ is a root subsystem of the root system $\Delta$ of $\g$, $\g_\alpha$ is the root subspace of $\g$ corresponding to root $\alpha$.
Moreover, by Theorem C in [@C] $s$ can be represented as a product of two involutions, $$\label{inv}
s=s^1s^2,$$ where $s^1=s_{\gamma_1}\ldots s_{\gamma_n}$, $s^2=s_{\gamma_{n+1}}\ldots s_{\gamma_{l'}}$, the roots in each of the sets $\gamma_1, \ldots \gamma_n$ and ${\gamma_{n+1}}\ldots {\gamma_{l'}}$ are positive and mutually orthogonal, and the roots $\gamma_1, \ldots \gamma_{l'}$ form a linear basis of ${{\mathfrak h}'}^*$, in particular $l'$ is the rank of $\g'$. Recall that $\gamma_1, \ldots , \gamma_{l'}$ form a basis of a subspace ${{\mathfrak h}'}^*\subset {\mathfrak h}^*$ on which $s$ acts without fixed points. We shall study the matrix elements of the Cayley transform of the restriction of $s$ to ${{\mathfrak h}'}^*$ with respect the basis $\gamma_1, \ldots , \gamma_{l'}$.
[**([@S9], Lemma 6.2)**]{}\[tmatrel\] Let $P_{{{\mathfrak h}'}^*}$ be the orthogonal projection operator onto ${{{\mathfrak h}'}^*}$ in ${\mathfrak h}^*$, with respect to the Killing form. Then the matrix elements of the operator ${1+s \over 1-s }P_{{{\mathfrak h}'}^*}$ in the basis $\gamma_1, \ldots , \gamma_{l'}$ are of the form: $$\label{matrel}
\left( {1+s \over 1-s }P_{{{\mathfrak h}'}^*}\gamma_i , \gamma_j \right)=
\varepsilon_{ij}(\gamma_i,\gamma_j),$$ where $$\varepsilon_{ij} =\left\{ \begin{array}{ll}
-1 & i <j \\
0 & i=j \\
1 & i >j
\end{array}
\right . .$$
Let $\gamma_i^*$, $i=1,\ldots, l'$ be the basis of ${\mathfrak h}'^*$ dual to $\gamma_i$, $i=1,\ldots, l'$ with respect to the restriction of the bilinear form $(\cdot,\cdot)$ to ${\mathfrak h}'^*$. Since the numbers $(\gamma_i,\gamma_j)$ are integer each element $\gamma_i^*$ has the form $\gamma_i^*=\sum_{j=1}^{l'}m_{ij}\gamma_j$, where $m_{ij}\in \mathbb{Q}$. Therefore by the previous lemma the numbers $$\begin{aligned}
p_{ij}=\frac{1}{d_j}\left( {1+s \over 1-s }P_{{{\mathfrak h}'}^*}\alpha_i,\alpha_j\right)= \qquad \qquad \qquad \qquad \qquad \qquad \qquad \label{pij} \\ \qquad \qquad =\frac{1}{d_j}\sum_{k,l,p,q=1}^{l'}\gamma_k(\alpha_i)\gamma_l(\alpha_j)\left( {1+s \over 1-s }P_{{{\mathfrak h}'}^*}\gamma_p,\gamma_q\right)m_{kp}m_{lq},~i,j=1,\ldots,l \nonumber\end{aligned}$$ are rational. Let $d$ be a positive integer such that $p_{ij}\in \frac{1}{d}\mathbb{Z}$ for any $i<j$ (or $i>j$), $i,j=1,\ldots ,l$.
Now we suggest a new realization of the quantum group $U_\varepsilon({\frak g})$ associated to $s\in W$. Fix a positive integer number $n\in \mathbb{N},~n>0$. Assume that $\varepsilon^{2d_i}\neq 1$. Let $U_\varepsilon^{s}({\frak g})$ be the associative algebra over ${\Bbb C}$ generated by elements $e_i , f_i , L_i^{\pm 1},~i=1, \ldots l$ subject to the relations: $$\label{sqgr}
\begin{array}{l}
[L_i,L_j]=0,~~L_iL_i^{-1}=L_i^{-1}L_i=1,~~ L_ie_j L_i^{-1}=\varepsilon_i^{\delta_{ij}}e_j,~~ L_if_j L_i^{-1}=\varepsilon_i^{-\delta_{ij}}f_j,~~\varepsilon_i=\varepsilon^{d_i}, \\
\\
e_i f_j -\varepsilon^{ c_{ij}} f_j e_i = \delta _{i,j}{K_i -K_i^{-1} \over \varepsilon_i -\varepsilon_i^{-1}} , c_{ij}=nd\left( {1+s \over 1-s }P_{{{\mathfrak h}'}^*}\alpha_i , \alpha_j \right),\\
\\
\mbox{where }K_i=\prod_{j=1}^lL_j^{a_{ji}}, \\
\\
\sum_{r=0}^{1-a_{ij}}(-1)^r \varepsilon^{r c_{ij}}
\left[ \begin{array}{c} 1-a_{ij} \\ r \end{array} \right]_{\varepsilon_i}
(e_i )^{1-a_{ij}-r}e_j (e_i)^r =0 ,~ i \neq j , \\
\\
\sum_{r=0}^{1-a_{ij}}(-1)^r \varepsilon^{r c_{ij}}
\left[ \begin{array}{c} 1-a_{ij} \\ r \end{array} \right]_{\varepsilon_i}
(f_i )^{1-a_{ij}-r}f_j (f_i)^r =0 ,~ i \neq j .
\end{array}$$
\[newreal\] Assume that $\varepsilon^{2d_i}\neq 1$. For every solution $n_{ij}\in {\Bbb Z},~i,j=1,\ldots ,l$ of equations $$\label{eqpi}
d_jn_{ij}-d_in_{ji}=c_{ij}$$ there exists an algebra isomorphism $\psi_{\{ n\}} : U_\varepsilon^{s}({\frak g}) \rightarrow
U_\varepsilon({\frak g})$ defined by the formulas: $$\begin{array}{l}
\psi_{\{ n\}}(e_i)=X_i^+ \prod_{p=1}^lL_p^{n_{ip}},\\
\\
\psi_{\{ n\}}(f_i)=\prod_{p=1}^lL_p^{-n_{ip}}X_i^- , \\
\\
\psi_{\{ n\}}(L_i^{\pm 1})=L_i^{\pm 1} .
\end{array}$$
The proof of this theorem is similar to the proof of Theorem 4.1 in [@S8].
The general solution of equation (\[eqpi\]) is given by $$\label{eq3}
n_{ij}=\frac 1{2d_j} (c_{ij} + {s_{ij}}),$$ where $s_{ij}=s_{ji}$. If $p_{ij}\in \frac{1}{d}\mathbb{Z}$ for any $i<j$, we put $$s_{ij} =\left\{ \begin{array}{ll}
c_{ij} & i <j \\
0 & i=j \\
-c_{ij} & i >j
\end{array}
\right . .$$ Then $$n_{ij} =\left\{ \begin{array}{ll}
\frac 1{d_j}c_{ij} & i <j \\
0 & i=j \\
0 & i >j
\end{array}
\right . .$$ By the choice of $c_{ij}$ and $d$ we have $\frac 1{d_j}c_{ij}=\frac {nd}{d_j}\left( {1+s \over 1-s }P_{{{\mathfrak h}'}^*}\alpha_i, \alpha_j \right)=ndp_{ij}\in n\mathbb{Z}$ for $i<j$, $i,j=1,\ldots ,l$. Therefore $n_{ij}\in \mathbb{Z}$ for any $i,j=1,\ldots ,l$, and integer valued solutions to equations (\[eqpi\]) exist if $p_{ij}\in \frac{1}{d}\mathbb{Z}$ for any $i<j$. A similar consideration shows that if $p_{ij}\in \frac{1}{d}\mathbb{Z}$ for any $i>j$ integer valued solutions to equations (\[eqpi\]) exist as well.
We call the algebra $U_\varepsilon^{s}({\frak g})$ the realization of the quantum group $U_\varepsilon({\frak g})$ corresponding to the element $s\in W$.
\[auts\] Let $n_{ij}\in {\Bbb Z}$ be a solution of the homogeneous system that corresponds to (\[eqpi\]), $$\label{homeq}
d_in_{ji}-d_jn_{ij}=0.$$ Then the map defined by $$\label{sautom}
\begin{array}{l}
X_i^+ \mapsto X_i^+ \prod_{p=1}^lL_p^{n_{ip}},\\
\\
X_i^- \mapsto \prod_{p=1}^lL_p^{-n_{ip}}X_i^- , \\
\\
L_i^{\pm 1} \mapsto L_i^{\pm 1}
\end{array}$$ is an automorphism of $U_\varepsilon({\frak g})$. Therefore for given element $s\in W$ the isomorphism $\psi_{\{ n\}}$ is defined uniquely up to automorphisms (\[sautom\]) of $U_\varepsilon({\frak g})$.
Now we shall study the algebraic structure of $U_\varepsilon^{s}({\frak g})$. Denote by $U_\varepsilon^{s}({\frak n}_\pm) $ the subalgebra in $U_\varepsilon^{s}({\frak g})$ generated by $e_i ~(f_i) ,i=1, \ldots l$. Let $U_\varepsilon^{s}({\frak h})$ be the subalgebra in $U_\varepsilon^{s}({\frak g})$ generated by $L_i^{\pm 1},~i=1,\ldots ,l$.
We shall construct a Poincaré–Birkhoff-Witt basis for $U_\varepsilon^{s}({\frak g})$.
\[rootss\] (i) For any integer valued solution of equation (\[eqpi\]) and any normal ordering of the root system $\Delta_+$ the elements $e_{\beta}=\psi_{\{ n\}}^{-1}(X_{\beta}^+\prod_{i,j=1}^lL_j^{c_in_{ij}})$ and $f_{\beta}=\psi_{\{ n\}}^{-1}(\prod_{i,j=1}^lL_j^{-c_in_{ij}}X_{\beta}^-),~\beta=\sum_{i=1}^lc_i\alpha_i \in \Delta_+$ lie in the subalgebras $U_\varepsilon^{s}({\frak n}_+)$ and $U_\varepsilon^{s}({\frak n}_-)$, respectively. The elements $f_{\beta},~\beta \in \Delta_+$ satisfy the following commutation relations $$\label{erel}
f_{\alpha}f_{\beta} - \varepsilon^{(\alpha,\beta)+nd({1+s \over 1-s}P_{{{\mathfrak h}'}^*}\alpha,\beta)}f_{\beta}f_{\alpha}= \sum_{\alpha<\delta_1<\ldots<\delta_n<\beta}C'(k_1,\ldots,k_n)
f_{\delta_n}^{k_n}f_{\delta_{n-1}}^{k_{n-1}}\ldots f_{\delta_1}^{k_1},~\alpha <\beta,$$ where $C'(k_1,\ldots,k_n)\in \mathbb{C}$.
\(ii) Moreover, the elements $(e)^{\bf r}=(e_{\beta_1})^{r_1}\ldots (e_{\beta_D})^{r_D},~~(f)^{\bf t}=(f_{\beta_D})^{t_D}\ldots (f_{\beta_1})^{t_1}$ and $L^{\bf s}=L_1^{s_1}\ldots L_l^{s_l}$ for ${\bf r},~{\bf t}\in {\Bbb N}^l$, ${\bf s}\in {\Bbb Z}^l$ form bases of $U_\varepsilon^{s}({\frak n}_+),~U_\varepsilon^{s}({\frak n}_-)$ and $U_\varepsilon^{s}({\frak h})$, and the products $(f)^{\bf t}L^{\bf s}(e)^{\bf r}$ form a basis of $U_\varepsilon^{s}({\frak g})$. In particular, multiplication defines an isomorphism of vector spaces, $$U_\varepsilon^{s}({\frak n}_-)\otimes U_\varepsilon^{s}({\frak h})\otimes U_\varepsilon^{s}({\frak n}_+)\rightarrow U_\varepsilon^{s}({\frak g}).$$
\(iii) The subalgebra $Z_0\subset U_\varepsilon^{s}({\frak g})$ is the polynomial algebra with generators $e_{\alpha}^m$, $f_{\alpha}^m$, $\alpha \in \Delta_+$ and $l_i^{\pm 1}$, $i=1,\ldots,l$.
\(iv) $U_\varepsilon^s({\frak g})$ is a free $Z_0$–module with basis the set of monomials $(f)^{\bf r}L^{\bf s}(e)^{\bf t}$ for which $0\leq r_k,t_k,s_i<m$ for $i=1,\ldots ,l$, $k=1,\ldots ,D$.
The proof of this proposition is similar to the proof of Proposition 4.2 in [@S9].
Nilpotent subalgebras and quantum groups {#nilpuq}
========================================
In this section we define the subalgebras of $U_\varepsilon({\frak g})$ which resemble nilpotent subalgebras in ${\frak g}$ and possess nontrivial characters. We start by recalling the definition of certain normal orderings of root systems associated to Weyl group elements (see [@S9], Section 5 for more details). The definition of subalgebras of $U_\varepsilon(\g)$ having nontrivial characters will be given in terms of root vectors associated to such normal orderings.
Let $s$, as in the previous section, be an element of the Weyl group $W$ of the pair $(\g,{\mathfrak h})$ and ${\mathfrak h}_{\mathbb{R}}$ the real form of ${\mathfrak h}$, the real linear span of simple coroots in ${\mathfrak h}$. The set of roots $\Delta$ is a subset of the dual space ${\mathfrak h}_\mathbb{R}^*$.
The Weyl group element $s$ naturally acts on ${\mathfrak h}_{\mathbb{R}}$ as an orthogonal transformation with respect to the scalar product induced by the Killing form of $\g$. Using the spectral theory of orthogonal transformations we can decompose ${\mathfrak h}_{\mathbb{R}}$ into a direct orthogonal sum of $s$–invariant subspaces, $$\label{hdec}
{\mathfrak h}_\mathbb{R}=\bigoplus_{i=0}^{K} {\mathfrak h}_i,$$ where we assume that ${\mathfrak h}_0$ is the linear subspace of ${\mathfrak h}_{\mathbb{R}}$ fixed by the action of $s$, and each of the other subspaces ${\mathfrak h}_i\subset {\mathfrak h}_\mathbb{R}$, $i=1,\ldots, K$, is either two–dimensional or one–dimensional and the Weyl group element $s$ acts on it as rotation with angle $\theta_i$, $0<\theta_i\leq\pi$ or as reflection with respect to the origin (which also can be regarded as rotation with angle $\pi$). Note that since $s$ has finite order $\theta_i=\frac{2\pi}{m_i}$, $m_i\in \mathbb{N}$.
Since the number of roots in the root system $\Delta$ is finite one can always choose elements $h_i\in {\mathfrak h}_i$, $i=0,\ldots, K$, such that $h_i(\alpha)\neq 0$ for any root $\alpha \in \Delta$ which is not orthogonal to the $s$–invariant subspace ${\mathfrak h}_i$ with respect to the natural pairing between ${\mathfrak h}_{\mathbb{R}}$ and ${\mathfrak h}_{\mathbb{R}}^*$.
Now we consider certain $s$–invariant subsets of roots $\overline{\Delta}_i$, $i=0,\ldots, K$, defined as follows $$\label{di}
{\overline{\Delta}}_i=\{ \alpha\in \Delta: h_j(\alpha)=0, j>i,~h_i(\alpha)\neq 0 \},$$ where we formally assume that $h_{K+1}=0$. Note that for some indexes $i$ the subsets ${\overline{\Delta}}_i$ are empty, and that the definition of these subsets depends on the order of terms in direct sum (\[hdec\]).
Now consider the nonempty $s$–invariant subsets of roots $\overline{\Delta}_{i_k}$, $k=0,\ldots, M$. For convenience we assume that indexes $i_k$ are labeled in such a way that $i_j<i_k$ if and only if $j<k$. According to this definition $\overline{\Delta}_{i_0}=\{\alpha \in \Delta: s\alpha=\alpha\}$ is the set of roots fixed by the action of $s$. Observe also that the root system $\Delta$ is the disjoint union of the subsets $\overline{\Delta}_{i_k}$, $$\Delta=\bigcup_{k=0}^{M}\overline{\Delta}_{i_k}.$$
Now assume that $$\label{cond}
|h_{i_k}(\alpha)|>|\sum_{l\leq j<k}h_{i_j}(\alpha)|, ~{\rm for~any}~\alpha\in \overline{\Delta}_{i_k},~k=0,\ldots, M,~l<k.$$ Condition (\[cond\]) can be always fulfilled by suitable rescalings of the elements $h_{i_k}$.
Consider the element $$\label{hwb}
\bar{h}=\sum_{k=0}^{M}h_{i_k}\in {\mathfrak h}_\mathbb{R}.$$
From definition (\[di\]) of the sets $\overline{\Delta}_i$ we obtain that for $\alpha \in \overline{\Delta}_{i_k}$ $$\label{dech}
\bar{h}(\alpha)=\sum_{j\leq k}h_{i_j}(\alpha)=h_{i_k}(\alpha)+\sum_{j< k}h_{i_j}(\alpha)$$ Now condition (\[cond\]), the previous identity and the inequality $|x+y|\geq ||x|-|y||$ imply that for $\alpha \in \overline{\Delta}_{i_k}$ we have $$|\bar{h}(\alpha)|\geq ||h_{i_k}(\alpha)|-|\sum_{j< k}h_{i_j}(\alpha)||>0.$$ Since $\Delta$ is the disjoint union of the subsets $\overline{\Delta}_{i_k}$, $\Delta=\bigcup_{k=0}^{M}\overline{\Delta}_{i_k}$, the last inequality ensures that $\bar{h}$ belongs to a Weyl chamber of the root system $\Delta$, and one can define the subset of positive roots $\Delta_+$ and the set of simple positive roots $\Gamma$ with respect to that chamber. From condition (\[cond\]) and formula (\[dech\]) we also obtain that a root $\alpha \in \overline{\Delta}_{i_k}$ is positive if and only if $$\label{wc}
h_{i_k}(\alpha)>0.$$ We denote by $(\overline{\Delta}_{i_k})_+$ the set of positive roots contained in $\overline{\Delta}_{i_k}$, $(\overline{\Delta}_{i_k})_+=\Delta_+\bigcap \overline{\Delta}_{i_k}$.
We shall also need a parabolic subalgebra $\p$ of $\g$ associated to the semisimple element $\bar{h}_0=\sum_{k=1}^{M}h_{i_k}\in {\mathfrak h}_\mathbb{R}$. This subalgebra is defined with the help of the linear eigenspace decomposition of $\g$ with respect to the adjoint action of $\bar{h}_0$ on $\g$, $\g=\bigoplus_{m}(\g)_m$, $(\g)_m=\{ x\in \g \mid [\bar{h}_0,x]=mx\}$, $m \in \mathbb{R}$. By definition $\p=\bigoplus_{m\geq 0}(\g)_m$ is a parabolic subalgebra in $\g$, $\n=\bigoplus_{m>0}(\g)_m$ and ${\mathfrak l}=\{x\in \g \mid [\bar{h}_0,x]=0\}$ are the nilradical and the Levi factor of $\p$, respectively. Note that we have natural inclusions of Lie algebras $\p\supset{\mathfrak b}_+\supset\n$, where ${\mathfrak b}_+$ is the Borel subalgebra of $\g$ corresponding to the system $\Gamma$ of simple roots, and $\Delta_{i_0}$ is the root system of the reductive Lie algebra ${\mathfrak l}$. We also denote by $\opn$ the nilpotent subalgebra opposite to $\n$, $\opn=\bigoplus_{m<0}(\g)_m$.
For every element $w\in W$ one can introduce the set $\Delta_w=\{\alpha \in \Delta_+: w(\alpha)\in -\Delta_+\}$, and the number of the elements in the set $\Delta_w$ is equal to the length $l(w)$ of the element $w$ with respect to the system $\Gamma$ of simple roots in $\Delta_+$.
Now recall that $s$ can be represented as a product of two involutions, $$s=s^1s^2,$$ where $s^1=s_{\gamma_1}\ldots s_{\gamma_n}$, $s^2=s_{\gamma_{n+1}}\ldots s_{\gamma_{l'}}$, the roots in each of the sets $\gamma_1, \ldots \gamma_n$ and ${\gamma_{n+1}}\ldots {\gamma_{l'}}$ are positive and mutually orthogonal, and the roots $\gamma_1, \ldots \gamma_{l'}$ form a linear basis of ${\mathfrak h}'$, in particular $l'$ is the rank of a regular subalgebra $\g'\subset \g$ (see formula (\[inv\])).
\[pord\][**([@S9], Proposition 5.1)**]{} Let $s\in W$ be an element of the Weyl group $W$ of the pair $(\g,{\mathfrak h})$, $\Delta$ the root system of the pair $(\g,{\mathfrak h})$ and $\Delta_+$ the system of positive roots defined with the help of element (\[hwb\]), $\Delta_+=\{\alpha \in \Delta|\bar{h}(\alpha)>0\}$.
Then there is a normal ordering of the root system $\Delta_+$ of the following form $$\begin{aligned}
\beta_1^1,\ldots, \beta_t^1,\beta_{t+1}^1, \ldots,\beta_{t+\frac{p-n}{2}}^1, \gamma_1,\beta_{t+\frac{p-n}{2}+2}^1, \ldots , \beta_{t+\frac{p-n}{2}+n_1}^1, \gamma_2, \nonumber \\
\beta_{t+\frac{p-n}{2}+n_1+2}^1 \ldots , \beta_{t+\frac{p-n}{2}+n_2}^1, \gamma_3,\ldots, \gamma_n, \beta_{t+p+1}^1,\ldots, \beta_{l(s^1)}^1,\ldots, \label{NO} \\
\beta_1^2,\ldots, \beta_q^2, \gamma_{n+1},\beta_{q+2}^2, \ldots , \beta_{q+m_1}^2, \gamma_{n+2}, \beta_{q+m_1+2}^2,\ldots , \beta_{q+m_2}^2, \gamma_{n+3},\ldots, \nonumber \\
\gamma_{l'},\beta_{q+m_{l(s^2)}+1}^2, \ldots,\beta_{2q+2m_{l(s^2)}-(l'-n)}^2, \beta_{2q+2m_{l(s^2)}-(l'-n)+1}^2,\ldots, \beta_{l(s^2)}^2, \nonumber \\
\beta_1^0, \ldots, \beta_{D_0}^0, \nonumber\end{aligned}$$ where $$\begin{aligned}
\{\beta_1^1,\ldots, \beta_t^1,\beta_{t+1}^1, \ldots,\beta_{t+\frac{p-n}{2}}^1, \gamma_1,\beta_{t+\frac{p-n}{2}+2}^1, \ldots , \beta_{t+\frac{p-n}{2}+n_1}^1, \gamma_2, \nonumber \\
\beta_{t+\frac{p-n}{2}+n_1+2}^1 \ldots , \beta_{t+\frac{p-n}{2}+n_2}^1, \gamma_3,\ldots, \gamma_n, \beta_{t+p+1}^1,\ldots, \beta_{l(s^1)}^1\}=\Delta_{s^1},\end{aligned}$$ $$\begin{aligned}
\{\beta_{t+1}^1, \ldots,\beta_{t+\frac{p-n}{2}}^1, \gamma_1,\beta_{t+\frac{p-n}{2}+2}^1, \ldots , \beta_{t+\frac{p-n}{2}+n_1}^1, \gamma_2, \nonumber \\
\beta_{t+\frac{p-n}{2}+n_1+2}^1 \ldots , \beta_{t+\frac{p-n}{2}+n_2}^1, \gamma_3,\ldots, \gamma_n\}=\{\alpha\in \Delta_+|s^1(\alpha)=-\alpha\},\end{aligned}$$ $$\begin{aligned}
\{\beta_1^2,\ldots, \beta_q^2, \gamma_{n+1},\beta_{q+2}^2, \ldots , \beta_{q+m_1}^2, \gamma_{n+2}, \beta_{q+m_1+2}^2,\ldots , \beta_{q+m_2}^2, \gamma_{n+3},\ldots, \nonumber \\
\gamma_{l'},\beta_{q+m_{l(s^2)}+1}^2, \ldots,\beta_{2q+2m_{l(s^2)}-(l'-n)}^2, \beta_{2q+2m_{l(s^2)}-(l'-n)+1}^2,\ldots, \beta_{l(s^2)}^2\}=\Delta_{s^2},\end{aligned}$$ $$\begin{aligned}
\{\gamma_{n+1},\beta_{q+2}^2, \ldots , \beta_{q+m_1}^2, \gamma_{n+2}, \beta_{q+m_1+2}^2,\ldots , \beta_{q+m_2}^2, \gamma_{n+3},\ldots, \nonumber \\
\gamma_{l'},\beta_{q+m_{l(s^2)}+1}^2, \ldots,\beta_{2q+2m_{l(s^2)}-(l'-n)}^2\}=\{\alpha\in \Delta_+|s^2(\alpha)=-\alpha\},\end{aligned}$$ $$\{\beta_1^0, \ldots, \beta_{D_0}^0\}=\Delta_0=\{\alpha\in \Delta_+|s(\alpha)=\alpha\},$$ and $s^1,s^2$ are the involutions entering decomposition (\[inv\]), $s^1=s_{\gamma_1}\ldots s_{\gamma_n}$, $s^2=s_{\gamma_{n+1}}\ldots s_{\gamma_{l'}}$, the roots in each of the sets $\gamma_1, \ldots, \gamma_n$ and ${\gamma_{n+1}},\ldots, {\gamma_{l'}}$ are positive and mutually orthogonal.
The length of the ordered segment $\Delta_{\m_+}\subset \Delta$ in normal ordering (\[NO\]), $$\begin{aligned}
\Delta_{\m_+}=\gamma_1,\beta_{t+\frac{p-n}{2}+2}^1, \ldots , \beta_{t+\frac{p-n}{2}+n_1}^1, \gamma_2, \beta_{t+\frac{p-n}{2}+n_1+2}^1 \ldots , \beta_{t+\frac{p-n}{2}+n_2}^1, \nonumber \\
\gamma_3,\ldots, \gamma_n, \beta_{t+p+1}^1,\ldots, \beta_{l(s^1)}^1,\ldots, \beta_1^2,\ldots, \beta_q^2, \label{dn} \\
\gamma_{n+1},\beta_{q+2}^2, \ldots , \beta_{q+m_1}^2, \gamma_{n+2}, \beta_{q+m_1+2}^2,\ldots , \beta_{q+m_2}^2, \gamma_{n+3},\ldots, \gamma_{l'}, \nonumber\end{aligned}$$ is equal to $$\label{dimm}
D-(\frac{l(s)-l'}{2}+D_0),$$ where $D$ is the number of roots in $\Delta_+$, $l(s)$ is the length of $s$ and $D_0$ is the number of positive roots fixed by the action of $s$.
Moreover, for any two roots $\alpha, \beta\in \Delta_{\m_+}$ such that $\alpha<\beta$ the sum $\alpha+\beta$ can not be represented as a linear combination $\sum_{k=1}^qc_k\gamma_{i_k}$, where $c_k\in \mathbb{N}$ and $\alpha<\gamma_{i_1}<\ldots <\gamma_{i_q}<\beta$.
Now we can define the subalgebras of $U_\varepsilon({\frak g})$ which resemble nilpotent subalgebras in ${\frak g}$ and possess nontrivial characters.
\[qnil\] Let $s\in W$ be an element of the Weyl group $W$ of the pair $(\g,{\mathfrak h})$, $\Delta$ the root system of the pair $(\g,{\mathfrak h})$. Fix a decomposition (\[inv\]) of $s$ and let $\Delta_+$ be the system of positive roots associated to $s$. Assume that $\varepsilon^{2d_i}\neq 1$ and that $\varepsilon^{nd-1}=1$, where $d$ and $n$ are defined in Section \[wqreal\]. Let $U_\varepsilon^{s}({\frak g})$ be the realization of the quantum group $U_\varepsilon({\frak g})$ associated to $s$. Let $f_\beta\in U_\varepsilon^{s}({\n_-})$, $\beta \in \Delta_+$ be the root vectors associated to the corresponding normal ordering (\[NO\]) of $\Delta_+$.
Then elements $f_\beta\in U_\varepsilon^{s}({\n_-})$, $\beta \in \Delta_{\m_+}$, where $\Delta_{\m_+}\subset \Delta$ is ordered segment (\[dn\]), generate a subalgebra $U_\varepsilon^{s}({\frak m}_-)\subset U_\varepsilon^{s}({\frak g})$ The elements $f^{\bf r}=f_{\beta_D}^{r_D}\ldots f_{\beta_1}^{r_1}$, $r_i\in \mathbb{N}$, $i=1,\ldots D$ and $r_i$ can be strictly positive only if $\beta_i\in \Delta_{\m_+}$, form a linear basis of $U_\varepsilon^{s}({\frak m}_-)$.
Moreover the map $\chi^s:U_\varepsilon^{s}({\frak m}_-)\rightarrow \mathbb{C}$ defined on generators by $$\label{char}
\chi^s(f_\beta)=\left\{ \begin{array}{ll} 0 & \beta \not \in \{\gamma_1, \ldots, \gamma_{l'}\} \\ c_i & \beta=\gamma_i, c_i\in \mathbb{C}
\end{array}
\right .$$ is a character of $U_\varepsilon^{s}({\frak m}_-)$.
The first statement of the theorem follows straightforwardly from commutation relations (\[erel\]) and Proposition \[rootss\].
In order to prove that the map $\chi^s:U_\varepsilon^{s}({\frak m}_-)\rightarrow \mathbb{C}$ defined by (\[char\]) is a character of $U_\varepsilon^{s}({\frak m}_-)$ we show that all relations (\[erel\]) for $f_\alpha,~f_\beta$ with $\alpha,\beta \in \Delta_{\m_+}$, which are obviously defining relations in the subalgebra $U_\varepsilon^{s}({\frak m}_-)$, belong to the kernel of $\chi^s$. By definition the only generators of $U_\varepsilon^{s}({\frak m}_-)$ on which $\chi^s$ does not vanish are $f_{\gamma_i}$, $i=1,\ldots,l'$. By the last statement in Proposition \[pord\] for any two roots $\alpha, \beta\in \Delta_{\m_+}$ such that $\alpha<\beta$ the sum $\alpha+\beta$ can not be represented as a linear combination $\sum_{k=1}^qc_k\gamma_{i_k}$, where $c_k\in \mathbb{N}$ and $\alpha<\gamma_{i_1}<\ldots <\gamma_{i_k}<\beta$. Hence for any two roots $\alpha, \beta\in \Delta_{\m_+}$ such that $\alpha<\beta$ the value of the map $\chi^s$ on the l.h.s. of the corresponding commutation relation (\[erel\]) is equal to zero.
Therefore it suffices to prove that $$\chi^s(f_{\gamma_i}f_{\gamma_j} - \varepsilon^{(\gamma_i,\gamma_j)+nd({1+s \over 1-s}P_{{{\mathfrak h}'}^*}\gamma_i,\gamma_j)}f_{\gamma_j}f_{\gamma_j})=c_ic_j(1-\varepsilon^{(\gamma_i,\gamma_j)+nd({1+s \over 1-s}P_{{{\mathfrak h}'}^*}\gamma_i,\gamma_j)})=0,~i<j.$$
Since $\varepsilon^{nd-1}=1$ and $({1+s \over 1-s}P_{{{\mathfrak h}'}^*}\gamma_i,\gamma_j)$ are integer numbers for any $i,j=1,\ldots ,l'$, the last identity holds provided $(\gamma_i,\gamma_j)+({1+s \over 1-s}P_{{{\mathfrak h}'}}^*\gamma_i, \gamma_j)=0$ for $i<j$. As we saw in the Lemma \[tmatrel\] this is indeed the case. This completes the proof.
Some facts about the geometry of the conjugation action {#crossect}
=======================================================
In this section we collect some results on the geometry of the conjugation action that will be used later. Let $r\in {\rm End}~{\frak g}$ be a linear operator on $\g$ satisfying the classical modified Yang–Baxter equation, $$\left[ rX,rY\right] -r\left( \left[ rX,Y\right] +\left[ X,rY\right] \right)
=-\left[ X,Y\right] ,\;X,Y\in {\frak g}.$$ One can check that if we define operators $r_\pm \in {\rm End}\ {\frak g}$ by $$r_{\pm }=\frac 12\left( r\pm id\right)$$ then the linear subspace $\g^*\subset {\frak g\oplus {\g}}$, $\g^*=\{(X_{+},~X_{-}),~~~X_{\pm
}~=~r_{\pm }X, X\in \g\}$ is a Lie subalgebra in ${\frak g\oplus {\g}}$ (see, for instance, [@dual]). We denote by $G^*$ the corresponding subgroup in $G\times G$.
Let $r^0, r^s\in {\rm End}~{\frak g}$ be the linear operators on $\g$ defined by $$r^0=P_+-P_-,~~r^{s}=P_--P_++{1+s \over 1-s}P_{{{\mathfrak h}'}},$$ where $P_+,P_-$ and $P_{{{\mathfrak h}'}}$ are the projection operators onto ${\frak n}_+,{\frak
n}_-$ and ${\frak h}'$ in the direct sum $$\label{ddd}
{\frak g}={\frak n}_+ +{\frak h}'+{{\mathfrak h}'}^\perp + {\frak n}_-,$$ and ${{\mathfrak h}'}^\perp$ is the orthogonal complement to ${\mathfrak h}'$ in ${\mathfrak h}$ with respect to the Killing form. One can check that both $r^0$ and $r^s$ satisfy the classical modified Yang–Baxter equation. Therefore one can define the corresponding subgroups $G_0^*,G^*_s\subset G\times G$
Note also that $$r^{s}_+=P_+ + {1 \over 1-s}P_{{{\mathfrak h}'}}+\frac{1}{2}P_{{{\mathfrak h}'}^\perp},~~r^{s}_-=-P_- + {s \over 1-s}P_{{{\mathfrak h}'}}-\frac{1}{2}P_{{{\mathfrak h}'}^\perp},$$ where $P_{{{\mathfrak h}'}^\perp}$ is the projection operator onto ${{\mathfrak h}'}^\perp$ in direct sum (\[ddd\]). Hence every element $(L_+,L_-)\in G^*_s$ may be uniquely written as $$\label{fact}
(L_+,L_-)=(h_+,h_-)(n_+,n_-),$$ where $n_\pm \in N_\pm$, $h_+=exp(({1 \over 1-s}P_{{{\mathfrak h}'}}+\frac{1}{2}P_{{{\mathfrak h}'}^\perp})x),~h_-=exp(({s \over 1-s}P_{{{\mathfrak h}'}}-\frac{1}{2}P_{{{\mathfrak h}'}^\perp})x),~x\in
{\frak h}$. In particular, $G^*_s$ is a solvable algebraic subgroup in $G\times G$.
Similarly we have $$r^{0}_+=P_+ +\frac{1}{2}P_{{{\mathfrak h}}},~~r^{0}_-=-P_- -\frac{1}{2}P_{{{\mathfrak h}}},~~P_{{{\mathfrak h}}}=P_{{{\mathfrak h}'}}+P_{{{\mathfrak h}'}^\perp},$$ and hence every element $(L_+',L_-')\in G^*_0$ may be uniquely written as $$(L_+',L_-')=(h_+',h_-')(n_+',n_-'),~n_\pm \in N_\pm,~h_+'=exp(\frac{1}{2}x'),~h_-'=exp(-\frac{1}{2}x'),~x'\in
{\frak h}.$$ In particular, $G^*_0$ is also a solvable algebraic subgroup in $G\times G$.
We shall need an isomorphism of varieties $\phi: G^*_0\rightarrow G^*_s$ which is uniquely defined by the requirement that if $\phi(L_+',L_-')=(L_+,L_-)$ then $$\label{piso}
L=tL't^{-1},~L'=L_-'(L_+')^{-1}, L=L_-L_+^{-1},~t=e^{Ax'},$$ where $A\in {\rm End}~{{\mathfrak h}}$ is the endomorphism of ${\mathfrak h}$ defined by $$\label{Kdef}
AH_i=\frac{1}{2nd}\sum_{j=1}^l{n_{ij} \over d_i}Y_j,~i=1,\ldots ,l,$$ $n_{ij}$ are solutions to equations (\[eqpi\]), and $$Y_i=\sum_{j=1}^l d_i(a^{-1})_{ij}H_j,$$ are the weight–type generators of ${\mathfrak h}$ (see [@S9] for more detail).
In fact (\[piso\]) is an isomorphism of Poisson manifolds if $G^*_0$ is regarded as the dual Poisson–Lie group to the Poisson–Lie group $G$ equipped with the standard Sklyanin bracket, and $G^*_s$ is regarded as the dual Poisson–Lie group to the Poisson–Lie group $G$ equipped with the Sklyanin bracket associated to the r–matrix $r^{s}$ (see [@S9], Section 10). We shall not need this fact in this paper.
Formula (\[fact\]) and decomposition of $N_+$ into products of one–dimensional subgroups corresponding to roots also imply that every element $L_-$ may be represented in the form $$\label{lm}
\begin{array}{l}
L_- = exp\left[ \sum_{i=1}^lb_i({s \over 1-s}P_{{{\mathfrak h}'}}-\frac{1}{2}P_{{{\mathfrak h}'}^\perp})H_i\right]\times \\
\prod_{\beta}
exp[b_{-\beta}X_{-\beta}],~b_i,b_{-\beta}\in {\Bbb C},
\end{array}$$ where the product over roots is taken in the same order as in (\[NO\]), and the root vectors $X_{-\beta}$ are constructed as in (\[rootvectg\]) using the normal ordering of $\Delta_+$ opposite to (\[NO\]).
Let $M_\pm$ be the subgroups in $N_\pm$ corresponding to the Lie subalgebras $\m_\pm \subset \n_\pm$ which are generated by root vectors $X_{\pm \beta}$, $\beta\in \Delta_{{\mathfrak{m}}_+}$. Now define a map $\mu_{M_+}:G^*_s \rightarrow M_-$ by $$\label{mun}
\mu_{M_+}(L_+,L_-)=m_-,$$ where for $L_-$ given by (\[lm\]) $m_-$ is defined as follows $$\label{mm}
m_-=\prod_{\beta\in \Delta_{\m_+}}
exp[b_{-\beta}X_{-\beta}],$$ and the product over roots is taken in the same order as in the normally ordered segment $\Delta_{\m_+}$.
By definition $\mu_{M_+}$ is a morphism of algebraic varieties.
Let $u$ be the element defined by $$\label{defu}
u=\prod_{i=1}^{l'}exp[t_{i} X_{-\gamma_i}]~ \in M_-,t_{i}\in \mathbb{C},$$ where the product over roots is taken in the same order as in the normally ordered segment $\Delta_{\m_+}$.
Let $X_\alpha(t)=\exp(tX_\alpha)\in G$, $t\in \mathbb{C}$ be the one–parametric subgroup in the algebraic group $G$ corresponding to root $\alpha\in \Delta$. Recall that for any $\alpha \in \Delta_+$ and any $t\neq 0$ the element $s_\alpha(t)=X_\alpha(-t)X_{-\alpha}(t)X_\alpha(-t)\in G$ is a representative for the reflection $s_\alpha$ corresponding to the root $\alpha$. Denote by $s\in G$ the following representative of the Weyl group element $s\in W$, $$\label{defrep}
s=s_{\gamma_1}(t_1)\ldots s_{\gamma_{l'}}(t_{l'}),$$ where the numbers $t_{i}$ are defined in (\[defu\]), and we assume that $t_i\neq 0$ for any $i$.
Let $Z$ be the subgroup of $G$ corresponding to the Lie subalgebra $\z$ generated by the semisimple part $\m$ of the Levi subalgebra ${\mathfrak l}$ and by the centralizer of $s$ in ${\mathfrak h}$. Denote by $N$ the subgroup of $G$ corresponding to the Lie subalgebra $\n$ and by $\overline{N}$ the opposite unipotent subgroup in $G$ with the Lie algebra $\overline{\n}=\bigoplus_{m<0}(\g)_m$. By definition we have that $N_+\subset ZN$.
Now assume that the roots $\gamma_1, \ldots , \gamma_n$ are simple or the set $\gamma_1, \ldots , \gamma_n$ is empty. In that case the complementary subset to $\Delta_{\m_+}$ in $\Delta_+$ is a segment $\Delta_{\m_+}^0$ with respect to normal ordering (\[NO\]).
Let $q:G^*_s\rightarrow G$ be the map defined by, $$q(L_+,L_-)=L_-L_+^{-1}.$$
Consider the space $\mu_{M_+}^{-1}(u)$ which can be explicitly described as follows $$\label{mun1}
\mu_{M_+}^{-1}(u)=\{(h_+n_+,s(h_+)ux) | n_+ \in N_+ , h_+ \in H, x\in M_-^0 \},$$ where $M_-^0$ is the subgroup of $G$ generated by the one–parametric subgroups corresponding to the roots from the segment $-\Delta_{\m_+}^0$ . Therefore $$\label{dva}
q(\mu_{M_+}^{-1}(u))=
\{ s(h_+)uxn_+^{-1}h_+^{-1}| n_+ \in N_+ , h_+ \in H, x\in M_-^0 \}.$$
\[constrt\] [**([@S9], Proposition 12.1)**]{} Let $q:G^*_s\rightarrow G$ be the map defined by, $$q(L_+,L_-)=L_-L_+^{-1}.$$ Assume that the roots $\gamma_1, \ldots , \gamma_n$ are simple or the set $\gamma_1, \ldots , \gamma_n$ is empty. Suppose also that the numbers $t_{i}$ defined in (\[defu\]) are not equal to zero for all $i$. Then $q(\mu_{M_+}^{-1}(u))$ is a subvariety in $NsZN$ which consists of elements of the form $s(h_+)x''skh_+^{-1}$ with arbitrary $k\in ZN$, $h_+\in H$ and $x''$ given by $$\begin{aligned}
\label{x''}
x''=X_{\gamma_1}(u_1)X_{s_{\gamma_1}\gamma_2}(u_2)\ldots X_{s_{\gamma_1}\ldots s_{\gamma_{l'-1}}\gamma_{l'}}(u_{l'})sx's^{-1}\in N,~x'\in \overline{N},~sx's^{-1}\in N,\end{aligned}$$ where $u_i$ are arbitrary nonzero complex numbers.
The closure $\overline{q(\mu_{M_+}^{-1}(u))}$ of $q(\mu_{M_+}^{-1}(u))$ is obtained by adding elements of the same form with some $u_i$ equal to $0$. The closure $\overline{q(\mu_{M_+}^{-1}(u))}$ is also contained in $NsZN$.
Elements of the form $x''sk$, where $k\in ZN$ and $$\begin{aligned}
\label{x''1}
x''=X_{\gamma_1}(t_1)X_{s_{\gamma_1}\gamma_2}(t_2)\ldots X_{s_{\gamma_1}\ldots s_{\gamma_{l'-1}}\gamma_{l'}}(t_{l'})sx's^{-1}\in N,~x'\in \overline{N},~sx's^{-1}\in N\end{aligned}$$ can be represented as follows $x''sk=uxn_+^{-1}\in q(\mu_{M_+}^{-1}(u))$, $n_+ \in N_+,~ x\in M_-^0$.
\[crosssect\] [**([@S6], Propositions 2.1 and 2.2)**]{} Let $N_s=\{ v \in N|svs^{-1}\in \overline{N} \}$. Then the conjugation map $$\label{cross}
N\times sZN_s\rightarrow NsZN$$ is an isomorphism of varieties. Moreover, the variety $\Sigma_s=sZN_s$ is a transversal slice to the set of conjugacy classes in $G$.
Now we prove a short technical lemma which will play the key role in the proof of the main statement of this paper.
\[Hcomp\] Assume that the roots $\gamma_1, \ldots , \gamma_n$ are simple or the set $\gamma_1, \ldots , \gamma_n$ is empty. Suppose also that the numbers $t_{i}$ defined in (\[defu\]) are not equal to zero for all $i$. For any $\eta\in NsZN$ and $h\in H$ one can find $n\in N$ such that $n\eta n^{-1}\in {q(\mu_{M_+}^{-1}(u))}$, and $n\eta n^{-1}=s(h)uxn_+^{-1}h^{-1}$ for some $n_+ \in N_+,~ x\in M_-^0$.
Let $u_i$, $i=1,\ldots,l'$ be nonzero complex numbers such that $$s(h)X_{\gamma_1}(t_1)X_{s_{\gamma_1}\gamma_2}(t_2)\ldots X_{s_{\gamma_1}\ldots s_{\gamma_{l'-1}}\gamma_{l'}}(t_{l'})s(h^{-1})=X_{\gamma_1}(u_1)X_{s_{\gamma_1}\gamma_2}(u_2)\ldots X_{s_{\gamma_1}\ldots s_{\gamma_{l'-1}}\gamma_{l'}}(u_{l'}).$$ Obviously for any $\eta \in NsZN$ one can find $n\in N$ such that $$\begin{aligned}
n\eta n^{-1}=X_{\gamma_1}(u_1)X_{s_{\gamma_1}\gamma_2}(u_2)\ldots X_{s_{\gamma_1}\ldots s_{\gamma_{l'-1}}\gamma_{l'}}(u_{l'})sy= \qquad \qquad \qquad \\
=s(h)X_{\gamma_1}(t_1)X_{s_{\gamma_1}\gamma_2}(t_2)\ldots X_{s_{\gamma_1}\ldots s_{\gamma_{l'-1}}\gamma_{l'}}(t_{l'})s\widetilde{y}h^{-1},~~y,\widetilde{y}\in ZN.\end{aligned}$$
Now by Proposition \[constrt\] $n\eta n^{-1}$ can be represented in the form $n\eta n^{-1}=s(h)uxn_+^{-1}h^{-1}$ for some $n_+ \in N_+,~ x\in M_-^0$. This completes the proof.
Consider the restriction of the action of $G$ on itself by conjugations to the subgroup $M_+$. Denote by $\pi_q:G\rightarrow G/M_+$ the canonical projection onto the quotient with respect to this action.
\[var\][**([@S9], Theorem 12.3)**]{} Assume that the roots $\gamma_1, \ldots , \gamma_n$ are simple or the set $\gamma_1, \ldots , \gamma_n$ is empty. Suppose also that the numbers $t_{i}$ defined in (\[defu\]) are not equal to zero for all $i$. Then $sZN_s\cap G^0\subset {q(\mu_{M_+}^{-1}(u))}$, $sZN_s\subset \overline{q(\mu_{M_+}^{-1}(u))}$, the (locally defined) conjugation action of $M_+$ on $q(\mu_{M_+}^{-1}(u))$ is (locally) free, the quotient $\pi_q(\overline{q(\mu_{M_+}^{-1}(u))})$ is a smooth variety and the algebra of regular functions on $\pi_q(\overline{q(\mu_{M_+}^{-1}(u))})$ is isomorphic to the algebra of regular functions on the slice $sZN_s$.
Statements similar to Propositions \[constrt\], \[var\] and Lemma \[Hcomp\] can be proved in case when the roots $\gamma_{n+1}, \ldots , \gamma_{l'}$ are simple or the set $\gamma_{n+1}, \ldots , \gamma_{l'}$ is empty. In that case instead of the map $q:G^*\rightarrow G$ one should use another map $q':G^*\rightarrow G$, $q'(L_+,L_-)=L_-^{-1}L_+$ which has the same properties as $q$ (see [@dual], Section 2).
Whittaker vectors {#WHITT}
=================
In this section we introduce the notion of Whittaker vectors for modules over quantum groups at roots of unity and prove an analogue of Engel theorem for them. We start by studying some properties of quantum groups at roots of unity.
From now on we fix an element $s\in W$ and a representation (\[inv\]) for $s$. We also fix positive integers $n$ and $d$ such that $p_{ij}\in \frac{1}{d}\mathbb{Z}$ for any $i<j$ (or $i>j$), $i,j=1,\ldots ,l$, where the numbers $p_{ij}$ are defined by formula (\[pij\]). We shall always assume that $\varepsilon^{2d_i}\neq 1$, $i=1,\ldots ,l$ and that $\varepsilon^{nd-1}=1$. We fix an integer valued solution $n_{ij}$ to equations (\[eqpi\]) and identify the algebra $U_\varepsilon^{s^{-1}}({\frak g})$ associated to the Weyl group element $s^{-1}$ with $U_\varepsilon({\frak g})$ using Theorem \[newreal\] and the solution $-n_{ij}-\delta_{ij}$ to equations (\[eqpi\]) (a motivation for adding the extra term $-\delta_{ij}$ to $n_{ij}$ will be given later; as it was explained in Remark \[auts\] this term is a solution to homogeneous equations (\[homeq\]) and corresponds to an automorphism of $U_\varepsilon({\frak g})$). Using this identification $U_\varepsilon^{s^{-1}}({\frak m}_-)$ can be regarded as a subalgebra in $U_\varepsilon({\frak g})$. Therefore for every character $\eta: Z_0 \rightarrow \mathbb{C}$ one can define the corresponding subalgebra in $U_\eta({\frak g})$. We denote this subalgebra by $U_\eta({\frak m}_-)$.
First we study some properties of the finite dimensional algebras $U_\eta({\frak g})$ and $U_\eta({\frak m}_-)$. We remind that a finite dimensional algebra is called Frobenius if its left regular representation is isomorphic to the dual of the right regular representation. Thus any free module over a Frobenius algebra is also injective and projective.
\[frob\] For any character $\eta: Z_0 \rightarrow \mathbb{C}$ the algebra $U_\eta({\frak g})$ and its subalgebra $U_\eta({\frak m}_-)$ are Frobenius algebras.
The proof of this proposition is parallel to the proof of a similar statement for Lie algebras over fields of prime characteristic (see Proposition 1.2 in [@FP]). By Theorem 61.3 in [@CR] it suffices to show that there is a non–degenerate bilinear form $B_\eta:U_\eta({\frak g}) \times U_\eta({\frak g})\rightarrow \mathbb{C}$ which restricts to a non–degenerate bilinear form $B_\eta:U_\eta({\frak m}_-) \times U_\eta({\frak m}_-)\rightarrow \mathbb{C}$ and which is associative in the sense that $$B_\eta(ab,c)=B_\eta(a,bc),~~a,b,c\in U_\eta({\frak g}).$$
Consider the free $Z_0$–basis of $U_\varepsilon({\frak g})$ introduced in part (iv) of Proposition \[rootss\]. This basis consists of the monomials $x_I=(f)^{\bf r}L^{\bf s}(e)^{\bf t}$, $I=(r_1,\ldots,r_D,s_1,\ldots,s_l,t_1,\ldots,t_D)$ for which $0\leq r_k,t_k,s_i<m$ for $i=1,\ldots ,l$, $k=1,\ldots ,D$. Set $c(x_I)=\sum_{k=1}^D r_k+\sum_{k=1}^D t_k+\sum_{i=1}^l s_i$, $I'=(m-1-r_1,\ldots,m-1-r_D,m-1-s_1,\ldots,m-1-s_l,m-1-t_1,\ldots,m-1-t_D)$ and $P=(m-1,\ldots,m-1)$.
Let $\Phi: U_\varepsilon({\frak g})\rightarrow Z_0$ be the $Z_0$–linear map defined on the basis $x_I$ of monomials by $$\Phi(x_I)=\left\{ \begin{array}{ll}
1 & I = P \\
0 & {\rm otherwise}
\end{array}
\right . .$$
Using commutation relations (\[sqgr\]), (\[erel\]) and similar relations for generators $f_\alpha$ one can check that $\Phi(x_I x_J)=0$ if $c(x_I)+c(x_J)\leq (m-1)(l+2N)$ and $J\neq I'$, while $\Phi(x_I x_{I'})=c_I\neq 0$. Now by the argument given in the proof of Proposition 1.2 in [@FP] the discriminant of the associative $Z_0$–bilinear pairing $B:U_\varepsilon({\frak g})\otimes_{Z_0} U_\varepsilon({\frak g}) \rightarrow Z_0$, $B(x,y)=\Phi(xy)$ is a unit and the associative bilinear form $B_\eta:U_\eta({\frak g}) \times U_\eta({\frak g})\rightarrow \mathbb{C}$, $B_\eta(x,y)=\eta(B(x,y))$ is non–degenerate. By construction the restriction of $B_\eta$, $B_\eta:U_\eta({\frak m}_-) \times U_\eta({\frak m}_-)\rightarrow \mathbb{C}$ is non–degenerate and associative as well. This completes the proof.
In order to define Whittaker vectors for quantum groups at roots of unity we shall need some axillary notions that we are going to discuss now.
Consider the isomorphism of varieties $$\phi\circ \widetilde{\pi}: {\rm Spec}(Z_0)\rightarrow G^*_s$$ constructed with the help of the normal ordering of the positive root system $\Delta_+$ opposite to (\[NO\]) and with the help of the solution $n_{ij}$ of equations (\[eqpi\]). We shall need some property of elements $\eta\in {\rm Spec}(Z_0)$ such that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$. To describe this property we observe that a straightforward calculation using the explicit form of the isomorphism $\psi_{\{-n_{ij}-\delta_{ij}\}}$ shows that the $n_-$–component ${\bf Y}^-$ of the map $\phi\circ \widetilde{\pi}$ in the image $G^*_s$ with respect to factorization (\[fact\]) has the form $${\bf Y}^-: {\rm Spec}(Z_0) \rightarrow N_-,$$ $$\label{ym}
{\bf Y}^-=\exp(y_{\beta_D}^-X_{-\beta_D})\exp(y_{\beta_{D-1}}^-X_{-\beta_{D-1}})\ldots \exp(y_{\beta_1}^-X_{-\beta_1}),$$ where $y_{\alpha}^-=k_\alpha f_{\alpha}^m$, for some $k_\alpha\in \mathbb{C}$, $k_\alpha\neq 0$, and $y_{\alpha}^-$ should be regarded as complex-valued functions on ${\rm Spec}(Z_0)$. Note that the elements $f_\alpha \in U_\varepsilon^{(s^{-1})}({\frak m}_-)$ are constructed using the normal ordering opposite to (\[NO\]), so the order of terms corresponding to roots in the product (\[ym\]) coincides with the order of roots in normal ordering (\[NO\]).
The following property of elements $\eta\in {\rm Spec}(Z_0)$ , $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$ is a direct consequence of formula (\[ym\]) and of the definition of the variety $\mu_{M_+}^{-1}(u)$ in terms of the map $\mu_{M_+}$ (see formulas (\[mun\]), (\[mm\]) and (\[defu\])).
\[charn\] Let $\eta$ be an element of ${\rm Spec}(Z_0)$. Assume that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$. Then for $\beta \in \Delta_{\m_+}$ we have $$\label{etachar}
\eta(f_{\beta}^m)=\left\{ \begin{array}{ll}
d_i=\frac{t_i}{k_{\gamma_i}} & \beta=\gamma_i,~i=1,\ldots,l' \\
0 & \beta \not\in \{\gamma_1,\ldots,\gamma_{l'}\}
\end{array}
\right . .$$
Finally consider the subalgebra $U_{\eta}({\frak h})\subset U_{\eta}({\frak g})$ generated by $L_1,\ldots,L_l$. Since $\eta(L_i^m)\neq 0$, $i=1,\ldots ,l$ the elements $L_1,\ldots,L_l$ act on any finite–dimensional $U_{\eta}({\frak g})$–module $V$ as mutually commuting semisimple automorphisms. Therefore if by a weight we mean an $l$–tuple ${ \omega}=(\omega_1,\ldots ,\omega_l)\in (\mathbb{C}^*)^l$, the space $V$ has a weight decomposition with respect to the action of $U_{\eta}({\frak h})$, $$V=\bigoplus_{{ \omega}\in (\mathbb{C}^*)^l}V_{{ \omega}},$$ where $$V_{{ \omega}}=\{v\in V, L_iv=\omega_iv,~\omega_i\in \mathbb{C}^*,~i=1,\ldots ,l\}$$ is the weight space corresponding to weight ${ \omega}$.
Observe that by Proposition \[pord\] for any two roots $\alpha, \beta\in \Delta_{\m_+}$ such that $\alpha<\beta$ the sum $\alpha+\beta$ can not be represented as a linear combination $\sum_{k=1}^qc_k\gamma_{i_k}$, where $c_k\in \mathbb{N}$ and $\alpha<\gamma_{i_1}<\ldots <\gamma_{i_k}<\beta$, and hence from commutation relations (\[erel\]) one can deduce that the elements $f_\beta\in U_{\eta}({\frak m}_-)$, $\beta\in \Delta_{\m_+}$, $\beta\not\in \{\gamma_1, \ldots ,\gamma_{l'}\}$ generate an ideal $\mathcal{J}$ in $U_{\eta}({\frak m}_-)$.
Let $\eta$ be an element of ${\rm Spec}(Z_0)$. Assume that $t_i\neq 0$, $i=1,\ldots ,l'$ in formula (\[defu\]) and $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$, so that $\eta(f_{\gamma_i}^m)=d_i\neq 0$, $i=1,\ldots ,l'$ and $f_\beta^m=0$ for $\beta\in \Delta_{\m_+}$, $\beta\not\in \{\gamma_1, \ldots ,\gamma_{l'}\}$. Then the ideal $\mathcal{J}$ is the Jacobson radical of $U_{\eta}({\frak m}_-)$ and $U_{\eta}({\frak m}_-)/\mathcal{J}$ is isomorphic to the truncated polynomial algebra $$\mathbb{C}[f_{\gamma_1},\ldots,f_{\gamma_{l'}}]/\{f_{\gamma_i}^m=d_i\}_{ i=1,\ldots ,l'}$$.
Since $f_\beta^m=0$, $\beta\in \Delta_{\m_+}$, $\beta\not\in \{\gamma_1, \ldots ,\gamma_{l'}\}$ we have $\mathcal{J}^m=0$, and hence the ideal $\mathcal{J}$ is nilpotent. We deduce that $\mathcal{J}$ is contained in the Jacobson radical of $U_{\eta}({\frak m}_-)$.
Using commutation relations (\[erel\]) we also have (see the proof of Theorem \[qnil\]) $$f_{\gamma_i}f_{\gamma_j} - f_{\gamma_j}f_{\gamma_j}\in \mathcal{J}.$$ Therefore the quotient algebra $U_{\eta}({\frak m}_-)/\mathcal{J}$ is isomorphic to the truncated polynomial algebra $$\mathbb{C}[f_{\gamma_1},\ldots,f_{\gamma_{l'}}]/\{f_{\gamma_i}^m=d_i\}_{ i=1,\ldots ,l'}$$ which is semisimple. Therefore $\mathcal{J}$ coincides with the Jacobson radical of $U_{\eta}({\frak m}_-)$.
Next, commutation relations (\[erel\]) and part (iv) of Proposition \[rootss\] also imply the following lemma.
\[mepbw\] Let $\beta_1, \ldots, \beta_D$ be the normal ordering of $\Delta_+$ opposite to (\[NO\]). Then for any character $\eta: Z_0 \rightarrow \mathbb{C}$ the elements $f_{\beta_D}^{r_D}\ldots f_{\beta_1}^{r_1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}$, where $r_i, n_j\in \mathbb{N}$, $0\leq r_i,n_j\leq m-1$, $i=1,\ldots D$, $j=1,\ldots, l'$, and $r_i$ can be strictly positive only if $\beta_i\in \Delta_{\m_+}$, $\beta_i\not\in \{\gamma_1, \ldots ,\gamma_{l'}\}$, form a linear basis of $U_\eta({\frak m}_-)$.
The elements $f_{\beta_D}^{r_D}\ldots f_{\beta_1}^{r_1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}$ $r_i, n_j\in \mathbb{N}$, $0\leq r_i,n_j\leq m-1$, $i=1,\ldots D$, $j=1,\ldots, l'$, $r_i$ can be strictly positive only if $\beta_i\in \Delta_{\m_+}$, $\beta_i\not\in \{\gamma_1, \ldots ,\gamma_{l'}\}$, and at least one $r_i$ is strictly positive, form a linear basis of $\mathcal{J}$.
In Theorem \[qnil\] we constructed some characters of the algebra $U_\varepsilon^{s^{-1}}({\frak m}_-)$. Now we show that the algebra $U_\eta({\frak m}_-)$ has a unique up to isomorphism irreducible representation which is one–dimensional.
Let $\eta$ be an element of ${\rm Spec}(Z_0)$. Assume that $t_i\neq 0$, $i=1,\ldots ,l'$ in formula (\[defu\]) and $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$, so that $\eta(f_{\gamma_i}^m)=d_i\neq 0$, $i=1,\ldots ,l'$. Then all nonzero irreducible representations of the algebra $U_\eta({\frak m}_-)$ are one–dimensional and have the form $$\label{chichar}
\chi(f_\beta)=\left\{ \begin{array}{ll} 0 & \beta \not \in \{\gamma_1, \ldots, \gamma_{l'}\} \\ c_i & \beta=\gamma_i
\end{array}
\right .,$$ where complex numbers $c_i$ satisfy the conditions $c_i^m=d_i$, $i=1,\ldots ,l'$. Moreover, all irreducible representations $\mathbb{C}_\chi$ of $U_\eta({\frak m}_-)$ are isomorphic to each other.
Let $V$ be a nonzero finite–dimensional irreducible $U_{\eta}({\frak m}_-)$–module. By Lemma \[charn\] elements of the ideal $\mathcal{J}\subset U_{\eta}({\frak m}_-)$ act by nilpotent transformations on $V$. Therefore from Engel theorem one can deduce that the subspace $V_\mathcal{J}=\{v\in V | xv=0 ~~\forall x\in \mathcal{J}\}$, $V_\mathcal{J}\subset V$, is nonzero.
Using commutation relations (\[erel\]) we have (see the proof of Theorem \[qnil\]) $$f_{\gamma_i}f_{\gamma_j} - f_{\gamma_j}f_{\gamma_j}\in \mathcal{J}.$$ These relations and the fact that $\mathcal{J}$ is an ideal in $U_{\eta}({\frak m}_-)$ imply that the elements $f_{\gamma_1},\ldots,f_{\gamma_{l'}}$ act on $V_\mathcal{J}$ by mutually commuting endomorphisms. Note that by Lemma \[charn\] $\eta(f_{\gamma_i}^m)=d_i\neq 0$, $i=1,\ldots ,l'$ and hence elements $f_{\gamma_i}$ act on $V_\mathcal{J}$ and on $V$ by semisimple automorphisms.
Let $v\in V_\mathcal{J}$ be a common eigenvector in $V_\mathcal{J}$ for the mutually commuting semisimple automorphisms generated by the action of $f_{\gamma_1},\ldots,f_{\gamma_{l'}}$, $f_{\gamma_i}v=c_iv$, $c_i\neq 0$, $i=1,\ldots ,l'$. By construction the one–dimensional subspace generated by $v$ in $V$ is a submodule. Since $V$ is irreducible this subspace must coincide with $V$. Thus $V$ is one–dimensional. If we denote by $\chi:U_{\eta}({\frak m}_-) \rightarrow \mathbb{C}$ the character of $U_{\eta}({\frak m}_-)$ such that $$\chi(f_\beta)=\left\{ \begin{array}{ll} 0 & \beta \not \in \{\gamma_1, \ldots, \gamma_{l'}\} \\ c_i & \beta=\gamma_i
\end{array}
\right .$$ and by $\mathbb{C}_{\chi}$ the corresponding one–dimensional representation of $U_{\eta}({\frak m}_-)$ then we have $V=\mathbb{C}_{\chi}$.
Now we have to prove that the representations $\mathbb{C}_{\chi}$ are isomorphic for different characters $\chi$. Note that $\eta(f_{\gamma_i}^m)=d_i\neq 0$, $i=1,\ldots ,l'$ and hence we have the following relations in $U_{\eta}({\frak m}_-)$: $f_{\gamma_i}^m=d_i$, $i=1,\ldots ,l'$. These relations imply that $\chi(f_{\gamma_i}^m)=c_i^m=d_i$, $i=1,\ldots ,l'$. Therefore for given $\eta$ such that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$ there are only finitely many possible characters $\chi$.
If $\chi$ and $\chi'$ are two such characters such that $\chi(f_{\gamma_i})=c_i$, $i=1,\ldots ,l'$ and $\chi'(f_{\gamma_i})=c_i'$, $i=1,\ldots ,l'$ then the relations $c_i^m={c_i'}^m=d_i$, $i=1,\ldots ,l'$ imply that $c_i'=\varepsilon^{m_i}c_i$, $0\leq m_i \leq m-1$, $m_i\in \mathbb{Z}$, $i=1,\ldots ,l'$.
Now observe that for any $h\in {\mathfrak h}$ the map defined by $f_\alpha \mapsto \varepsilon^{\alpha(h)}f_\alpha$, $\alpha \in \Delta_{\m_+}$ is an automorphism of the algebra $U_\varepsilon^{s^{-1}}({\frak m}_-)$ generated by elements $f_\alpha$, $\alpha \in \Delta_{\m_+}$ with defining relations (\[erel\]), and if in addition $\varepsilon^{m\gamma_i(h)}=1$, $i=1,\ldots ,l'$ the above defined map gives rise to an automorphism $\varsigma$ of $U_{\eta}({\frak m}_-)$. Indeed in that case $(\varepsilon^{\gamma_i(h)}f_{\gamma_i})^m=f_{\gamma_i}^m$, $i=1,\ldots ,l'$ and all the remaining defining relations $\eta(f_{\gamma_i}^m)=d_i\neq 0$, $i=1,\ldots ,l'$, $\eta(f_{\beta}^m)=0$, $\beta \in \Delta_{\m_+}$, $\beta \not \in \{\gamma_1, \ldots, \gamma_{l'}\}$ of the algebra $U_{\eta}({\frak m}_-)$ are preserved by the action of the above defined map $\varsigma$.
Now fix $h\in {\mathfrak h}$ such that $\gamma_i(h)=m_i$, $i=1,\ldots ,l'$. Obviously we have $\varepsilon^{mm_i}=1$, $i=1,\ldots ,l'$. We claim that the representation $\mathbb{C}_{\chi}$ twisted by the corresponding automorphism $\varsigma$ coincides with $\mathbb{C}_{\chi'}$. Indeed, we obtain $$\chi(\varsigma f_{\gamma_i})=\chi(\varepsilon^{m_i}f_{\gamma_i})=\varepsilon^{m_i}c_i=c_i',~i=1,\ldots ,l'.$$ This establishes the isomorphism $\mathbb{C}_{\chi}\simeq \mathbb{C}_{\chi'}$ and completes the proof of the proposition.
Let $V$ be a $U_{\eta}({\frak g})$–module, $\eta$ be an element of ${\rm Spec}(Z_0)$ such that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$. Assume that $t_i\neq 0$, $i=1,\ldots ,l'$ in formula (\[defu\]). Let $\chi:U_\eta({\frak m}_-)\rightarrow \mathbb{C}$ be a character defined in the previous proposition, $\mathbb{C}_{\chi}$ the corresponding one–dimensional $U_\eta({\frak m}_-)$–module. Then the space $V_\chi={\rm Hom}_{U_\eta({\frak m}_-)}(\mathbb{C}_{\chi},V)$ is called the space of Whittaker vectors of $V$. Elements of $V_\chi$ are called Whittaker vectors.
The following proposition is an analogue of Engel theorem for quantum groups at roots of unity.
\[Whitt\] Assume that $t_i\neq 0$, $i=1,\ldots ,l'$ in formula (\[defu\]). Suppose also that $Y_j(\sum_{i=1}^{l'}m_i\gamma_i)\neq mp$ for any $m_i\in \{0,\ldots ,m-1\}$, where at least one of the numbers $m_i$ is nonzero, $p\in \mathbb{Z}$ and $j=1,\ldots ,l$. Let $\eta$ be an element of ${\rm Spec}(Z_0)$ such that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$. Let $\chi:U_\eta({\frak m}_-)\rightarrow \mathbb{C}$ be a character defined in the previous proposition. Then any nonzero finite–dimensional $U_{\eta}({\frak g})$–module contains a nonzero Whittaker vector.
Consider the subalgebra $U_\eta({\frak m}_-+{\mathfrak h})$ in $U_{\eta}({\frak g})$ generated by the elements of $U_\eta({\frak m}_-)$ and by $L_i^{\pm 1}$, $i=1,\ldots ,l$. Let $\mathcal{I}$ be the ideal in $U_\eta({\frak m}_-+{\mathfrak h})$ generated by $\mathcal{J}$.
\[fh\] Assume that $t_i\neq 0$, $i=1,\ldots ,l'$ in formula (\[defu\]). Suppose also that $Y_j(\sum_{i=1}^{l'}m_i\gamma_i)\neq mp$ for any $m_i\in \{0,\ldots ,m-1\}$, where at least one of the numbers $m_i$ is nonzero, $p\in \mathbb{Z}$ and $j=1,\ldots ,l$. Let $\eta$ be an element of ${\rm Spec}(Z_0)$ such that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$. Let $V_0$ be a nonzero finite–dimensional $U_\eta({\frak m}_-+{\mathfrak h})/\mathcal{I}$–module. Then $V_0$ is free over the subalgebra $\mathcal{A}$ of $U_\eta({\frak m}_-+{\mathfrak h})/\mathcal{I}$ generated by the classes of the elements $f_{\gamma_i}$, $i=1,\ldots, l'$ in $U_\eta({\frak m}_-+{\mathfrak h})/\mathcal{I}$, and one can choose a weight $\mathcal{A}$–basis in $V_0$. Fix numbers $c_i$, $i=1,\ldots, l'$ such that $c_i^m=d_i$, $i=1,\ldots, l'$, where $d_i$ are defined by (\[etachar\]). Then the rank of $V_0$ over $\mathcal{A}$ is equal to the dimension of the subspace of $V_0$ which consists of elements $v$ such that $f_{\gamma_i}v=c_iv$, $i=1,\ldots, l'$
Denote the classes of $f_{\gamma_i}$, $i=1,\ldots, l'$ and of $L_i^{\pm 1}$, $i=1,\ldots ,l$ in $U_\eta({\frak m}_-+{\mathfrak h})/\mathcal{I}$ by the same letters. Then $U_\eta({\frak m}_-+{\mathfrak h})/\mathcal{I}$ has generators $f_{\gamma_i}$, $i=1,\ldots, l'$ and $L_i^{\pm 1}$, $i=1,\ldots ,l$, and relations $$L_iL_i^{-1}=1,~L_iL_j=L_jL_i, L_i^m=\eta(L_i),~ f_{\gamma_i}f_{\gamma_j}=f_{\gamma_j}f_{\gamma_i},~ f_{\gamma_i}^m=d_i,~ L_if_{\gamma_j}=\varepsilon^{Y_i(\gamma_j)}f_{\gamma_j}L_i.$$
From the relations $L_i^m=\eta(L_i)\neq 0$ and $f_{\gamma_i}^m=d_i$ we obtain that the elements $f_{\gamma_1},\ldots,f_{\gamma_{l'}}$ and $L_1,\ldots,L_l$ act on $V_0$ by semisimple automorphisms. In particular, $V_0$ has a weight space decomposition for the action of the commutative subalgebra generated by the $L_i$. If $v\in V_0$ is a vector of weight $\omega$ then $$\label{bI}
L_j f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v=\varepsilon^{Y_j(\sum_{i=1}^{l'}n_i\gamma_i)}\omega_jf_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v.$$ Since $Y_j(\sum_{i=1}^{l'}m_i\gamma_i)\neq mp$ for any $m_i\in \{0,\ldots ,m-1\}$, where at least one of the numbers $m_i$ is nonzero, $p\in \mathbb{Z}$, $j=1,\ldots ,l$ and elements $f_{\gamma_i}$ act on $V_0$ by semisimple automorphisms, (\[bI\]) implies that the nonzero vectors $f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v$ have different weights for different $l'$–tuples $(n_1,\ldots ,n_{l'})$, $0\leq n_i\leq m-1$, and hence they are linearly independent in $V_0$.
This implies that one can choose linearly independent weight vectors $v_k\in V_0$, $k=1,\ldots , M$ such that $$V_0=\bigoplus_{k=1}^MV_0^k~~{\rm (direct ~sum ~of}~\mathcal{A}-{\rm modules)},$$ where $V_0^k$ is the free $\mathcal{A}$–module with the linear basis $f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v_k$, for $0\leq n_i\leq m-1$, $i=1,\ldots ,l'$.
One check directly that the vectors $$\prod_{i=1}^{l'}\sum_{j=0}^{m-1}c_i^{m+1-j}\varepsilon^{-(j+1)p_i}f_{\gamma_i}^jv_k,~0\leq p_i\leq m-1$$ form another linear basis of $V_0^k$, and the vector $$\label{WW}
w_k=\prod_{i=1}^{l'}\sum_{j=0}^{m-1}c_i^{m+1-j}f_{\gamma_i}^jv_k$$ is the only vector in $V_0^k$ satisfying the conditions $f_{\gamma_i}w_k=c_iw_k$, $i=1,\ldots, l'$. Thus the rank $M$ of $V_0$ over $\mathcal{A}$ is equal to the dimension of the subspace of such vectors.
Now recall that by Lemma \[charn\] elements of the ideal $\mathcal{I}\subset U_{\eta}({\frak m}_-+{\mathfrak h})$ act by nilpotent transformations on $V$. Therefore from Engel theorem one can deduce that the subspace $V_{\mathcal{I}}=\{v\in V | xv=0 ~~\forall x\in \mathcal{I}\}$, $V_{\mathcal{I}}\subset V$, is nonzero. Now the statement of Proposition \[Whitt\] follows from Lemma \[fh\] applied to the $U_\eta({\frak m}_-+{\mathfrak h})/\mathcal{I}$–module $V_{\mathcal{I}}$ and the definition of Whittaker vectors..
Some properties of finite–dimensional modules over quantum groups at roots of unity {#FREE}
===================================================================================
This section is central in the paper. We shall prove that finite–dimensional modules over quantum groups at roots of unity are free over certain subalgebras. More precisely, we have the following theorem.
\[fdfree\] Let $\zeta$ be an element of ${\rm Spec}(Z_0)$. Assume that $Y_j(\sum_{i=1}^{l'}m_i\gamma_i)\neq mp$ for any $m_i\in \{0,\ldots ,m-1\}$, where at least one $m_i$ is nonzero, $p\in \mathbb{Z}$ and $j=1,\ldots ,l$. Assume that the roots $\gamma_1,\ldots, \gamma_n$ (or $\gamma_{n+1},\ldots, \gamma_{l'}$) are simple or one of the sets $\gamma_1,\ldots, \gamma_n$ or $\gamma_{n+1},\ldots, \gamma_{l'}$ is empty. Suppose also that $t_i\neq 0$, $i=1,\ldots ,l'$ in formula (\[defu\]) and that there exists a quantum coadjoint transformation $\widetilde{g}'$ such that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$, where $\eta=\widetilde{g}'\zeta$. Then there exists a quantum coadjoint transformation $\widetilde{g}\in \mathcal{G}$ such that $\phi\circ \widetilde{\pi}(\widetilde{g}\eta)\in \mu_{M_+}^{-1}(u)$ and any nonzero finite–dimensional $U_{\widetilde{g}\eta}({\frak g})$–module $V$ is free over $U_{\widetilde{g}\eta}({\frak m}_-)$ of rank equal to the dimension of the space of Whittaker vectors $V_{\chi}$, where $\chi$ is a character of $U_{\widetilde{g}\eta}({\frak m}_-)$, and hence any nonzero finite–dimensional $U_{\zeta}({\frak g})$–module is free over $\widetilde{g}'^{-1}\widetilde{g}^{-1}U_{\widetilde{g}\eta}({\frak m}_-)$.
Let $\zeta$ be an element of ${\rm Spec}(Z_0)$ satisfying the conditions imposed in the formulation of Theorem \[fdfree\], $\eta=\widetilde{g}'\zeta$ and $\widetilde{g}\in \mathcal{G}$ an arbitrary quantum coadjoint transformation such that $\phi\circ \widetilde{\pi}(\widetilde{g}\eta)\in \mu_{M_+}^{-1}(u)$. Let $V$ be a finite–dimensional nonzero $U_{\widetilde{g}\eta}({\frak g})$–module.
In the proof we shall use the notation of Lemma \[fh\]. By Lemma \[charn\] elements of the ideal $\mathcal{I}\subset U_{\widetilde{g}\eta}({\frak m}_-+{\mathfrak h})$ act by nilpotent transformations on $V$. Therefore from Engel theorem one can deduce that the subspace $V_{\mathcal{I}}=\{v\in V | xv=0 ~~\forall x\in \mathcal{I}\}$, $V_{\mathcal{I}}\subset V$, is nonzero.
By Lemma \[fh\] $V_{\mathcal{I}}$ is free over the algebra $\mathcal{A}$ with a weight basis $v_k$, $k=1,\ldots ,M$. As in Lemma \[fh\] we denote by $V_{\mathcal{I}}^k$ the free $\mathcal{A}$–submodule in $V_{\mathcal{I}}$ generated by $v_k$.
Since the elements $f_{\gamma_i}$ act on $V_{\mathcal{I}}$ by semisimple automorphisms the $m$-th powers of which are multiplications by nonzero numbers we can assume that if $V_{\mathcal{I}}^k$ and $V_{\mathcal{I}}^{k'}$ contain vectors of the same weight then the weight of $v_k$ is equal to the weight of $v_{k'}$.
Let $V_{\mathcal{I}}'$ be the linear space with the linear basis $v_k\in V$, $k=1,\ldots , M$. Fix a basis $B$ of $V$ which consists of weight vectors and contains all elements $f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v_k$, for $0\leq n_i\leq m-1$, $i=1,\ldots ,l'$, $k=1,\ldots , M$. Let $\rho:V\rightarrow V_{\mathcal{I}}'$ be the linear projection such that $\rho v=0$ for $v\in B$, $v\neq v_k$ for some $k$. Obviously $\rho$ sends weight vectors to weight vectors.
Consider the left $U_{\widetilde{g}\eta}({\frak m}_-)$–module ${\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')$ with the left $U_{\widetilde{g}\eta}({\frak m}_-)$–action induced by multiplication in $U_{\widetilde{g}\eta}({\frak m}_-)$ from the right. Note that since by Proposition \[frob\] the algebra $U_{\widetilde{g}\eta}({\frak m}_-)$ is Frobenius and the space $V_{\mathcal{I}}'$ is finite–dimensional we have a $U_{\widetilde{g}\eta}({\frak m}_-)$–module isomorphism ${\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')\simeq U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$. Therefore ${\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')$ is a free $U_{\widetilde{g}\eta}({\frak m}_-)$–module.
Now let $\sigma:V\rightarrow {\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')$ be the homomorphism of $U_{\widetilde{g}\eta}({\frak m}_-)$–modules defined by $\sigma(v)(x)=\rho(xv)$, $x\in U_{\widetilde{g}\eta}({\frak m}_-)$, $v\in V$. We claim that $\sigma$ is an isomorphism when $\widetilde{g}$ is chosen in an appropriate way. Since ${\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')$ is a free $U_{\widetilde{g}\eta}({\frak m}_-)$–module this will imply that $V$ is free over $U_{\widetilde{g}\eta}({\frak m}_-)$ of rank equal to the dimension of $V_{\mathcal{I}}'$. By Lemma \[fh\] that dimension is equal to the dimension of the space of Whittaker vectors in $V$.
First we show that $\sigma$ is injective. Indeed, the kernel ${\rm Ker}~\sigma$ of $\sigma$ is a $U_{\widetilde{g}\eta}({\frak m}_-)$–submodule of $V$, and hence, by Engel theorem, if ${\rm Ker}~\sigma\neq \{0\}$ it must contain a nonzero element $v$ annihilated by the nilpotent transformations $f_\beta$, $\beta\in \Delta_{\mathfrak{m}_+}$, $\beta \not\in \{\gamma_1,\ldots ,\gamma_{l'}\}$. Thus by definition $v\in V_{\mathcal{I}}$. Since $V_{\mathcal{I}}$ is free over $\mathcal{A}$ with basis $v_k$, $v$ can be uniquely represented as a linear combination of elements $f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v_k$, for $0\leq n_i\leq m-1$, $i=1,\ldots ,l'$, $k=1,\ldots , M$, $$v=\sum_{0\leq n_i\leq m-1,k=1,\ldots , M} c_{n_1\ldots n_{l'}}^{k}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v_k.$$
Recall that elements $f_{\gamma_i}$ act on $V$ by automorphisms the $m$-th powers of which are multiplications by nonzero numbers. Therefore if $c_{n_1'\ldots n_{l'}'}^{k'}\neq 0$ then the element $w=f_{\gamma_{l'}}^{m-n_{l'}'}\ldots f_{\gamma_1}^{m-n_1'}v$ can be represented in the form $$w=\sum_{0\leq n_i\leq m-1,k=1,\ldots , M} d_{n_1\ldots n_{l'}}^{k}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v_k,$$ where $d_{0\ldots 0}^{k'}\neq 0$.
Now we have $$\sigma(w)(1)=\rho(w)=\sum_{k=1,\ldots , M} d_{0\ldots 0}^{k}v_k,$$ where at least one coefficient $d_{0\ldots 0}^{k'}\neq 0$. Since the elements $v_k$ are linearly independent we deduce that $\sigma(w)(1)\neq 0$, and hence $\sigma(w)\neq 0$.
On the other hand if $v\in {\rm Ker}~\sigma$ then we also have $w=f_{\gamma_{l'}}^{m-n_{l'}'}\ldots f_{\gamma_1}^{m-n_1'}v\in {\rm Ker}~\sigma$ since ${\rm Ker}~\sigma$ is a $U_{\widetilde{g}\eta}({\frak m}_-)$–submodule of $V$. Thus we arrive at a contradiction, and hence $\sigma$ is injective.
Now we prove that $\sigma$ is surjective. We start with the following lemma.
Let $\eta$ be an element of ${\rm Spec}(Z_0)$. Assume that the roots $\gamma_1,\ldots, \gamma_n$ (or $\gamma_{n+1},\ldots, \gamma_{l'}$) are simple or one of the sets $\gamma_1,\ldots, \gamma_n$ or $\gamma_{n+1},\ldots, \gamma_{l'}$ is empty. Suppose also that $t_i\neq 0$, $i=1,\ldots ,l'$ in formula (\[defu\]) and that $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$. Then there exists a quantum coadjoint transformation $\widetilde{g}\in \mathcal{G}$ such that $\phi\circ \widetilde{\pi}(\widetilde{g}\eta)\in \mu_{M_+}^{-1}(u)$ and for any $\alpha \in \Delta_{{\frak m}_+}$, $\alpha \not\in \{ \gamma_1,\ldots ,\gamma_{l'}\}$ any nonzero finite–dimensional $U_{\widetilde{g}\eta}({\frak g})$–module $V$ is free over the subalgebra $U_{\widetilde{g}\eta}(f_\alpha)$ of $U_{\widetilde{g}\eta}({\frak g})$ generated by the unit and by $f_\alpha$.
Recall that by Lemma \[charn\] for $\alpha \in \Delta_{{\frak m}_+}$, $\alpha \not\in \{ \gamma_1,\ldots ,\gamma_{l'}\}$ we have $f_\alpha^m=0$. Hence $U_{\widetilde{g}\eta}(f_\alpha)$ is isomorphic to the truncated polynomial algebra $U_{\widetilde{g}\eta}(f_\alpha)=\mathbb{C}[f_\alpha]/\{f_\alpha^m=0\}$, and in order to show that $V$ is $U_{\widetilde{g}\eta}(f_\alpha)$–free it suffices to verify that all Jordan blocks of the nilpotent endomorphism given by the action of $f_\alpha$ in $V$ have size $m$.
Denote by $V_{f_\alpha}$ the kernel of $f_\alpha$ in $V$. Since $f_\alpha^m=0$ the subspace $V_{f_\alpha}$ is not trivial. Let $w_i$, $i=1,\ldots ,P$ be a linear basis of $V_{f_\alpha}$. We show that the vectors $e_\alpha^kw_i$, $k=0,\ldots ,m-1$, $i=1,\ldots ,P$ are linearly independent.
Assume that they are linearly dependant. Then there are nonzero elements $w_k^1\in V_{f_\alpha}$, $k=0,\ldots ,Q$, $Q\leq m-1$ such that $$\label{ld}
\sum_{k=0}^Q a_ke_\alpha^kw_k^1=0,$$ where $a_k\in \mathbb{C}$ and $a_Q\neq 0$.
Now from formula (13), Sect. 9.3 in [@ChP] we deduce the following commutation relations $$\label{ef}
f_\alpha e_\alpha^k=e_\alpha^kf_\alpha-[k]_{\varepsilon_\alpha}e_\alpha^{k-1}
\frac{\varepsilon_\alpha^{k-1}K_\alpha-\varepsilon_\alpha^{1-k}K_\alpha^{-1}}{\varepsilon_\alpha-\varepsilon_\alpha^{-1}},$$ where if $\alpha=s_{i_1}\ldots s_{i_{p-1}}\alpha_{i_p}$ then $K_\alpha=T_{i_1}\ldots T_{i_{p-1}}K_{i_p}$.
Applying $f_\alpha$ to relation (\[ld\]), using commutation relations (\[ef\]) and the fact that $f_\alpha v=0$ for ant $v\in V_{f_\alpha}$ we obtain that $$\label{1ap}
\sum_{k=1}^Q a_ke_\alpha^{k-1}[k]_{\varepsilon_\alpha}\frac{\varepsilon_\alpha^{k-1}K_\alpha-\varepsilon_\alpha^{1-k}K_\alpha^{-1}}{\varepsilon_\alpha-\varepsilon_\alpha^{-1}}w_k^1=0.$$
Now observe that commutation relations $L_if_{\alpha}=\varepsilon^{Y_i(\alpha)}f_{\alpha}L_i$ and the definition of $K_\alpha$ imply that $V_{f_\alpha}$ is an invariant subspace for the action of $K_\alpha^{\pm 1}$. Therefore in (\[1ap\]) $[k]_{\varepsilon_\alpha}\frac{\varepsilon_\alpha^{k-1}K_\alpha-\varepsilon_\alpha^{1-k}K_\alpha^{-1}}{\varepsilon_\alpha-\varepsilon_\alpha^{-1}}w_k^1\in V_{f_\alpha}$. We claim that one can choose a quantum coadjoint transformation $\widetilde{g}\in \mathcal{G}$ such that $\phi\circ \widetilde{\pi}(\widetilde{g}\eta)\in \mu_{M_+}^{-1}(u)$ and vectors $\frac{\varepsilon_\alpha^{k-1}K_\alpha-\varepsilon_\alpha^{1-k}K_\alpha^{-1}}{\varepsilon_\alpha-\varepsilon_\alpha^{-1}}w_k^1$ are all nonzero in (\[1ap\]).
Let $H'$ be the subgroup of $H$ which corresponds to the Lie subalgebra ${\mathfrak h}'\subset {\mathfrak h}$ and ${H'}^\perp\subset H$ be the subgroup corresponding to the Lie subalgebra ${{\mathfrak h}'}^\perp\subset {\mathfrak h}$ so that $H=H'{H'}^\perp$ (direct product of subgroups). Since any $\alpha \in \Delta_{{\frak m}_+}$, $\alpha \not\in \{ \gamma_1,\ldots ,\gamma_{l'}\}$ has a nonzero projection onto ${\mathfrak h}'$ one can find $h\in H'$ such that $(h(K_\alpha^m))^{\frac{2}{m}}\neq \varepsilon_\alpha^{2(1-k)}$ for all $\alpha \in \Delta_{{\frak m}_+}$, $\alpha \not\in \{ \gamma_1,\ldots ,\gamma_{l'}\}$, $k=1,\ldots ,m-1$, and for all roots $(h(K_\alpha^m))^{\frac{2}{m}}$ of degree $\frac{2}{m}$ of $h(K_\alpha^m)$. By Lemma \[Hcomp\] for $\eta\in {\rm Spec}(Z_0)$, $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$ one can find a quantum coadjoint transformation $\widetilde{g}\in \mathcal{G}$ such that $\phi\circ \widetilde{\pi}(\widetilde{g}\eta)\in \mu_{M_+}^{-1}(u)$ and the $H={\rm Spec}(Z_0^0)$–component $\widetilde{g}\eta_0$ of $\widetilde{g}\eta$ in ${\rm Spec}(Z_0^+)\times {\rm Spec}(Z_0^0) \times {\rm Spec}(Z_0^-)$ is equal to $h$. Thus we have $(\widetilde{g}\eta(K_\alpha^m))^{\frac{2}{m}}=(\widetilde{g}\eta_0(K_\alpha^m))^{\frac{2}{m}}=(h(K_\alpha^m))^{\frac{2}{m}}\neq \varepsilon_\alpha^{2(1-k)}$ for all $\alpha \in \Delta_{{\frak m}_+}$, $\alpha \not\in \{ \gamma_1,\ldots ,\gamma_{l'}\}$, $k=1,\ldots ,m-1$, and for all roots $(\widetilde{g}\eta(K_\alpha^m))^{\frac{2}{m}}$ of degree $\frac{2}{m}$ of $\widetilde{g}\eta(K_\alpha^m)$.
Since $K_\alpha^m= \widetilde{g}\eta(K_\alpha^m)$ in $U_{\widetilde{g}\eta}({\frak g})$ the numbers $(\widetilde{g}\eta(K_\alpha^m))^{\frac{2}{m}}$ exhaust all possible eigenvalues of $K_\alpha^2$ in $V$, and hence the operators $K_\alpha^2-\varepsilon_\alpha^{2(1-k)}$ acting in $V$ are invertible for all $\alpha \in \Delta_{{\frak m}_+}$, $\alpha \not\in \{ \gamma_1,\ldots ,\gamma_{l'}\}$, $k=1,\ldots ,m-1$. Therefore the operators $$\frac{\varepsilon_\alpha^{k-1}K_\alpha-\varepsilon_\alpha^{1-k}K_\alpha^{-1}}{\varepsilon_\alpha-\varepsilon_\alpha^{-1}}=
\frac{\varepsilon_\alpha^{k-1}K_\alpha^{-1}(K_\alpha^{2}-\varepsilon_\alpha^{2(1-k)})}{\varepsilon_\alpha-\varepsilon_\alpha^{-1}}$$ are invertible as well.
Thus vectors $\frac{\varepsilon_\alpha^{k-1}K_\alpha-\varepsilon_\alpha^{1-k}K_\alpha^{-1}}{\varepsilon_\alpha-\varepsilon_\alpha^{-1}}w_k^1$ are all nonzero in (\[1ap\]), and from (\[1ap\]) we obtain the following relation $$\sum_{k=1}^Q a_ke_\alpha^{k-1}w_k^2=0,$$ where $w_k^2\in V_{f_\alpha}$ are nonzero vectors.
Applying successively $f_\alpha$ to the above relation $Q-1$ times and using similar arguments we obtain that $$a_Qw_k^{Q-1}=0$$ for a nonzero vector $w_k^{Q-1}\in V_{f_\alpha}$. This is a contradiction. Thus the vectors $e_\alpha^kw_i$, $k=0,\ldots ,m-1$, $i=1,\ldots ,P$ are linearly independent. The last assertion implies that ${\dim}~V\geq m~{\rm dim}~V_{f_\alpha}$. Since the Jordan blocks of $f_\alpha$ in $V$ have size at most $m$ we also have the opposite inequality, ${\dim}~V\leq m~{\rm dim}~V_{f_\alpha}$. Thus ${\dim}~V= m~{\rm dim}~V_{f_\alpha}$, and hence all Jordan blocks of $f_\alpha$ in $V$ have size $m$. This completes the proof.
From now on we assume that $\widetilde{g}\in \mathcal{G}$ is fixed as in the previous lemma. Recall that we already proved that the $U_{\widetilde{g}\eta}({\frak m}_-)$–module homomorphism $$\sigma:V\rightarrow {\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')\simeq U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$$ is an imbedding. Thus $V$ is a submodule of the free $U_{\widetilde{g}\eta}({\frak m}_-)$–module $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$.
Let $\beta_1, \ldots, \beta_D$ be the normal ordering of $\Delta_+$ opposite to (\[NO\]). Then by Lemma \[mepbw\] the elements $f_{\beta_D}^{r_D}\ldots f_{\beta_1}^{r_1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}$, where $r_i, n_j\in \mathbb{N}$, $0\leq r_i,n_j\leq m-1$, $i=1,\ldots D$, $j=1,\ldots, l'$, and $r_i$ can be strictly positive only if $\beta_i\in \Delta_{\m_+}$, $\beta_i\not\in \{\gamma_1, \ldots ,\gamma_{l'}\}$, form a linear basis of $U_\eta({\frak m}_-)$. Hence the elements $$\label{Homb}
f_{\beta_{i_L}}^{r_L}\ldots f_{\beta_{i_1}}^{r_1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k,$$ $\{\beta_{i_1},\ldots ,\beta_{i_L}\}=\Delta_{\m_+}\setminus \{\gamma_1, \ldots ,\gamma_{l'}\}$, $\beta_{i_1}<\ldots <\beta_{i_L}$ form a linear basis of $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$.
Our aim now is to show that the image of $\sigma$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ contains a subspace spanned by generating vectors. This will justify that $\sigma$ is surjective. First we describe the image of the subspace $V_{\mathcal{I}}$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ under the homomorphism $\sigma$.
\[sim\] The image of the subspace $V_{\mathcal{I}}\subset V$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ under the homomorphism $\sigma$ is the linear subspace $X$ with the linear basis $$\label{linb}
f_{\beta_{i_L}}^{m-1}\ldots f_{\beta_{i_1}}^{m-1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k, 0\leq n_j\leq m-1, j=1,\ldots, l',k=1,\ldots ,M,$$ $\{\beta_{i_1},\ldots ,\beta_{i_L}\}=\Delta_{\m_+}\setminus \{\gamma_1, \ldots ,\gamma_{l'}\}$, $\beta_{i_1}<\ldots <\beta_{i_L}$.
By the construction of the isomorphism $\Phi$ given in the proof of Proposition \[frob\], commutation relations (\[erel\]) and the fact that $\mathcal{J}$ is an ideal in $U_{\widetilde{g}\eta}({\frak m}_-)$, the image of $X$ under the isomorphism $$U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}' \simeq {\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')$$ is the linear subspace with the basis $$\label{linb1}
f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}v_k, 0\leq n_j\leq m-1, j=1,\ldots, l',k=1,\ldots ,M,$$ where $v_k$ are regarded as images of the corresponding elements of $V$ under $\sigma$. By the first part of the proof of this proposition elements (\[linb1\]), where $v_k$ are regarded as elements of $V$, form a linear basis of $V_{\mathcal{I}}$. Hence elements (\[linb1\]) form a linear basis of the image of $V_{\mathcal{I}}$ in ${\rm Hom}_{\mathbb{C}}(U_{\widetilde{g}\eta}({\frak m}_-),V_{\mathcal{I}}')$ under $\sigma$, and elements (\[linb\]) form a linear basis of the image of the image of $V_{\mathcal{I}}$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ under the homomorphism $\sigma$.
Now we show that the image of $\sigma$ contains some special elements which in fact generate $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$.
\[sim1\] The image of $\sigma$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ contains elements of the form $$\label{linb2}
f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k+x, 0\leq n_j\leq m-1, j=1,\ldots, l', k=1,\ldots,M,$$ where $
x\in {\rm Im}~\mathcal{J}.
$
Recall that using injective homomorphism $\sigma$ the module $V$ can be regarded as a free $U_{\widetilde{g}\eta}(f_{\beta_{i_L}})$–submodule of $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$. Elements (\[linb\]) belong to that submodule and each of elements (\[linb\]) is annihilated by $f_{\beta_{i_L}}$. Since all Jordan blocks of $f_{\beta_{i_L}}$ in $V$ have size $m$ the image of $V$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ must also contain elements which are mapped to elements (\[linb\]) under the action of $f_{\beta_{i_L}}^{m-1}$. Such elements have the form $$f_{\beta_{i_{L-1}}}^{m-1}\ldots f_{\beta_{i_1}}^{m-1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k+x_L, 0\leq n_j\leq m-1, j=1,\ldots, l', k=1,\ldots ,M,~x_L\in {\rm Im}~f_{\beta_{i_L}}.$$
Now we proceed by induction. Assume that for some $0<p<L$ the image of $\sigma$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ contains elements of the form $$\label{linb3}
f_{\beta_{i_p}}^{m-1}\ldots f_{\beta_{i_1}}^{m-1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k +x_{p+1}, 0\leq n_j\leq m-1, j=1,\ldots, l',k=1,\ldots ,M,$$ where $
x_{p+1}\in {\rm Im}~\mathcal{J}_p,
$ and $\mathcal{J}_p$ is the ideal in $U_{\widetilde{g}\eta}({\frak m}_-)$ generated by the elements $f_{\beta_{i_{p+1}}}, \ldots, f_{\beta_{i_L}}$.
By commutation relations (\[erel\]) and by the fact that $\mathcal{J}$ is an ideal, the image $V_p$ of $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ under the action of the ideal $\mathcal{J}_p$ is invariant under the action of $f_{\beta_{i_p}}$. Moreover, for the same reasons and by the Poincaré–Birkhoff–Witt theorem for $U_{\widetilde{g}\eta}({\frak m}_-)$ both $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ and the subspace $V_p$ are free modules over $U_{\widetilde{g}\eta}(f_{\beta_{i_p}})$, and there is a Jordan basis $f_{\beta_{i_p}}^nw_t$, $n=1,\ldots , m-1$, $t=1,\ldots ,S$ for the action of $f_{\beta_{i_p}}$ on $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ such that $f_{\beta_{i_p}}^nw_t$, $n=1,\ldots , m-1$, $t=1,\ldots ,R\leq S$ is a Jordan basis of $V_p$. Actually as $w_t$ one can take the following elements $$f_{\beta_{i_L}}^{r_L}\ldots f_{\beta_{i_{p+1}}}^{r_{p+1}}f_{\beta_{i_{p-1}}}^{r_{p-1}}\ldots f_{\beta_{i_1}}^{r_1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k,~0\leq r_i~,n_i\leq m-1,k=1,\ldots ,M,$$ and for the elements $w_t$ with $1\leq t\leq R$ at least one $r_i$ is nonzero for $p+1\leq i\leq L$.
Now recall that $V$ can be regarded as a free $U_{\widetilde{g}\eta}(f_{\beta_{i_p}})$–submodule of the free $U_{\widetilde{g}\eta}(f_{\beta_{i_p}})$–module $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$. Hence there is an $f_{\beta_{i_p}}$–Jordan basis of $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ such that the image of $V$ under $\sigma$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ consists of Jordan blocks of that basis. An arbitrary $f_{\beta_{i_p}}$–Jordan basis of $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ has the form $$\begin{aligned}
w_{0s}=\sum_{r=1}^{m-1}\sum_{t=1}^Sa_r^{st}f_{\beta_{i_p}}^rw_t, \nonumber \\
w_{1s}=\sum_{r=1}^{m-2}\sum_{t=1}^Sa_r^{st}f_{\beta_{i_p}}^{r+1}w_t, \label{linb4} \\
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \nonumber \\
w_{m-1s}=\sum_{t=1}^Sa_0^{st}f_{\beta_{i_p}}^{m-1}w_t, \nonumber\end{aligned}$$ where $s=1,\ldots , S$, $a_r^{st}\in \mathbb{C}$, and ${\rm det}~a_0^{st}~\neq 0$.
Assume that the coefficients $a_r^{st}$ are chosen in such a way that $w_{qs}$ for $q=1,\ldots, m-1$ and $s=1,\ldots ,K\leq S$ form a linear basis of the image of $V$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ under $\sigma$.
Since $V_p\subset U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ is a $U_{\widetilde{g}\eta}(f_{\beta_{i_p}})$–submodule and by the construction of the elements $w_t$ the quotient $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'/V_p$ is a $U_{\widetilde{g}\eta}(f_{\beta_{i_p}})$–module spanned by the classes of the elements $w_{qs}$ for $q=1,\ldots, m-1$ and $s=1,\ldots ,S$.
Let $A$ be the rank of the matrix $a_0^{st}$, $s=1,\ldots ,S$, $t=R+1, \ldots ,S$. One can find indexes $s_i$, $i=1,\ldots ,A$ such that the classes of the elements $w_{qs_i}$ for $q=1,\ldots, m-1$ and $i=1,\ldots ,A$ form an $f_{\beta_{i_p}}$–Jordan basis of $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'/V_p$. Thus by the construction of the basis the quotient $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'/V_p$ is a free $U_{\widetilde{g}\eta}(f_{\beta_{i_p}})$–module, and the image of $V$ in it has the $f_{\beta_{i_p}}$–Jordan basis $w_{qs_i}$, $q=1,\ldots, m-1$, $s_i\in \{1,\ldots ,K\}$.
Therefore the image of $V$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'/V_p$ is a free $U_{\widetilde{g}\eta}(f_{\beta_{i_p}})$–module. This image contains the classes of elements (\[linb3\]) which are nonzero by construction and which are annihilated by the action of $f_{\beta_{i_p}}$. Hence that image must also contain the classes of elements $$f_{\beta_{i_{p-1}}}^{m-1}\ldots f_{\beta_{i_1}}^{m-1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k +x_{p}', 0\leq n_j\leq m-1, j=1,\ldots, l',k=1,\ldots ,M, x_{p}'\in {\rm Im}~f_{\beta_{i_p}}$$ which are mapped to the classes of elements (\[linb3\]) under the action of $f_{\beta_{i_p}}^{m-1}$, and the image of $V$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ must contain elements
$$f_{\beta_{i_{p-1}}}^{m-1}\ldots f_{\beta_{i_1}}^{m-1}f_{\gamma_1}^{n_1}\ldots f_{\gamma_{l'}}^{n_{l'}}\otimes v_k +x_{p}, 0\leq n_j\leq m-1, j=1,\ldots, l',k=1,\ldots ,M,$$ where $
x_{p}\in {\rm Im}~\mathcal{J}_{p-1}.
$ This completes the proof.
Now using the relations $f_{\gamma_i}^m=d_i\neq 0$, the fact that $\mathcal{J}$ is an ideal in $U_{\widetilde{g}\eta}({\frak m}_-)$ and applying appropriate products of powers of elements $f_{\gamma_i}$ to elements (\[linb2\]) we deduce that the image of $\sigma$ in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ contains elements of the form $$\label{linb5}
y_k=1\otimes v_k+x, 0\leq n_j\leq m-1, j=1,\ldots, l', k=1,\ldots,M,$$ where $
x\in {\rm Im}~\mathcal{J}.
$
Elements (\[linb5\]) are linearly independent over $U_{\widetilde{g}\eta}({\frak m}_-)$ and generate $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ over $U_{\widetilde{g}\eta}({\frak m}_-)$.
Assume that elements (\[linb5\]) are linearly dependent over $U_{\widetilde{g}\eta}({\frak m}_-)$. Let $$\label{relt1}
\sum_{k=1}^Mz_ky_k=0, z_k\in U_{\widetilde{g}\eta}({\frak m}_-)$$ be a relation between them in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$.
Consider the corresponding relation in $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'/\mathcal{J}\otimes V_{\mathcal{I}}'$, $$\sum_{k=1}^Mz_k^0\otimes v_k=0,$$ where $z_k^0$ are the classes of the elements $z_k$ in $U_{\widetilde{g}\eta}({\frak m}_-)/\mathcal{J}$. The last relation obviously implies $z_k^0=0$, and hence $z_k\in \mathcal{J}$. Therefore $z_ky_k\in \mathcal{J}\otimes V_{\mathcal{I}}'$.
Now from (\[relt1\]) we derive the following relation in $\mathcal{J}\otimes V_{\mathcal{I}}'/\mathcal{J}^2\otimes V_{\mathcal{I}}'$, $$\label{relt2}
\sum_{k=1}^Mz_k^1\otimes v_k=0,$$ where $z_k^1$ are the classes of the elements $z_k$ in $\mathcal{J}/\mathcal{J}^2$. Clearly, (\[relt2\]) yields $z_k^1=0$, and hence $z_k\in \mathcal{J}^2$.
Finally simple induction and the fact that $\mathcal{J}^m=0$ imply that $z_k=0$. Thus elements (\[linb5\]) are linearly independent over $U_{\widetilde{g}\eta}({\frak m}_-)$. The number of elements (\[linb5\]) is equal to the rank of $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ over $U_{\widetilde{g}\eta}({\frak m}_-)$. Hence elements (\[linb5\]) generate $U_{\widetilde{g}\eta}({\frak m}_-)\otimes V_{\mathcal{I}}'$ over $U_{\widetilde{g}\eta}({\frak m}_-)$. This completes the proof of the lemma
By the previous lemma $\sigma$ is surjective. This completes the proof of the theorem.
From the previous theorem and the fact that ${\rm dim}~U_{\widetilde{g}\eta}({\frak m}_-)=m^{{\rm dim}~\mathfrak{m}_-}$ we immediately obtain the following corollary.
\[divis\] Assume that the conditions of Theorem \[fdfree\] are satisfied. Then the dimension of any finite–dimensional $U_{\eta}({\frak g})$–module $V$ is divisible by $m^{{\rm dim}~\mathfrak{m}_-}$, and ${\rm dim}~V=m^{{\rm dim}~\mathfrak{m}_-}{\rm dim}~V_\chi$.
Proposition \[var\] implies that $2{\rm dim}~\mathfrak{m}_-+{\rm dim}~\Sigma_s={\rm dim}~G$. Therefore ${\rm dim}~\mathfrak{m}_-=\frac{1}{2}({\rm dim}~G-{\rm dim}~\Sigma_s)$. By Proposition \[crosssect\] $\Sigma_s$ is transversal to the set of conjugacy classes in $G$. Therefore by Proposition \[qcoadj\] and by the definition of maps $\phi$ and $\widetilde{\pi}$ for $\eta \in {\rm Spec}(Z_0)$, $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$ we have ${\rm dim}~G-{\rm dim}~\Sigma_s\leq {\rm dim}~\mathcal{O}_\eta$, where $\mathcal{O}_\eta$ is the $\mathcal{G}$–orbit of $\eta$. In [@DKP1] De Concini, Kac and Procesi formulated the following conjecture.
[**(De Concini, Kac and Procesi (1992))**]{} The dimension of any finite–dimensional irreducible $U_{\eta}({\frak g})$–module $V$ is divisible by $m^{\frac{1}{2}{\rm dim}~\mathcal{O}_\eta}$.
By Proposition \[except\] it suffices to verify this conjecture in case of elements $\eta \in {\rm Spec}(Z_0)$ such that $\pi\eta\in G^0$ is exceptional. Recall that by the discussion above for $\eta \in {\rm Spec}(Z_0)$, $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$ the dimension of any finite–dimensional $U_{\eta}({\frak g})$–module $V$ is divisible by $m^{\frac{1}{2}({\rm dim}~G-{\rm dim}~\Sigma_s)}$. Remind also that the map $\phi$ is induced by the conjugation action. Combining these facts with the description of the quantum coadjoint action orbits in Proposition \[qcoadj\] in terms of the finite covering $\pi$ we deduce that for $\eta \in {\rm Spec}(Z_0)$ such that $\pi\eta\in G^0$ is conjugate to an element from $q\mu_{M_+}^{-1}(u)$ the dimension of any $U_{\eta}({\frak g})$–module $V$ is divisible by $m^{{\rm dim}~\mathfrak{m}_-}$ by Corollary \[divis\]. Recalling that $\Sigma_s\cap G^0\subset {q(\mu_{M_+}^{-1}(u))}$ and $\Sigma_s\subset \overline{q(\mu_{M_+}^{-1}(u))}$ (see Proposition \[var\]), so ${\rm dim}~\Sigma_s={\rm dim}~\Sigma_s\cap G^0$, one obtains that De Concini–Kac–Procesi conjecture follows from the following statement. 0.3cm [**Statement**]{} [*For the conjugacy class $\mathcal{O}_g$ of every exceptional element $g\in G^0$ there exists a Weyl group element $s\in W$ such that the roots $\gamma_1,\ldots, \gamma_n$ (or $\gamma_{n+1},\ldots, \gamma_{l'}$) appearing in decomposition (\[inv\]) of $s$ are simple, with respect to a system of positive roots associated to $s$ in Section \[nilpuq\], or one of the sets $\gamma_1,\ldots, \gamma_n$ or $\gamma_{n+1},\ldots, \gamma_{l'}$ is empty, $Y_j(\sum_{i=1}^{l'}m_i\gamma_i)\neq mp$ for any $m_i\in \{0,\ldots ,m-1\}$, where at least one $m_i$ is nonzero, $p\in \mathbb{Z}$ and $j=1,\ldots ,l$, and $\mathcal{O}_g$ is strictly transversal to the transversal slice $\Sigma_s\cap G^0$ in the sense that $\mathcal{O}_g$ intersects $\Sigma_s\cap G^0$ and ${\rm dim}~G-{\rm dim}~\Sigma_s={\rm dim}~\mathcal{O}_g$.*]{}
This statement will be proved in a subsequent paper.
A categorial equivalence {#SKR}
========================
In this section we establish an equivalence between categories of finite–dimensional representations of quantum groups and of q-W algebras at roots of unity. This is a version of Skryabin equivalence for quantum groups at roots of unity (see [@Pr]).
In this section we assume that the conditions of Theorem \[fdfree\] are satisfied. We shall also use the notation introduced in that theorem. For given $\eta\in {\rm Spec}(Z_0)$, $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$ we assume that a quantum coadjoint transformation $\widetilde{g}\in \mathcal{G}$ is fixed as in Theorem \[fdfree\] and denote $\xi=\widetilde{g}\eta\in {\rm Spec}(Z_0)$. Let $\chi$ be a character of $U_{\xi}({\frak m}_-)$, $\mathbb{C}_\chi$ the corresponding representation of $U_{\xi}({\frak m}_-)$. Denote by $Q_\chi$ the induced left $U_{\xi}(\g)$–module, $Q_\chi=U_{\xi}(\g)\otimes_{U_{\xi}({\frak m}_-)}\mathbb{C}_\chi$. Let $W^s_\varepsilon(G)={\rm End}_{U_{\xi}(\g)}(Q_\chi)^{opp}$ be the algebra of $U_{\xi}(\g)$–endomorphisms of $Q_\chi$ with the opposite multiplication. The algebra $W^s_\varepsilon(G)$ is called a q-W algebra associated to $s\in W$. Denote by $U_{\xi}(\g)-{\rm mod}$ the category of finite–dimensional left $U_{\xi}(\g)$–modules and by $W^s_\varepsilon(G)-{\rm mod}$ the category of finite–dimensional left $W^s_\varepsilon(G)$–modules. Observe that if $V\in U_{\xi}(\g)-{\rm mod}$ then the algebra $W_\varepsilon^s(G)$ naturally acts on the finite–dimensional space $V_\chi={\rm Hom}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_\chi,V)={\rm Hom}_{U_{\xi}(\g)}(Q_\chi,V)$ by compositions of homomorphisms.
The functor $E\mapsto Q_\chi\otimes_{W_\varepsilon^s(G)}E$ establishes an equivalence of the category of finite–dimensional left $W_\varepsilon^s(G)$–modules and the category $U_{\xi}(\g)-{\rm mod}$. The inverse equivalence is given by the functor $V\mapsto V_\chi$. In particular, the latter functor is exact, and every finite–dimensional $U_{\xi}(\g)$–module is generated by Whittaker vectors.
Let $E$ be a finite–dimensional $W_\varepsilon^s(G)$–module. First we observe that by the definition of the algebra $W_\varepsilon^s(G)$ we have $W^s_\varepsilon(G)={\rm End}_{U_{\xi}(\g)}(Q_\chi)^{opp}={\rm Hom}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_\chi,Q_\chi)=(Q_\chi)_\chi$ as a linear space, and hence $(Q_{\chi}\otimes_{W_\varepsilon^s(G)}E)_\chi=E$. Therefore to prove the theorem it suffices to check that for any $V\in U_{\xi}(\g)-{\rm mod}$ the canonical map $f:Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi \rightarrow V$ is an isomorphism.
Indeed, $f$ is injective because otherwise its kernel would contain a nonzero Whittaker vector by Proposition \[Whitt\]. But all Whittaker vectors of $Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi$ belong to the subspace $1\otimes V_\chi$, and the restriction of $f$ to $1\otimes V_\chi$ induces an isomorphism of the spaces of Whittaker vectors of $Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi$ and of $V$.
In order to prove that $f$ is surjective we consider the exact sequence $$0\rightarrow Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi \rightarrow V \rightarrow W \rightarrow 0,$$ where $W$ is the cokernel of $f$, and the corresponding long exact sequence of cohomology, $$0\rightarrow {\rm Ext}^{0}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_{\chi},
Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi)\rightarrow {\rm Ext}^{0}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_{\chi},
V)\rightarrow {\rm Ext}^{0}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_{\chi},
W)\rightarrow$$ $$\rightarrow {\rm Ext}^{1}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_{\chi},
Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi)\rightarrow \ldots .$$
Now recall that $f$ induces an isomorphism of the spaces of Whittaker vectors of $Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi$ and of $V$. By Theorem \[fdfree\] the finite–dimensional $U_{\xi}(\g)$–module $Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi$ is free over $U_{\xi}({\frak m}_-)$. Since $U_{\xi}({\frak m}_-)$ is Frobenius $Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi$ is also injective over $U_{\xi}({\frak m}_-)$, and hence\
${\rm Ext}^{1}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_{\chi},
Q_{\chi}\otimes_{W_\varepsilon^s(G)}V_\chi)=0$. Therefore the initial part of the long exact cohomology sequence takes the form $$0\rightarrow V_\chi \rightarrow V_\chi \rightarrow W_\chi \rightarrow 0,$$ where the second map in the last sequence is an isomorphism. Using the last exact sequence we deduce that $W_\chi=0$. But if $W$ is not trivial it will contain a nonzero Whittaker vector by Proposition \[Whitt\]. Thus $W=0$, and $f$ is surjective. This completes the proof of the theorem
Next we study some further properties of q-W algebras at roots of unity and of the module $Q_\chi$. First we prove the following lemma.
The left $U_{\xi}(\g)$–module $Q_\chi$ is projective in the category $U_{\xi}(\g)-{\rm mod}$.
We have to show that the functor ${\rm Hom}_{U_{\xi}(\g)}(Q_\chi,\cdot)$ is exact. Let $V^\bullet$ be an exact complex of finite–dimensional $U_{\xi}(\g)$–modules. Since by Theorem \[fdfree\] objects of $U_{\xi}(\g)-{\rm mod}$ are $U_{\xi}({\frak m}_-)$–free, and $U_{\xi}({\frak m}_-)$ is Frobenius we have $$V^\bullet=U_{\xi}({\frak m}_-)\otimes \overline{V}^\bullet\simeq U_{\xi}({\frak m}_-)^*\otimes \overline{V}^\bullet,$$ where $\overline{V}^\bullet$ is an exact complex of vector spaces and the action of $U_{\xi}({\frak m}_-)$ on $U_{\xi}({\frak m}_-)^*$ is induced by multiplication from the right on $U_{\xi}({\frak m}_-)$.
Now by Frobenius reciprocity we have obvious isomorphisms of complexes, $$\begin{aligned}
{\rm Hom}_{U_{\xi}(\g)}(Q_\chi,V^\bullet)\simeq{\rm Hom}_{U_{\xi}(\g)}(Q_\chi,U_{\xi}({\frak m}_-)^*\otimes \overline{V}^\bullet)={\rm Hom}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_\chi,U_{\xi}({\frak m}_-)^*\otimes \overline{V}^\bullet)\simeq \\ \simeq{\rm Hom}_\mathbb{C}(U_{\xi}({\frak m}_-)\otimes_{U_{\xi}({\frak m}_-)}\mathbb{C}_\chi,\overline{V}^\bullet)=\overline{V}^\bullet,\end{aligned}$$ where the last complex is exact. Therefore the functor ${\rm Hom}_{U_{\xi}({\frak m}_-)}(Q_\chi,\cdot)$ is exact.
The following proposition is an analogue of Theorem 2.3 in [@Pr] for quantum groups at roots of unity.
Let $\eta\in {\rm Spec}(Z_0)$, $\phi\circ \widetilde{\pi}(\eta)\in \mu_{M_+}^{-1}(u)$ and assume that a quantum coadjoint transformation $\widetilde{g}\in \mathcal{G}$ is fixed as in Theorem \[fdfree\]. Denote $\xi=\widetilde{g}\eta\in {\rm Spec}(Z_0)$ and $d=m^{{\rm dim}~\mathfrak{m}_-}$. Let $\chi$ be a character of $U_{\xi}({\frak m}_-)$, $\mathbb{C}_\chi$ the corresponding representation of $U_{\xi}({\frak m}_-)$. Then $Q_\chi^d\simeq U_{\xi}(\g)$ as left $U_{\xi}(\g)$–modules, $U_{\xi}(\g)\simeq {\rm Mat}_d(W_\varepsilon^s(G))$ as algebras and $Q_\chi\simeq (W_\varepsilon^s(G)^{opp})^d$ as right $W_\varepsilon^s(G)$–modules.
Let $E_i$, $i=1,\ldots ,S$ be the simple finite–dimensional modules over the finite–dimensional algebra $U_{\xi}(\g)$. Denote by $P_i$ the projective cover of $E_i$. Since by Theorem \[fdfree\] the dimension of $E_i$ is divisible by $d$ we have ${\rm dim}~E_i=dr_i$, $r_i\in \mathbb{N}$, where $r_i$ is the rank of $E_i$ over $U_{\xi}({\frak m}_-)$ equal to the dimension of the space of Whittaker vectors in $E_i$. By Proposition 2.1 in [@Pr] $$U_{\xi}(\g)={\rm Mat}_d({\rm End}_{U_{\xi}(\g)}(P)^{opp}),$$ where $P=\bigoplus_{i=1}^SP_i^{r_i}$. Therefore to prove the second statement of the proposition it suffices to show that $P\simeq Q$. Since by the previous lemma $Q_\chi$ is projective we only need to verify that $$r_i={\rm dim}~{\rm Hom}_{U_{\xi}(\g)}(P,E_i)={\rm dim}~{\rm Hom}_{U_{\xi}(\g)}(Q_\chi,E_i).$$
Indeed, by Frobenius reciprocity we have $${\rm dim}~{\rm Hom}_{U_{\xi}(\g)}(Q_\chi,E_i)={\rm dim}~{\rm Hom}_{U_{\xi}({\frak m}_-)}(\mathbb{C}_\chi,E_i)=r_i.$$
This proves the second statement of the proposition. From Proposition 2.1 in [@Pr] we also deduce that $P^d\simeq U_{\xi}(\g)$ as left $U_{\xi}(\g)$–modules. Together with the isomorphism $P\simeq Q$ this gives the first statement of the proposition.
Using results of Section 6.4 in [@Pie] and the fact that $Q_\chi$ is projective one can find an idempotent $e\in U_{\xi}(\g)$ such that $Q_\chi\simeq U_{\xi}(\g)e$ as modules and $W_\varepsilon^s(G)\simeq eU_{\xi}(\g)e$ as algebras.
By the first two statements of this proposition one can also find idempotents $e=e_1,e_2,\ldots ,e_d\in U_{\xi}(\g)$ such that $e_1+\ldots +e_d=1$, $e_ie_j=0$ if $i\neq j$ and $e_iU_{\xi}(\g)=eU_{\xi}(\g)$ as right $U_{\xi}(\g)$–modules. Therefore $e_iU_{\xi}(\g)e=eU_{\xi}(\g)e$ as right $eU_{\xi}(\g)e$–modules, and $$Q_\chi\simeq U_{\xi}(\g)e=\bigoplus_{i=1}^de_iU_{\xi}(\g)e\simeq (eU_{\xi}(\g)e)^d\simeq (W_\varepsilon^s(G)^{opp})^d$$ as right $W_\varepsilon^s(G)$–modules. This completes the proof of the proposition
The algebra $W_\varepsilon^s(G)$ is finite–dimensional, and ${\rm dim}~W_\varepsilon^s(G)=m^{{\rm dim}~\Sigma_s}$.
By Proposition \[var\] $2{\rm dim}~\mathfrak{m}_-+{\rm dim}~\Sigma_s={\rm dim}~G$. Therefore by the definition of $Q_\chi$ we have ${\rm dim}~Q_\chi=m^{{\rm dim}~G-{\rm dim}~\mathfrak{m}_-}=m^{{\rm dim}~\mathfrak{m}_-+{\rm dim}~\Sigma_s}$. Finally from the last statement of the previous theorem one obtains that ${\rm dim}~W_\varepsilon^s(G)={\rm dim}~Q_\chi/m^{{\rm dim}~\mathfrak{m}_-}=m^{{\rm dim}~\Sigma_s}$.
[99]{} Bourbaki, N., Groupes et algebras de Lie, Chap. 4,5,6, Paris, Hermann (1968).
Belavin, A.A., Drinfeld, V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, [*Funct. Anal. Appl.*]{} [**16**]{} (1981), 159-180.
Bezrukavnikov, R., Mirkovic, I., Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic. With an appendix by Bezrukavnikov and Simon Riche, [*Ann. of Math.*]{} [**167**]{} (2008), 945–-991.
Cantarini, N., Carnovale, G., Costantini, M., Spherical orbits and representations of ${U}_\varepsilon({\g})$, [*Transform. Groups*]{} [**10**]{} (2005), 29–-62.
Cantarini, N., The quantized enveloping algebra $U_q(sl(n))$ at the roots of unity, [*Comm. Math. Phys.*]{} [**211**]{} (2000), 207–-230.
Cantarini, N., Spherical orbits and quantized enveloping algebras, [*Comm. Algebra*]{} [**27**]{} (1999), 3439–3458.
Cantarini, N., Mod-p reduction for quantum groups, [*J. Algebra*]{} [**202**]{} (1998), 357–-366.
Carter, R. W., Conjugacy classes in the Weyl group, [*Compositio Math.*]{} [**25**]{} (1972), 1–59.
Chari, V., Pressley, A., A guide to quantum groups, Cambridge Univ. Press (1994).
De Concini, C., Kac, V. G., Representations of quantum groups at roots of $1$. Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), [*Progr. Math.*]{} [**92**]{}, Birkhäuser Boston, Boston, MA (1990), 471–506.
De Concini, C., Kac, V. G., Procesi, C., Quantum coadjoint action, [*J. Amer. Math. Soc.*]{} [**5**]{} (1992), 151–-189.
De Concini, C., Kac, V. G., Representations of quantum groups at roots of 1: reduction to the exceptional case, [*Int. J. Mod. Phys. A*]{} [**7**]{} [em Suppl. 1A]{} (1992), 141–149.
De Concini, C., Kac, V. G., Procesi, C., Some remarkable degenerations of quantum groups, [*Comm. Math. Phys.*]{} [**157**]{} (1993), 405–-427.
Curtis, C. W., Reiner, I., Representation theory of finite groups and associativfe algebras, Interscience Publishers (1962).
Drupieski, C. M., On injective modules and support varieties for the small quantum group, [*Int. Math. Res. Not.*]{} [**2011**]{}, 2263–-2294.
Friedlander, E. M., Parshall, B. J., Support varieties for restricted Lie algebras [*Invent. Math.*]{} [**86**]{} (1986), 553–-562.
Friedlander, E. M., Parshall, B. J., Geometry of $p$-unipotent Lie algebras, [*J. Algebra*]{} [**109**]{} (1987), 25–-45.
Friedlander, E. M., Parshall, B. J., Modular representation theory of Lie algebras, [*Amer. J. Math.*]{} [**110**]{} (1988), 1055–1093.
Ginzburg, V., Kumar, S., Cohomology of quantum groups at roots of unity, [*Duke Math. J.* ]{} [**69**]{} (1993), 179–-198.
He, X., Lusztig, G., A generalization of Steinberg’s cross-section, [*J. Amer. Math. Soc.*]{} [**25**]{} (2012), 739–-757.
Kac, V. G., Weisfeiler, B., The irreducible representations of Lie $p$-algebras, [*Funkcional. Anal. i Priložen.*]{} [**5**]{} (1971), no. 2, 28–-36.
Khoroshkin, S.M., Tolstoy, V.N., Universal R–matrix for quantized (super)algebras, [*Comm. Math. Phys.*]{} [**141**]{} (1991), 599–617.
Khoroshkin, S. M., Tolstoy, V. N., The Cartan-Weyl basis and the universal R-matrix for quantum Kac-Moody algebras and superalgebras. Quantum symmetries (Clausthal, 1991), 336–351, World Sci. Publ., River Edge, NJ (1993).
Kremnizer, K., Proof of the De Concini-Kac-Procesi conjecture, arXiv:math/0611236.
Ostrik, V. V., Cohomological supports for quantum groups, [*Funct. Anal. Appl.*]{} [**32**]{} (1999), 237–-246.
Pierce, R.S., Associative algebras. [*Graduate Texts in Mathematics*]{} [**88**]{} [*Studies in the History of Modern Science*]{} [**9**]{}, Springer-Verlag, New York-Berlin (1982).
Premet, A., Skryabin, S., Representations of restricted Lie algebras and families of associative $\mathcal{L}$-algebras, [*J. Reine Angew. Math.*]{} [**507**]{} (1999), 189–-218.
Premet, A., Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture, [*Invent. Math.*]{} [**121**]{} (1995), 79–-117.
Premet, A., Support varieties of non-restricted modules over Lie algebras of reductive groups, [*J. London Math. Soc.*]{} (2) [**55**]{} (1997), 236–-250.
Premet, A., Special transverse slices and their enveloping algebras. With an appendix by Serge Skryabin, [*Adv. Math.*]{} [**170**]{} (2002), 1–55.
Semenov-Tian-Shansky, M.A., Poisson Lie groups, quantum duality principle and the quantum double, [*Contemporary Math.*]{} [**175**]{}, 219–248.
Sevostyanov, A. [ The Whittaker model of the center of the quantum group and Hecke algebras.]{} PhD thesis, Uppsala (1999).
Sevostyanov, A., Algebraic group analogues of Slodowy slices and deformations of Poisson W–algebras, [*Int. Math. Res. Not.*]{} [ **2011** ]{} (2011), 1880–1925.
Sevostyanov, A., Conjugacy classes in Weyl groups and q-W algebras, [*Adv. Math*]{} [**228**]{} (2011), 1315–1376.
Zelobenko, D.P., Compact Lie groups and their representations, Providence (1973).
Zelobenko, D.P., Extremal cocycles on Weyl groups, [*Funk. Anal. i.ego Pril.*]{} [**21**]{}, no. 3 (1987), 11–21.
|
---
abstract: 'This paper derives a population sizing relationship for genetic programming (GP). Following the population-sizing derivation for genetic algorithms in , it considers building block decision making as a key facet. The analysis yields a GP-unique relationship because it has to account for bloat and for the fact that GP solutions often use subsolutions multiple times. The population-sizing relationship depends upon tree size, solution complexity, problem difficulty and building block expression probability. The relationship is used to analyze and empirically investigate population sizing for three model GP problems named [ ORDER]{}, [ON-OFF]{} and [LOUD]{}. These problems exhibit bloat to differing extents and differ in whether their solutions require the use of a building block multiple times.'
author:
- |
Kumara Sastry\
Illinois Genetic Algorithms Laboratory (IlliGAL), and\
Department of Material Science & Engineering\
University of Illinois at Urbana-Champaign\
[[email protected]]{}
- |
Una-May O’Reilly\
Computer Science & Artificial Intelligence Laboratory\
Massachusetts Institute of Technology, Cambridge, MA, USA\
[[email protected]]{}
- |
David E. Goldberg\
Illinois Genetic Algorithms Laboratory (IlliGAL), and\
Department of General Engineering\
University of Illinois at Urbana-Champaign\
[[email protected]]{}
bibliography:
- 'GPbib.bib'
- 'myBib.bib'
title: Population Sizing for Genetic Programming Based Upon Decision Making
---
\
\
\
\
\
\
\
\
\
\
IlliGAL Report No. 2004028\
April, 2004\
Introduction
============
The growth in application of genetic programming (GP) to problems of practical and scientific importance is remarkable [@keijzer:2004:GP; @RioloWorzel:2003; @GECCO2003-PartI; @GECCO2003-PartII]. Yet, despite this increasing interest and empirical success, GP researchers and practitioners are often frustrated—sometimes stymied—by the lack of theory available to guide them in selecting key algorithm parameters or to help them explain empirical findings in a systematic manner. For example, GP population sizes run from ten to a million members or more, but at present there is no practical guide to knowing when to choose which size.
To continue addressing this issue, this paper builds on a previous paper [@sastry:2003:GPTP] wherein we considered the building block supply problem for GP. In this earlier step, we asked what population size is required to ensure the presence of all raw building blocks for a given tree size (or size distribution) in the initial population. The building-block supply based population size is conservative because it does not guarantee the growth in the market share of good substructures. That is, while ensuring the building-block supply is important for a selecto-recombinative algorithm’s success, ensuring a growth in the market share of good building blocks by correctly deciding between competing building blocks is also critical [@Goldberg:2002:DOI]. Furthermore, the population sizing for GA success is usually bounded by the population size required for making good decisions between competing building blocks. Our results herein show this to be the case, at least for the [ORDER]{} problem.
Therefore, the purpose of this paper is to derive a population-sizing model to ensure good decision making between competing building blocks. Our analytical approach is similar to that used by for developing a population-sizing model based on decision-making for genetic algorithms (GAs). In our population-sizing model, we incorporate factors that are common to both GP and GAs, as well as those that are unique to GP. We verify the populations-sizing model on three different test problem that span the dimension of building block [*expression*]{}—thus, modeling the phenomena of bloat at various degrees. Using [ORDER]{}, with [UNITATION]{} as its fitness function, provides a model problem where, per tree, a building block can be expressed only once despite being present multiple times. At the opposite extreme, our [LOUD]{} problem models a building block being expressed each time it is present in the tree. In between, the [ON-OFF]{} problem provides tunability of building block expression. A parameter controls the frequency with which a ‘function’ can suppress the expression of the subtrees below it, thus effecting how frequently a tree expresses a building block. This series of experiments not only validates the population-sizing relationship, but also empirically illustrates the relationship between population size and problem difficulty, solution complexity, bloat and tree structure.
We proceed as follows: The next section gives a brief overview of past work in developing facetwise population-sizing models in both GAs and GP. In Section \[sec:ga-popsize\], we concisely review the derivation by [@Goldberg:1992:ComplexSystems] of a population sizing equation for GAs. Section \[sec:defns\] provides GP-equivalent definitions of building blocks, competitions (a.k.a partitions), trials, cardinality and building-block size. In Section \[sec:gp-popsize\] we follow the logical steps of [@Goldberg:1992:ComplexSystems] while factoring in GP perspectives to derive a general GP population sizing equation. In Section \[sec:examples\], we derive and empirically verify the population sizes for model problems that span the range variable BB presence and its expressive probability. Finally, section \[sec:conclusions\] summarizes the paper and provides key conclusions of the study.
Background
==========
One of the key achievements of GA theory is the identification of the building-block decision making to be a statistical one [@Holland:1973:SIAM]. illustrated this using a 2$^k$-armed bandit model. Based on Holland’s work, proposed equations for the 2-armed bandit problem without using Holland’s assumption of foresight. He recognized the importance of noise in the decision-making process. He also proposed a population-sizing model based on the signal and noise characteristics of a problem. ’s suggestion went unimplemented till the study by . computed the fitness variance using Walsh analysis and proposed a population-sizing model based on the fitness variance.
A subsequent work [@Goldberg:1992:ComplexSystems] proposed an estimate of the population size that controlled decision-making errors. Their model was based on deciding correctly between the best and the next best BB in a partition in the presence of noise arising from adjoining BBs. This noise is termed as [*collateral noise*]{} [@Goldberg:1991:ComplexSystems]. The model proposed by yielded practical population-sizing bounds for selectorecombinative GAs. More recently refined the population-sizing model proposed by . proposed a tighter bound on the population size required for selectorecombinative GAs. They incorporated both the initial BB supply model and the decision-making model in the population-sizing relation. They also eliminated the requirement that only a successful decision-making in the first generation results in the convergence to the optimum. To eliminate this requirement, they modeled the decision-making in subsequent generations using the well known gambler’s ruin model [@Feller:1970]. extended the population-sizing model for noisy environments and applied it for parallel GAs.
While, population-sizing in genetic algorithms has been successfully studied with the help of facetwise and dimensional models, similar efforts in genetic programming are still in the early stages. Recently, we developed a population sizing model to ensure the presence of all raw building blocks in the initial population size. We first derived the exact population size to ensure adequate supply for a model problem named [ORDER]{}. [ORDER]{} has an expression mechanism that models how a primitive in GP is expressed depending on its spatial context. We empirically validated our supply-driven population size result for [ORDER]{} under two different fitness functions: [UNITATION]{} where each primitive is a building block with uniform fitness contribution, and [DECEPTION]{} where each of $m$ subgroups, each subgroup consisting of $k$ primitives, has its fitness computed using a deceptive trap function.
After dealing specifically with [ORDER]{} in which, per tree, a building block can be expressed at most once, we considered the general case of ensuring an adequate building block supply where every building block in a tree is always expressed. This is analogous to the instance of a GP problem that exhibits no bloat. In this case, the supply equation does not have to account for subtrees that are present yet do not contribute to fitness. This supply-based population size equation is: $$\label{eqn:supply-ps}
n = \frac{1}{{\lambda}{}}2^k\kappa\left(\log\kappa - \log\epsilon\right).$$ where $\kappa$ enumerates the partition or building block competition, $k$ is the building-block size, $\epsilon$ is supply error and [$\lambda$]{} is average tree size.
In the context of supply, to finally address the reality of bloat, we noted that the combined probability of a building block being present in the population and its probability of being expressed must be computed and amalgamated into the supply derivation. This would imply that Equation \[eqn:supply-ps\], though conservative under the assumed condition that every raw building block must be present in the initial population, is an underestimate in terms of accounting for bloat. Overall, the building block supply analysis yielded insight into how two salient properties of GP: building block expression and tree structure influence building block supply and thus influence population size. Building block expression manifests itself in ‘real life’ as the phenomena of bloat in GP. Average tree size in GP typically increases as a result of the interaction of selection, crossover and program degeneracy.
As a next step, this study derives a decision-making based population-sizing model. We employ the methodology of used for deriving a population sizing relationship for GA. In this method, the population size is chosen so that the population contains enough competing building blocks that decisions between two building blocks can be made with a pre-specified confidence. Compared to the GA derivation, there are two significant differences. First, the collateral noise in fitness, arises from a variable quantity of expressed BBs. Second, the number of trials of a BB, rather than one per individual in the GA case, depends on tree structure and whether a BB that is present in a tree is expressed. In the GP case, the variable, $\kappa$ related to cardinality (e.g. the binary alphabet of a simple GA) and building block defining length, is considerably larger because GP problems typically use larger primitive sets. It is incorporated into the relationship by considering BB expression and presence.
Before presenting the decision-making model for GP, we briefly discuss the population-sizing model of in the following section.
GA Population Sizing from the Perspective of Competing Building Blocks {#sec:ga-popsize}
======================================================================
The derivational foundation for our GP population sizing equation is the 1992 result for the selecto-recombinative GA by [@Goldberg:1992:ComplexSystems] entitled “Genetic Algorithms, Noise and the Sizing of Populations”. The paper considers how the GA can derive accurate estimates of BB fitness in the presence of detrimental noise. It recognizes that, while selection is the principal decision maker, it distinguishes among individuals based on fitness and not by considering BBs. Therefore, there is a possibility that an inferior BB gets selected over a better BB in a competition due to noisy observed contributions from adjoining BBs that are also engaged in competitions.
To derive a relation for the probability of deciding correctly between competing BBs, the authors considered two individuals, one with the best BB and the other with the second best BB in the same competition. [@Goldberg:1992:ComplexSystems].
Let $i_1$ and $i_2$ be these two individuals with $m$ non-overlapping BBs of size $k$ as shown in figure \[fig:bbdecision\]. Individual $i_1$ has the best BB, $H_{1}$ ($111\cdots 111$ in figure \[fig:bbdecision\]) and individual $i_2$ has the second best BB, $H_{2}$ ($000\cdots 000$ in figure \[fig:bbdecision\]). The fitness values of $i_1$ and $i_2$ are $f_{H_1}$ and $f_{H_2}$ respectively. To derive the probability of correct decision making, we have to first recognize that the fitness distribution of the individuals containing $H_{1}$ and $H_{2}$ is Gaussian since we have assumed an additive fitness function and the central limit theorem applies. Two possible fitness distributions of individuals containing BBs $H_{1}$ and $H_{2}$ are illustrated in figure \[fig:bbfitdist\].
The distance between the mean fitness of individuals containing $H_1$, $\overline{f}_{H_1}$, and the mean fitness of individuals containing $H_2$, $\overline{f}_{H_2}$, is the [*signal*]{}, $d$. That is $$d = \overline{f}_{H_1} - \overline{f}_{H_2}.$$
Recognize that the probability of correctly deciding between $H_1$ and $H_2$ is equivalent to the probability that $f_{H_1}-f_{H_2} >
0$. Also, since $f_{H_1}$ and $f_{H_2}$ are normally distributed, $f_{H_1}-f_{H_2}$ is also normally distributed with mean $d$ and variance $\sigma^2_{H_1} + \sigma^2_{H_2}$, where $\sigma^2_{H_1}$ and $\sigma^2_{H_2}$ are the fitness variances of individuals containing $H_1$ and $H_2$ respectively. That is, $$f_{H_1}-f_{H_2} \sim {\mathcal{N}}(d,\sigma^2_{H_1}+\sigma^2_{H_2}).$$ The probability of correct decision making, $p_{dm}$, is then given by the cumulative density function of a unit normal variate which is the signal-to-noise ratio : $$p_{dm} = \Phi\left({d \over \sqrt{\sigma^2_{H_1} + \sigma^2_{H_2}}}\right).$$ Alternatively, the probability of making an error on a single trial of each BB can estimated by finding the probability $\alpha$ such that $$z^2(\alpha) = \frac{d^2}{\sigma^2_{H_1} + \sigma^2_{H_2}}$$ where $z(\alpha)$ is the ordinate of a unit, one-sided normal deviate. Notationally $z(\alpha)$ is shortened to $z$.
Now, consider the BB variance, $\sigma^2_{H_1}$ (and $\sigma^2_{H_2}$): since it is assumed the fitness function is the sum of $m$ independent subfunctions each of size $k$, $\sigma^2_{H_1}$ (and similarly $\sigma^2_{H_2}$) is the sum of the variance of the adjoining $m-1$ subfunctions. Also, since it is assumed that the $m$ partitions are uniformly scaled, the variance of each subfunction is equal to the average BB variance, $\sigma^2_{bb}$. Therefore, $$\mbox{GA BB Variance:\hspace{0.25in}}
\sigma^2_{H_1} = \sigma^2_{H_2} = (m-1)\sigma^2_{bb}.\label{eqn:ga-bb-variance}$$ A population-sizing equation was derived from this error probability by recognizing that as the number of trials, ${\tau}{}$, increases, the variance of the fitness is decreased by a factor equal to the trial quantity: $$z^2(\alpha) = \frac{d^2}{\frac{(m-1)\sigma_{bb}}{{\tau}{}}}$$
To derive the quantity of trials, ${\tau}{}$, assume a uniformly random population (of size $n$). Let $\chi$ represent the cardinality of the alphabet (2 for the GA) and $k$ the building-block size. For any individual, the probability of $H_1$ is $1/\kappa$ where $\kappa =
\chi^{k}$. There is exactly one instance per individual of the competition, $\phi = 1$. Thus,
$$\label{eqn:trials}
{\tau}{} = n \cdot p_{BB} \cdot \phi = n \cdot 1/\kappa \cdot 1 = n/\kappa$$
By rearrangement and calling $z^2$ the coefficient $c$ (still a function of $\alpha$) a fairly general population-sizing relation was obtained: $$n = 2c\chi^k (m-1)\frac{\sigma^2_{bb}}{d^2}\label{eqn:ga-popsize}$$ To summarize, the decision-making based population sizing model in GAs consists of the following factors:
- [**Competition complexity**]{}, quantified by the total number of competing building blocks, $\chi^k$.
- [**Subcomponent Complexity**]{}, quantified by the number of building blocks, $m$.
- [**Ease of decision making**]{}, quantified by the signal-to-noise ratio, $d/\sigma^2_{bb}$.
- [**Probabilistic safety factor**]{}, quantified by the coefficient $c$.
GP Definitions for a Population Sizing Derivation {#sec:defns}
=================================================
Most GP implementations reported in the literature use parse trees to represent candidate programs in the population [@langdon:fogp]. We have assumed this representation in our analysis. To simplify the analysis further, we consider the following:
1. A primitive set of the GP tree is $\mathcal{F} \cup \mathcal{T}$ where $\mathcal{F}$ denotes the set of functions (interior nodes to a GP parse tree) and $\mathcal{T}$ denotes the set of terminals (leaf nodes in a GP parse tree).
2. The cardinality of $\mathcal{F} = \chi_{f}$ and the cardinality of $\mathcal{T} =\chi_{t}$.
3. The arity of all functions in the primitive set is two: All functions are binary and thus the GP parse trees generated from the primitive set are binary.
We believe that our analysis could be extended to primitive sets containing functions with arity greater than two (non-binary trees). We also note that our assumption closely matches a common GP benchmark, symbolic regression, which frequently has arithmetic functions of arity two.
As in our BB supply paper [@sastry:2003:GPTP], our analysis adopts a definition of a GP schema (or similarity template) called a “tree fragment”. A tree fragment is a tree with at least one leaf that is a “don’t care” symbol. This “don’t care” symbol can be matched by any subtree (including degenerate leaf only trees). As before, we are most interested in only the small set set of tree fragments that are defined by three or fewer nodes. See Figure \[fig:partitions\] for this set.
The defining length of a tree fragment is the sum of its quantities of function symbols, $\mathcal{F}$, and terminal symbols, $\mathcal{T}$: $$k = N_f + N_t$$ Because a tree fragment is a similarity template, it also represents a competition. Since this paper is concerned with decision making, we will therefore use “competition” instead of a “tree fragment”. The size of a competition (i.e. how many BBs compete) is $$\kappa = \chi_{f}^{N_f}*\chi_{t}^{N_t}
\label{NumCompetSchema}$$ As mentioned in [@sastry:2003:GPTP], because a tree fragment is defined without any positional anchoring, it can appear multiple times in a single tree. We denote the number of instances of a tree fragment that are present in a tree of size ${\lambda}{}$, (a.k.a the quantity of a tree fragment in a tree) as $\phi$. This is equivalent to the instances of a competition as $\phi$ is used in the GA case (see Equation \[eqn:trials\]). For full binary trees: $$\phi \approx 2^{-k}{\lambda}{}$$ Later, we will explain how $\phi$ describes [*potential*]{} quantity of per tree” of a BB.
GP Population Sizing based on Decision Making {#sec:gp-popsize}
=============================================
We now proceed to derive a GP population sizing relationship based on building block decision making. Preliminarily, unless noted, we make the same assumptions as the GA derivation of Section \[sec:ga-popsize\].
The first way the GP population size derivation diverges from the GA case is how BB fitness variance (i.e. $\sigma^2_{H_1}$ and $\sigma^2_{H_2}$) is estimated (for reference, see Equation \[eqn:ga-bb-variance\]). Recall that for the GA the source of a BB’s fitness variance was collateral noise from the $ (m-1)$ competitions of its adjoining BBs. In GP, the source of collateral noise is the average number of adjoining BBs present and expressed in each tree, denoted as $\bar{q}$. Thus: $$\mbox{GP BB Variance:\hspace{0.25in}}
\sigma^2_{H_1} = \sigma^2_{H_2} = [\bar{q}_{BB}^{expr}(m,{\lambda}{}) -1]\sigma^2_{bb}.
\label{eqn:gp-bb-variance}$$
Thus, the probability of making an error on a single trial of the BB can be estimated by finding the probability $\alpha$ such that $$z^2(\alpha) = \frac{d^2}{2[\bar{q}_{BB}^{expr} -1]\sigma^2_{bb}}$$
The second way the GP population size derivation diverges from the GA case is in how the number of trials of a BB is estimated (for reference, see Equation \[eqn:trials\]). As with the GA, for GP we assume a uniformly distributed population of size $n$. In GP the probability of a trial of a particular BB must account for it being both present, $1/\kappa$, [*and*]{} expressed in an individual (or tree), which we denote as $p_{BB}^{expr}$. So, in GP: $$\label{eqn:gp-alpha}
{\tau}{} = {1\over\kappa} \cdot p_{BB}^{expr} \cdot \phi \cdot n$$
Thus, the population size relationship for GP is: $$n = 2c\frac{\sigma^2_{bb}}{d^2}\kappa\left[\bar{q}_{BB}^{expr} -1\right]
\frac{1}{p_{BB}^{expr}\phi}
\label{eqn:gp-popsize}$$ where $c = z^2(\alpha)$ is the square of the ordinate of a one-sided standard Gaussian deviate at a specified error probability $\alpha$. For low error values, $c$ can be obtained by the usual approximation for the tail of a Gaussian distribution: $\alpha \approx
\exp(-c/2)/(\sqrt{2c})$.
Obviously, it is not always possible to factor the real-world problems in the terms of this population sizing model. A practical approach would first approximate $\phi = 2^{-k}({\lambda}{})$ trials per tree (the full binary tree assumption). Then, estimate the size of the shortest program that will solve the problem, (one might regard this as the Kolomogorov complexity of the problem, ${\lambda}{}_k$), and choose a multiple of this for ${\lambda}{}$ in the model. In this case, $\bar{q} = c_k m_k$. To ensure an initial supply of building blocks that is sufficient to solve the problem, the initial population should be initialized with trees of size ${\lambda}{}$. Therefore, the population sizing in this case can be written as $$n = c\frac{\sigma^2_{bb}}{d^2}\kappa{\left(c_k m_k -1\right)2^{k+1} \over p_{BB}^{expr}{\lambda}{}}
\label{eqn:gp-popsize1}$$
Similar to the GA population sizing model, the decision-making based population sizing model in GP consists of the following factors:
- [**Competition complexity**]{}, quantified by the total number of competing building blocks, $\kappa$.
- [**Ease of decision making**]{}, quantified by the signal-to-noise ratio, $d/\sigma^2_{bb}$.
- [**Probabilistic safety factor**]{}, quantified by the coefficient $c$.
- [**Number of subcomponents**]{}, which unlike GA population-sizing, depends not only on the minimum number of building blocks, required to solve the problem $m_k$, but also tree size ${\lambda}{}$, the size of the problem primitive set and how bloat factors into trees. (quantified by $p_{BB}^{expr}$).
Sizing Model Problems {#sec:examples}
=====================
This section derives the components of the population-sizing model (Equation \[eqn:gp-popsize\]) for three test problems, [ORDER]{}, [LOUD]{}, and [ON-OFF]{}. We develop the population-sizing equation for each of theses problems and verify them with empirical results. In all experiments we assume that $\alpha = 1/m$ and thus derive $c$. Table \[tab:mandc\] shows some of these values. For all empirical experiments the the initial population is randomly generated with either full trees or by the ramped half-and-half method. The trees were allowed to grow up to a maximum size of 1024 nodes. We used a tournament selection with tournament size of 4 in obtaining the empirical results. We used subtree crossover with a crossover probability of 1.0 and retained 5% of the best individuals from the previous population. A GP run was terminated when either the best individual was obtained or when a predetermined number of generations were exceeded. The average number of BBs correctly converged in the best individuals were computed over 50 independent runs. The minimum population size required such that $m-1$ BBs converge to the correct value is determined by a bisection method [@Sastry:2002:Masters]. That is the error tolerance, $\alpha =
1/m$. The results of population size and convergence time was averaged over 30 such bisection runs, while the results for the number of function evaluations was averaged over 1500 independent runs. We start with population sizing for [ORDER]{}, where a building block can be expressed at most once in a tree.
$m$ 8 16 32 64 128
----- ----- ------ ------ ------ ------
$c$ .97 1.76 2.71 3.77 4.89
: Values of $c = z^2(\alpha)$ used in population sizing equation.
\[tab:mandc\]
[ORDER]{}: At most one expression per building block per tree
-------------------------------------------------------------
[ORDER]{} is a simple, yet intuitive expression mechanism which makes it amenable to analysis and modeling [@goldberg:1998:good; @oreilly:1998:fssaGP]. The primitive set of [ORDER]{} consists of the primitive [JOIN]{} of arity two and complimentary primitive pairs $\left(X_i,\bar{X}_i\right)$, $i =
0,1,\cdots,m$ of arity one. A candidate solution of the [ORDER]{} problem is a binary tree with [JOIN]{} primitive at the internal nodes and either $X_i$’s or $\bar{X}_i$’s at its leaves. The candidate solution’s expression is determined by parsing the program tree inorder (from left to right). The program expresses the value $X_i$ if, during the inorder parse, a $X_i$ leaf is encountered before its complement $\bar{X}_i$. Furthermore, only unique primitives are expressed in [ORDER]{} during the inorder parse.
For each $X_i$ (or $\bar{X}_i$) that is expressed, an equal unit of fitness value is accredited. That is, $$f_1(x_i) = \left\{ \begin{array}{ll} 1& {\mathrm{if}}~x_i \in
\{X_1,X_2,\cdots,X_m\}\\
0 & otherwise\end{array} \right. .$$ The fitness function for [ORDER]{} is then defined as $$F({\mathbf{x}}) = \sum_{i = 1}^{m} f_1\left(x_i\right),$$ where ${\mathbf{x}}$ is the set of primitives expressed by the tree. The output for optimal solution of a $2m$-primitive [ORDER]{} problem is $\{X_1,X_2,\cdots,X_m\}$, and its fitness value is $m$. The building blocks in [ORDER]{} are the primitives, $X_i$, that are part of the subfunctions that reduce error (alternatively improve fitness). The shortest perfect program is ${\lambda}{}_k=2m-1$.
For example, consider a candidate solution for a 4-primitive [ ORDER]{} problem as shown in figure \[fig:treeOrder\]. The sequence of leaves for the tree is $\{X_1,$ $\bar{X}_1,$ $\bar{X}_1,$ $X_4,$ $X_1,$ $\bar{X}_2\}$, the expression during inorder parse is $\{X_1,$ $\bar{X}_2,$ $X_4\}$, and its fitness is 2. For more details, motivations, and analysis of the [ORDER]{} problem, the interested reader should refer elsewhere [@goldberg:1998:good; @oreilly:1998:fssaGP].
For the [ORDER]{} problem, we can easily see that $\sigma^2_{bb} =
0.25$, $d = 1$, and $\phi = 1$. From , we know that $$\label{eqn:pbbExpOrder}p_{BB}^{expr} \approx \exp\left[-k\cdot e^{-{{\lambda}{}\over 2m}}\right].$$
Additionally, for [ORDER]{}, $\bar{q}_{BB}^{expr}$ is given by $$\label{eqn:nbbExpOrder} \bar{q}_{BB}^{expr} = 1 + \sum_{i =
0}^{m-1}\left(\begin{array}{c}m-1\\i\end{array}\right) i \sum_{j =
0}^{i}\left(\begin{array}{c}i\\j\end{array}\right)(-1)^j\left({i-j+1
\over m}\right)^{n_l-1},$$ where, $n_l$ is the average number of leaf nodes per tree in the population. The derivation of the above equation was involved and detailed. It is provided in Appendix \[sec:orderDERIVATION\]).
Substituting the above relations (Equations \[eqn:pbbExpOrder\] and \[eqn:nbbExpOrder\]) in the population-sizing model (Equation \[eqn:gp-popsize\]) we obtain the following population-sizing equation for [ORDER]{}: $$\label{eqn:popSizeOrder}
n = 2^{k-1} z^2(\alpha)\left({\sigma^2_{bb} \over
d^2}\right)\left[\bar{q}_{BB}^{expr}
-1\right]\exp\left[k\cdot e^{-{{\lambda}{}\over 2m}}\right].$$
The above population-sizing equation is verified with empirical results in Figure \[fig:OrderPopSize\]. The initial population was randomly generated with either full trees or by the ramped half-and-half method with trees of heights, $h \in
[h_k-1,h_k+1]$, where, $h_k$ is the minimum tree height with an average of $2m$ leaf nodes.
As shown in Figure \[fig:OrderConvTime\], we empirically observed that the convergence time and the number of function evaluations scale linearly and cubically with the program size of the most compact solution, ${\lambda}{}_k$, respectively. From this empirical observation, we can deduce that the population size for [ORDER]{} scales quadratically with the program size of the most-compact solution. For [ORDER]{}, ${\lambda}{}_k = 2m -1$.
To summarize for the [ORDER]{} problem, where a building block is expressed at most once per individual, the population size scales as $n =
{\mathcal{O}}\left(2^k{\lambda}{}_k^2\right)$, the convergence time scales as $t_c = {\mathcal{O}}\left({\lambda}{}_k\right)$, and the total number of function evaluations required to obtain the optimal solution scales as $n_{fe} = {\mathcal{O}}\left(2^k{\lambda}{}_k^3\right)$.
[LOUD]{}: Every building block in a tree is expressed
-----------------------------------------------------
In [ORDER]{}, a building block could be expressed at most once in a tree, however, in many GP problems a building block can be expressed multiple times in an individual. Indeed, an extreme case is when every building block occurrence is expressed. One such problem is a modified version of a test problem proposed by (see also [@soule:2002:GPEM; @soule:2003:GPTP]), which we call as [LOUD]{}.
In [LOUD]{}, the primitive set consists of an “add” function of arity two, and three constant terminal 0, 1 and 4. The objective is to find an optimal number of fours and ones. That is, for an individual with $i$ 4s and $j$ 1s, the fitness function is given by $$F({\mathbf{x}}) = \left|i - m_4\right| + \left|j - m_1\right|$$ Therefore, even though a zero is expressed it does not contribute to fitness. Furthermore, a 4 or 1 is expressed each time it appears in an individual and each occurrence contributes to the fitness value of the individual. Moreover, the problem size, $m = m_4 + m_1$ and${\lambda}_k = 2m-1$ .
For the [LOUD]{} problem the building blocks are “4” and “1”. It is easy to see that for [LOUD]{}, $\sigma^2_{BB} = 0.25$, $d = 1$, $\phi = {\lambda}/2$, and $p_{BB}^{expr} = 1/3$. Furthermore, the average number of building blocks expressed is given by $\bar{q}_{BB}^{expr} =
2n_l/3 \approx {\lambda}{}/3$. Substituting these values in the population-sizing model (Equation \[eqn:gp-popsize\]) we obtain $$\label{eqn:loudPopSize}n = 2\cdot3^{k} z^2(\alpha)\left({\sigma^2_{bb} \over
d^2}\right)\left[\frac{1}{3}{\lambda}{} -1\right] \cdot \left({2\over {\lambda}{}}\right).$$
The above population-sizing equation is verified with empirical results in Figure \[fig:loudPopSize\]. The initial population was randomly generated by the ramped half-and-half method with trees of heights, $h \in [2,7]$ yielding an average tree size of 4.1 (this value is analytically 4.5).
We empirically observed that the convergence time was constant with respect to the problem size, and the number of function evaluations scales sub-linearly with the program size of the most-compact solution, ${\lambda}{}_k$. From this empirical observation, we can deduce that the population size for [LOUD]{} scales sub-linearly with the program size of the most-compact solution. For [LOUD]{} ${\lambda}{}_k = 2m-1$.
To summarize for the [LOUD]{} problem, where a building block is expressed each time it occurs in an individual, the population size scales as $n = {\mathcal{O}}\left(3^k{\lambda}{}_k^{0.5}\right)$, the convergence time is almost constant with the problem size, and, and the total number of function evaluations required to obtain the optimal solution scales as $n_{fe} =
{\mathcal{O}}\left(3^k{\lambda}{}_k^{0.5}\right)$.
[ON-OFF]{}: Tunable building block expression
---------------------------------------------
In the previous sections we considered two extreme cases, one where a building block could be expressed at most once in an individual and the other where every building block occurrence is expressed. However, usually in GP problems, some of the building blocks are expressed and others are not. For example, a building block in a non-coded segment is neither expressed nor contributes to the fitness. Empirically, [@luke:2000:cgnci] calculates the percentage of inviable nodes in runs of the 6 and 11 bit multiplexer problems and symbolic regression over the course of a run. This value is seen to vary between problems and change over generations. Therefore, the third test function, which we call [ON-OFF]{}, is one in which the probability of a building block being expressed is tunable.
In [ON-OFF]{}, the primitive set consists of two functions [EXP]{} and $\overline{{\mathtt{EXP}}}$ of arity two and terminal [X]{}$_1$, and [X]{}$_2$. The function [EXP]{} expresses its child nodes, while $\overline{{\mathtt{EXP}}}$ suppresses its child nodes. Therefore a leaf node is expressed only when all its parental nodes have the primitive [EXP]{}. This function can potentially approximate some bloat scenarios of standard GP problems such as symbolic-regression and multiplexer problems where invalidators are responsible for nullifying a building block’s effect [@luke:2000:cgnci]. The probability of expressing a building block can be tuned by controlling the frequency of selecting [EXP]{} for an internal node in the initial tree.
Similar to [LOUD]{}, the objective for [ON-OFF]{} is to find an optimal number of fours and ones. That is, for an individual with $i$ [X]{}$_1$s and $j$ [X]{}$_2$s, the fitness function is given by $$F({\mathbf{x}}) = \left|i - m_{X_1}\right| + \left|j - m_{X_2}\right|$$ The problem size, $m = m_{X_1} + m_{X_2}$ and ${\lambda}{}_k = 2m -1$.
For example, consider a candidate solution for the [LOUD]{} problem as shown in figure \[fig:treeOnOff\]. The terminals that are expressed are $\{X_1$, $X_1$, $X_1$, $X_2\}$ and the fitness is given by $|3-m_{x_1}| + |1 - m_{x_2}|$.
For the [ON-OFF]{} problem the building blocks are $X_1$ and $X_2$, $\sigma^2_{BB} = 0.25$, $d = 1$, $\phi = {\lambda}{}/2$, and $p_{BB}^{expr} =
p_{EXP}^h$. Here, $p_{EXP}$ is the probability of a node being the primitive [EXP]{}. The average number of building blocks expressed is given by $\bar{q}_{BB}^{expr} = n_l\cdot p_{EXP}^h \approx
\frac{s}{2}\cdot p_{EXP}^h$. Substituting these values in the population-sizing model (Equation \[eqn:gp-popsize\]) we obtain $$\label{eqn:onOffPopSize}n = 2^{k+1} z^2(\alpha)\left({\sigma^2_{bb} \over
d^2}\right)\left[\frac{{\lambda}{}}{2}p_{EXP}^h -1\right] \cdot \left({2\over {\lambda}{} p_{EXP}^h}\right).$$
The above population-sizing equation is verified with empirical results in Figure \[fig:onOffPopSize\]. The initial population was randomly generated by the ramped half-and-half method with trees of heights, $h \in [h_k-1,h_k+1]$, where $h_k$ is the minimum tree height with an average of $m$ leaf nodes. We empirically observed that the convergence time was linear with respect to the problem size, and the number of function evaluations scales sub-quadratically with the program size of the most-compact solution, ${\lambda}{}_k$. From this empirical observation, we can deduce that the population size for [On-Off]{} scales sub-linearly with the program size of the most-compact solution (${\lambda}{}_k = 2m-1$).
To summarize for the [On-Off]{} problem, where a building block expression is tunable, the population size scales as $n =
{\mathcal{O}}\left(2^k{\lambda}{}_k^{0.5}/p_{exp}\right)$, the convergence time scales linearly as $t_c =
{\mathcal{O}}\left(2^k{\lambda}{}_k/p_{exp}\right)$, and the total number of function evaluations required to obtain the optimal solution scales as $n_{fe} = {\mathcal{O}}\left(2^k{\lambda}{}_k^{1.5}/p_{exp}^2\right)$.
Conclusions {#sec:conclusions}
===========
This contribution is a second step towards a reliable and accurate model for sizing genetic programming populations. In the first step the model estimated the minimum population size required to ensure that every building block was present with a given certainty in the initial population. We accepted this conservative model (i.e. it oversized the population) because in the process of deriving it, we gained valuable insight into a) what makes GP different from a GA in the sizing context and b) the implications of these differences. The difference of GP’s larger alphabet, while influential in implying GP needs larger population sizes, was not a difficult factor to handle while bloat and the variable length individuals in GP are more complicated.
Moving to the second step, by considering a decision making model (which is less conservative than the BB supply model), we extended the GA decision making model along these dimensions: first, our model retains a term describing collateral noise from competing BBs ($\bar{q}[m,{\lambda}{}]$) but it recognizes that the quantity of these competitors depends on tree size and the likelihood that the BB is present and expresses itself (rather than behave as an intron). Second, our model, like its GA counterpart, assumes that trials decrease BB fitness variance, however, what was simple in a GA – there is one trial per population member, for the GP case is more involved. That is, the probability that a BB is present in a population member depends both on the likelihood that it is present in lieu of another BB [*and*]{} expresses itself, [*plus*]{} the number of potential trials any BB has in each population member.
The model shows that, to ensure correct decision making within an error tolerance, population size must go up as the probability of error decreases, noise increases, alphabet cardinality increases, the signal-to-noise ratio decreases [*and*]{} tree size decreases and bloat frequency increases. This obviously matches intuition. There is an interesting critical trade-off with tree size with respect to determining population size: pressure for larger trees comes from the need to express all correct BBs in the solution while pressure for smaller trees comes from the need to reduce collateral noise from competing BBs.
The model is conservative because “it assumes that decisions are made irrevocably during any given generation. It sizes the population to ensure that the correct decision is made on average in a single generation” [@Goldberg:2002:DOI]. In this way, it is similar to the Schema Theorem. A more accurate and different model would account for how correct decision making accumulates over the course of a run, and how, over the course of a run, improper decision making can be rectified.
The fact that the model is based on statistical decision making means that crossover does not have to be incorporated. In GAs crossover solely acts as a mixer or combiner of BBs. Interestingly, in GP, crossover also interacts with selection with the potential result that programs’ size grows and structure changes. When this happens, the frequency of bloat can also change (see [@luke:2000:cgnci; @luke:dissertation] for examples of this with multiplexer and symbolic regression). These changes in size, structure and bloat frequency imply a much more complex model if one were to attempt to account for decision making throughout a run. They also suggest that when using the model as a rule of thumb to size an initial population it may prove more accurate if the practitioner overestimates bloat in anticipation of subsequent tree growth causing more than the bloat seen in the initial population, given its average tree size.
It appears difficult to use this model with real problems where, among the GP particular factors, the most compact solution and BB size is not known and the extent of bloat can not be estimated. In the case of the GA model, the estimation of model factors has been addressed by [@Reed:2000:WRR]. They estimated variance with the standard deviation of the fitness of a large random population. In the GP case, this sampling population should be controlled for average tree size. If a practitioner were willing to work with crude estimates of bloat, BB size and most compact solution size, a multiple of the size of the most compact solution could be plugged in, and bloat could be used with that size to estimate the probability that a BB is expressed and present and the average number of BBs of the same size present and expressed, on average, in each tree. In the future we intend to experiment with the model and well known toy GP problems (e.g. multiplexer, symbolic regression) where bloat frequency and most compact problem size are obtainable, and simple choices for BB size exist to see whether the ideal population size scales with problem size within the order of complexity the model predicts.
Population sizing has been important to GAs and is now important to GP, because it is the principle factor in controlling ultimate solution quality. Once the quality-size relation is understood, populations can be sized to obtain a desired quality and only two things can happen in empirical trials. The quality goal can be equaled or exceeded in which case, all is well with the design of the algorithm, or (as is more likely) the quality target can be missed, in which case there is some other obstacle to be overcome in the algorithm design. Moreover, once population size is understood in this way it can be combined with an understanding of run duration (citation), thereby yielding first estimates of GP run complexity, a key milestone in making our understanding of these processes more rigorous.
Acknowledgments {#acknowledgments .unnumbered}
===============
We gratefully acknowledge the organizers and reviewers of the 2004 GP Theory and Practice Workshop.
This work was sponsored by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant F49620-00-0163 and F49620-03-1-0129, the National Science Foundation under ITR grant DMR-99-76550 (at Materials Computation Center), and ITR grant DMR-0121695 (at CPSD), and the Dept. of Energy under grant DEFG02-91ER45439 (at Fredrick Seitz MRL). The U.S. Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation thereon.
The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research, the National Science Foundation, or the U.S. Government.
Derivation of the Average Number of Expressed Building Blocks for the Problem {#sec:orderDERIVATION}
=============================================================================
The following derivation provides expression for the average number of expressed building blocks (BBs) (best or second best) in other partitions, given that a best BB or second best BB is already expressed in a particular partition. For example, I assume that either $X_1$ or $\bar{X}_1$ is expressed in a tree. Therefore the total number of leaf nodes available to potential express other BBs is $n_l - 1$.
Given that the problem has $m$ building blocks, the total number of terminals, $\chi_t = 2m$ (Recall that the terminal set, ${\mathcal{T}}
\equiv \{X_1, \bar{X}_1, X_2, \bar{X}_2, \cdots, X_m,
\bar{X}_m\}$). Therefore, the total possible terminal sequences, given $n_l - 1$ leaf nodes, $N_{{\mathrm{tot}}}$, is $$N_{{\mathrm{tot}}} = \left(2m\right)^{n_l-1}.$$
The number of building blocks that expressed in $n_l -1$ nodes vary from 0 to $m-1$ (note that we assume that one building block is already expressed). That is, if either $X_1$ or $\bar{X}_1$ are present in the remaining $n_l-1$ leaf nodes, the number of expressed building blocks other than $X_1$ or $\bar{X}_1$ is zero. Similarly if there is at least one copy of one of the $m-1$ complementary primitives present in $n_l-1$ leaf nodes, then the number of BBs expressed other than $X_1$ or $\bar{X}_1$ is $m-1$. For brevity, in the reminder of this report, the number of expressed BBs refer to only the BBs expressed in $n_l-1$ leaf nodes.
Before proceeding with the derivation itself, we develop few identities that will be used throughout the derivation. $$\label{eqn:identity1} \sum_{j = 0}^{n}
\left(\begin{array}{c}n\\j\end{array}\right) = 2^n$$
$$\begin{aligned}
\nonumber
\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)a^{n-j} &=&
a^n\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)\left({1\over
a}\right)^j\\
\nonumber &=&
a^n\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)\left({1\over
a}\right)^j\cdot 1^{n-j}\\
\nonumber &=&
a^n\left(1 + \frac{1}{a}\right)^n\\
\label{eqn:identity2} \sum_{j = 0}^{n}
\left(\begin{array}{c}n\\j\end{array}\right)a^{n-j} &=& (a+1)^n\end{aligned}$$
where $a \ge 2$ is an integer.
$$\begin{aligned}
\nonumber \sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)j
&=&
2^{n}\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)j\left(\frac{1}{2}\right)^j\left(\frac{1}{2}\right)^{n-j}\\
\nonumber &=& 2^n\left[n\cdot\frac{1}{2}\right]\\
\label{eqn:identity3} \sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)j
&=& n\cdot 2^{n-1}\end{aligned}$$
$$\begin{aligned}
\nonumber \sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)j^2
&=&
2^{n}\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)j^2\left(\frac{1}{2}\right)^j\left(\frac{1}{2}\right)^{n-j}\\
\nonumber &=& 2^n\left[\sigma^2_{\mathrm{Binomial}} +
\mu^2_{\mathrm{Binomial}}\right]\\
\nonumber &=& 2^n\left[n\cdot\frac{1}{2}\cdot\frac{1}{2} + n^2\cdot\frac{1}{4}\right]\\
\label{eqn:identity4} \sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)j^2
&=& n\cdot(n+1)\cdot 2^{n-2}\end{aligned}$$
$$\begin{aligned}
\nonumber
\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)ja^{n-j}
\nonumber &=&(a+1)^n
\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)j\left(\frac{1}{a+1}\right)^j\left(\frac{a}{a+1}\right)^{n-j}\\
\nonumber &=&
(a+1)^n\left[\mu_{\mathrm{Binomial}}\left(n,\frac{1}{a+1}\right)\right]\\
\nonumber &=& (a+1)^n\left[n\cdot\frac{1}{a+1}\right] \\
\label{eqn:identity5}
\sum_{j=0}^{n}\left(\begin{array}{c}n\\j\end{array}\right)ja^{n-j} &=& n\cdot(a+1)^{n-1} \end{aligned}$$
Here again, $a \ge 2$ is an integer.
Number of expressed BBs = 0.
: The number of ways either $X_1$ or $\bar{X}_1$ is present in $n_l - 1$ nodes is $$\begin{aligned}
N\left(n_{{\mathrm{BB}}}^{\exp} = 0\right) &=& \sum_{j
= 0}^{n_l-1} {\left(n_l-1\right)! \over j!\left(n_l-1-j\right)!}\\
&=& \sum_{j = 0}^{n_l-1} \left(\begin{array}{c}n_l-1\\ i\end{array}\right)\\
&=& 2^{n_l-1}\end{aligned}$$
Number of expressed BBs = 1.
: Here the terminals that can be present in the $n_l-1$ nodes are $X_1$ or $\bar{X_1}$ or exactly one of the other complementary pairs. Therefore, we begin by counting the number of ways of having at least one copy of either $X_2$ or $\bar{X}_2$ in $n_l-1$ nodes. In other words, we count the number of ways in which only $X_2$ or its complement, $\bar{X}_2$ can be expressed. $$\begin{aligned}
\nonumber &~& N\left({\mathrm{Terminals}}~{\mathrm{present}} =
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2\right)\\
&=& \sum_{j = 0}^{n_l-2}\sum_{k = 0}^{n_l-2-j}\sum_{q = 0}^{n_l-1-j-k} {\left(n_l-1\right)! \over j!k!q!\left(n_l-1-j-k-q\right)!}\\
&=& \sum_{j = 0}^{n_l-2} \left(\begin{array}{c}n_l-1\\
j\end{array}\right)\sum_{k =
0}^{n_l-2-j}\left(\begin{array}{c}n_l-1-j
\\k\end{array}\right)\sum_{q =
0}^{n_l-1-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
&=& \sum_{j = 0}^{n_l-2}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\sum_{k =
0}^{n_l-2-j}\left(\begin{array}{c}n_l-1-j\\
k\end{array}\right)2^{n_l-1-j-k}\\
\label{eqn:midStep0}&=& \sum_{j = 0}^{n_l-2}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[\sum_{k =
0}^{n_l-1-j}\left\{\left(\begin{array}{c}n_l-1-j\\
k\end{array}\right)2^{n_l-1-j-k}\right\} - 1\right]\\
\label{eqn:midStep1}&=& \sum_{j = 0}^{n_l-2}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[3^{n_l-1-j} - 1\right]\\
&=& \sum_{j = 0}^{n_l-1}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[3^{n_l-1-j} - 1\right]\\
&=& 3^{n_l-1}\left({4 \over 3}\right)^{n_l-1} - 2^{n_l-1}\\
&=& 4^{n_l-1} - 2^{n_l-1}\end{aligned}$$ In arriving at Equation \[eqn:midStep1\] from Equation \[eqn:midStep0\] we use the identity given by Equation \[eqn:identity1\].
Note that we chose $X_2$ (or equivalently its complement, $\bar{X}_2$) as an example. In fact there are $\left(\begin{array}{c}m-1\\1\end{array}\right)$ alternatives to choose from. Therefore, the total number of ways in which only one BB gets expressed in $n_l-1$ nodes is given by $$\begin{aligned}
N\left(n_{BB}^{\exp} = 1\right) &=& \left(\begin{array}{c} m-1\\
1\end{array}\right)N\left({\mathrm{Terminals}}~{\mathrm{present}}
=
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2\right)\\
&=& (m-1)\left[4^{n_l-1} - 2^{n_l-1}\right]\end{aligned}$$
Number of expressed BBs = 2.
: Here the terminals that can be present in the $n_l-1$ nodes are $X_1$ or $\bar{X_1}$ or exactly two other complementary pairs. Therefore, we begin by counting the number of ways of having at least one copy of either $X_2$ or $\bar{X}_2$ and at least one copy of either $X_3$ or $\bar{X}_3$ in $n_l-1$ nodes. In other words, we count the number of ways in which only $X_2$ or its complement, $\bar{X}_2$, and $X_3$ or its complement $\bar{X}_3$ can be expressed. $$\begin{aligned}
\nonumber &~& N\left({\mathrm{Terminals}}~{\mathrm{present}} =
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3\right)\\
\nonumber &=&
\sum_{j=0}^{n_l-3}\sum_{k=0}^{n_l-3-j}\sum_{q=0}^{n_l-2-j-k}\sum_{r=0}^{n_l-2-j-k-q}\sum_{s=0}^{n_l-1-j-k-q-r}
{\left(n_l-1\right)! \over j!k!q!r!s!\left(n_l-1-j-k-q-r-s\right)!}\\
&~&~ -
\sum_{j=0}^{n_l-3}\sum_{k=0}^{n_l-3-j}\sum_{s=0}^{n_l-1-j-k}{\left(n_l-1\right)!
\over j!k!s!\left(n_l-1-j-k-s\right)!}\end{aligned}$$ The second summation removes the extra counting of the case when neither $X_2$ or its complement, $\bar{X}_2$ are present in the $n_l-1$ nodes. In other words, it ensures the presence of at least one copy of either $X_2$ or $\bar{X}_2$. $$\begin{aligned}
\nonumber &~& N\left({\mathrm{Terminals}}~{\mathrm{present}} =
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3\right)\\
\nonumber &=& \left[\sum_{j=0}^{n_l-3}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
\nonumber &~&~\left.\sum_{r=0}^{n_l-2-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{s=0}^{n_l-1-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\s\end{array}\right)\right]\\
&~&~ -
\label{eqn:twoBBexp} \left[\sum_{j =
0}^{n_l-3}\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{s=0}^{n_l-1-j-k}\left(\begin{array}{c}n_l-1-j-k\\s\end{array}\right)\right]\end{aligned}$$ Consider the sum $$S_2 = \left[\sum_{j =
0}^{n_l-3}\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{s=0}^{n_l-1-j-k}\left(\begin{array}{c}n_l-1-j-k\\s\end{array}\right)\right],$$ which can be written as $$\begin{aligned}
S_2 &=& \sum_{j = 0}^{n_l-3}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\sum_{k =
0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\
k\end{array}\right)2^{n_l-1-j-k}\\
&=& \sum_{j = 0}^{n_l-3}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[\sum_{k =
0}^{n_l-1-j}\left\{\left(\begin{array}{c}n_l-1-j\\
k\end{array}\right)2^{n_l-1-j-k}\right\} - 1 - 2\left(n_l-1-j\right)\right]\\
&=& \sum_{j = 0}^{n_l-3}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[3^{n_l-1-j} - 1 - 2\left(n_l-1-j\right)\right]\\
&=& \sum_{j = 0}^{n_l-1}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[3^{n_l-1-j} - 1 - 2\left(n_l-1-j\right)\right]\\
&=& 4^{n_l-1} - 2^{n_l-1} -
2\left(n_l-1\right)2^{n_l-1} + 2\left(n_l-1\right)2^{n_l-2}\\
\label{eqn:s2}&=& 4^{n_l-1} - n_l2^{n_l-1}\end{aligned}$$ The second last step in the above derivation uses the identity given by Equation \[eqn:identity3\].
Now consider the sum $$\begin{aligned}
\nonumber S_1 &=& \left[\sum_{j=0}^{n_l-3}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
&~&~\left.\sum_{r=0}^{n_l-2-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{s=0}^{n_l-1-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\s\end{array}\right)\right]\\[2mm]
\nonumber &=& \left[\sum_{j=0}^{n_l-3}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
&~&~\left.\sum_{r=0}^{n_l-2-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)2^{n_l-1-j-k-q-r}\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-3}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\\
&~&~\sum_{q=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\left[3^{n_l-1-j-k-q}
- 1\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-3}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-3-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left[4^{n_l-1-j-k}
- 2^{n_l-1-j-k}\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-3}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left[5^{n_l-1-j} -
3^{n_l-1-j} - 2\left(n_l-1-j\right)\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-1}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left[5^{n_l-1-j} -
3^{n_l-1-j} - 2\left(n_l-1-j\right)\right]\\[2mm]
\nonumber &=& 6^{n_l-1} - 4^{n_l-1} - 2\left(n_l-1\right)2^{n_l-1} +
2\left(n_l-1\right)2^{n_l-2} \\[2mm]
\label{eqn:s1} S_1 &=& 6^{n_l-1} - 4^{n_l-1} - \left(n_l-1\right)2^{n_l-1}\end{aligned}$$
Using Equations \[eqn:s1\] and \[eqn:s2\], we get $$\begin{aligned}
\nonumber &~& N\left({\mathrm{Terminals}}~{\mathrm{present}} =
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3\right)
= S_1 - S_2\\
&=& \left[6^{n_l-1} - 4^{n_l-1} -
\left(n_l-1\right)2^{n_l-1}\right]-\left[4^{n_l-1} -
n_l2^{n_l-1}\right]\\
&=& 6^{n_l-1} - 2\cdot4^{n_l-1} + 2^{n_l-1}.\end{aligned}$$
Note that we chose $X_2$ and $X_3$ (or equivalently their complement, $\bar{X}_2$ and $\bar{X}_3$) as an example. In fact there are $\left(\begin{array}{c}m-1\\2\end{array}\right)$ alternative pairs to choose from. Therefore, the total number of ways in which only one BB gets expressed in $n_l-1$ nodes is given by $$\begin{aligned}
N\left(n_{BB}^{\exp} = 2\right) &=& \left(\begin{array}{c} m-1\\
2\end{array}\right)N\left({\mathrm{Terminals}}~{\mathrm{present}}
=
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3\right)\\
&=& \frac{1}{2}(m-1)(m-2)\left[6^{n_l-1} - 2\cdot4^{n_l-1} + 2^{n_l-1}\right]\end{aligned}$$
Number of expressed BBs = 3.
: Here the terminals that can be present in the $n_l-1$ nodes are $X_1$ or $\bar{X_1}$ or exactly three other complementary pairs. Therefore, we begin by counting the number of ways of having at least one copy of either $X_2$ or $\bar{X}_2$, at least one copy of either $X_3$ or $\bar{X}_3$, and at least one copy of either $X_4$ or $\bar{X}_4$ in $n_l-1$ nodes. In other words, we count the number of ways in which only $X_2$ or its complement, $\bar{X}_2$, $X_3$ or its complement $\bar{X}_3$, $X_4$ or its complement $\bar{X}_4$ can be expressed. $$\begin{aligned}
\nonumber &~& N\left({\mathrm{Terminals}}~{\mathrm{present}} =
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3~{\mathrm{or}}~X_4~{\mathrm{or}}~\bar{X}_4\right)\\
\nonumber &=&
\sum_{j=0}^{n_l-4}\sum_{k=0}^{n_l-4-j}\sum_{q=0}^{n_l-3-j-k}\sum_{r=0}^{n_l-3-j-k-q}\sum_{s=0}^{n_l-2-j-k-q-r}\\
\nonumber &~&~\sum_{t=0}^{n_l-2-j-k-q-r-s}\sum_{u=0}^{n_l-1-j-k-q-r-s-t}
{\left(n_l-1\right)! \over j!k!q!r!s!t!u!\left(n_l-1-j-k-q-r-s-t-u\right)!}\\
&~&~ -
\nonumber \sum_{j=0}^{n_l-4}\sum_{k=0}^{n_l-4-j}\sum_{q=0}^{n_l-3-j-k}\sum_{r=0}^{n_l-3-j-k-q}\sum_{u=0}^{n_l-1-j-k-q-r}{\left(n_l-1\right)!
\over j!k!q!r!u!\left(n_l-1-j-k-q-r-u\right)!}\\
&~&~ -
\nonumber \sum_{j=0}^{n_l-4}\sum_{k=0}^{n_l-4-j}\sum_{s=0}^{n_l-2-j-k}\sum_{t=0}^{n_l-2-j-k-s}\sum_{u=0}^{n_l-1-j-k-s-t}{\left(n_l-1\right)!
\over j!k!s!t!u!\left(n_l-1-j-k-s-t-u\right)!}\\
&~&~+
\sum_{j=0}^{n_l-4}\sum_{k=0}^{n_l-4-j}\sum_{u=0}^{n_l-1-j-k}{\left(n_l-1\right)!
\over j!k!u!\left(n_l-1-j-k-u\right)!}\end{aligned}$$
The above equation can be rewritten as $$\begin{aligned}
\nonumber &~& N\left({\mathrm{Terminals}}~{\mathrm{present}} =
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3~{\mathrm{or}}~X_4~{\mathrm{or}}~\bar{X}_4\right)\\
\nonumber &=& \left[\sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
\nonumber
&~&~~\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{s=0}^{n_l-2-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\s\end{array}\right)\\
\nonumber
&~&~~\sum_{t=0}^{n_l-2-j-k-q-r-s}\left(\begin{array}{c}n_l-1-j-k-q-r-s\\t\end{array}\right)\\
\nonumber
&~&~~\left.\sum_{u=0}^{n_l-1-j-k-q-r-s-t}\left(\begin{array}{c}n_l-1-j-k-q-r-s-t\\u\end{array}\right)\right]\\[2mm]
\nonumber
&~&~ - \left[\sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
\nonumber
&~&~~\left.\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{u=0}^{n_l-1-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\u\end{array}\right)\right]\\[2mm]
\nonumber &~&~ - \left[\sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{s=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\s\end{array}\right)\right.\\
\nonumber
&~&~~\left.\sum_{t=0}^{n_l-2-j-k-s}\left(\begin{array}{c}n_l-1-j-k-s\\t\end{array}\right)\sum_{u=0}^{n_l-1-j-k-s-t}\left(\begin{array}{c}n_l-1-j-k-s-t\\u\end{array}\right)\right]\\[2mm]
&~&~ +
\left[\sum_{j = 0}^{n_l-4}\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{u=0}^{n_l-1-j-k}\left(\begin{array}{c}n_l-1-j-k\\u\end{array}\right)\right]\end{aligned}$$ Consider the sum $$S_4 = \left[\sum_{j =
0}^{n_l-4}\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{u=0}^{n_l-1-j-k}\left(\begin{array}{c}n_l-1-j-k\\u\end{array}\right)\right],$$ which can be written as $$\begin{aligned}
S_4 &=& \sum_{j = 0}^{n_l-4}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\sum_{k =
0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\
k\end{array}\right)2^{n_l-1-j-k}\\
\nonumber &=& \sum_{j = 0}^{n_l-4}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[\sum_{k =
0}^{n_l-1-j}\left\{\left(\begin{array}{c}n_l-1-j\\
k\end{array}\right)2^{n_l-1-j-k}\right\} - 1\right.\\
&~&~~ \left.- 2\left(n_l-1-j\right) - 2\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\
&=& \sum_{j = 0}^{n_l-4}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[3^{n_l-1-j} - 1 - 2\left(n_l-1-j\right) - 2\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\
&=& \sum_{j = 0}^{n_l-1}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[3^{n_l-1-j} - 1 - 2\left(n_l-1-j\right) - 2\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\
&=& \sum_{j = 0}^{n_l-1}\left(\begin{array}{c}n_l-1\\
j\end{array}\right)\left[3^{n_l-1-j} -
\left(2\left(n_l-1\right)\left(n_l-2\right) + 2n_l - 1\right) +
4\left(n_l-1\right)j - 2j^2\right]\\
&=& 4^{n_l-1} - \left[2\left(n_l-1\right)\left(n_l-2\right) + 2n_l -
1\right]2^{n_l-1} + 4\left(n_l-1\right)^22^{n_l-2} - 2n_l\left(n_l-1\right)2^{n_1-3}\\
\label{eqn:s4a}&=& 4^{n_l-1} - 2^{n_l-1} - n_l\left(n_l-1\right)2^{n_l-2}\end{aligned}$$
$$\begin{aligned}
\nonumber S_3 &=& \left[\sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{s=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
&~&~\left.\sum_{t=0}^{n_l-2-j-k-s}\left(\begin{array}{c}n_l-1-j-k-s\\t\end{array}\right)\sum_{u=0}^{n_l-1-j-k-s-t}\left(\begin{array}{c}n_l-1-j-k-s-t\\u\end{array}\right)\right]\\[2mm]
\nonumber &=& \left[\sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{s=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\s\end{array}\right)\right.\\
&~&~\left.\sum_{t=0}^{n_l-2-j-k-s}\left(\begin{array}{c}n_l-1-j-k-s\\t\end{array}\right)2^{n_l-1-j-k-s-t}\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\\
&~&~\sum_{s=0}^{n_l-2-j-k}\left(\begin{array}{c}n_l-1-j-k\\s\end{array}\right)\left[3^{n_l-1-j-k-s}
- 1\right]\\[2mm]
&=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left[4^{n_l-1-j-k}
- 2^{n_l-1-j-k}\right]\\[2mm]
&=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left[5^{n_l-1-j} -
3^{n_l-1-j} - 2\left(n_l-1-j\right) - 6\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\[2mm]
&=& \sum_{j=0}^{n_l-1}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left[5^{n_l-1-j} -
3^{n_l-1-j} - 2\left(n_l-1-j\right) -
6\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\[2mm]
&=& 6^{n_l-1} - 4^{n_l-1} - \sum_{j=0}^{n_l-1}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left[6\left(n_l-1\right)^2
- 4\left(n_l-1\right) - 12\left(n_l-1\right)j + 4j +
6j^2\right]\\[2mm]
\nonumber &=& 6^{n_l-1} - 4^{n_l-1} - 6\left(n_l-1\right)^22^{n_l-1} +
4\left(n_l-1\right)2^{n_l-1} + 12\left(n_l-1\right)^22^{n_l-2}\\
&~&~~ -
4\left(n_l-1\right)2^{n_l-2} - 6n_l\left(n_l-1\right)2^{n_l-3}\\[2mm]
\label{eqn:s3a} S_3 &=& 6^{n_l-1} - 4^{n_l-1} +
2\left(n_l-1\right)2^{n_l-1} - 3n_l\left(n_l-1\right)2^{n_l-2}\end{aligned}$$
$$\begin{aligned}
\nonumber S_2 &=& \left[\sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
&~&~\left.\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{u=0}^{n_l-1-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\u\end{array}\right)\right]\\[2mm]
\nonumber &=& \left[\sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
&~&~\left.\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)2^{n_l-1-j-k-q-r}\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\\
&~&~\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\s\end{array}\right)\left[3^{n_l-1-j-k-q}
- 1 - 2\left(n_l-1-j-k-q\right)\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left[4^{n_l-1-j-k}
- \left(2n_l-1-2j-2k\right)2^{n_l-1-j-k}\right.\\
&~&~~\left. + 2\left(n_l-1-j-k\right)2^{n_l-2-j-k}\right]\\[2mm]
&=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left[4^{n_l-1-j-k}
- \left(n_l-j-k\right)2^{n_l-1-j-k}\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left\{\sum_{k=0}^{n_l-1-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left[4^{n_l-1-j-k}
- \left(n_l-j-k\right)2^{n_l-1-j-k}\right]\right.\\
&~&~~\left. - 2\left(n_l-1-j\right)\left(n_l-2-j\right)\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left[5^{n_l-1-j} -
\left(n_l-j\right)3^{n_l-1-j} +
\left(n_l-1-j\right)3^{n_l-2-j}\right.\\
&~&~~\left. - 2\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-1}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left[5^{n_l-1-j} -
\left(n_l-j\right)3^{n_l-1-j} +
\left(n_l-1-j\right)3^{n_l-2-j}\right.\\
&~&~~\left. - 2\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\[2mm]
\nonumber &=& 6^{n_l-1} - n_l4^{n_1-1} +
\frac{1}{3}\left(n_l-1\right)4^{n_l-1} -
2\left(n_l-1\right)\left(n_l-2\right)2^{n_l-1} +\\
&~&~~ 2\left(2n_l-3\right)\sum_{j=0}^{n_l-1}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)j - 2\sum_{j=0}^{n_l-1}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)j^2 + \frac{2}{3} \sum_{j=0}^{n_l-1}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)j3^{n_l-1-j}\\[2mm]
\nonumber &=& 6^{n_l-1} - n_l4^{n_1-1} +
\frac{1}{3}\left(n_l-1\right)4^{n_l-1} -
2\left(n_l-1\right)\left(n_l-2\right)2^{n_l-1} +\\
&~&~~ 2\left(2n_l-3\right)\left(n_l-1\right)2^{n_l-2} -
2n_l\left(n_l-1\right)2^{n_l-3} +
\frac{2}{3}\left(n_l-1\right)4^{n_l-2}\\[2mm]
\label{eqn:s2a} S_2 &=& 6^{n_l-1} - \frac{1}{2}\left(n_l+1\right)4^{n_1-1} +
\left(n_l-1\right)2^{n_l-1} - n_l\left(n_l-1\right)2^{n_l-2}\end{aligned}$$
$$\begin{aligned}
\nonumber S_1 &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
\nonumber
&~&~~\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{s=0}^{n_l-2-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\s\end{array}\right)\\
\nonumber
&~&~~\sum_{t=0}^{n_l-2-j-k-q-r-s}\left(\begin{array}{c}n_l-1-j-k-q-r-s\\t\end{array}\right)\\
&~&~~\sum_{u=0}^{n_l-1-j-k-q-r-s-t}\left(\begin{array}{c}n_l-1-j-k-q-r-s-t\\u\end{array}\right)\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
\nonumber
&~&~~\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{s=0}^{n_l-2-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\s\end{array}\right)\\
&~&~~\sum_{t=0}^{n_l-2-j-k-q-r-s}\left(\begin{array}{c}n_l-1-j-k-q-r-s\\t\end{array}\right)
2^{n_l-1-j-k-q-r-s-t}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
\nonumber
&~&~~\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{s=0}^{n_l-2-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\s\end{array}\right)\\
&~&~~\left\{3^{n_l-1-j-k-q-r-s} - 1\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
\nonumber
&~&~~\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\sum_{s=0}^{n_l-1-j-k-q-r}\left(\begin{array}{c}n_l-1-j-k-q-r\\s\end{array}\right)\\
&~&~~\left\{3^{n_l-1-j-k-q-r-s} - 1\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
&~&~~\sum_{r=0}^{n_l-3-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\left\{4^{n_l-1-j-k-q-r}
- 2^{n_l-1-j-k-q-r}\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left[\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\right.\\
\nonumber &~&~~\sum_{r=0}^{n_l-1-j-k-q}\left(\begin{array}{c}n_l-1-j-k-q\\r\end{array}\right)\left\{4^{n_l-1-j-k-q-r}
- 2^{n_l-1-j-k-q-r}\right\}\\
&~&\left. - 2\left(n_l-1-j-k-q\right)\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-3-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
&~&\left\{5^{n_l-1-j-k-q} - 3^{n_l-1-j-k-q} - 2\left(n_l-1-j-k-q\right)\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\sum_{q=0}^{n_l-1-j-k}\left(\begin{array}{c}n_l-1-j-k\\q\end{array}\right)\\
&~&\left\{5^{n_l-1-j-k-q} - 3^{n_l-1-j-k-q} -
2\left(n_l-1-j-k-q\right)\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left\{6^{n_l-1-j-k}
- 4^{n_l-1-j-k}\right.\\
&~&~~\left. -2\left(n_l-1-j-k\right)2^{n_l-1-j-k} + 2\left(n_l-1-j-k\right)2^{n_l-2-j-k}\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-4-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left\{6^{n_l-1-j-k}
- 4^{n_l-1-j-k}\right.\\
&~&~~\left.- \left(n_l-1-j-k\right)2^{n_l-1-j-k}\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}\left[
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\sum_{k=0}^{n_l-1-j}\left(\begin{array}{c}n_l-1-j\\k\end{array}\right)\left\{6^{n_l-1-j-k}
- 4^{n_l-1-j-k}\right.\right.\\
&~&~~\left.\left.- \left(n_l-1-j-k\right)2^{n_l-1-j-k}\right\} - 6\left(n_l-1-j\right)\left(n_l-2-j\right)\right]\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}
\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left\{7^{n_l-1-j} -
5^{n_l-1-j} - \left(n_l-1-j\right)3^{n_l-1-j} \right.\\
&~&~~+ \left.\left(n_l-1-j\right)3^{n_l-2-j} - 6\left(n_l-1-j\right)\left(n_l-2-j\right)\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-4}\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left\{7^{n_l-1-j} -
5^{n_l-1-j} - \frac{2}{3}\left(n_l-1-j\right)3^{n_l-1-j} \right.\\
&~&~~\left.-6\left(n_l-1-j\right)\left(n_l-2-j\right)\right\}\\[2mm]
\nonumber &=& \sum_{j=0}^{n_l-1}\left(\begin{array}{c}n_l-1\\j\end{array}\right)\left\{7^{n_l-1-j} -
5^{n_l-1-j} - \frac{2}{3}\left(n_l-1-j\right)3^{n_l-1-j} \right.\\
&~&~~\left. - 6\left(n_l-1-j\right)\left(n_l-2-j\right)\right\}\\[2mm]
\nonumber &=& 8^{n_l-1-j} - 6^{n_l-1-j} -
\frac{2}{3}\left(n_l-1\right)4^{n_l-1} +
\frac{2}{3}\left(n_l-1\right)4^{n_l-2} -
6\left(n_l-1\right)\left(n_l-2\right)2^{n_l-1}\\
&~&~~ +
6\left(2n_l-3\right)\left(n_l-1\right)2^{n_l-2} -
6n_l\left(n_l-1\right)2^{n_l-3}\\[2mm]
&=& 8^{n_l-1-j} - 6^{n_l-1-j} -
\frac{1}{2}\left(n_l-1\right)4^{n_l-1} +
3\left(n_l-1\right)2^{n_l-1} - 3n_l\left(n_l-1\right)2^{n_l-2}\\[2mm]
\label{eqn:s1a} S_1 &=& 8^{n_l-1} - 6^{n_l-1} + 4^{n_l-1} -
\frac{1}{2}\left(n_l+1\right)4^{n_l-1} + 3\left(n_l-1\right)2^{n_l-1} - 3n_l\left(n_l-1\right)2^{n_l-2}\end{aligned}$$
Using Equations \[eqn:s4a\] – \[eqn:s1a\], we get $$\begin{aligned}
\nonumber && N\left({\mathrm{Terminals}}~{\mathrm{present}} =
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3~{\mathrm{or}}~X_4~{\mathrm{or}}~\bar{X}_4\right)
= S_1 - S_2 - S_3 + S4\\[2mm]
\nonumber &=& \left[8^{n_l-1} - 6^{n_l-1} + 4^{n_l-1} -
\frac{1}{2}\left(n_l+1\right)4^{n_l-1} +
3\left(n_l-1\right)2^{n_l-1} -
3n_l\left(n_l-1\right)2^{n_l-2}\right]\\
\nonumber &~&~~-\left[6^{n_l-1} - \frac{1}{2}\left(n_l+1\right)4^{n_1-1} +
\left(n_l-1\right)2^{n_l-1} -
n_l\left(n_l-1\right)2^{n_l-2}\right]\\
\nonumber &~&~~-\left[6^{n_l-1} - 4^{n_l-1} +
2\left(n_l-1\right)2^{n_l-1} - 3n_l\left(n_l-1\right)2^{n_l-2}\right]\\
&~&~~+\left[4^{n_l-1} - 2^{n_l-1} - n_l\left(n_l-1\right)2^{n_l-2}\right]\\[2mm]
&=& 8^{n_l-1} - 3\cdot6^{n_l-1} + 3\cdot4^{n_l-1} - 2\cdot2^{n_l-1}.\end{aligned}$$
Note that we chose $X_2$, $X_3$, and $X_4$ (or equivalently their complement, $\bar{X}_2$, $\bar{X}_3$, and $\bar{X}_4$) as an example. In fact there are $\left(\begin{array}{c}m-1\\3\end{array}\right)$ alternative pairs to choose from. Therefore, the total number of ways in which only one BB gets expressed in $n_l-1$ nodes is given by $$\begin{aligned}
N\left(n_{BB}^{\exp} = 3\right) &=& \left(\begin{array}{c} m-1\\
3\end{array}\right)N\left({\mathrm{Terminals}}~=
X_1~{\mathrm{or}}~\bar{X}_1~{\mathrm{or}}~X_2~{\mathrm{or}}~\bar{X}_2~{\mathrm{or}}~X_3~{\mathrm{or}}~\bar{X}_3~{\mathrm{or}}~X_4~{\mathrm{or}}~\bar{X}_4\right)\\
&=& \left(\begin{array}{c} m-1\\
3\end{array}\right)\left[8^{n_l-1} - 3\cdot6^{n_l-1} + 3\cdot4^{n_l-1} - 2^{n_l-1}\right]\end{aligned}$$
From the above cases we can generalize the number of ways of expressing $i$ BBs in $n_l-1$ nodes is given by $$N\left(n_{BB}^{\exp} = i\right) = \left(\begin{array}{c} m-1\\
i\end{array}\right)\sum_{j = 0}^{i}\left(\begin{array}{c} i\\
j\end{array}\right)\left(-1\right)^j\left[2\left(i-j+1\right)\right]^{n_l-1}$$
Recall that the total number of ways of arranging the $2m$ terminals in $n_l-1$ nodes is given by $$N_{{\mathrm{tot}}} = \left(2m\right)^{n_l-1}$$
Therefore, the probability of expressing $i$ BBs is given by $$\begin{aligned}
p\left(n_{BB}^{\exp} = i\right) &=& {N\left(n_{BB}^{\exp} = i\right)
\over N_{{\mathrm{tot}}}}\\
&=& \left(\begin{array}{c} m-1\\
i\end{array}\right)\sum_{j = 0}^{i}\left(\begin{array}{c} i\\
j\end{array}\right)\left(-1\right)^j\left({i-j+1 \over m}\right)^{n_l-1}\end{aligned}$$
The average number of expressed building blocks other than the one that decision is being made on $$\bar{n}_{BB}^{\exp} = \sum_{i = 0}^{m-1}\left(\begin{array}{c} m-1\\
i\end{array}\right)i\sum_{j = 0}^{i}\left(\begin{array}{c} i\\
j\end{array}\right)\left(-1\right)^j\left({i-j+1 \over m}\right)^{n_l-1}$$
The variance in the number of expressed building blocks other than the one that decision is being made on $$\sigma^2_{{n}_{BB}^{\exp}} = \sum_{i = 0}^{m-1}\left(\begin{array}{c} m-1\\
i\end{array}\right)i^2\sum_{j = 0}^{i}\left(\begin{array}{c} i\\
j\end{array}\right)\left(-1\right)^j\left({i-j+1 \over
m}\right)^{n_l-1} - \left[\bar{n}_{BB}^{\exp}\right]^2$$
Estimating Tree Sizes
=====================
We start with defining two utility procedures that generate a non-full tree and full tree respectively. We have named them accordingly and they correspond in common GP parlance to GROW and FULL. These procedures are called by RAMPED-FULL, RAMPED-GROW and RAMPED-HALF-HALF.
Both algorithms are parameterized by:
- $maxHeight$ : the maximum allowable height of the tree
- $q$: the probability with which the terminal set is used to draw a new tree node
Often $q$ is implicitly set as the frequency of terminal nodes in the primitive set and GPr’s simply set maxHeight. However, sometimes (like we do in the ORDER problem) a bias between functions and terminals is introduced. We note that Luke [@luke:2000:2ftcaGP] has similar versions of these algorithms without $q$ explicitized.
[1]{} Algorithm I: create-tree-not-necessarily-full (q, maxHeight) // create trees of more than 1 node root = get-function() height = 1 left-child = create-subtree(q, maxHeight-1, height) right-child = create-subtree(q, maxHeight-1,height) return make-tree(root, left-child,right-child)
procedure create-subtree(q, maxHeight, current-height) if current-height = (maxHeight-1) then return get-terminal() else if rand(0,1) < q then return get-terminal() else return create-tree-not-necessarily-full(q, maxHeight-1)
The [*create-tree-not-necessarily-full*]{} algorithm creates a GP tree of height between 2 and maxHeight, not allowing a single leaf to be generated as a tree. The tree is not necessarily full. After drawing a function for the tree’s root node, it uses $q$ to decide between making each child subtree of the root a function or a terminal, [*except*]{} when the tree’s height is equal to $(maxHeight -1)$. When the tree’s height is equal to $maxHeight -1$, a terminal is alway generated as the child subtree. This ensures that no tree has height greater than $maxHeight$.
[1]{} Algorithm II: create-tree-full (q, maxHeight) // create full trees of more than 1 node root = get-function() height = 1 left-child = create-full-subtree(q, maxHeight-1, height) right-child = create-tree-full(q, height(left-child)) return make-tree(root, left-child,right-child)
procedure create-full-subtree(q, maxHeight, current-height) if current-height = (maxHeight-1) then return get-terminal() else if rand(0,1) < q then return get-terminal() else return create-tree-full(q, maxHeight-1)
The [*create-tree-full*]{} algorithm creates a GP tree of height between 2 and maxHeight, not allowing a single leaf to be generated as a tree. The tree is always full. After drawing a function for the tree’s root node, it uses $q$ to decide between making the left child subtree of the root a function or a terminal, [*except*]{} when the tree’s height is equal to $(maxHeight -1)$. When the tree’s height is equal to $maxHeight -1$, a terminal is alway generated as the child subtree. This ensures that no tree has height greater than $maxHeight$. The right child subtree of the root is generated by calling [*create-tree-full*]{} with the maxHeight parameter taking the value of the height of the left child subtree. (UM: But I haven’t checked my pseudocode carefully at all)
Usually these procedures are subsumed by procedures that create an initial population with random [*fitness*]{} but predetermined expected GP tree structure. The procedures are:
- ramped full. Create subsamples of trees for each height, h, between height 1 and maxHeight. Each subsample has full trees of height up to h.
- ramped not-necessarily-full. Create subsamples of trees for each height, h, between 1 and maxHeight. Each subsample has not-necessarily full trees of height up to h.
- ramped half-half (implying half full and half not necessarily full). Create two subsamples for each height, h, between height 1 and maxHeight. One subsample has full trees of height up to h and one subsample has not-necessarily full trees of height up to h.
Assuming any of these algorithms is executed to generate a tree of size $s$, because the tree is binary, the following is known,:
1. the number of leaves (terminals), $t_s = \frac{s+1}{2}$
2. the number of internal nodes (functions), $f_s = \frac{s-1}{2}$
The average size of a tree created using Algorithm [ *create-tree-not-necessarily-full*]{} can be estimated as follows:
1. a tree of size h has a range of possible sizes from $s_{\min} = 2h+1$ to $s_{\max} = 2^{h+1}-1$. This range is $s_{\min}, s_{\min}+2, \dots,s_{\max}$.
2. the probability of a tree of size $s$ given it has height $h$ and $h < h_{\max}$, where $h_{\max}$ is maxHeight: $$p\left(s|h;h<h_{\max}\right) = (1-q)^{f_s-1}q^{t_s}$$
3. the probability of a tree of $h < h_{\max}$: $$p\left(h<h_{\max}\right) = \sum_{h=1}^{h_{\max}-1}\sum_{s=s_{\min} by 2}^{s=s_{\max}} p(h|s)$$
4. the average size of a tree of height h: $$\overline{s}(h) =
\sum_{s=s_{\min} by 2}^{s=s_{\max}} p(s|h)s$$
5. the average size of trees of height $h < h_{\max}$ $$\overline{s}(h;h<h_{\max}) = \sum_{h=1}^{h_{\max}-1} \overline{s}(h)\|p(h|s)\|$$
6. the estimated average size of a tree of height, $h = h_{\max}$ can be estimated conservatively (underestimation): $$\hat{s}\left(h = h_{\max}\right) =$$
7. the probability of a tree of height = $maxHeight$: $$p\left(h = h_{\max}\right) = 1 - p\left(h < h_{\max}\right)$$
8. the estimated average size of any tree: $$\hat{s} = p\left(h = h_{\max}\right)\hat{s}\left(h = h_{\max}\right) +
p\left(h < h_{\max}\right)\overline{s}\left(h < h_{\max}\right)$$
The average size of a tree created using Algorithm [ *create-tree-full*]{} can be estimated as follows:
1. the probability of a tree of height h when $h < maxHeight$: $$p(h) = (1-q)^{h-1}q$$
2. the probability of any tree of height, $h$, that is less than the maxHeight, $h_{\max}$: $$\begin{aligned}
\nonumber p\left(h<h_{\max}\right) &=& \sum_{h=1}^{h_{\max}-1}p(h),\\
\nonumber &=& \sum_{h=1}^{h_{\max}-1}q(1-q)^{h-1},\\
\nonumber &=& 1 - (1-q)^{h_{\max}-1}.\end{aligned}$$
3. the probability of a tree of $h = h_{\max}$: $$\begin{aligned}
\nonumber p\left(h=h_{\max}\right) &=& 1 - p\left(h<h_{\max}\right),\\
&=& (1-q)^{h_{\max}-1}.\end{aligned}$$
4. the size of a tree of height h, $s(h) = 2^{h+1}-1$
5. the average size of a tree of height, $h < h_{\max}$: $$\begin{aligned}
\nonumber \overline{s}\left(h < h_{\max}\right) &=&
\sum_{h=1}^{h_{\max}-1}\|p(h)\|s(h),\\
\nonumber &=& \left[{1 \over 1 - (1-q)^{h_{\max}-1}}\right]\sum_{h=1}^{h_{\max}-1}
\left[\left(2^{h+1}-1\right)q(1-q)^(h-1)\right],\\
&=& \left({4q \over 2q-1}\right)\left[{1 -
\left(2(1-q)\right)^{h_{\max}-1} \over 1 - (1-q)^{h_{\max}-1}}\right] - 1.\end{aligned}$$
6. the average size of a tree of height, $h = h_{\max}$, $$\begin{aligned}
\nonumber \overline{s}\left(h = h_{\max}\right) &=&
\|p\left(h=h_{\max}\right)\|s\left(h = h_{\max}\right), \\
&=& 2^{h_{\max}+1}-1.\end{aligned}$$
7. the average size of a tree $\overline{s}$ is $$\begin{aligned}
\nonumber \overline{s} &=& \overline{s}\left(h=h_{\max}\right)p\left(h =
h_{\max}\right) + \overline{s}\left(h<h_{\max}\right)p\left(h <
h_{\max}\right),\\
\nonumber &=& \left[2^{h_{\max}+1}-1\right](1-q)^{h_{\max}-1} +\\
\nonumber && \left[\left({4q \over 2q-1}\right)\left[{1 -
\left(2(1-q)\right)^{h_{\max}-1} \over 1 -
(1-q)^{h_{\max}-1}}\right] - 1\right]\left[1 -
(1-q)^{h_{\max}-1}\right],\\
&=& {2q - 2\cdot\left[2(1-q)\right]^{h_{\max}} + 1 \over 2q-1}\end{aligned}$$
The average size of a tree created using ramped full, ramped not-full or ramped half-half can now be easily calculated. I have done this but don’t have time to write out the derivation here! (I feel a bit like Fermat ;0)
Hence, given a $q$, $maxHeight$ and GP tree initialization algorithm, using the equations about, we can derive an estimate of average GP tree size, $\hat{s}$.
|
---
author:
- |
Brijnesh Jain and David Schultz\
Technische Universität Berlin, Germany\
e-mail: [email protected]
title: On the Existence of a Sample Mean in Dynamic Time Warping Spaces
---
=1
#### Abstract. {#abstract. .unnumbered}
The concept of sample mean in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems and generalize centroid-based clustering algorithms. Its existence has neither been proved nor challenged. This article presents sufficient conditions for existence of a sample mean in DTW spaces. The proposed result justifies prior work on approximate mean algorithms, sets the stage for constructing exact mean algorithms, and is a first step towards a statistical theory of DTW spaces.
Introduction {#sec:intro}
============
Time series are time-dependent observations that vary in length and temporal dynamics (speed). Examples of time series data include acoustic signals, electroencephalogram recordings, electrocardiograms, financial indices, and internet traffic data.
Time series averaging aims at finding a typical time series that “best” represents a given sample of time series. First works on time series averaging started in the 1970ies with speech recognition as the prime application [@Rabiner1979]. Since then, research predominantly focused on devising averaging algorithms for improving pattern recognition systems and generalizing centroid-based clustering algorithms [@Abdulla2003; @Gupta1996; @Hautamaki2008; @Kruskal1983; @Niennattrakul2009; @Petitjean2011; @Petitjean2014; @Petitjean2016; @Sathianwiriyakhun2016; @Soheily-Khah2016]. In contrast to averaging points in a Euclidean space, averaging time series is a non-trivial task, because the sample time series can vary in length and speed. To filter out these variations, the above cited averaging algorithms apply dynamic time warping (DTW).
The most promising direction poses time series averaging as an optimization problem [@Cuturi2017; @Hautamaki2008; @Petitjean2011; @Schultz2017; @Soheily-Khah2016]: Suppose that $\S{T}$ is the set of all time series of finite length and $\S{X} = \args{x^{(1)}, \ldots, x^{(N)}}$ is a sample of $N$ time series $x^{(k)} \in \S{T}$. Then time series averaging amounts in minimizing the Fréchet function [@Frechet1948] $$\begin{aligned}
\label{eq:Frechet-Function}
F: \S{T}_m \rightarrow \R, \quad x \, \mapsto \; \frac{1}{N}\sum_{k=1}^N \delta^2\!\args{x, x^{(k)}},\end{aligned}$$ where the solution space $\S{T}_m\subset \S{T}$ is the subset of all time series of length $m$ and $\delta(x,y)$ denotes the DTW distance between time series $x$ and $y$ [@Sakoe1978]. The global minimizers $z \in \S{T}_m$ of the Fréchet function $F(x)$ are the restricted sample means of sample $\S{X}$. The restriction refers to confining the length $m$ of the candidate solutions.
Using Fréchet functions, the notion of a “typical time series that best represents a sample” has a precise meaning. A typical time series is any global minimizer of the Fréchet function. If a global minimum exists, it best represents a sample in the sense that it deviates least from all sample time series. The Fréchet function is motivated by the property that an arithmetic mean of real numbers minimizes the mean squared error from the sample numbers. Following Fréchet [@Frechet1948], we can use this property to generalize the concept of sample mean to arbitrary distance spaces for which a well-defined addition is unknown and therefore an arithmetic mean can not be computed in closed form by a weighted sum.
Research on the Fréchet function as defined in Eq. has the following shortcomings:
1. Existence of a global minimum of the Fréchet function has neither been proved nor challenged. Existence of an optimal solution depends on the particular choice of DTW distance and loss function. A restricted sample mean trivially exists if the DTW distance between two time series is always zero. Conversely, there are DTW spaces for which a sample mean does not always exist (cf. Example \[example:non-existence\]). Thus it is unclear whether existing heuristics indeed approximate a typical time series or unknowingly search for a phantom.
2. In experiments, the solution space $\S{T}_m$ is heuristically chosen. For example, if all sample time series are of length $m$ then a common choice is $\S{T}_m$. The intuition behind this choice is that the length of a restricted sample mean should “best” represent the lengths of the sample time series. This intuition may lead to solutions that fail to capture the characteristic properties of the sample time series as illustrated by Figure \[fig:ex\_unrestricted\_mean\].
3. The Fréchet function only generalizes the sample mean. Neither weighted means nor other important measures of central location such as the sample median are captured by Eq. .
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
restricted sample mean $z_3$ restricted sample mean $z_4$ variance
![Dependence of the variance (global minimum of the Fréchet function) on the parameter $m$ of the solutions space $\S{T}_m$. Plot (a) shows two time series (red) of length $3$ together with a restricted sample mean $z_3$ of length $3$ (blue). The restricted sample mean $z_3$ fails to properly capture the peak of one and the valley of the other sample time series. Plot (b) shows the same time series (red) warped onto the time axis of a restricted sample mean $z_4$ of length $4$ (blue). Warping increases the length of the red sample time series to length $4$. For the upper (lower) sample time series the first (third) element is copied. In contrast to $z_3$, the restricted sample mean $z_4$ captures the peak of one and the valley of the other sample time series. Plot (c) shows the variance $F(z_m)$ depending on the parameter $m \in \cbrace{1, \ldots, m}$ of the solution space $\S{T}_m$. We see that $F(z_4) \leq F(z_m)$ for all $m$ and in particular $F(z_4) < F(z_3)$. This shows that the better representational properties of $z_4$ are reflected by a lower variance.[]{data-label="fig:ex_unrestricted_mean"}](./ex_mean01.png "fig:"){width="30.00000%"} ![Dependence of the variance (global minimum of the Fréchet function) on the parameter $m$ of the solutions space $\S{T}_m$. Plot (a) shows two time series (red) of length $3$ together with a restricted sample mean $z_3$ of length $3$ (blue). The restricted sample mean $z_3$ fails to properly capture the peak of one and the valley of the other sample time series. Plot (b) shows the same time series (red) warped onto the time axis of a restricted sample mean $z_4$ of length $4$ (blue). Warping increases the length of the red sample time series to length $4$. For the upper (lower) sample time series the first (third) element is copied. In contrast to $z_3$, the restricted sample mean $z_4$ captures the peak of one and the valley of the other sample time series. Plot (c) shows the variance $F(z_m)$ depending on the parameter $m \in \cbrace{1, \ldots, m}$ of the solution space $\S{T}_m$. We see that $F(z_4) \leq F(z_m)$ for all $m$ and in particular $F(z_4) < F(z_3)$. This shows that the better representational properties of $z_4$ are reflected by a lower variance.[]{data-label="fig:ex_unrestricted_mean"}](./ex_mean02.png "fig:"){width="30.00000%"} ![Dependence of the variance (global minimum of the Fréchet function) on the parameter $m$ of the solutions space $\S{T}_m$. Plot (a) shows two time series (red) of length $3$ together with a restricted sample mean $z_3$ of length $3$ (blue). The restricted sample mean $z_3$ fails to properly capture the peak of one and the valley of the other sample time series. Plot (b) shows the same time series (red) warped onto the time axis of a restricted sample mean $z_4$ of length $4$ (blue). Warping increases the length of the red sample time series to length $4$. For the upper (lower) sample time series the first (third) element is copied. In contrast to $z_3$, the restricted sample mean $z_4$ captures the peak of one and the valley of the other sample time series. Plot (c) shows the variance $F(z_m)$ depending on the parameter $m \in \cbrace{1, \ldots, m}$ of the solution space $\S{T}_m$. We see that $F(z_4) \leq F(z_m)$ for all $m$ and in particular $F(z_4) < F(z_3)$. This shows that the better representational properties of $z_4$ are reflected by a lower variance.[]{data-label="fig:ex_unrestricted_mean"}](./ex_mean03.png "fig:"){width="30.00000%"}
(a) (b) (c)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
To address all three issues, we consider a more general formulation of the Fréchet function as given by $$\begin{aligned}
%\label{eq:General-Frechet-Function}
F: \S{U} \rightarrow \R, \quad x \mapsto \sum_{k=1}^N h_k\Big(\delta\args{x, x^{(k)}}\Big),\end{aligned}$$ where $\S{U} \subseteq \S{T}$ is the solution space and the $h_k:\R \rightarrow \R$ are loss functions. We recover the standard Fréchet function given in Eq. by setting $h_k(u) = u^2$ for all $k$. To average the sum of squared DTW distances, we define $h_k(u) = u^2/N$, where $1/N$ is the uniform weight. To obtain a weighted version of Eq. , we demand that $h_k(u) = w_k\cdot u^2$, where $w_k \in \R_+$ is a positive weight. The sample median is obtained by setting $h_k(u) = u$ for all $k$. Regardless of the choice of loss function, we refer to global minimizers of the general Fréchet function as sample means as umbrella term.
We focus on two forms of solution sets $\S{U}$: (i) $\S{U} = \S{T}$ is the set of all time series of finite length and (ii) $\S{U} = \S{T}_m$ is the subset of time series of length $m$. We call $(i)$ the unrestricted and $(ii)$ the restricted form. Note that restrictions refer to the solution space $\S{U}$ only. Sample time series $x^{(k)}$ are always from $\S{T}$ and therefore may have arbitrary length. We assume no restrictions on the elements of the time series. The elements can be real values, feature vectors, symbols, trees, graphs, and mixtures thereof.
This contribution presents sufficient conditions for existence of a sample mean in restricted and unrestricted form. We show that common DTW distances mentioned in the literature satisfy the proposed sufficient conditions. A key result is the Reduction Theorem stating that there is a sample-dependent bound $\rho$ on the length beyond which the Fréchet function can not be further decreased. For the two sample time series in Figure \[fig:ex\_unrestricted\_mean\] the bound is $\rho = 4$. Hence, the restricted sample mean $z_4$ is also an unrestricted sample mean.
This contribution has the following implications: Existence of a sample mean together with the necessary conditions of optimality proposed in [@Schultz2017] enable the formulation of exact mean algorithms [@Brill2017]. Existence of restricted sample means theoretically justify prior work [@Cuturi2017; @Hautamaki2008; @Petitjean2011; @Rabiner1979; @Schultz2017; @Soheily-Khah2016] in the sense that the concept of a restricted sample mean is not a phantom but does in fact exist. Existence of the weighted mean justfies the soft-DTW approach proposed by [@Cuturi2017]. Finally, this contribution is a first step towards a statistical theory of DTW spaces in the spirit of a statistical theory of shape, tree, and graph spaces [@Dryden1998; @Feragen2013; @Ginestet2012; @Huckemann2010; @Jain2016; @Kendall1984; @Marron2014].
The rest of this paper is structured as follows: Section 2 states the main results of this contribution and Section 3 concludes with a summary of the main findings and an outlook to further research. All proofs are delegated to the appendix.
Existence of a Sample Mean via the Reduction Theorem {#sec:reduction-theorem+implications}
====================================================
This section first introduces the DTW-distance and Fréchet functions. Then the Reduction Theorem is stated and its implications are presented. Finally, sufficient conditions of existence of a sample mean are proposed.
#### Notations. {#notations. .unnumbered}
We write $\R_{\geq 0}$ for the set of non-negative reals. By $\N$ we denote the set of positive integers. We write $[n]$ to denote the set $\cbrace{1, \ldots, n}$ for a given $n \in \N$. Finally, $\S{S}^N = \S{S} \times \cdots \times \S{S}$ is the $N$-fold Cartesian product of the set $\S{S}$, where $N \in \N$.
The Dynamic Time Warping Distance
---------------------------------
Suppose that $\S{A}$ is an attribute set. A *time series* $x$ of *length* $\ell(x) = m$ is a sequence $x = (x_1, \ldots, x_m)$ consisting of *elements* $x_i \in \S{A}$ for every *time point* $i \in [m]$. By $\S{T}_n$ we denote the set of all time series of length $n \in \N$ with elements from $\S{A}$. Then $$\S{T} = \bigcup_{n \in \N} \S{T}_n$$ is the set of all time series of finite length with elements from $\S{A}$.
Without further mention, we assume that the attribute set $\S{A}$ is given. Since we do not impose restrictions on the attribute set $\S{A}$, the above definition of time series covers a broad range of sequential data structures. For example, to represent real-valued univariate and multivariate time series, we use $\S{A} = \R$ and $\S{A} = \R^d$, resp., as attribute set. For text strings and biological sequences, the set $\S{A}$ is an alphabet consisting of a finite set of symbols. Further examples are time series of satellite images and time series of graphs as studied in anomaly detection.
Time series vary in length and speed. To filter out these variations, we introduce the technique of dynamic time warping.
\[definition:warping-path\] Let $m, n \in \N$. A *warping path* of order $m \times n$ is a sequence $p = (p_1 , \dots, p_L)$ of $L$ points $p_l = (i_l,j_l) \in [m] \times [n]$ such that
1. $p_1 = (1,1)$ and $p_L = (m,n)$ *(*boundary conditions*)*
2. $p_{l+1} - p_{l} \in \cbrace{(1,0), (0,1), (1,1)}$ for all $l \in [L-1]$ *(*step condition*)*
The set of all warping paths of order $m \times n$ is denoted by $\S{P}_{m,n}$. A warping path of order $m \times n$ can be thought of as a path in a $[m] \times [n]$ grid, where rows are ordered top-down and columns are ordered left-right. The boundary condition demands that the path starts at the upper left corner and ends in the lower right corner of the grid. The step condition demands that a transition from on point to the next point moves a unit in exactly one of the following directions: down, diagonal, and right.
A warping path $p = (p_1, \ldots, p_L)\in \S{P}_{m,n}$ defines an alignment (or warping) between time series $x = (x_1, \ldots, x_m)$ and $y = (y_1, \ldots, y_n)$. Every point $p_l = (i_l,j_l)$ of warping path $p$ aligns element $x_{i_l}$ to element $y_{j_l}$. The *cost* of aligning time series $x$ and $y$ along warping path $p$ is defined by $$%\label{eq:cost}
c_p(x,y) = \sum_{l=1}^L d\args{x_{i_l}, y_{j_l}},$$ where $d: \S{A} \times \S{A} \rightarrow \R$ is a *local distance function* on $\S{A}$. We demand that the local distance $d$ satisfies the following properties:
1. $d(a, a') \geq 0$
2. $d(a, a) = 0$
3. $d(a, a') = d(a', a)$
for all $a, a'\in \S{A}$. As with the attribute set $\S{A}$ we tacitly assume that the local distance $d$ is given without further mention.
Now we are in the position to define the DTW-distance. We obtain the DTW-distance between two time series $x$ and $y$ by minimizing the cost $c_p(x,y)$ over all possible warping paths.
\[def:DTW-distance\] Let $f:\R_{\geq 0} \rightarrow \R$ be a monotonous function. Let $x$ and $y$ be two time series of length $m$ and $n$, respectively. The *DTW-distance* between $x$ and $y$ is defined by $$%\label{eq:def_dtw}
\dtw(x,y) = \min \cbrace{f\args{c_p(x,y)} \,:\, p \in \S{P}_{m,n}}.$$ An *optimal warping path* is any warping path $p \in \S{P}_{m,n}$ satisfying $\dtw(x, y) = f\args{c_p(x,y)}$.
The next example presents a common and widely applied DTW-distance in order to illustrates all components of Definition \[def:DTW-distance\].
\[ex:Euclidean-DTW\] The *Euclidean DTW-distance* is specified by the attribute set $\S{A} = \R^d$, the squared Euclidean distance $d(x, y) = \normS{x-y}{^2}$ for all $x, y \in \S{A}$, and the square root function $f(x) = \sqrt{x}$ for all $x \in \R_{\geq 0}$.
Even if the underlying local distance function $d$ is a metric, the induced DTW-distance is generally only a pseudo-semi-metric satisfying
1. $\dtw(x, y) \geq 0$
2. $\dtw(x, x) = 0$
for all $x, y \in \S{T}$. Computing the DTW-distance and deriving an optimal warping path is usually solved by applying techniques from dynamic programming [@Sakoe1978].
A *DTW-space* is a pair $\args{\S{T}, \dtw}$ consisting of a set of time series of finite length and a DTW-distance $\delta$ defined on $\S{T}$. For the sake of convenience, we occasionally write $\S{T}$ to denote a DTW-space and tacitly assume that $\dtw$ is the underlying DTW-distance.
Fréchet Functions
-----------------
Let $\args{\S{T}, \dtw}$ be a DTW-space. A *loss function* is a monotonously increasing function of the form $h: \R_{\geq 0} \rightarrow \R$. A typical example of a loss function is the squared loss $h(u) = u^2$ for all $u \geq 0$.
\[definition:Frechet-Function\] Let $\S{X} = \args{x^{(1)}, \ldots, x^{(N)}} \in \S{T}^N$ be a sample of $N$ time series $x^{(k)}$ with corresponding loss function $h_k: \R_{\geq 0} \rightarrow \R$ for all $k \in [N]$. Then the function $$\begin{aligned}
%\label{eq:frechet}
F: \S{T} \rightarrow \R, \quad x \mapsto \frac{1}{N}\sum_{i = 1}^N h_k\args{\dtw\args{x,x^{(k)}}}\end{aligned}$$ is the *Fréchet function* of sample $\S{X}$ corresponding to the loss functions $h1, \ldots, h_N$.
We omit explicitly mentioning the corresponding loss functions of a Fréchet function if no confusion can arise. Note that Definition \[definition:Frechet-Function\] refers to the unrestricted form as described in Section \[sec:intro\]. We present some examples and assume that $\S{X} = \args{x^{(1)}, \ldots, x^{(N)}} \in \S{T}^N$ is a sample of $N$ time series.
\[ex:Fp\] Let $p \geq 1$. The Fréchet function of $\S{X}$ corresponding to the loss functions $h_k(u) = u^p$ takes the form $$\begin{aligned}
F^p(x) = \sum_{k = 1}^N \dtw^p\args{x,x^{(k)}}.\end{aligned}$$ For $p = 1$ *($p=2$)* the Fréchet function $F^p$ generalizes the concept of sample median (sample mean) in Euclidean spaces.
\[ex:wFp\] Let $w_k > 0$. The Fréchet function of $\S{X}$ corresponding to the loss functions $h_k(u) = w_k \cdot u^p$ is of the form $$\begin{aligned}
F^p(x) = \sum_{k = 1}^N w_k\dtw^p\args{x,x^{(k)}}.\end{aligned}$$ The function $F^p(x)$ is a weighted average of the sum of $p$-distances $\delta^p$. In the special case of $w_k = 1/N$, the function $F^p(x)$ averages the sum of $p$-distances.
Next, we consider the global minimizers – if exist – of a Fréchet function.
The *sample mean set* of $\S{X}$ is the (possibly empty) set defined by $$\S{F} = \cbrace{z \in \S{T} \,:\, F(z) \leq F(x) \text{ for all } x \in \S{T}}.$$ The elements of $\S{F}$ are the *sample means* of $\S{X}$.
A sample mean is a time series that minimizes the Fréchet function $F$. It can happen that the corresponding set $\S{F}$ is empty. In this case, a sample mean does not exist. Existence of a sample mean depends on the choice of DTW-distance and loss function. Moreover, if a sample mean exists, it may not be uniquely determined. In contrast, the *sample variance*[^1] $$F^* = \inf_{x \in \S{T}} F(x)$$ exists and is uniquely determined, because the DTW-distance is bounded from below and the loss is monotonously increasing. Thus, existence of a sample mean means that the Fréchet function $F$ attains its infimum.
The next example presents a DTW-space for which a sample mean may not always exist. This example is inspired by the edit distance for sequences and drastically simplified to directly convey the main idea.
\[example:non-existence\] Let $\S{A} = \R$ be the attribute set with local cost function $d$ of the form $$d(a, a') =
\begin{cases}
\argsS{a-a'}{^2} & a \neq 0 \text{ \emph{and} } a' \neq 0\\
1 & a = 0 \text{ \emph{xor} } a' = 0\\
0 & a = 0 \text{ \emph{and} } a' = 0
\end{cases}$$ for all $a, a' \in \S{A}$. We assume that the function $f$ corresponding to the DTW distance is the identity such that $$\dtw(x, y) = \min \cbrace{c_p(x,y) \,:\, p \in \S{P}_{m,n}},$$ for all time series $x$ and $y$ of length $m$ and $n$, respectively. Consider the sample $\S{X} = (x, y)$ consisting of two time series $x = (1, 1)$ and $y = (1, -1)$. As indicated by Figure \[fig:ex\_nonexistence\], the Fréchet function $$F(z) = \frac{1}{2}\delta(x, z) + \frac{1}{2}\delta(y,z)$$ never attains its infimum $0.5$. Thus $F(x)$ has no global minimum and therefore $\S{X}$ has no sample mean.
![Example of non-existence of a sample mean. Plot (a) shows a sample $\S{X} = \args{x^{(1)}, x^{(2)}}$ of two time series and a candidate solution $x$ of the Fréchet function $F$. The black lines indicate optimal warping paths between the sample time series and $x$. Any warping path must satisfy the boundary condition. Therefore, it is sufficient to consider the case that $x$ is of length $2$. Plot (b) depicts the sample time series and candidate solution $x$ of length $2$ as points in the vector space $\R^2$. Plot (c) shows the Fréchet function $F(x)$ of $\S{X}$ corresponding to the DTW distance of Example \[example:non-existence\]. Suppose $(z_i)$ is a sequence of candidate solutions starting at $z_1 = x$ and converging to $x$ on a straight line as indicated by the dashed arrow in plot (b). Then the induced sequence $(F(z_i))$ is strictly monotonously decreasing and converges to but never attains its infimum $0.5$. Hence, the Fréchet function $F$ has not a global minimum, which implies that $\S{X}$ has not a sample mean.[]{data-label="fig:ex_nonexistence"}](./ex_nonexistence.png){width="99.00000%"}
Restricted and Unrestricted Fréchet Functions
---------------------------------------------
The Fréchet function $F:\S{T} \rightarrow \R$ of Definition \[definition:Frechet-Function\] is in unrestricted form, because it is defined on the entire set $\S{T}$ and imposes no restrictions on the length of the sample mean. In restricted form, the function $$F_m: \S{T}_m \rightarrow \R, \quad x \mapsto F(x).$$ is the Fréchet function of $\S{X}$ restricted to the subset $\S{T}_m \subset \S{T}$ of all time series of length $m$. It is important to note that the lengths of the sample time series in $\S{X}$ may vary, but the length of the independent variable $x$ of $F_m(x)$ is fixed beforehand to value $m$. In line with the unrestricted form, the set $$\S{F}_m = \cbrace{z \in \S{T}_m \,:\, F_m(z) \leq F_m(x) \text{ for all } x \in \S{T}_m}$$ is the *restricted sample mean set* of $\S{X}$ restricted to the subset $\S{T}_m$. Occasionally, we call the elements of $\S{F}_m$ restricted sample means and the elements of $\S{F}$ the (unrestricted) sample means. As for the unrestricted form, existence of a restricted sample mean depends on the choice of DTW-distance and loss function.
The Reduction Theorem
---------------------
This section presents sufficient conditions for existence of a sample mean in restricted and unrestricted form. The approach is as follows: First, we present sufficient conditions for existence of restricted sample means. Second, under these assumptions we infer that an unrestricted sample mean also exists. The main tool for the second step is the Reduction Theorem. Proofs are delegated to the appendix.
Suppose that $\S{X} = \args{x^{(1)}, \ldots, x^{(N)}} \in \S{T}^N$ is a sample of $N$ time series. The Reduction Theorem is based on the notion of *reduction bound* $\rho(\S{X})$ of sample $\S{X}$. The exact definition of $\rho(\S{X})$ requires some technicalities and is fully spelled out in Section \[subsec:glued-graphs\]. Here, we present a simpler definition that conveys the main idea and covers the relevant use cases in pattern recognition. For this, we assume that every sample time series $x^{(k)}$ of sample $\S{X}$ has length $\ell(x^{(k)}) \geq 2$. Then the *reduction bound* of $\S{X}$ is defined by $$\begin{aligned}
\label{eq:reduction-bound-simple}
\rho(\S{X}) = \sum_{k=1}^N \ell\args{x^{(k)}} - \; 2(N-1).\end{aligned}$$ In contrast to Eq. , the exact definition of $\rho(\S{X})$ admits samples that contain trivial time series of length one. Equation shows that the reduction bound of a sample increases linearly with the sum of the lengths of the sample time series. The following results hold for arbitrary samples and assume the exact definition of a reduction bound as provided in Section \[subsec:glued-graphs\].
\[theorem:reduction\] Let $F$ be the Fréchet function of a sample $\S{X}\in \S{T}^N$. Then for every time series $x \in \S{T}$ of length $\ell(x) > \rho(\S{X})$ there is a time series $x' \in \S{T}$ of length $\ell(x') = \ell(x) -1$ such that $F(x') \leq F(x)$.
The Reduction Theorem deserves some explanations. To illustrate the following comments we refer to Figures \[fig:ex\_reduce01\] – \[fig:ex\_longmean\] with the following specifications: In these figures, we assume univariate time series with real values. The underlying distance is the Euclidean DTW-distance of Example \[ex:Euclidean-DTW\]. The Fréchet functions of the different samples $\S{X} = \args{x^{(1)}, x^{(2)}}$ are given by $$F(z) = \frac{1}{2}\delta\args{x^{(1)}, z} + \,\frac{1}{2}\delta\args{x^{(1)}, z}.$$ The figures show warping paths by black lines connecting aligned elements of the time series to be compared. We make the following observations:
#### {#section .unnumbered}
From the proof of the Reduction Theorem follows that every candidate solution $x$ whose length exceeds the reduction bound has an element that can be removed without increasing the value $F(x)$. Such elements are said to be *redundant*. Figure \[fig:ex\_reduce01\] schematically characterizes redundant elements of a time series.
#### {#section-1 .unnumbered}
In general, removing a redundant element does not increase the Fréchet function. Figure \[fig:ex\_reduce01\] shows that removing a redundant element can even decrease the value of the Fréchet function.
#### {#section-2 .unnumbered}
The reduction bound of the sample in Figure \[fig:ex\_reduce01\] is given by $$\rho(\S{X}) = \ell\args{x^{(1)}}+\;\ell\args{x^{(2)}} - \;2(N-1) = 4 + 4 - 2 = 6.$$ The length of time series $x$ is only $\ell(x) = 5 < \rho(\S{X})$. This shows that short candidate solutions whose lengths are bounded by the reduction bound may also have redundant elements that can be removed without increasing the value of the Fréchet function. Existence of a redundant element depends on the choice of warping path between $x$ and the sample time series. For short time series $x$, we can always find warping paths such that $x$ has no redundant elements. In contrast, long time series whose lengths exceed the reduction bound always have a redundant element, regardless which warping paths we consider.
#### {#section-3 .unnumbered}
Removing a non-redundant element of a candidate solution can increase the value of the Fréchet function. Figure \[fig:ex\_reduce02\] presents an example.
#### {#section-4 .unnumbered}
The Reduction Theorem does not exclude existence of sample means whose lengths exceed the reduction bound of a sample. Figure \[fig:ex\_longmean\] presents an example for which a sample mean can have almost any length.
The Reduction Theorem and observations 1–4 form the basis for existence proofs in unrestricted form and point to a technique to improve algorithms for approximating a sample mean (if exists). Statements on the existence of a sample mean are presented in the next section. From observations 1–3 follows that a candidate solution $x$ of any length could be improved or at least shortened by detecting and removing redundant elements of $x$. This observation is not further explored in this article and left for further research.
![Schematic depiction of redundant elements. Shown are two sample time series $x^{(1)}$ and $x^{(2)}$ (red) and two further time series $x$ and $x'$ (blue). The third element of $x$ is redundant (enclosed by a circle). Redundant elements are characterized by the following property: An element $x_i$ of time series $x$ is redundant if every element of the sample time series connected to $x_i$ is also connected to another element of $x$. The elements of the sample time series connected to the third element of $x$ are enclosed by a square. Both squared elements are connected to two elements of $x$. The time series $x'$ is obtained from $x$ by removing its redundant element. The DTW-distances of the sample time series from $x$ are both one and from $x'$ are both zero (see main text for a specification of the DTW distance). Then we obtain $F(x) > F(x')$. This shows that removing the redundant element in $x$ does not increase the value of the Fréchet function. In this particular case, the value of the Fréchet function is even decreased.[]{data-label="fig:ex_reduce01"}](./ex_reduce01.png){width="80.00000%"}
------------------------------------------------------------------------
![Removing a non-redundant element can increase the Fréchet function. The time series $x'$ is obtained from $x$ by removing the first element. The first element of $x$ is not redundant according to the characterization of redundant elements given in Figure \[fig:ex\_reduce01\]. The DTW-distances of the sample time series from $x$ are both zero and from $x'$ are both one (see main text for a specification of the DTW distance). This shows that $F(x) < F(x')$.[]{data-label="fig:ex_reduce02"}](./ex_reduce02.png){width="75.00000%"}
------------------------------------------------------------------------
![Restricted sample means of almost any length (see main text for a specification of the DTW distance). The time series $x$ is a sample mean of $\S{X}$, because $F(x) = 0$ is the lowest possible value. The time series $x'$ is obtained from the sample mean $x$ by appending arbitrarily many time points with element $3$. From $F(x')=0$ follows that $x'$ is also a sample mean of $\S{X}$.[]{data-label="fig:ex_longmean"}](./ex_longmean.png){width="65.00000%"}
Sufficient Conditions of Existence
----------------------------------
In this section, we derive sufficient conditions of existence of a sample mean in restricted and unrestricted form.
The Reduction Theorem guarantees existence of a sample mean in unrestricted form if sample means exist in restricted forms. Thus, existence proofs in the general unrestricted form reduce to existence proofs in the simpler restricted form. This statement is proved in Corollary \[cor:existence:restricted-to-unrestricted\].
\[cor:existence:restricted-to-unrestricted\] Let $\S{X}\in \S{T}^N$ be a sample and let $\rho \in \N$ be the reduction bound of $\S{X}$. Suppose that $\S{F}_m \neq \emptyset$ for every $m \in \bracket{\rho}$. Then $\S{X}$ has a sample mean.
It is not self-evident that existence of a class of restricted sample mean implies existence of a sample mean. To see this, we define the restricted (sample) variance by $$F_m^* = \inf_{x \in \S{T}_m} F_m(x).$$ If $F_m$ attains its infimum, then $\S{X}$ has a restricted sample mean. Suppose that $\S{X}$ has a restricted sample mean for every $m \in \N$. Then $$v_m = \min_{l \leq m} F_m^*.$$ is the smallest restricted variance over all lengths $l \in [m]$. The sequence $(v_m)_{m \in\N}$ is bounded from below and monotonously decreasing. Therefore, the sequence $(v_m)_{m \in\N}$ converges to the unrestricted sample variance $F_*$. Then $\S{X}$ has a sample mean only if the sequence $(v_m)_{m \in\N}$ attains its infimum $F_*$. Corollary \[cor:existence:restricted-to-unrestricted\] guarantees that the sequence $(v_m)_{m \in\N}$ indeed attains its infimum $F^*$ latest at $m = \rho(\S{X})$.
Next, we present sufficient conditions of existence. The first result proposes sufficient conditions of existence of a restricted and unrestricted sample mean for time series (sequences) with discrete attribute values.
\[prop:restricted-existence-discrete-case\] Let $\S{X} \in \S{T}^N$ be a sample. Suppose that $\S{A}$ is a finite attribute set. Then the following statements hold:
1. $\S{F}_m \neq \emptyset$ for every $m\in \N$.
2. $\S{F}\neq \emptyset$.
The second result proposes sufficient conditions of existence of a restricted and unrestricted sample mean of uni- and multivariate time series with elements from $\S{A} = \R^d$.
\[prop:sufficient-conditions-of-existence\] Let $\S{X} \in \S{T}^N$ be a sample. Suppose that the following assumptions hold:
1. $\args{\S{A}, d}$ is a metric space of the form $\args{\R^d, \norm{\cdot}}$, where $\norm{\cdot}$ is a norm on $\R^d$.
2. The loss functions $h_1, \ldots, h_N$ are continuous and strictly monotonously increasing.
Then the following statements hold:
1. $\S{F}_m \neq \emptyset$ for every $m\in \N$.
2. $\S{F}\neq \emptyset$.
The attribute set $\S{A}$ covers the case of univariate ($d=1$) and the case of multivariate ($d>1$) time series. The local cost function $d$ on $\S{A}$ is a metric induced by a norm on $\R^d$. Loss functions $h:\R_{\geq 0} \rightarrow \R$ of the form $h(u) = w\cdot u^p$ are continuous and strictly monotonously increasing for $w > 0$ and $p \geq 1$. Thus, the sufficient conditions of Prop. \[prop:sufficient-conditions-of-existence\] cover customary DTW-spaces. We conclude this section with a remark on weighted means.
\[cor:existence-of-weighted-mean\] Proposition \[prop:sufficient-conditions-of-existence\] holds when we replace the loss functions $h_k$ by the loss functions $h'_k = w_k h_k$ with $w_k \in \R_{\geq 0}$ for all $k \in [N]$.
Note that only loss functions $h'_k = w_k h_k$ with positive weights $w_k > 0$ are strictly monotonously increasing. Hence, assumption (2) of Prop. \[prop:sufficient-conditions-of-existence\] is violated for loss functions $h_k'$ with zero-weights $w_k = 0$. Corollary \[cor:existence-of-weighted-mean\] relaxes the condition of strictly positive weights to non-negative weights. For a proof of Remark \[cor:existence-of-weighted-mean\] we refer to Section \[subsec:proofs\].
Conclusion
==========
This article presents sufficient conditions for the existence of a sample mean in DTW spaces in restricted and unrestricted form. The sufficient conditions hold for common DTW distances reported in the literature. Key result is the Reduction Theorem stating that time series whose lengths exceed the reduction bound can be reduced to shorter time series without increasing the value of the Fréchet function. This result guarantees existence of a sample mean in unrestricted form if sample means exist in restricted form. The proof of the Reduction Theorem is framed into the theory of warping graphs. The existence proofs theoretically justify existing mean-algorithms and related pattern recognition applications in retrospect. The Reduction Theorem sets the stage for studying the unrestricted sample mean problem. Finally, existence of the sample mean sets the stage for constructing exact algorithms and a statistical theory of DTW spaces. The next step towards such a theory consists in studying under which conditions a sample mean is a consistent estimator of a population mean.
#### ***Acknowledgements.*** {#acknowledgements. .unnumbered}
B. Jain was funded by the DFG Sachbeihilfe `JA 2109/4-1`.
[00]{}
W.H. Abdulla, D. Chow, and G. Sin. Cross-words reference template for DTW-based speech recognition systems. *Conference on Convergent Technologies for Asia-Pacific Region*, 2003.
M. Brill, T. Fluschnik, V. Froese, B. Jain, R. Niedermeier, D. Schultz. Exact Mean Computation in Dynamic Time Warping Spaces. *arXiv*, arXiv:1710.08937, 2017.
M. Cuturi and M. Blondel. Soft-DTW: a differentiable loss function for time-series. , 2017.
I.L. Dryden and K.V. Mardia. *Statistical shape analysis*, Wiley, 1998.
A. Feragen, P. Lo, M. De Bruijne, M. Nielsen, and F. Lauze. Toward a theory of statistical tree-shape analysis. *IEEE Transaction of Pattern Analysis and Machine Intelligence*, 35:2008–2021, 2013.
M. Fréchet. Les éléments aléatoires de nature quelconque dans un espace distancié. *Annales de l’institut Henri Poincaré*, 215–310, 1948.
C.E. Ginestet. Strong Consistency of Fréchet Sample Mean Sets for Graph-Valued Random Variables. arXiv: 1204.3183, 2012.
L. Gupta, D. Molfese, R. Tammana, and P.G. Simos. Nonlinear alignment and averaging for estimating the evoked potential. *IEEE Transactions on Biomedical Engineering*, 43(4):348–356, 1996.
V. Hautamaki, P. Nykanen, P. Franti. Time-series clustering by approximate prototypes. *International Conference on Pattern Recognition*, 2008.
S. Huckemann, T. Hotz, and A. Munk. Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric Lie group actions. *Statistica Sinica*, 20:1–100, 2010.
B.J. Jain. Statistical Analysis of Graphs. *Pattern Recognition*, 60:802–812, 2016.
D.G. Kendall. Shape manifolds, procrustean metrics, and complex projective spaces. *Bulletin of the London Mathematical Society*, 16:81–121, 1984.
J.B. Kruskal and M. Liberman. The symmetric time-warping problem: From continuous to discrete. *Time warps, string edits and macromolecules: The theory and practice of sequence comparison*, pp. 125–161, 1983.
J.S. Marron and A.M. Alonso. Overview of object oriented data analysis. *Biometrical Journal*, 56(5):732–753, 2014.
V. Niennattrakul and C.A. Ratanamahatana. Shape averaging under time warping. *IEEE International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology*, 2009.
F. Petitjean, A. Ketterlin, and P. Gancarski. A global averaging method for dynamic time warping, with applications to clustering. *Pattern Recognition* 44(3):678–693, 2011.
F. Petitjean, G. Forestier, G.I. Webb, A.E. Nicholson, Y. Chen, and E. Keogh. Dynamic Time Warping Averaging of Time Series Allows Faster and More Accurate Classification. *IEEE International Conference on Data Mining*, 2014.
F. Petitjean, G. Forestier, G.I. Webb, A.E. Nicholson, Y. Chen, and E. Keogh. Faster and more accurate classification of time series by exploiting a novel dynamic time warping averaging algorithm. *Knowledge and Information Systems*, 47(1):1–26, 2016.
L.R. Rabiner and J.G. Wilpon. Considerations in applying clustering techniques to speaker-independent word recognition. *The Journal of the Acoustical Society of America*, 66(3): 663–673, 1979.
H. Sakoe and S. Chiba. Dynamic programming algorithm optimization for spoken word recognition. *IEEE Transactions on Acoustics, Speech, and Signal Processing*, 26(1):43–49, 1978.
P. Sathianwiriyakhun, T. Janyalikit, and C.A. Ratanamahatana. Fast and accurate template averaging for time series classification. *IEEE International Conference on Knowledge and Smart Technology*, 2016.
D. Schultz and B. Jain. Nonsmooth analysis and subgradient methods for averaging in dynamic time warping spaces. *Pattern Recognition*, 74:340–358, 2017.
S. Soheily-Khah, A. Douzal-Chouakria, and E. Gaussier. Generalized k-means-based clustering for temporal data under weighted and kernel time warp. *Pattern Recognition Letters*, 75:63–69, 2016.
Theory of Warping Graphs
========================
This appendix develops a theory of warping graphs to prove the Reduction Theorem and all other results in this article. The line of argument follows a bottom-up approach. First, we introduce warping chains to model abstract warping paths as given in Definition \[definition:warping-path\] and derive their relevant local properties. Then we proceed to warping graphs that model the alignment of two time series by a warping path and derive global properties from local properties. We enhance warping graphs with node labels and define the notion of weight of a warping graph to model the DTW-distance. Finally, we glue warping graphs to model Fréchet functions, derive the Reduction Theorem, and prove the statements presented in Section \[sec:reduction-theorem+implications\].
Warping Chains
--------------
The basic constituents of a Fréchet function are time series and optimal warping paths. This section represents the linear order of time series by chains. Then we introduce the notion of warping chain to model abstract warping paths and study its local properties.
Let $\S{V}$ be a partially ordered set with partial order $\leq_{\S{V}}$. Suppose that $i,j \in \S{V}$ are two elements with $i \leq_{\S{V}} j$. We write $i <_{\S{V}} j$ to mean $i \leq_{\S{V}} j$ and $i \neq j$. A linear order $\leq_{\S{V}}$ on $\S{V}$ is a partial order such that any two elements in $\S{V}$ are comparable: For all $i, j \in \S{V}$, we have either $i \leq_{\S{V}} j$ or $j \leq_{\S{V}} i$.
A *chain* is a linearly ordered set.
A chain $\S{V}$ models the order of a time series $x = (x_1, \ldots, x_m)$, where element $i \in \S{V}$ refers to the positions of element $x_i$ in $x$. For the sake of convenience, we assume that the explicit notation of a chain $\S{V} = \cbrace{i_1, \ldots, i_m}$ always lists its elements in linear order $i_1 <_{\S{V}} \cdots <_{\S{V}} i_m$. We call $i_1$ the first and $i_m$ the last element in $\S{V}$. The first and last element of a chain are the boundary elements. Any element of chain $\S{V}$ that is not a boundary element is called an inner element of $\S{V}$.
A subset $\S{V}' \subseteq \S{V}$ is a subchain of $\S{V}$. Note that any subset of a chain is again a chain by transitivity of the linear order. Suppose that $i_p, i_q \in \S{V}$ such that $i_p \leq_{\S{V}} i_q$. Then the subchain $\S{V}' = \cbrace{i \in \S{V} \,:\, i_p \leq_{\S{V}} i \leq_{\S{V}} i_q}$ is said to be contiguous.
Let $\S{V} = \cbrace{i_1, \ldots, i_m}$ be a chain and let $\S{V}^* = \S{V} \cup \cbrace{*}$, where $*$ is a distinguished symbol denoting the void element. The successor $i_l^+$ and predecessor $i_l^-$ of element $i_l \in \S{V}$ are defined by $$\begin{aligned}
i_l^+ = \begin{cases}
i_{l+1} & 1 \leq l < L \\
* & l = L
\end{cases}
& \qquad \text{ and } \qquad
i_l^- = \begin{cases}
i_{l-1} & 1 < l \leq L \\
* & l = 1
\end{cases}.\end{aligned}$$
We assume that $\S{W}$ is another chain. The chains $\S{V}$ and $\S{W}$ induce a partial order on the product $\S{U} = \S{V} \times \S{W}$ by $$(i,j) \leq_{\S{U}} (r, s) \; \Leftrightarrow \; i \leq_{\S{V}} r \text{ and } j \leq_{\S{W}} s$$ for all $(i,j), (r, s) \in \S{U}$.
Let $\S{U} = \S{V} \times \S{W}$ be the product of chains $\S{V}$ and $\S{W}$. The *successor map* on $\S{U}$ is a point-to-set map $$S_{\S{U}}: \S{U} \rightarrow 2^{\S{U}}, \quad (i,j) \;\mapsto\;
\cbrace{\args{i^+, j}, \args{i, j^+}, \args{i^+, j^+}} \; \cap \; \big(\S{V} \times \S{W}\big),$$ where $2^{\S{U}}$ denotes the set of all subsets of $\S{U}$.
The successor map models the set of feasible warping steps for a given element $(i,j) \in \S{U}$. Intersection of the successor map with $\S{V} \times \S{W}$ ensures that elements with $i^+ = *$ or $j^+=*$ are excluded. The successor map sends $(i,j)$ to the empty set if $i$ and $j$ are the last elements of the respective chains $\S{V}$ and $\S{W}$. The next result shows that the successor map preserves the partial product order $\leq_{\S{U}}$ as well as the linear orders $\leq_{\S{V}}$ and $\leq_{\S{W}}$.
\[lemma:order-preserving\] Let $\S{U} = \S{V} \times \S{W}$ be the product of chains $\S{V}$ and $\S{W}$. Suppose that $e = (i,j) \in \S{U}$ is an element with $S_{\S{U}}(e) \neq \emptyset$. Then the following order preserving properties hold:
1. $e \leq_{\S{U}} e'$ for all $e'\in S_{\S{U}}(e)$.
2. $i \leq_{\S{V}} r \text{ and } j \leq_{\S{W}} s$ for all $(r,s)\in S_{\S{U}}(i,j)$.
Directly follows from the definitions of $\leq_{\S{U}}$ and $S_{\S{U}}$.
\[lemma:warping-chain\] Let $\S{U} = \S{V} \times \S{W}$ be the product of chains $\S{V}$ and $\S{W}$. Let $\S{E} \subseteq \S{U}$ be a subset consisting of $L$ elements $e_1, \ldots, e_L \in \S{E}$ such that $e_{l+1} \in S_{\S{U}}(e_l)$ for all $l \in [L-1]$. Then $\S{E}$ is a chain.
The successor map is order preserving with respect to the product order $\leq_{\S{U}}$ according to Lemma \[lemma:order-preserving\]. The assertion follows, because any order is transitive.
The chain $\S{E} \subseteq \S{U}$ in Lemma \[lemma:warping-chain\] is compatible with the successor map $S_{\S{U}}$. We call such a chain a warping chain.
Let $\S{U} = \S{V} \times \S{W}$ be the product of chains $\S{V}$ and $\S{W}$. A *warping chain* in $\S{U}$ is a chain $\S{E} = \cbrace{e_1, \ldots, e_L} \subseteq \S{W}$ such that $e_{l+1} \in S_{\S{U}}(e_l)$ for all $l \in [L-1]$.
The next result shows that warping chains preserve the order of the factor chains.
\[prop:order-preserving-factors\] Let $\S{V}$ and $\S{W}$ be chains. Let $\S{E}$ be a warping chain in $\S{V} \times \S{W}$. Then any pair of elements $(i, j), (r, s) \in \S{E}$ satisfies $$\begin{aligned}
\label{eq:prop:order-preserving-factors}
\args{i \leq_{\S{V}} r \; \wedge \; j \leq_{\S{W}} s} \; \vee \; \args{r \leq_{\S{V}} i \; \wedge \; s \leq_{\S{W}} j}.\end{aligned}$$
Suppose that $\S{E} = \cbrace{e_1, \ldots, e_L}$. Then there are indices $p, q \in [L]$ such that $e_p = (i,j)$ and $e_q = (r,s)$. Without loss of generality, we assume that $p \leq q$. Let $u = q - p$. Repeatedly applying Lemma \[lemma:order-preserving\] yields $$e_p \leq_{\S{U}} \cdots \leq_{\S{U}} e_{p+u} = e_q,$$ where $\S{U} = \S{V} \times \S{W}$. Since any order is transitive, we have $e_p \leq_{\S{U}} e_q$. Then the assertion directly follows from the definition of the product order $\leq_{\S{U}}$.
Equation is the order-preserving property (or non-crossing property) of a warping chain. Note that the order preserving property does not hold for all subsets of $\S{V} \times \S{W}$.
Warping Graphs
--------------
This section introduces the notion of warping graph that models the alignment of two time series by a warping path and studies its local and global structure.
A graph is a pair $G = \args{\S{V}, \S{E}}$ consisting of a finite set $\S{V} \neq \emptyset$ of nodes and a set $\S{E} \subseteq \S{V} \times \S{V}$ of edges. A node $i \in \S{V}$ is incident with an edge $e \in \S{E}$, if there is a node $j\in \S{V}$ such that $e = (i,j)$ or $e = (j,i)$. Similarly, an edge $(i,j) \in \S{E}$ is said to be incident to node $i$ and to node $j$. The neighborhood of node $i \in \S{V}$ is the subset of nodes defined by $\S{N}(i) = \cbrace{j \in \S{V} \,:\, (i, j) \in \S{E} \text{ or } (j,i) \in \S{E}}$. The elements of $\S{N}(i)$ are the neighbors of $i$. The degree $\deg(i) = \abs{\S{N}(i)}$ of node $i$ in $G$ is the number of neighbors of $i$.
A subgraph of graph $G = \args{\S{V}, \S{E}}$ is a graph $G' = \args{\S{V}', \S{E}'}$ such that $\S{V}' \subseteq \S{V}$ and $\S{E}' \subseteq \S{E}$. We write $G' \subseteq G$ to denote that $G'$ is a subgraph of $G$. A graph $G$ is connected, if for any two nodes $i, j \in \S{V}$ there is a sequence $i = u_1, u_2, \ldots, u_n = j$ of nodes in $G$ such that $u_{k+1} \in \S{N}(u_k)$ for all $k \in [n-1]$. A component $C$ of graph $G$ is a connected subgraph $C \subseteq G$ such that $C \subseteq C'$ implies $C = C'$ for every connected subgraph $C' \subseteq G$.
A graph $G = \args{\S{U}, \S{E}}$ is bipartite, if $\S{U}$ can be partitioned into two disjoint and non-empty subsets $\S{V}$ and $\S{W}$ such that $\S{E} \subseteq \S{V} \times \S{W}$. We write $G = (\S{V}, \S{W}, \S{E})$ to denote a bipartite graph with node partitions $\S{V}$ and $\S{W}$. Note that the order of the node partitions $\S{V}$ and $\S{W}$ in a bipartite graph $G = \args{\S{U}, \S{E}}$ matters. A bipartite chain graph is a bipartite graph whose node partitions are chains.
A bipartite chain graph $G = \args{\S{V}, \S{W}, \S{E}}$ with node partitions $\S{V} = \cbrace{i_1, \ldots, i_m}$ and $\S{W} = \cbrace{j_1, \ldots, j_n}$ is a *warping graph* of size $m \times n$ if
1. $\args{i_1,j_1}, \args{i_m,j_n} \in \S{E}$ *(*boundary condition*)*
2. $\S{E}$ is a warping chain in $\S{V} \times \S{W}$ *(*step condition*)*,
The set of all warping graphs of size $m \times n$ is denoted by $\S{G}_{m,n}$. If $G = \args{\S{V}, \S{W}, \S{E}}$ is a warping graph, we briefly write $S_G$ to denote the successor map $S_{\S{V} \times \S{W}}$ and $\leq_G$ to denote the induced product order $\leq_{\S{V}\times \S{W}}$. The following result is a direct consequence of the boundary and step conditions:
\[prop:isolated-nodes\] Every node in a warping graph has a neighbor.
We show that the neighborhood of a node of one partition of a warping graph is a contiguous chain of the other partition.
\[prop:contiguous-neighborhood\] Let $G$ be a warping graph with node partitions $\S{Z}$ and $\S{Z}'$. Suppose that $i \in \S{Z}$ is a node. Then the neighborhood $\S{N}(i)$ of a node in $i \in \S{Z}$ is a contiguous subchain of $\S{Z}'$.
Suppose that the warping graph is of the form $G = \args{\S{V}, \S{W}, \S{E}}$. Without loss of generality, we assume that $\S{Z} = \S{V}$ and $\S{Z}' = \S{W}$. Then we have $i \in \S{V}$. The assertion trivially holds for $\abs{\S{N}(i)} = 1$. Suppose that $\abs{\S{N}(i)} > 1$. We assume that $\S{N}(i)$ is not contiguous. Then there are elements $j', j'' \in \S{N}(i)$ and $j \in \S{W} \setminus \S{N}(i)$ such that $j' \leq_G j \leq_G j''$. From Prop. \[prop:isolated-nodes\] follows that there is a node $i' \in \S{V}\setminus \cbrace{i}$ such that $(i',j) \in \S{E}$.
Two cases can occur: (1) $i' <_{\S{V}} i$ and (2) $i <_{\S{V}} i'$. It is sufficient to consider the first case $i' <_{\S{V}} i$. The proof of the second case is analogue. By construction, there are edges $(i',j)$ and $(i,j')$ such that $i' <_{\S{V}} i$ and $j' <_{\S{W}} j$. These relationships violate the order preserving property of a warping chain given in Eq. of Prop. \[prop:order-preserving-factors\]. Hence, $\S{N}(i)$ is contiguous.
We introduce compact warping graphs that represent warping paths of minimal length.
A warping graph $G \in \S{G}_{m,n}$ is *compact* if there is no warping graph $G' \in \S{G}_{m,n}$ such that $G'$ is a proper subgraph of $G$.
A warping graph is compact if no edge can be deleted without violating the boundary or step conditions. Figure \[fig:ex\_compact\_wg\] shows an example of a non-compact warping graph and its compactification.
![Example of a non-compact warping graph $G$ (a) and its compactification $H$ (b).[]{data-label="fig:ex_compact_wg"}](./ex_compact_wg.png){width="60.00000%"}
\[prop:characterization-compactness\] Let $G$ be a warping graph with edge set $\S{E} = \cbrace{e_1, \ldots, e_L}$. Then the following statements are equivalent:
1. $G$ is compact.
2. Let $2 \leq k < L$. Then $e_{l+k} \notin S_G(e_l)$ for all $l \in [L-k]$.
We first prove the following Lemma:
Let $\S{V}$ and $\S{W}$ be two chains, let $\S{E}= \cbrace{e_1, \ldots, e_L} \subseteq \S{V} \times \S{W}$ be a warping chain, and let $3 \leq k < L$. Then $e_{l+k} \notin S_{\S{V} \times \S{W}}(e_l)$ for every $l \in [L-k]$.
Let $\S{C} = \cbrace{i_1, \ldots, i_m}$ be a chain and let $i_k, i_l \in \S{C}$ be two elements of $\S{C}$. Then we define the distance $$\Delta_{\S{C}}\args{i_k,i_l} = \abs{l-k}+1.$$ Suppose that $e_l = (i,j)$ for some $l \in [L-1]$. Let $e'=(r,s) \in S_{\S{V} \times \S{W}}(s_l)$ be an arbitrary successor of $e_l$. Then by definition of the successor map, we have $\Delta_{\S{V}}(i,r), \Delta_{\S{W}}(j,s) \in \cbrace{0,1}$. Let $l \in [L-k]$. Suppose that $e_l = (i,j)$ and $e_{l+k} = (r,s)$. Then by induction, we have $$\Delta_{\S{V}}(i,r) + \Delta_{\S{W}}(j,s) \geq k.$$ From $k \geq 3$ follows $\Delta_{\S{V}}(i,r) \geq 2$ or $\Delta_{\S{W}}(j,s) \geq 2$. Hence, $\Delta_{\S{V}}(i,r) \notin \cbrace{0,1}$ or $\Delta_{\S{W}}(j,s) \notin \cbrace{0,1}$. This shows the assertion $e_{l+k} \notin S_{\S{V} \times \S{W}}(e_l)$.
From the above Lemma follows that the case $k > 2$ is impossible for a warping chain. Therefore it is sufficient to consider the case $k = 2$.
Let $G$ be compact. We assume that there is an $l \in [L-2]$ such that $e_{l+2} \in S_G(e_l)$. By construction, the edge $e_{l+1}$ is an inner element of the chain $\S{E}$. From $e_{l+2} \in S_G(e_l)$ follows that removing $e_{l+1}$ neither violates the boundary conditions nor the step condition. This contradicts compactness of $G$ and shows that a compact warping graph $G$ implies the second statement.
Next, we show the opposite direction. Suppose that $e_{l+2} \notin S_G(e_l)$ for all $l \in [L-2]$. We assume that $G$ is not compact. Then there is an edge $e_k \in \S{E}$ that can be removed without violating the boundary and step conditions. Not violating the boundary condition implies that $1 < k < L$. Hence, $e_{k-1}$ and $e_{k+1}$ are edges in $\S{E}$. We set $l = k-1$. Then we obtain the contradiction that $e_{l} \in S_G(e_{l+2})$. Hence, $G$ is compact.
Suppose that $G = \args{\S{V}, \S{W}, \S{E}}$ is a warping graph. By $\S{V} \sqcup \S{W}$ we denote the disjoint union of the node partitions. If $i \in \S{V} \sqcup \S{W}$ is a node of one partition, then its neighborhood $\S{N}(i)$ is a subset of the other node partition. Hence, $\S{N}(i)$ is a chain and has boundary and eventually inner nodes. Let $\S{N}^\circ(i)$ denote the possibly empty subset of inner nodes of chain $\S{N}(i)$. We show that inner nodes of $\S{N}(i)$ always have degree one.
\[lemma:star-graph-decomposition:1\] Let $G = \args{\S{V}, \S{W}, \S{E}}$ be a warping graph. Suppose that $i \in \S{V} \sqcup \S{W}$ is a node with neighborhood $\S{N}(i)$. Then $\deg(j) = 1$ for all $j \in \S{N}^\circ(i)$.
Without loss of generality, we assume that $i \in \S{V}$. Then $\S{N}(i) \subseteq \S{W}$ is a chain. The assertion holds for $\abs{\S{N}(i)} \leq 2$, because in this case $\S{N}(i)$ has no inner node. Suppose that $\abs{\S{N}(i)} > 2$. Let $j \in \S{N}(i)$ be an inner node. We assume that $\deg(j) > 1$. Then there is a node $i' \in \S{V} \setminus \cbrace{i}$ such that $(i', j) \in \S{E}$. Since $\S{V}$ is a chain, we find that either $i' <_{\S{V}} i$ or $i <_{\S{V}} i'$.
We only consider the first case $i' <_{\S{V}} i$. The proof of the second case is analogue. Observe that $j$ is an inner node of $\S{N}(i)$ and $\S{N}(i)$ is contiguous by Prop. \[prop:contiguous-neighborhood\]. Then $\S{N}(i)$ contains the predecessor $j' = j^-$ of node $j$. By construction $e = (i, j')$ and $e' = (i',j)$ are edges of $\S{E}$ such that $i' <_{\S{V}} i$ and $j' <_{\S{W}} j$. Thus, the edges $e$ and $e'$ violate the order preserving property of a warping chain given in Eq. of Prop. \[prop:order-preserving-factors\]. This contradicts our assumption that $\S{E}$ is a warping chain. Hence, we have $\deg(j) = 1$. Since the inner node $j$ was chosen arbitrarily, the assertion follows.
\[lemma:star-graph-decomposition:2\] Let $G = \args{\S{V}, \S{W}, \S{E}}$ be a warping graph and let $i \in \S{V} \sqcup \S{W}$ be a node with neighborhood $\S{N}(i)$. Suppose that $\abs{\S{N}(i)} \geq 2$ and $j \in \S{N}(i)$ is a boundary node with $deg(j) \geq 2$. Then the following properties hold:
1. If $j$ is the first node in $\S{N}(i)$, then $i^- \in \S{V}$ exists and $(i^-, j) \in \S{E}$.
2. If $j$ is the last node in $\S{N}(i)$, then $i^+ \in \S{V}$ exists and $(i^+, j) \in \S{E}$.
We show the second assertion. The proof of the first assertion is analogue. Since $\abs{\S{N}(i)} \geq 2$ and $j$ is the last node of $\S{N}(i)$, we find that $j' = j^- \in \S{N}(i)$ and therefore $(i,j') \in \S{E}$ exists. From $\deg(j) \geq 2$ follows that there is a node $i' \in \S{V}$ such that $(i',j) \in \S{E}$. We assume that $i' \in \S{N}(j)$ satisfies $i' <_{\S{V}} i$ . This implies that $e = (i,j')$ and $e' = (i',j)$ are two edges of $\S{E}$ such that $i' <_{\S{V}} i$ and $j' < _{\S{W}} j$. Thus, the edges $e$ and $e'$ violate the order preserving property of a warping chain given in Eq. of Prop. \[prop:order-preserving-factors\]. This contradicts our assumption that $i' <_{\S{V}} i$. Therefore, we have $i <_{\S{V}} i'$. This in turn shows that $i^+ \in \S{V}$ exists. From $i, i' \in \S{N}(j)$ and $i <_{\S{V}} i'$ follows $i^+ \in \S{N}(j)$, because $\S{N}(j)$ is a contiguous subchain of $\S{V}$ by Prop. \[prop:contiguous-neighborhood\]. This shows $(i^+, j) \in \S{E}$ and completes the proof.
A bipartite graph $G = (\S{V}, \S{W}, \S{E})$ is complete if $\S{E} = \S{V} \times \S{W}$. Let $r \in \N$. A complete bipartite graph $G = (\S{V}, \S{W}, \S{E})$ is a star graph of the form $K_{1,r}$, if $\abs{\S{V}} = 1$ and $\abs{\S{W}} = r$. Similarly, $G$ is a star graph of the form $K_{r,1}$, if $\abs{\S{V}} = r$ and $\abs{\S{W}} = 1$. By definition, a star graph has at least two nodes. A star forest is a graph whose components are star graphs.
\[prop:star-graph-decomposition\] A compact warping graph is a star forest.
Let $G = \args{\S{V}, \S{W}, \S{E}}$ be a compact warping graph. Suppose that $C = \args{\S{V}', \S{W}', \S{E}'}$ is a component of $G$. From Prop. \[prop:isolated-nodes\] follows that $C$ has at least two nodes connected by an edge.
We assume that $C$ is not a star. Then $C$ has two nodes $i, j \in \S{V}' \sqcup \S{W}'$ with degree larger than one. Without loss of generality, we assume that $i \in \S{V}'$. Then $\S{N}(i) \subseteq \S{W}'$ has at least two elements. Suppose that all nodes from $\S{N}(i)$ have degree one. Since component $C$ is bipartite, we find that $C$ is isomorphic to the star $K_{1,r}$, where $r = \deg(i) > 1$. This contradicts our assumption that $C$ is not a star. Hence, there is a node $j \in \S{N}(i) \subseteq \S{W}'$ with $\deg(j) > 1$.
From Lemma \[lemma:star-graph-decomposition:1\] follows that node $j$ is a boundary node of $\S{N}(i)$. We show the assertion for the case that $j$ is the last node in $\S{N}(i)$. The proof for the case that $j$ is the first node in $\S{N}(i)$ is analogue. Since $j$ is the last node in $\S{N}(i)$ and $\abs{\S{N}(i)} \geq 2$, we have $j^- \in \S{N}(i)$ and therefore $(i,j^-) \in \S{E}$. Applying Lemma \[lemma:star-graph-decomposition:2\] yields that $i^+ \in \S{V}$ exists and $(i^+, j) \in \S{E}$.
By construction, we have $\args{i, j^-}, \args{i, j}, \args{i^+, j} \in \S{E}$. This shows that $(i,j)$ is not a boundary edge in $\S{E}$. Since $\args{i^+, j} \in S_{\S{V} \times \S{W}}\args{i, j^-}$, we can remove $(i,j)$ without violating the step condition. Then the subgraph $G' = \args{\S{V}, \S{W}, \S{E}\setminus \cbrace{(i,j)}}$ of $G$ is a warping graph. This contradicts our assumption that $G$ is compact. Hence, $C$ is a star.
An immediate consequence of the proof of Prop. \[prop:star-graph-decomposition\] is the following corollary.
\[cor:prop:star-graph-decomposition\] Let $G = \args{\S{V}, \S{W}, \S{E}}$ be a compact warping graph in $\S{G}_{m,n}$. Then every component of $G$ is a star with at least one node in $\S{V}$ and one node in $\S{W}$.
\[prop:2-star-graph-distribution\] Let $G \in \S{G}_{m,n}$ be a compact warping graph with $m > n$. Then we have:
1. $G$ has at most $n-1$ components of the form $K_{1,1}$.
2. $G$ has a component of the form $K_{r,1}$ with $r > 1$.
Let $G = \args{\S{V}, \S{W}, \S{E}}$ with $\abs{\S{V}} = m$ and $\abs{\S{W}} = n$. To show the first assertion, we assume that $G$ has $n' > n-1$ components of the form $K_{1,1}$. This is only possible for $n' = n$, because every component of $G$ has at least one node in $\S{W}$ by Corollary \[cor:prop:star-graph-decomposition\].
Let $C_1, \ldots, C_n$ be $n$ components of $G$ of the form $K_{1,1}$. Suppose that $\S{C}_k = \args{\cbrace{i_k}, \cbrace{j_k}, \cbrace{\args{i_k, j_k}}}$ for all $k \in [n]$. The union of the first node partitions over the $n$ components $C_k$ gives $\S{V}' = \cbrace{i_1, \ldots, i_n}$. From $m > n$ follows $\S{V}' \subsetneq \S{V}$. Then there is a node $i \in \S{V}\setminus\S{V}'$. From Prop. \[prop:star-graph-decomposition\] follows that there is a component $C$ of $G$ is a star of the form $K_{r,s}$ that contains node $i$. Since $s \geq 1$ by definition of a star, component $C$ has a node $j \in \S{W}$. Then there is a $k \in [n]$ such that $j = j_k$ is a node in component $\S{C}_k$. Since $i \neq i_k$ by construction, the graph $H = \args{\cbrace{i, i_k}, \cbrace{j_k}, \cbrace{\args{i,j_k}, \args{i_k,j_k}}}$ is a connected subgraph of $G$ that includes component $C_k$ as a proper subgraph. This contradicts our assumption that $C_k$ is a maximal connected subgraph of $G$. Consequently, $G$ cannot have more than $n-1$ components of the form $K_{1,1}$.
Next, we show the second assertion. Suppose that $C_1, \ldots, C_q$ are all components of $G$ that are of the form $K_{1,1}$. Let $\S{V}' = \cbrace{i_1, \ldots, i_q} \subseteq \S{V}$ and $\S{W}' = \cbrace{j_1, \ldots, j_q} \subseteq \S{W}$ be the subsets covered by the $q$ components $C_k$. From the first part of this proof follows that $q < n$ and by assumption, we have $n < m$. Then $\S{V}'' = \S{V}\setminus\S{V}'$ and $\S{W}'' = \S{W}\setminus\S{W}'$ are non-empty. By $m'' = \abs{\S{V}''} = m-q$ and $n'' = \abs{\S{W}''} = n-q$ we denote the respective number of nodes not contained in any of the $q$ components $C_k$. From $q < n < m$ follows that $1 \leq n'' < m''$. The pigeonhole principle states that there is at least one node $j \in \S{W}''$ that is connected to at least two nodes $i, i' \in \S{V}''$. Let $C$ be the component of $G$ containing the three nodes $i$, $i'$, and $j$. From Prop. \[prop:star-graph-decomposition\] follows that $C$ is a star of the form $K_{r,s}$. We find that $r \geq 2$, because $C$ contains at least the two nodes $i$ and $i'$ from $\S{V}'' \subset \S{V}$. Then $s = 1$, because $C$ is a star. This shows the second assertion.
A node $i$ of a compact warping graph $G$ is *redundant* if $\deg(j) \geq 2$ for all $j \in \S{N}(i)$.
Let $i$ be a node in $G$. Then $G-\cbrace{i}$ is the subgraph of $G$ obtained by deleting node $i$ and its incident edges. The next result shows that deleting a redundant node of a compact warping graph is again a compact warping graph.
\[prop:reduce-redundant-node\] Let $i$ be a redundant node of a compact warping graph $G$. Then $G-\cbrace{i}$ is a compact warping graph.
Without loss of generality, we assume that $i \in \S{V}$. Let $G = \args{\S{V}, \S{W}, \S{E}}$ with warping chain $\S{E} = \cbrace{e_1, \ldots, e_L}$. Suppose that $G' = G-\cbrace{i} = \args{\S{V}', \S{W}, \S{E}'}$, where $\S{V}' = \S{V} \setminus \cbrace{i}$ and $\S{E}'$ is the chain obtained from $\S{E}$ by removing all edges incident to node $i$.
We first show that $\S{N}(i) \subseteq \S{W}$ consists of a singleton. Let $C$ be the component of $G$ that contains node $i$. Then $C$ also includes the neighborhood $\S{N}(i)$. Since $G$ is compact, we can apply Prop. \[prop:star-graph-decomposition\] and find that component $C$ is a star of the form $K_{r,1}$ or $K_{1,r}$, where $r \geq 1$. Observe that $r = \deg(j) \geq 2$ for every neighbor $j \in \S{N}(i)$, because node $i$ is redundant. Hence, $C$ is a star of the form $K_{r,1}$ and therefore $\S{N}(i) = \cbrace{j}$.
From $\deg(j) \geq 2$ follows that there is a node $i' \in \S{V} \setminus \cbrace{i}$ such that $(i',j) \in \S{E}$. We distinguish between two cases: (1) $i' <_{\S{V}} i$ and (2) $i <_{\S{V}} i'$. We only consider the first case $i' <_{\S{V}} i$. The proof of the second case is analogue. From Prop. \[prop:contiguous-neighborhood\] follows that $\S{N}(j)$ is a contiguous subchain of $\S{W}$ with $i', i \in \S{N}(j)$. Then $i^- \in \S{N}(j)$. We distinguish between two cases: (1) $e_L = (i,j)$, and (2) $e_l = (i,j)$ for some $1 < l < L$. Note that the case $e_1 = (i,j)$ cannot occur due to existence of $i^- \in \S{V}$.
*Case 1:* From $e_L = (i,j)$ follows that the edge set of $G'$ is of the form $\S{E}' = \cbrace{e_2, \ldots, e_{L-1}}$. In addition, $i$ is the last node in $\S{V}$ and therefore $i^-$ is the last node in $\S{V}'$. Furthermore, we find that $j \in \S{W}$ is the last node in $\S{W}$. We show that $\S{E}'$ is a warping chain. The first boundary condition is satisfied by $e_1 \in \S{E}'$. From $i^- \in \S{N}(j)$ follows that $(i^-, j) \in \S{E}$ is an edge that satisfies the second boundary condition connecting the last nodes of $\S{V}'$ and $\S{W}$. Finally, from $e_{l+1} \in S_G(e_l)$ for all $l \in [L-1]$ follows that the step condition remains valid in $G'$. Therefore, the edge set $\S{E}'$ is a warping chain of length $L' = L-1$. It remains to show that $G'$ is compact. For this, we assume that $G'$ is not compact. Then from Prop. \[prop:characterization-compactness\] follows that there is an index $l \in [L'-2]$ such that $e_{l+2} \in S_{G'}(e_l)$. This implies that $e_{l+2} \in S_{G}(e_l)$ contradicting the assumption that $G$ is compact. Hence, $G'$ is a compact warping graph.
*Case 2:* There is an index $1 < l < L$ such that $e_l = (i,j)$. Hence, $\S{E}$ has at least three edges and the edge set of $G'$ is of the form $\S{E}' = \cbrace{e_1, \ldots,e_{l-1}, e_{l+1}, \ldots, e_L}$. To show that $\S{E}'$ is a warping chain, we assume that $e_{l-1} = (i',j')$ and $e_{l+1} = (i'',j'')$. From $\S{N}(i) = \cbrace{j}$ and the step condition follows that $i' = i^-$ and $i'' = i^+$. This shows that $i$ is neither the first nor last node in $\S{V}$. Hence, $e_1$ and $e_L$ satisfy the boundary conditions in $\S{E}'$.
We show that $\S{E}'$ satisfies the step condition. Since $\S{E}$ is a warping chain, we have $e_{k+1} = S_G(e_k)$ for all $k \in [L-1]$. Since $S_{G'} = S_G$ on $\S{E}'$, it is sufficient to show that $e_{l+1}\in S_{G'}(e_l)$. According to the previous parts of the proof, we have $(i^-,j) \in \S{E}$. The step condition together with $i' = i^-$ imply that $e_{l-1} = (i',j') = (i^-, j') = (i^-,j)$ and therefore $j' = j$. Again, from the step condition follows that either $j'' = j$ or $j'' = j^+$. Observe that $i'^{+} = i''$ in $\S{V}'$. Then we have $(i'', j) \in S_{G'}(i',j)$ and $(i'', j^+) \in S_{G'}(i',j)$. This shows that $\S{E}'$ satisfies the step condition in either of both cases $j'' = j$ or $j'' = j^+$.
It remains to show that $G'$ is compact. For this, we assume that $G'$ is not compact. Suppose that $\S{I} = [L]\setminus \cbrace{l}$ is the index set of $\S{E}'$. Then from Prop. \[prop:characterization-compactness\] follows that there is an index $k \in \S{I}\setminus\cbrace{L-1,L}$ such that $e_{k+2} \in S_{G'}(e_k)$. This implies that $e_{k+2} \in S_{G}(e_k)$ contradicting the assumption that $G$ is compact. Hence, $G'$ is a compact warping graph.
Glued Warping Graphs {#subsec:glued-graphs}
--------------------
This section glues warping graphs to model Fréchet function. Then the graph-theoretic foundation of the Reduction Theorem is presented.
A graph $G = \args{\S{U}, \S{E}}$ is a centered $N$-partite graph if the set $\S{U}$ can be partitioned into $N+1$ disjoint and non-empty subsets $\S{V}, \S{W}_1 \ldots, \S{W}_N$ such that $$\S{E} \subseteq \bigcup_{k=1}^N \S{V} \times \S{W}_k.$$
Let $G_1, \ldots, G_N$ be compact warping graphs of the form $G_k = \args{\S{V}, \S{W}_k, \S{E}_k}$ for all $k \in [N]$. The *glued graph* with *splice* $\S{V}$ and *particles* $G_1, \ldots, G_N$ is a centered $N$-partite graph $G = \args{\S{V}, \S{W}_1, \ldots, \S{W}_N, \S{E}}$ with edge set $\S{E} = \S{E}_1 \sqcup \cdots \sqcup \S{E}_N$.
Note that a particle of a glued graph is always a compact warping graph. The definition of a glued graph assumes that all $N$ particles $G_1, \ldots, G_N$ share a common node partition $\S{V}$ and that any two particles $G_k$ and $G_l$ have disjoint node partitions $\S{W}_k$ and $\S{W}_l$, respectively. Then the glued graph with splice $\S{V}$ is obtained by taking the disjoint union of the particles $G_1, \ldots, G_N$, but by identifying the nodes from $\S{V}$. A special case of a glued graph is any compact warping graph $G = \args{\S{V}, \S{W}, \S{E}}$ with the first partition $\S{V}$ as its splice.
Let $G$ be a glued graph with splice $\S{V}$ and particles $G_1, \ldots, G_N$. Node $i \in \S{V}$ is *redundant* in $G$, if it is redundant in $G_k$ for every $k \in [N]$.
\[prop:reduced-glued-graph\] Let $G$ be a glued graph with splice $\S{V}$ and particles $G_1, \ldots, G_N$. Suppose that $i \in \S{V}$ is redundant. Then $G-\cbrace{i}$ is a glued graph with splice $\S{V}\setminus\cbrace{i}$ and particles $G_1-\cbrace{i}, \ldots, G_N-\cbrace{i}$.
Let $k \in [N]$. Then particle $G_k = \args{\S{V}, \S{W}_k, \S{E}_k}$ is a compact warping graph by definition of a glued graph. Since splice node $i \in \S{V}$ is redundant in $G$, it is redundant in $G_k$. Then from Prop. \[prop:reduce-redundant-node\] follows that $G_k' = G_k - \cbrace{i} = \args{\S{V}', \S{W}_k, \S{E}'}$ is a compact warping graph with $\S{V}' = \S{V}\setminus\cbrace{i}$ and $\S{E}_k' = \S{E}_k\setminus \S{E}_k(i)$, where $\S{E}_k(i) \subseteq \S{E}_k$ is the subset of edges in $G_k$ incident to node $i$.
The graph $G' = G-\cbrace{i} = \args{\S{V}', \S{W}_1, \ldots, \S{W}_N, \S{E}'}$ has an edge set of the form $\S{E}' = \S{E}\setminus\S{E}(i)$, where $\S{E}(i) \subseteq \S{E}$ is the subset of edges in $G$ incident to node $i$. Since $\S{E} = \S{E}_1 \sqcup \cdots \sqcup \S{E}_N$, we have $$\S{E}(i) = \S{E}_1(i) \sqcup \cdots \sqcup \S{E}_N(i) = \S{E}_1'\sqcup \cdots \sqcup \S{E}_N'.$$ This shows that $G-\cbrace{i}$ is a glued graph of particles $G_1', \ldots, G_N'$ along splice $\S{V}'$.
Suppose that $G$ is a glued graph with splice $\S{V}$ and particles $G_1, \ldots, G_N$. A particle is said to be *trivial* if it is a star of the form $K_{m,1}$. By $\S{I}_G \subseteq [N]$ we denote the subset of all indices $k \in [N]$ for which particle $G_k$ is non-trivial. We call $\S{I}_G$ the *core index set* (core) of $G$.
Let $G$ be a glued graph with splice $\S{V}$ and $N$ particles $G_k = \args{\S{V}, \S{W}_k, \S{E}_k}$ for all $k \in [N]$. Suppose that $\S{I}_G$ is the core index set of $G$. Then $$\rho(G) = \begin{cases}
\displaystyle \sum_{k \in \S{I}_G} \abs{\S{W}_k} - \;2\args{\abs{\S{I}_G}-1} & \S{I}_G \neq \emptyset \\
1 & \S{I}_G = \emptyset
\end{cases}$$ is the *reduction bound* of $G$.
Now we present the graph-theoretic foundation of the Reduction Theorem.
\[prop:existence-redundant-node\] Let $G$ be a glued graph with splice $\S{V}$ such that $\rho(G) < \abs{\S{V}}$. Then $\S{V}$ has a redundant node.
Suppose that $G_1, \ldots, G_N$ are the particles of $G$ with $G_k = \args{\S{V}, \S{W}_k, \S{E}_k}$ for all $k \in [N]$. Let $m = \abs{\S{V}}$, $n_k = \abs{\S{W}_k}$, and $\S{N}_k(i) = \S{N}(i) \cap \S{W}_k$ for every $k \in [N]$.
We first consider the special case that $\S{I}_G = \emptyset$. Then all $N$ particles are trivial, that is $n_k = 1$ for every $k \in [N]$. The reduction bound is of the form $\rho(G) = 1$. By assumption, we have $m > \rho(G)$. Hence, every particle $G_k$ is a star of the form $K_{m,1}$, where $m > 1$. This shows that every splice node is redundant.
Next, we assume that $\S{I}_G \neq \emptyset$. We set $N' = \abs{\S{I}_G}$. Obviously, we have $N' \geq 1$. We say, $G_k$ supports node $i \in \S{V}$, if there is a node $j \in \S{N}_k(i) \subseteq \S{W}_k$ with $\deg(j) = 1$. It is sufficient to show that $\S{V}$ has a node not supported by any of the $N$ particles $G_1, \ldots, G_N$. The proof proceeds in four steps.
#### {#section-5 .unnumbered}
We show that $n_k < m$ for any $k \in \S{I}_G$. Suppose that $\S{J} = \S{I}_G\setminus\cbrace{k}$. Then from $\S{I}_G \neq \emptyset$ follows $$\begin{aligned}
\rho(G) = \sum_{l \in \S{I}_G} n_l - 2(N'-1) = n_k + \sum_{l \in \S{J}} n_l - 2(N'-1). \end{aligned}$$ From $l \in \S{I}_G$ follows $n_l \geq 2$. This together with $\abs{\S{J}} = N'-1$ yields $$\rho(G)
\geq n_k + \sum_{l \in \S{J}} 2 -2(N'-1)
= n_k + 2\abs{\S{J}} -2(N'-1)
= n_k.$$ Then from $m > \rho(G)$ follows $m > n_k \geq 2$.
#### {#section-6 .unnumbered}
We bound the number of splice nodes that can be supported by any non-trivial particle. For any $k \in \S{I}_G$ let $\S{W}_k'\subseteq \S{W}_k$ be the subset of nodes in $G_k$ that support a splice node. We define a map $\phi_k:\S{W}_k' \rightarrow \S{V}$ such that $(\phi_k(j), j) \in \S{E}_k$. Such a map exists due to the boundary and step conditions of warping graph $G_k$. Moreover, the map $\phi_k$ is uniquely determined, because $\deg(j) = 1$ for any node $j \in \S{W}_k'$. This shows that $\S{V}_k = \phi_k\!\args{\S{W}_k'}$ is the set of splice nodes supported by $G_k$. Since $\phi_k$ is bijective, we have $\abs{\S{V}_k} = \abs{\S{W}_k'}$.
From step 1 of this proof follows that $G_k$ is a compact warping graph with $n_k < m$. Therefore, we can apply Prop. \[prop:2-star-graph-distribution\] and obtain that $G_k$ has at most $n_k-1$ components of the form $K_{1,1}$ and at least once component of the form $K_{r,1}$ with $r > 1$. This shows that $\abs{\S{W}_k'} = \abs{\S{V}_k} \leq n_k-1$.
#### {#section-7 .unnumbered}
We show that there is a splice node not supported by any non-trivial particle. For this, we define the set $$\S{U} = \bigcup_{k\in \S{I}_G} \S{V}_k$$ of all splice nodes that are supported by at least one non-trivial particle of $G$. Then it is sufficient to show that $m > \abs{\S{U}}$. We consider three cases: (1) $N' = 1$, (2) $N' = 2$, and (3) $N' > 2$.
*Case 1: $N' = 1$.* Suppose that $\S{I}_G = \cbrace{u}$. Since $\S{I}_G \neq \emptyset$, the reduction bound is of the form $$\rho(G) = n_u - 2(N'-1) = n_u \geq 2.$$ According to step 2, we have $n_u-1 \geq \abs{\S{V}_u}$. By using $\S{U} = \S{V}_u$ we find that $$m > \rho(G) > \abs{\S{V}_u} = \abs{\S{U}}.$$
*Case 2: $N' = 2$.* Suppose that $\S{I}_G = \cbrace{u, v}$. Since $\S{I}_G \neq \emptyset$, the reduction bound takes the form $$\rho(G) = n_u + n_v - 2(N'-1) = n_u + n_v - 2 = (n_u -1)+(n_v-1).$$ According to step 2, we have $n_u-1 \geq \abs{\S{V}_u}$ and $n_v-1 \geq \abs{\S{V}_v}$. By using $\S{U} = \S{V}_u \cup \S{V}_v$ we obtain $$m > \rho(G) \geq \abs{\S{V}_u} + \abs{\S{V}_v} \geq \abs{\S{U}}.$$
*Case 3: $N' > 2$.* Suppose that the slice $\S{V}$ is a chain of the form $\S{V} = \cbrace{i_1, \ldots, i_m}$ with boundary nodes $\bd(\S{V}) = \cbrace{i_1, i_m}$. We assume that $\abs{\S{U}} = m$. Then there are (not necessarily distinct) indices $u,v \in \S{I}_G$ such that $i_1 \in \S{V}_u$ and $i_m \in \S{V}_v$. From the boundary and step conditions follows that the boundary nodes of any $W_k$ ($k \in \S{I}_G$) can only support the respective boundary nodes of $\S{V}$ and not any other splice node. Then the first node of $\S{W}_u$ only supports $i_1 \in \S{V}$ and the last node of $\S{W}_v$ only supports $i_m \in \S{V}$.
Let $\S{J} = \cbrace{u,v}$, let $\S{J}' = \S{I}_G \setminus \S{J}$, and let $\S{V}_k' = \S{V}_k \setminus \args{\S{V}_u \cup \S{V}_v}$ for all $k \in \S{J}'$. The set $\S{V}_k'$ consists of all splice nodes supported by $G_k$ but not by $G_u$ and $G_v$. Hence, the boundary nodes of $\S{V}$ are not contained in $\S{V}_k'$. Then both boundary nodes of $\S{W}_k$ do not support any node in $\S{V}_k'$. This implies $\abs{\S{V}_k'} \leq n_k-2$. From the cardinality of the set union follows $$\begin{aligned}
\abs{\S{U}}
&\leq \sum_{l \in \S{J}} \args{n_l - 1} + \sum_{k\in \S{J}'} \args{n_k-2}
= \abs{\S{J}} + \sum_{l \in \S{J}} \args{n_l - 2} + \sum_{k\in \S{J}'} \args{n_k-2}
= \sum_{k \in \S{I}_G} n_k - 2N' + \abs{\S{J}}.\end{aligned}$$ Since $\abs{\S{J}} \leq 2$, we obtain $$\abs{\S{U}} \leq \sum_{k \in \S{I}_G} n_k - 2(N'-2) = \rho(G) < m.$$ This contradicts the assumption $\abs{\S{U}} = m$ and shows that $\abs{\S{U}} < m$ holds.
All three cases show that $\abs{\S{U}} < m$. Hence, $G$ has a splice node not supported by any of the non-trivial particles.
#### {#section-8 .unnumbered}
We show that $G$ has a splice node not supported by any of the trivial and non-trivial particles. The non-trivial part follows from step 3. Therefore, it is sufficient to consider trivial particles only. Since $\S{I}_G \neq \emptyset$ by assumption, there is a $k \in \S{I}_G$. From $m > n_k$ and $n_k \geq 2$ follows $m > 2$. This implies that the trivial particles of $G$ do not support any of the splice nodes in $\S{V}$. This completes the proof.
Labeled Warping Graphs
----------------------
This section labels the nodes of warping graphs with the attributes of the corresponding time series to be aligned. Then we introduce the weight of a labeled warping graph for modeling the cost of aligning two time series along a warping path.
We assume that $\S{A}$ is an attribute set and $d: \S{A} \times \S{A} \rightarrow \R$ is a non-negative distance function on $\S{A}$.
A *labeled warping graph* $H = \args{G, \lambda}$ consists of a warping graph $G = \args{\S{V}, \S{W}, \S{E}}$ and a labeling function $\lambda: \S{V} \sqcup \S{W} \rightarrow \S{A}$.
The labeling function $\lambda$ assigns an attribute $\lambda(i) \in \S{A}$ to any node $i \in \S{V} \sqcup \S{W}$. Thus, the nodes correspond to time points and the attributes to the elements at every time point.
The set of all labeled warping graphs of order $m \times n$ with label function $\lambda$ is denoted by $\S{G}_{m,n}^\lambda$. Since the set $\S{G}_{m,n}^\lambda$ fixes both node partitions and the label function, the graphs in $\S{G}_{m,n}^\lambda$ differ only in their edge sets. Thus, $\S{G}_{m,n}^\lambda$ describes the set of all possible warping paths that align time series $x = (x_1, \ldots, x_m)$ and $y = (y_1, \ldots, y_n)$ whose elements $x_i = \lambda(i)$ and $y_j = \lambda(j)$ are specified by the labeling function $\lambda$.
Let $H = \args{G, \lambda}$ be a labeled warping graph with edge set $\S{E}$. The *weight* of $H$ is defined by $$\omega\args{H} = \sum_{(i,j) \in \S{E}} d(\lambda(i),\lambda(j))$$
The weight of a labeled warping graph corresponds to the cost of aligning two time series along a warping path. A DTW-graph is a labeled warping graph with minimal weight.
A graph $H \in \S{G}_{m,n}^\lambda$ is a *DTW-graph*, if $$\omega(H) = \min \cbrace{\omega(H') \,:\, H' \in \S{G}_{m,n}^\lambda}.$$
The weight of a DTW-graph is the DTW-distance between the time series represented by the labeled node partitions.
Proofs of Results from Section \[sec:reduction-theorem+implications\] {#subsec:proofs}
---------------------------------------------------------------------
#### Proof of Theorem \[theorem:reduction\]. {#proof-of-theorem-theoremreduction. .unnumbered}
Let $F$ be the Fréchet function of a sample $\S{X}\in \S{T}^N$. Then for every time series $x \in \S{T}$ of length $\ell(x) > \rho(\S{X})$ there is a time series $x' \in \S{T}$ of length $\ell(x') = \ell(x) -1$ such that $F(x') \leq F(x)$.
Let $\S{X} = \args{x^{(1)}, \ldots, x^{(k)}}$, $m = \ell(x)$, and $n_k = \ell\args{x^{(k)}}$ for all $k \in [N]$. By assumption, we have $m > \rho(\S{X})$.
For every $k \in [N]$ there is an optimal warping path $p^{(k)} \in \S{P}_{m, n_k}$ aligning $x$ and $x^{(k)}$. Let $H_k = (G_k, \lambda_k)$ be the DTW-graph representing $p^{(k)}$. Then $\omega(H_k) = \dtw(x,x^{(k)})$ and $G_k =\args{\S{V}, \S{W}_k, \S{E}_k} \in \S{G}_{m,n_k}$ is a warping graph with $m = \abs{\S{V}}$ and $n_k = \abs{\S{W}_k}$. Then we have $$F(x) = \frac{1}{N} \sum_{k=1}^N h_k(\omega(H_k)),$$ where $h_1, \ldots, h_N$ are the corresponding loss functions.
Suppose that $G_k$ is non-compact and $G'_k \subseteq G_k$ is compact. Since $H_k$ is a DTW-graph, we have $\omega(H_k') = \omega(H_k)$, where $H_k' = (G_k', \lambda_k')$. Hence, without loss of generality we can assume that $G_k$ is compact for all $k \in [N]$. Let $G$ be the glued graph with splice $\S{V}$ and particles $G_1, \ldots, G_N$. Since $m > \rho(G)$, we can apply Prop. \[prop:existence-redundant-node\] and obtain that $G$ has a redundant splice node $i \in \S{V}$. Applying Prop. \[prop:reduced-glued-graph\] yields that $G' = G-\cbrace{i}$ is a glued graph with splice $\S{V}' = \S{V}\setminus\cbrace{i}$ and particles $G_1', \ldots, G_N'$. The particles $G_k'$ are of the form $G_k' = G_k-\cbrace{i} = \args{\S{V}', \S{W}_k, \S{E}_k'}$, where the edge set $\S{E}_k'$ is obtained from $\S{E}_k$ by removing all edges incident to splice node $i \in \S{V}$.
The redundant node $i\in \S{V}$ refers to element $x_i$ of time series $x = (x_1,\ldots, x_m)$. By $$x' = (x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_m)$$ we denote the time series obtained from $x$ by removing element $x_i$. Let $H_k' = (G_k', \lambda_k')$ be the resulting labeled warping graph, where $\lambda_k'$ denotes the labeling function obtained by restricting $\lambda_k$ to the subset $\S{V}' \sqcup \S{W}_k$ for all $k \in [N]$. Then the labeled warping graphs $H_k'$ represent warping paths $q^{(k)}$ that align time series $x'$ with sample time series $x^{(k)}$. By construction and definition of the weight function $\omega$, we find that $\omega(H_k') \leq \omega(H_k)$. Since the loss functions $h_k$ are monotonously increasing, we obtain $$F(x') = \frac{1}{N} \sum_{k=1}^N h_k\args{\omega(H_k')} \leq \frac{1}{N} \sum_{k=1}^N h_k\args{\omega(H_k)} = F(x).$$ By construction, we have $\ell(x') = \ell(x)-1$. This completes the proof.
#### Proof of Corollary \[cor:existence:restricted-to-unrestricted\]. {#proof-of-corollary-corexistencerestricted-to-unrestricted. .unnumbered}
Let $\S{X}\in \S{T}^N$ be a sample and let $\rho \in \N$ be the reduction bound of $\S{X}$. Suppose that $\S{F}_m \neq \emptyset$ for every $m \in \bracket{\rho}$. Then $\S{X}$ has a sample mean.
For every $m\in \bracket{\rho}$ let $F_m^*$ denote the restricted sample variance. We assume that $\S{F} = \emptyset$. Then there is a time series $x \in \S{T}$ of length $\ell(x) = p$ such that $F(x) = F_p(x) < F_m^*$ for all $m \in \bracket{\rho}$. This implies $p > \rho$, because otherwise we obtain the contradiction that $F_p(x) < F_p^*$. Let $q = p - \rho(\S{X})$. By applying Theorem \[theorem:reduction\] exactly $q$-times, we obtain a time series $x' \in \S{T}$ of length $\ell(x') = \rho$ such that $F_{\rho}^* \leq F(x') \leq F(x)$. This contradicts our assumption that $F(x) < F_{\rho}^*$. Hence, $\S{F}$ is non-empty.
#### Proof of Proposition \[prop:restricted-existence-discrete-case\]. {#proof-of-proposition-proprestricted-existence-discrete-case. .unnumbered}
Let $\S{X} \in \S{T}^N$ be a sample. Suppose that $\S{A}$ is a finite attribute set. Then the following statements hold:
1. $\S{F}_m \neq \emptyset$ for every $m\in \N$.
2. $\S{F}\neq \emptyset$.
Let $m \in \N$ be arbitrary. Since $\S{A}$ is finite, the set subset $\S{T}_m$ is also finite and consists of $m^{\abs{\S{A}}}$ time series. Then the set $F\args{\S{T}^N}$ is a finite set. Hence, the restricted sample mean set $\S{F}_m$ is non-empty and finite. Since $m$ was chosen arbitrarily, the first assertion follows. The second assertion follows from Corollary \[cor:existence:restricted-to-unrestricted\].
#### Proof of Proposition \[prop:sufficient-conditions-of-existence\]. {#proof-of-proposition-propsufficient-conditions-of-existence. .unnumbered}
Let $\S{X} \in \S{T}^N$ be a sample. Suppose that the following assumptions hold:
1. $\args{\S{A}, d}$ is a metric space of the form $\args{\R^d, \norm{\cdot}}$, where $\norm{\cdot}$ is a norm on $\R^d$.
2. The loss functions $h_1, \ldots, h_N$ are continuous and strictly monotonously increasing.
Then the following statements hold:
1. $\S{F}_m \neq \emptyset$ for every $m\in \N$.
2. $\S{F}\neq \emptyset$.
The proof of Prop. \[prop:sufficient-conditions-of-existence\] uses the notion of coercive function. A continuous function $f:\R^q \rightarrow \R$ is coercive if $$\lim_{\norm{x} \to \infty} f(x) = + \infty,$$ where $\norm{\cdot}$ is a norm on $\R^q$.
We first consider the Euclidean norm $\norm{\cdot}_{2}$ on some real-valued vector space $\R^q$. The Euclidean norm is coercive. Since $h_k$ is continuous and strictly monotonously increasing on $\R_{\geq 0}$, the composition $h_k\args{\norm{x}_2}$ is coercive and continuous for all $k \in [N]$. Every norm $\norm{\cdot}$ on $\R^q$ is equivalent to the Euclidean norm. Therefore, we can find constants $0 < c \leq C$ such that $$c\normS{x}{_2} \leq \normS{x} \leq C\normS{x}{_2}$$ for all $x \in \R^q$. Hence, $h_k\args{\norm{x}}$ is coercive and continuous for every norm on $\R^q$.
Suppose that $\S{X} = \args{x^{(1)},\dots,x^{(N)}} \in \S{T}^N$ is a proper sample of $N$ time series $x^{(k)}$ of length $\ell\args{x^{(k)}} = n_k \geq 2$ for all $k \in [N]$. Let $m \in \N$ be arbitrary. Expanding the definition of the restricted Fréchet function $F_m$ gives $$\begin{aligned}
F_m(x) = \frac{1}{N}\sum_{k=1}^N h_k\args{\dtw\args{x, x^{(k)}}} = \frac{1}{N}\sum_{k=1}^N \min \cbrace{h_k\args{c_p\args{x, x^{(k)}}} \,:\, p \in \S{P}},\end{aligned}$$ where $c_p(x, y)$ is the cost of aligning time series $x$ and $y$ along warping path $p$. Since $\S{T}_m = \S{A}^m = \R^{d \times m} = \R^q$, we can define the function $$g_{p^{(k)}}:\R^q \rightarrow \R, \quad x \mapsto c_{p^{(k)}}\args{x, x^{(k)}} = \sum_{l=1}^{L_k} h_k\args{\norm{x_{i_l} - x_{j_l}^{(k)}}},$$ where $p^{(k)} \in \S{P}_{m, n_k}$ is a warping path with $L_k$ elements aligning $x$ and $x^{(k)}$. The function $g_{p^{(k)}}$ is continuous and coercive as a sum of non-negative continuous and coercive functions. Then $g_{p^{(k)}}$ has a global minimum.
We define the set $\S{P}_m = \S{P}_{m,n_1} \times \cdots \times \S{P}_{m,n_N}$. Then every element of $\S{P}_m$ is of the form $\S{C} = \args{p^{(1)}, \ldots, p^{(N)}}$, where $p^{(k)}$ is associated to time series $x^{(k)}$ for all $k \in [N]$. Then we can equivalently rewrite the restricted Fréchet function $F_m(x)$ as $$F_m(x) = \min \cbrace{F_{\S{C}}(x) \,:\, \S{C} \in \S{P}_m},$$ where the component functions $F_{\S{C}}: \R^{d \times m} \rightarrow \R$ are functions of the form $$F_{\S{C}}(x) = \frac{1}{N}\sum_{k=1}^N g_{p^{(k)}}(x).$$ This shows that $F_{\S{C}}(x)$ has a minimum. Let $F_{\S{C}}^*$ denote the minimum value of $F_{\S{C}}(x)$. From $$\min_x F_m(x) = \min_x \min_{\S{C}} F_{\S{C}}(x) = \min_{\S{C}} \min_x F_{\S{C}}(x)$$ follows $$\min_x F_m(x) = \min_{\S{C} \in \S{P}_m} F_{\S{C}}^*.$$ Since $\S{P}_m$ is a finite set, we obtain that $F_m$ has a minimum. This shows the first assertion, because $m$ was arbitrary. The second assertion follows from Corollary \[cor:existence:restricted-to-unrestricted\].
#### Proof of Remark \[cor:existence-of-weighted-mean\]. {#proof-of-remark-corexistence-of-weighted-mean. .unnumbered}
Proposition \[prop:sufficient-conditions-of-existence\] holds when we replace the loss functions $h_k$ by the loss functions $h'_k = w_k h_k$ with $w_k \in \R_{\geq 0}$ for all $k \in [N]$.
Suppose that all weights are zero. Then the Fréchet function corresponding to the loss functions $h'_k = 0$ is zero. Hence, every time series $z \in \S{T}$ is an optimal solution and the assertion follows.
We assume that at least one weight is non-zero. Without loss of generality, let $r \in [N]$ such that $w_1, \ldots, w_r > 0 $ and $w_{r+1} = \cdots w_N = 0$. Then the loss functions $h'_1, \ldots, h'_r$ are continuous and strictly monotonously increasing. Moreover, the Fréchet function $F(z)$ of sample $\S{X}$ corresponding to the loss functions $h'_1, \ldots h'_N$ coincides with the Fréchet function $F'(z)$ of sample $x^{(1)}, \ldots, x^{(r)}$ corresponding to the loss functions $h'_1, \ldots h'_r$. Then the assertion follows from Prop. \[prop:sufficient-conditions-of-existence\].
[^1]: The sample variance $F^*$ corresponding to loss $h(u) = u^2$ generalizes the sample variance in Euclidean spaces.
|
---
abstract: 'We consider the nonlinear Hartree equation for an interacting gas containing infinitely many particles and we investigate the large-time stability of the stationary states of the form $f(-\Delta)$, describing an homogeneous Fermi gas. Under suitable assumptions on the interaction potential and on the momentum distribution $f$, we prove that the stationary state is asymptotically stable in dimension 2. More precisely, for any initial datum which is a small perturbation of $f(-\Delta)$ in a Schatten space, the system weakly converges to the stationary state for large times.'
address:
- 'CNRS & Université de Cergy-Pontoise, Mathematics Department (UMR 8088), F-95000 Cergy-Pontoise, France'
- 'Université de Cergy-Pontoise, Mathematics Department (UMR 8088), F-95000 Cergy-Pontoise, France'
author:
- Mathieu LEWIN
- Julien SABIN
title: 'The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D'
---
[^1]
Introduction
============
This article is the continuation of the previous work [@LewSab-13a] where we considered the nonlinear Hartree equation for infinitely many particles (but the main result of the present article does not rely on [@LewSab-13a]).
The Hartree equation can be written using the formalism of density matrices as $$\label{eq:Hartree}
\left\{\begin{array}{rcl}
i\partial_t\gamma & = & \big[-\Delta+w*\rho_\gamma,\gamma\big], \\
\gamma(0) & = & \gamma_0.
\end{array}\right.$$ Here $\gamma(t)$ is the one-particle density matrix of the system, which is a bounded non-negative self-adjoint operator on $L^2({{\ensuremath {\mathbb R} }}^d)$ with $d{\geqslant}1$, and $\rho_\gamma(t,x)=\gamma(t,x,x)$ is the density of particles in the system at time $t$. On the other hand $w$ is the interaction potential between the particles, which we assume to be smooth and fastly decaying at infinity.
The starting point of [@LewSab-13a] was the observation that has many stationary states. Indeed, if $f\in L^{\infty}({{\ensuremath {\mathbb R} }}_+,{{\ensuremath {\mathbb R} }})$ is such that $$\int_{{{\ensuremath {\mathbb R} }}^d}|f(|k|^2)|\,dk<+{\infty},$$ then the operator $$\gamma_f:=f(-\Delta)$$ (the Fourier multiplier by $k\mapsto f(|k|^2)$) is a bounded self-adjoint operator which commutes with $-\Delta$ and whose density $$\rho_{\gamma_f}(x)=(2\pi)^{-d} \int_{{{\ensuremath {\mathbb R} }}^d}f(|k|^2)\,dk,\qquad \forall x\in{{\ensuremath {\mathbb R} }}^d,$$ is constant. Hence, for $w\in L^1({{\ensuremath {\mathbb R} }}^d)$, $w\ast\rho_{\gamma_f}$ is also constant, and $[w\ast \rho_{\gamma_f},\gamma_f]=0$. Therefore $\gamma(t)\equiv\gamma_f$ is a stationary solution to . The purpose of [@LewSab-13a] and of this article is to investigate the stability of these stationary states, under “local perturbations”. We do not necessarily think of small perturbations in norm, but we typically think of $\gamma(0)-\gamma_f$ being compact.
The simplest choice is $f\equiv0$ which corresponds to the vacuum case. We are interested here in the case of $f\neq0$, describing an infinite, homogeneous gas containing infinitely many particles and with positive constant density $\rho_{\gamma_f}>0$. Four important physical examples are the
$\bullet$ *Fermi gas at zero temperature*: $$\gamma_f={{\ensuremath {\mathds 1} }}(-\Delta{\leqslant}\mu),\qquad\mu>0;
\label{eq:Fermi-gas-zero-temp}$$
$\bullet$ *Fermi gas at positive temperature $T>0$*: $$\gamma_f=\frac{1}{e^{(-\Delta-\mu)/T}+1},\qquad \mu\in{{\ensuremath {\mathbb R} }};
\label{eq:Fermi-gas-positive-temp}$$
$\bullet$ *Bose gas at positive temperature $T>0$*: $$\gamma_f=\frac{1}{e^{(-\Delta-\mu)/T}-1},\qquad \mu<0;
\label{eq:Bose-gas-positive-temp}$$
$\bullet$ *Boltzmann gas at positive temperature $T>0$*: $$\gamma_f=e^{(\Delta+\mu)/T},\qquad \mu\in{{\ensuremath {\mathbb R} }}.
\label{eq:Boltzmann-gas-positive-temp}$$ In the density matrix formalism, the number of particles in the system is given by ${{\rm Tr}}\,\gamma$. It is clear that ${{\rm Tr}}\,\gamma_f=+{\infty}$ in the previous examples since $\gamma_f$ is a translation-invariant (hence non-compact) operator. Because they contain infinitely many particles, these systems also have an infinite energy. In [@LewSab-13a], we proved the existence of global solutions to the equation in the defocusing case $\widehat{w}{\geqslant}0$, when the initial datum $\gamma_0$ has a finite *relative energy* counted with respect to the stationary states $\gamma_f$ given in –, in dimensions $d=1,2,3$. We also proved the orbital stability of $\gamma_f$.
In this work, we are interested in the *asymptotic stability* of $\gamma_f$. As usual for Schrödinger equations, we cannot expect strong convergence in norm and we will rather prove that $\gamma(t){\rightharpoonup}\gamma_f$ weakly as $t\to\pm{\infty}$, if the initial datum $\gamma_0$ is small enough. Physically, this means that a small defect added to the translation-invariant state $\gamma_f$ disappears for large times due to dispersive effects, and the system locally relaxes towards the homogeneous gas. More precisely, we are able to describe the exact behavior of $\gamma(t)$ for large times, by proving that $$e^{-it\Delta}\big(\gamma(t)-\gamma_f\big)e^{it\Delta}\underset{t\to\pm {\infty}}{\longrightarrow} Q_{\pm}$$ strongly in a Schatten space (hence for instance for the operator norm). This nonlinear scattering result means that the perturbation $\gamma(t)-\gamma_f$ of the homogeneous gas evolves for large times as in the case of free particles: $$\gamma(t)-\gamma_f\underset{t\to\pm {\infty}}{\simeq} e^{it\Delta}Q_{\pm}e^{-it\Delta}\underset{t\to\pm {\infty}}{{\rightharpoonup}}0.$$
If $f\equiv0$ and $\gamma_0=|u_0\rangle\langle u_0|$ is a rank-one orthogonal projection, then reduces to the well-known Hartree equation for one function $$\label{eq:Hartree-u}
\begin{cases}
i\partial_t u=(-\Delta+w*|u|^2)u,\\
u(0)=u_0.
\end{cases}$$ There is a large literature about scattering for the nonlinear equation , see for instance [@GinVel-80; @Strauss-81; @HayTsu-87; @Mochizuki-89; @GinVel-00c; @Nakanishi-99]. The intuitive picture is that the nonlinear term is negligible for small $u$, since $w*|u|^2u$ is formally of order $3$. It is important to realize that this intuition does not apply in the case $f\neq0$ considered in this paper. Indeed the nonlinear term is not small and it behaves linearly with respect to the small parameter $\gamma-\gamma_f$: $$\big[w*\rho_\gamma,\gamma\big]=\big[w*\rho_{\gamma-\gamma_f},\gamma\big]\simeq \big[w*\rho_{\gamma-\gamma_f},\gamma_f\big]\neq0.
\label{eq:linear_intro}$$ One of the main purpose of this paper is to rigorously study the linear response of the homogeneous Hartree gas $\gamma_f$ (the last term in ), which is a very important object in the physical literature, called the Lindhard function (see [@Lindhard-54] and [@GiuVig-05 Chap. 4]). For a general $f$, our main result requires that the interaction potential $w$ is small enough, in order to control the linear term. Under the natural assumption that $f$ is strictly decreasing (as it is in the three physical examples –), the condition can be weakened in the defocusing case $\widehat{w}{\geqslant}0$.
The paper is organized as follows. In the next section we state our main result and make several comments. In Section \[sec:linear-response\] we study the linear response in detail, before turning to the higher order terms in the expansion of the wave operator in Section \[sec:higher-order\]. Apart from the linear response, our method requires to treat separately the next $d-1$ terms of this expansion, in spacial dimension $d$. Even if all the other estimates are valid in any dimension, in this paper we only deal with the second order in dimension $d=2$.
Main result
===========
In the whole paper, we denote by ${\mathcal{B}}({\mathfrak{H}})$ the space of bounded operators on the Hilbert space ${\mathfrak{H}}$. The corresponding operator norm is $\|A\|$. We use the notation ${\mathfrak{S}}^p({\mathfrak{H}})$ for the Schatten space of all the compact operators $A$ on ${\mathfrak{H}}$ such that ${{\rm Tr}}|A|^p<{\infty}$, with $|A|=\sqrt{A^*A}$, and use the norm ${ \left| \! \left| A \right| \! \right| }_{{\mathfrak{S}}^p({\mathfrak{H}})}:=({{\rm Tr}}|A|^p)^{1/p}$. We refer to [@Simon-77] for the properties of Schatten spaces. The spaces ${\mathfrak{S}}^2({\mathfrak{H}})$ and ${\mathfrak{S}}^1({\mathfrak{H}})$ correspond to Hilbert-Schmidt and trace-class operators. We often use the shorthand notation ${\mathcal{B}}$ and ${\mathfrak{S}}^p$ when the Hilbert space ${\mathfrak{H}}$ is clear from the context.
Our main result is the following.
\[thm:main\] Let $f:{{\ensuremath {\mathbb R} }}_+\to{{\ensuremath {\mathbb R} }}$ be such that $$\int_{0}^{\infty}(1+r^{\frac{k}2})|f^{(k)}(r)|\,dr<{\infty}\quad\text{for $k=0,...,4$}
\label{eq:derivees_f}$$ and $\gamma_f:=f(-\Delta)$. Denote by $\check{g}$ the Fourier inverse on ${{\ensuremath {\mathbb R} }}^2$ of $g(k)=f(|k|^2)$. Let $w\in W^{1,1}({{\ensuremath {\mathbb R} }}^2)$ be such that $${ \left| \! \left| \check{g} \right| \! \right| }_{L^1({{\ensuremath {\mathbb R} }}^2)}{ \left| \! \left| {\widehat}w \right| \! \right| }_{L^{\infty}({{\ensuremath {\mathbb R} }}^2)}<4\pi
\label{eq:condition_linear_response}$$ or, if $f'<0$ a.e. on ${{\ensuremath {\mathbb R} }}_+$, such that $$\max\left({\varepsilon}_g \widehat{w}(0)_+\;,\; { \left| \! \left| \check{g} \right| \! \right| }_{L^1({{\ensuremath {\mathbb R} }}^2)}{ \left| \! \left| ({\widehat}w)_- \right| \! \right| }_{L^{\infty}({{\ensuremath {\mathbb R} }}^2)}\right)<4\pi
\label{eq:condition_negative_part}$$ where $({\widehat}w)_-$ is the negative part of ${\widehat}w$ and $0{\leqslant}{\varepsilon}_g{\leqslant}{ \left| \! \left| \check{g} \right| \! \right| }_{L^1({{\ensuremath {\mathbb R} }}^2)}$ is a constant depending only on $g$ (defined later in Section \[sec:linear-response\]).
Then, there exists a constant ${\varepsilon}_0>0$ (depending only on $w$ and $f$) such that, for any $\gamma_0\in\gamma_f+{\mathfrak{S}}^{4/3}$ with $${ \left| \! \left| \gamma_0-\gamma_f \right| \! \right| }_{{\mathfrak{S}}^{4/3}}{\leqslant}{\varepsilon}_0,$$ there exists a unique $\gamma\in \gamma_f + C^0_t({{\ensuremath {\mathbb R} }},{\mathfrak{S}}^{2})$ solution to the Hartree equation with initial datum $\gamma_0$, such that $$\rho_\gamma-\rho_{\gamma_f}\in L^2_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^2).$$ Furthermore, $\gamma(t)$ scatters around $\gamma_f$ at $t=\pm{\infty}$, in the sense that there exists $Q_\pm\in{\mathfrak{S}}^4$ such that $$\begin{gathered}
\lim_{t\to\pm{\infty}}{ \left| \! \left| e^{-it\Delta}(\gamma(t)-\gamma_f)e^{it\Delta}-Q_\pm \right| \! \right| }_{{\mathfrak{S}}^4}\\
=\lim_{t\to\pm{\infty}}{ \left| \! \left| \gamma(t)-\gamma_f-e^{it\Delta}Q_\pm e^{-it\Delta} \right| \! \right| }_{{\mathfrak{S}}^4}=0.
\label{eq:scattering}\end{gathered}$$
Before explaining our strategy to prove Theorem \[thm:main\], we make some comments.
First we notice that the gases at positive temperature , and are all covered by the theorem with the condition , since the corresponding $f$ is smooth, strictly decreasing and exponentially decaying at infinity. Our result does not cover the Fermi gas at zero temperature , however. We show in Section \[sec:linear-response\] that its linear response is unbounded and it is a challenging task to better understand its dynamical stability.
The next remark concerns the assumption which says that the interactions must be small or, equivalently, that the gas must contain few particles having a small momentum (if $\check{g}{\geqslant}0$, then the condition can be written $f(0){ \left| \! \left| \widehat w \right| \! \right| }_{L^{\infty}({{\ensuremath {\mathbb R} }}^2)}<2$ and hence $f(|k|^2)$ must be small for small $k$). Our method does not work without the condition if no other information on $w$ and $f$ is provided. However, under the natural additional assumption that $f$ is strictly decreasing, we can replace the condition by the weaker condition . The latter says that the negative part of $\widehat{w}$ and the value at zero of the positive part should be small (with a better constant for the latter). We will explain later where the condition comes from, but we mention already that we are not able to deal with an arbitrary large potential $\widehat{w}$ in a neighborhood of the origin, even in the defocusing case. We also recall that the focusing or defocusing character of our equation is governed by the sign of $\widehat{w}$ and not of $w$, as it is for . This is seen from the sign of the nonlinear term $$\int_{{{\ensuremath {\mathbb R} }}^d}\int_{{{\ensuremath {\mathbb R} }}^d}w(x-y)\rho_{\gamma-\gamma_f}(x)\rho_{\gamma-\gamma_f}(y)\,dx\,dy=(2\pi)^{d/2}\int_{{{\ensuremath {\mathbb R} }}^d}{\widehat}{w}(k)|{\widehat}{\rho_{\gamma-\gamma_f}}(k)|^2\,dk$$ which appears in the relative energy of the system (see [@LewSab-13a Eq. (9)–(10)]).
Let us mention that our results hold for *small* initial data, where the smallness is not only qualitative (meaning that $\gamma_0-\gamma_f\in {\mathfrak{S}}^{4/3}$ for instance), but also quantitative since we need that ${ \left| \! \left| \gamma_0-\gamma_f \right| \! \right| }_{{\mathfrak{S}}^{4/3}}$ be small enough. This is a well-known restriction, coming from our method of proof, based on a fixed point argument. The literature on nonlinear Schrödinger equations suggests that, in order to remove this smallness assumption, one would need some assumption on $w$ like $\widehat{w}{\geqslant}0$, as well as some additional (almost) conservation laws [@Cazenave]. Our study of the linear response operator however indicates that the situation is involved and more information on the momentum distribution $f$ is certainly also necessary.
We finally note that in our previous article [@LewSab-13a], we proved the existence of global solutions under the assumption that the initial state $\gamma_0$ has a finite relative entropy with respect to $\gamma_f$ (and for $f$ being one of the physical examples –). By the Lieb-Thirring inequality (see [@FraLewLieSei-11; @FraLewLieSei-12] and [@LewSab-13a]), this implies that $\rho_{\gamma(t)}-\rho_{\gamma_f}\in L^{\infty}_t(L^2_x)$. By interpolation we therefore get that $\rho_{\gamma(t)}-\rho_{\gamma_f}\in L^p_t(L^2_x)$ for every $2{\leqslant}p{\leqslant}{\infty}$. This requires of course that the initial perturbation $\gamma_0-\gamma_f$ be small in ${\mathfrak{S}}^{4/3}$. Our method does not allow to replace this condition by the fact that $\gamma_0$ has a small relative entropy with respect to $\gamma_f$.
We now explain our strategy for proving Theorem \[thm:main\]. The idea of the proof relies on a fixed point argument, in the spirit of [@LewSab-13a Sec. 5]. If we can prove that $\rho_\gamma-\rho_{\gamma_f}\in L^{2}_{t,x}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^2)$, then we deduce from [@Yajima-87; @FraLewLieSei-13] that there exists a family of unitary operators $U_V(t)\in C^0_t({{\ensuremath {\mathbb R} }}_+,{\mathcal{B}})$ on $L^2({{\ensuremath {\mathbb R} }}^2)$ such that $$\gamma(t)=U_V(t)\gamma_0U_V(t)^*,$$ for all $t\in{{\ensuremath {\mathbb R} }}_+$. We furthermore have $$U_V(t)=e^{it\Delta}{\mathcal{W}}_V(t),$$ where ${\mathcal{W}}_V(t)$ is the wave operator. By iterating Duhamel’s formula, the latter can be expanded in a series as $${\mathcal{W}}_V(t)=1+\sum_{n{\geqslant}1}{\mathcal{W}}_V^{(n)}(t)
\label{eq:expansion_wave_op}$$ with $$\begin{gathered}
{\mathcal{W}}_V^{(n)}(t):=(-i)^n\int_0^t\,dt_n\int_0^{t_n}\,dt_{n-1}\cdots\int_0^{t_2}\,dt_1\times\\
\times e^{-it_n\Delta}V(t_n)e^{i(t_n-t_{n-1})\Delta}\cdots e^{i(t_2-t_1)\Delta}V(t_1)e^{it_1\Delta}.
\end{gathered}$$ The idea is to find a solution to the nonlinear equation $$\label{eq:rhoQ1}
\rho_Q(t)=\rho\left[e^{it\Delta}{\mathcal{W}}_{w*\rho_Q}(t)(\gamma_f+Q_0){\mathcal{W}}_{w*\rho_Q}(t)^*e^{-it\Delta}\right]-\rho_{\gamma_f},$$ by a fixed point argument on the variable $\rho_Q\in L^2_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^2)$, where $Q:=\gamma-\gamma_f$ and $Q_0=\gamma_0-\gamma_f$.
Inserting the expansion of the wave operator ${\mathcal{W}}_V$, the nonlinear equation may be written as $$\label{eq:rhoQ}
\rho_Q=\rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right]-{\mathcal{L}}(\rho_Q)+{\mathcal{R}}(\rho_Q),$$ where ${\mathcal{L}}$ is linear and ${\mathcal{R}}(\rho_Q)$ contains higher order terms. The sign convention for ${\mathcal{L}}$ is motivated by the stationary case [@FraLewLieSei-12]. The linear operator ${\mathcal{L}}$ can be written $${\mathcal{L}}={\mathcal{L}}_1+{\mathcal{L}}_2$$ where $${\mathcal{L}}_1(\rho_Q)=-\rho\left[e^{it\Delta}({\mathcal{W}}^{(1)}_{w*\rho_Q}(t)\gamma_f+\gamma_f{\mathcal{W}}^{(1)}_{w*\rho_Q}(t)^*)e^{-it\Delta}\right]$$ and $${\mathcal{L}}_2(\rho_Q)=-\rho\left[e^{it\Delta}({\mathcal{W}}^{(1)}_{w*\rho_Q}(t)Q_0+Q_0{\mathcal{W}}^{(1)}_{w*\rho_Q}(t)^*)e^{-it\Delta}\right].$$ Note that ${\mathcal{L}}_2$ depends on $Q_0$ and it can always be controlled by adding suitable assumptions on $Q_0$. On the other hand, the other linear operator ${\mathcal{L}}_1$ does not depend on the studied solution, it only depends on the functions $w$ and $f$.
In Section \[sec:linear-response\], we study the linear operator ${\mathcal{L}}_1$ in detail, and we prove that it is a space-time Fourier multiplier of the form $\widehat{w}(k)m_f(\omega,k)$ where $m_f$ is a famous function in the physics literature called the *Lindhard function* [@Lindhard-54; @Mihaila-11; @GiuVig-05]), which only depends on $f$ and $d$. We particularly investigate when ${\mathcal{L}}_1$ is bounded on $L^p_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^2)$ and we show it is the case when $w$ and $f$ are sufficiently smooth. For the Fermi sea , we prove that ${\mathcal{L}}_1$ is unbounded on $L^2_{t,x}$.
The next step is to invert the linear part by rewriting the equation in the form $$\label{eq:rhoQ2}
\rho_Q=(1+{\mathcal{L}})^{-1}\Big(\rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right]+{\mathcal{R}}(\rho_Q)\Big)$$ and to apply a fixed point method. In the time-independent case, a similar technique was used for the Dirac sea in [@HaiLewSer-05a]. In order to be able to invert the Fourier multiplier ${\mathcal{L}}_1$, we need that $$\boxed{\phantom{\int}\min_{(\omega,k)\in{{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^2}\left|\widehat{w}(k)m_f(\omega,k)+1\right|>0.\phantom{\int}}
\label{eq:condition_w_precise}$$ Then $1+{\mathcal{L}}=1+{\mathcal{L}}_1+{\mathcal{L}}_2$ is invertible if $Q_0$ is small enough. In Section \[sec:linear-response\] we prove the simple estimate $$|m_f(\omega,k)|{\leqslant}(4\pi)^{-1}{ \left| \! \left| \check{g} \right| \! \right| }_{L^1({{\ensuremath {\mathbb R} }}^2)}$$ and this leads to our condition . If $f$ is strictly decreasing, then we are able to prove that the imaginary part of $m_f(k,\omega)$ is never 0 for $k\neq0$ or $\omega\neq0$. Since $m_f(\omega,k)$ has a fixed sign for $\omega=0$ and $k=0$, everything boils down to investigating the properties of $m_f$ at $(\omega,k)=(0,0)$. At this point $m_f$ will usually not be continuous, and it can take both positive and negative values. We have $$\limsup_{(\omega,k)\to(0,0)}\Re \,m_f(\omega,k)=(4\pi)^{-1}{ \left| \! \left| \check{g} \right| \! \right| }_{L^1({{\ensuremath {\mathbb R} }}^2)}$$ and we denote $$\liminf_{(\omega,k)\to(0,0)}\Re \,m_f(\omega,k):=-(4\pi)^{-1}{\varepsilon}_g,$$ leading to our condition . It is well-known in the physics literature that the imaginary part of the Lindhard function plays a crucial role in the dynamics of the homogeneous Fermi gas. In our rigorous analysis it is used to invert the linear response operator outside of the origin. The behavior of $m_f(\omega,k)$ for $(\omega,k)\to(0,0)$ is however involved and $1+{\mathcal{L}}_1$ is not invertible if $\widehat{w}(0)>4\pi/{\varepsilon}_g$ or $\widehat{w}(0)<-4\pi/{ \left| \! \left| \check{g} \right| \! \right| }_{L^1({{\ensuremath {\mathbb R} }}^2)}$.
For the Fermi gas at zero temperature we will prove that the minimum in is always zero, except when $\widehat{w}$ vanishes sufficiently fast at the origin, this means that $1+{\mathcal{L}}_1$ is never invertible. It is an interesting open question to understand the asymptotic stability of the Fermi sea.
Once the linear response ${\mathcal{L}}$ has been inverted, it remains to study the zeroth order term $\rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right]$ and the higher order terms contained in ${\mathcal{R}}(\rho_Q)$. At this step we use a recent Strichartz estimate in Schatten spaces which is due to Frank, Lieb, Seiringer and the first author.
\[thm:est-Wn\] Let $d{\geqslant}1$, $1+d/2{\leqslant}q<{\infty}$, and $p$ such that $2/p+d/q=2$. Let also $0<{\varepsilon}<1/p$. Then, there exists $C=C(d,p,{\varepsilon})>0$ such that for any $V\in L^p_t({{\ensuremath {\mathbb R} }},L^q_x({{\ensuremath {\mathbb R} }}^d))$ and any $t\in{{\ensuremath {\mathbb R} }}$, we have the estimates $$\label{eq:est-W1}
\left\|{\mathcal{W}}^{(1)}_V(t)\right\|_{{\mathfrak{S}}^{2q}}{\leqslant}C\|V\|_{L^p_t L^q_x}$$ and $$\label{eq:est-Wn}
\forall n{\geqslant}2,\qquad \left\|{\mathcal{W}}^{(n)}_V(t)\right\|_{{\mathfrak{S}}^{2\left\lceil\frac{q}{n}\right\rceil}}{\leqslant}\frac{C^n}{(n!)^{\frac{1}{p}-{\varepsilon}}}\|V\|_{L^p_t L^q_x}^n.$$
The estimate is the dual version of $$\label{eq:Strichartz}
{ \left| \! \left| \rho_{e^{it\Delta}Ae^{-it\Delta}} \right| \! \right| }_{L^p({{\ensuremath {\mathbb R} }},L^q({{\ensuremath {\mathbb R} }}^d))}{\leqslant}C\|A\|_{{\mathfrak{S}}^{\frac{2q}{q+1}}},$$ for any $(p,q)$ such that $2/p+d/q=d$ and $1{\leqslant}q{\leqslant}1+2/d$, see [@FraLewLieSei-13 Thm. 1]. The estimate is useful to deal with the first order term involving $Q_0$ in , leading to the natural condition that $Q_0\in {\mathfrak{S}}^{4/3}$ in dimension $d=2$ with $p=q=2$.
In dimension $d$, it seems natural to prove that $\rho_Q\in L^{1+2/d}_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^d)$. The estimate turns out to be enough to deal with the terms of order ${\geqslant}d+1$ but it does not seem to help for the terms of order ${\leqslant}d$, because the wave operators ${\mathcal{W}}^{(n)}_V$ with small $n$ belong to a Schatten space with a too large exponent. Apart from the linear response, we are therefore left with $d-1$ terms for which a more detailed computation is necessary. We are not able to do this in any dimension (the number of such terms grows with $d$), but we can deal with the second order term in dimension $d=2$, $$\rho\left[e^{it\Delta}({\mathcal{W}}^{(2)}_{w*\rho_Q}(t)\gamma_f+{\mathcal{W}}^{(1)}_{w*\rho_Q}(t)\gamma_f{\mathcal{W}}^{(1)}_{w*\rho_Q}(t)^*+\gamma_f{\mathcal{W}}^{(2)}_{w*\rho_Q}(t)^*)e^{-it\Delta}\right],$$ which then finishes the proof of the theorem in this case. The second-order term is the topic of Section \[sec:second-order\].
Even if our final result only covers the case $d=2$, we have several estimates in any dimension $d{\geqslant}2$. With the results of this paper, only the terms of order $2$ to $d$ remain to be studied to obtain a result similar to Theorem \[thm:main\] (with $\rho_\gamma-\rho_{\gamma_f}\in L^{1+2/d}_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^d)$) in dimensions $d{\geqslant}3$.
Linear response theory {#sec:linear-response}
======================
Computation of the linear response operator
-------------------------------------------
As we have explained before, we deal here with the linear response ${\mathcal{L}}_1$ associated with the homogeneous state $\gamma_f$. The first order in Duhamel’s formula is defined by $$Q_1(t):=-i\int_0^t e^{i(t-t')\Delta}[w*\rho_{Q(t')},\gamma_f]e^{i(t'-t)\Delta}\,dt'.$$ We see that it is a linear expression in $\rho_Q$, and we compute its density as a function of $\rho_Q$.
\[prop:linear-response\] Let $d{\geqslant}1$, $f\in L^{\infty}({{\ensuremath {\mathbb R} }}_+,{{\ensuremath {\mathbb R} }})$ such that $\int_{{{\ensuremath {\mathbb R} }}^d}|f(k^2)|\,dk<+{\infty}$, and $w\in L^1({{\ensuremath {\mathbb R} }}^d)$. Then, the linear operator ${\mathcal{L}}_1$ defined for all ${\varphi}\in{\mathcal{D}}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)$ by $${\mathcal{L}}_1({\varphi})(t):=-\rho\big[Q_1(t)\big]=\rho\left[i\int_0^t e^{i(t-t')\Delta}[w*{\varphi}(t'),\gamma_f]e^{i(t'-t)\Delta}\,dt'\right]$$ is a space-time Fourier multiplier by the kernel $K^{(1)}=\widehat{w}(k)\,m_f(\omega,k)$, where $$\boxed{
\left[{\mathcal{F}}^{-1}_\omega m_f\right](t,k):=2{{\ensuremath {\mathds 1} }}_{t{\geqslant}0}\sqrt{2\pi}\sin(t|k|^2)\check{g}(2tk)
}$$ (we recall that $g(k):=f(k^2)$ and that $\check{g}$ is its Fourier inverse). This means that for all ${\varphi}\in{\mathcal{D}}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)$, we have $${\mathcal{F}}_{t,x}\left[{\mathcal{L}}_1({\varphi})\right](\omega,k)={\widehat}{w}(k)m_f(\omega,k)\left[{\mathcal{F}}_{t,x}{\varphi}\right](\omega,k),\quad\forall(\omega,k)\in{{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^d$$ where ${\mathcal{F}}_{t,x}$ is the space-time Fourier transform. Furthermore, if $\int_0^{\infty}|x|^{2-d}|\check{g}(x)|\,dx<{\infty}$, then $m_f\in L^{\infty}_{\omega,k}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^d)$ and we have the explicit estimates $$\|m_f\|_{L^{\infty}_{\omega,k}}{\leqslant}\frac{1}{2|\mathbb{S}^{d-1}|}\left(\int_{{{\ensuremath {\mathbb R} }}^d}\frac{|\check{g}(x)|}{|x|^{d-2}}\,dx\right)
\label{eq:estim_m_f}$$ and $$\|{\mathcal{L}}_1\|_{L^2_{t,x}\to L^2_{t,x}}{\leqslant}\frac{{ \left| \! \left| {\widehat}{w} \right| \! \right| }_{L^{\infty}}}{2|\mathbb{S}^{d-1}|}\left(\int_{{{\ensuremath {\mathbb R} }}^d}\frac{|\check{g}(x)|}{|x|^{d-2}}\,dx\right).
\label{eq:estim_L_1}$$
Let ${\varphi}\in{\mathcal{D}}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)$. In order to compute ${\mathcal{L}}({\varphi})$, we use the relation $$\int_0^{\infty}{{\rm Tr}}[W(t,x)Q_1(t)]\,dt=\int_0^{\infty}\int_{{{\ensuremath {\mathbb R} }}^d}W(t,x)\rho_{Q_1}(t,x)\,dx\,dt,$$ valid for any function $W\in {\mathcal{D}}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)$. This leads to $$\begin{gathered}
\int_0^{\infty}\int_{{{\ensuremath {\mathbb R} }}^d}W(t,x)\rho_{Q_1}(t,x)\,dx\,dt=\frac{-i}{(2\pi)^{d}}\int_0^{\infty}\int_0^t\int_{{{\ensuremath {\mathbb R} }}^d}\int_{{{\ensuremath {\mathbb R} }}^d} e^{-2i(t-t')k\cdot\ell}\times\\
\times{\widehat}{W}(t,-k){\widehat}{V}(t',k)(g(\ell-k/2)-g(\ell+k/2)) d\ell\,dk\,dt'\,dt,\end{gathered}$$ where $g(k):=f(k^2)$ and $V=w*{\varphi}$. Computing the $\ell$-integral gives $$\begin{gathered}
\int_{{{\ensuremath {\mathbb R} }}^d}e^{-2i(t-t')k\cdot\ell}(g(\ell-k/2)-g(\ell+k/2))\,d\ell\\=-(2\pi)^{d/2}2i\sin((t-t')|k|^2)\check{g}(2(t-t')k). \end{gathered}$$ Hence, using that ${\widehat}{V}=(2\pi)^{d/2}{\widehat}{w}{\widehat}{{\varphi}}$, we find that $$\begin{gathered}
\int_0^{\infty}\int_{{{\ensuremath {\mathbb R} }}^d}W(t,x)\rho_{Q_1}(t,x)\,dx\,dt\\
=-2\int_0^{\infty}\int_0^t\int_{{{\ensuremath {\mathbb R} }}^d}\sin((t-t')|k|^2)\check{g}(2(t-t')k)\widehat{w}(k){\widehat}{W}(t,-k){\widehat}{{\varphi}}(t',k)\,dk\,dt'\,dt.\end{gathered}$$ Since $g$ is radial, then $\check{g}$ is also radial and we have $$\begin{aligned}
|m_f(\omega,k)| & {\leqslant}& 2\int_0^{\infty}|\sin(t|k|^2)||\check{g}(2t|k|)|\,dt\\
& {\leqslant}& 2\int_0^{\infty}\frac{|\sin(t|k|)|}{|k|}|\check{g}(2t)|\,dt\\
& {\leqslant}& \frac{1}{2}\int_0^{\infty}r|\check{g}(r)|\,dr.\end{aligned}$$ This ends the proof of the proposition.
We now make several remarks about the previous result.
First, the physical examples for $g$ are $$g(k)=\begin{cases}
\displaystyle {{\ensuremath {\mathds 1} }}(|k|^2{\leqslant}\mu), & \mu>0,\\
\displaystyle e^{-(|k|^2-\mu)/T}, &T>0,\ \mu\in{{\ensuremath {\mathbb R} }},\\
\displaystyle \frac{1}{e^{(|k|^2-\mu)/T}+1}, & T>0,\ \mu\in{{\ensuremath {\mathbb R} }},\\
\displaystyle \frac{1}{e^{(|k|^2-\mu)/T}-1}, & T>0,\ \mu<0.
\end{cases}$$ In the last three choices, $g$ is a Schwartz function hence $\check{g}\in L^1({{\ensuremath {\mathbb R} }}^d)$. For the first choice of $g$ (Fermi sea at zero temperature), we have $\check{g}(r)\sim r^{-1}\sin r$, which obviously does not verify $r\check{g}(r)\in L^1(0,+{\infty})$.
Then, we remark that is optimal without more assumptions on $f$. Indeed, for $\omega=0$ and small $k$ we find $$\begin{aligned}
m_f(0,k)&=2\int_0^{\infty}\sin(t|k|^2)\check{g}(2tk)\,dt\\
&\underset{k\to0}\longrightarrow \frac{1}2\int_0^{\infty}r\check{g}(r)\,dr=\frac{1}{2|\mathbb{S}^{d-1}|}\int_{{{\ensuremath {\mathbb R} }}^d}\frac{\check{g}(x)}{|x|^{d-2}}\,dx.\end{aligned}$$ We conclude that is optimal if $\check{g}$ has a constant sign (for instance $f$ is decreasing, as in the physical examples –). Similarly, is optimal if both $\check{g}$ and $w$ have a constant sign (then $|\widehat{w}(0)|={ \left| \! \left| \widehat{w} \right| \! \right| }_{L^{\infty}}$).
In general, the function $m_f$ is complex-valued and it is not an easy task to determine when ${\widehat}{w}(k)m_f(\omega,k)$ stays far from $-1$. Since the stationary linear response is real, $\Im m_f(0,k)\equiv0$, the condition should at least involve the maximum or the minimum of $m_f$ on the set $\{\omega=0\}$, depending on the sign of $\widehat{w}$. Even if the function $m_f$ is bounded on ${{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^d$ by , it will usually not be continuous at the point $(0,0)$. Under the additional condition that $f$ is strictly decreasing, we are able to prove that $$\{\Im m_f(\omega,k)=0\}=\{\omega=0\}\cup\{k=0\}$$ and this can be used to replace the assumption on $\widehat{w}$ by one on $(\widehat{w})_-$ and $\widehat{w}(0)_+$. In order to explain this, we first compute $m_f$ in the case of a Fermi gas at zero temperature, $f(k^2)={{\ensuremath {\mathds 1} }}(|k|^2{\leqslant}\mu)$.
Let $d{\geqslant}1$ and $\mu>0$. Then, for the Fermi sea at zero temperature $\gamma_f={{\ensuremath {\mathds 1} }}(-\Delta{\leqslant}\mu)$, the corresponding Fourier multiplier $m_f(\omega,k):=m^{\rm F}_{d}(\mu,\omega,k)$ of the linear response operator in dimension $d$ is given by $$\begin{gathered}
m_1^{\rm F}(\mu,\omega,k)=\frac{1}{2\sqrt{2\pi}|k|}\log\left|\frac{(|k|^2+2|k|\sqrt{\mu})^2-\omega^2}{(|k|^2-2|k|\sqrt{\mu})^2-\omega^2}\right|\\ + i\frac{\sqrt{\pi}}{2\sqrt{2}|k|}\bigg\{{{\ensuremath {\mathds 1} }}\left(|\omega+|k|^2|{\leqslant}2\sqrt\mu|k|\right)-{{\ensuremath {\mathds 1} }}\left(|\omega-|k|^2|{\leqslant}2\sqrt\mu|k|\right)\bigg\}
\label{eq:m_f_1D}\end{gathered}$$ for $d=1$, by $$\begin{gathered}
m_2^{\rm F}(\mu,\omega,k)=\frac 14\bigg\{2-\frac{{\rm sgn}(|k|^2+\omega)}{|k|^2}\Big((|k|^2+\omega)^2-4\mu |k|^2\Big)^{\frac12}_+\\
- \frac{{\rm sgn}(|k|^2-\omega)}{|k|^2}\Big((|k|^2-\omega)^2-4\mu |k|^2\Big)^{\frac12}_+\bigg\}\\
+i\frac{1}{2|k|^2}\bigg\{\Big((|k|^2+\omega)^2-4\mu |k|^2\Big)^{\frac12}_- - \Big((|k|^2-\omega)^2-4\mu |k|^2\Big)^{\frac12}_-\bigg\}.
\label{eq:m_f_2D}\end{gathered}$$ for $d=2$, and by $$\begin{aligned}
m_d^{\rm F}(\mu,\omega,k)&=\frac{|\mathbb{S}^{d-2}|\mu^{\frac{d-1}{2}}}{(2\pi)^{\frac{d-1}2}}\int_{0}^1 m^{\rm F}_1\big(\mu(1-r^2),\omega,k\big) r^{d-2}\,dr,&\text{for $d{\geqslant}2$,}\nonumber\\
&=\frac{|\mathbb{S}^{d-3}|\mu^{\frac{d-2}{2}}}{(2\pi)^{\frac{d-2}2}}\int_{0}^1 m^{\rm F}_2\big(\mu(1-r^2),\omega,k\big) r^{d-3}\,dr,&\text{for $d{\geqslant}3$.}\label{eq:m_f_3D}\end{aligned}$$
The formula for $m^{\rm F}_{d}$ is well known in the physics literature (see [@Lindhard-54], [@Mihaila-11] and [@GiuVig-05 Chap. 4]). It is also possible to derive an explicit expression for $m_{3}^{\rm F}(\mu,\omega,k)$, see [@GiuVig-05 Chap. 4]. We remark that $m_{d}^{\rm F}(\mu,0,k)$ coincides with the time-independent linear response computed in [@FraLewLieSei-12 Thm 2.5].
From the formulas we see that the real part of $m^{\rm F}_{d}$ can have both signs. It is always positive for $\omega=0$ and it can take negative values for $\omega\neq0$. For instance, in dimension $d=2$, on the curve $\omega=|k|^2+2\sqrt{\mu}|k|$ the imaginary part vanishes and we get $$m_{2}^{\rm F}\big(\mu,|k|^2+2\sqrt{\mu}|k|,k\big)=\frac 12\left(1-\sqrt{1+\frac{2\sqrt\mu}{|k|}}\right)\underset{k\to0}{\longrightarrow}-{\infty}.
\label{eq:curve_omega_k}$$ In particular, if ${\widehat}{w}(k)/\sqrt{|k|}\to+{\infty}$ when $k\to0$, then $\widehat{w}(k)m_f(|k|^2+2\sqrt{\mu}|k|,k)\to-{\infty}$ when $|k|\to 0$. Since on the other hand $\widehat{w}(k)m_f(|k|^2+2\sqrt{\mu}|k|,k)\to0$ when $|k|\to{\infty}$, we conclude that the function must cross $-1$, and $(1+{\mathcal{L}}_1)^{-1}$ is not bounded.
An important feature of $m_d^{\rm F}$ which we are going to use in the positive temperature case, is that the imaginary part $\Im m_{d}^{\rm F}(\mu,\omega,k)$ has a constant sign on $\{\omega>0\}$ and on $\{\omega<0\}$. Before we discuss this in detail, we provide the proof of the proposition.
First, a calculation shows that the Fourier inverse $ \check{g}_1$ of the radial $g$ in dimension $d=1$ is given by $$\check{g}_1(\mu,x)=\sqrt{\frac{2}{\pi}}\frac{\sin(\sqrt{\mu}|x|)}{|x|}.
\label{eq:g_Fourier_inverse_1D}$$ In dimension $d{\geqslant}2$ we can write $$\begin{aligned}
\check{g}_d(\mu,|x|)=&\frac{1}{(2\pi)^{d/2}}\int_{{{\ensuremath {\mathbb R} }}^d}{{\ensuremath {\mathds 1} }}(|k|^2{\leqslant}\mu) e^{i k\cdot x}\nonumber\\
=&\frac{1}{(2\pi)^{d/2}}\int_{{{\ensuremath {\mathbb R} }}}dk_1\int_{{{\ensuremath {\mathbb R} }}^{d-1}}dk_\perp {{\ensuremath {\mathds 1} }}(|k_1|^2{\leqslant}\mu-|k_\perp|^2) e^{i k_1|x|}\nonumber\\
=&\frac{|\mathbb{S}^{d-2}|\mu^{\frac{d-1}{2}}}{(2\pi)^{d/2}}\int_{{{\ensuremath {\mathbb R} }}}dk_1\int_{0}^1 {{\ensuremath {\mathds 1} }}\big(|k_1|^2{\leqslant}\mu(1-r^2)\big) e^{i k_1|x|}r^{d-2}dr\nonumber\\
=&\frac{|\mathbb{S}^{d-2}|\mu^{\frac{d-1}{2}}}{(2\pi)^{\frac{d-1}2}}\int_{0}^1 \check{g}_1\big(\mu(1-r^2),|x|\big)r^{d-2}\,dr\nonumber\\
=&\frac{2|\mathbb{S}^{d-2}|}{(2\pi)^{d/2}}\frac{\mu^{\frac{d-1}{2}}}{|x|}\int_0^1\sin(\sqrt{\mu}|x|\sqrt{1-r^2})r^{d-2}\,dr.\label{eq:g_Fourier_inverse_2D}\end{aligned}$$ Similarly, we have in dimension $d{\geqslant}3$ $$\check{g}_d(\mu,|x|)=\frac{|\mathbb{S}^{d-3}|\mu^{\frac{d-2}{2}}}{(2\pi)^{\frac{d-2}2}}\int_{0}^1 \check{g}_2\big(\mu(1-r^2),|x|\big)r^{d-3}\,dr.
\label{eq:g_Fourier_inverse_3D}$$ Now we can compute the multiplier $m^{\rm F}_d(\mu,\omega,k)$ for $d=1,2$. We start with $d=1$ for which we have $$[{\mathcal{F}}^{-1}_\omega m_{f,1}](t,k)=4{{\ensuremath {\mathds 1} }}_{t{\geqslant}0}\frac{\sin(t|k|^2)\sin(2\sqrt{\mu}t|k|)}{2t|k|}.$$ There remains to compute the time Fourier transform. We use the formula valid for any $a,b\in{{\ensuremath {\mathbb R} }}$, $$\begin{gathered}
\int_0^{\infty}\frac{\sin(at)\sin(bt)}{t}e^{-it\omega}{{\ensuremath{\,\text{d}t}}}=\frac{1}{4}\log\left|\frac{(a+b)^2-\omega^2}{(a-b)^2-\omega^2}\right|\\
+i\frac{\pi}{8}\left({{\rm sgn}}(a-b-\omega)-{{\rm sgn}}(a+b-\omega)+{{\rm sgn}}(a+b+\omega)-{{\rm sgn}}(a-b+\omega)\right), \end{gathered}$$ and obtain . To provide the more explicit expression in dimension 2, we use this time the formula $$\forall a\in{{\ensuremath {\mathbb R} }},\qquad \frac{1}{a}\int_0^1\log\frac{|a+2\sqrt{1-r^2}|}{|a-2\sqrt{1-r^2}|}{{\ensuremath{\,\text{d}r}}}=\frac{\pi}{2}-\frac{\pi}{2}\left(1-\frac{4}{a^2}\right)_+^{1/2},$$ which leads to the claimed form of $m_{2}^{\rm F}(\mu,\omega,k)$.
Now we will use the imaginary part of $m_d^{\rm F}$ to show that $1+{\mathcal{L}}_1$ is invertible with bounded inverse when $\widehat{w}{\geqslant}0$ with $\widehat{w}(0)$ not too large, and when $f$ is strictly decreasing.
\[cor:linear-response\_defocusing\] Let $d{\geqslant}1$ and $f\in L^{\infty}({{\ensuremath {\mathbb R} }}_+,{{\ensuremath {\mathbb R} }})$ such that $\int_0^{\infty}(r^{d/2-1}|f(r)|+|f'(r)|)\,dr<{\infty}$ and $f'(r)<0$ for all $r>0$. Assume furthermore that $\int_{{{\ensuremath {\mathbb R} }}^d}|x|^{2-d} |\check{g}(x)|\,dx<{\infty}$ with $g(k)=f(|k|^2)$. If $w\in L^1({{\ensuremath {\mathbb R} }}^d)$ is an even function such that $${ \left| \! \left| ({\widehat}{w})_- \right| \! \right| }_{L^{\infty}}\left(\int_{{{\ensuremath {\mathbb R} }}^d}\frac{|\check{g}(x)|}{|x|^{d-2}}\,dx\right)<2|\mathbb{S}^{d-1}|,
\label{eq:condition_negative_part_bis}$$ and such that $${\varepsilon}_g {\widehat}{w}(0)_+<2|\mathbb{S}^{d-1}|,\quad\text{where}\quad {\varepsilon}_g:=-\liminf_{(\omega,k)\to(0,0)}\frac{\Re m_f(\omega,k)}{2|\mathbb{S}^{d-1}|},
\label{eq:condition_negative_part_bis2}$$ then we have $$\min_{(\omega,k)\in{{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^d}|\widehat{w}(k)m_f(\omega,k)+1|>0$$ and $(1+{\mathcal{L}}_1)$ is invertible on $L^2_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^d)$ with bounded inverse.
First we recall that $m_f$ is uniformly bounded by . Therefore we only have to look at the set $$A=\left\{k\in{{\ensuremath {\mathbb R} }}^d\ :\ |\widehat{w}(k)|{\geqslant}\frac{1}{4|\mathbb{S}^{d-1}|}\left(\int_{{{\ensuremath {\mathbb R} }}^d}\frac{|\check{g}(x)|}{|x|^{d-2}}\,dx\right)\right\}.$$ On the complement of $A$, we have $|\widehat{w}\,m_f+1|{\geqslant}1/2$. Since $\widehat{w}(k)\to0$ when $|k|\to{\infty}$, then $A$ is a compact set. Next, from the integral formula $$f(|k|^2)=-\int_0^{\infty}{{\ensuremath {\mathds 1} }}(|k|^2{\leqslant}s) f'(s)\,ds,$$ we infer that $$m_f(\omega,k)=-\int_0^{\infty}m_d^{\rm F}(s,\omega,k) f'(s)\,ds.$$ This integral representation can be used to prove that $m_f$ is continuous on ${{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}_+\setminus\{(0,0)\}$. In general, the function $m_f$ is not continuous at $(0,0)$, however.
Since $m_d^{\rm F}(s,0,k){\geqslant}0$ for all $k$ and $s{\geqslant}0$, we conclude that $m_f(0,k){\geqslant}0$ and that $$m_f(0,k)\widehat{w}(k){\geqslant}-m_f(0,k)\widehat{w}(k)_-{\geqslant}-{ \left| \! \left| \widehat{w}_- \right| \! \right| }_{L^{\infty}}\frac{1}{2|\mathbb{S}^{d-1}|}\left(\int_{{{\ensuremath {\mathbb R} }}^d}\frac{|\check{g}(x)|}{|x|^{d-2}}\,dx\right),$$ due to . In particular, $$|m_f(0,k)\widehat{w}(k)+1|{\geqslant}1-{ \left| \! \left| \widehat{w}_- \right| \! \right| }_{L^{\infty}}\frac{1}{2|\mathbb{S}^{d-1}|}\left(\int_{{{\ensuremath {\mathbb R} }}^d}\frac{|\check{g}(x)|}{|x|^{d-2}}\,dx\right)>0$$ due to our assumption on $(\widehat{w})_-$. Similarly, we have $m_f(\omega,0)=0$ for all $\omega\neq0$ and therefore $m_f(\omega,0)\widehat{w}(0)+1=1$ is invertible on $\{k=0,\omega\neq0\}$.
Now we look at $k\neq0$ and $\omega>0$ and we prove that the imaginary part of $m_f$ never vanishes. We write the argument for $d=1$, as it is very similar for $d{\geqslant}2$, using the integral representation . We have $$\begin{gathered}
\Im m_f(\omega,k)
=\frac{\sqrt{\pi}}{2\sqrt{2}|k|}\times\\
\times\int_0^{\infty}\bigg\{{{\ensuremath {\mathds 1} }}\left((\omega-|k|^2)^2{\leqslant}4s|k|^2\right)-{{\ensuremath {\mathds 1} }}\left((\omega+|k|^2)^2{\leqslant}4s|k|^2\right)\bigg\} f'(s)\,ds.\end{gathered}$$ The difference of the two Heaviside functions is always ${\geqslant}0$ for $\omega>0$. Furthermore, it is equal to 1 for all $s$ in the interval $$\frac{(\omega-|k|^2)^2}{4|k|^2}{\leqslant}s{\leqslant}\frac{(\omega+|k|^2)^2}{4|k|^2}.$$ Therefore we have $$\Im m_f(\omega,k){\leqslant}\frac{\sqrt{\pi}}{2\sqrt{2}|k|}\int_{\frac{(\omega-|k|^2)^2}{4|k|^2}}^{\frac{(\omega+|k|^2)^2}{4|k|^2}}
f'(s)\,ds<0$$ for all $\omega>0$ and $k\neq0$. For $\omega<0$ we can simply use that $\Im m_f(\omega,k)=-\Im m_f(-\omega,k)$ and this concludes the proof that the imaginary part does not vanish outside of $\{k=0\}\cup\{\omega=0\}$.
From the previous argument, we see that everything boils down to understanding the behavior of $\Re m_f$ in a neighborhood of $(0,0)$. At this point the maximal value is $\frac{1}2\int_0^{\infty}r\check{g}(r)\,dr$ and the minimal value is $-{\varepsilon}_g2|\mathbb{S}^{d-1}|$ by definition, hence the result follows.
We remark that $$\Re\,m_f(\omega,k)\underset{\substack{k\to0\\ \omega\to0}}\simeq \frac12\int_{0}^{\infty}t\check{g}(t)\cos\left(\frac{\omega}{2|k|} t\right)\,dt$$ and therefore we can express $$-{\varepsilon}_g:=\frac{1}{4|\mathbb{S}^{d-1}|}\min_{a\in{{\ensuremath {\mathbb R} }}}\int_{0}^{\infty}t\check{g}(t)\cos(a t)\,dt.$$
In the three physical cases –, the function $f$ satisfies the assumptions of the corollary, and therefore $1+{\mathcal{L}}_1$ is invertible with bounded inverse when $w$ satisfies and . Numerical computations show that ${\varepsilon}_g$ is always $>0$, but usually smaller than the maximum, by a factor 2 to 10. As an illustration, we display the function $\Re m_f(\omega,k)$ for $T=100$ and $\mu=1$ in Figure \[fig:m\_f\] below.
![Plot of $\Re\, m_f(\omega,k)$ in the fermionic case for $d=2$, $T=100$ and $\mu=1$\[fig:m\_f\]](Fermi.eps){width="8cm"}
Boundedness of the linear response in $L^p_{t,x}$
-------------------------------------------------
We have studied the boundedness of ${\mathcal{L}}_1$ from $L^2_{t,x}$ to $L^2_{t,x}$. This is useful in dimension $d=2$, where the density $\rho_Q$ naturally belongs to $L^2_{t,x}$. However, in other space dimensions, we would like to prove that $\rho_Q$ belongs to $L^{1+2/d}_{t,x}$ and hence, it makes sense to ask whether ${\mathcal{L}}_1$ is bounded from $L^p_{t,x}$ to $L^p_{t,x}$. This is the topic of this section. The study of Fourier multipliers acting on $L^p$ is a classical subject in harmonic analysis. We use theorems of Stein and Marcinkiewicz to infer the required boundedness.
Let $w\in L^1({{\ensuremath {\mathbb R} }}^d)$ be such that $|x|^{d+2}w\in L^1({{\ensuremath {\mathbb R} }}^d)$ and such that $$\left(\prod_{i\in I}|k_{i}|^2\prod_{j\in J}\partial_{k_{j}}\right){\widehat}{w}(k)\in L^{\infty}_k({{\ensuremath {\mathbb R} }}^d),\,\,\forall I\subset\{1,...,d\},\,\,\forall J\subset I.$$ Let also $h:{{\ensuremath {\mathbb R} }}^d\to{{\ensuremath {\mathbb R} }}$ be an even function such that $$\forall\alpha\in{{\ensuremath {\mathbb N} }}^d,\, |\alpha|{\leqslant}d+3,\quad \int_{{{\ensuremath {\mathbb R} }}^d}(1+|k|^{d+4})|\partial^\alpha h(k)|{{\ensuremath{\,\text{d}k}}}<+{\infty}$$ and $$\left(\prod_{i\in I}\partial_{k_{i}}\right)h\in L^{\infty}_k({{\ensuremath {\mathbb R} }}^d),\qquad\text{for all $I\subset\{1,...,d\}$}.$$ Then the Fourier multiplier $${\mathcal{F}}_t\big\{{{\ensuremath {\mathds 1} }}(t{\geqslant}0)\sin(t|k|^2)h(2tk)\big\}$$ defines a bounded operator from $L^p_{t,x}$ to itself, for every $1<p<{\infty}$.
The conditions on $h$ are fulfilled if for instance $h$ is a Schwartz function, hence they are fulfilled for our physical examples –, where we take $h=\check{g}$.
We define $$m_1(t,k)={{\ensuremath {\mathds 1} }}(t{\geqslant}1){\widehat}{w}(k)\sin(t|k|^2)h(2tk)$$ and $$m_2(t,k)={{\ensuremath {\mathds 1} }}(0{\leqslant}t{\leqslant}1){\widehat}{w}(k)\sin(t|k|^2)h(2tk),$$ and use a different criterion for these two multipliers.
To show that $m_1$ defines a bounded operator on $L^p$, we use the criterion of Stein [@Stein-70 Thm. 1, II §2]. We write $m_1(t,k)={\widehat}{w}(k){\widetilde}{m_1}(t,k)$. We first prove estimates on ${\widetilde}{m_1}$, which then imply that $m_1$ defines a bounded Fourier multiplier on $L^p$ by Stein’s theorem. Computing the inverse Fourier transform of ${\widetilde}{m_1}$, one has $$M_1(t,x):=[{\mathcal{F}}^{-1}_k{\widetilde}{m_1}](t,x)={{\ensuremath {\mathds 1} }}(t{\geqslant}1)(2\pi)^{-d/2}\int_{{{\ensuremath {\mathbb R} }}^d}\sin(t|k|^2)h(2tk)e^{ix\cdot k}{{\ensuremath{\,\text{d}k}}}.$$ Then, we have $$\label{eq:nablaxm1NLSgeneral}
\nabla_x M_1(t,x)={{\ensuremath {\mathds 1} }}(t{\geqslant}1)\frac{(2\pi)^{-d/2}}{(2t)^{d+1}}i\int_{{{\ensuremath {\mathbb R} }}^d}kh(k)\sin\left(\frac{|k|^2}{4t}\right)e^{i\frac{x\cdot k}{2t}}\,dk.$$ From this formula, we see that for all $(t,x)$, $$\label{eq:est-t-nabla-x-M1}
t^{d+2}|\nabla_x M_1(t,x)|{\leqslant}C\int_{{{\ensuremath {\mathbb R} }}^d}|k|^3|h(k)|\,dk.$$ Next, let $1{\leqslant}j{\leqslant}d$ and notice that $$x_j^{d+2}e^{i\frac{x\cdot k}{2t}}=\frac{d^{d+2}}{dk_j^{d+2}}(2t)^{d+2}(-i)^{d+2}e^{i\frac{x\cdot k}{2t}},$$ and hence by an integration by parts we obtain $$\begin{gathered}
x_j^{d+2}\nabla_x M_1(t,x)\\
={{\ensuremath {\mathds 1} }}(t{\geqslant}1)(2\pi)^{-d/2}2ti(-i)^{d+2}\int_{{{\ensuremath {\mathbb R} }}^d}\frac{d^{d+2}}{dk_j^{d+2}}\left[kh(k)\sin\left(\frac{|k|^2}{4t}\right)\right]e^{i\frac{x\cdot k}{2t}}\,dk.
\end{gathered}$$ When the $k_j$-derivative hits at least once $\sin(|k|^2/4t)$, one gains at least $1/4t$ compensating the $2t$ before the integral; the only term for which we have to prove that it is bounded in $t$ is when all the $k_j$-derivatives hit the term $kh(k)$, which is $${{\ensuremath {\mathds 1} }}(t{\geqslant}1)(2\pi)^{-d/2}2it(-i)^{d+2}\int_{{{\ensuremath {\mathbb R} }}^d}\frac{d^{d+2}}{dk_j^{d+2}}\left[kh(k)\right]\sin\left(\frac{|k|^2}{4t}\right)e^{i\frac{x\cdot k}{2t}}\,dk.$$ It is also bounded since $|\sin(|k|^2/4t)|{\leqslant}|k|^2/4t$. We deduce that for all $(t,x)$, $$\label{eq:est-x-nabla-x-M1}
|x|^{d+2}|\nabla_xM_1(t,x)|{\leqslant}C\sup_{\substack{\alpha\in{{\ensuremath {\mathbb N} }}^d \\ |\alpha|{\leqslant}d+2}}\int_{{{\ensuremath {\mathbb R} }}^d}(1+|k|^{d+3})|\partial^\alpha h(k)|\,dk.$$ For the time derivative we use the form $$M_1(t,x)={{\ensuremath {\mathds 1} }}(t{\geqslant}1)(2\pi)^{-d/2}\int_{{{\ensuremath {\mathbb R} }}^d}h(2tk)\sin(t|k|^2)\cos(x\cdot k)\,dk$$ to infer that for $t\neq1$, $$\begin{aligned}
\partial_t M_1(t,x) =& 2{{\ensuremath {\mathds 1} }}(t{\geqslant}1)(2\pi)^{-d/2}\int_{{{\ensuremath {\mathbb R} }}^d}k\cdot\nabla_kh(2tk)\sin(t|k|^2)\cos(x\cdot k){{\ensuremath{\,\text{d}k}}}\nonumber\\
&+{{\ensuremath {\mathds 1} }}(t{\geqslant}1)(2\pi)^{-d/2}\int_{{{\ensuremath {\mathbb R} }}^d}|k|^2h(2tk)\cos(t|k|^2)\cos(x\cdot k)\,dk \nonumber\\
=& \frac{2{{\ensuremath {\mathds 1} }}(t{\geqslant}1)}{(2t)^{d+1}}(2\pi)^{-d/2}\int_{{{\ensuremath {\mathbb R} }}^d}k\cdot\nabla_kh(k)\sin\left(\frac{|k|^2}{4t}\right)\cos\left(\frac{x\cdot k}{2t}\right)\,dk\nonumber\\
&+\frac{{{\ensuremath {\mathds 1} }}(t{\geqslant}1)}{(2t)^{d+2}}(2\pi)^{-d/2}\int_{{{\ensuremath {\mathbb R} }}^d}|k|^2h(k)\cos\left(\frac{|k|^2}{4t}\right)\cos\left(\frac{x\cdot k}{2t}\right)\,dk.\label{eq:dtm1NLSgeneral}
\end{aligned}$$ By the same method as before, we infer $$\label{eq:est-t-x-nabla-t-M1}
\|(t,x)\|^{d+2}|\partial_t M_1(t,x)|{\leqslant}C\sup_{\substack{\alpha\in{{\ensuremath {\mathbb N} }}^d \\ |\alpha|{\leqslant}d+3}}\int_{{{\ensuremath {\mathbb R} }}^d}(1+|k|^{d+4})|\partial^\alpha h(k)|\,dk.$$ Now let us go back to the multiplier $m_1$. We have $${\mathcal{F}}_x^{-1}m_1(t,x)=(2\pi)^{d/2}(w\star M_1(t,\cdot))(x),$$ and hence $$\nabla_{t,x}{\mathcal{F}}_x^{-1}m_1(t,x)=(2\pi)^{d/2}(w\star\nabla_{t,x}M_1(t,\cdot))(x).$$ First we have $$|t^{d+2}\nabla_{t,x}{\mathcal{F}}_x^{-1}m_1(t,x)|{\leqslant}C\|w\|_{L^1_x}\|t^{d+2}\nabla_{t,x}M_1(t,x)\|_{L^{\infty}_{t,x}},$$ which is finite thanks to , , and . Next, $$\begin{gathered}
|x|^{d+2}|\nabla_{t,x}{\mathcal{F}}_x^{-1}m_1(t,x)|{\leqslant}C\||\cdot|^{d+2}w\|_{L^1_x}\|\nabla_{t,x}M_1(t,x)\|_{L^{\infty}_{t,x}}\\
+C\|w\|_{L^1_x}\||x|^{d+2}\nabla_{t,x}M_1(t,x)\|_{L^{\infty}_{t,x}}.
\end{gathered}$$ The second term is finite also from and , while the first term is finite by the expressions and . As a consequence, we can apply Stein’s theorem to $m_1$ and we deduce that the corresponding operator is bounded on $L^p_{t,x}$ for all $1<p<{\infty}$.
The multiplier $m_2$ is treated differently. We show that $$m_2\in L^1_t({{\ensuremath {\mathbb R} }},{\mathcal{B}}(L^p_x\to L^p_x)),$$ which is enough to show that $m_2$ defines a bounded operator on $L^p_{t,x}$. Indeed, for any ${\varphi}\in L^p_{t,x}$, define the Fourier multiplication operator $T_{m_2}$ by $$(T_{m_2}{\varphi})(t,x)=\int_{{\ensuremath {\mathbb R} }}{\mathcal{F}}_x^{-1}\left[m_2(t-t',\cdot)({\mathcal{F}}_x{\varphi})(t',\cdot)\right](x)\,dt'.$$ Then, we have $$\begin{aligned}
\|T_{m_2}{\varphi}(t)\|_{L^p_x} & {\leqslant}& \int_{{\ensuremath {\mathbb R} }}\|{\mathcal{F}}_x^{-1}[m_2(t-t',\cdot)({\mathcal{F}}_x{\varphi})(t',\cdot)]\|_{L^p_x}\,dt'\\
& {\leqslant}& \int_{{\ensuremath {\mathbb R} }}\|m_2(t-t')\|_{{\mathcal{B}}(L^p_x\to L^p_x)}\|{\varphi}(t')\|_{L^p_x}\,dt',\\
\end{aligned}$$ and hence $$\|T_{m_2}{\varphi}\|_{L^p_{t,x}}{\leqslant}\|m_2\|_{L^1_t({{\ensuremath {\mathbb R} }},{\mathcal{B}}(L^p_x\to L^p_x))}\|{\varphi}\|_{L^p_{t,x}}.$$ Hence, let us show that $\|m_2\|_{L^p_x\to L^p_x}\in L^1_t$. We estimate $\|m_2\|_{L^p_x\to L^p_x}$ by the Marcinkiewicz theorem [@Grafakos-book Cor. 5.2.5]. Namely, we have to show that for all $1{\leqslant}i_1,\ldots,i_\ell {\leqslant}d$ all different indices, we have $$k_{i_1}\cdots k_{i_\ell}\partial_{k_{i_1}}\cdots\partial_{k_{i_\ell}}m_2(t,k)\in L^{\infty}_k,$$ and if so the Marcinkiewicz theorem tells us that $$\|m_2(t)\|_{L^p_x\to L^p_x}{\leqslant}C\sup_{i_1,\ldots,i_\ell}\|k_{i_1}\cdots k_{i_\ell}\partial_{k_{i_1}}\cdots\partial_{k_{i_\ell}}m_2(t,k)\|_{L^{\infty}_k}.$$ A direct computation shows that $$\begin{gathered}
|k_{i_1}\cdots k_{i_\ell}\partial_{k_{i_1}}\cdots\partial_{k_{i_\ell}}m_2(t,k)|\\
{\leqslant}C{{\ensuremath {\mathds 1} }}_{0{\leqslant}t{\leqslant}1}\sum_{I\subset\{i_1,\ldots,i_\ell\}}\sum_{J\subset I}|k_{i_1}|^2\cdots|k_{i_\ell}|^2|\partial_I{\widehat}{w}(k)|(\partial_Jh)(2tk)|,
\end{gathered}$$ where we used the notation $\partial_Jh:=\prod_{j\in J}\partial_{k_j}h$. Hence, $$\begin{gathered}
\|k_{i_1}\cdots k_{i_\ell}\partial_{k_{i_1}}\cdots\partial_{k_{i_\ell}}m_2(t,k)\|_{L^{\infty}_k}\\
{\leqslant}C{{\ensuremath {\mathds 1} }}_{0{\leqslant}t{\leqslant}1}\sup_{I,J\subset\{i_1,\ldots,i_\ell\}}\||k_{i_1}|^2\cdots|k_{i_\ell}|^2|\partial_I{\widehat}{w}(k)|\|_{L^{\infty}_k}\|\partial_Jh\|_{L^{\infty}_k},
\end{gathered}$$ which is obviously a $L^1_t$–function.
Higher order terms {#sec:higher-order}
==================
In this section, we explain how to treat the higher order terms in . We recall the decomposition of the solution for all $t{\geqslant}0$: $$\rho_Q(t)=\rho\left[e^{it\Delta}{\mathcal{W}}_{w*\rho_Q}(t)(\gamma_f+Q_0){\mathcal{W}}_{w*\rho_Q}(t)^*e^{-it\Delta}\right]-\rho_{\gamma_f}.$$ We first estimate the terms involving $Q_0$, in dimension 2.
\[lemma:higher-order-Q0\] Let $Q_0\in{\mathfrak{S}}^{4/3}(L^2({{\ensuremath {\mathbb R} }}^2))$ and $V\in L^2_{t,x}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^2)$. Then, we have the following estimate for all $n,m{\geqslant}0$ $$\begin{gathered}
{ \left| \! \left| \rho\left[e^{it\Delta}{\mathcal{W}}_V^{(n)}(t)Q_0{\mathcal{W}}_V^{(m)}(t)^*e^{-it\Delta}\right] \right| \! \right| }_{L^2_{t,x}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^2)}\\
{\leqslant}C{ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}\frac{C^{n+m}{ \left| \! \left| V \right| \! \right| }_{L^2_{t,x}}^{n+m}}{(n!)^{\frac 14}(m!)^{\frac 14}},\end{gathered}$$ for some $C>0$ independent of $Q_0$, $n$, $m$, and $V$.
Defining ${\mathcal{W}}^{(0)}_V(t):=1$, for $n,m{\geqslant}0$, the density of $$e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)Q_0{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}$$ is estimated by duality in the following fashion. Let $U\in L^2_{t,x}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^2)$. The starting point is the formula $$\begin{gathered}
\int_0^{\infty}\int_{{{\ensuremath {\mathbb R} }}^2}U(t,x)\rho\left[e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)Q_0{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}\right](t,x)\,dx\,dt\\
=\int_0^{\infty}{{\rm Tr}}\left[U(t,x)e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)Q_0{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}\right]\,dt.
\end{gathered}$$ By cyclicity of the trace, we have $$\begin{gathered}
{{\rm Tr}}\left[U(t,x)e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)Q_0{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}\right]\\
={{\rm Tr}}\left[{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}U(t,x)e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)Q_0\right]
\end{gathered}$$ A straightforward generalization of Theorem \[thm:est-Wn\] shows that we have $$\begin{gathered}
{ \left| \! \left| \int_0^{\infty}{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}U(t,x)e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)\,dt \right| \! \right| }_{{\mathfrak{S}}^4}\\
{\leqslant}{ \left| \! \left| U \right| \! \right| }_{L^2_{t,x}}\frac{C^n{ \left| \! \left| V \right| \! \right| }_{L^2_{t,x}}^n}{(n!)^{1/4}}\frac{C^m{ \left| \! \left| V \right| \! \right| }_{L^2_{t,x}}^m}{(m!)^{1/4}},
\end{gathered}$$ and hence using that $Q_0\in{\mathfrak{S}}^{4/3}$ and Hölder’s inequality, we infer that $${ \left| \! \left| \rho\left[e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)Q_0{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}\right] \right| \! \right| }_{L^2_{t,x}}{\leqslant}{ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}\frac{C^n{ \left| \! \left| V \right| \! \right| }_{L^2_{t,x}}^n}{(n!)^{1/4}}\frac{C^m{ \left| \! \left| V \right| \! \right| }_{L^2_{t,x}}^m}{(m!)^{1/4}}.$$ This concludes the proof of the lemma.
When $d{\geqslant}2$, the corresponding result is
Let $d{\geqslant}2$, $Q_0\in{\mathfrak{S}}^{\frac{d+2}{d+1}}(L^2({{\ensuremath {\mathbb R} }}^d))$, $1<q{\leqslant}1+2/d$ and $p$ such that $2/p+d/q=d$. Let $V\in L^{p'}_tL^{q'}_x({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)$. Then, we have the following estimate for any $n,m{\geqslant}0$ $$\begin{gathered}
{ \left| \! \left| \rho\left[e^{it\Delta}{\mathcal{W}}_V^{(n)}(t)Q_0{\mathcal{W}}_V^{(m)}(t)^*e^{-it\Delta}\right] \right| \! \right| }_{L^{p}_tL^{q}_x({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)}\\
{\leqslant}C{ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{\frac{d+2}{d+1}}}\frac{C^{n+m}{ \left| \! \left| V \right| \! \right| }_{L^{p'}_tL^{q'}_x}^{n+m}}{(n!)^{\frac{1}{2q'}}(m!)^{\frac{1}{2q'}}},\end{gathered}$$ for some $C>0$ independent of $Q_0$, $n$, $m$, and $V$.
The proof follows the same lines as in $d=2$, and relies on the following estimate for any $n,m$ $$\begin{gathered}
{ \left| \! \left| \int_0^{\infty}{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}U(t,x)e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)\,dt \right| \! \right| }_{{\mathfrak{S}}^{d+2}}\\
{\leqslant}{ \left| \! \left| U \right| \! \right| }_{L^{p'}_tL^{q'}_x}\frac{C^n{ \left| \! \left| V \right| \! \right| }_{L^{p'}_tL^{q'}_x}^n}{(n!)^{\frac{1}{2q'}}}\frac{C^m{ \left| \! \left| V \right| \! \right| }_{L^{p'}_tL^{q'}_x}^m}{(m!)^{\frac{1}{2q'}}}.
\end{gathered}$$
We see that the terms involving $Q_0$ can be treated in any dimension, provided that $Q_0$ is in an adequate Schatten space. This is not the case for the terms involving $\gamma_f$, for which we can only deal with the higher orders.
\[lemma:higher-order-gamma\] Let $d{\geqslant}1$, $g:{{\ensuremath {\mathbb R} }}^d\to{{\ensuremath {\mathbb R} }}$ such that $\check{g}\in L^1({{\ensuremath {\mathbb R} }}^d)$, $1<q{\leqslant}1+2/d$ and $p$ such that $2/p+d/q=d$. Let $V\in L^{p'}_tL^{q'}_x({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)$. Then, for all $n,m$ such that $$n+m+1{\geqslant}2q',$$ we have $$\begin{gathered}
{ \left| \! \left| \rho\left[e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)\gamma_f{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}\right] \right| \! \right| }_{L^p_tL^q_x}\\
{\leqslant}C\|\check{g}\|_{L^1}\frac{C^n{ \left| \! \left| V \right| \! \right| }_{L^{p'}_tL^{q'}_x}^n}{(n!)^{\frac{1}{2q'}}}\frac{C^m{ \left| \! \left| V \right| \! \right| }_{L^{p'}_tL^{q'}_x}^m}{(m!)^{\frac{1}{2q'}}}.
\end{gathered}$$ where $\gamma_f=g(-i\nabla)$.
We again argue by duality. Let $U\in L^{p'}_tL^{q'}_x$. Without loss of generality, we can assume that $U,V{\geqslant}0$. Then, we evaluate $$\begin{gathered}
\int_0^{\infty}{{\rm Tr}}\left[U(t,x)e^{it\Delta}{\mathcal{W}}^{(n)}_V(t)\gamma_f{\mathcal{W}}^{(m)}_V(t)^*e^{-it\Delta}\right]\,dt\\
=(-i)^ni^m\int_0^{\infty}\,dt\int_{0{\leqslant}s_1{\leqslant}\cdots{\leqslant}s_m{\leqslant}t}\,ds_1\cdots ds_m\int_{0{\leqslant}t_1{\leqslant}\cdots{\leqslant}t_n{\leqslant}t}\,dt_1\cdots dt_n\times\\
\times{{\rm Tr}}\left[V(s_1,x-2is_1\nabla)\cdots V(s_m,x-2is_m\nabla)U(t,x-2it\nabla)\times\right.\\
\left.\times V(t_n,x-2it_n\nabla)\cdots V(t_1,x-2it_1\nabla)\gamma_f\right],\end{gathered}$$ where we used the relation $$e^{-it\Delta}W(t,x)e^{it\Delta}=W(t,x-2it\nabla).$$ In the spirit of [@FraLewLieSei-13], we gather the terms using the cyclicity of the trace as $$\begin{gathered}
\label{eq:bigtrace}
{{\rm Tr}}\left[V(s_1,x-2is_1\nabla)\cdots V(s_m,x-2is_m\nabla)U(t,x-2it\nabla)\times\right.\\
\left.\times V(t_n,x-2it_n\nabla)\cdots V(t_1,x-2it_1\nabla)\gamma_f\right]\\
={{\rm Tr}}\left[V(s_1,x-2is_1\nabla)^{\frac 12}V(s_2,x-2is_2\nabla)^{\frac 12}\cdots\right.\times\\
V(s_m,x-2is_m\nabla)^{\frac 12}U(t,x-2it\nabla)^{\frac 12}U(t,x-2it\nabla)^{\frac 12}V(t_n,x-2it_n\nabla)^{\frac 12}\times\\
\left.\times\cdots V(t_1,x-2it_1\nabla)^{\frac 12}\gamma_fV(s_1,x-2is_1\nabla)^{\frac 12}\right]. \end{gathered}$$ The first ingredient to estimate this trace is [@FraLewLieSei-13 Lemma 1], which states that $${ \left| \! \left| {\varphi}_1(\alpha x-i\beta\nabla){\varphi}_2(\gamma x-i\delta\nabla) \right| \! \right| }_{{\mathfrak{S}}^r}{\leqslant}\frac{{ \left| \! \left| {\varphi}_1 \right| \! \right| }_{L^r({{\ensuremath {\mathbb R} }}^d)}{ \left| \! \left| {\varphi}_2 \right| \! \right| }_{L^r({{\ensuremath {\mathbb R} }}^d)}}{(2\pi)^{\frac dr}|\alpha\delta-\beta\gamma|^{\frac dr}},\quad\forall r{\geqslant}2.
\label{eq:gKSS}$$ The second ingredient, to treat the term with $\gamma_f$, is a generalization of this inequality involving $\gamma_f$.
\[lemma:KSS-gamma\] There exists a constant $C>0$ such that for all $t,s\in{{\ensuremath {\mathbb R} }}$ we have $${ \left| \! \left| {\varphi}_1(x+2it\nabla)g(-i\nabla) {\varphi}_2(x+2is\nabla) \right| \! \right| }_{{\mathfrak{S}}^r}{\leqslant}\frac{\|\check{g}\|_{L^1({{\ensuremath {\mathbb R} }}^d)}}{(2\pi)^{\frac{d}{2}}}\frac{{ \left| \! \left| {\varphi}_1 \right| \! \right| }_{L^r({{\ensuremath {\mathbb R} }}^d)}{ \left| \! \left| {\varphi}_2 \right| \! \right| }_{L^r({{\ensuremath {\mathbb R} }}^d)}}{(2\pi)^{\frac{d}{r}}|t-s|^{\frac dr}}
\label{eq:estim_gamma_f}$$ for all $r{\geqslant}2$.
We remark that reduces to when $g=1$ and $\check{g}=(2\pi)^{d/2}\delta_0$. We postpone the proof of this lemma, and use it to estimate in the following way: $$\begin{gathered}
\left|{{\rm Tr}}\left[V(s_1,x-2is_1\nabla)\cdots V(s_m,x-2is_m\nabla)U(t,x-2it\nabla)\times\right.\right.\\
\left.\left.\times V(t_n,x-2it_n\nabla)\cdots V(t_1,x-2it_1\nabla)\gamma_f\right]\right|\\
{\leqslant}C{ \left| \! \left| \check{g} \right| \! \right| }_{L^1}\frac{{ \left| \! \left| V(s_1) \right| \! \right| }_{L^{q'}}\cdots{ \left| \! \left| V(s_m) \right| \! \right| }_{L^{q'}}{ \left| \! \left| U(t) \right| \! \right| }_{L^{q'}}{ \left| \! \left| V(t_n) \right| \! \right| }_{L^{q'}}{ \left| \! \left| V(t_1) \right| \! \right| }_{L^{q'}}}{|s_1-t_1|^{\frac{d}{2q'}}\cdots|s_m-t|^{\frac{d}{2q'}}|t-t_n|^{\frac{d}{2q'}}\cdots|t_2-t_1|^{\frac{d}{2q'}}}.\end{gathered}$$ Here, we have used the condition $n+m+1{\geqslant}2q'$ to ensure that the operator inside the trace is trace-class by Hölder’s inequality. From this point the proof is identical to the proof of [@FraLewLieSei-13 Thm. 3].
The inequality is immediate if $r={\infty}$. Hence, by complex interpolation, we only have to prove it for $r=2$. We have $$\begin{aligned}
\|{\varphi}_1(t,x+2it\nabla) & g(-i\nabla) {\varphi}_2(s,x+2is\nabla)\|_{{\mathfrak{S}}^2}^2\\
&= {{\rm Tr}}\left[{\varphi}_1(x)^2e^{i(t-s)\Delta}g(-i\nabla){\varphi}_2(x)^2e^{i(s-t)\Delta}g(-i\nabla)\right]\\
&= \frac{(2\pi)^{-2d}}{|t-s|^d}\iint {\varphi}_1(x)^2\left|\left(\check{g}*e^{-i\frac{|\cdot|^2}{4(t-s)}}\right)(x-y)\right|^2{\varphi}_2(y)^2\,dx\,dy\\
&{\leqslant}\frac{(2\pi)^{-2d}}{|t-s|^d}{ \left| \! \left| \check{g} \right| \! \right| }_{L^1}^2{ \left| \! \left| {\varphi}_1 \right| \! \right| }_{L^2}^2{ \left| \! \left| {\varphi}_2 \right| \! \right| }_{L^2}^2.
\end{aligned}$$
In dimension $d$, we want to prove that $\rho_Q$ belongs to $L^{1+2/d}_{t,x}$, hence we consider $q=1+2/d$ and $q'=1+d/2$. The previous result estimates the terms of order $n+m+1{\geqslant}d+2$, that is $n+m{\geqslant}d+1$. The case $n+m=1$ corresponds exactly to the linear response studied in the previous section. In dimension $d=2$, we see that we are still lacking the case $n+m=2$, which is what we call the second order. The next section is devoted to this order. We are not able to treat the terms with $1<n+m{\leqslant}d$ in other dimensions.
Second order in 2D {#sec:second-order}
==================
The study of the linear response is not enough to prove dispersion for the Hartree equation in 2D. We also have to estimate the second order term, that we first compute explicitly in any dimension, and then study only in dimension 2.
Exact computation in any dimension
----------------------------------
Define the second order term in the Duhamel expansion of $Q(t)$, $$\begin{gathered}
Q_2(t):=\\
(-i)^2\int_0^t{{\ensuremath{\,\text{d}s}}}\int_0^s{{\ensuremath{\,\text{d}t_1}}}e^{i(t-s)\Delta}[V(s),e^{i(s-t_1)\Delta}[V(t_1),\gamma_f]e^{i(t_1-s)\Delta}]e^{i(s-t)\Delta},
\end{gathered}$$ where we again used the notation $V=w*\rho_Q$. We compute explicitly its density. To do so, we let $W\in {\mathcal{D}}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^d)$ and use the relation $$\int_0^{\infty}\int_{{{\ensuremath {\mathbb R} }}^d}W(t,x)\rho_{Q_2}(t,x)\,dx\,dt=\int_0^{\infty}{{\rm Tr}}[W(t)Q_2(t)]\,dt.$$ For any $(p,q)\in{{\ensuremath {\mathbb R} }}^d\times{{\ensuremath {\mathbb R} }}^d$ we have $$\begin{gathered}
{\widehat}{Q_2}(t,p,q)=-\frac{1}{(2\pi)^d}\int_0^tds\int_0^sdt_1\int_{{{\ensuremath {\mathbb R} }}^d}dq_1\,e^{i(t-s)(q^2-p^2)}\times\\
\times\left[{\widehat}{V}(s,p-q_1)e^{i(s-t_1)(q^2-q_1^2)}{\widehat}{V}(t_1,q_1-q)(g(q)-g(q_1))\right.\\
\left.-{\widehat}{V}(s,q_1-q)e^{i(s-t_1)(q_1^2-p^2)}{\widehat}{V}(t_1,p-q_1)(g(q_1)-g(p))\right].
\end{gathered}$$ Using that $${{\rm Tr}}[W(t)Q_2(t)]=\frac{1}{(2\pi)^{d/2}}\int_{{{\ensuremath {\mathbb R} }}^d\times{{\ensuremath {\mathbb R} }}^d}{\widehat}{W}(t,q-p){\widehat}{Q_2}(t,p,q)\,dp\,dq,$$ we arrive at the formula $$\begin{gathered}
\int_0^{\infty}\int_{{{\ensuremath {\mathbb R} }}^d}W(t,x)\rho_{Q_2}(t,x)\,dx\,dt = \int_0^{\infty}\int_0^{\infty}\int_0^{\infty}\int_{{{\ensuremath {\mathbb R} }}^d\times{{\ensuremath {\mathbb R} }}^d}\,dt\,ds\,dt_1\,dk\,d\ell\times\\
\times K^{(2)}(t-s,s-t_1;k,\ell){\widehat}{W}(t,-k){\widehat}{\rho_Q}(s,k-\ell){\widehat}{\rho_Q}(t_1,\ell),
\end{gathered}$$ with $$\begin{gathered}
K^{(2)}(t,s;k,\ell)\\
={{\ensuremath {\mathds 1} }}_{t{\geqslant}0}{{\ensuremath {\mathds 1} }}_{s{\geqslant}0}\frac{4{\widehat}{w}(\ell){\widehat}{w}(k-\ell)}{(2\pi)^{d/2}}\sin(tk\cdot(k-\ell))\sin(\ell\cdot(tk+s\ell))\check{g}(2(tk+s\ell)).
\end{gathered}$$
Estimates in 2D
---------------
\[prop:second-order\] Assume that $g\in L^1({{\ensuremath {\mathbb R} }}^2)$ is such that $|x|^{a}|\check{g}(x)|\in L^{\infty}({{\ensuremath {\mathbb R} }}^2)$ for some $a>3$. Assume also that $w$ is such that $(1+|k|^{1/2})|{\widehat}{w}(k)|\in L^{\infty}({{\ensuremath {\mathbb R} }}^2)$. Then, if $\rho_Q\in L^2_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^2)$ we have $$\|\rho_{Q_2}\|_{L^2_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^2)}{\leqslant}C{ \left| \! \left| (1+|\cdot|^2)^{a/2}\check{g} \right| \! \right| }_{L^{\infty}}{ \left| \! \left| (1+|\cdot|^{1/2}){\widehat}{w} \right| \! \right| }_{L^{\infty}}\|\rho_Q\|_{L^2_{t,x}({{\ensuremath {\mathbb R} }}\times{{\ensuremath {\mathbb R} }}^2)}^2,$$ for some constant $C(g,w)$ only depending on $g$ and $w$.
First, we have the estimate $$\left|\int_{{{\ensuremath {\mathbb R} }}^3}G(t_1-t_2,t_2-t_3)f_1(t_1)f_2(t_2)f_3(t_3)\,dt_1\,dt_2\,dt_3\right|{\leqslant}C\|G\|_{L^2L^1}\prod_{i=1}^3\|f_i\|_{L^2},$$ for any $G$, and hence $$\begin{gathered}
\left|\int_{{{\ensuremath {\mathbb R} }}^3}\!\!K^{(2)}(t_1-t_2,t_2-t_3;k,\ell){\widehat}{W}(t_1,-k){\widehat}{\rho_Q}(t_2,k-\ell){\widehat}{\rho_Q}(t_3,\ell)\,dt_1\,dt_2\,dt_3\right|\\
{\leqslant}\|K^{(2)}(t,s;k,\ell)\|_{L^2_tL^1_s}\|{\widehat}{W}(\cdot,-k)\|_{L^2}\|{\widehat}{\rho_Q}(\cdot,k-\ell)\|_{L^2}\|{\widehat}{\rho_Q}(\cdot,\ell)\|_{L^2}.\end{gathered}$$ Let us thus estimate $\|K^{(2)}(t,s;k,\ell)\|_{L^2_tL^1_s}$. To do so, we use $|\sin(tk\cdot(k-\ell))|{\leqslant}1$, $|\sin(\ell\cdot(tk+s\ell))|{\leqslant}|\ell||tk+s\ell|$ and get $$\begin{gathered}
\|K^{(2)}(t,s;k,\ell)\|_{L^2_tL^1_s}^2\\ {\leqslant}\frac{16{\widehat}{w}(\ell)^2{\widehat}{w}(k-\ell)^2}{(2\pi)^{d}}\ell^2\int_{{{\ensuremath {\mathbb R} }}}\,dt\left|\int_{{{\ensuremath {\mathbb R} }}}\,ds|tk+s\ell||\check{g}(2(tk+s\ell))|\right|^2. \end{gathered}$$ We let $$u=\ell s+t\frac{k\cdot\ell}{\ell}\quad \text{and}\quad v=\sqrt{k^2-\frac{(k\cdot\ell)^2}{\ell^2}}t$$ and notice that $$|tk+s\ell|=\left(\ell^2 \left(s+t\frac{k\cdot\ell}{\ell^2}\right)^2 +\left(k^2-\frac{(k\cdot\ell)^2}{\ell^2}\right)t^2\right)^{1/2}=\sqrt{u^2+v^2}.$$ Since $\check{g}$ is a radial function we find that $$\begin{gathered}
\ell^2\int_{{{\ensuremath {\mathbb R} }}}\,dt\left|\int_{{{\ensuremath {\mathbb R} }}}\,ds|tk+s\ell||\check{g}(2(tk+s\ell))|\right|^2\\
=\frac{|\ell|}{\left(k^2\ell^2-(k\cdot\ell)^2\right)^{1/2}}\int_{{{\ensuremath {\mathbb R} }}}\,dv\left|\int_{{{\ensuremath {\mathbb R} }}}\,du\sqrt{u^2+v^2}|\check{g}(2\sqrt{u^2+v^2})|\right|^2.\end{gathered}$$ The double integral on the right is finite under some mild decay assumptions on $\check{g}$, for instance it is finite if $|\check{g}(r)|{\leqslant}C(1+r^2)^{-a/2}$, for some $a>3$. Noticing that $\left(k^2\ell^2-(k\cdot\ell)^2\right)^{1/2}=|\det(k,\ell)|$, we thus have $$\begin{gathered}
|\langle W,\rho_{Q_2}\rangle|{\leqslant}C{ \left| \! \left| (1+|\cdot|^2)^{a/2}\check{g} \right| \! \right| }_{L^{\infty}}\int_{{{\ensuremath {\mathbb R} }}^{2d}}\,dk\,d\ell\times\\
\times\frac{\|{\widehat}{W}(\cdot,-k)\|_{L^2}|{\widehat}{w}(k-\ell)|\|{\widehat}{\rho_Q}(\cdot,k-\ell)\|_{L^2}|\ell|^{1/2}|{\widehat}{w}(\ell)|\|{\widehat}{\rho_Q}(\cdot,\ell)\|_{L^2}}{|\det(k,\ell)|^{1/2}}.\end{gathered}$$ We prove the following inequality of Hardy-Littlewood-Sobolev type:
\[lemma:det\] For any functions $f,g,h$ we have $$\left|\int_{{{\ensuremath {\mathbb R} }}^2\times{{\ensuremath {\mathbb R} }}^2}\frac{f(k)g(k-\ell)h(\ell)}{|\det(k,\ell)|^{1/2}}{{\ensuremath{\,\text{d}k}}}{{\ensuremath{\,\text{d}\ell}}}\right|{\leqslant}C\|f\|_{L^2}\|g\|_{L^2}\|h\|_{L^2}.$$
Since $\det(k,\ell)=k_1\ell_2-k_2\ell_1$, we first fix $k_1\neq0$, $\ell_1\neq0$, $k_1\neq\ell_1$ and estimate $$\begin{gathered}
\left|\int_{{{\ensuremath {\mathbb R} }}^2}\frac{f(k_1,k_2)g(k_1-\ell_1,k_2-\ell_2)h(\ell_1,\ell_2)}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2\right| \\
{\leqslant}\left(\int_{{{\ensuremath {\mathbb R} }}^2}\frac{|f(k_1,k_2)|^{3/2}|g(k_1-\ell_1,k_2-\ell_2)|^{3/2}}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2\right)^{1/3}\times\\
\times\left(\int_{{{\ensuremath {\mathbb R} }}^2}\frac{|f(k_1,k_2)|^{3/2}|h(\ell_1,\ell_2)|^{3/2}}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2\right)^{1/3}\times\\
\times\left(\int_{{{\ensuremath {\mathbb R} }}^2}\frac{|g(k_1-\ell_1,k_2-\ell_2)|^{3/2}|h(\ell_1,\ell_2)|^{3/2}}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2\right)^{1/3}.
\end{gathered}$$ We then have $$\begin{aligned}
\int_{{{\ensuremath {\mathbb R} }}^2}&\frac{|f(k_1,k_2)|^{3/2}|g(k_1-\ell_1,k_2-\ell_2)|^{3/2}}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2 \\
&= \int_{{{\ensuremath {\mathbb R} }}^2}\frac{|f(k_1,k_2)|^{3/2}|g(k_1-\ell_1,\ell_2)|^{3/2}}{|k_2(k_1-\ell_1)-\ell_2k_1|^{1/2}}\,dk_2\,d\ell_2\\
&= \frac{1}{|k_1||k_1-\ell_1|}\int_{{{\ensuremath {\mathbb R} }}^2}\frac{|f(k_1,k_2/(k_1-\ell_1))|^{3/2}|g(k_1-\ell_1,\ell_2/k_1)|^{3/2}}{|k_2-\ell_2|^{1/2}}\,dk_2\,d\ell_2\\
&{\leqslant}\frac{C}{|k_1||k_1-\ell_1|}\|f(k_1,\cdot/(k_1-\ell_1))\|_{L^2}^{3/2}\|g(k_1-\ell_1,\cdot/k_1)\|_{L^2}^{3/2}\\
&{\leqslant}\frac{C}{|k_1|^{1/4}|k_1-\ell_1|^{1/4}}\|f(k_1,\cdot)\|_{L^2}^{3/2}\|g(k_1-\ell_1,\cdot)\|_{L^2}^{3/2},
\end{aligned}$$ and in the same fashion $$\int_{{{\ensuremath {\mathbb R} }}^2}\!\!\frac{|f(k_1,k_2)|^{3/2}|h(\ell_1,\ell_2)|^{3/2}}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2{\leqslant}\frac{C}{|k_1|^{\frac 14}|\ell_1|^{\frac 14}}\|f(k_1,\cdot)\|_{L^2}^{3/2}\|h(\ell_1,\cdot)\|_{L^2}^{3/2},$$ $$\begin{gathered}
\int_{{{\ensuremath {\mathbb R} }}^2}\frac{|g(k_1-\ell_1,k_2-\ell_2)|^{3/2}|h(\ell_1,\ell_2)|^{3/2}}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2\\
{\leqslant}\frac{C}{|\ell_1|^{1/4}|k_1-\ell_1|^{1/4}}\|g(k_1-\ell_1,\cdot)\|_{L^2}^{3/2}\|h(\ell_1,\cdot)\|_{L^2}^{3/2}.
\end{gathered}$$ As a consequence, we have $$\begin{gathered}
\left|\int_{{{\ensuremath {\mathbb R} }}^2}\frac{f(k_1,k_2)g(k_1-\ell_1,k_2-\ell_2)h(\ell_1,\ell_2)}{|k_1\ell_2-k_2\ell_1|^{1/2}}\,dk_2\,d\ell_2\right|\\
{\leqslant}C\frac{\|f(k_1,\cdot)\|_{L^2}\|g(k_1-\ell_1,\cdot)\|_{L^2}\|h(\ell_1,\cdot)\|_{L^2}}{|k_1|^{1/6}|\ell_1|^{1/6}|k_1-\ell_1|^{1/6}}.
\end{gathered}$$ We now need a multilinear Hardy-Littlewood-Sobolev-type inequality. Integrating over $(k_1,\ell_1)$ we find that $$\begin{aligned}
\left|\int_{{{\ensuremath {\mathbb R} }}^2\times{{\ensuremath {\mathbb R} }}^2}\right. & \left.\frac{f(k)g(k-\ell)h(\ell)}{|\det(k,\ell)|^{1/2}}\,dk\,d\ell \right| \\
&{\leqslant}C\int_{{{\ensuremath {\mathbb R} }}^2}\frac{\|f(k_1,\cdot)\|_{L^2}\|g(k_1-\ell_1,\cdot)\|_{L^2}\|h(\ell_1,\cdot)\|_{L^2}}{|k_1|^{1/6}|\ell_1|^{1/6}|k_1-\ell_1|^{1/6}}\,dk_1\,d\ell_1\\
&{\leqslant}C\left(\frac{\|g(k_1-\ell_1,\cdot)\|_{L^2}^{3/2}\|h(\ell_1,\cdot)\|_{L^2}^{3/2}}{|k_1|^{1/2}}\,dk_1\,d\ell_1\right)^{1/3}\times\\
&\quad\times\left(\frac{\|f(k_1,\cdot)\|_{L^2}^{3/2}\|g(k_1-\ell_1,\cdot)\|_{L^2}^{3/2}}{|\ell_1|^{1/2}}\,dk_1\,d\ell_1\right)^{1/3}\times\\
&\quad\times\left(\frac{\|f(k_1,\cdot)\|_{L^2}^{3/2}\|h(\ell_1,\cdot)\|_{L^2}^{3/2}}{|k_1-\ell_1|^{1/2}}\,dk_1\,d\ell_1\right)^{1/3}\\
&{\leqslant}C\|f\|_{L^2}\|g\|_{L^2}\|h\|_{L^2}
\end{aligned}$$ where in the last line we have used the 2D Hardy-Littlewood-Sobolev inequality.
From the lemma, we deduce that $$|\langle W,\rho_{Q_2}\rangle|{\leqslant}C{ \left| \! \left| (1+|\cdot|^2)^{a/2}\check{g} \right| \! \right| }_{L^{\infty}}{ \left| \! \left| (1+|\cdot|^{1/2}){\widehat}{w} \right| \! \right| }_{L^{\infty}}\|\rho_Q\|_{L^2_{t,x}}^2,$$ which ends the proof of the proposition.
Proof of the main theorem {#sec:proof-thm}
=========================
Let $T>0$. Assume also that ${ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}{\leqslant}1$. We solve the equation $$\begin{aligned}
\rho_Q(t)&=\rho\left[e^{it\Delta}{\mathcal{W}}_{w*\rho_Q}(t)(\gamma_f+Q_0){\mathcal{W}}_{w*\rho_Q}(t)^*e^{-it\Delta}\right]-\rho_{\gamma_f}\\
&=\rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right]-{\mathcal{L}}(\rho_Q)+{\mathcal{R}}(\rho_Q)\end{aligned}$$ by a fixed-point argument. Here ${\mathcal{L}}={\mathcal{L}}_1+{\mathcal{L}}_2$ where ${\mathcal{L}}_1$ was studied in Section \[sec:linear-response\] and $${\mathcal{L}}_2(\rho_Q)=-\rho\left[e^{it\Delta}({\mathcal{W}}^{(1)}_{w*\rho_Q}(t)Q_0+Q_0{\mathcal{W}}^{(1)}_{w*\rho_Q}(t)^*)e^{-it\Delta}\right].$$ As explained in Proposition \[prop:linear-response\] and in Corollary \[cor:linear-response\_defocusing\], under the assumption (or when $f$ is strictly decreasing), $(1+{\mathcal{L}}_1)$ is invertible with bounded inverse on $L^2_{t,x}$. The operator $1+{\mathcal{L}}=1+{\mathcal{L}}_1+{\mathcal{L}}_2$ is invertible with bounded inverse when $${ \left| \! \left| {\mathcal{L}}_2 \right| \! \right| }<\frac{1}{{ \left| \! \left| (1+{\mathcal{L}}_1)^{-1} \right| \! \right| }}.$$ By Lemma \[lemma:higher-order-Q0\], we have $${ \left| \! \left| {\mathcal{L}}_2 \right| \! \right| }{\leqslant}C{ \left| \! \left| w \right| \! \right| }_{L^1}{ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}$$ and therefore a sufficient condition can be expressed as $${ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}<\frac{1}{C{ \left| \! \left| w \right| \! \right| }_{L^1}{ \left| \! \left| (1+{\mathcal{L}}_1)^{-1} \right| \! \right| }}.$$ Then we can write $$\rho_Q(t)=(1+{{\mathcal{L}}})^{-1}\left(\rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right]+{\mathcal{R}}(\rho_Q)\right).$$ For any ${\varphi}\in L^2_{t,x}([0,T]\times{{\ensuremath {\mathbb R} }}^2)$, define $$F({\varphi})(t)=\rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right]+{\mathcal{R}}({\varphi}).$$ We apply the Banach fixed-point theorem on the map $(1+{\mathcal{L}})^{-1}F$. To do so, we expand $F$ as $$\begin{gathered}
F({\varphi})(t)=\rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right]+\sum_{n+m{\geqslant}2}\rho\left[e^{it\Delta}{\mathcal{W}}_{w*{\varphi}}(t)Q_0{\mathcal{W}}_{w*{\varphi}}(t)^*e^{-it\Delta}\right]\\
+\sum_{n+m=2}\rho\left[e^{it\Delta}{\mathcal{W}}_{w*{\varphi}}^{(n)}(t)\gamma_f{\mathcal{W}}_{w*{\varphi}}^{(m)}(t)^*e^{-it\Delta}\right]\\
+\sum_{n+m{\geqslant}3}\rho\left[e^{it\Delta}{\mathcal{W}}_{w*{\varphi}}^{(n)}(t)\gamma_f{\mathcal{W}}_{w*{\varphi}}^{(m)}(t)^*e^{-it\Delta}\right].\end{gathered}$$ By the Strichartz estimate , we have $${ \left| \! \left| \rho\left[e^{it\Delta}Q_0e^{-it\Delta}\right] \right| \! \right| }_{L^2_{t,x}}{\leqslant}C{ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}.$$ By Lemma \[lemma:higher-order-Q0\], we have $$\begin{gathered}
{ \left| \! \left| \sum_{n+m{\geqslant}2}\rho\left[e^{it\Delta}{\mathcal{W}}_{w*{\varphi}}(t)Q_0{\mathcal{W}}_{w*{\varphi}}(t)^*e^{-it\Delta}\right] \right| \! \right| }_{L^2_{t,x}}\\
{\leqslant}C{ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}\sum_{n+m{\geqslant}2}\frac{C^{n+m}{ \left| \! \left| w*{\varphi}\right| \! \right| }_{L^2_{t,x}}^{n+m}}{(n!)^{\frac 14}(m!)^{\frac 14}}.\end{gathered}$$ By Proposition \[prop:second-order\], we have $$\begin{gathered}
{ \left| \! \left| \sum_{n+m=2}\rho\left[e^{it\Delta}{\mathcal{W}}_{w*{\varphi}}^{(n)}(t)\gamma_f{\mathcal{W}}_{w*{\varphi}}^{(m)}(t)^*e^{-it\Delta}\right] \right| \! \right| }_{L^2_{t,x}}\\
{\leqslant}C{ \left| \! \left| (1+|\cdot|^2)^{a/2})\check{g} \right| \! \right| }_{L^{\infty}}{ \left| \! \left| (1+|\cdot|^{1/2}){\widehat}{w} \right| \! \right| }_{L^{\infty}}{ \left| \! \left| {\varphi}\right| \! \right| }_{L^2_{t,x}}^2.\end{gathered}$$ Finally, by Lemma \[lemma:higher-order-gamma\] we have $$\begin{gathered}
{ \left| \! \left| \sum_{n+m{\geqslant}3}\rho\left[e^{it\Delta}{\mathcal{W}}_{w*{\varphi}}^{(n)}(t)\gamma_f{\mathcal{W}}_{w*{\varphi}}^{(m)}(t)^*e^{-it\Delta}\right] \right| \! \right| }_{L^2_{t,x}}\\
{\leqslant}C\|\check{g}\|_{L^1}\sum_{n+m{\geqslant}3}\frac{C^{n+m}{ \left| \! \left| w*{\varphi}\right| \! \right| }_{L^2_{t,x}}^{n+m}}{(n!)^{\frac{1}{4}}(m!)^{\frac{1}{4}}}.\end{gathered}$$ We deduce that for all ${\varphi}\in L^2_{t,x}([0,T]\times{{\ensuremath {\mathbb R} }}^2)$, we have the estimate $${ \left| \! \left| (1+{\mathcal{L}})^{-1}F({\varphi}) \right| \! \right| }_{L^2_{t,x}}{\leqslant}C{ \left| \! \left| (1+{\mathcal{L}})^{-1} \right| \! \right| }\left({ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4/3}}+A({ \left| \! \left| {\varphi}\right| \! \right| }_{L^2_{t,x}})\right),$$ where we used the notation $$\begin{gathered}
A(z)=C\sum_{n+m{\geqslant}2}\frac{C^{n+m}\left({ \left| \! \left| w \right| \! \right| }_{L^1}z\right)^{n+m}}{(n!)^{\frac 14}(m!)^{\frac 14}}\\
+C{ \left| \! \left| (1+|\cdot|^2)^{a/2})\check{g} \right| \! \right| }_{L^{\infty}}{ \left| \! \left| (1+|\cdot|^{1/2}){\widehat}{w} \right| \! \right| }_{L^{\infty}}z^2\\
+C\|\check{g}\|_{L^1}\sum_{n+m{\geqslant}3}\frac{C^{n+m}\left({ \left| \! \left| w \right| \! \right| }_{L^1}z\right)^{n+m}}{(n!)^{\frac{1}{4}}(m!)^{\frac{1}{4}}}.\end{gathered}$$ We have $A(z)=O(z^2)$ as $z\to0$. As a consequence, there exists $C_0,z_0>0$ only depending on ${ \left| \! \left| w \right| \! \right| }_{L^1}$, ${ \left| \! \left| (1+|\cdot|^2)^{a/2})\check{g} \right| \! \right| }_{L^{\infty}}{ \left| \! \left| (1+|\cdot|^{1/2}){\widehat}{w} \right| \! \right| }_{L^{\infty}}$, and $\|\check{g}\|_{L^1}$ such that $$|A(z)|{\leqslant}C_0z^2,$$ for all $|z|{\leqslant}z_0$. Choosing $$R=\min\left(z_0,\frac{1}{2C_0{ \left| \! \left| (1+{\mathcal{L}})^{-1} \right| \! \right| }}\right)$$ and $${ \left| \! \left| Q_0 \right| \! \right| }_{{\mathfrak{S}}^{4,3}}{\leqslant}\min\left(1,\frac{R}{2C{ \left| \! \left| (1+{\mathcal{L}})^{-1} \right| \! \right| }}\right),$$ leads to the estimate $${ \left| \! \left| (1+{\mathcal{L}})^{-1}F({\varphi}) \right| \! \right| }_{L^2_{t,x}}{\leqslant}R,$$ for all ${ \left| \! \left| {\varphi}\right| \! \right| }_{L^2_{t,x}}{\leqslant}R$, independently of the maximal time $T>0$. Similar estimates show that $F$ is also a contraction on this ball, up to diminishing $R$ if necessary. The Banach fixed point theorem shows that there exists a solution for any time $T>0$, with a uniform estimate with respect to $T$. Having built this solution ${\varphi}_0\in L^2_{t,x}({{\ensuremath {\mathbb R} }}_+\times{{\ensuremath {\mathbb R} }}^2)$, we define the operator $\gamma$ as $$\gamma(t)=e^{it\Delta}{\mathcal{W}}_{w*{\varphi}_0}(t)(\gamma_f+Q_0){\mathcal{W}}_{w*{\varphi}_0}(t)^*e^{-it\Delta}.$$ We have ${\varphi}_0=\rho_\gamma-\rho_{\gamma_f}$ by definition. Uniqueness of solutions independently of whether they belong to the ball where we performed the fixed point argument follows from the same arguments as in the proof of [@LewSab-13a Thm. 5].
From [@FraLewLieSei-13 Thm. 3] we know that ${\mathcal{W}}_{w\ast{\varphi}_0}-1\in C^0_t({{\ensuremath {\mathbb R} }}_+,{\mathfrak{S}}^4)$ and that ${\mathcal{W}}_{w\ast{\varphi}_0}-1$ admits a strong limit in ${\mathfrak{S}}^4$ when $t\to{\infty}$, which gives that $\gamma-\gamma_f\in C^0({{\ensuremath {\mathbb R} }}_+,{\mathfrak{S}}^{4})$ and our scattering result . Next, we remark that since $w\in W^{1,1}({{\ensuremath {\mathbb R} }}^2)\subset L^2({{\ensuremath {\mathbb R} }}^2)$, we have $w\ast{\varphi}_0\in L^2_t(L^{\infty}_x\cap L^2_x)$. From [@LewSab-13a Lemma 7] and the fact that $g\in L^2({{\ensuremath {\mathbb R} }}^2)$ (due to ), we deduce that $({\mathcal{W}}_{w\ast{\varphi}_0}(t)-1)\gamma_f\in C^0({{\ensuremath {\mathbb R} }}_+,{\mathfrak{S}}^{2})$. This now shows that $\gamma-\gamma_f\in C^0({{\ensuremath {\mathbb R} }}_+,{\mathfrak{S}}^{2})$. Of course, we can perform the same procedure for negative times and this finishes the proof of Theorem \[thm:main\].
**Acknowledgements.** This work was partially done while the authors were visiting the Centre Émile Borel at the Institut Henri Poincaré in Paris. The authors acknowledge financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement MNIQS 258023), and from the French ministry of research (ANR-10-BLAN-0101).
[10]{}
, [*Semilinear [S]{}chrödinger equations*]{}, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003.
, [*Energy [C]{}ost to [M]{}ake a [H]{}ole in the [F]{}ermi [S]{}ea*]{}, Phys. Rev. Lett., 106 (2011), p. 150402.
height 2pt depth -1.6pt width 23pt, [*A positive density analogue of the [L]{}ieb-[T]{}hirring inequality*]{}, Duke Math. J., 162 (2012), pp. 435–495.
height 2pt depth -1.6pt width 23pt, [*Strichartz inequality for orthonormal functions*]{}, J. Eur. Math. Soc. (JEMS), in press (2013).
, [*On a class of nonlinear [S]{}chr[ö]{}dinger equations with nonlocal interaction*]{}, Math. Z., 170 (1980), pp. 109–136.
, [*Scattering theory in the energy space for a class of hartree equations*]{}, Contemp. Math., 263 (2000), pp. 29–60.
, [*Quantum Theory of the Electron Liquid*]{}, Cambridge University Press, 2005.
, [*Classical Fourier analysis*]{}, Springer-Verlag New York, 2008.
, [*Existence of a stable polarized vacuum in the [B]{}ogoliubov-[D]{}irac-[F]{}ock approximation*]{}, Commun. Math. Phys., 257 (2005), pp. 515–562.
, [*Scattering theory for [H]{}artree type equations*]{}, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), pp. 187–213.
, [*The [H]{}artree equation for infinitely many particles. [I]{}. [W]{}ell-posedness theory*]{}. in preparation, 2013.
, Dan. Mat. Fys. Medd., 28 (1954), p. 1.
, [*[Lindhard function of a d-dimensional Fermi gas]{}*]{}, ArXiv e-prints, (2011).
, [*On small data scattering with cubic convolution nonlinearity*]{}, J. Math. Soc. Japan, 41 (1989), pp. 143–160.
, [*Energy scattering for [H]{}artree equations*]{}, Math. Res. Lett., 6 (1999), pp. 107–118.
, [*Geometric methods in multiparticle quantum systems*]{}, Commun. Math. Phys., 55 (1977), pp. 259–274.
, [*Singular Integrals and Differentiability Properties of Functions*]{}, Princeton University Press, 1970.
, [*Nonlinear scattering theory at low energy*]{}, J. Funct. Anal., 41 (1981), pp. 110–133.
, [*Existence of solutions for [S]{}chrödinger evolution equations*]{}, Comm. Math. Phys., 110 (1987), pp. 415–426.
[^1]: ©2013 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
|
---
abstract: 'In the framework of the Chern-Simons gravity proposed by Witten, a transition amplitude of a torus universe in 2+1 dimensional quantum gravity is computed. This amplitude has the desired properties as a probability amplitude of the quantum mechanics of a torus universe, namely, it has a peak on the “classical orbit” and it satisfies the Shrödinger equation of the 2+1 dimentional gravity. The discussion is given that the classical orbit dominance of the amplitude is not altered by taking the modular invariance into account and that this amplitude can serve as a covariant transition amplitude in a particular sense. A set of the modular covariant wavefunctions is also constructed and they are shown to be equivalent with the weight-$1/2$ Maass forms.'
---
OU-HET/179\
hep-th/9305170\
May 1993
\
Kiyoshi Ezawa[^1]
Department of Physics\
Osaka University, Toyonaka, Osaka 560, Japan\
Introduction
============
For more than ten years, attention has been paid to the 2+1 dimensional gravity [@deser] [@marti] as a useful toy model which gives insights into the 3+1 dimensional quantum gravity. In 2+1 dimensions, the vacuum Einstein equation with vanishing cosmological constant requires the spacetime manifolds to be locally flat. The 2+1 dimensional Einstein gravity is therefore described by a finite number of global degrees of freedom [@deser]. It has been shown by Witten that this 2+1 dimensional Einstein gravity is equivalent to the ISO(2,1) gauge theory represented by the Chern-Simons action [@witte]. Many people have studied this “ Chern-Simons gravity ” [@regge] since then. Some of them investigated relations between the C-S gravity and the 2+1 dimensional ADM formalism [@carli] [@ander].
Most of these results are, however, formal in the sense that it is hard to extract physical pictures from them. There are some works which deal with physical processes by topological ideas on path integrals in the WKB approximation [@fujiw]. There appears to be no work which determines the quantum evolution of moduli in a fixed spatial topology.
In this paper, we compute a transition amplitude in the 2+1 dimensional quantum gravity that describes the evolution of moduli of the space with a definite topology ( here, we investigate the most tractable topology, i.e. a torus $T^{2}$ ). Our strategy is the following. The ADM formalism [@moncr] [@hosoy]is suitable for following the evolution of space manifolds. The complicated Hamiltonian, however, makes its quantization very hard. While the C-S gravity can be quantized in a simple way, it is difficult to give physical interpretations. We exploit both formalisms. By using relations between these two formalisms studied by Carlip [@carli] and by solving the Schrödinger equation of the quantum C-S gravity, we compute the transition amplitude describing a time-evolution of the moduli parameters, which are basic configuration variables of the ADM formalism.
This transition amplitude has a divergent peak on the classical orbit. It satisfies the Schrödinger equation of the ADM formalism, and possesses required properties as a probability amplitude of the 2+1dimensional quantum gravity. In particular it turns out that our amplitude serves as a modular covariant transition amplitude when the integration region of the inner product is extended to the upper-half plane.
Besides, a set of modular covariant wavefunctions, which are the eigenfunctions of the volume operator, are constructed. It is shown that these wavefunctions are related with the weight-$1/2$ Maass forms [@iwani] by way of the integral transformation.
In §2 we solve the Hamilton equations in the ADM formalism to investigate the classical evolution of a torus universe. There, the classical orbit is also found out. §3 is devoted to the review of the classical and quantum relations between the ADM formalism and the C-S gravity studied by Carlip. From these results in §3, the transition amplitude of a torus universe is computed in §4. Its properties are examined and we show that it can be interpreted as a probability amplitude. The main result in §5 is the equivalence of eq.(\[eq:covev\]) with eq.(\[eq:covev1\]), owing to which our amplitude can be regarded to be modular covariant in a sense. We will also show that physics is not modified after taking the modular invariance into account. The set of covariant wavefunctions is also given in this section. In §6, after summarizing our results, we discuss a possibility to generalize our method to more general contexts.
The classical orbit of a torus universe
=======================================
The 2+1 dimensional ADM formalism on a torus is formulated by Moncrief [@moncr] and Hosoya and Nakao [@hosoy] [@nakao]. In this section, we briefly review this ADM formalism on a torus, and look into the classical behavior of the torus universe by solving the Hamilton equations given by the reduced ADM action.
In the 2+1 dimensional ADM formalism, the spacetime manifold M is assumed to be homeomorphic to $R^{1}\times\Sigma$, where $\Sigma$ is a two dimensional space called time-slice. Then, the ADM action is given by $$I_{ADM} = \int dt\int_{\Sigma}d^{2}x(\pi^{ab}\dot{g}_{ab}-N^{a}{\cal H}_{a}-
N{\cal H}), \label{eq:action}$$ where $g_{ab}$ is the induced metric on $\Sigma$ , $\pi^{ab}$ is its conjugate momentum, $N^{a}$ is the shift vector, N is the lapse function, and ${\cal H}_{a}$ and ${\cal H}$ are respectively the momentum and Hamiltonian constraint. Here, we further assume that the time-slice $\Sigma$ has the topology of a torus $T^{2}$. On taking York’s time-slice, we find that eq.(\[eq:action\]) reduces to $$I^{*}_{ADM}=\int dt(p_{1}\dot{m}_{1}+p_{2}\dot{m}_{2}+\tau\dot{v}-N'{\cal H}'),
\label{eq:red.ac.}$$ where $m_{A}$ and $p_{A}$ (A = 1,2) are the moduli parameters and their conjugate momenta respectively, $N'\equiv N/2v$, and $${\cal H}'=m_{2}^{2}\{(p_{1})^{2}+(p_{2})^{2}\}-v^{2}\tau^{2}.$$
With York’s time-slice, the mean curvature $\tau\equiv\sqrt{g}^{-1}g_{ab}\pi^{ab}$ is constant everywhere on $\Sigma$. The conformal factor ( the “volume” ) $v\equiv\sqrt{\det(g_{ab})}$ is a global scalar quantity independent of spatial coordinates.
This reduced ADM action (\[eq:red.ac.\]) has a first class constraint ${\cal H}'\approx 0$. We may use this reduced action to see the classical behavior of the torus universe. Alternatively, we use the gauge-fixed system. We impose a coordinate condition $$\tau=t(=\mbox{time coordinate}),$$ and solve the constraint ${\cal H}'\approx 0$ explicitly. We find $$I^{**}_{ADM}=\int d\tau(p_{1}\dot{m}_{1}+p_{2}\dot{m}_{2}-H),
\label{eq:ADMaction}$$ where the dot denotes a derivative with respect to $\tau$, and the Hamiltonian $H$ of this system is $$H\equiv v=\frac{m_{2}}{\tau}\sqrt{(p_{1})^{2}+(p_{2})^{2}}.
\label{eq:H.ADM}$$
Let us use eqs.(\[eq:ADMaction\]) and (\[eq:H.ADM\]) to investigate the time evolution of the torus universe.
First, we solve the Hamilton equations: $$\frac{d}{d\tau}(m_{A},p_{A})=\{(m_{A},p_{A}),H\}_{P.B.},(A=1,2).$$ The results are $$\begin{aligned}
&&m_{1}=\frac{\frac{L+E}{\tau_{0}}+\frac{\tau_{0}(L-E)}{\tau^{2}}}{\frac{p_{1}}
{\tau_{0}}+\frac{p_{1}\tau_{0}}{\tau^{2}}}
\nonumber \\*
&&m_{2}=\frac{p_{1}}{|p_{1}|}\frac{2\frac{E}{\tau}}{\frac{p_{1}}{\tau_{0}}+
\frac{p_{1}\tau_{0}}{\tau^{2}}}
\nonumber \\*
&&p_{1}=\mbox{constant with respect to }\tau
\nonumber \\*
&&p_{2}=-\frac{\tau}{2}(\frac{|p_{1}|}{\tau_{0}}-\frac{|p_{1}|\tau_{0}}
{\tau^{2}}) ,
\label{eq:phase space}\end{aligned}$$ where $E$, $L$, $\tau_{0}$, and $p_{1}$ are the constants of motion.
We will explicitly construct a spacetime metric in 2+1 dimensions from this solution. Let us determine the lapse function and the shift vector. Following Moncrief [@moncr] , the lapse $N$ is found to be obtained from the Hamilton equation in the constrained system $ 1\equiv \frac{\partial \tau}{\partial t}=N'\{\tau,{\cal H}'\}_{P.B.}
=N\tau^{2} $; $$N=\frac{1}{\tau^{2}}.\label{eq:lapse}$$ The shift $N^{a}$ can be absorbed by a redefinition of the origin of each time-slice. Henceforth, we put $$N^{a}=0.\label{eq:shift}$$ From eqs.(\[eq:phase space\])–(\[eq:shift\]), we can compute the spacetime metric as $$\begin{aligned}
ds^{2}&=&-(Nd\tau)^{2}+\frac{v}{m_{2}}\{(dx+m_{1}dy)^{2}+(m_{2}dy)^{2}\}
\nonumber \\*
&=&-\frac{d\tau^{2}}{\tau^{4}}+\frac{1}{\tau^{2}}(\sqrt{\frac{|p_{1}|\tau_{
0}}{2}}dx+\sqrt{\frac{|p_{1}|\tau_{0}}{2}}\frac{L-E}{p_{1}}dy)^{2}
\nonumber \\*
& & +(\sqrt{\frac{|p_{1}|}{2\tau_{0}}}dx+\sqrt{\frac{|p_{1}|}{2\tau_{0}}}
\frac{L+E}{p_{1}}dy)^{2}, \label{eq:metric}\end{aligned}$$ where $(x,y)$ is the coordinate on time-slice $\Sigma$ with the periodic condition $$(x,y)\sim(x+1,y)\sim(x,y+1).$$
The spacetime equipped with this metric turns out to be embedded into the Minkowski space. To see this explicitly, we transform the coordinates in two steps. First, $$\left\{ \begin{array}{ll}
\tau =\tau \\
\theta=\sqrt{\frac{|p_{1}|\tau_{0}}{2}}x+\sqrt{\frac{|p_{1}|\tau_{0}}{2}}
\frac{L-E}{p_{1}}y\\
Y=-\frac{p_{1}}{|p_{1}|}(\sqrt{\frac{|p_{1}|}{2\tau_{0}}}x+\sqrt{\frac
{|p_{1}|}{2\tau_{0}}}\frac{L+E}{p_{1}}y). \end{array} \right.$$ Second, $$\left\{ \begin{array}{ll}
T=\frac{1}{\tau}\cosh\theta \\ X=\frac{1}{\tau}\sinh\theta \\
Y=Y . \end{array} \right.$$ The metric (\[eq:metric\]) becomes $$ds^{2}=-\frac{d\tau^{2}}{\tau^{4}}+\frac{1}{\tau^{2}}d\theta^{2}+dY^{2}=
-dT^{2}+dX^{2}+dY^{2}.$$ Thus, the torus universe $M$ is regarded as a quotient space $$M={\cal F}/G .\label{eq:Quot.S.}$$ We denote by ${\cal F}=\{(T,X,Y)|\mbox{ }T>|X|\}$ the “fundamental region” in the Minkowski space, and by $G$ a discrete subgroup of ISO(2,1) generated by the following two transformations. $$\begin{aligned}
&&\Lambda_{1}:(T,X,Y)\rightarrow(T\cosh a+X\sinh a,T\sinh a+X\cosh a,Y+u)
\nonumber \\*
&&\Lambda_{2}:(T,X,Y)\rightarrow(T\cosh b+X\sinh b,T\sinh b+X\cosh b,Y+w),
\label{eq:Poincare}\end{aligned}$$ where $$\begin{aligned}
&&(a,u)\equiv(\sqrt{\frac{\tau_{0}|p_{1}|}{2}},-\frac{p_{1}}{|p_{1}|}\sqrt{\frac{|p_{1}|}{2\tau_{0}}})
\nonumber \\*
&&(b,w)\equiv(\frac{L-E}{p_{1}}\sqrt{\frac{\tau_{0}|p_{1}|}{2}},
-\frac{L+E}{|p_{1}|}\sqrt{\frac{|p_{1}|}{2\tau_{0}}}).
\label{eq:C-S P.S.}\end{aligned}$$ Therefore, the evolution of the torus universe can be visualized as illustrated in Fig.1, and is determined by the four time-independent parameters $(a,b,u,w)$.
Next, we determine the classical orbit. Here we mean by “classical orbit” a curve drawn by the point $(m_{1}',m_{2}')$ in the moduli space, which the torus reaches at time $\tau_{2}$ by way of classical trajectories, assuming that the torus left the point $(m_{1},m_{2})$ at time $\tau_{1}$. It will be useful in §4 when we discuss the properties of the transition amplitude in the quantum theory. To this end, we first express the moduli parameters in terms of the parameters $(a,b,u,w)$ using eqs. (\[eq:phase space\]), (\[eq:C-S P.S.\]). $$m_{1}(\tau)=\frac{uw+\frac{ab}{\tau^{2}}}{u^{2}+\frac{a^{2}}{\tau^{2}}},
m_{2}(\tau)=\frac{\frac{-aw+bu}{\tau}}{u^{2}+\frac{a^{2}}{\tau^{2}}},
\label{eq:Mod.sp.}
\label{eq:class.rel.}$$ where we have shown explicitly that the moduli parameters are the $\tau$-dependent functions.
In general, we have four free parameters $(a,b,u,w)$ in (\[eq:Mod.sp.\]). Two of them are fixed when we impose an initial condition $$m_{1}(\tau_{1})=m_{1},m_{2}(\tau_{1})=m_{2}.$$ We solve this set of equations for parameters $(b,w)$ and find $$b=am_{1}+\tau_{1}um_{2},
w=um_{1}-\frac{a}{\tau_{1}}m_{2}. \label{eq:bypass}$$ Substituting eq.(\[eq:bypass\]) into $$m_{1}(\tau_{2})=m_{1}'\mbox{ and }m_{2}(\tau_{2})=m_{2}',$$ we finally obtain $$\begin{aligned}
&&m_{1}'=m_{1}+\frac{m_{2}}{2}(\frac{\tau_{1}}{\tau_{2}}-\frac{\tau_{2}}
{\tau_{1}})\frac{2x}{1+x^{2}},
\nonumber \\*
&&m_{2}'=\frac{m_{2}}{2}(\frac{\tau_{2}}{\tau_{1}}+\frac{\tau_{1}}{\tau_{2}})+
\frac{m_{2}}{2}(\frac{\tau_{1}}{\tau_{2}}-\frac{\tau_{2}}{\tau_{1}})
\frac{1-x^{2}}{1+x^{2}},\end{aligned}$$ where $x\equiv a/(u\tau_{2})$. It is easy to see that, when $x$ varies in the range $(-\infty,+\infty)$, the point $(m_{1}',m_{2}')$ in the moduli space moves on the curve: $$C_{c.o.}:(m_{1}'-m_{1})^{2}+\{m_{2}'-\frac{m_{2}}{2}(\frac{\tau_{2}}{\tau_{1}}
+\frac{\tau_{1}}{\tau_{2}})\}^{2}=\{\frac{m_{2}}{2}(\frac{\tau_{2}}
{\tau_{1}}-\frac{\tau_{1}}{\tau_{2}})\}^{2}. \label{eq:Cl.or.}$$ As is illustrated in Fig.2, this classical orbit is a circle in the moduli space[^2], expanding with time $\tau_{2}$ . In the limit $\tau_{2}\rightarrow\infty$(or $0$), the circle approaches the $m_{1}'$-axis plus infinity.
The relations between Chern-Simons gravity and ADM formalism
============================================================
The classical and the quantum equivalence of the C-S gravity to the ADM formalism is investigated by Carlip [@carli]. In this section, we review his work, putting an emphasis on the quantum relation.
According to Witten [@witte], the physical phase space of the C-S gravity is the moduli space ${\cal M}$ of flat ISO(2,1) connections on $\Sigma$. When $\Sigma$ has the topology of a torus $T^{2}$, the moduli space ${\cal M}$ is parametrized by two commuting elements of ISO(2,1) up to conjugation. This phase space is known to have three disconnected sectors [@husai] which are denoted by $$\mbox{the timelike sector }{\cal M}_{t} ,\mbox{ the null sector } {\cal
M}_{n} \mbox{ and the spacelike sector } {\cal M}_{s}.$$ These three sectors are characterized by the restriction of their ISO(2,1) transformations to SO(2,1), which are two spatial rotations, null rotations and Lorentz boosts respectively. Here we pick the spacelike sector ${\cal M}_{s}$ which is physically most relevant. Taking a proper conjugation, two ISO(2,1) transformations which coordinatize the phase space can be expressed as follows; $$\begin{aligned}
&&\Lambda_{1}:(T,X,Y)\rightarrow(T\cosh a+X\sinh a,T\sinh a+X\cosh a,Y+u)
\nonumber \\*
&&\Lambda_{2}:(T,X,Y)\rightarrow(T\cosh b+X\sinh b,T\sinh b+X\cosh b,Y+w).\end{aligned}$$ This is precisely the same expression as eq.(\[eq:Poincare\]). Thus in the C-S gravity, given a point in ${\cal M}_{s}$, a spacetime manifold $M$ can be constructed as in eq.(\[eq:Quot.S.\]). Moreover, a detailed analysis shows the free parameters $(a,b;w,u)$ to be canonical coordinates of the C-S gravity on a torus [@carli]. With these facts, we can write the basic variables $(m_{1},m_{2};p_{1},p_{2})$ in the ADM in terms of $(a,b;w,u)$. The moduli parameters are given in eq.(\[eq:class.rel.\]) and the conjugate momenta are $$p_{1}=-2au,p_{2}=-\tau(u^{2}-\frac{a^{2}}{\tau^{2}}).
\label{eq:Conj.mom.}$$ The ADM Hamiltonian eq.(\[eq:H.ADM\]) is written as $$H=\frac{-aw+bu}{\tau}.$$
Using eqs.(\[eq:class.rel.\]) and (\[eq:Conj.mom.\]), we can relate the C-S gravity to the ADM formalism by a time-dependent canonical transformation $$p_{1}dm_{1}+p_{2}dm_{2}-Hd\tau=-2udb+2wda+dF,$$ where $$F(m_{1},m_{2},a,b;\tau)=\frac{1}{m_{2}\tau}[(b-m_{1}a)^{2}+(m_{2}a)^{2}]$$ is the generating function. From this relation, we see the Hamiltonian $H_{C-S}$ which generates the time-evolution in the C-S gravity vanishes, $$H_{C-S}\equiv H+\frac{\partial}{\partial \tau}F=0.$$ Here we should note that while a point in the phase space of the C-S gravity is stable under the $\tau$-evolution, it determines the evolution of the torus universe with $\tau$ as in Fig. 1.
Let us now see the quantum relation. We construct operators which represent the ADM-variables in the Hilbert space of the C-S gravity. The fundamental operators in the C-S gravity are the self-adjoint ones $\hat{a}$, $\hat{b}$, $\hat{w}$ and $\hat{u}$ corresponding to the canonical variables, whose canonical commutation relations are$$[\hat{a},\hat{w}]=-[\hat{b},\hat{u}]=\frac{i}{2}.$$ We can construct the basic operators in the ADM by replacing eqs.(\[eq:class.rel.\]) and (\[eq:Conj.mom.\]) by the corresponding operator relations $$\begin{aligned}
&&\hat{m}\equiv\hat{m}_{1}+i\hat{m}_{2}=[\hat{u}+i\frac{\hat{a}}{\tau}]^{-1}
[\hat{w}+i\frac{\hat{b}}{\tau}],\label{eq:mod.op.} \\
&&\hat{p}\equiv\hat{p}_{1}+i\hat{p}_{2}=-i\tau[\hat{u}-i\frac{\hat{a}}{\tau}]
^{2}.\end{aligned}$$ The ADM Hamiltonian is expressed as $$\hat{H}=\frac{-\hat{a}\hat{w}+\hat{u}\hat{b}}{\tau}.$$
Here, we adopt the $(\hat{a},\hat{b})$-diagonalized representation, where the wave function $\chi$ is a function of $a$ and $b$. We adopt a natural inner product proposed by Ashtekar et.al. [@husai] $$<\chi_{1}|\chi_{2}>=\int\int dadb\overline{\chi_{1}(a,b)}\chi_{2}(a,b),
\label{eq:inner-product}$$ where the bar denotes complex conjugate. The fundamental operators act on the wave function as $$\begin{aligned}
\hat{a}\chi &=a\cdot\chi\mbox{ },\mbox{ }\hat{b}\chi &=b\cdot\chi \\
\hat{w}\chi &=-\frac{i}{2}\frac{\partial}{\partial a}\chi,
\hat{u}\chi &=\frac{i}{2}\frac{\partial}{\partial b}\chi.\end{aligned}$$
In this representation, the eigenfunction of the moduli operators $(\hat{m},\hat{m}^{\dagger})$ eq.(\[eq:mod.op.\]) is known to be $$K(m,\bar{m};a,b,\tau)=\frac{b-ma}{\pi\tau\sqrt{2m_{2}}}
\exp(-\frac{i}{m_{2}\tau}|b-ma|^{2}),$$ which satisfies $$\hat{m}K=m\cdot K\mbox{ \ and \ }\hat{m}^{\dagger}K=\bar{m}\cdot K.$$ This $K$ possesses the following properties; $$\begin{aligned}
\mbox{orthogonality }:& \int\int dadb\overline{K(m',\bar{m'};a,b,\tau)}
K(m,\bar{m};a,b,\tau)
\nonumber \\*
& =m_{2}^{2}\delta^{2}(m-m'), \label{eq:orth.} \\
\mbox{time dependence}:& -i\frac{\partial}{\partial\tau}K(m,\bar{m};a,b,\tau)
=\hat{H}K(m,\bar{m};a,b,\tau). \label{eq:inv-Sch.}\end{aligned}$$
Using this $K$ as a kernel, we can define the integral transformation from the $(\hat{m},\hat{m}^{\dagger})$-diagonal representation $\tilde{\chi}
(m,\bar{m})$ to the $(\hat{a},\hat{b})$-diagonal one $\chi(a,b)$ $$\chi(a,b)=\int\int\frac{d^{2}m}{m_{2}^{2}}K(m,\bar{m};a,b,\tau)
\tilde{\chi}(m,\bar{m}) \label{eq:int.trf.},$$ where the domain of integration is taken to be the upper-half complex $m$-plane. Inserting eq.(\[eq:int.trf.\]) into eq.(\[eq:inner-product\]) and using the orthogonality (\[eq:orth.\]), we obtain the modular-invariant inner-product in the $(\hat{m},\hat{m}^{\dagger})$-diagonal representation $$<\tilde{\chi}_{1}|\tilde{\chi}_{2}>(\equiv<\chi_{1}|\chi_{2}>)
=\int\int\frac{d^{2}m}{m_{2}^{2}}
\overline{\tilde{\chi}(m,\bar{m})}\tilde{\chi}(m,\bar{m}).
\label{eq:Inn.pdt.}$$
We can set up a quantization of the ADM formalism using $\tilde{\chi}$ which transforms as the “spinor representation” under the modular transformation [@carli]. In particular, the ADM Hamiltonian operator takes the form, $$\hat{H}=\frac{1}{\tau}[m_{2}(\hat{p}^{\dagger}\hat{p})^{1/2}-\frac{1}{2}(\hat{p}^{\dagger}\hat{p})^{-1/2}\hat{p}] =\tau^{-1}(\Delta_{1/2})^{1/2},
\label{eq:Qt.Ham.}$$ where $$\Delta_{1/2}\equiv -m_{2}^{2}(\frac{\partial^{2}}{\partial m_{1}^{2}}
+\frac{\partial^{2}}{\partial m_{2}^{2}})
+im_{2}\frac{\partial}{\partial m_{1}}-\frac{1}{4}$$ is the Maass Laplacian [@fay] for automorphic forms of weight $\frac{1}{2}$. With this Hamiltonian, the time evolution of the wave function $\tilde{\chi}$ is described by the Schrödinger equation $$i\frac{\partial}{\partial\tau}\tilde{\chi}(m,\bar{m};\tau)=
\tau^{-1}(\Delta_{1/2})^{1/2}\tilde{\chi}(m,\bar{m};\tau).
\label{eq:ADM evn.}$$ On the other hand, the following Schrödinger equation holds in the C-S gravity: $$i\frac{\partial}{\partial\tau}\chi(a,b)=0, \label{eq:C-S evn.}$$ since we have vanishing Hamiltonian. Eq.(\[eq:inv-Sch.\]) is necessary for $\tilde{\chi}$ and $\chi$ which are related via eq.(\[eq:int.trf.\]) to satisfy eqs. (\[eq:ADM evn.\]) and (\[eq:C-S evn.\]) respectively. Thus, we can, at least formally, show that for a solution $\chi(a,b)$ of eq. (\[eq:C-S evn.\]), its inverse transform $$\tilde{\chi}(m,\bar{m})=\int\int dadb \overline{K(m,\bar{m};a,b,\tau)}
\chi(a,b) \label{eq:inv.trf.}$$ solves the Schrödinger equation (\[eq:ADM evn.\]).
The transition amplitude of a torus universe
============================================
In the last section, we have given a framework of the quantum C-S gravity on a torus. Let us now calculate the transition amplitude of the torus universe.
Equation (\[eq:C-S evn.\]) tells us that the Schrödinger wave function in the C-S gravity is a $\tau$-independent function of configuration variables $a,b$. Therefore, the eigenfunction of moduli parameters $K(m,\bar{m};a,b,\tau)$ is not a solution to eq.(\[eq:C-S evn.\]). However, the wave function $$K(m,\bar{m};a,b,\tau_{1})=\frac{b-ma}{\pi\tau_{1}\sqrt{2m_{2}}}
\exp(-\frac{i}{m_{2}\tau_{1}}|b-ma|^{2})$$ obtained by replacing $\tau$ with a (positive) constant $\tau_{1}$, has no time dependence and becomes a Schrödinger wave function. It is interpreted as a state under which the moduli parameters $\hat{m}$, $\hat{m}^{\dagger}$ have an eigenvalue $m$, $\bar{m}$ [*at the moment $\tau=\tau_{1}$*]{}, and keeps its form during the evolution. Similarly, the wave function $K(m',\bar{m'};a,b,\tau_{2})$ is the state for which the eigenvalues of the moduli parameters at $\tau=\tau_{2}$ are $m'$ and $\bar{m'}$.
If we regard the inner product (\[eq:inner-product\]) to be a probability amplitude as in the standard quantum mechanics, the transition amplitude from moduli $m$ at $\tau=\tau_{1}$ to moduli $m'$ at $\tau=\tau_{2}$ is written as $$\begin{aligned}
&& <m,\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}> =\int \int dadb \overline
{K(m',\bar{m'};a,b,\tau_{2})}K(m,\bar{m};a,b,\tau_{1})
\nonumber \\*
&& =\int_{-\infty}^{\infty}da\int_{-\infty}^{\infty}db
\frac{(b-\bar{m}'a)(b-ma)}{2\pi^{2}\tau_{1}\tau_{2}\sqrt{m_{2}m_{2}'}
}\exp\{\frac{i}{m_{2}'\tau_{2}}|b-m'a|^{2}-\frac{i}{m_{2}\tau_{1}}
|b-ma|^{2}\}.\label{eq:tr.am.}\end{aligned}$$ This expression involves the integral of the Fresnel-type $\int dxx^{2}
\exp(i\alpha x^{2})$, and we regularize it by the $i\epsilon$-prescription: $$\int_{-\infty}^{\infty}dxx^{2}e^{i\alpha x^{2}}=\frac{i}{2\alpha}
\sqrt{\frac{i\pi}{\alpha +i\epsilon}}
\mbox{ \ , \ \ } \epsilon>0.$$ With this regularization, eq.(\[eq:tr.am.\]) becomes $$\begin{aligned}
&& <m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}> \nonumber \\*
&& = \frac{1}{4\pi}\frac{m_{2}m_{2}'}{\sqrt{\tau_{1}\tau_{2}}}
\frac{(\tau_{1}-\tau_{2})(\bar{m'}-m)}{\{m_{2}'^{2}+m_{2}^{2}-m_{2}m_{2}'
(\frac{\tau_{1}}{\tau_{2}}+\frac{\tau_{2}}{\tau_{1}})+(m_{1}'-m_{2})^{2}
\mp i\epsilon\}^{3/2}},\label{eq:Tr. Am.}\end{aligned}$$ where $-i\epsilon$ $(+i\epsilon)$ corresponds to the case of $m_{2}\tau_{1}>
m_{2}'\tau_{2}$ $(m_{2}\tau_{1}<m_{2}'\tau_{2})$ . This is the desired transition amplitude, which enjoys the following properties.
1. The “orthogonality” $$\lim_{\tau_{2}\rightarrow\tau_{1}}<m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}>=
m_{2}^{2}\delta^{2}(m'-m). \label{eq:limit}$$
2. The divergent-peak on the circle $$C_{\infty}:(m_{1}'-m_{1})^{2}+m_{2}'^{2}+m_{2}^{2}-m_{2}m_{2}'
(\frac{\tau_{2}}{\tau_{1}}+\frac{\tau_{1}}{\tau_{2}})=0.$$
3. The Schrödinger equation in ( the spinor representation of ) the ADM formalism $$i\frac{\partial}{\partial\tau_{2}}<m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}>
=\hat{H}(m',\bar{m'})<m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}>.
\label{eq:dyn.eq.}$$
The orthogonality (\[eq:limit\]) is necessary for the amplitude to be consistent with eq.(\[eq:orth.\]).
The circle $C_{\infty}$, where the amplitude becomes divergent, exactly coincides with the classical orbit $C_{c.o.}$ in eq.(\[eq:Cl.or.\]). Therefore, the amplitude in the quantum theory is distributed around the classical orbit. This result is reasonable from the quantum mechanical viewpoint. This, together with the positive definiteness of eq.(\[eq:inner-product\]) and the fact that it is conserved for the solutions of eq.(\[eq:C-S evn.\]), gives a support to the interpretation of the natural inner product (\[eq:inner-product\]) as a probability amplitude.
The Schrödinger equation (\[eq:dyn.eq.\]) is formally a direct consequence of the fact that eq.(\[eq:inv.trf.\]) solves eq.(\[eq:ADM evn.\]). After lengthy calculations, we see explicitly that the equation $$(i\tau_{2}\frac{\partial}{\partial\tau_{2}})^{2}<m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}>=\Delta_{1/2}(m',\bar{m'})<m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}>$$ holds. Therefore, the amplitude obtained here can be regarded as that in the spinor representation of the ADM formalism. This follows from the equivalence between the (weight-$1/2$) spinor representation of the ADM formalism and the canonical quantization of the C-S gravity, which are related via the transformation (\[eq:int.trf.\]) as is mentioned in the previous section.
The modular invariance
======================
While we have not considered so far, there is a problem concerning with the modular invariance. Though at first glance it does not appear that our amplitude (\[eq:Tr. Am.\]) exhibits the desired transformation property, we will show in this section that this is the case.
Under a modular transformation $\gamma\in\Gamma$ ( $\Gamma$ is the modular group $SL(2,{\bf Z})/{\bf Z}_{2}$ );$$\gamma:m\rightarrow\gamma m\equiv\frac{xm+y}{zm+w},
\quad where \quad \left( \begin{array}{cc} x & y \\ z & w \end{array}\right)
\in SL(2,{\bf Z}),$$ the wave function $\tilde{\chi}$ should transform as a weight-$\frac{1}{2}$ automorphic form [@carli], namely spinor: $$\gamma:\tilde{\chi}(m,\bar{m})\rightarrow
\tilde{\chi}(\gamma m,\overline{\gamma m})= e^{i\phi_{\gamma}}
\left(\frac{zm+w}{z\bar{m}+w}\right)^{1/2}\tilde{\chi}(m,\bar{m}),
\label{eq:modcov}$$ where $\phi_{\gamma}$ is a constant phase factor which represents an abelianization of $\Gamma$. For such $\tilde{\chi}$’s, the integration region of the inner product (\[eq:Inn.pdt.\]) can be restricted to the fundamental region: $$F=\{m|\Im m>0, |\Re m|\leq 1/2, |m|\geq 1 \}.$$ Let us construct the modular covariant transition amplitude. First we should be aware that the transformation law of our amplitude can be written as: $$\begin{array}{ll}
\left(\frac{z\bar{m'}+w}{zm'+w}\right)^{1/2}&
<\gamma m',\overline{\gamma m'};\tau_{2}|m,\bar{m};\tau_{1}> \\
&=\left(\frac{-zm+x}{-z\bar{m}+x}\right)^{1/2}
<m',\bar{m'};\tau_{2}|\gamma^{-1}m,\overline{\gamma^{-1}m};\tau_{1}>.
\end{array}\label{eq:tfmlaw}$$ If we take the infinite sum $$\begin{aligned}
\ll m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}\gg &\equiv &
\sum_{\gamma\in\Gamma}e^{-i\phi_{\gamma}}
\left(\frac{z\bar{m'}+w}{zm'+w}\right)^{1/2}
<\gamma m',\overline{\gamma m'};\tau_{2}|m,\bar{m};\tau_{1}> \nonumber \\
&= & \sum_{\gamma\in\Gamma}e^{i\phi_{\gamma}}
\left(\frac{zm+w}{z\bar{m}+w}\right)^{1/2}
<m',\bar{m'};\tau_{2}|\gamma m,\overline{\gamma m};\tau_{1}>,
\label{eq:invamp}\end{aligned}$$ the obtained $\ll m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}\gg$ behaves as a weight-$1/2$ (or $-1/2$) automorphic form under the modular transformation of the argument $m'$ (or $m$). We can therefore regard this infinite sum as a covariant amplitude.
Using this covariant amplitude (\[eq:invamp\]) we can express the evolution of the covariant wave function $\tilde{\chi}$ as: $$\tilde{\chi}(m,\bar{m};\tau_{2})=\int\int_{F}\frac{d^{2}m'}{(m'_{2})^{2}}
\ll m,\bar{m};\tau_{2}|m',\bar{m'};\tau_{1}\gg
\tilde{\chi}(m',\bar{m'};\tau_{1}).\label{eq:covev1}$$ This equation can be rewritten as follows. We substitute the definition (\[eq:invamp\]) of the covariant amplitude into (\[eq:covev1\]), and exploit the covariance (\[eq:modcov\]) of $\tilde{\chi}$, the modular invariance of the integration measure, and the mathematical fact [@serr]: $$\bigcup_{\gamma\in\Gamma}\gamma\cdot F\quad=\quad H,$$ where $H$ is the upper-half plane. Then eq.(\[eq:covev1\]) is reexpressed in terms of the original amplitude (\[eq:Tr. Am.\]): $$\tilde{\chi}(m,\bar{m};\tau_{2})=\int\int_{H}\frac{d^{2}m'}{(m'_{2})^{2}}
<m,\bar{m};\tau_{2}|m',\bar{m'};\tau_{1}>\tilde{\chi}(m',\bar{m'};\tau_{1}).
\label{eq:covev}$$ We should remark that the covariance of $\tilde{\chi}$ is preserved under the evolution (\[eq:covev\]) owing to the transformation law (\[eq:tfmlaw\]) of the amplitude. We can thus consider that our amplitude (\[eq:Tr. Am.\]) serves as a covariant amplitude when the integration region is extended to the upper-half plane.
Here we will claim that the physical content is not modified by taking the sum (\[eq:invamp\]). We should notice that for every point $m\in H$ (except fixed points $m=i,e^{i\pi/3},e^{2i\pi/3}$) there exists a [*unique*]{} $\gamma\in\Gamma$ such that $\gamma m\in F$ [@serr]. So we can use proper modular transformations to confine the trajectories or the classical orbits into the fundamental region $F$. These “confined” trajectories or orbits are uniquely determined from the original trajectories or orbits. Morover, two trajectories (or classical orbits) which give the same confined trajectory (or orbit) cannot be distinguished even at the classical stage because two points in the upper-half plane which can transform with each other by an element of the modular group give the same torus. Thus we conclude that taking the infinite sum (\[eq:invamp\]) does not modify any of the physics except that it confines the physical process into the fundamental region $F$.
Since we have obtained the modular covariant amplitude, the issue of the modular invariance is reduced to the problem of searching for the modular covariant wave functions. A set of covariant wave functions can be obtained as follows. We first diagonalize operators $\hat{E}\equiv\tau\hat{H}$ and $\hat{p}_{1}$ in the C-S gravity. Their eigenstates are $$\psi^{(E,p_{1})}(a,b)=a^{-2iE-1}\exp(ip_{1}\frac{b}{a}), \label{eq:vef}$$ which satisfy $$\hat{E}\psi^{(E,p_{1})}=E\cdot\psi^{(E,p_{1})} \mbox{ \ and \ }
\hat{p}_{1}\psi^{(E,p_{1})}=p_{1}\cdot\psi^{(E,p_{1})}.$$ We sum up these $\psi$’s with the same $E$ as $$\Psi^{(E)}(a,b)=\sum_{p_{1}=\phi_{T}+2\pi n,n\in{\bf Z}}\rho_{E}(p_{1})\cdot
\psi^{(E,p_{1})},
\label{eq:V}$$ where $\rho$’s are weights properly chosen so that the $\Psi$’s are invariant under the modular transformation [^3] $$\gamma:(b,a)\rightarrow(xb+ya,zb+wa).$$ Such wavefunctions should be related with the weight-$1/2$ Maass forms [@fay] [@iwani] via the integral transformation (\[eq:int.trf.\]). We will show below that this is indeed the case.
We first perform the inverse transformation (\[eq:inv.trf.\]) of the wavefunction (\[eq:vef\]) and find $$\tilde{\psi}^{(E,p_{1})}(m,\bar{m})=
N(E,p_{1})\tau^{-iE}e^{ip_{1}m_{1}}f_{E,p_{1}}(2|p_{1}|m_{2}),\label{eq:vadm}$$ where $N(E,p_{1})$ is an overall constant and $f_{E,p_{1}}$ is given by $$\begin{aligned}
f_{E,p_{1}}(z)\quad&=& ze^{-z/2}\int_{0}^{\infty}dx\mbox{ }x^{-2iE-1}
(x+\frac{p_{1}}{|p_{1}|}\frac{1}{x})
\exp\{-\frac{z}{4}(x-\frac{1}{x})^{2}\} \nonumber \\
=e^{-z/2}
z^{iE+1/2}& \int_{0}^{\infty}dt&(t+\sqrt{t^{2}+z})^{-2iE}
\{(1-\frac{p_{1}}{|p_{1}|})\frac{t}{\sqrt{t^{2}+z}}+
(1+\frac{p_{1}}{|p_{1}|})\}e^{-t^{2}} \nonumber \\
+e^{-z/2} z^{-iE+1/2}&\int_{0}^{\infty}dt&(t+\sqrt{t^{2}+z})^{2iE}
\{-(1-\frac{p_{1}}{|p_{1}|})\frac{t}{\sqrt{t^{2}+z}}+
(1+\frac{p_{1}}{|p_{1}|})\}e^{-t^{2}},
\label{eq:intrep}\end{aligned}$$ which is well-defined in the region $\Re z>0$. We find that this $f_{E,p_{1}}$ is reduced to the following form $$\begin{array}{l}
f_{E,p_{1}}(z)=2\sqrt{\pi}(-iE)^{\frac{|p_{1}|-p_{1}}{2|p_{1}|}}
W_{\frac{p_{1}}{2|p_{1}|},iE}(z) \\ \mbox{ } \\
\equiv e^{-z/2} \left[\begin{array}{l}
2^{-2iE}\Gamma(-iE+1/2)z^{iE+1/2}
F(iE+\frac{|p_{1}|-p_{1}}{2|p_{1}|},1+2iE,z) \\
\quad +\frac{p_{1}}{|p_{1}|} 2^{2iE}\Gamma(iE+1/2)z^{-iE+1/2}
F(-iE+\frac{|p_{1}|-p_{1}}{2|p_{1}|},1-2iE,z) \end{array}
\right], \end{array} \label{eq:whitf}$$ where $F(\alpha,\beta,z)$ denotes the confluent hypergeometric function and $W_{\mu,\kappa}(z)$ is the Whittaker function [@fay]. The proof goes as follows: i) eq.(\[eq:intrep\]) satisfies Whittaker’s differential equation $$\frac{d^{2}}{dz^{2}}f_{E,p_{1}}(z)
=\left[\frac{-E^{2}-1/4}{z^{2}}-\frac{p_{1}}{2|p_{1}|z}+\frac{1}{4}\right]
f_{E,p_{1}}(z) \quad ; \label{eq:white}$$ ii) the asymptotic behavior of eq.(\[eq:intrep\]) as $z\rightarrow +\infty$ precisely coincides with that of eq.(\[eq:whitf\]), i.e. $$f_{E,p_{1}}(z)\longrightarrow 2\sqrt{\pi}
(-2iE)^{\frac{|p_{1}|-p_{1}}{2|p_{1}|}}e^{-z/2}z^{\frac{p_{1}}{2|p_{1}|}}
\qquad as \quad z\rightarrow +\infty \quad ;\label{eq:asym}$$ and iii) the local behavior of eq.(\[eq:intrep\]) in the immediate neighbourhood of $z=0$ is exactly the same as that of eq.(\[eq:whitf\]).
Using the above result, we see that the inverse transform of eq.(\[eq:V\]) is expressed as follows: $$\begin{aligned}
\tilde{\Psi}^{(E)}(m,\bar{m})\quad=\qquad \tau^{-iE}
\sum_{\scriptsize \begin{array}{c}p_{1}=\phi_{T}+2\pi n \\
n\in{\bf Z} \end{array}}
\tilde{\rho}_{E}(p_{1})\cdot
W_{\frac{p_{1}}{2|p_{1}|},iE}(2|p_{1}|m_{2})e^{ip_{1}m_{1}},\nonumber \\
\left(\quad \tilde{\rho}_{E}(p_{1})\equiv 2\sqrt{\pi}
(-iE)^{\frac{|p_{1}|-p_{1}}{2|p_{1}|}}N(E,p_{1})\rho_{E}(p_{1})\quad\right).
\label{eq:covwf}\end{aligned}$$ This is the very expression that gives the weight-$1/2$ Maass forms [@fay]. We should notice that these covariant wave function $\tilde{\Psi}$’s satisfy the Schrödinger equation (\[eq:ADM evn.\]) owing to Whittaker’s equation (\[eq:white\])[^4] and that each term in the sum (\[eq:covwf\]) damps exponentially as $m_{2}\rightarrow \infty$ thanks to eq.(\[eq:asym\]). The latter property is similar to that of the wavefunctions obtained by Hosoya and Nakao [@hosoy], which are weight-$0$ Maass forms, and so the probability of the occurrence of the singular universes is expected to be extremely small.
In fact, the $\hat{E}$-operator is modular invariant. We do not know whether the physical operators should be modular invariant or not. If that is the case, however, the $\hat{E}$-operator and its eigenfunction $\Psi^{(E)}$ play an important role. Classically, $E$ is connected with the volume $v$ of the time-slice $\Sigma$ ( see eq.(\[eq:H.ADM\]) ). Thus the state $\Psi^{(E)}$ represents the torus universe with a given “volume” $E$.
Finally we remark that our set of covariant wave functions involves the wave function of the “static universe with a zero-volume” which is obtained by Carlip [@carli]. Setting $E=0$ and $\phi_{T}=\pi/6$ in the sum (\[eq:covwf\]) and noting the fact: $$W_{\frac{p_{1}}{2|p_{1}|},E=0}(z)=\left\{\begin{array}{lll}
e^{-z/2}z^{1/2}& for & p_{1}>0\\
0 & for & p_{1}<0, \end{array}\right.$$ we have the desired result $$\tilde{\Psi}^{(0)}(m,\bar{m})=m_{2}^{\mbox{ }1/2}\eta^{2}(m),$$ where $\eta$ denotes the Dedekind $\eta$ function.
Summary and Discussion
======================
We have computed the transition amplitude in the 2+1 dimensional Einstein gravity, which describes the evolution of the moduli of a torus $T^{2}$, by way of the C-S gravity. It has a peak on the classical orbit and enables us to interpret the inner product (\[eq:inner-product\]) in the C-S gravity as a probability amplitude.
We should notice that this amplitude is obtained [*via the C-S gravity*]{}. In principle, we can deduce this result entirely in the ADM’s framework, by means of the wave-packet or path integrals. In practice, however, the ADM Hamiltonian (\[eq:Qt.Ham.\]) is too complicated to carry out the computation. The Schrödinger equation (\[eq:C-S evn.\]) is trivial and we can easily compute the amplitude in the C-S gravity.
Putting our results and Carlip’s [@carli]together, we find an answer to the long-standing problems in the quantum gravity, namely the issue of time and how to interpret the wave function, in the 2+1 dimentional gravity on a torus. That is, if we use the York-time $\tau\equiv
\pi^{ab}g_{ab}\sqrt{g}^{-1}$ as time, and the natural inner product (\[eq:inner-product\]) as a probability amplitude, such an amplitude conserves under the evolution in $\tau$, so we can give the probability interpretation to a wave function, which, in turn, describes the quantum evolution of a time-slice $\Sigma$. Besides, the transition amplitude which we have obtained can be regarded as modular covariant in the sense explained in §5 and thus the issue of the modular invariance is reduced to the problem of constructing the modular covariant wavefunctions. Though we have shown that a set of covariant wave functions are obtained using the weight-$1/2$ Maass forms, construction of a modular covariant wave function in general requires a considerable understanding of the non-holomorphic modular forms [@iwani] and is left to the future investigation.
We have studied the 2+1 dimensional quantum gravity in the simplest case, in which the spatial hypersurface has the topology of a torus $T^{2}$. The extension to the case of arbitrary topology is an interesting problem. Since the classical equivalence of the C-S gravity and the ADM formalism has already been proved [@mess], we expect that this equivalence enables us to compute the transition amplitudes for the case of more complicated topologies. However in the case of a complicated topology ( e.g. a Riemann surface of genus $g\geq 2$ ), it is hard to extract relevant variables which parametrize the phase space. We need a further study to compute such a higher genus amplitude.
Our original motivation to this work was to study the physical meaning of the loop functionals in the 2+1 dimensional quantum gravity [@husai]. It turns out that there is a certain difficulty in carrying it out. As is mentioned in §3, the phase space of the C-S gravity on a torus is composed of three disconnected sectors. Among them, the loop representation, which Ashtekar et.al. constructed [@husai], is well-defined in the timelike sector. The equivalence of the two formalisms has been established in the spacelike sector alone. Thus the interpretation of the loop functionals in the context of the ADM formalism appears to be formidable.[^5]
In the case of the 3+1 dimensional gravity, the phase space of the Ashtekar formalism [@ashte] naturally includes that of the ADM formalism because the former contains the singularity of the latter i.e. $\det(g_{ab})=0$ . The “quantum tunneling” occurring in the Ashtekar formalism, which is confirmed in the minisuperspace model [@kodama], is due to such a situation.
Similarly, in the 2+1 dimensional case, the phase space of the C-S gravity is in fact larger than that of the ADM formalism. Therefore, if we reconstruct the theory so that we can handle all disconnected sectors, possibly the tunneling can appear from the Euclidean to the Lorentzian spacetime. The physical interpretation of the loop representation will also be given. This interpretation, in turn, may give some useful insights into the 3+1 dimensional loop representation proposed by Rovelli and Smolin [@rovel], and therefore to the quantum gravity in 3+1 dimensions.
2.5cm
Acknowledgments
I would like to thank Prof. K. Kikkawa and H. Kunitomo for useful discussions and helpful comments on the manuscript. I am also grateful to Prof. H. Itoyama for his encouragement and a careful reading of the manuscript.
[9]{} S. Deser, R. Jackiw and G. ’t Hooft, Ann. Phys. 152 (1984) 220;\
S. Deser and R. Jackiw, Ann. Phys. 153 (1984) 405 E. Martinec, Phys. Rev. D30 (1984) 1198 E. Witten, Nucl. Phys. B311 (1988) 46 S. Carlip, Nucl. Phys. B324 (1989) 106;\
J.E. Nelson and T. Regge, Nucl. Phys. B328 (1989) 190 S. Carlip, Phys. Rev. D42 (1990) 2647; D45 (1992) 3584 A. Anderson, preprint Imperial/TP/92-93/02, gr-qc/9210007 E. Witten, Nucl. Phys. B323 (1989) 113;\
Y. Fujiwara, S. Higuchi, A. Hosoya, T. Mishima and M. Siino, Phys. Rev. D44 (1991) 1756 and 1763;\
S. Carlip, preprint UCD- 92-16, hep-th/9206103 V. Moncrief, J. Math. Phys. 30 (1989) 2907 A. Hosoya and K. Nakao, Prog. Theor. Phys. 84 (1990) 739 A. Hosoya and K. Nakao, Class. Quant. Grav. 7 (1990) 163 A. Ashtekar, V. Husain, C. Rovelli, J. Samuel and L. Smolin, Class. Quant. Grav. 6 (1989) L185. J.D. Fay, J. Reine Angew. Math. 293 (1977) 143. J. P. Serre, A. Course in Arithmetic ( Springer-Verlag, New-York, Heidelberg, Verlin ) Chap. VII. Modular Forms, [*ed. by*]{} R.A. Rankin ( Ellis Horwood, Chichester, 1984 ). G. Mess, preprint IHES/M/90/28;\
S. Carlip, Class. Quant. Grav. 8 (1991) 5. A. Ashtekar, Phys. Rev. D36 (1987) 295;\
T. Jacobson and L. Smolin, Nucl. Phys. B299 (1988) 295. H. Kodama, Phys. Rev. D42 (1990) 2548. C. Rovelli and L. Smolin, Nucl. Phys. B331 (1990) 80. J. Louko and D. M. Marolf, preprint SU-GP-93/7-6, gr-qc/9308018.
Fig.1a
: A torus universe embedded into the Minkowski space.\
We take the $T$-axis as a baseline B and draw lines which are obtained by transforming the baseline by $\Lambda_{1}$, $\Lambda_{2}$ and $\Lambda_{1}
\Lambda_{2}$ ( see eq.(\[eq:Poincare\]) ), The torus universe is expressed by the region inside these lines, with the boundary surface identified by the above transformations.\
The case of $(a,u)=(1,0)$, $(b,w)=(0,1)$ is shown explicitly. The torus universe is the shadowed region inside the bold lines, where the two dotted lines and the two dashed lines are respectively identified.
Fig.1b
: An evolution of the time-slice developed into the $(Y,
\theta/\tau)$-plane.\
The baseline B is the $T$-axis, i.e. $Y=\theta=0$. Three parallelograms (drawn by bold lines) intersecting with B at $T=0,1$ and $2$ represent time-slices at the time $\tau=\infty,1$ and $1/2$, respectively. On each parallelogram, opposite sides are regarded to be identified. The figure shows how a “wide” torus shrinks to the wire-like singularity as time elapses.\
From this picture, if we wish, we can deduce the trajectory of the moduli parameters, which becomes a semicircle centered on the $m_{1}$-axis ( see ref.[@nakao]). Note that a point $(a,b;w,u)$ in the C-S phase space corresponds to a $\tau$-evolution of the torus universe.
Fig.2
: The “classical orbit” $C_{c.o.}$ of moduli parameters.\
$C_{c.o.}$ is a circle in the upper-half plane. We can see that the orbit expands as $\tau_{2}$ gets larger (or smaller), starting from $\tau_{1}$.
[^1]: e-mail address: [email protected]
[^2]: More precisely, this circle is in the upper-half plane $\{(m_{1}',m_{2}')|m_{2}'>0\}$. The true moduli space is a quotient space of this upper-half plane modulo the action of the modular group $\Gamma$.
[^3]: Note that $p_{1}$ takes the values which are a constant $\phi_{T}$ plus $2\pi$ times integrals follows from the requirement that the $\Psi$’s are invariant up to a constant phase factor under the Dehn twist $T:(a,b)\rightarrow(a,a+b)$.
[^4]: Due to the symmetry of the Whittaker function: $$W_{\mu,\kappa}=W_{\mu,-\kappa}\quad ,$$ we can take $E$ as non-negative provided that the norm of $\tilde{\Psi}$ is well-defined. Though the spectrum of $E$ is continuous for each constituent $\tilde{\psi}$, it is discretized by imposing the covariance on $\tilde{\Psi}$ [@fay].
[^5]: Recently, Louko and Marolf [@louko] have proposed the quantum theory which is defined on almost the whole phase space ${\cal M}$ of the C-S gravity. Using this, it is possible to elucidate the relation between the ADM formalism and the loop representation.
|
---
abstract: |
[Let $(T,M)$ be a complete local domain containing the integers. Let $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ be a chain of nonmaximal prime ideals of $T$ such that $T_{p_{n}}$ is a regular local ring. We construct a chain of excellent local domains $A_{n} \subseteq A_{n-1} \subseteq \cdots \subseteq A_{1}$ such that for each $1 \leq i \leq n$, the completion of $A_{i}$ is $T$, the generic formal fiber of $A_{i}$ is local with maximal ideal $p_{i}$, and if $I$ is a nonzero ideal of $A_{i}$ then $A_{i}/I$ is complete. Consequently, if in addition $T$ is a UFD, we can construct a chain of excellent local UFDs $A_{n} \subseteq A_{n-1} \subseteq \cdots \subseteq
A_{1}$ satisfying the same conditions. ]{}
author:
- '**'
---
****
====
We begin with some basic definitions. Let $A$ be a local ring with maximal ideal $M$. We use $\hat{A}$ to denote the completion of $A$ in the $M$-adic topology. If $P$ is a prime ideal of $A$, then the formal fiber ring of $A$ at $P$ is defined to be $\hat{A} \otimes_A k(P)$, where $k(P) =
A_P/PA_P$. When $A$ is a domain, we refer to the formal fiber ring at $(0)$ as the generic formal fiber ring of $A$. If $p \in {\mathrm{Spec}}\hat{A}$ and $K$ is the quotient field of $A$, then when $\hat{A}
\otimes_A K$ is local with maximal ideal $p \otimes_A K$, we say that $A$ has a local generic formal fiber with maximal ideal $p$. It is worthwhile to note that it is unusual for an integral domain $A$ to have a local generic formal fiber, and normally there exist many nonzero ideals $I$ of $A$ such that $A/I$ is not complete. In this paper, we construct chains of excellent rings that do not satisfy these “usual” conditions.
In [@LR02], Loepp and Rotthaus showed that for $T$ a complete local domain containing the integers with maximal ideal $M$ such that $T/M$ is at least the cardinality of the reals, and $p$ a nonmaximal prime ideal of $T$ such that $T_p$ is a regular local ring, there exists an excellent local domain $A$ such that the completion of $A$ is $T$, the generic formal fiber of $A$ is local with maximal ideal $p$ and for any nonzero ideal $I$ of $A$, $A/I$ is complete.
In this paper, we improve the result from [@LR02] in two ways. We are first able to eliminate the condition that $T/M$ has to have the cardinality at least that of the reals. In addition, we extend the result to a chain of excellent local rings. In particular, we show that if $T$ is a complete local domain containing the integers and $p_1
\subseteq p_2 \subseteq \cdots \subseteq p_n$ a chain of nonmaximal prime ideals of $T$ such that $T_{p_n}$ is a regular local ring, then there exists a chain of excellent local domains $A_n \subseteq A_{n-1}
\subseteq \cdots \subseteq A_1$ such that for each $i$, the completion of $A_i$ is $T$, the generic formal fiber of $A_i$ is local with maximal ideal $p_i$, and if $I$ is a nonzero ideal of $A_i$ then $A_i/I$ is complete. As a corollary, we show that if, in addition, $T$ is a UFD, then we may construct a chain of UFDs that satisfy the same conditions.
The construction of each $A_i$ is similar to that in [@LR02]. In order to guarantee that the completion of each $A_i$ is $T$, we make use of the result from [@H94] (Proposition \[p1\] below), and show that $IT \cap A_i=I$ for every finitely generated ideal $I$ of $A_i$ and $A_i \rightarrow T/M^2$ is onto. We also show that $p_i \cap A_i = (0)$ and if $q_i$ is a prime ideal of $T$ not contained in $p_i$, then $q_i \cap A_i \not= (0)$. This gives that the generic formal fiber of $A_i$ is local with maximal ideal $p_i$. We also need to ensure that $A_i/I$ is complete for each nonzero ideal $I$ of $A_i$.
To extend the result to a chain of excellent local rings, we take an approach similar to that in [@SMALL98], and break the proof into two parts. First we construct a chain such that the conditions hold for the first ring of the chain. Then we use induction to refine the rest of the chain to obtain the desired rings.
In the construction, we maintain a chain of subrings of $T$ that satisfy some “nice” properties. These are similar to the $p$-subrings in [@LR02]. But in order to get rid of the condition that $T/M$ has to have at least the cardinality of the reals, we take care to ensure that each subring in the chain is a [*Small $C$ Avoiding*]{} ($SCA$) subring as defined in [@CL03]. We strengthen a Lemma from [@CL03] and use $SCA$-subrings in place of $p$-subrings in all of our proofs.
Throughout this paper, all rings are commutative with unity. When we say a ring is local, Noetherian is implied. And we call a ring with one maximal ideal that is not necessarily Noetherian quasi-local. We use $(T,M)$ to denote a quasi-local ring $T$ with maximal ideal $M$, and we use $c$ to denote the cardinality of the reals.
****
====
We use the following proposition from [@H94] to ensure the the completion of each $A_i$ is $T$.
\[p1\] If $(R, M \cap R)$ is a quasi-local subring of a complete local ring $(T, M)$, the map $R \longrightarrow T/M^{2}$ is onto and $IT \cap R = I$ for every finitely generated ideal $I$ of $R$, then R is Noetherian and the natural homomorphism $\hat{R} \longrightarrow T$ is an isomorphism.
In our construction, we build subrings of $T$ satisfying some “nice” properties. We follow the definition in [@CL03] and call them $SCA$-subrings.
Let $(T, M)$ be a complete local ring and $C$ a set of prime ideals of $T$. Suppose that $(R, R \cap R)$ is a quasi-local subring of $T$ such that $|R|<|T|$ and $R \cap P = (0)$ for every $P
\in C$. Then we call $R$ a small $C$-avoiding subring of $T$ and will denote it by $SCA$-subring.
Lemma \[l7\] is well-known, and a proof can be found in [@CL03]. We implicitly use this Lemma in a couple of cardinality arguments.
\[l7\] Let $(T,M)$ be a complete local ring of dimension at least one. Let $P$ be a nonmaximal prime ideal of $T$. Then $|T/P| = |T| \geq c$.
The following lemma from [@CL03] will help us find transcendental elements.
\[l1\] Let $(T, M)$ be a complete local ring such that $\dim T \geq 1$, $C$ a finite set of nonmaximal prime ideals such that no ideal in $C$ is contained in another ideal of $C$, and $D$ a subset of $T$ such that $|D|<|T|$. Let $I$ be an ideal of $T$ such that $I \not\subseteq P$ for all $P \in C$. Then $I \not\subseteq \bigcup
\lbrace r + P | r \in D, P \in C \rbrace$.
We now prove a stronger version of the above lemma which will be immediately applicable to our construction. Here we weaken the conditions on $C$ and only require it to have finitely many maximal elements.
\[c9\] Let $(T,M)$ be a complete local ring such that $\dim T \geq 1$, $C$ a set of nonmaximal prime ideals with finitely many maximal elements, and $D$ a subset of $T$ such that $|D|<|T|$. Let $I$ be an ideal of $T$ such that $I \not\subseteq P$ for all $P \in C$. Then $I
\not\subseteq \bigcup \{r+P|r \in D, P \in C\}$.
Let $C'$ be the set of all maximal elements of $C$. So $C'$ is finite, and no ideal in $C'$ is contained in another ideal of $C'$. Let $I$ be an ideal of $T$ such that $I \not\subseteq P$ for all $P \in
C$. So $I \not\subseteq P'$ for all $P' \in C'$. Then by Lemma \[l1\], $I \not\subseteq \bigcup \{r+P'| r \in D, P' \in C'\}$. Now assume, for a contradiction, that $I \subseteq \bigcup \{r+P | r\in
D, P \in C\}$. Then if $x \in I$, we have $x \in r+P$, for some $r \in D,
P \in C$. But $P \subseteq P'$ for some $P' \in C'$, so $x \in r+P'$. Therefore, $I \subseteq \bigcup \{r+P'| r \in D, P' \in C'\}$, a contradiction. It follows that $I
\not\subseteq \bigcup \{r+P|r \in D, P \in C\}$.
Recall that to satisfy the hypothesis of Proposition \[p1\], we need the map $A_i \rightarrow T/M^2$ to be onto. We also want that $A_i/I$ is complete for every nonzero ideal $I$ of $A_i$. Note that if $T$ is the completion of $A_i$, $I$ is a nonzero ideal of $A_i$ and $A_i \rightarrow T/IT$ is onto, then $\ker (A_i \rightarrow T/IT)= A_i
\cap IT = I$. So $\frac{A_i}{I} \cong \frac{T}{IT} \cong \frac{\hat{A_i}}{I\hat{A_i}}
\cong \widehat{\frac{A_i}{I}}$ is complete. To satisfy this condition and ensure that $A_i \rightarrow T/M^2$ is onto, we construct $A_i$ such that $A_i \rightarrow T/J_i$ is onto for every ideal $J_i$ of $T$ that is not contained in the ideal $p_i$. It turns out that this condition will also help us to show that $A_i$ is excellent.
For $J_i$ an ideal of $T$ not contained in $p_i$, and $\overline{u}
\in T/J_i$, we use the following 2 lemmas to construct $SC_iA$-subrings of $T$ satisfying certain properties. These lemmas will be used to guarantee that $A_i \rightarrow T/J_i$ is onto.
Lemma \[l2\] is our first attempt to generalize Lemma 3 in [@LR02] to the chain version. Here we show that the desired properties hold for “one end” of the chain. Later we further extend this result by induction. This approach is similar to that in [@SMALL98].
\[l2\] Let $(T, M)$ be a complete local ring with $\dim T \geq 1$, $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ a chain of nonmaximal prime ideals of $T$ and $J$ an ideal of $T$ with $J \not\subseteq p_{n}$, and $J \not\subseteq Q, \forall Q \in {\mathrm{Ass}}T$. Let $C_i = {\mathrm{Ass}}T \cup
\{p_i\}$. Let $R_{n} \subseteq R_{n-1} \subseteq \cdots \subseteq R_{1}$ be a chain of subrings of $T$, where $R_{i}$ is an $SC_iA$-subring for each $1 \leq i \leq n$. Let $\overline{u} \in T/J$. Then there exists a chain of infinite subrings $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$ of $T$ with $R_{i} \subseteq S_{i} \subseteq T$, where $S_{i}$ is an $SC_iA$-subring for each $1 \leq i \leq n$, and $\overline{u} \in Img(S_{n} \longrightarrow T/J)$. Moreover, if $\overline{u} =
\overline{0}$, then $S_{n} \cap J \not= (0)$.
Let $C={\mathrm{Ass}}T \cup \{p_{1}, \ldots, p_{n}\}$ and let $P \in C$. Define $D_{(P)}$ to be a full set of coset representatives of the cosets $t + P$ that make $(u+t)+P$ algebraic over $\frac{R_{1}}{R_1 \cap P}$ as an element of $T/P$. Then $|D_{(P)}| \leq \max(|R_{1}|,\aleph_{0})$. Since $R_{1}$ is an $SC_1A$-subring, we have $|R_{1}| < |T|$. It follows that $|D_{(P)}| < |T|$. Clearly, $J \not\subseteq P, \forall P \in
C$. Let $D = \bigcup_{P \in C}D_{(P)}$. Note that $|D| < |T|$. Use Corollary [\[c9\]]{} to find $x \in J$ such that $x \not\in \bigcup\{r+P|r \in D, P \in C\}$. We define $S_{i} = R_{i}[x+u]$ localized at $R_{i}[x+u] \cap M$. Since we are adjoining the same element, clearly $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$. And note that each $S_i$ is infinite. We claim the $S_{i}$’s are the desired $SC_iA$-subrings.
Since $|R_{i}| < |T|$ and $S_{i} = max(|R_{i}|, \aleph_{0})$, we have $|S_{i}| < |T|$. Suppose $P_i \in C_i$. Then if $f_{i} \in R_{i}[u+x] \cap P_i$, we have $$f_{i} = r_{i,n}(u+x)^n + \cdots + r_{i,1}(u+x) + r_{i,0} \in P_i.$$ So $f_{i} \equiv 0$ modulo $P_i$. Since $(u+x)+P_i$ is transcendental over $\frac{R_{1}}{R_1 \cap P_i}$, it is transcendental over $\frac{R_{i}}{R_i \cap P_i} \cong R_i$. Thus $r_{i,j} \equiv 0$ modulo $P_i$ for each $j$. Then $r_{i,j} \in P_i$, and $r_{i,j} \in P_i \cap R_{i} = (0)$. Hence, $S_{i} \cap P_i = (0)$ and it follows that $S_{i}$ is a $SC_iA$-subring. Note that under the map $S_{n} \longrightarrow T/J$, $u+x$ is mapped to $u+J$. Hence $\overline{u} \in Img(S_{n} \longrightarrow T/J)$. Also, if $\overline{u} = \overline{0}$, then $u+x \in J$. Since $(u+x)+p_{n}$ is transcendental over $R_{n}$ as an element of $T/p_{n}$, we have $u+x \not= 0$. It follows that $S_{n} \cap J \not= (0)$.
Now we are ready to give the full chain version of Lemma 3 in [@LR02]. We argue inductively from Lemma \[l2\].
\[l3\] Let $(T, M)$ be a complete local ring with $\dim T \geq 1$, $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ a chain of nonmaximal prime ideals of $T$ and $J_{1}$, $J_{2}$ $\ldots$ $J_{n}$ $n$ ideals of $T$ with $J_{i} \not\subseteq p_{i}$ for each $1 \leq i \leq n$, and $J_i \not\subseteq Q, \forall Q \in {\mathrm{Ass}}T$. Let $C_i = {\mathrm{Ass}}T
\cup \{p_i\}$. Let $R_{n} \subseteq R_{n-1} \subseteq \cdots \subseteq
R_{1}$ be a chain of subrings of $T$, where $R_{i}$ is an $SC_iA$-subring for each $1 \leq i \leq n$. Let $\overline{u}_{i} \in T/J_{i}$ for each $i$. Then there exists a chain of infinite subrings $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$ of $T$ with $R_{i} \subseteq S_{i} \subseteq T$, where $S_{i}$ an $SC_iA$-subring for each $1 \leq i \leq n$, and $\overline{u}_{i} \in Img(S_{i}
\longrightarrow T/J_{i})$ for every $i$. Moreover, if $\overline{u}_{i} = \overline{0}$, then $S_{i} \cap J_{i} \not= (0)$.
We will induct on $n$. The base case is clear if we let $n=1$ in Lemma [\[l2\]]{}. Assume given a chain of $n-1$ prime ideals we can find a chain of $n-1$ desired $SC_iA$-subrings. Now consider the case with $n$ prime ideals. Use Lemma [\[l2\]]{} to find a chain of $n$ subrings $S'_{n} \subseteq S'_{n-1} \subseteq \cdots \subseteq S'_{1}$ such that $S'_{i}$ is an $SC_iA$-subring, $R_{i} \subseteq S'_{i} \subseteq T$, $\overline{u}_{n} \in Img(S_{n} \longrightarrow T/J_{n})$, and if $\overline{u}_{n} = \overline{0}$, then $S_{n} \cap J_{n} \not= (0)$.
Next use the inductive hypothesis on the chain $S'_{n-1} \subseteq S'_{n-2} \subseteq \cdots \subseteq S'_{1}$ to find $S_{n-1} \subseteq S_{n-2} \subseteq \cdots \subseteq S_{1}$ such that for $1 \leq i \leq n-1$, $S_{i}$ is an $SC_iA$-subring, $R_{i} \subseteq S'_{i} \subseteq S_{i} \subseteq T$, $\overline{u}_{i} \in Img(S_{i} \longrightarrow T/J_{i})$, and if $\overline{u}_{i} = \overline{0}$, then $S_{i} \cap J_{i} \not= (0)$.
Finally, let $S_{n} = S'_{n}$. We have $S_{n} = S'_{n} \subseteq S'_{n-1} \subseteq S_{n-1}$. Hence, we have constructed the chain: $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$ as desired. By induction the lemma holds.
The following two lemmas are the generalization of Lemma 2.6 from [@CL03]. We need this to show that $I_iT \cap A_i = I_i$ for each finitely generated ideal $I_i$ of $A_i$. The proof again is broken up into two steps similar to the ones in the previous two lemmas.
\[l8\] Let $(T, M)$ be a complete local ring with $M \not\in {\mathrm{Ass}}T$ and $\dim T \geq 1$. Let $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ be a chain of nonmaximal prime ideals of $T$. Let $C_i = {\mathrm{Ass}}T \cup \{p_i\}$ and $R_{n} \subseteq R_{n-1} \subseteq \cdots \subseteq R_{1}$ a chain of subrings of $T$ where $R_{i}$ is an $SC_iA$-subring for each $1 \leq i \leq n$. Suppose $I$ is a finitely generated ideal of $R_n$ and $c \in IT \cap R_n$. Then there exists a chain of subrings $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$ of $T$ such that for each $1 \leq i \leq n$, $R_{i} \subseteq S_{i} \subseteq T$, $S_{i}$ is an $SC_iA$-subring and $c \in IS_{n}$.
We will induct on the number of generators of $I$. If $I$ is principal, $I = aR_n$, for some $a \in R_n$. If $a=0$, then $I=(0)$ and $c \in IT \cap R_n$ implies $c=0$. So $S_i = R_i$ gives the desired $SC_iA$-subrings. So we consider the case when $a \not= 0$. In this case, $c = au$ for some $u \in T$. We claim that $S_i =
R_i[u]_{(R_i[u]\cap M)}$ are the desired $SC_iA$-subrings. To see this, first note that $|S_i|<|T|$ for each $i = 1,2,\ldots,n$. Now let $P_i \in C_i$ and suppose $f_i = r_nu^n + \cdots + r_1u+r_0 \in R_i[u]
\cap P_i$. Multiplying through by $a^n$, we obtain $a^nf_i = r_n(au)^n + \cdots + r_1a^{n-1}(au)+r_0a^n$. Since $a \in R_n \subseteq R_i$, and $R_i \cap P_i = (0)$ for each $P_i \in C_i$, it follows that $a^nf_i = r_nc^n + \cdots + r_1a^{n-1}c+r_0a^n \in P_i \cap R_i =
(0).$ Since $C_n$ contains all associated primes of $T$, and $a \in
R_n$ which is a $SC_nA$-subring, we know $a$ is not a zero divisor in $T$. Then it must be the case that $f=0$. Thus $S_i$ is an $SC_iA$-subring and this proves the base case.
Now suppose the Lemma holds for all $I$ generated by $m-1$ elements. Consider the case when $I$ has $m$ generators. Let $I = (y_1,\ldots,y_m)R_n$. Since $c \in IT \cap R_n$, $c = y_1t_1 + \cdots + y_mt_m$ for some $t_1, \ldots, t_m \in T$. Note that by adding $0$, for any $t \in T$, $c = y_1t_1 + y_1y_2t-y_1y_2t+y_2t_2
+ \cdots + y_mt_m = y_1(t_1+y_2t) + y_2(t_2-y_1t)+y_3t_3 + \cdots
y_mt_m$. Let $x_1 = t_1 + y_2t$ and $x_2 = t_2 - y_1t$, where we will choose $t$ later. Let $C ={\mathrm{Ass}}T \cup \{p_1,\ldots,p_n\}$ and $P \in
C$. Note that $y_2 \not\in P$ since $y_2 \in R_n$, $y_2 \not=0$, and $R_n \cap P = (0)$. Thus, $t+P\not=t'+P$ implies $(t_1+y_2t)+P \not=
(t_1+y_2t')+P$. Let $D_{(P)}$ be a full set of coset representatives of the cosets $t+P$ that makes $x_1+P$ algebraic over $\frac{R_1}{R_1
\cap P}$. Let $D =
\bigcup_{P \in C} D_{(P)}$. Note that $|D| < |T|$. We use Corollary \[c9\] with $I=T$ to find an element $t\in T$ such that $x_1+P$ is transcendental over $\frac{R_1}{R_1 \cap P}$ for every $P \in C$. Then $x_1+P_i$ is transcendental over $\frac{R_i}{R_i \cap P_i} \cong R_i$ for every $P_i \in C_i$. As in the proof of Lemma \[l2\], we may show that $R'_i = R_i[x_1]_{(R_i[x_1]\cap M)}$ are $SC_iA$-subrings. Let $I' = (y_2,\ldots,y_m)R'_n$ and $c^* = c-y_1x_1$. So $c^* \in I'T
\cap R'_n$. Then by our inductive hypothesis, there exists a chain of subrings $S_n \subseteq \cdots \subseteq S_1$ such that $R'_i
\subseteq S_i \subseteq T$, $S_i$ is an $SC_iA$-subring, and $c^* \in
I'S_n$. Thus $c^* = y_2s_2 + \cdots + y_ms_m$ for $s_1,\ldots,s_m \in
S_n$. It follows that $c=y_1x_1+y_2s_2+\cdots+y_ms_m \in IS_n$. And since $R_i \subseteq R'_i$, we conclude that $R_i \subseteq S_i \subseteq T$ and the $S_i$’s are the desired $SC_iA$-subrings.
\[l4\] Let $(T, M)$ be a complete local ring with $M \not\in {\mathrm{Ass}}T$ and $\dim T \geq 1$. Let $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ be a chain of nonmaximal prime ideals of $T$. Let $C_i = {\mathrm{Ass}}T \cup \{p_i\}$ and $R_{n} \subseteq R_{n-1} \subseteq \cdots \subseteq R_{1}$ a chain of subrings of $T$ where $R_{i}$ is an $SC_iA$-subring for each $1 \leq i \leq n$. Let $I_{1}$, $I_{2}$ $\ldots$ $I_{n}$ and $c_{1}$, $c_{2}$ $\ldots$ $c_{n}$ be such that $I_{i}$ is a finitely generated ideal of $R_{i}$ with $c_{i} \in I_{i}T \cap R_{i}$. Then there exists a chain of subrings $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$ of $T$ such that for each $1 \leq i \leq n$, $R_{i} \subseteq S_{i} \subseteq T$, $S_{i}$ is an $SC_iA$-subring and $c_{i} \in I_{i}S_{i}$.
We will induct on $n$. The base case clearly holds if we let $n=1$ in Lemma \[l8\]. Assume that given a chain of $n-1$ prime ideals and the corresponding chain of subrings, we can find a chain of $n-1$ desired $SC_iA$-subrings. Now consider the case with $n$ prime ideals. Use Lemma \[l8\] to construct a chain of subrings $S'_n \subseteq S'_{n-1} \subseteq \cdots \subseteq S'_1$ such that for each $i$, $R_i \subseteq S'_i \subseteq T$, $S'_i$ is a $SC_iA$-subring, and $c_n \in I_nS'_n$. We let $S_n = S'_n$ and use our inductive hypothesis on the chain $S'_{n-1} \subseteq \cdots
\subseteq S'_1$ to find a chain of subrings $S_{n-1} \subseteq \cdots
\subseteq S_1$ such that $R_i \subseteq S'_i \subseteq S_i \subseteq
T$, $S_i$ is $SC_iA$-subring, and $c_i \in I_iS_i$. Since $S_n = S'_n
\subseteq S'_{n-1} \subseteq S_{n-1}$, we have constructed a chain of subrings $S_n \subseteq \cdots \subseteq S_1$ satisfying all the conditions. By induction, our lemma holds.
The following definition is from [@H93].
Let $\Omega$ be a well-ordered set and $\alpha \in \Omega$. We define $\gamma (\alpha) = sup \lbrace \beta \in \Omega | \beta < \alpha \rbrace$.
Lemma \[l5\] is the generalization of Lemma 5 from [@LR02]. Here we construct a chain of $SC_iA$-subrings that simultaneously satisfy many of the desired properties. Condition ($iii$) and ($v$) will help us to show that the completion of $A_i$ is $T$. Condition ($iv$) is needed to ensure that the generic formal fiber of $A_i$ is local with maximal ideal $p_i$.
\[l5\] Let $(T, M)$ be a complete local ring with $\dim T \geq 1$, $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ a chain of nonmaximal prime ideals of $T$ and $J_{1}$, $J_{2}$ $\ldots$ $J_{n}$ $n$ ideals of $T$ with $J_{i} \not\subseteq p_{i}$ for each $1 \leq i \leq n$. Let $C_i = {\mathrm{Ass}}T \cup \{p_i\}$. Let $R_{n} \subseteq R_{n-1} \subseteq \cdots \subseteq R_{1}$ be a chain of subrings of $T$, where $R_{i}$ is an $SC_iA$-subring. Let $\overline{u}_{i} \in
T/J_{i}$ for every $i$. Then there exists a chain of subrings $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$ of $T$ such that for every $1 \leq i \leq n$
[()]{} [ ]{}
$S_{i}$ is an $SC_iA$-subring,
$R_{i} \subseteq S_{i} \subseteq T$,
$\overline{u}_{i} \in Img(S_{i} \longrightarrow T/J_{i})$,
if $\overline{u}_{i} = \overline{0}$, then $S_{i} \cap J_{i} \not= (0)$, and
For every finitely generated ideal $I$ of $S_{i}$, we have $IT \cap S_{i} = I$.
Use Lemma [\[l3\]]{} to find a chain of infinite subrings $R_{n,0} \subseteq R_{n-1,0} \subseteq \cdots \subseteq R_{1,0}$ such that for each $1 \leq i \leq n$, $R_{i,0}$ is an $SC_iA$-subring, $R_{i} \subseteq R_{i,0} \subseteq T$, $\overline{u}_{i} \in
Img(R_{i,0} \longrightarrow T/J_{i})$ and if $\overline{u}_{i} =
\overline{0}$, then $R_{i,0} \cap J_{i} \not= (0)$. We construct $S_{i}$ to contain $R_{i,0}$, so conditions $(ii)-(iv)$ will follow automatically. Now for each $1 \leq i \leq n$, define $$\Omega_{i} = \{(I_{i}, c_{i}) | I_{i} \mbox{ is a finitely generated ideal
of } R_{i,0} \mbox{ and } c_{i} \in I_{i}T \cap R_{i,0}\}.$$ Since $I_{i}$ can be $R_{i,0}$, we have $|R_{i,0}| \leq |\Omega_{i}|$, and since $|R_{i,0}|$ is infinite and there are $|R_{i,0}|$ finite subsets of $R_{i,0}$, we have $|\Omega_{i}| \leq |R_{i,0}|$. So $|\Omega_{i}| = |R_{i,0}|$. Well-order each $\Omega_{i}$ so that it does not have a maximal element. Note that $((0),0) \in \Omega_i$ for all $i$. Suppose $\Omega_I$ is a set with the maximal cardinality, let $\Psi$ be its index set. We use $\Psi$ to index each $\Omega_i$, and when $|\Omega_i| < |\Omega_I|$, we simply append extra $((0),0)$’s. Now define $$\Omega = \Delta\{(\Omega_{1} \times \Omega_{2} \times \cdots
\times \Omega_{n})\}.$$ where $\Delta$ denotes the diagonal. We can naturally well-order $\Omega$ using $\Psi$. Next, we recursively define $n$ families of $SC_iA$-subrings, starting with $R_{n,0} \subseteq R_{n-1,0} \subseteq \cdots \subseteq R_{1,0}$. We take care to preserve the ascending chain structure at each step. Let $\alpha \in \Omega$. Assume $R_{n,\beta} \subseteq R_{n-1,\beta} \subseteq \cdots \subseteq R_{1,\beta}$ has been defined for each $\beta < \alpha$. $\gamma(\alpha) = ((I_{1},c_{1}), (I_{2},c_{2}), ..., (I_{n},c_{n}))$ for some finitely generated ideal $I_{i}$ of $R_{i,0}$ and $c_{i} \in I_{i}T \cap R_{i,0}$. If $\gamma(\alpha) < \alpha$, use Lemma [\[l4\]]{} to find $R_{n,\alpha} \subseteq R_{n-1,\alpha} \subseteq \cdots \subseteq
R_{1,\alpha}$ such that $R_{i,\alpha}$ is an $SC_iA$-subring, $R_{i,\gamma(\alpha)} \subseteq R_{i, \alpha} \subseteq T$ and $c_{i} \in I_{i}R_{i,\alpha}$. If $\gamma(\alpha) = \alpha$, define $R_{i,\alpha} = \bigcup_{\beta<\alpha}R_{i,\beta}$. We still need to show that the chain condition holds in this case. Let $1 \leq a < b \leq n$, and $\zeta \in R_{b, \alpha}$. Hence $\zeta \in R_{b, \beta}$ for some $\beta < \alpha$. Since each of $R_{i, \beta}$ has been defined to preserve the chain, we have $\zeta \in R_{a, \beta} \subseteq R_{a, \alpha}$. It follows that $R_{b,\alpha} \subseteq R_{a,\alpha}$, as desired. Now, define $R_{i,1} = \bigcup_{\alpha \in \Omega}R_{i,\alpha}$. Since $R_{n,\alpha} \subseteq R_{n-1,\alpha} \subseteq \cdots \subseteq
R_{1,\alpha}$ holds for each $\alpha \in \Omega$, it follows that $R_{n,1} \subseteq R_{n-1,1} \subseteq \cdots \subseteq R_{1,1}$. It is easy to verify that $R_{i,1}$ is an $SC_iA$-subring, for each $i=1,2,\ldots, n$. Also, if $I_{i}$ is a finitely generated ideal of $R_{i,0}$, and $c_{i} \in I_{i}T \cap R_{i,0}$, then $((I_{1}, c_{1}), \ldots, (I_{i}, c_{i}), \ldots (I_{n}, c_{n}))
= \gamma(\alpha)$ for some $\alpha \in \Omega$ with $\gamma(\alpha) < \alpha$. So, $c_{i} \in I_{i}R_{i,\alpha} \subseteq I_{i}R_{i,1}$. Hence $I_{i}T \cap R_{i,0} \subseteq I_{i}R_{i,1}$.
Repeat this process to obtain a chain of $SC_iA$-subrings $R_{n,2} \subseteq R_{n-1,2} \subseteq \cdots \subseteq R_{1,2}$ such that $I_{i}T \cap R_{i,1} \subseteq I_{i}R_{i,2}$ Continue to construct an ascending chain $R_{i,0} \subseteq R_{i,1} \subseteq \cdots$ such that for each $j$, $R_{n,j} \subseteq R_{n-1,j} \subseteq \cdots \subseteq R_{1,j}$ is a chain of $SC_iA$-subrings, and for each finitely generated ideal $I_{i}$ of $R_{i,m}$, $I_{i}T \cap R_{i,m} \subseteq I_{i}R_{i,m+1}$. Then $S_{i} = \bigcup_{j=1}^{\infty}R_{i,j}$ is an $SC_iA$-subring. Clearly $S_{n} \subseteq S_{n-1} \subseteq \cdots \subseteq S_{1}$. If $I$ is a finitely generated ideal of $S_{i}$, then some $R_{i,n}$ contains a generating set for $I$, say $\{y_{1}, \ldots, y_{k}\} \subseteq R_{i,n}$. Clearly $I \subseteq IT \cap S_{i}$. Now let $c \in IT \cap S_{i}$, then $c \in R_{i,m}$ for some $m \geq n$. Hence $c \in (y_{1}, \ldots, y_{k})R_{m+1} \subseteq I$. It follows that $I = IT \cap S_{i}$.
The following lemma is a generalization of Lemma 6 in [@LR02]. Here we construct a chain of rings $A_i$ satisfying all conditions we desire except that $A_i$ might not be excellent.
\[l6\] Let $(T, M)$ be a complete local ring with $\dim T \geq 1$ and such that no integer of $T$ is a zero-divisor. Let $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ be a chain of nonmaximal prime ideals of $T$, such that $p_{1}$ contains all the associated prime ideals of $T$. Suppose that $p_{n}$ intersected with the prime subring of $T$ is the zero ideal. Then there exists a chain of local domains $A_{n} \subseteq A_{n-1} \subseteq \cdots \subseteq A_{1}$ such that for each $1 \leq i \leq n$
[()]{} [ ]{}
$\hat{A}_{i} = T$,
if $P_{i}$ is a nonzero prime ideal of $A_{i}$, then $T \otimes_{A_{i}} k(P_{i}) \cong k(P_{i})$, where $k(P_i)=\frac{A_{P_i}}{P_iA_{P_i}}$,
the generic formal fiber of $A_{i}$ is local with maximal ideal $p_{i}$, and
if $I$ is a nonzero ideal of $A_{i}$, then $A_{i}/I$ is complete.
Let $C_i = {\mathrm{Ass}}T \cup \{p_i\}$. Define $n$ sets $\Omega_{1}$, $\Omega_{2}$, $\ldots$, $\Omega_{n}$ to be such that $$\Omega_{i} = \lbrace u+J_{i} \in T/J_{i} | J_{i} \mbox{ is an ideal
of } T \mbox{ with } J_{i} \not\subseteq p_{i}\rbrace.$$ Since $T$ is Noetherian, then each ideal of $T$ is finitely generated. Hence $|\{J_{i} | J_{i}$ is ideal of $T$ and $J_{i} \not\subseteq p_{i}\}|
\leq |T|$. Now, if $J$ is an ideal of $T$, then $|T/J| \leq |T|$. It follows that for each $1 \leq i \leq n$, $|\Omega_{i}| \leq |T|$. Well order $\Omega_{i}$ so that each element has fewer than $|\Omega_{i}|$ predecessors. Note that $0+M \in \Omega_i$ for each $i$. By using the same technique as in the proof of Lemma \[l5\], we use the same index set for all $n$ orderings, call this index set $\Psi$. Next define $$\Omega = \Delta\{(\Omega_{1} \times \Omega_{2} \times \cdots
\times \Omega_{n})\}.$$ We can naturally well order $\Omega$ using $\Psi$. Let $0$ denote the first element of $\Omega$. Let $R'_{0}$ be the prime subring of $T$ and for each $1 \leq i \leq n$, let $R_{i,0} = {R'_{0}}_{(R'_{0} \cap
M)}$. Note that $R_{i,0}$ is an $SC_iA$-subring and the condition $R_{n,0} \subseteq R_{n-1,0} \subseteq \cdots \subseteq R_{1,0}$ holds trivially.
Next, we recursively define $n$ families of $SC_iA$-subrings and take care to preserve the chain structure of the $SC_iA$-subrings at each step. $R_{n,0} \subseteq R_{n-1,0} \subseteq \cdots \subseteq R_{1,0}$ is already defined. Let $\lambda \in \Omega$. Assume $R_{n,\beta} \subseteq R_{n-1,\beta} \subseteq \cdots \subseteq R_{1,\beta}$ has been defined for each $\beta < \lambda$. Then $\gamma(\lambda) = ((u_{1}+J_{1}), (u_{2}+J_{2}), \ldots, (u_{n}+J_{n}))$ for some ideals $J_{i}$ of $T$ with $J_{i} \not\subseteq p_{i}$. If $\gamma(\lambda) < \lambda$, use Lemma [\[l5\]]{} to obtain a chain of $SC_iA$-subrings $R_{n,\lambda} \subseteq R_{n-1,\lambda} \subseteq \cdots \subseteq
R_{1,\lambda}$ such that $R_{i,\gamma(\lambda)} \subseteq R_{i,\lambda} \subseteq T$, and $u_{i}+J_{i} \in Img(R_{i, \lambda} \longrightarrow T/J_{i})$. Moreover, if $u_{i}+J_{i} = 0+J_{i}$, then $R_{i, \lambda} \cap J_{i}
\not= (0)$, and for every finitely generated ideal $I$ of $R_{i, \lambda}$, we have $IT \cap R_{i, \lambda} = I$. If $\gamma(\lambda) = \lambda$, define $R_{i, \lambda} = \bigcup_{\beta < \lambda}R_{i, \beta}$. We still need to show the chain condition holds in this case. Let $1 \leq a < b \leq n$, and $\zeta \in R_{b, \lambda}$. Hence $\zeta \in R_{b, \beta}$ for some $\beta < \lambda$. Since each of $R_{i, \beta}$ has been defined to preserve the chain, we have $\zeta \in R_{a, \beta} \subseteq R_{a, \lambda}$. Thus $R_{n,\lambda} \subseteq R_{n-1,\lambda} \subseteq \cdots \subseteq R_{1,\lambda}$ holds for each $\lambda \in \Omega$ and $R_{i, \lambda}$ is an $SC_iA$-subring. Let $A_{i} = \bigcup_{\lambda \in \Omega}R_{i,\lambda}$, clearly $A_{n} \subseteq A_{n-1} \subseteq \cdots \subseteq A_{1}$ and we claim this is the chain of the desired domains.
We first prove condition $(iii)$. Since each $R_{i, \lambda}$ is a $SC_iA$-subring, we have $R_{i, \lambda} \cap p_{i} = (0)$. Hence, $A_{i} \cap p_{i} = (0)$. Next, let $J_{i}$ be an ideal of $T$ with $J_{i} \not\subseteq p_{i}$, then $0+J_{i} \in \Omega_{i}$. So there exists $\lambda \in \Omega$ with $\gamma(\lambda)<\lambda$ and $\gamma(\lambda) = ((u_{1}+J_{1}), (u_{2}+J_{2}), \ldots,
(0+J_{i}), \ldots, (u_{n}+J_{n}))$. By our construction, $R_{i,\lambda} \cap J_{i} \not= (0)$. It follows that $A_{i} \cap J_{i} \not= (0)$, $\forall J_{i}$ an ideal of $T$ with $J_{i} \not\subseteq p_{i}$. Thus, the generic formal fiber of $A_{i}$ is local with maximal ideal $p_{i}$.
To show $(i)$, we make use of Proposition [\[p1\]]{}. Since $p_{i}$ is nonmaximal, we have $M^{2} \not\subseteq p_{i}$. Thus by our construction, the map $A_{i} \longrightarrow T/M^{2}$ is onto. For every finitely generated ideal $I$ of $A_{i}$, clearly $I \subseteq IT \cap A_{i}$. Let $I = (y_{1}, y_{2}, \ldots, y_{k})$ be a finitely generated ideal of $A_i$ and $c \in IT \cap A_{i}$. We have $(c, y_{1}, \ldots, y_{k}) \subseteq
R_{i, \lambda}$ for some $\lambda \in \Omega$ with $\gamma(\lambda) < \lambda$. By construction, $(y_{1}, \ldots, y_{k})T \cap R_{i, \lambda} =
(y_{1}, \ldots, y_{k})R_{i, \lambda}$. Hence, $c \in (y_{1}, \ldots, y_{k})T \cap R_{i, \lambda}
= (y_{1}, \ldots, y_{k})R_{i, \lambda} \subseteq I$. We thus have $IT \cap A_{i} = I$. It follows from Proposition [\[p1\]]{} that $A_{i}$ is Noetherian and the completion of $A_{i}$ is $T$.
Next, let $I$ be a nonzero ideal of $A_{i}$. Let $J_{i} = IT$. Suppose $J_{i} \subseteq p_{i}$, then $I \subseteq J_{i} \cap A_{i}
\subseteq p_{i} \cap A_{i} = (0)$, a contradiction. Hence, $J_{i} \not\subseteq p_{i}$. It follows that the map $A_{i} \longrightarrow T/IT$ is surjective. The kernel of this map is $A_{i} \cap IT = I$. Hence $A_{i}/I \cong T/IT$, which implies that $A_{i}/I$ is complete.
Finally, if $P_i$ is a nonzero prime ideal of $A_i$, then by the above paragraph, we have that $A_{i}/P_{i} \cong T/P_{i}T$. It then follows that $T \otimes_{A_{i}} k(P_{i})\cong (T/P_{i}T)_{\overline{A_{i}-P_{i}}}
\cong (A_{i}/P_{i})_{\overline{A_{i}-P_{i}}} \cong
{A_{i}}_{P_{i}}/P_i{A_{i}}_{P_{i}} = k(P_{i})$.
And this completes the proof.
Now we prove that each $A_i$ is excellent and thus conclude our main theorem.
\[t1\] Let $(T, M)$ be a complete local domain containing the integers. Let $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ be a chain of nonmaximal prime ideals of $T$ such that $T_{p_{n}}$ is a regular local ring. Then there exists a chain of excellent local domains $A_{n} \subseteq A_{n-1} \subseteq \cdots
\subseteq A_{1}$ such that for each $1 \leq i \leq n$, the completion of $A_{i}$ is $T$, the generic formal fiber of $A_{i}$ is local with maximal ideal $p_{i}$ and if $I$ is a nonzero ideal of $A_{i}$, then $A_{i}/I$ is complete.
Notice that since $T$ is a local domain, in order for the chain $p_1
\subseteq \cdots \subseteq p_n$ to exist, we must have $\dim T \geq 1$. We use Lemma [\[l6\]]{} to construct the chain of local domains $A_{n} \subseteq A_{n-1} \subseteq \cdots \subseteq A_{1}$. Then we only need to show that for each $1 \leq i \leq n$, $A_{i}$ is excellent. Since $A_{i}$ is local (and thus Noetherian), to show $A_{i}$ is universally catenary, we only need to show $A_{i}$ is formally equidimensional. Hence, it suffices to show that $\hat{A_{i}} = T$ is equidimensional. This is true since $T$ is domain.
Next, we show $A_{i}$ is $G-ring$ for each $i$. Let $P_{i}$ be a nonzero prime ideal of $A_{i}$. By Lemma [\[l6\]]{}, $T \otimes_{A_{i}} k(P_{i}) \cong k(P_{i})$ is regular. Let $L$ be a finite field extension of $k(P_{i})$. We want to show $T \otimes_{A_{i}} L$ is regular. Here we make use of a fact that if $M$ is an $R$-module, then $M \otimes_{R} R \cong M$. Hence we have: $T \otimes_{A_{i}} L \cong T \otimes_{A_{i}} (k(P_{i})
\otimes_{k(P_{i})} L) = (T \otimes_{A_{i}} k(P_{i}))
\otimes_{k(P_{i})} L \cong k(P_{i}) \otimes_{k(P_{i})} L
\cong L$ is regular.
Now the only thing left to show is that the generic formal fiber of $A_{i}$ is geometrically regular. First, recall that if $R$ is a regular local ring, then $R_P$ is a regular local ring $\forall P \in {\mathrm{Spec}}R$. So $T_{p_n}$ being a RLR implies that $T_{p_i} \cong (T_{p_n})_{p_iT_{p_n}}$ is a RLR. Then, $T \otimes_{A_{i}} k(0) \cong T_{p_{i}}$ is regular. Next, we want to show that $T \otimes_{A_{i}} L$ is regular, $\forall$ finite field extension $L$ of $k(0)$. Since $k(0) = {A_{i}}_{(0)}/(0){A_{i}}_{(0)}$ is the quotient field of $A_{i}$, it is a field of characteristic zero. Thus by Rotthaus’ notes in [@Ro97], we may assume that $L$ is purely inseparable. Since $k(0)$ is characteristic zero, it must be that $L = k(0)$. Hence, it follows that $A_{i}$ is excellent for each $1 \leq i \leq n$.
Finally, the following corollary extends the results to UFDs.
\[c3\] Let $(T, M)$ be a complete local unique factorization domain containing the integers. Let $p_{1} \subseteq p_{2} \subseteq \cdots \subseteq p_{n}$ be a chain of nonmaximal prime ideals of $T$ such that $T_{p_{n}}$ is a regular local ring. Then there exists a chain of excellent local UFDs $A_{n} \subseteq A_{n-1} \subseteq \cdots
\subseteq A_{1}$ such that for each $1 \leq i \leq n$, the completion of $A_{i}$ is $T$, the generic formal fiber of $A_{i}$ is local with maximal ideal $p_{i}$ and if $I$ is a nonzero ideal of $A_{i}$, then $A_{i}/I$ is complete.
We use Theorem \[t1\] to construct a chain of excellent local domains $A_n \subseteq A_{n-1} \subseteq \cdots \subseteq A_1$. It remains to show that each $A_i$ is a UFD. But this follows immediately from the fact (see Exercise 20.4 in [@Ma86]) that for a local ring $A$, if $\hat{A}$ is a UFD, then so is $A$.
[**Acknowledgements**]{}
I thank S. Loepp for her helpful advice and editorial comments. I also thank D. Jensen for pointing out a useful fact which immediately leads to Corollary \[c3\].
P. Charters and S. Loepp. *Semilocal Generic Formal Fibers*. preprint.
M. Florenz, D. Kunvipusilkul and J. Yang. *Constructing Chains of Excellent Rings with Local Generic Formal Fibers*. Communications in Algebra, 30(8) (2002), 3569-3587.
R. Heitmann. *Characterization of Completions of Unique Factorization Domains*. Trans. Amer. Math. Soc, 337 (1993), 379-387.
R. Heitmann. *Completions of Local Rings with an Isolated Singularity*. J. Algebra 163 (1994), 538-567.
S. Loepp and C. Rotthaus. *On the Completeness of Factor Rings*. Proc. Amer. Math. Soc, 130 (2002), 2189-2195.
H. Matsumura. *Commutative Ring Theory*. Cambridge Univ. Press, Cambridge, 1986.
C. Rotthaus. *Excellent Rings, Henselian Rings, and the Approximation Property*. Rocky Mountain J. Math. 27 (1997), no.1, 317-334.
|
---
abstract: 'In this work, the crack nucleation under fretting loading is investigated experimentally with a damage tolerant 2024 aluminium alloy. A new method is introduced to determine its condition with respect to all loading parameters including the number of fretting cycles. Further work deals with the prediction of this threshold using the Smith-Watson-Topper criterion. New developments are presented, in particular a process volume of variable size is introduced in the computations of the fretting crack initiation.'
author:
- 'H. Proudhon'
- 'S. Fouvry'
- 'G. R. Yantio'
bibliography:
- 'fretting\_swt.bib'
title: 'Determination and prediction of the fretting crack initiation: introduction of the (P,Q,N) representation and definition of a variable process volume'
---
[^1]
Introduction {#intro.sec}
============
Fretting damage has been recognized as a problem in several industrial applications for years now. The first studies defined the concept of fretting [@Waterhouse1981], while further work introduced more rationalization with precise concepts such as fretting cycles and fretting maps [@Vincent1992; @Fouvry1996]. Depending on the contact parameters and especially the displacement amplitude, it has been shown that two main regimes can be defined. First, In the large displacement range, the full sliding regime exists, which can induce wear of the surfaces in contact. Second, In the low displacement amplitude range, the contact causes a partial slip condition and severe stress gradients at the contact border. This can induce very rapid crack nucleation compared to classical fatigue testing. In this paper we focus on the latter regime and particularly on the crack nucleation condition. This has been recognized as a critical issue for industry in the past ten years.
Thanks to recent work, different methods are now available to model the propagation of fretting fatigue cracks using for example the “*Crack analogue*” model, fracture mechanics [@Giannakopoulos1998; @Ciavarella2001] and also weight functions [@Navarro2003]. But due to the possible drastic reduction of the fatigue limit induced by fretting [@Lindley1992], a great part of the work has been devoted to determining the fretting crack initiation conditions in various types of materials [@Szolwinski1996; @Fouvry1998; @Bernardo2004]. This process is quite complex due to high stress and strain gradients, oxidation phenomena and other tribological phenomenon (TTS formation [@Sauger2000]), surface deformation and wear, and debris formation. Until now the main method used to describe quantitatively crack nucleation deals only with the mechanical state under the fretting conditions and relies on computing a multiaxial fatigue criterion (this was first attempted by Petiot et al. [@Petiot1995]). This can be done either by using analytical solutions of the stress/strain state under fretting contact (as it can be found in [@Hills1994]), or more recently by numerous author with finite elements [@Naboulsi2003; @Bernardo2004; @Sum2005]. The latter method is more constrained but allows one to test more general geometries and more complex loading conditions. Due to the very strong gradients located at the contact border, one must use a process volume approach, which consists of averaging the stress/strain state in a micro-volume of material prior applying the multiaxial criterion. This method has popularized and has been investigated with various fatigue criteria [@Fouvry1998; @Naboulsi2003; @Proudhon2005]. These investigations have been capable of predicting, in some cases, the crack nucleation threshold ; but, the significance of the process volume is not yet fully understood, and as a consequence cannot be truly estimated without doing experiments. For completeness, it must be pointed out that some alternative methods exists such as the asymptotic stress intensity analysis [@Mugadu2002]. In this kind of approach, the stress singularity induced by the contact configuration is captured by asymptotic analysis in order to deduce a stress intensity factor at the contact edge.
In this paper we restrict the analysis to the prediction of the crack nucleation condition in the fretting wear configuration by the Smith-Watson-Topper criterion. We present an new vision of the process volume and apply the method to a complete experiment series with various fretting wear conditions. This work attempts to clarify the physical meaning of the process volume and to set up a systematical way of characterizing the crack nucleation condition with respect to parameters relevant for the industrial applications (contact pressure, shear stress amplitude and number of cycles).
Experimental work {#exp_N.sec}
=================
Part of the experimental work has been presented elsewhere. The reader is referred to [@Proudhon2005] for more details on the experimental setup, though general information is summarized here. A general fretting wear apparatus is used in cylinder/plane configuration and under partial slip regime. A normal force P is applied on the counter-body to maintain the surface in contact causing an elliptic pressure $p(x)$ over the contact zone $|x|<a$ ; a cyclic relative displacement is imposed to the interface leading to a classical shear stress $q(x)$ which exhibits its maximum at the stick zone limit $x=\pm c$ (cf. fig. \[fretting\_schema\_stress.fig\]). The flat specimen being maintained in a fixed position, this displacement gives rise to a cyclic shear force[^2] of magnitude $Q$ which is measured during the test via a force sensor. Regarding the materials, an aluminium/aluminium contact is studied, with plane samples in a 2024T351 alloy (see table \[alu\_mecha.tab\] for mechanical properties) and a cylindrical counter body of 49 mm radius made of a Al7075T6 alloy.
$E$ (GPa) $\nu$ $\sigma_{0.2\%}$ (MPa) $\sigma_d$ (MPa)
----------- ------- ------------------------ ------------------
72 0.33 325 140
: Mechanical properties of the studied Al2024
\[alu\_mecha.tab\]
![Schematic of the fretting wear cylinder/plane configuration.[]{data-label="fretting_schema_stress.fig"}](figure_01)
The fretting crack nucleation has been investigated in the following way (cf. fig. \[fretting\_damage.fig\]):
- fretting test is conducted on the sample for chosen loading conditions (fig. \[fretting\_damage.fig\]a).
- fretting crack presence is investigated by cutting the sample in the middle of the scars and polishing the newly created face (fig. \[fretting\_damage.fig\]b,c).
- chemical etching can be performed after a first optical micrograph observation in order to avoid a possible blurring of the crack by the polishing process (fig. \[fretting\_damage.fig\]d).
- depending on the previous investigations, new fretting conditions are determined to refine or confirm the results.
![Experimental method to investigated cracking after a fretting test ; see the text for details.[]{data-label="fretting_damage.fig"}](figure_02){width="75mm"}
A previous study (with the same geometry and material) conducted to 50000 fretting cycles was described in [@Proudhon2005]. According to these experiments, the crack nucleation condition was found to be mainly independent of the normal load (see fig. \[prediction\_r\_variable.fig\] where the experimental results have been replotted), leading to the crack nucleation threshold in terms of the tangential load $Q_{th}\simeq 240$ N/mm.
Though this value defines a boundary between safe and a crack risk zones, the number of cycle needed to be more deeply investigated. Indeed a rapid calculation of the stress state at the contact border does show that the contact is subjected to loading conditions equivalent to low cycle fatigue. We consider the simple analysis of the two-dimensional plane strain cylinder/flat contact with the Hertz theory from which we can write the peak contact pressure $p_0$:
$$p_0=\frac{2P}{\pi a}=\left(\frac{PE^*}{\pi R}\right)^{1/2}
\label{po.eqn}$$
$R$ being the radius of the cylinder and $E^*$ the effective Young modulus.
The stress field is known in partial slip conditions from the analytical solutions (see for instance [@Hills1994]). At the contact border ($x=a,y=0$), $\sigma_{yy}=\sigma_{xy}=0$ so the stress tensor matches the principal stress state ($\sigma_1,\sigma_2,\sigma_3$) which is biaxial (the contact pressure vanishes):
$$\begin{aligned}
\label{biaxial.eqn}
\sigma_1 & =\sigma_{xx}=2p_0\sqrt{\frac{\mu Q}{P}} \nonumber\\
\sigma_2 & =\sigma_{yy}=0 \\
\sigma_3 & =\sigma_{zz}=\nu\sigma_{xx} \nonumber
%\end{empheq}\end{aligned}$$
with $\mu$ the friction coefficient in partial slip condition ($\mu$ was determined to be close to 1,1 in our case [@Proudhon2005]). The equivalent Von Mises stress $\sigma_e$ can be easily expressed:
$$\begin{aligned}
\sigma_e & =\frac1{\sqrt{2}}\left((\sigma_1-\sigma_3)^2+\sigma_1^2+\sigma_3^2\right)^{1/2} \nonumber\\
& =\frac1{\sqrt{2}}\left(2\sigma_1^2+2\sigma_3^2-2\sigma_1\sigma_3\right)^{1/2} \nonumber\\
& = \sigma_1\,\left(1+(\frac{\sigma_3}{\sigma_1})^2-\frac{\sigma_3}{\sigma_1}\right)^{1/2}
\label{VM_expr.eqn}
%\end{empheq}\end{aligned}$$
combining with equation \[biaxial.eqn\]:
$$\begin{aligned}
%\begin{empheq}[]{align}
\sigma_e(x=a,y=0) & = 2p_0\sqrt{\frac{\mu Q}{P}}\,(\nu^2-\nu+1)^{1/2} \nonumber \\
& = 2p_0\sqrt{\frac{\mu E^* Q}{\pi R}}\,(\nu^2-\nu+1)^{1/2}
\label{VM_max.eqn}
%\end{empheq}\end{aligned}$$
Thus the shear force leading to the conventional yield at 0.2% of deformation $\sigma_Y$, in the slip zone, can be written as:
$$Q_Y=\frac{\pi R}{4\mu E^*}\frac{\sigma_Y^2}{\nu^2-\nu+1}
\label{pression_plast.eqn}$$
With $\sigma_Y=\sigma_{0.2\%}$, this expression gives $Q_Y=117$ N/mm. It is clear that an aluminium/aluminium contact with a friction coefficient greater than unity (like in our case) will induce very high contact stresses and it can be seen that all the tests carried out in this study operates under low cycle fatigue range ($Q>Q_Y$). It is therefore expected that the number of fretting cycles must have an influence on the crack nucleation condition.
To fully determine the crack nucleation condition, the experimental study was completed as follows: for a chosen normal pressure, additional fretting tests were conducted with different numbers of cycles. The initiation threshold is then determined for each new number of fretting cycles. The crack nucleation boundary is then expressed as a function of the three loading parameters $P$, $Q$ and $N$. The whole experimental procedure is described in fig. \[fretting\_nucleation.fig\].
![Methodology used to determine the crack nucleation condition with respect to the $,Q,N$ parameters, a) tests at constant N, b) tests at constant P, c) whole (P,Q,N) representation.[]{data-label="fretting_nucleation.fig"}](figure_03){width="75mm"}
Here, new tests have been performed with a constant normal load $P=400$ N/mm. Two different numbers of fretting cycles have been investigated: $5.10^5$ and $10^6$ cycles. The corresponding crack nucleation threshold are determined according to fig. \[fretting\_nucleation.fig\]a ; values have been reported in table \[fretting\_amorcage\_N.tab\] and plotted on fig. \[fretting\_influence\_N.fig\].
Number of cycles Threshold (N/mm)
------------------ ------------------ --
$5.10^4$ $240$
$5.10^5$ $190$
$1.10^6$ $170$
: Influence of the number of fretting cycles on the crack nucleation threshold ($P=400$ N/mm).
\[fretting\_amorcage\_N.tab\]
![Evolution of the critical tangential force $Q_{th}$ to initiate a fretting crack (determined as shown in fig. \[fretting\_nucleation.fig\]a), with the number of fretting cycles ($P=400$ N/mm).[]{data-label="fretting_influence_N.fig"}](figure_04)
As expected, the number of applied fretting cycles has an influence on $Q_{th}$. The more fretting cycles applied, the less the threshold value. Moreover, the evolution of $Q_{th}(N)$ shows first a rapid reduction with increasing $N$, and then tends to a saturation value around 160–170 N/mm ,although more experiment are needed to establish the exact value. In particular, experiments with $N>10^6$ cycles, which are very time consuming, are required. In addition, it has been shown elsewhere that with this material in this configuration, $10^6$ cycle does correspond to a stabilized condition [@Munoz2006].
Assuming a homotetic behaviour with respect to $P$, the whole crack nucleation condition can be plotted in a 3D diagram $f(P,Q,N)$ as shown in figure \[fretting\_PQN.fig\].
![Representation of the crack nucleation condition in the (P,Q,N) space.[]{data-label="fretting_PQN.fig"}](figure_05){width="75mm"}
This defines completely both the safe and cracking domains by the boundary $f(P,Q,N)$. The advantage is the strong physical meaning of these parameters, which can be extended to other test configurations and more generally to any structure. In fretting fatigue tests for instance, the definition of $P$ is unchanged but there is an additional load (bulk stress) in the material; the parameter $Q$ must take that bulk stress into account and $N$ would be the number of fatigue cycles. In a more complicated case like an industrial component, first step will be to locate the contact area and then calculate the contact pressure $P$, for example by finite elements (integrating the normal stresses over the contact surface). Then $Q$ can be defined as the maximum shear force transmitted through the contact during a cycle and should be estimated in the same way, integrating this time the shear stresses over the contact surface (this will, however, require a detailed contact analysis which must capture the partial slip contact conditions). $N$ can then be identified as the number of cycles associated to the load responsible of the relative displacement of the surfaces. To conclude, this new representation aims to narrow the gap between ideal test configuration and actual structures encountered in real industrial problems.
Prediction of the crack nucleation condition
============================================
The purpose of this section is to use a multiaxial fatigue criterion to predict the crack nucleation condition. The stress/strain state is computed from the analytical solutions of the cylinder plane contact ; these fields are averaged[^3] on a circular shaped process volume to soften the contact gradients and then the cracking risk is evaluated. Among the numerous studies on the subject, it has been shown that no criterion is able to predict the location of nucleation, the crack angle and load leading to crack initiation together.
In this study we consider the SWT parameter despite the fact that it has been shown to be unable to predict the crack initiation angle (the investigated cracks are very often inclined toward the centre of the contact). The hypothesis to identify the crack initiation plane as the critical plane of the criterion is not obvious and the crack initiation process may be much more complicated as recently pointed out by another study [@Proudhon2007]. On the other hand the SWT criterion has a low cycle fatigue behaviour which has been shown to be effective for finite life [@Fouvry2002] and also was used recently be Fridrici et al. to tackle the same kind of problems in titanium alloys [@Fridrici2005]. In addition, the material coefficients for this criterion are given by Szolwinski et al. [@Szolwinski1996].
Prediction of the crack nucleation boundary at 50000 fretting cycles {#50000.ssc}
--------------------------------------------------------------------
The use of the SWT criterion for fretting is now well documented (see for example [@Sum2005; @Proudhon2005; @Fridrici2005]). Thus, only the general equation is reproduced here. According to this model, the initiation is likely to occur on the plane where the SWT parameter is maximum. The SWT parameter $\Gamma$ is evaluated in the current plane by the product between the amplitude of the strain $\varepsilon _a$ and the maximum stress normal to this plane $\sigma_{max}$.
$$\Gamma=\sigma_{max}\times\varepsilon_a=\frac{(\sigma _f')^2}{E}(2N)^{2b'}+\sigma _f'\varepsilon _f'(2N)^{b'+c'}
\label{SWT.eqn}$$
where $\sigma_f'$ is the fatigue strength coefficient, $b'$ is the fatigue strain exponent, $\varepsilon_f'$ is the fatigue ductility coefficient and $c'$ is the fatigue ductility exponent. The mechanical and fatigue properties of the studied alloy are listed in tables \[alu\_mecha.tab\] and \[alu\_fatigue.tab\] respectively.
$\sigma_f'$ (MPa) $b'$ $\varepsilon_f'$ $c'$
------------------- -------- ------------------ --------
714 -0.078 0.166 -0.538
: Fatigue properties of the studied Al2024 (from [@Szolwinski1996])
\[alu\_fatigue.tab\]
This approach has been previously used to study the crack nucleation at 50000 cycles [@Proudhon2005]. The results can be summarized as follow:
- the crack nucleation threshold predicted by the analysis of the local analytical stresses is very far from the experimental result ($\simeq100$ N/mm to be compared to $Q_{th}=240$ N/mm).
- performing size effect calculations allowed for fitting of the average experimental result but the precise behaviour (i.e. the effect of the loading pressure) cannot be captured.
- the average process volume showing the best correlation was found to be correlated to the mean grain radius (which was measured by Electron BackScattered Diffraction analysis giving 75 m).
The physical meaning of the process volume size is still an open question. The most often quoted significance is the grain size. Indeed it is argued here that the stress/strain state inducing initiation must be sufficiently widespread to make the very short crack propagate in the adjacent grain. Looking at several studies on different materials, a large range of sizes have been used for the process volume. 5 m in steel [@Fouvry2002], 30 m in titanium alloy [@Fridrici2005] and 80 m in aluminium alloy [@Proudhon2005], each time correlated to the grain size. On the other hand, the grain size cannot be the only significant parameter due to other cases where no correlation has been found. Moreover, this parameter does not take into account any mechanical quantity such as for example the slip amplitude which is certainly relevant for the crack initiation condition.
We consider here a new approach using a variable process volume. This idea comes from the fact that the crack initiation may be strongly monitored by the stress state in the slip zone only. More precisely, the severity of the stress gradient located in the slip zone may be responsible of the crack initiation risk. This is illustrated by figure \[s11\_plot\_P1P2.fig\] where two distributions of the surface traction[^4] ($\sigma_{xx}$) are plotted for the same conditions but with different normal loads ($P_1<P_2$).
![Illustration of the radius of the process volume needed to average the stress gradient with two different normal loads $P_1$ and $P_2$.[]{data-label="s11_plot_P1P2.fig"}](figure_06){width="75mm"}
It is clear that the severity of the stress gradient can be related to the slip zone width $a-c$ due to the quasi-linear evolution of $\sigma_{xx}$. Keeping the process volume size constant would introduce a strong effect of the pressure, which can actually be seen looking at the solid line prediction replotted on fig. \[prediction\_r\_variable.fig\] for the calculation with $r=80$ m. From here the radius of the process volume zone is no longer constant. We introduce a new variable $\gamma$ defined as the ratio of the process volume radius $r$ and the slip zone width $a-c$:
$$\gamma=\frac{r}{a-c}$$
The value of $\gamma$ is identified once for all for the conditions which gave r=80m. For the corresponding loading conditions ($P=318$ N/mm, $Q=240$ N/mm) one can calculate $a=700$ m and $c=393$ m. This leads to the value of $\gamma=0.26$. The crack nucleation threshold is then computed through this new approach according to the flowchart shown in fig. \[flowchart.fig\].
![Flowchart of the different steps required to predict the critical load $Q_{th}$ for each pressure level $P_i$ ; the third step has been added to the conventional constant process volume approach.[]{data-label="flowchart.fig"}](figure_07)
Prediction of the crack nucleation boundary for 50000 fretting cycles with $\gamma=0.26$ is presented in fig. \[prediction\_r\_variable.fig\]. This approach gives, a very good correlation with the experimental results. In particular, the pressure effect is well described.
![Prediction of the crack nucleation boundary at 50000 fretting cycles with the variable process volume approach.[]{data-label="prediction_r_variable.fig"}](figure_08)
This result highlights the reliability of the approach and shows that the process volume size, in addition to be related to some microstructural characteristic of the material, must be linked to the mechanical state as well, such as the slip zone width, to be able to predict the crack nucleation precisely.
Prediction of the low cycle fretting behaviour
----------------------------------------------
The same model is applied to test the experimental results obtained for $5.10^5$ and $10^6$ fretting cycles (see §\[exp\_N.sec\]). The $\gamma$ parameter is kept constant at the same value identified for 50000 cycles ($\gamma=0.26$, see §\[50000.ssc\]). The prediction is done in the same way by varying the number of cycles in equation \[SWT.eqn\]. The results of these computations are gathered in figure \[fretting\_SWT\_vs\_N.fig\].
![Prediction of the evolution of $Q_{th}$ with the number of cycles, by application of the SWT criterion suited with a variable process volume size.[]{data-label="fretting_SWT_vs_N.fig"}](figure_09){width="75mm"}
The experimental behaviour appears well correlated with the prediction of the SWT parameter. In particular one can see a very rapid decrease of the critical tangential force needed to nucleate a fretting crack, as the number of cycles increases. Around $10^6$ cycles a plateau is reached, corresponding to endurance conditions. This behaviour is consistent with another study on the same alloy and another 7xxx series alloy [@Munoz2006] where $10^6$ fretting cycles are clearly identified as a stabilized condition.
Conclusion
==========
Two main results have been presented in this paper. On the experimental point of view, a new representation of the crack nucleation condition is introduced through the $(P,Q,N)$ diagram, leading to a concrete and complete description of the material fretting resistance to initiation and this with a limited, although still quite important, number of tests. In order to predict this initiation condition, the SWT criterion is used and the classical computation is extended with a variable process volume size. This further opens the question of the significance of this parameter as it appears not to be only related to a microstructure characteristic length. The use of the slip zone width to determine the process volume radius clearly identifies a *mechanical* significance. Eventually the final answer may require a mix of the microstructural/mechanical behaviour. This could at last be answered by extending the approach to different materials as for instance steels and titanium alloys.
[^1]: Corresponding author. Tel.: +33-472-186-562; fax: +33-472-433-383.
[^2]: note that in fretting wear experiments, there is no bulk load imposed
[^3]: this process is sometimes referred to as a size effect
[^4]: One should note that the $\sigma_{xx}$ component is dominant in the slip zone, compared to the other stress values, (they vanish when $x\mapsto a$ where the stress state becomes purely uniaxial)
|
---
abstract: 'ANAIS (Annual Modulation with NaI’s) is an experiment planned to investigate seasonal modulation effects in the signal of galactic WIMPs using up to 107 kg of NaI(Tl) in the Canfranc Underground Laboratory (Spain). A prototype using one single crystal (10.7 kg) is being developed before the installation of the complete experiment; the first results presented here show an average background level of 1.2 counts/(keV kg day) from threshold ($E_{thr} \sim 4$ keV) up to 10 keV.'
address: 'Laboratory of Nuclear and High Energy Physics, University of Zaragoza, 50009 Zaragoza, Spain'
author:
- 'S. Cebrián[^1], J. Amaré, J. M. Carmona, E. García, I. G. Irastorza[^2], G. Luzón, A. Morales, J. Morales, A. Ortiz de Solórzano, J. Puimedón, M.L. Sarsa, J. A. Villar'
title: Status and preliminary results of the ANAIS experiment at Canfranc
---
INTRODUCTION
============
There is a substancial evidence to conclude that most of the matter in the Universe must be dark and that it consists mainly of cold non-baryonic particles. Weak Interacting Massive Particles (WIMPs) are favourite candidates to such non-baryonic components. A convicing proof of the detection WIMPs, which are supposedly filling the galactic halo, would be to find unique signatures in the data, like seasonal asymmetries. ANAIS (Annual Modulation with NaI’s) is a large mass experiment intended to investigate the annual modulation effect which would be produced in the signal of galactic WIMPs due to the variations in the relative velocity between the Earth and the halo [@Freese88]. It will be installed in the Canfranc Underground Laboratory, located in an old railway tunnel in the Spanish Pyrenees with an overburden of 2450 m.w.e., using up to 10 NaI(Tl) hexagonal crystals (10.7 kg each) as an improved scaled-up version of a previuos experiment [@Sarsa97]. Before setting-up the whole experiment, a prototype is being developed in Canfranc in an attempt to obtain the best energy threshold and lowest radioactive background in the low energy region (2 to 50 keV), as well as to check the stability of the environmental conditions which influence on the detector response.
THE ANAIS PROTOTYPE
===================
One single detector has been used in the ANAIS prototype; it consists of a NaI(Tl) crystal encapsulated inside 0.5-mm-thick stainless steel and coupled to a PMT through a quartz window. Some components of the photomultiplier have been removed because of their radioimpurities. The scintillator has been placed in a shielding consisting of 10 cm of archaeological lead (of less than 9 mBq/kg of $^{210}$Pb) followed by 20 cm of low activity lead, a sealed box in PVC (maintained at overpressure to prevent the intrusion of radon), 2-mm-thick cadmium sheets, and finally, 40 cm of polyethylene and tanks of borated water. An active veto made of plastic scintillators is covering the set-up.
The data acquisition system, based on standard NIM and CAMAC electronics, has two different parts following the two output signals implemented from the PMT; the fast signal is recorded using a digital oscilloscope while the slow signal is routed through a linear amplifier and analog-to-digital converters controlled by a PC through parallel interfaces, to register the energy of events up to $\sim 1.7$ MeV.
Parameters of the pulses used to reject the noise produced by PMT in the various previous NaI experiments are the mean amplitude [@Gerbier99], a ratio of area portions [@Bernabei99], etc. In the present work the filtering of noise uses the squared deviation of the digitalized pulse from the well-known theoretical shape of a scintillation event of the same area. To reject the noise, a safe cut at $3\sigma$ from the center of the gaussian distribution of this parameter for calibration events from 4 to 10 keV has been used.
By comparing the data recorded from December 2000 to August 2001 with the Monte Carlo simulations, it was possible to identify the main sources of background in the region of interest. The $^{210}$Pb 46.5 keV line as well as a peak due to the escape of X-rays of I at $\sim$ 16 keV seen in the spectrum, may be caused by the presence of radioimpurities in the stainless steel can and/or in the PMT. The area of the 1460.8 keV peak is compatible with an activity of 15 mBq/kg from $^{40}$K in the NaI crystal; these impurities produce an almost flat background in the low energy region due to their beta emission. A comparison between the spectra recorded with and without the neutron shielding does not show noticeable differences.
A pulse shape analysis has been carried out with the purpose of investigate the possible appearance of the so-called “anomalous” or “bump” events found in other NaI experiments [@Liubarsky00; @Gerbier00]. No evidence of such anomaly has been found in the distributions for background events, neither following the method of the UKDMC (fitting integrated pulses to calculate the decay time constant) nor using other parameters (like the first momemtum of the pulse). Monitoring and stabilisation control of the environmental conditions (radon levels, $N_{2}$ flux, temperature in the laboratory and in the inner enceinte, photomultiplier working voltage, …) is underway. Using the data of the prototype, collected along almost 6000 hours, the stability of some parameters has been checked. The fluctuations of the ADC channels for the different peaks used to perform the energy calibration range from 1 to 1.5 %. With respect to the counting rates, the gaussian distributions of the deviations from the mean values have a sigma of 1.27 for the rate integrated above 6 keV and 1.47 for the rate above 100 keV.
FIRST RESULTS
=============
The results presented here correspond to an exposure of 1225.4 kg$\times$day. Fig. \[spectrum\] shows the raw spectrum and the spectrum after the noise rejection up to 100 keV. The energy threshold is of $\sim 4$ keV and the background level registered from the threshold up to 10 keV is about 1.2 counts/(keV kg day).
We have used this region to derive the corresponding limits for the WIMP-nucleon cross sections. The galactic halo is supposed to be isotropic, isothermal and non-rotating, assuming a density of $\rho$=0.3 GeV/cm$^{3}$, a Maxwellian velocity distribution with $v_{rms}$=270 km/s (with an upper cut corresponding to an escape velocity of 650 km/s) and a relative Earth-halo velocity of $v_{r}$=230 km/s. The Helm parameterization [@Engel91] is used for the coherent form factor, while the approximation from [@Lewin96] is considered for the SD case. Spin factors ($\lambda_{p}J(J+1)$) 0.089 and 0.126 are assumed for Na and I respectively. Fig. \[plot\] shows, in addition to the limits derived from the prototype results (solid lines), the estimates considering a flat background of 1 count/(keV kg day) from 2 to 8 keV after an exposure of 107 kg$\times$y both for raw data (dotted lines) and assuming PSD (dashed lines). The plots show the contour lines for each nucleus, Na and from I, as well as the NaI case. That is shown both for SI scalar interactions and SD WIMP-proton interactions. It should be noted that for SI interactions and using PSD techniques, ANAIS will be able to explore the region of WIMPs singled out by the possible annual modulation effect reported by the DAMA collaboration [@Bernabei00].
FUTURE PROSPECTS
================
The next steps in the development of the prototype of ANAIS, according to these first results, are the removal of the present PMT and the steel can and to install, instead, two ultra-low background PMT and a 1-cm-thick teflon enclosure filled with special mineral oil, as in the NAIAD experiment [@Spooner00]. A program of measurements to select high radiopurity materials is in course in Canfranc, using an ultra-low background Ge detector. The program includes the removal of components when neccesary to reduce, as much as possible, the various sources of background, to diminish the noise by using anticoincidence read-out (lowering so also the energy threshold) and improving the collection of the scintillation light.
Acknowledgements {#acknowledgements .unnumbered}
================
The Canfranc Astroparticle Underground Laboratory is operated by the University of Zaragoza under contract No. AEN99-1033. This research was funded by the Spanish Commission for Science and Technology (CICYT) and the Government of Aragón.
[9]{}
A. K. Drukier, K. Freese and D. N. Spergel. Phys. Rev. D 33 (1986) 3495. K. Freese, J. Frieman and A. Gould. Phys. Rev. D 37 (1988) 3388. M. L. Sarsa [*et al*]{}. Phys. Rev. D 56 (1997) 1856. G. Gerbier [*et al*]{}. Astrop. Phys. 11 (1999) 287. R. Bernabei [*et al*]{}. Il Nuovo Cimento A 112 (1999) 545. I. Liubarsky [*et al*]{}. Nucl. Phys. B (Proc. Suppl.) 87 (2000) 64. G. Gerbier [*et al*]{}. Nucl. Phys. B (Proc. Suppl.) 87 (2000) 61.
J. Engel. Phys. Lett. B 264 (1991) 114.
J. D. Lewin y P. F. Smith. Astrop. Phys. 6 (1996) 87. R. Bernabei [*et al*]{}. Phys. Lett. B 480 (2000) 23. N. J. C. Spooner [*et al*]{}. Phys. Lett. B 473 (2000) 330.
[^1]: Attending speaker: [email protected]
[^2]: Present address: CERN, EP Division, CH-1211 Geneva 23, Switzerland
|
---
author:
- 'Vakhid A. Gani'
- Aliakbar Moradi Marjaneh
- Alidad Askari
- Ekaterina Belendryasova
- Danial Saadatmand
date: 'Received: date / Revised version: date'
title: 'Scattering of the double sine-Gordon kinks'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#sec:Introduction}
============
The $(1+1)$-dimensional field-theoretical models possessing the topologically non-trivial solutions — kinks — are of special interest for modern physics. They arise in a vast variety of models in quantum and classical field theory, high energy physics, cosmology, condensed matter physics, and so on, [@Rajaraman.book.1982]–[@Vachaspati.book.2006]. Firstly, the $(1+1)$-dimensional models can be investigated analytically and numerically much easier than $(2+1)$ or $(3+1)$-dimensional. Because of that, some general properties of topological defects can be studied within the $(1+1)$-dimensional setups. Secondly, many physical systems can be effectively described by the one-dimensional structures. For example, a plane domain wall — a wall, which separates regions with different vacuum states — in the direction orthogonal to it, presents a kink. Surely, the topological defects arise and in more complex models with two, three or more fields. For example, in [@Bazeia.PLA.2013]–[@Alonso.2017.2] the kink-like structures were studied in models with two interacting real scalar fields, for further information see also [@Lensky.JETP.2001.eng]–[@Ashcroft.JPA.2016]. Kink-antikink collisions, as well as interactions of kinks with impurities (spatial inhomogeneities), are of growing interest since 1970s [@Kudryavtsev.UFN.1997.eng; @Kudryavtsev.UFN.1997.rus]. Nevertheless, today it is a very fast developing area of research. For investigating of the kink-antikink interactions various approximate methods are widely used. Among them, the collective coordinate approximation [@GaKuLi]–[@GaKu.SuSy.2001.rus]. Withing this method a real field system “kink+antikink” is approximately described as a system with one or several degrees of freedom. For instance, the kink-antikink separation can be used as the only (translational) degree of freedom. The more complicated modifications of the method have also been elaborated, which include other degrees of freedom (in particular, vibrational), see, [*e.g.*]{}, [@Weigel.cc.2014; @Weigel.cc.2016; @Demirkaya.cc.2017].
Another approximate method for investigating the kinks interactions is Manton’s method [@Manton.book.2004 Ch. 5], [@perring62]–[@KKS.PRE.2004]. This method is based on using of the kinks asymptotics, it enables to estimate the force between the kink and the antikink at large separations.
On the other hand, recently the numerical simulation has become a powerful tool for studying the dynamics of the one-dimensional field systems. Using various numerical methods, many important results were obtained. In particular, the resonance phenomena — escape windows and quasi-resonances — have been found and investigated in the kinks’ scattering [@GaKuLi], [@Gani.PRE.1999]–[@Saadatmand.PRD.2015]. Many important results have been obtained for the models with polynomial potentials of fourth, sixth, eighth, and higher degree self-interaction [@GaKuLi], [@Dorey.PRL.2011]–[@Moradi.JHEP.2017], [@Belendryasova.arXiv.2017; @Belendryasova.conf.2017], [@lohe]–[@Snelson.arXiv.2016]. One should note interesting results on the long-range interaction between kink and antikink [@Radomskiy; @Belendryasova.arXiv.2017; @Belendryasova.conf.2017], [@Guerrero.PRE.1997]–[@Gomes.PRD.2012]. The models with non-polynomial potentials are also widely discussed in the literature, for example, the modified sine-Gordon [@Peyrard.msG.1983], the multi-frequency sine-Gordon [@Delfino.NPB.1998], the double sine-Gordon [@Gani.PRE.1999], [@Campbell.1986]–[@Malomed.PLA.1989], and a number of models, which can be obtained by using the deformation procedure [@Bazeia.arXiv.2017.sinh; @Bazeia.arXiv.2017.sinh.conf; @Bazeia.PRD.2006; @Bazeia.PRD.2002; @Bazeia.PRD.2004].
The impressive progress is achieved in the investigation of domain walls, bubbles, vortices, strings, [@Campanelli.IJMPD.2004]–[@Gani.YaF.2001.rus], as well as the embedded topological defects, [*e.g.*]{}, a Q-lump on a domain wall, a skyrmion on a domain wall, etc. [@nitta1]–[@Bazeia.fermion.2017]. Besides that, we have to mention various configurations of the type of Q-balls [@Schweitzer.PRD.2012.1]–[@Dzhunushaliev.PRD.2016]. Topologically non-trivial field configurations could also lead to a variety of phenomena in the early Universe [@GaKiRu; @GaKiRu.conf].
In this paper we study the kink-antikink collisions within the double sine-Gordon model [@Gani.PRE.1999], [@Campbell.dsG.1986]–[@Malomed.PLA.1989]. There is a critical value of the initial velocity of the colliding kinks, $v_\mathrm{cr}^{}$, which separates different regimes of the collision. At $v_\mathrm{in}^{}>v_\mathrm{cr}^{}$ the kinks pass through each other and escape to infinities, while at $v_\mathrm{in}^{}<v_\mathrm{cr}^{}$ the kinks’ capture and a complex picture of the so-called escape windows are observed, see, [*e.g.*]{}, [@Kudryavtsev.UFN.1997.eng; @Kudryavtsev.UFN.1997.rus; @Campbell.dsG.1986]. We performed a detailed study of the kink-antikink scattering at various values of the model parameter $R$. We have found a series of local maxima of the dependence of $v_\mathrm{cr}^{}$ on $R$, which has not been reported up to now. Besides that, at some initial velocities of the colliding kinks we observed final configuration of the type of two oscillons, which form a bound state or could escape to spatial infinities with some final velocities.
Our paper is organized as follows. In section \[sec:Topological\_defects\] we give some general information about the $(1+1)$-dimensional models with one real scalar field. Section \[sec:DSG\_model\] introduces the double sine-Gordon model, describes its potential, kinks, and their main properties. In section \[sec:Scattering\] we study the scattering of the kink and antikink. In this section we present our main results related to the kink-antikink collisions. Finally, we summarize and formulate prospects for future works in section \[sec:Conclusion\].
Topological defects in $(1+1)$ dimensions {#sec:Topological_defects}
=========================================
Consider a field-theoretical model in $(1+1)$-dimensional space-time with a real scalar field $\phi(x,t)$. The dynamics of the field $\phi$ is described by the Lagrangian density $$\label{eq:Largangian}
\mathcal{L} = \frac{1}{2}\left(\frac{\partial\phi}{\partial t}\right)^2 - \frac{1}{2}\left(\frac{\partial\phi}{\partial x}\right)^2 - V(\phi),$$ where the potential $V(\phi)$ defines self-interaction of the field $\phi$. We assume that the potential is a non-negative function of $\phi$, which has a set of minima $$\mathcal{V}=\left\{\phi_1^{\mathrm{(vac)}}, \phi_2^{\mathrm{(vac)}}, \phi_3^{\mathrm{(vac)}},\dots\right\},$$ which is a vacuum manifold of the model, and $V(\phi)=0$ for all $\phi\in\mathcal{V}$. The energy functional corresponding to the Lagrangian is $$\label{eq:energy}
E[\phi] = \int\limits_{-\infty}^{\infty}\left[\frac{1}{2}\left(\frac{\partial\phi}{\partial t}\right)^2 + \frac{1}{2}\left(\frac{\partial\phi}{\partial x}\right)^2 + V(\phi)\right]dx.$$ The Lagrangian yields the equation of motion for the field $\phi(x,t)$: $$\label{eq:eqmo}
\frac{\partial^2\phi}{\partial t^2} - \frac{\partial^2\phi}{\partial x^2} + \frac{dV}{d\phi} = 0.$$ In the static case $\phi=\phi(x)$, $\displaystyle\frac{\partial\phi}{\partial t}=0$, and we obtain $$\label{eq:eqmo_static}
\frac{d^2\phi}{dx^2} = \frac{dV}{d\phi}.$$ This equation can be reduced to the first order ordinary differential equation $$\label{eq:bps}
\frac{d\phi}{dx} = \pm\sqrt{2V(\phi)}.$$ In order for the energy of the static configuration to be finite, it is necessary that $$\label{eq:minus_infty}
\phi(-\infty) = \lim_{x \to -\infty} \phi (x) = \phi_i^{\mathrm{(vac)}}$$ and $$\label{eq:plus_infty}
\phi(+\infty) = \lim_{x \to +\infty} \phi (x) = \phi_j^{\mathrm{(vac)}},$$ where $\phi_i^{\mathrm{(vac)}},\phi_j^{\mathrm{(vac)}}\in\mathcal{V}$. If these two equalities hold, the second and the third terms in square brackets in fall off at $x\to\pm\infty$ (the first term turns to zero for all static configurations), and the integral in can be convergent.
If the vacuum manifold $\mathcal{V}$ consists of more than one point, i.e the potential $V(\phi)$ possesses two or more degenerate minima, the set of all static configurations with finite energy can be split into disjoint equivalence classes (or topological sectors) according to the asymptotic behaviour of the configuration at $x\to\pm\infty$. Configurations with $\phi_i^{\mathrm{(vac)}}\neq\phi_j^{\mathrm{(vac)}}$ in eqs. and are called topological, while those with $\phi_i^{\mathrm{(vac)}}=\phi_j^{\mathrm{(vac)}}$ — non-topological. A configuration belonging to one equivalence class (topological sector) can not be transformed into a configuration from another class (topological sector) through a continuous deformation, that is via a sequence of configurations with finite energies.
To describe the topological properties of the configurations, one can introduce a conserved topological current, [*e.g.*]{}, $$\label{eq:top_current}
j_\mathrm{top}^\mu = \frac{1}{2}\varepsilon^{\mu\nu}\partial_\nu\phi,$$ here $\varepsilon^{\mu\nu}$ stands for the Levi-Civita symbol, the indices $\mu$ and $\nu$ take values 0 and 1 for a $(1+1)$-dimensional configuration, and $\partial_0\phi\equiv\displaystyle\frac{\partial\phi}{\partial t}$, $\partial_1\phi\equiv\displaystyle\frac{\partial\phi}{\partial x}$. The corresponding topological charge does not depend on the behaviour of the field at finite $x$, $$\label{eq:top_charge}
Q_\mathrm{top} = \int\limits_{-\infty}^{\infty}j_\mathrm{top}^0dx = \frac{1}{2}\left[ \phi(+\infty)-\phi(-\infty)\right].$$ The value of $Q_\mathrm{top}$ is determined only by the asymptotics , of the field. The topological charge is conserved during the evolution of the configuration. Nevertheless, configurations from different topological sectors may have the same topological charge. At the same time, configurations with different topological charges necessarily belong to different topological sectors.
Further, for the non-negative potential $V(\phi)$ we can introduce the superpotential — a smooth (continuously differentiable) function $W(\phi)$ of the field $\phi$: $$\label{eq:dwdfi}
V(\phi) = \frac{1}{2}\left(\frac{dW}{d\phi}\right)^2.$$ Using this representation of the potential, the energy of a static configuration can be written as $$\label{eq:static_energy_with_bps}
E = E_\mathrm{BPS}^{} + \frac{1}{2}\int_{-\infty}^{\infty}\left(\frac{d\phi}{dx}\pm\frac{dW}{d\phi}\right)^2dx,$$ where $$\label{eq:static_energy_bps}
E_\mathrm{BPS}^{} = \big|W[\phi(+\infty)]-W[\phi(-\infty)]\big|.$$ Here the subscript “BPS” stands for Bogomolny, Prasad, and Sommerfield [@BPS1.eng; @BPS1.rus; @BPS2]. From eq. one can see that, firstly, the energy of any static configuration is bounded from below by $E_\mathrm{BPS}^{}$, $$\label{eq:bound_BPS}
E \ge E_\mathrm{BPS}^{},$$ and, secondly, the static configuration, which satisfies the equation $$\label{eq:bps_with_superpotential}
\displaystyle\frac{d\phi}{dx} = \pm\frac{dW}{d\phi},$$ saturates the inequality , i.e has the minimal energy among all the configurations within a given topological sector. The solutions of eq. are called BPS (or BPS saturated) configurations. Note that eq. coincides with eq. .
A kink is a BPS saturated topological solution $\phi_\mathrm{k}^{}(x)$ of eq. , which connects two neighboring vacua of the model, [*i.e.*]{} for the kink solution the values $\phi_i^{\mathrm{(vac)}}$ and $\phi_j^{\mathrm{(vac)}}$ in , are adjacent minima of the potential $V(\phi)$. Below we use the terms “kink” and “antikink” for solutions with $\phi_j^{\mathrm{(vac)}}>\phi_i^{\mathrm{(vac)}}$ and $\phi_j^{\mathrm{(vac)}}<\phi_i^{\mathrm{(vac)}}$, respectively. Nevertheless, in some cases we use “kink” for both solutions, just to be brief.
The double sine-Gordon model {#sec:DSG_model}
============================
Consider the double sine-Gordon (DSG) model. The potential of the DSG model can be written in several different forms. Below in this section we briefly recall two of them, and after that we give a detailed introduction to the properties of the DSG model employed by us.
The $\eta$-parameterized potential
----------------------------------
Recall that, [*e.g.*]{}, in papers [@Gani.PRE.1999; @Campbell.dsG.1986] the following parameterization has been used: $$\label{eq:potential_eta}
V_\eta^{}(\phi) = \frac{4}{1+4|\eta|}\left(\eta\:(1-\cos\phi)+1+\cos\frac{\phi}{2}\right),$$ where $\eta$ is a real parameter, $-\infty<\eta<+\infty$. It is easy to see that $$V_\eta^{}(\phi) =
\begin{cases}
\cos\phi - 1\quad \mbox{for}\quad \eta\to-\infty,\\
1 - \cos\phi\quad \mbox{for}\quad \eta\to+\infty,\\
4\left(1+\cos\displaystyle\frac{\phi}{2}\right)\quad \mbox{for}\quad \eta=0,
\end{cases}$$ [*i.e.*]{} the potential $V(\phi)$ reduces to a sine-Gordon form for the field $\phi$ at $\eta\to\pm\infty$, and for the field $\phi/2$ at $\eta=0$.
The shape of the potential crucially depends on the parameter $\eta$. Following [@Campbell.dsG.1986], we can split all values of $\eta$ into four regions: $\eta<-\displaystyle\frac{1}{4}$, $-\displaystyle\frac{1}{4}<\eta<0$, $0<\eta<\displaystyle\frac{1}{4}$, and $\eta>\displaystyle\frac{1}{4}$.
1\. At $\eta<-\displaystyle\frac{1}{4}$ the potential has two distinct types of minima, $\phi_n^{\mathrm{(vac)}}=4\pi n+\arccos\displaystyle\frac{1}{4\eta}$ and $\phi_m^{\mathrm{(vac)}}=4\pi m-\arccos\displaystyle\frac{1}{4\eta}$, $n,m=0,\pm 1,\pm 2,\dots$, degenerate in energy, $V(\phi_n^{\mathrm{(vac)}})=V(\phi_m^{\mathrm{(vac)}})=0$, which are separated by inequivalent barriers.
2\. At $-\displaystyle\frac{1}{4}<\eta<0$ the potential has a single type of minima at $\phi_n^{\mathrm{(vac)}}=(2n+1)2\pi$ with $V(\phi_n^{\mathrm{(vac)}})=0$.
3\. At $0<\eta<\displaystyle\frac{1}{4}$ the potential is structurally similar to the previous region, with the same set of minima.
4\. At $\eta>\displaystyle\frac{1}{4}$ the minima of the potential are $\phi_n^{}=4n\pi$ with $V(\phi_n^{})=\displaystyle\frac{8}{1+4|\eta|}$, and $\phi_m^{\mathrm{(vac)}}=(2m+1)2\pi$ with $V(\phi_m^{\mathrm{(vac)}})=0$, $n,m=0,\pm 1,\pm 2,\dots$.
This our paper deals with the DSG model with the positive values of $\eta$. In this case, as we show below, it is convenient to introduce another positive parameter.
The $R$-parameterized potential
-------------------------------
For positive values of $\eta$, it is convenient to introduce another positive parameter, $R$, such that $$\eta = \frac{1}{4}\sinh^2 R.$$ In terms of the parameter $R$ the potential of the DSG model reads: $$\label{eq:potential_R}
V_\mathrm{R}^{}(\phi) = \tanh^2R\:(1-\cos\phi) + \frac{4}{\cosh^2R}\left(1+\cos\frac{\phi}{2}\right).$$ Depending on the parameter $R$ the shape of the potential looks different, see fig. \[fig:PotentialPhi\].
![The potential as a function of $\phi$ for various $R$.[]{data-label="fig:PotentialPhi"}](PotentialPhi.pdf){width="45.00000%"}
At $R=0$ we have a sine-Gordon potential for the field $\phi/2$, while at $R\to+\infty$ the potential reduces to a sine-Gordon form for the field $\phi$: $$\label{eq:potential_R_limits}
V_\mathrm{R}^{}(\phi) =
\begin{cases}
4\left(1+\cos\displaystyle\frac{\phi}{2}\right)\quad \mbox{for}\quad R=0,\\
4-\left(1-\cos\displaystyle\frac{\phi}{2}\right)^2\quad \mbox{for}\quad R=\mbox{arcsinh}\:1,\\
1 - \cos\phi\quad \mbox{for}\quad R\to+\infty.
\end{cases}$$
The double sine-Gordon kinks
----------------------------
In terms of the parameter $R$ the static kink ($+$) and antikink ($-$) solutions can be written in a simple form, $$\label{eq:DSG_kinks_1}
\phi_{\mathrm{k}(\mathrm{\bar k})}(x) = 4\pi n \pm 4\arctan\frac{\sinh x}{\cosh R}.$$ The DSG kink (antikink) can also be expressed as a superposition of two sine-Gordon solitons, $$\label{eq:DSG_kinks_2}
\phi_{\mathrm{k}(\mathrm{\bar k})}^{}(x) = 4\pi n \pm \left[\phi_\mathrm{SGK}^{}(x+R)-\phi_\mathrm{SGK}^{}(R-x)\right],$$ or $$\label{eq:DSG_kink}
\phi_\mathrm{k}^{}(x) = 2\pi(2n-1) + \left[\phi_\mathrm{SGK}^{}(x+R)+\phi_\mathrm{SGK}^{}(x-R)\right]$$ and $$\label{eq:DSG_antikink}
\phi_\mathrm{\bar k}^{}(x) = 2\pi(2n+1) - \left[\phi_\mathrm{SGK}^{}(x+R)+\phi_\mathrm{SGK}^{}(x-R)\right],$$ where $\phi_\mathrm{SGK}^{}(x)=4\arctan\exp (x)$ is the sine-Gordon soliton. According to eqs. –, the DSG kink can be viewed as the superposition of two sine-Gordon solitons, which are separated by the distance $2R$ and centered at $x=\pm R$, see fig. \[fig:SolitonSolutions\].
\
The energy of the static DSG kink (antikink) is a function of the parameter $R$, $$\label{eq:DSG_kink_energy}
E(R) = 16\left(1+\frac{2R}{\sinh 2R}\right),$$ this dependence is shown in fig. \[fig:kink\_energy\_vs\_R\].
![The energy of the static DSG kink (antikink) as a function of the parameter $R$.[]{data-label="fig:kink_energy_vs_R"}](EnergyR.pdf){width="45.00000%"}
Below we study the collisions of the DSG kinks. In such processes the kink’s internal modes may be very important. Therefore, now we investigate the spectrum of small localized excitations of the DSG kink (antikink) using a standard method. Namely, we add a small perturbation $\delta\phi(x,t)$ to the static DSG kink $\phi_\mathrm{k}^{}(x)$: $$\phi(x,t) = \phi_\mathrm{k}^{}(x) + \delta\phi(x,t), \quad |\delta\phi| \ll |\phi_\mathrm{k}^{}|.$$ Substituting this $\phi(x,t)$ into the equation of motion , and linearizing in $\delta\phi$, we obtain the partial differential equation for $\delta\phi(x,t)$: $$\label{eq:eq_for_delta_phi}
\frac{\partial^2\delta\phi}{\partial t^2} - \frac{\partial^2\delta\phi}{\partial x^2} + \left.\frac{d^2V}{d\phi^2}\right|_{\phi_\mathrm{k}^{}(x)}\cdot\delta\phi = 0.$$ Looking for $\delta\phi$ in the form $$\label{eq:delta_phi}
\delta\phi(x,t) = \sum_n\eta_n^{}(x)\cos\:\omega_n^{} t,$$ we obtain the Schrödinger-like eigenvalue problem $$\label{eq:Schrodinger}
\hat{H}\eta_n^{}(x) = \omega_n^2\eta_n^{}(x)$$ with the operator $\hat{H}$ (the Hamiltonian) $$\label{eq:Schrod_Hamiltonian}
\hat{H} = -\frac{d^2}{dx^2} + U(x).$$ Here the quantum-mechanical potential is $$\label{eq:Schrod_potential}
U(x) = \left.\frac{d^2V}{d\phi^2}\right|_{\phi_\mathrm{k}^{}(x)}.$$ It can be easily shown that the discrete spectrum in the potential always possesses a zero mode $\omega_0^{}=0$. Differentiating eq. with respect to $x$, and taking into account that $\phi_\mathrm{k}^{}(x)$ is a solution of eq. , we see that $$-\frac{d^2}{dx^2}\frac{d\phi_\mathrm{k}^{}}{dx} + \left.\frac{d^2V}{d\phi^2}\right|_{\phi_\mathrm{k}^{}(x)}\cdot\frac{d\phi_\mathrm{k}^{}}{dx} = 0,$$ or, in other words, $$\hat{H}\cdot\frac{d\phi_\mathrm{k}^{}}{dx} = 0\cdot\frac{d\phi_\mathrm{k}^{}}{dx}.$$ So $\displaystyle\frac{d\phi_\mathrm{k}^{}}{dx}$ is really an eigenfunction of the Hamiltonian associated with the zero frequency.
The potential $U(x)$ for the double sine-Gordon kink (antikink) can be obtained by substituting eqs. and in , $$U(x) = \frac{8\tanh^2 R}{(1+\mbox{sech}^2R\sinh^2x)^2}$$ $$\label{eq:Schrod_potential_DSG}
+\frac{2(3-4\cosh^2R)}{\cosh^2R}\frac{1}{1+\mbox{sech}^2R\sinh^2x}+1.$$ The shape of the potential crucially depends on the parameter $R$, see fig. \[fig:quantum-mechanical\_potential\].
For $R=0$ equation gives the Pöschl-Teller potential, $$\label{eq:Schrod_potential_DSG_zero}
U_0^{}(x) = 1-\frac{2}{\cosh^2x},$$ which corresponds to the case of the sine-Gordon model. On the other hand, for $R\gg 1$ from eq. we obtain $$\label{eq:Schrod_potential_DSG_infty}
U_\infty^{}(x) \approx \begin{cases}
1-\displaystyle\frac{2}{\cosh^2(x-R)}\ \mbox{for } ||x|-R|\lesssim 1,\\
\qquad\quad 1 \qquad\qquad\ \mbox{for } ||x|-R|\gg 1.
\end{cases}$$ The discrete spectrum in the potential well for arbitrary value of $R$ can be obtained numerically by using a modification of the shooting method, see, [*e.g.*]{}, [@Gani.JHEP.2015; @Belendryasova.arXiv.2017; @Belendryasova.conf.2017]. First of all, for all $R$ there is the zero mode $\omega_0^{}=0$. Apart from that, we have found the vibrational mode $\omega_1^{}$ with the frequency that depends on $R$, see fig. \[fig:Omega1R\].
At $R\to 0$ the frequency $\omega_1^{}$ goes to the boundary of the continuum, which corresponds to the sine-Gordon case. With increasing $R$ the frequency $\omega_1^{}$ decreases to zero. At large $R$’s the levels $\omega_0^{}$ and $\omega_1^{}$ are the result of splitting of the zero mode of each of the two potential wells .
Collisions of the double sine-Gordon kinks {#sec:Scattering}
==========================================
We studied the collision of the DSG kink and antikink. In order to do this, we used the initial configuration in the form of the DSG kink and the DSG antikink, centered at $x=-\xi$ and $x=\xi$, respectively, and moving towards each other with the initial velocities $v_\mathrm{in}^{}$. We solved the partial differential equation with the $R$-parameterized potential numerically, extracting the values of $\phi(x,0)$ and $\displaystyle\frac{\partial\phi(x,0)}{\partial t}$ from the following initial configuration: $$\phi(x,t) = \phi_\mathrm{k}^{}\left(\frac{x+\xi-v_\mathrm{in}^{}t}{\sqrt{1-v_\mathrm{in}^2}}\right) + \phi_\mathrm{\bar k}^{}\left(\frac{x-\xi+v_\mathrm{in}^{}t}{\sqrt{1-v_\mathrm{in}^2}}\right)-2\pi$$ $$= 4\arctan \left[\frac{1}{\cosh R}\sinh \left(\frac{x+\xi-v_{in}^{}t}{\sqrt{1-v_\mathrm{in}^2}} \right)\right]$$ $$\label{eq:incond}
- 4\arctan \left[\frac{1}{\cosh R}\sinh\left(\frac{x-\xi+v_{in}^{}t}{\sqrt{1-v_\mathrm{in}^2}}\right)\right]-2\pi.$$
We discretized space and time using a grid with the spatial step $h$, and the time step $\tau$. We used the following discrete expressions for the second derivatives of the field: $$\frac{\partial^2\phi}{\partial t^2} = \frac{11\phi_{n,j+1} -20\phi_{n,j}+6\phi_{n,j-1}+4\phi_{n,j-2}-\phi_{n,j-3}}{12\tau^2},$$ $$\frac{\partial^2\phi}{\partial x^2} = \frac{-\phi_{n-2,j} +16\phi_{n-1,j}-30\phi_{n,j}+16\phi_{n+1,j}-\phi_{n+2,j}}{12h^2},$$ where $\phi_{n,j}=\phi(nh,j\tau)$, $n=0,\pm 1,\pm 2,\dots$, and\
$j=-3,-2,-1,0,1,2,\dots$.
We performed the numerical simulations for the steps $h=0.025$ and $\tau=0.005$, respectively, and for two different $\xi$: 10 and 20. We have also checked the stability of the results with respect to decrease of the steps. Fixed boundary conditions were used.
In the kink-antikink collisions there is a critical value of the initial velocity, $v_\mathrm{cr}^{}$, which separates two different regimes of the collisions. At the initial velocities above the critical value, $v_\mathrm{in}^{}>v_\mathrm{cr}^{}$, the DSG kinks pass through each other and escape to infinities after one collision, see fig. \[fig:F3DR1V02500\].
\
At $v_\mathrm{in}^{}<v_\mathrm{cr}^{}$ one observes the kinks’ capture and formation of their long-living bound state — a bion, see fig. \[fig:F3DR1V02100\]. At the same time, in the range $v_\mathrm{in}^{}<v_\mathrm{cr}^{}$ the so-called “escape windows” have been found. An escape window is a narrow interval of the initial velocities, at which the kink and the antikink escape after two, three, or more collisions, see figs. \[fig:F3DR1V02340\], \[fig:F3DR1V02380\].
The $R$-dependence of the critical velocity
-------------------------------------------
First of all, we found the dependence of the critical velocity $v_\mathrm{cr}^{}$ on the parameter $R$. Our results are shown in fig. \[fig:critical\_velocity\_vs\_R\].
![The critical velocity $v_\mathrm{cr}^{}$ as a function of the parameter $R$. (The initial half-separation is $\xi=20$ in these calculations.)[]{data-label="fig:critical_velocity_vs_R"}](criticalvelocity.pdf){width="55.00000%"}
One can see a series of peaks on the curve $v_\mathrm{cr}^{}(R)$, see also table \[tab:Table1\].
$n$ $R_n^\mathrm{(max)}$ $v_\mathrm{cr}^{}(R_n^\mathrm{(max)})$
----- ---------------------- ----------------------------------------
1 1.0 0.2409
2 2.1 0.1743
3 3.0 0.0296
4 3.5 0.0135
5 3.8 0.0081
6 4.0 0.0061
7 4.2 0.0047
: Positions of the local maxima of the dependence $v_\mathrm{cr}^{}(R)$, which is shown in fig. \[fig:critical\_velocity\_vs\_R\].[]{data-label="tab:Table1"}
Note that at this point we have some discrepancy with the results of [@Campbell.dsG.1986]. The authors of [@Campbell.dsG.1986] report only one maximum of the dependence $v_\mathrm{cr}^{}(R)$ at $R\approx 1$. Probably it can be a consequence of small amount of experimental points in [@Campbell.dsG.1986].
From fig. \[fig:critical\_velocity\_vs\_R\] one can see that the critical velocity turns to zero at $R=0$, which corresponds to the integrable sine-Gordon model, see eq. . Besides that, $v_\mathrm{cr}^{}$ decreases to zero with increasing $R$ at large $R$. Remind here, that the limit $R\to+\infty$ also corresponds to the case of the integrable sine-Gordon model, as one can see from eq. . Therefore it is quite natural that the critical velocity has a maximum at some $R$ and tends to zero at $R\to 0$ and $R\to+\infty$. The presence of a [*series*]{} of local maxima on the curve is an interesting fact that is observed for the first time. Apparently we can assume that one of the maxima is the main (probably, $R_1^\mathrm{(max)}$), while the other ones appear due to some change of the kink-antikink interaction in the collision process with increasing $R$. At large values of $R$ the DSG kink splits into two sine-Gordon solitons. Therefore, we can assume that in the DSG kink-antikink collision the four sine-Gordon solitons interact pairwise. This transition from the simple collision of the DSG kinks to more complicated pairwise interaction of the sine-Gordon solitons can lead, in particular, to the non-monotonicity of the dependence $v_\mathrm{cr}^{}(R)$ at $R>R_1^\mathrm{(max)}$.
Two oscillons in the final state
--------------------------------
In the kink-antikink collisions below the critical velocity we observed a phenomenon, which, to the best of our knowledge, has not been reported for the double sine-Gordon model before. At some initial velocities of the colliding kinks we observed final configuration in the form of two escaping oscillons. At the same time, at some initial velocities we found formation of the configuration, which we can classify as a bound state of two oscillons. In fig. \[fig:two\_Breathers\_in\_the\_final\_state\] we show some typical scenarios of that kind.
For example, at the initial velocity $v_\mathrm{in}^{}=0.1847$ we observe formation of a bound state of the kink and the antikink (a bion), which then evolves into two oscillons. These two oscillons are moving from each other, then stop, and start moving back to the collision point. This repeats several times, and after that the oscillons escape to infinities with the final velocity $v_\mathrm{f}^{}\approx 0.10$, see fig. \[fig:F3DR1V01847\].
Formation of the bound state of oscillons and the escape of oscillons are extremely sensitive to changes of the initial velocity of the colliding kinks. For example, at the initial velocity $v_\mathrm{in}^{}=0.18467$ the oscillons escape to infinities after fewer number of collisions, see fig. \[fig:F3DR1V018467\]. The final velocity of the escaping oscillons also differs substantially: at $v_\mathrm{in}^{}=0.1847$ we obtain $v_\mathrm{f}^{}\approx 0.10$ (fig. \[fig:F3DR1V01847\]), while at $v_\mathrm{in}^{}=0.18467$ we have $v_\mathrm{f}^{}\approx 0.03$ (fig. \[fig:F3DR1V018467\]), and at $v_\mathrm{in}^{}=0.18470001$ we obtain $v_\mathrm{f}^{}\approx 0.15$ (fig. \[fig:F3DR1V018470001\]). At the initial velocity of the colliding kinks $v_\mathrm{in}^{}=0.18470003$, fig. \[fig:F3DR1V018470003\], the final velocity of the escaping oscillons is $v_\mathrm{f}^{}\approx 0.19$. In the kink-antikink collision at the initial velocity $v_\mathrm{in}^{}=0.18472$, fig. \[fig:F3DR1V018472\], we observe formation of two oscillons, which are moving apart from each other, then approach and collide. After that we have final configuration in the form of the oscillating configuration of the type of bion at the origin. Apparently this final configuration can be viewed as a bound state of two oscillons, which oscillate around each other with small amplitude.
At the initial velocity $v_\mathrm{in}^{}=0.18473$, fig. \[fig:F3DR1V018473\], we observe even more complicated picture. The kinks collide, form a bion, which, in turn, decays into two oscillons. These oscillons escape at some distance and then collide again. After that, for some time we observe the bound state of oscillons — small amplitude oscillations of oscillons around each other. Finally, the oscillons escape at some valuable distance, collide for the last time, and escape to infinities with the final velocities $v_\mathrm{f}^{}\approx 0.07$. The obtained results show that in the DSG kink-antikink scattering we found new phenomenon — formation of the pair of oscillons, which can form a bound state or escape to spatial infinities. Note that similar behaviour has been observed recently in the collisions of kinks of another model with non-polynomial potential [@Bazeia.arXiv.2017.sinh; @Bazeia.arXiv.2017.sinh.conf].
Conclusion {#sec:Conclusion}
==========
We have studied the scattering of kinks of the double sine-Gordon model. Several different parameterizations of this model are known in the literature. We used the so-called $R$-parameterization, in which the potential of the model depends on the positive parameter $R$, see eq. .
The scattering of the DSG kink and antikink looks as follows. There is a critical value of the initial velocity $v_\mathrm{cr}$ such that at $v_\mathrm{in}>v_\mathrm{cr}$ the kinks pass through each other and then escape to infinities. At $v_\mathrm{in}<v_\mathrm{cr}$ one observes formation of a bound state of the kinks — a bion. Besides that, at some narrow intervals of the initial velocity (which are called “escape windows”) from the range $v_\mathrm{in}<v_\mathrm{cr}$ the kinks escape to infinities after two or more collisions.
We have obtained the dependence of the critical velocity $v_\mathrm{cr}$ on the parameter $R$. The curve $v_\mathrm{cr}(R)$ has several well-seen local maxima, see fig. \[fig:critical\_velocity\_vs\_R\] and table \[tab:Table1\]. Note some discrepancy between our results and the results of [@Campbell.dsG.1986]. The authors of [@Campbell.dsG.1986] reported only one maximum of the curve $v_\mathrm{cr}(R)$. This could be a consequence of small number of experimental points between $R=1.8$ and $R=2.4$ presented in [@Campbell.dsG.1986].
Apart from the previously known bions and escape windows, in the range $v_\mathrm{in}<v_\mathrm{cr}$ in our numerical experiments we observed a new phenomenon, which could be classified as formation of a bound state of two oscillons, and their escape in some cases. So at some initial velocities of the colliding kinks, in the final state we observed two oscillons escaping from the collision point. The time between the first kinks impact and the beginning of the oscillons escaping can be rather big. The field evolution during this time is quite complicated. First, we observe formation of a bion. After a short time, this bion evolves into a configuration, which can be identified as a bound state of two oscillons oscillating around each other. The amplitude of these oscillations can vary substantially. After that the oscillons either remain bound or escape to spatial infinities, depending on the initial velocity of the colliding kinks. It is interesting that formation of a bound state of two oscillons, as well as escape of oscillons, has been found recently in the collisions of kinks of the sinh-deformed $\varphi^4$ model [@Bazeia.arXiv.2017.sinh; @Bazeia.arXiv.2017.sinh.conf]. We think that this new phenomenon can be a part of new interesting physics within a wide class of non-linear models.
We can assume that the escape of oscillons is a kind of resonance phenomena, [*i.e.*]{} it is a consequence of the resonant energy exchange between oscillon’s kinetic energy and its internal vibrational degree(s) of freedom. A detailed study of such exchange could be a subject of future work.
In conclusion, we would like to mention several issues that we think could become a subject of future study.
- First, it would be interesting to explain the behaviour of the dependence $v_\mathrm{cr}(R)$ with a series of local maxima. This non-monotonicity could be a consequence of the kink’s shape changing with increasing of the parameter $R$. So at large $R$’s the interaction of the DSG kinks could be reduced to pairwise interaction of the subkinks, which are the sine-Gordon solitons separated by the distance $2R$. Note that the authors of [@Demirkaya.cc.2017] observed the non-monotonic dependence of $v_\mathrm{cr}$ on the model parameter in the parametrically modified $\varphi^6$ model. In order to explain the phenomenon, they applied the collective coordinate approach. We believe that similar analysis could be applied to the double sine-Gordon kink-antikink system.
- Second, the oscillons escape in the final state, as well as formation of a bound state of two oscillons, are new interesting phenomena, which have to be explained qualitatively and probably quantitatively.
- Third, it would be very interesting to study multikink collisions within the DSG model in the spirit of [@Moradi.JHEP.2017]. Due to complex internal structure of the DSG kinks, the multikink collisions could result in a rich variety of new phenomena.
Answers to these questions would substantially improve our understanding of the DSG kinks dynamics.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by the MEPhI Academic Excellence Project (contract No. 02.a03.21.0005, 27.08.2013).
[99]{}
R. Rajaraman, [*Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory*]{}, North-Holland, Amsterdan (1982).
A. Vilenkin and E.P.S. Shellard, [*Cosmic Strings and Other Topological Defects*]{}, Cambridge University Press, Cambridge U.K. (2000).
N. Manton and P. Sutcliffe, [*Topological Solitons*]{}, Cambridge University Press, Cambridge U.K. (2004).
T. Vachaspati, [*Kinks and Domain Walls: An Introduction to Classical and Quantum Solitons*]{}, Cambridge University Press, Cambrifge U.K. (2006).
D. Bazeia, L. Losano and J.R.L. Santos, [*Kinklike structures in scalar field theories: From one-field to two-field models*]{}, [[*Phys. Lett.*]{} [**A 377**]{} (2013) 1615](https://doi.org/10.1016/j.physleta.2013.04.047) \[[arXiv:1304.6904](https://arxiv.org/abs/1304.6904)\].
A. Alonso-Izquierdo, D. Bazeia, L. Losano and J. Mateos Guilarte, [*New Models for Two Real Scalar Fields and Their Kink-Like Solutions*]{}, [[*Adv. High Energy Phys.*]{} [**2013**]{} (2013) 183295](https://doi.org/10.1155/2013/183295) \[[arXiv:1308.2724](https://arxiv.org/abs/1308.2724)\].
H. Katsura, [*Composite-kink solutions of coupled nonlinear wave equations*]{}, [[*Phys. Rev.*]{} [**D 89**]{} (2014) 085019](https://doi.org/10.1103/PhysRevD.89.085019) \[[arXiv:1312.4263](https://arxiv.org/abs/1312.4263)\].
R.A.C. Correa, A. de Souza Dutra and M. Gleiser, [*Information-entropic measure of energy-degenerate kinks in two-field models*]{}, [[*Phys. Lett.*]{} [**B 737**]{} (2014) 388](https://doi.org/10.1016/j.physletb.2014.09.002) \[[arXiv:1409.0029](https://arxiv.org/abs/1409.0029)\].
D. Saadatmand, A. Moradi Marjaneh and M. Heidari, [*Dynamics of coupled field solitons: A collective coordinate approach*]{}, [[*Pramana - J Phys* ]{} [**83**]{} (2014) 505](https://doi.org/10.1007/s12043-014-0797-3).
A. Alonso-Izquierdo, [*Kink dynamics in a system of two coupled scalar fields in two space-time dimensions*]{}, [[*Physica*]{} [**D 365**]{} (2018) 12](https://doi.org/10.1016/j.physd.2017.10.006) \[[arXiv:1711.08784](https://arxiv.org/abs/1711.08784)\].
A. Alonso-Izquierdo, [*Reflection, transmutation, annihilation and resonance in two-component kink collisions*]{}, [[*Phys. Rev.*]{} [**D 97**]{} (2018) 045016](https://doi.org/10.1103/PhysRevD.97.045016) \[[arXiv:1711.10034](https://arxiv.org/abs/1711.10034)\].
V.A. Lensky, V.A. Gani and A.E. Kudryavtsev, [*Domain walls carrying a U(1) charge*]{}, [[*Sov. Phys. JETP*]{} [**93**]{} (2001) 677](https://doi.org/10.1134/1.1420436) \[[hep-th/0104266](https://arxiv.org/abs/hep-th/0104266)\].
V.A. Lensky, V.A. Gani and A.E. Kudryavtsev, [*Domain walls carrying a U(1) charge*]{}, [*Zh. Eksp. Teor. Fiz.*]{} [**120**]{} (2001) 778 \[[hep-th/0104266](https://arxiv.org/abs/hep-th/0104266)\].
V.A. Gani, N.B. Konyukhova, S.V. Kurochkin and V.A. Lensky, [*Study of stability of a charged topological soliton in the system of two interacting scalar fields*]{}, [*Comput. Math. Math. Phys.*]{} [**44**]{} (2004) 1968 \[[arXiv: 0710.2975](https://arxiv.org/abs/0710.2975)\].
V.A. Gani, N.B. Konyukhova, S.V. Kurochkin and V.A. Lensky, [*Study of stability of a charged topological soliton in the system of two interacting scalar fields*]{}, [*Zh. Vychisl. Mat. Mat. Fiz.*]{} [**44**]{} (2004) 2069 \[[arXiv: 0710.2975](https://arxiv.org/abs/0710.2975)\].
D. Bazeia, A.S. Lobão Jr., L. Losano and R. Menezes, [*First-order formalism for twinlike models with several real scalar fields*]{}, [[*Eur. Phys. J.*]{} [**C 74**]{} (2014) 2755](https://doi.org/10.1140/epjc/s10052-014-2755-0) \[[arXiv:1312.1198](https://arxiv.org/abs/1312.1198)\].
V.A. Gani, M.A. Lizunova and R.V. Radomskiy [*Scalar triplet on a domain wall: an exact solution*]{}, [[*JHEP*]{} [**04**]{} (2016) 043](https://doi.org/10.1007/JHEP04(2016)043) \[[arXiv:1601.07954](https://arxiv.org/abs/1601.07954)\].
V.A. Gani, M.A. Lizunova and R.V. Radomskiy, [*Scalar triplet on a domain wall*]{}, [[*J. Phys.: Conf. Ser.*]{} [**675**]{} (2016) 012020](https://doi.org/10.1088/1742-6596/675/1/012020) \[[arXiv:1602.04446](https://arxiv.org/abs/1602.04446)\].
S. Akula, C. Balázs and G.A. White, [*Semi-analytic techniques for calculating bubble wall profiles*]{}, [[*Eur. Phys. J.*]{} [**C 76**]{} (2016) 681](https://doi.org/10.1140/epjc/s10052-016-4519-5) \[[arXiv:1608.00008](https://arxiv.org/abs/1608.00008)\].
J. Ashcroft et al., [*Head butting sheep: kink collisions in the presence of false vacua*]{}, [[*J. Phys. A: Math. Theor.*]{} [**49**]{} (2016) 365203](https://doi.org/10.1088/1751-8113/49/36/365203) \[[arXiv:1604.08413](https://arxiv.org/abs/1604.08413)\].
T.I. Belova and A.E. Kudryavtsev, [*Solitons and their interactions in classical field theory*]{}, [[*Phys. Usp.*]{} [**40**]{} (1997) 359](https://doi.org/10.1070/PU1997v040n04ABEH000227).
T.I. Belova and A.E. Kudryavtsev, [*Solitons and their interactions in classical field theory*]{}, [[*Usp. Fiz. Nauk*]{} [**167**]{} (1997) 377](https://doi.org/10.3367/UFNr.0167.199704b.0377).
V.A. Gani, A.E. Kudryavtsev and M.A. Lizunova, [*Kink interactions in the (1+1)-dimensional $\varphi^6$ model*]{}, [[*Phys. Rev.*]{} [**D 89**]{} (2014) 125009](https://doi.org/10.1103/PhysRevD.89.125009) \[[arXiv:1402.5903](https://arxiv.org/abs/1402.5903)\].
H. Weigel, [*Kink-Antikink Scattering in $\varphi^4$ and $\phi^6$ Models*]{}, [[*J. Phys.: Conf. Ser.*]{} [**482**]{} (2014) 012045](https://doi.org/10.1088/1742-6596/482/1/012045) \[[arXiv:1309.6607](https://arxiv.org/abs/1309.6607)\].
I. Takyi and H. Weigel, [*Collective coordinates in one-dimensional soliton models revisited*]{}, [[*Phys. Rev.*]{} [**D 94**]{} (2016) 085008](https://doi.org/10.1103/PhysRevD.94.085008) \[[arXiv:1609.06833](https://arxiv.org/abs/1609.06833)\].
A. Demirkaya et al., [*Kink Dynamics in a Parametric $\phi^6$ System: A Model With Controllably Many Internal Modes*]{}, [[*JHEP*]{} [**12**]{} (2017) 071](https://doi.org/10.1007/JHEP12(2017)071) \[[arXiv:1706.01193](https://arxiv.org/abs/1706.01193)\].
H.E. Baron, G. Luchini and W.J. Zakrzewski, [*Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model*]{}, [[*J. Phys. A: Math. and Theor.*]{} [**47**]{} (2014) 265201](https://doi.org/10.1088/1751-8113/47/26/265201) \[[arXiv:1308.4072](https://arxiv.org/abs/1308.4072)\].
K. Javidan, [*Collective coordinate variable for soliton-potential system in sine-Gordon model*]{}, [[*J. Math. Phys.*]{} [**51**]{} (2010) 112902](https://doi.org/10.1063/1.3511337) \[[arXiv:0910.3058](https://arxiv.org/abs/0910.3058)\].
I. Christov and C.I. Christov, [*Physical dynamics of quasi-particles in nonlinear wave equations*]{}, [[*Phys. Lett.*]{} [**A 372**]{} (2008) 841](https://doi.org/10.1016/j.physleta.2007.08.038) \[[nlin/0612005](https://arxiv.org/abs/nlin/0612005)\].
V.A. Gani and A.E. Kudryavtsev, [*Collisions of domain walls in a supersymmetric model*]{}, [[*Phys. Atom. Nucl.*]{} [**64**]{} (2001) 2043](https://doi.org/10.1134/1.1423755) \[[hep-th/9904209](https://arxiv.org/abs/hep-th/9904209), [hep-th/9912211](https://arxiv.org/abs/hep-th/9912211)\].
V.A. Gani and A.E. Kudryavtsev, [*Collisions of domain walls in a supersymmetric model*]{}, [*Yad. Fiz.*]{} [**64**]{} (2001) 2130 \[[hep-th/9904209](https://arxiv.org/abs/hep-th/9904209), [hep-th/9912211](https://arxiv.org/abs/hep-th/9912211)\].
J.K. Perring and T.H.R. Skyrme, [*A model unified field equation*]{}, [[*Nucl. Phys*]{} [**31**]{} (1962) 550](https://doi.org/10.1016/0029-5582(62)90774-5).
R. Rajaraman, [*Intersoliton forces in weak-coupling quantum field theories*]{}, [[*Phys. Rev.*]{} [**D 15**]{} (1977) 2866](https://doi.org/10.1103/PhysRevD.15.2866).
N.S. Manton, [*An effective Lagrangian for solitons*]{}, [[*Nucl. Phys.*]{} [**B 150**]{} (1979) 397](https://doi.org/10.1016/0550-3213(79)90309-2).
P.G. Kevrekidis, A. Khare and A. Saxena, [*Solitary wave interactions in dispersive equations using Manton’s approach*]{}, [[*Phys. Rev.*]{} [**E 70**]{} (2004) 057603](https://doi.org/10.1103/PhysRevE.70.057603) \[[nlin/0410045](https://arxiv.org/abs/nlin/0410045)\].
V.A. Gani and A.E. Kudryavtsev, [*Kink-antikink interactions in the double sine-Gordon equation and the problem of resonance frequencies*]{}, [[*Phys. Rev.*]{} [**E 60**]{} (1999) 3305](https://doi.org/10.1103/PhysRevE.60.3305) \[[cond-mat/9809015](https://arxiv.org/abs/cond-mat/9809015)\].
P. Dorey, K. Mersh, T. Romanczukiewicz and Y. Shnir, [*Kink-Antikink Collisions in the $\phi^6$ Model*]{}, [[*Phys. Rev. Lett.*]{} [**107**]{} (2011) 091602](https://doi.org/10.1103/PhysRevLett.107.091602) \[[arXiv:1101.5951](https://arxiv.org/abs/1101.5951)\].
V.A. Gani, V. Lensky and M.A. Lizunova, [*Kink excitation spectra in the (1+1)-dimensional $\varphi^8$ model*]{}, [[*JHEP*]{} [**08**]{} (2015) 147](https://doi.org/10.1007/JHEP08(2015)147) \[[arXiv:1506.02313](https://arxiv.org/abs/1506.02313)\].
V.A. Gani, V. Lensky, M.A. Lizunova and E.V. Mrozovskaya, [*Excitation spectra of solitary waves in scalar field models with polynomial self-interaction*]{}, [[*J. Phys.: Conf. Ser.*]{} [**675**]{} (2016) 012019](https://doi.org/10.1088/1742-6596/675/1/012019) \[[arXiv:1602.02636](https://arxiv.org/abs/1602.02636)\].
R.V. Radomskiy, E.V. Mrozovskaya, V.A. Gani and I.C. Christov, [*Topological defects with power-law tails*]{}, [[*J. Phys.: Conf. Ser.*]{} [**798**]{} (2017) 012087](https://doi.org/10.1088/1742-6596/798/1/012087) \[[arXiv:1611.05634](https://arxiv.org/abs/1611.05634)\].
A. Moradi Marjaneh, D. Saadatmand, Kun Zhou, S.V. Dmitriev and M.E. Zomorrodian, [*High energy density in the collision of $N$ kinks in the $\phi^4$ model*]{}, [[*Commun. Nonlinear Sci. Numer. Simulat.*]{} [**49**]{} (2017) 30](https://doi.org/10.1016/j.cnsns.2017.01.022) \[[arXiv:1605.09767](https://arxiv.org/abs/1605.09767)\].
A. Moradi Marjaneh, V.A. Gani, D. Saadatmand, S.V. Dmitriev and K. Javidan, [*Multi-kink collisions in the $\phi^6$ model*]{}, [[*JHEP*]{} [**07**]{} (2017) 028](https://doi.org/10.1007/JHEP07(2017)028) \[[arXiv:1704.08353](https://arxiv.org/abs/1704.08353)\].
A. Moradi Marjaneh, A. Askari, D. Saadatmand and S.V. Dmitriev, [*Extreme values of elastic strain and energy in sine-Gordon multi-kink collisions*]{}, [[*Eur. Phys. J*]{} [**B 91**]{} (2018) 22](https://doi.org/10.1140/epjb/e2017-80406-y) \[[arXiv:1710.10159](https://arxiv.org/abs/1710.10159)\].
E. Belendryasova and V.A. Gani, [*Scattering of the $\varphi^8$ kinks with power-law asymptotics*]{}, [arXiv:1708.00403](https://arxiv.org/abs/1708.00403).
E. Belendryasova and V.A. Gani, [*Resonance phenomena in the $\varphi^8$ kinks scattering*]{}, [[*J. Phys.: Conf. Ser.*]{} [**934**]{} (2017) 012059](https://doi.org/10.1088/1742-6596/934/1/012059) \[[arXiv:1712.02846](https://arxiv.org/abs/1712.02846)\].
D. Bazeia, E. Belendryasova and V.A. Gani, [*Scattering of kinks of the sinh-deformed $\varphi^4$ model*]{}, [[*Eur. Phys. J.*]{} [**C 78**]{} (2018) 340](https://doi.org/10.1140/epjc/s10052-018-5815-z) \[[arXiv:1710.04993](https://arxiv.org/abs/1710.04993)\].
D. Bazeia, E. Belendryasova and V.A. Gani, [*Scattering of kinks in a non-polynomial model*]{}, [[*J. Phys.: Conf. Ser.*]{} [**934**]{} (2017) 012032](https://doi.org/10.1088/1742-6596/934/1/012032) \[[arXiv:1711.07788](https://arxiv.org/abs/1711.07788)\].
D. Saadatmand, S.V. Dmitriev and P.G. Kevrekidis, [*High energy density in multisoliton collisions*]{}, [[*Phys. Rev.*]{} [**D 92**]{} (2015) 056005](https://doi.org/10.1103/PhysRevD.92.056005) \[[arXiv:1506.01389](https://arxiv.org/abs/1506.01389)\].
M.A. Lohe, [*Soliton structures in $P(\varphi)_2$*]{}, [[*Phys. Rev.*]{} [**D 20**]{} (1979) 3120](https://doi.org/10.1103/PhysRevD.20.3120).
A. Khare, I.C. Christov and A. Saxena, [*Successive phase transitions and kink solutions in $\phi^8$, $\phi^{10}$, and $\phi^{12}$ field theories*]{}, [[*Phys. Rev.*]{} [**E 90**]{} (2014) 023208](https://doi.org/10.1103/PhysRevE.90.023208) \[[arXiv:1402.6766](https://arxiv.org/abs/1402.6766)\].
H. Weigel, [*Emerging Translational Variance: Vacuum Polarization Energy of the $\varphi^6$ kink*]{}, [[*Adv. High Energy Phys.*]{} [**2017**]{} (2017) 1486912](https://doi.org/10.1155/2017/1486912) \[[arXiv:1706.02657](https://arxiv.org/abs/1706.02657)\].
H. Weigel, [*Vacuum polarization energy for general backgrounds in one space dimension*]{}, [[*Phys. Lett.*]{} [**B 766**]{} (2017) 65](https://doi.org/10.1016/j.physletb.2016.12.055) \[[arXiv:1612.08641](https://arxiv.org/abs/1612.08641)\].
P. Dorey et al., [*Boundary scattering in the $\phi^4$ model*]{}, [[*JHEP*]{} [**05**]{} (2017) 107](https://doi.org/10.1007/JHEP05(2017)107) \[[arXiv:1508.02329](https://arxiv.org/abs/1508.02329)\].
D. Bazeia, M.A. González León, L. Losano and J. Mateos Guilarte, [*Deformed defects for scalar fields with polynomial interactions*]{}, [[*Phys. Rev.*]{} [**D 73**]{} (2006) 105008](https://doi.org/10.1103/PhysRevD.73.105008) \[[hep-th/0605127](https://arxiv.org/abs/hep-th/0605127)\].
S. He, Y. Jiang and J. Liu, [*Toda chain from the kink-antikink lattice*]{}, [arXiv:1605.06867](https://arxiv.org/abs/1605.06867).
S. Snelson, [*Asymptotic stability for odd perturbations of the the stationary kink in the variable-speed $\varphi^4$ model*]{}, [arXiv:1603.07344](https://arxiv.org/abs/1603.07344).
L.E. Guerrero, E.López-Atencio and J.A. González, [*Long-range self-affine correlations in a random soliton gas*]{}, [[*Phys. Rev.*]{} [**E 55**]{} (1997) 7691](https://doi.org/10.1103/PhysRevE.55.7691).
B.A. Mello, J.A. González, L.E. Guerrero and E. López-Atencio, [*Topological defects with long-range interactions*]{}, [[*Phys. Lett.*]{} [**A 244**]{} (1998) 277](https://doi.org/10.1016/S0375-9601(98)00213-8).
L.E. Guerrero and J.A. González, [*Long-range interacting solitons: pattern formation and nonextensive thermostatistics*]{}, [[*Physica*]{} [**A 257**]{} (1998) 390](https://doi.org/10.1016/S0378-4371(98)00165-4) \[[patt-sol/9905010](https://arxiv.org/abs/patt-sol/9905010)\].
A.R. Gomes, R. Menezes and J.C.R.E. Oliveira, [*Highly interactive kink solutions*]{}, [[*Phys. Rev.*]{} [**D 86**]{} (2012) 025008](https://doi.org/10.1103/PhysRevD.86.025008) \[[arXiv:1208.4747](https://arxiv.org/abs/1208.4747)\].
M. Peyrard and D.K. Campbell, [*Kink-antikink interactions in a modified sine-Gordon model*]{}, [[*Physica*]{} [**D 9**]{} (1983) 33](https://doi.org/10.1016/0167-2789(83)90290-7).
G. Delfino and G. Mussardo, [*Non-integrable aspects of the multi-frequency sine-Gordon model*]{}, [[*Nucl. Phys.*]{} [**B 516**]{} (1998) 675](https://doi.org/10.1016/S0550-3213(98)00063-7) \[[hep-th/9709028](https://arxiv.org/abs/hep-th/9709028)\].
D.K. Campbell and M. Peyrard, [*Solitary wave collisions revisited*]{}, [[*Physica*]{} [**D 18**]{} (1986) 47](https://doi.org/10.1016/0167-2789(86)90161-2).
D.K. Campbell, M. Peyrard and P. Sodano, [*Kink-antikink interactions in the double sine-Gordon equation*]{}, [[*Physica*]{} [**D 19**]{} (1986) 165](https://doi.org/10.1016/0167-2789(86)90019-9).
Yu.S. Kivshar and B.A. Malomed, [*Radiative and inelastic effects in dynamics of double sine-Gordon solitons*]{}, [[*Phys. Lett.*]{} [**A 122**]{} (1987) 245](https://doi.org/10.1016/0375-9601(87)90815-2).
B.A. Malomed, [*Dynamics and kinetics of solitons in the driven damped double Sine-Gordon equation*]{}, [[*Phys. Lett.*]{} [**A 136**]{} (1989) 395](https://doi.org/10.1016/0375-9601(89)90422-2).
D. Bazeia, L. Losano and J.M.C. Malbouisson, [*Deformed defects*]{}, [[*Phys. Rev.*]{} [**D 66**]{} (2002) 101701](https://doi.org/10.1103/PhysRevD.66.101701) \[[hep-th/0209027](https://arxiv.org/abs/hep-th/0209027)\].
C.A. Almeida, D. Bazeia, L. Losano and J.M.C. Malbouisson, [*New results for deformed defects*]{}, [[*Phys. Rev.*]{} [**D 69**]{} (2004) 067702](https://doi.org/10.1103/PhysRevD.69.067702) \[[hep-th/0405238](https://arxiv.org/abs/hep-th/0405238)\].
L. Campanelli, P. Cea, G.L. Fogli and L. Tedesco, [*Charged domain walls*]{}, [[*Int. J. Mod. Phys.*]{} [**D 13**]{} (2004) 65](https://doi.org/10.1142/S021827180400369X) \[[astro-ph/0307211](https://arxiv.org/abs/astro-ph/0307211)\].
V.A. Gani, V.G. Ksenzov and A.E. Kudryavtsev, [*Example of a self-consistent solution for a fermion on domain wall*]{}, [[*Phys. Atom. Nucl.*]{} [**73**]{} (2010) 1889](https://doi.org/10.1134/S1063778810110104) \[[arXiv:1001.3305](https://arxiv.org/abs/1001.3305)\].
V.A. Gani, V.G. Ksenzov and A.E. Kudryavtsev, [*Example of a self-consistent solution for a fermion on domain wall*]{}, [*Yad. Fiz.*]{} [**73**]{} (2010) 1940 \[[arXiv:1001.3305](https://arxiv.org/abs/1001.3305)\].
V.A. Gani, V.G. Ksenzov and A.E. Kudryavtsev, [*Stable branches of a solution for a fermion on domain wall*]{}, [[*Phys. Atom. Nucl.*]{} [**74**]{} (2011) 771](https://doi.org/10.1134/S1063778811050085) \[[arXiv:1009.4370](https://arxiv.org/abs/1009.4370)\].
V.A. Gani, V.G. Ksenzov and A.E. Kudryavtsev, [*Stable branches of a solution for a fermion on domain wall*]{}, [*Yad. Fiz.*]{} [**74**]{} (2011) 797 \[[arXiv:1009.4370](https://arxiv.org/abs/1009.4370)\].
V.A. Gani et al, [*On decay of bubble of disoriented chiral condensate*]{}, [*Phys. Atom. Nucl.*]{} [**62**]{} (1999) 895 \[[hep-ph/9712526](https://arxiv.org/abs/hep-ph/9712526)\].
V.A. Gani et al, [*On decay of bubble of disoriented chiral condensate*]{}, [*Yad. Fiz.*]{} [**62**]{} (1999) 956 \[[hep-ph/9712526](https://arxiv.org/abs/hep-ph/9712526)\].
T.I. Belova, V.A. Gani and A.E. Kudyavtsev, [*Decay of a large-amplitude bubble of a disoriented chiral condensate*]{}, [[*Phys. Atom. Nucl.*]{} [**64**]{} (2001) 140](https://doi.org/10.1134/1.1344952) \[[hep-ph/0003308](https://arxiv.org/abs/hep-ph/0003308)\].
T.I. Belova, V.A. Gani and A.E. Kudyavtsev, [*Decay of a large-amplitude bubble of a disoriented chiral condensate*]{}, [*Yad. Fiz.*]{} [**64**]{} (2001) 143 \[[hep-ph/0003308](https://arxiv.org/abs/hep-ph/0003308)\].
M. Nitta, [*Josephson vortices and the Atiyah-Manton construction*]{}, [[*Phys. Rev.*]{} [**D 86**]{} (2012) 125004](https://doi.org/10.1103/PhysRevD.86.125004) \[[arXiv:1207.6958](https://arxiv.org/abs/1207.6958)\].
M. Nitta, [*Correspondence between Skyrmions in 2+1 and 3+1 dimensions,*]{} [[*Phys. Rev.*]{} [**D 87**]{} (2013) 025013](https://doi.org/10.1103/PhysRevD.87.025013) \[[arXiv:1210.2233](https://arxiv.org/abs/1210.2233)\].
M. Nitta, [*Matryoshka Skyrmions*]{}, [[*Nucl. Phys.*]{} [**B 872**]{} (2013) 62](https://doi.org/10.1016/j.nuclphysb.2013.03.003) \[[arXiv:1211.4916](https://arxiv.org/abs/1211.4916)\].
M. Kobayashi and M. Nitta, [*Sine-Gordon kinks on a domain wall ring*]{}, [[*Phys. Rev.*]{} [**D 87**]{} (2013) 085003](https://doi.org/10.1103/PhysRevD.87.085003) \[[arXiv:1302.0989](https://arxiv.org/abs/1302.0989)\].
P. Jennings and P. Sutcliffe, [*The dynamics of domain wall Skyrmions*]{}, [[*J. Phys.*]{} [**A 46**]{} (2013) 465401](https://doi.org/10.1088/1751-8113/46/46/465401) \[[arXiv:1305.2869](https://arxiv.org/abs/1305.2869)\].
S.B. Gudnason and M. Nitta, [*Domain wall Skyrmions*]{}, [[*Phys. Rev.*]{} [**D 89**]{} (2014) 085022](https://doi.org/10.1103/PhysRevD.89.085022) \[[arXiv:1403.1245](https://arxiv.org/abs/1403.1245)\].
N. Blyankinshtein, [*Q-lumps on a domain wall with a spin-orbit interaction*]{}, [[*Phys. Rev.*]{} [**D 93**]{} (2016) 065030](https://doi.org/10.1103/PhysRevD.93.065030) \[[arXiv:1510.07935](https://arxiv.org/abs/1510.07935)\].
A.Yu. Loginov, [*Q kink of the nonlinear O(3) $\sigma$ model involving an explicitly broken symmetry*]{}, [[*Phys. Atom. Nucl.*]{} [**74**]{} (2011) 740](https://doi.org/10.1134/S1063778811040107).
A.Yu. Loginov, [*Q kink of the nonlinear O(3) $\sigma$ model involving an explicitly broken symmetry*]{}, [*Yad. Fiz.*]{} [**74**]{} (2011) 766.
D. Bazeia, A. Mohammadi, [*Fermionic bound states in distinct kinklike backgrounds*]{}, [[*Eur. Phys. J.*]{} [**C 77**]{} (2017) 203](https://doi.org/10.1140/epjc/s10052-017-4778-9) \[[arXiv:1702.00891](https://arxiv.org/abs/1702.00891)\].
D. Bazeia, A. Mohammadi and D.C. Moreira, [*Fermion bound states in geometrically deformed backgrounds*]{}, [arXiv:1706.04406](https://arxiv.org/abs/1706.04406).
M. Mai and P. Schweitzer, [*Energy momentum tensor, and the D-term of Q-balls*]{}, [[*Phys. Rev.*]{} [**D 86**]{} (2012) 076001](https://doi.org/10.1103/PhysRevD.86.076001) \[[arXiv:1206.2632](https://arxiv.org/abs/1206.2632)\].
M. Mai and P. Schweitzer, [*Radial excitations of Q-balls, and their D-term*]{}, [[*Phys. Rev.*]{} [**D 86**]{} (2012) 096002](https://doi.org/10.1103/PhysRevD.86.096002) \[[arXiv:1206.2930](https://arxiv.org/abs/1206.2930)\].
M. Cantara, M. Mai and P. Schweitzer, [*The energy-momentum tensor and D-term of Q-clouds*]{}, [[*Nucl. Phys.*]{} [**A 953**]{} (2016) 1](https://doi.org/10.1016/j.nuclphysa.2016.04.032) \[[arXiv:1510.08015](https://arxiv.org/abs/1510.08015)\].
I.E. Gulamov, E.Ya. Nugaev and M.N. Smolyakov, [*Analytic Q-ball solutions and their stability in a piecewise parabolic potential*]{}, [[*Phys. Rev.*]{} [**D 87**]{} (2013) 085043](https://doi.org/10.1103/PhysRevD.87.085043) \[[arXiv:1303.1173](https://arxiv.org/abs/1303.1173)\].
D. Bazeia, M.A. Marques and R. Menezes, [*Exact solutions, energy and charge of stable Q-balls*]{}, [[*Eur. Phys. J.*]{} [**C 76**]{} (2016) 241](https://doi.org/10.1140/epjc/s10052-016-4059-z) \[[arXiv:1512.04279](https://arxiv.org/abs/1512.04279)\].
D. Bazeia et al, [*Compact Q-balls*]{}, [[*Phys. Lett.*]{} [**B 758**]{} (2016) 146](https://doi.org/10.1016/j.physletb.2016.04.060) \[[arXiv:1604.08871](https://arxiv.org/abs/1604.08871)\].
D. Bazeia, L. Losano, M.A. Marques and R. Menezes, [*Split Q-balls*]{}, [[*Phys. Lett.*]{} [**B 765**]{} (2017) 359](https://doi.org/10.1016/j.physletb.2016.12.033) \[[arXiv:1612.04442](https://arxiv.org/abs/1612.04442)\].
V. Dzhunushaliev, A. Makhmudov and K.G. Zloshchastiev, [*Singularity-free model of electrically charged fermionic particles and gauged Q-balls*]{}, [[*Phys. Rev.*]{} [**D 94**]{} (2016) 096012](https://doi.org/10.1103/PhysRevD.94.096012) \[[arXiv:1611.02105](https://arxiv.org/abs/1611.02105)\].
V.A. Gani, A.A. Kirillov and S.G. Rubin, [*Classical transitions with the topological number changing in the early Universe*]{}, [[*JCAP*]{} [**04**]{} (2018) 042](https://doi.org/10.1088/1475-7516/2018/04/042) \[[arXiv:1704.03688](https://arxiv.org/abs/1704.03688)\].
V.A. Gani, A.A. Kirillov and S.G. Rubin, [*Transitions between topologically non-trivial configurations*]{}, [[*J. Phys.: Conf. Ser.*]{} [**934**]{} (2017) 012046](https://doi.org/10.1088/1742-6596/934/1/012046) \[[arXiv:1711.07700](https://arxiv.org/abs/1711.07700)\].
E.B. Bogomolny, [*Stability of Classical Solutions*]{}, [[*Sov. J. Nucl. Phys.*]{} [**24**]{} (1976) 449](http://inspirehep.net/record/101280).
E.B. Bogomolny, [*Stability of Classical Solutions*]{}, [[*Yad. Fiz.*]{} [**24**]{} (1976) 861](http://inspirehep.net/record/101280).
M.K. Prasad and C.M. Sommerfield, [*Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon*]{}, [[*Phys. Rev. Lett.*]{} [**35**]{} (1975) 760](https://doi.org/10.1103/PhysRevLett.35.760).
|
---
abstract: 'We study properties of a random walk in a generalized Sinai model (SM), in which a quenched random potential is a trajectory of a *fractional* Brownian motion with arbitrary Hurst parameter $H$, $0< H <1$, so that the random force field displays strong spatial correlations. In this case, the disorder-average mean-square displacement (MSD) grows in proportion to $\log^{2/H} (n)$, $n$ being time. We prove that moments of arbitrary order $k$ of the steady-state current $J_L$ through a finite segment of length $L$ of such a chain decay as $L^{-(1-H)}$, independently of $k$, which suggests that despite a logarithmic confinement the average current is much higher than its Fickian counterpart in homogeneous systems. Our results reveal a paradoxical behavior such that, for fixed $n$ and $L$, the MSD *decreases* when one varies $H$ from $0$ to $1$, while the average current *increases*. This counter-intuitive behavior is explained via an analysis of representative realizations of disorder.'
author:
- Gleb Oshanin
- Alberto Rosso
- Grégory Schehr
title: 'Anomalous fluctuations of currents in Sinai-type random chains with strongly correlated disorder'
---
Since the pioneering works [@kesten; @derrida; @sinai], random walks (RWs) in random media attracted a considerable attention. In part, due to a general interest in dynamics in disordered systems, but also because such RWs found many physical applications, including dynamics of the helix/coil phases boundary in a random heteropolymer [@pgg; @redner], a random-field Ising model [@bruinsma; @nattermann], dislocations in disordered crystals [@harth], mechanical unzipping of DNA [@walter], translocation of biomolecules through nanopores [@9] and molecular motors [@10]. Some functionals arising here, e.g., probability currents in finite samples, show up in mathematical finance [@alain; @greg]. Other examples can be found in [@comtet; @hughes; @sheinman].
In the discrete formulation, a RW evolves in a discrete time on a lattice. At each time step the walker jumps from site $X$ to either site $X + 1$ with the site-dependent probability $p_X = \frac{1}{2} (1 + \varepsilon \cdot s_X)$, or to the site $X - 1$ with the probability $q_X = 1-p_X$, where the amplitude $0 < \varepsilon < 1$ measures the strength of the disorder and $s_X$ are quenched, independent and identically distributed (i.i.d.) random variables (r.v.). One often assumes binomial r.v., i.e., $s_X = \pm 1$ with probabilities $p$ and $1-p$, respectively.
In case of no global bias ($p=1/2$), i.e. for the so-called Sinai model (SM), a remarkable result [@sinai] is that for a given environment $\{p_X\}$ the squared displacement $$\label{sinai}
X_n^2 \sim m(\{p_X\}) \, \ln^4(n) \,,$$ as $n \to \infty$ with probability almost $1$, where $m(\{p_X\})$ is a function of the environment only [@note2]. Another intriguing feature of the SM concerns transport properties. It was revealed by analysing the probability current $J_L$ through a finite Sinai chain of length $L$, that the disorder-average current decays as $1/\sqrt{L}$ [@gleb; @gleb1; @gleb2; @cecile]. Curiously enough, despite a logarithmic confinement (\[sinai\]), the disorder-average current appears to be anomalously high, so that such disordered chains offer on average less resistance with respect to transport of particles than homogeneous chains (all $p_X \equiv 1/2$) for which one finds Fick’s law $J_L \sim 1/L$. In absence of disorder, deviations from Fick’s law can also be found for Lévy walks [@dhar_derrida]. Full statistics of the current has been recently computed for ASEP model [@kirone].
It is well-known that a RW in uncorrelated random environment $\{p_X\}$ can be considered as a one in presence of a random potential $V_L$, which represents itself a RW in space. Indeed, on scales $L$ a RW “explores” the potential $$\label{walk}
V_L = \sum_{X = 1}^{L - 1} \ln\left(\frac{p_X}{q_X}\right) = \sigma \sum_{X = 0}^{L - 1} s_X \,, \sigma = \ln\left(\frac{1 + \varepsilon}{1 - \varepsilon}\right) \,,$$ which is just a RW trajectory with step length $\sigma$. Standard SM, in which the $s_X$’s are uncorrelated is now well understood. On the contrary, there hardly exist analytical results for the case where these r.v. are strongly correlated. Such correlations are important, e.g., for the dynamics of the helix-coil boundary in random heteropolymers, where the chemical units are usually strongly correlated [@gros3]. They are also currently studied in mathematical finance, improving the standard Black-Sholes-Merton (BSM) model [@fBm_finance]. Any exact result for such situations would thus be welcome.
In this Letter, we study properties of random walks in random environments in which the transition probabilities $\{p_X\}$ are strongly correlated so that the potential $V_L$ in (\[walk\]) is a fractional Brownian motion (fBm): $V_L$ is Gaussian, with $V_{L=0}=0$ and moments $$\label{def}
\mathbb{E}\left\{ V_L \right\} = 0 \,, \,\, \mathbb{E}\left\{ (V_L - V_{L'})^2 \right\} = \sigma^{2} |L - L'|^{2 H}\,,$$ where $\mathbb{E}\{\ldots\}$ here and henceforth denotes averaging over realizations of $V_L$ and $0 < H < 1$. The case $H = 1/2$ corresponds to the original SM. For $H < 1/2$ the potential is subdiffusive while for $H > 1/2 $ it is superdiffusive.
The mean squared displacement $\mathbb{E}\{X_n^2\}$ in a correlated random environment can be estimated as follows. Assuming Arrhenius’ law for the activated dynamics [@comtet], the time $n_L$ required for a particle to diffuse in a disordered potential $V_L$ over a scale $L$, is of order $n_L \sim e^{V_L^*}$, where $V_L^*$ is a typical energy barrier. For $V_L$ in (\[def\]), $V_L^* \sim \sigma L^H$, so that for sufficiently large times $n$ $$\begin{aligned}
\label{1}
\mathbb{E}\left\{X_n^2\right\} \sim \sigma^{-2/H} \ln^{2/H}(n) \,.\end{aligned}$$ Our focal interest here is in understanding the behavior of the disorder-average current $J_L$ through a finite sample (of length $L$) of such a disordered chain, of its moments of arbitrary order, and eventually, of the full probability density function (pdf) of $J_L$. We proceed to show that, while its typical value is exponentially small $J_{{\rm typ}} \sim \exp{(-L^H)}$, all its moments decay algebraically $$\label{2}
\mu_k(L) \equiv \mathbb{E}\left\{ \left({J_L}\right)^k \right\} \sim A_k {L}^{-\theta}\;, \; L \gg 1 \,,$$ where $\theta = 1 - H$ is the persistence exponent of the fBm [@krug; @molchan]. Recall that the persistence exponent associated to a stochastic process characterizes the algebraic decay of its survival probability $S(n)\sim n^{-\theta}$ [@majumdar; @aurzada_simon]. The $L$-independent constants $A_k$ depend in general, on the microscopic details (as the lattice discretization).
The result in (\[2\]) is rather astonishing: a) it states that $\mu_k(L)$ for arbitrary order $k$ decay in the same way. b) for arbitrary $H$, $0 < H < 1$, the disorder-average current in such random chains is larger than the Fickian current in homogeneous systems and c) on comparing (\[1\]) and (\[2\]) for fixed $n$ and $L$ sufficiently large, and varying $H$, one concludes that $\mathbb{E}\{X_n^2\}$ *increases* when $H$ goes from $1$ to $0$, while the disorder-average current *decreases*, which is an absolutely counter-intuitive and surprising behavior.
In what follows we prove (\[2\]) and explain this astonishing behavior using three complementary approaches: (i) a rigorous one, based on exact bounds, for the discrete RW in a fBm potential, (ii) scaling arguments for the continuous-space and -time version, which also allows to study the whole pdf of $J_L$ and (iii) via numerical simulations. We argue that (\[2\]) holds for any potential $V_L$, which is the trajectory of a stochastic process with persistence exponent $\theta$: as a matter of fact, such a behavior of $\mu_k(L)$ is dominated by the configurations of $V_L$ which drift to $-\infty$ without re-crossing the origin and occur with a probability $\sim L^{-\theta}$, yielding the $L$-dependence in (\[2\]).
![(Color online) $\mathbb{E}\left\{ J_L \right\} $ (squares) and $\mu_k(L)$ with $k=2$ (circles) and $k=3$ (triangles) vs $L$ for the fBm with $H =0.75$, $0.4$ and $0.25$ (from top to bottom). The solid line is $L^{-\theta}$ (\[2\]) with $\theta=1-H$. The temperature $T=0.25$ and averaging is performed over $10^5$ samples. We use arbitrary units \[a.u.\] because we vertically shift the data (by a factor $20$ for $H=3/4$, $5$ for $H=0.4$ and $1$ for $H=0.25$). In any case, the prefactors $A_k$ are non-universal and model dependent.[]{data-label="fig_moments"}](moments.eps){width="\linewidth"}
Consider first the discrete chain, take a finite segment of length $L$ and impose fixed concentrations of particles at the endpoints, $P_0$ and $P_L$. For a fixed environment $\{p_X\}$, the steady-state current is given by [@gleb; @gleb1] $$\begin{aligned}
\label{current}
J_L = \frac{D_0 P_0}{\tau_L} - \frac{D_0 P_L}{\tau^*_L},\end{aligned}$$ where $D_0 = 1/2$ is the diffusion coefficient of a homogeneous chain, $\tau_L$ is the so-called Kesten variable [@kesten2]: $$\begin{aligned}
\label{tau}
\tau_L = 1 + \frac{p_1 }{q_1} + \frac{p_1 p_2}{q_1 q_2} + \ldots + \frac{p_1 p_2 \ldots p_{L - 1}}{q_1 q_2 \ldots q_{L - 1}} \,,\end{aligned}$$ and $\tau_L^*$ is obtained from (\[tau\]) by replacements $p_k \to q_{L-k}$ and $q_k \to p_{L-k}$. Thinking of $L$ as “time”, one notices that $\tau_L$ and $\tau^*_L$ are time-averaged discretized geometric fractional Brownian motions (they can be thought of as the “prices” of Asian options within the framework of the fractional BSM model [@greg]). Note that in absence of a global bias $\mathbb{E}\{ 1/\tau_L\} = \mathbb{E}\{ 1/\tau^*_L\}$, and hence, without any loss of generality we set $P_L = 0$ in what follows. Thus, combining (\[walk\], \[current\], \[tau\]) and setting $P_0=1$ yields $$\begin{aligned}
\label{explicit_J}
J_L = \frac{1}{2} \left(1 + \sum_{l=1}^{L-1} \exp\left(V_l\right)\right)^{-1} \;.\end{aligned}$$ For typical realizations of $\{p_X\}$, the size of $|V_l|$ is $\mathcal{O}(l^H)$ so that the typical current $J_{{\rm typ}} $ is $J_{{\rm typ}} \sim \exp\left(- L^H\right)$.
To obtain an upper bound on $\mu_k(L)$, consider a given realization of the sequence $V_1,V_2, \ldots,V_{L-1}$ and denote the maximal among them as $V_{\max} = {\max}_{0\leq i \leq L-1}V_i$. From (\[tau\]) one has $\tau_L = (1 + \sum_{l=1}^{L-1} V_l)\geq \exp\left(V_{\max}\right)$, so that $J_L^k \leq \left( {1}/{2}\right)^k \exp\left( - k V_{\max}\right)$. Since $\exp\left( - k V_{\max}\right) \to 0$ as $L \to \infty$ (recall that $V_{\max} \sim L^H$) the average value of $\exp\left( - k V_{\max}\right)$ is dominated by configurations with $V_{\max} \to 0$. The asymptotic behavior of the pdf $P_L(V_{\rm max})$ for fixed $V_{\max}$ and large $L$ is known [@krug; @molchan], yielding $\ln P_L(V_{\max}) = \theta \ln L^{-1} + {\cal O}(1)$, where $\theta = 1 - H$ is the persistence exponent [@note]. Hence, we have $$\label{upper_2}
\mu_k(L)
%\mathbb{E}\left\{ J_L^k\right\}
\leq {B_k}{L^{H - 1}} \,, L \gg 1\,,$$ where $B_k$ is an $L$-independent constant.
To determine a lower bound on $\mu_k(L)$ we follow [@gleb; @gleb1; @cec] and make the following observation: averaging (\[explicit\_J\]) is to be performed over the entire set $\Omega$ of all possible trajectories $\{V_l\}_{1 \leq l \leq L}$. Since $\tau_L > 0$, a lower bound on $\mu_k(L)$ can be straightforwardly obtained if one averages instead over some finite subset $\Omega' \subset \Omega$ of trajectories with some prescribed properties, that is $\mu_k(L) \geq \mathbb{E}_{\Omega'}\left\{ J_L^k\right\}$. We choose $\Omega'$ as the set comprising all possible trajectories $\{V_l\}_{0 \leq l \leq L}$ which, starting at the origin at $l = 0$, never cross the deterministic curve $Y_l = Y_0 - \alpha \ln(1+l)$ with $Y_0 > 0$ and $\alpha > 1$. For any such trajectory $\tau_L = 1 + \sum_{l = 1}^{L - 1} \exp\left(V_l\right)$ is bounded from above by $\sum_{l = 0}^{L - 1} \exp\left(Y_l\right)$, which, in turn, is bounded from above by $\exp(Y_0) \zeta(\alpha)$, where $\zeta(\alpha)$ is the zeta-function. Hence, we have $\mu_k(L) \geq (\exp(Y_0) \zeta(\alpha)/2)^{-k} \, \mathbb{E}_{\Omega'}\left\{1\right\}$, where $\mathbb{E}_{\Omega'}\left\{1\right\}$ is, by definition, the survival probability, $S_L$ up to time $L$, for a fBm, starting at the origin in presence of a “moving trap” evolving via $Y_l = Y_0 - \alpha \ln(1 + l)$.
For standard Brownian motion ($H=1/2$) in presence of a trap which moves as $- l^z$, the leading large-$L$ behavior of the survival probability $S_L$ is exactly the same as in the case of an immobile trap, provided that $z < 1/2$ [@krap]. It is thus physically plausible to suppose that the same behavior holds for a more general Gaussian process such as a fBm. That is, one expects that for any $H > 0$ the leading large-$L$ behavior of $\mathbb{E}_{\Omega'}\left\{1\right\}$ will be exactly the same for an immobile trap and for a logarithmically moving trap, i.e., that $S_L = \mathbb{E}_{\Omega'}\left\{1\right\} \sim Y_0^{\theta/H}/L^{\theta}$ as $L \to \infty$ [@krug; @molchan], where $\theta = 1 - H$. In fact, this can be shown rigorously [@aurzada_fbm; @aurz]. Consequently, we find $$\label{lower}
\mu_k(L) \geq {D_k}{L^{H - 1}} \;, L \gg 1\,,$$ where $D_k$ is independent of $L$. Note that the bounds in (\[upper\_2\]) and (\[lower\]) show the same $L$-dependence and thus yields the exact result announced in (\[2\]).
We now turn to a continuous-time and -space dynamics in a disordered fBm potential. The position $x(t) \in [0,L]$ of a particle at time $t$ obeys a Langevin equation : $\dot x = - V'(x) + \eta(t)$, where $V'(x)$ is a quenched random force such that $V(x)$ is a fBm with Hurst exponent $H$ (\[def\]) and $\eta(t)$ is a Gaussian thermal noise of zero mean and covariance $\langle \eta(t) \eta(t')\rangle = 2 T \delta(t-t')$. The steady-state current and the concentration profile $C(x)$ can be obtained from the corresponding Fokker-Planck equation $$\begin{aligned}
\label{start_continuum}
&&J_L = {T}\left({\int_0^L \, \exp{[{V(x)}/{T}]}\, dx}\right)^{-1} \;, \\
&&C(x) = \frac{J_L}{T} \int_x^L dx' \, \exp{[(V(x')-V(x))/T]} \;, \nonumber\end{aligned}$$ \[see (\[current\], [\[explicit\_J\]]{}) with $D_0=T$ and $P_0=1$\] [@remark]. The total number of particles is then $N_L = \int_0^L C(x) \, dx$. We focus next on the moments and on the pdf of $J_L$ (\[start\_continuum\]).
Instead of $J_L/T$, which can be viewed as the inverse of the SM partition function, we study the pdf $\Pi_{T=0}(F)$ of the free energy $F = T \log(J_L/T)$. Consider first $T \equiv 0$, in which case $F=E_{\min} = \min_{0\leq x \leq L} V(x)$. Recalling that $V(0) = 0$ ($E_{\min} < 0$), the cumulative distribution $q_L(E) = {\Pr}(E_{\min} > - E)$, (with $E>0$), coincides with the probability that up to ’time’ $L$, $V(x)$ starting at $E$ at $x=0$ ’survives’ in presence of an absorbing boundary at $V=0$. For self-affine process, $q_L(E)$ takes the scaling form $q_L(E) = Q(E/L^H)$: for $L \gg E^{1/H}$, $q_L(E)$ behaves algebraically [@majumdar], $q_{L}(E) \sim {E^{\theta/H}}/{L^\theta}$ ($\theta = 1 - H$ for fBm [@krug; @molchan]), while for $L \ll E^{1/H}$, $q_{L}(E)$ is of order one. Hence, one has for $\Pi_{T = 0}(F) = \partial_E q_L(E)\big|_{E = -F}$ $$\begin{aligned}
\label{eq:dist_f}
\Pi_{T=0}(F) =
\begin{cases}
& 0 \;, \; F > 0\\
& L^{-\theta} |F|^{\theta/H-1} \;, \; - L^H \ll F < 0 \\
& \exp(-F^2/2 L^{2H}) \;, \; F \ll - L^H \;.
\end{cases}\end{aligned}$$ Lastly, the regime $F \ll - L^H$ corresponds to a fraction of paths $V(x)$ that propagate from $E$ to zero in a ’time’ $L$. In general, the tail of this probability coincides with the one of the free propagator, which is Gaussian for fBm. What happens at finite $T$ where $F = E - T S$ is now the balance between the energy $E$ and the entropy $S$ ? One expects a particle to be localized close to the minimum $E_{\min}$, which is of order ${\cal O}(L^H)$, while the maximum entropy $\sim {\cal O}(\ln L)$. Hence when $L \gg 1$ the main contribution to $F$ comes from $E_{\min}$ so that for a given sample at finite $T$, $F$ will be very close to $E_{\min}$. This is corroborated by numerical simulations (see Fig. \[fig2\]).
![(Color online) Pdf of the free energy $-F/L^H$ for different system sizes: $L=64 \,(\rm blue), 256 \,(\rm green), 1024 \, (\rm red)$. The black curve represents the distribution of $E_{\rm min}/L^H$ for $L=4096$. [**Inset:**]{} Pdf of $E_{\min} -F$ for different system sizes: $L=64 \,(\rm blue), 256 \,(\rm green), 1024 \, (\rm red)$. Histograms are computed using $10^5$ samples and setting $T=1$ and $H=3/4$. The dashed line corresponds to $(|F|/L^H)^{-2/3}$ (\[eq:dist\_f\]).[]{data-label="fig2"}](combinedfig2.eps){width="\linewidth"}
We now come back to the current distribution. Very small currents, $J_L \ll J_{\rm typ} \sim \exp(-L^H)$, correspond to $F \ll - L^H$ in (\[eq:dist\_f\]) and one obtains that $P(J_L)$ is log-normal, $\ln P(J_L) \propto {-{\ln^2(J_L/T)}}$. Within the opposite limit, $J_L \gg J_{\rm typ}$, one finds from (\[eq:dist\_f\]) that $$\begin{aligned}
\label{eq:inter_regime}
P(J_L) \sim \frac{[\log(J_L/T)]^{\theta/H-1}}{J_L L^\theta} \;.\end{aligned}$$ This power-law behavior holds up to a large cut-off value $J_{\max}$. At $T=0$ we have a sharp cut-off at $J_{\max}=1$ (\[eq:dist\_f\]), while at a finite $T$, $P(J_L)$ has a fast decay which depends on the fluctuations of $V(x)$ at a short length scale close to the origin $x=0$. For $L \gg 1$, $\mu_k(L)$ are dominated by the regime where $J_{\rm typ} \ll J_L < J_{\max} \sim {\cal O}(1)$ (\[eq:inter\_regime\]), such that one gets $\mu_k(L) \sim 1/{L^\theta}$ (\[2\]). This calculation shows that (rare) negative persistent potential leads to very large currents. We observe that these rare persistent profiles also exhibit large barriers, growing like $L^H$. These barriers stop the particle diffusion and are responsible for the subdiffusive behavior of the mean square displacement. One could expect that these barriers should also affect the behavior of the current. However, by looking at the steady state concentration profile $C(x)$ (\[start\_continuum\]), one can see that large barriers induce a very large number of particles in the system located in the deep valleys of the potential $V(x)$, which allows to sustain a large current.
In our numerical simulations we consider a discrete random potential $V_k$, $k=0,1,\ldots L-1$, with $\sigma^2=1$, which displays fBm correlations (\[def\]). We use a powerful algorithm [@rosso_fbm; @GRS10], which allows to generate very long samples of fBm paths. For each sample, we compute the current, $J_L = T [\sum_{k=0}^{L-1} \exp(-V_k/T)]^{-1}$, the free energy $F = T \ln (J_L/T)$ and the ground state energy $E_{\min} = \min_{k} V_k$. In Fig. \[fig\_moments\] we plot the first three moments as a function of $L$ for different values of $H$. These plots show a very good agreement with our analytical predictions in (\[2\]). In Fig. \[fig2\], we show that the pdf of the rescaled free energy, $F/L^H$, at finite temperature $T$ converges to the pdf of the rescaled ground state energy $E_{\min}/L^H$. The reason for this is that, for each sample, the difference between $F$ and $E_{\min}$ grows very slowly with $L$, probably logarithmically (inset of Fig. \[fig2\]). In the rescaled variables, this difference vanishes when $L \to \infty$.
![\[fig\] (Color online) The pdf in Eq. (\[omega\]) for $\sigma^{2} L^{2 H} = 1/2$ (blue), $\sigma^2 L^{2 H} = 2$ (red) and $\sigma^2 L^{2 H} = 5$ (green).](Fig3.eps){width="0.8\linewidth"}
We close with an observation that such chains show a transition to a diode-like behavior, when $\xi = \sigma^2 L^{2 H}$ exceeds some critical value $\xi_c$. Consider a chain in which at site $X = 0$ we maintain a fixed concentration $P_0 = 1$ of, say, “white” particles and place a sink for them at $X = L$. At $X = L$ we introduce a source which maintains concentration $1$ of “black” particles, and place a sink for them at site $X = 0$. The particles are mutually noninteracting. For a fixed $\{p_X\}$ we have counter-currents of white ($J_L^w$) and black ($J_L^b$) particles, which obviously obey, on average, $\mathbb{E}\left\{ (J_L^w)^k\right\} \equiv \mathbb{E}\left\{ (J_L^b)^k\right\}$ for any $k>0$.
Consider next the random variable: $\omega = {J_L^w}/{(J_L^w + J_L^b)} = {\tau_L^*}/{(\tau_L^* + \tau_L)}$, which probes the likelihood of an event that for a fixed $\{p_X\}$ one has $J_L^w = J_L^b$. The pdf of $\omega$ can be calculated exactly to give $$\begin{aligned}
\label{omega}
P(\omega) = \frac{1}{\sqrt{2 \pi} \omega (1-\omega) \sigma L^H} \exp\left(- \frac{\ln^2\left(\frac{1 - \omega}{\omega}\right)}{2 \sigma^{2 } L^{2 H}}\right)\,.\end{aligned}$$ Remarkably, $P(\omega)$ in (\[omega\]) changes the modality when $\xi$, (which defines the value of a typical barrier), exceeds a critical value $\xi_c =2$ (see Fig. \[fig\]). For short chains (or small $\sigma$) $P(\omega)$ is unimodal and centered at $\omega = 1/2$: any given sample is transmitting particles in both directions equally well and, most probably, $J_L^w = J_L^b$. For $\xi = \xi_c$ the pdf is nearly uniform (except for narrow regions at the edges) so that *any* relation between $J_L^w$ and $J_L^b$ is equally probable. Finally, for $\xi > \xi_c$ (sufficiently strong disorder and/or a long chain) the symmetry is broken and $P(\omega)$ becomes bimodal with a local minimum at $\omega = 1/2$ and two maxima close to $0$ and $1$. This means that a given sample is most likely permeable only in one direction.
We thank G. Biroli for a useful discussion. GO is partially supported by the ESF Research Network “Exploring the Physics of Small Devices”, AR - by ANR grant 09-BLAN-0097-02 and GS - by ANR grant 2011-BS04- 013-01 WALKMAT. This project was partially supported by the Indo-French Centre for the Promotion of Advanced Research under Project 4604-3.
[99]{}
H. Kesten, M. V. Kozlov and F. Spitzer, Compos. Math. [**30**]{}, 145 (1975). B. Derrida and Y. Pomeau, Phys. Rev. Lett. [**48**]{}, 627 (1982). Ya. G. Sinai, Theor. Probab. Appl. [**27**]{} 256, (1982). P. G. de Gennes, J. Stat. Phys. [**12**]{}, 463 (1975). G. Oshanin and S. Redner, EPL [**85**]{}, 10008 (2009). R. Bruinsma and G. Aeppli, Phys. Rev. Lett. [**52**]{}, 1547 (1984). T. Nattermann and J. Vilain, Phase Transit. [**11**]{}, 5 (1988). J. P. Harth and J. Lothe, [*The theory of dislocations*]{} (New York, McGraw-Hill, 1968). J.-C. Walter, A. Ferrantini, E. Carlon and C. Vanderzande, Phys. Rev. E [**85**]{}, 031120 (2012). J. Mathe et al., Biophys. J. [**87**]{}, 3205 (2004). Y. Kafri, D. K. Lubensky and D. R. Nelson, Biophys. J. [**86**]{}, 3373 (2004). A. Comtet, C. Monthus and M. Yor, J. Appl. Prob. [**35**]{}, 255 (1998). G. Oshanin and G. Schehr, Quant. Finance [**12**]{}, 1325 (2012). J.-P. Bouchaud, A. Comtet, A. Georges, and P. Le Doussal, Ann. Phys. [**201**]{}, 285 (1990). B. D. Hughes, [*Random walks and random environments*]{} Vol.[**2**]{} (Clarendon Press, Oxford, 1995). M. Sheinman, O. Bénichou, R. Voituriez and Y. Kafri, J. Stat. Mech. P09005 (2010). the probability being taken over all random walks of $n$ steps, starting at the same point. S. F. Burlatsky, G. Oshanin, A. Mogutov and M. Moreau, Phys. Rev. A [**45**]{}, R6955 (1992). G. Oshanin, S. F. Burlatsky, M. Moreau and B. Gaveau, Chem. Phys. [**177**]{}, 803 (1993). G. Oshanin, A. Mogutov and M. Moreau, J. Stat. Phys. [**73**]{}, 379 (1993). C. Monthus and A. Comtet, J. Phys. I France [**4**]{}, 635 (1994).
A. Dhar, K. Saito and B. Derrida, preprint arXiv:1207.1184.
M. Gorissen, A. Lazarescu, K. Mallick, C. Vanderzande, Phys. Rev. Lett. [**109**]{}, 170601 (2012).
E. N. Govorun et al., Phys. Rev. E [**64**]{}, 040903 (2001). C. Bender, T. Sottinen and E. Valkeila, in [*Advanced mathematical methods for finance*]{}, (ed. G. Di Nunno and B. Oksendal. Springer, Berlin), pp. 75 (2011), arXiv:1004.3106 G. M. Molchan, Comm. Math. Phys. [**205**]{}, 97 (1999). J. Krug et al., Phys. Rev. E [**56**]{}, 2702 (1997).
S. N. Majumdar, Current Science [**77**]{}, 370 (1999). F. Aurzada and T. Simon, preprint arXiv:1203.6554. F. Aurzada, Elect. Comm. in Probab. [**16**]{}, 392 (2011). F. Aurzada and C. Baumgarten, arXiv:1301.0424v1.
H. Kesten, Acta Math. [**131**]{}, 207 (1973). As a matter of fact, this result can be made more precise by writing the distribution $P_L(V_{\max})$ in a scaled form $P_L(V_{\max}) = L^{-H} f(y = V_{\max}/L^H)$. This implies that actually $P_L(V_{\max}) \sim V_{\max}^{(1-2H)/H}/L^{1 - H}$ as $L \to \infty$. C. Monthus, G. Oshanin, A. Comtet, and S. F. Burlatsky, Phys. Rev. E [**54**]{}, 231 (1996). We note that the functional in (\[start\_continuum\]) was studied within a different context in [@molchan; @MRZ09], and it was shown that its first moment decreases as $L^{-\theta}$ with $\theta = 1 - H$. See also different exact results [@gleb2; @cecile] for the $H = 1/2$ case. S. N. Majumdar, A. Rosso and A. Zoia, Phys. Rev. Lett. [**104**]{}, 020602 (2010).
K. Uchiyama, Z. Wahrsch. Verw. Gebiete [**54**]{}, 75 (1980); P. L. Krapivsky and S. Redner, Am. J. Phys. [**64**]{}, 546 (1996).
K. J. Wiese, S. N. Majumdar, A. Rosso, Phys. Rev. E [**83**]{}, 061141 (2011). R. Garcia-Garcia, A. Rosso and G. Schehr, Phys. Rev. E [**81**]{}, 010102(R) (2010)
|
---
abstract: 'To any $n \times n$ Latin square $L$, we may associate a unique sequence of mutually orthogonal permutation matrices $P = P_1, P_2, ..., P_n$ such that $L = L(P) = \sum kP_k$. Brualdi and Dahl (2018) described a generalisation of a Latin square, called an *alternating sign hypermatrix Latin-like square (ASHL)*, by replacing $P$ with an *alternating sign hypermatrix (ASHM)*. An ASHM is an $n \times n \times n$ (0,1,-1)-hypermatrix in which the non-zero elements in each row, column, and vertical line alternate in sign, beginning and ending with $1$. Since every sequence of $n$ mutually orthogonal permutation matrices forms the planes of a unique $n \times n \times n$ ASHM, this generalisation of Latin squares follows very naturally, with an ASHM $A$ having corresponding ASHL $L = L(A) =\sum kA_k$, where $A_k$ is the $k^{\text{th}}$ plane of $A$. This paper addresses some open problems posed in Brualdi and Dahl’s article, firstly by characterising how pairs of ASHMs with the same corresponding ASHL relate to one another and providing a tight lower bound on $n$ for which two $n \times n \times n$ ASHMs can correspond to the same ASHL, and secondly by exploring the maximum number of times a particular integer may occur as an entry of an $n \times n$ ASHL. A general construction is given for an $n \times n$ ASHL with the same entry occurring $\lfloor\frac{n^2 + 4n -19}{2}\rfloor$ times, improving considerably on the previous best construction, which achieved the same entry occuring $2n$ times.'
author:
- 'Cian O’Brien'
title: 'Alternating Sign Hypermatrix Decompositions of Latin-like Squares'
---
Introduction
============
An *alternating sign matrix (ASM)*, is a ${(0,1,-1)}$-matrix in which the non-zero elements in each row and column alternate in sign, beginning and ending with $1$.
The $n \times n$ *diamond ASM* is an ASM with the maximum number of non-zero entries for given $n$. For odd $n$, there is a unique diamond ASM $D_n$. For even $n$, there are two diamond ASMs $D_n$ and $D_n'$.
The following are the two $4 \times 4$ diamond ASMs $D_4$ and $D_4'$, and the $5 \times 5$ diamond ASM $D_5$.
{width="\textwidth"}
It is easily observed that permutation matrices are examples of ASMs, which arise naturally as the unique smallest lattice containing the permutation matrices of the *Bruhat order* [@lattice].
A *Latin square* of order $n$ is an $n \times n$ array containing $n$ symbols such that each symbol occurs exactly once in each row and column.
Any $n \times n$ Latin square $L$ with symbols $1, 2, \dots, n$ can be decomposed into a unique sequence $P$ of $n \times n$ mutually orthogonal permutation matrices $P = P_1, P_2, \dots, P_n$ by the following relation.
$$L = L(P) = \sum_k kP_k$$
For example, consider the following latin square.
$${{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1 \\
3 & 1 & 2\end{pmatrix}}}}
=
1{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0\end{pmatrix}}}}
+2{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1\end{pmatrix}}}}
+3{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0\end{pmatrix}}}}$$
This decomposition leads to a natural extension of the concept of a Latin square, first introduced by Brualdi and Dahl [@ashmbib], by replacing the sequence of permutation matrices with planes of an *alternating sign hypermatrix (ASHM)*. Before we define these objects, we must first define some features of a hypermatrix.
An $n \times n \times n$ hypermatrix $A = [a_{ijk}]$ has has $n^2$ lines of each of the $3$ following types. Each line has $n$ entries.
- *Row lines* $A_{*jk} = [a_{ijk} : i = 1, \dots, n]$, for given $1 \leq j,k \leq n$;
- *Column lines* $A_{i*k} = [a_{ijk} : j = 1, \dots, n]$, for given $1 \leq i,k \leq n$;
- *Vertical lines* $A_{ij*} = [a_{ijk} : k = 1, \dots, n]$, for given $1 \leq i,j \leq n$.
In this paper, we refer to a *plane* $P_k(A)$ of $A$ to be the horizontal plane $A_{**k} = [a_{ijk}:i,j=1,\dots,n]$, for given $1 \leq k \leq n$.
An *alternating sign hypermatrix (ASHM)* is a $(0,\pm 1)$-hypermatrix for which the non-zero entries in each row, column, and vertical line of the hypermatrix alternate in sign, starting and ending with $+1$.
For example, the following is a $3 \times 3$ ASHM.
$$A = {{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0\end{pmatrix}}}}
\hspace{-0.1cm}\nearrow\hspace{-0.1cm}{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
0 & 1 & 0 \\
1 & -1 & 1 \\
0 & 1 & 0\end{pmatrix}}}}
\hspace{-0.1cm}\nearrow\hspace{-0.1cm}{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1\end{pmatrix}}}}$$
The north-east arrow $P_k(A) \nearrow P_{k+1}(A)$ is used to denote that $P_k(A)$ is below $P_{k+1}$. For simplicity, for the remainder of this paper, we will omit the zero entries of all ASHMs, and represent all $\pm 1$ entries with $+$ or $-$.
An *ASHL* is an $n \times n$ matrix $L$ constructed from an $n \times n \times n$ ASHM $A$ by $$L = L(A) = \sum_k kP_k(A)\text{.}$$
From the previous example, we then have the following ASHL. $$L(A) = 1{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& & + \\
& + & \\
+ & & \end{pmatrix}}}}
+2{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& + & \\
+ & - & + \\
& + & \end{pmatrix}}}}
+3{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
+ & & \\
& + & \\
& & +\end{pmatrix}}}}
={{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
3 & 2 & 1 \\
2 & 2 & 2 \\
1 & 2 & 3\end{pmatrix}}}}$$
Brualdi and Dahl [@ashmbib] posed a number problems about ASHLs, the following of which are addressed in this paper.
- Given an $n \times n$ ASHL $L$, let $A_n(L)$ be the set of all $n \times n \times n$ ASHMs $A$ such that $L(A) = L$. Investigate $A_n(L)$.
- What is the maximum number of times an integer can occur as an entry of an $n \times n$ ASHL?
In this paper, the relationship between two ASHMs that generate the same ASHL is described, and a general construction is given for building an $n \times n$ ASHL containing $\lfloor\frac{n^2 + 4n -19}{2}\rfloor$ copies of one symbol. This improves on Brualdi and Dahl’s lower bound of $2n$ for the maximum number of times that one symbol can appear in an $n \times n$ ASHL.
ASHLs With Multiple ASHM-Decompositions
=======================================
In Brualdi and Dahl’s paper [@ashmbib], it was proven that for a Latin square $L$, if $L = L(A)$ for some ASHM $A$, then $A$ must be a permutation hypermatrix. The following question was then posed.
\[problem1\] Given an $n \times n$ ASHL $L$, let $A_n(L)$ be the set of all $n \times n \times n$ ASHMs $A$ such that $L(A) = L$. Investigate $A_n(L)$.
This is presented in [@ashmbib] as a completely open problem, with no examples of distinct ASHMs with the same corresponding ASHL given. This problem is motivated by the observation that in the case of a Latin square $L$, $A_n(L)$ contains exactly one ASHM whose planes form a set of mutually orthogonal permutation matrices. Before we discuss how two ASHMs in $A_n(L)$ relate to one another, it is useful to introduce the following definition, and more generally examine how any pair of $n \times n \times n$ ASHMs relate to one another.
A *T-block* $T_{i_1,j_1,k_1:\,i_2,j_2,k_2}$ is a hypermatrix $[t_{ijk}]$ such that $$t_{ijk} = \begin{cases}
\color{white}-\color{black}1 & (i,j,k) = (i_1,j_1,k_1), (i_2,j_2,k_1), (i_2,j_1,k_2), \text{or} (i_1,j_2,k_2) \\
-1 & (i,j,k) = (i_2,j_1,k_1), (i_1,j_2,k_1), (i_1,j_1,k_2), \text{or} (i_2,j_2,k_2) \\
\color{white}-\color{black}0 & \text{otherwise}
\end{cases}$$ where $i_1 < i_2, j_1 < j_2$, and $k_1 < k_2$.
A T-block can be most usefully visualised as an $n \times n \times n$ matrix containing the subhypermatrix $$\pm \left[{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
+ & -\\
- & +\end{pmatrix}}}}
\nearrow
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
- & +\\
+ & -\end{pmatrix}}}}\right]$$ such that these are the only non-zero entries. This is a 3-d extension of a concept defined by Brualdi, Kiernan, Meyer, and Schroeder [@brualdibib].
The following lemma will be needed to investigate $A_n(L)$.
\[cube\_decomp\_lemma\] Let $A$ and $B$ be two $n \times n \times n$ ASHMs. Then $A-B$ can be expressed as a sum of T-blocks.
If $A = B$, this is trivially true.
Assume $A \not = B$. In $A$ and $B$, the sum of the entries in any row, column, or vertical line is $1$. Therefore the sum of any row, column, or vertical line of $D = A-B$ is $0$. Now iterate the following step.
- Let $k_1$ be the least integer for which the plane $P_{k_1}(D)$ contains non-zero entries. For some positive entry $d_{i_1j_1k_1}$ of $D$, we can find entries $d_{i_2j_1k_1}$, $d_{i_1j_2k_1}$ and $d_{i_1j_1k_2}$ with negative sign, where $k_2 > k_1$. Let $D \leftarrow D - T_{i_1,j_1,k_1:\,i_2,j_2,k_2}$ and repeat this step if $D$ is not a $0$-hypermatrix.
Note that the sum of the absolute values of the entries of $P_{k_1}(D)$ is at least $2$ less than the previous step, and that all line sums of $D$ are $0$ after each step. We can therefore run this iterative process repeatedly, resulting in $P_{k_1}(D)$ becoming a 0-matrix for the current value of $k_1$, and eventually resulting in $D$ becoming a 0-hypermatrix. Therefore $A-B$ can be expressed as a sum of T-blocks.
Let $T$ be a T-block. The *depth* of $T$, $d(T)$, is defined as follows. $$d(T) = \begin{cases}
k_2 - k_1\text{,} & T = T_{i_1,j_1,k_1:\,i_2,j_2,k_2} \\
k_1 - k_2\text{,} & T = -T_{i_1,j_1,k_1:\,i_2,j_2,k_2}
\end{cases}$$ Two T-blocks, $T_1$ and $T_2$, have *opposite depth* if $d(T_1) = -d(T_2)$
\[ASHL\_difference\] Two ASHMs $A$ and $B$ satisfy $L(A) = L(B)$ if and only if any expression of $A-B$ as a sum of T-blocks satisfies that, in any vertical line $V$ of $A-B$, $$\sum_{T \in T_V} d(T) = 0\text{,}$$ where $T_V$ is the subset of these T-blocks with non-zero entries in $V$.
From Lemma \[cube\_decomp\_lemma\], we know that $A - B$ can be expressed as a sum of T-blocks. The entries of any vertical line $V$ in $A-B$ can be decomposed into pairs $(t_{k_1},t_{k_2})$, where $k_1 < k_2$, such that pairs are non-zero entries in the same T-block. Note that $t_{k_1} = -t_{k_2}$. Therefore $$\sum kV_k = \sum_{T \in T_V} k_1t_{k_1} + k_2t_{k_2} = \sum_{T \in T_V} \pm (k_1 - k_2) = \sum_{T \in T_V} d(T)\text{.}$$
- Suppose that, for any vertical line $V$ in $A - B$, $$\sum_{T \in T_V} d(T) = 0\text{.}$$ This means that $\sum kV_k = 0$, which means that $L(A-B) = 0$. Therefore $L(A) = L(B)$.
- Now suppose that $L(A) = L(B)$. Then $L(A-B) = 0$, so for any vertical line $V$ in $A-B$, we have $\sum kV_k = 0$. Therefore $$\sum_{T \in T_V} d(T) = 0\text{.}$$
This provides some progress on Problem \[problem1\], as an ASHM $B$ is contained in $A_n(L)$ if and only $A$ and $B$ satisfy Theorem \[ASHL\_difference\] for all $A \in A_n(L)$. In particular, Theorem \[ASHL\_difference\] provides a strategy for constructing an ASHM $B$ for which $L(B) = L(A)$ for some given ASHM $A$, by adding T-blocks to $A$ in such a way that satisfies $\sum_{T \in T_V} d(T) = 0$. This is by far our most successful method for generating pairs of ASHMs with the same corresponding ASHL. The relationship between elements of $A_n(L)$ can be characterised further, as shown in the following two examples.
\[uneven\_distance\] Here, $A$ and $B$ are two ASHMs for which $D=A-B$ has a very natural decomposition as the sum of three T-blocks with depths $1, 1, -2$, respectively, occupying the same vertical lines. $$A =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& \textbf{+} & & & & & & \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
& & & & & & \textbf{+} & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& \textbf{+} & & & & & & \\
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \\
& & & & & & \textbf{+} & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & \textbf{+} \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
& & \textbf{+} & & \textbf{-} & & \textbf{+} & \\
& \textbf{+} & & \textbf{-} & & \textbf{+} & & \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & \textbf{+} & & & & \\
& \textbf{+} & & & & & & \\
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \\
& & & & & & \textbf{+} & \\
& & & & \textbf{+} & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & & & & & & & \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
& \textbf{+} & & \textbf{-} & & \textbf{+} & & \\
& & \textbf{+} & & \textbf{-} & & \textbf{+} & \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
& & & & & & & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & \textbf{+} & & & \\
& & & & & & \textbf{+} & \\
& & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & \\
& & & \textbf{+} & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & \textbf{+} & \\
& & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & \end{pmatrix}}}}$$
$$B =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& \textbf{+} & & & & & & \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
& & & & & & \textbf{+} & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& \textbf{+} & & & & & & \\
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \\
& & & & & & \textbf{+} & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & \textbf{+} \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
& & \textbf{+} & \textbf{-} & & & \textbf{+} & \\
& \textbf{+} & & & \textbf{-} & \textbf{+} & & \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & \textbf{+} & & & & \\
& \textbf{+} & & & & & & \\
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \\
& & & & & & \textbf{+} & \\
& & & & \textbf{+} & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & & & & & & & \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
& \textbf{+} & & & \textbf{-} & \textbf{+} & & \\
& & \textbf{+} & \textbf{-} & & & \textbf{+} & \\
& & & \textbf{+} & & & & \\
& & & & \textbf{+} & & & \\
& & & & & & & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & \textbf{+} & & & \\
& & & & & & \textbf{+} & \\
& & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & \\
& & & \textbf{+} & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & \textbf{+} & \\
& & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & \end{pmatrix}}}}$$
$$D =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & \\
& \color{white}\textbf{+} & & & & & & \\
& & \color{white}\textbf{+} & & & & & \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & \\
& & & & & & \color{white}\textbf{+} & \\
& & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & \\
& \color{white}\textbf{+} & & & & & & \\
& & \color{white}\textbf{+} & & & & & \\
\color{black!70}\cdot& \color{black!70}\cdot &\color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
& & & & &\color{white}\textbf{+} & & \\
& & & & & &\color{white}\textbf{+} & \\
& & & & & & & \color{white}\textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & \\
& \color{white}\textbf{+} & & & & & & \\
& & \color{white}\textbf{+} & & & & & \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & \\
& & & & & & \color{white}\textbf{+} & \\
& & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & \\
& \color{white}\textbf{+} & & & & & & \\
& & \color{white}\textbf{+} & & & & & \\
& & & \textbf{-} & \textbf{+} & & & \\
& & & \textbf{+} & \textbf{-} & & & \\
& & & & & \color{white}\textbf{+} & & \\
& & & & & &\color{white}\textbf{+} & \\
& & & & & & & \color{white}\textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+}& & & & & & & \\
& \color{white}\textbf{+} & & & & & & \\
& &\color{white}\textbf{+} & & & & & \\
& & & \color{white}\textbf{+} & & & & \\
& & & &\color{white}\textbf{+} & & & \\
& & & & & \color{white}\textbf{+}& & \\
& & & & & &\color{white}\textbf{+} & \\
& & & & & & & \color{white}\textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+}& & & & & & & \\
& \color{white}\textbf{+} & & & & & & \\
& &\color{white}\textbf{+} & & & & & \\
\color{black!70}\cdot& \color{black!70}\cdot &\color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & \\
& & & & & &\color{white}\textbf{+} & \\
& & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & \\
& \color{white}\textbf{+} & & & & & & \\
& &\color{white}\textbf{+} & & & & & \\
& & & \color{white}\textbf{+} & & & & \\
& & & & \color{white}\textbf{+} & & & \\
& & & & & \color{white}\textbf{+} & & \\
& & & & & & \color{white}\textbf{+} & \\
& & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & \\
& \color{white}\textbf{+}& & & & & & \\
& & \color{white}\textbf{+} & & & & & \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & \\
& & & & & & \color{white}\textbf{+} & \\
& & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}$$
$$D = T_{4,4,1:\,5,5,2} + T_{4,4,3:\,5,5,4} - T_{4,4,6:\,5,5,8}$$
$$L(A) = L(B) =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
6 & 2 & 1 & 5 & 7 & 4 & 8 & 3 \\
1 & 5 & 2 & 6 & 3 & 8 & 7 & 4 \\
2 & 1 & 5 & 3 & 6 & 7 & 4 & 8 \\
5 & 6 & 3 & 3 & 3 & 6 & 3 & 7 \\
7 & 3 & 6 & 3 & 3 & 3 & 6 & 5 \\
4 & 8 & 7 & 6 & 3 & 5 & 1 & 2 \\
8 & 7 & 4 & 3 & 6 & 2 & 5 & 1 \\
3 & 4 & 8 & 7 & 5 & 1 & 2 & 6\end{pmatrix}}}}$$
$D$ can also be expressed as the sum of pairs of T-blocks with opposite depth occupying the same vertical lines.
$$D = (T_{4,4,1:\,5,5,2} - T_{4,4,7:\,5,5,8}) + (T_{4,4,3:\,5,5,4} - T_{4,4,6:\,5,5,7})$$
\[interlaced\] Here, $A$ and $B$ are two ASHMs for which $D=A-B$ has a very natural decomposition as the sum of three T-blocks such that each pair occupy exactly two of the same vertical lines. $$\hspace{-0.4cm}A =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \\
& & & & \textbf{+} & & & & \\
& & & \textbf{+} & & & & & \\
& & & & & \textbf{+} & & & \\
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \\
& & & & & \textbf{+} & & & \\
& & & & \textbf{+} & & & & \\
& & & \textbf{+} & & & & & \\
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \\
& & & \textbf{+} & & & & & \\
& & & & & \textbf{+} & & & \\
& & & & \textbf{+} & & & & \\
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
\textbf{+} & & & & \textbf{-} & & \textbf{+} & & \\
& \textbf{+} & & \textbf{-} & & & & \textbf{+} & \\
& & \textbf{+} & & & \textbf{-} & & & \textbf{+} \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& \textbf{+} & & & & \textbf{-} & & \textbf{+} & \\
& & \textbf{+} & & \textbf{-} & & & & \textbf{+} \\
\textbf{+} & & & \textbf{-} & & & \textbf{+} & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & \textbf{+} & \textbf{-} & & & & & \textbf{+} \\
\textbf{+} & & & & & \textbf{-} & \textbf{+} & & \\
& \textbf{+} & & & \textbf{-} & & & \textbf{+} & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \end{pmatrix}}}}$$
$$\hspace{-0.4cm}B =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
\textbf{+} & & & \textbf{-} & & & \textbf{+} & & \\
& \textbf{+} & & & \textbf{-} & & & \textbf{+} & \\
& & \textbf{+} & & & \textbf{-} & & & \textbf{+} \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& \textbf{+} & & & \textbf{-} & & & \textbf{+} & \\
& & \textbf{+} & & & \textbf{-} & & & \textbf{+} \\
\textbf{+} & & & \textbf{-} & & & \textbf{+} & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & \textbf{+} & & & \textbf{-} & & & \textbf{+} \\
\textbf{+} & & & \textbf{-} & & & \textbf{+} & & \\
& \textbf{+} & & & \textbf{-} & & & \textbf{+} & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & \textbf{+} & \\
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} & \\
& & & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & & \\
& & \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & & \\
& \textbf{+} & & & & & & & \end{pmatrix}}}}$$
$$\hspace{-0.4cm}D =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
& \color{white}\textbf{+} & & & & & & & \\
& &\color{white}\textbf{+} & & & & & & \\
\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot & \color{black!70}\cdot &\color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & & \color{white}\textbf{+} & & \\
& & & & & & &\color{white}\textbf{+} & \\
& & & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
&\color{white}\textbf{+} & & & & & & & \\
& & \color{white}\textbf{+} & & & & & & \\
\color{black!70}\cdot& \color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot &\color{black!70}\cdot & \color{black!70}\cdot\\
\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot &\color{black!70}\cdot & \textbf{+} & \textbf{-} &\color{black!70}\cdot &\color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & & \color{white}\textbf{+} & & \\
& & & & & & & \color{white}\textbf{+} & \\
& & & & & & & & \color{white}\textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
& \color{white}\textbf{+} & & & & & & & \\
& & \color{white}\textbf{+} & & & & & & \\
\color{black!70}\cdot & \color{black!70}\cdot &\color{black!70}\cdot & \textbf{+} & \color{black!70}\cdot & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \textbf{-} & \color{black!70}\cdot & \textbf{+} &\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot\\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & &\color{white}\textbf{+} & & \\
& & & & & & &\color{white}\textbf{+} & \\
& & & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
& \color{white}\textbf{+} & & & & & & & \\
& &\color{white}\textbf{+} & & & & & & \\
\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot & \color{black!70}\cdot &\color{black!70}\cdot & \textbf{-} & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & & \color{white}\textbf{+} & & \\
& & & & & & &\color{white}\textbf{+} & \\
& & & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
&\color{white}\textbf{+} & & & & & & & \\
& & \color{white}\textbf{+} & & & & & & \\
\color{black!70}\cdot& \color{black!70}\cdot& \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \textbf{-} & \color{black!70}\cdot &\color{black!70}\cdot & \color{black!70}\cdot\\
\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot &\color{black!70}\cdot & \textbf{-} & \textbf{+} &\color{black!70}\cdot &\color{black!70}\cdot & \color{black!70}\cdot \\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & & \color{white}\textbf{+} & & \\
& & & & & & & \color{white}\textbf{+} & \\
& & & & & & & & \color{white}\textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
& \color{white}\textbf{+} & & & & & & & \\
& & \color{white}\textbf{+} & & & & & & \\
\color{black!70}\cdot & \color{black!70}\cdot &\color{black!70}\cdot & \textbf{-} & \color{black!70}\cdot & \textbf{+} & \color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot \\
\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot & \textbf{+} & \color{black!70}\cdot & \textbf{-} &\color{black!70}\cdot & \color{black!70}\cdot & \color{black!70}\cdot\\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & &\color{white}\textbf{+} & & \\
& & & & & & &\color{white}\textbf{+} & \\
& & & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
& \color{white}\textbf{+} & & & & & & & \\
& & \color{white}\textbf{+} & & & & & & \\
& & & \color{white}\textbf{+} & & & & & \\
& & & & \color{white}\textbf{+} & & & & \\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & &\color{white}\textbf{+} & & \\
& & & & & & & \color{white}\textbf{+}& \\
& & & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
& \color{white}\textbf{+} & & & & & & & \\
& & \color{white}\textbf{+} & & & & & & \\
& & & \color{white}\textbf{+} & & & & & \\
& & & & \color{white}\textbf{+} & & & & \\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & &\color{white}\textbf{+} & & \\
& & & & & & & \color{white}\textbf{+}& \\
& & & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white}\textbf{+} & & & & & & & & \\
& \color{white}\textbf{+} & & & & & & & \\
& & \color{white}\textbf{+} & & & & & & \\
& & & \color{white}\textbf{+} & & & & & \\
& & & & \color{white}\textbf{+} & & & & \\
& & & & & \color{white}\textbf{+} & & & \\
& & & & & &\color{white}\textbf{+} & & \\
& & & & & & & \color{white}\textbf{+}& \\
& & & & & & & &\color{white}\textbf{+} \end{pmatrix}}}}$$
$$D = -T_{4,4,1:\,5,5,4} - T_{4,5,2:\,5,6,5} + T_{4,4,3:\,5,6,6}$$
$$L(A) = L(B) =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
3 & 1 & 2 & 6 & 4 & 5 & 9 & 7 & 8 \\
2 & 3 & 1 & 5 & 6 & 4 & 8 & 9 & 7 \\
4 & 5 & 6 & 6 & 5 & 4 & 4 & 5 & 6 \\
6 & 4 & 5 & 4 & 6 & 5 & 6 & 4 & 5 \\
5 & 6 & 4 & 5 & 4 & 6 & 5 & 6 & 4 \\
7 & 8 & 9 & 4 & 5 & 6 & 1 & 2 & 3 \\
9 & 7 & 8 & 6 & 4 & 5 & 3 & 1 & 2 \\
8 & 9 & 7 & 5 & 6 & 4 & 2 & 3 & 1\end{pmatrix}}}}$$
$D$ can also be expressed as the sum of pairs of T-blocks occupying the same vertical lines with opposite depth.
$$D = (T_{4,4,3:\,5,6,6} - T_{4,4,1:\,5,6,4}) + (T_{4,5,1:\,5,6,4} - T_{4,5,2:\,5,6,5})$$
These examples demonstrate an alternative characterisation of two ASHMs with the same corresponding ASHL.
\[T-block\_pairs\] Two ASHMs $A$ and $B$ satisfy $L(A) = L(B)$ if and only if $A-B$ can be expressed as a sum of pairs of T-blocks with opposite depth occupying the same vertical lines.
First assume that two ASHMs $A$ and $B$ satisfy $L(A) = L(B)$. From Theorem \[ASHL\_difference\], we know that $D = A-B$ can be decomposed into T-blocks such that in any vertical line $V$ of $D$, $$\sum_{T \in T_V} d(T) = 0\text{.}$$
Run the following iterative step.
- Let $k_1$ be the least integer for which the plane $P_{k_1}(D)$ contains non-zero entries. For some positive entry $d_{i_1j_1k_1}$ of $D$, we can find entries $d_{i_2j_1k_1}$, $d_{i_1j_2k_1}$ and $d_{i_1j_1k_2}$ with negative sign, where $k_2 > k_1$. Choose $k_2$ to be the least integer satisfying this condition.
As $\sum_{T \in T_{V_{i_1j_1}}} d(T) = 0$, there must also be another positive entry $d_{i_1j_1k_3}$ in $V_{i_1j_1}$. Choose $k_3$ to be the largest integer satisfying this condition. Let $k_4 = k_3 - (k_2-k_1)$, and let $D' = D - T_{i_1,j_1,k_1:\,i_2,j_2,k_2} + T_{i_1,j_1,k_4:\,i_2,j_2,k_3}$. Now repeat this step for $D'$.
Note that the sum of the absolute value of the entries of $P_{k_1}(D')$ is at least $2$ less than that of $P_{k_1}(D)$, and that all line sums of $D'$ are $0$. Note also that $k_4 > k_1$, because the sum of the absolute value of the negative entries in $V$ must equal the sum of the positive entries in $V$ and the weighted sums of each must also equal. If $k_4 \leq k_1$, this implies that all the negative entries of $V$ are positioned above all positive entries in $V$, which means that their weighted sum is negative. Therefore $k_4 > k_1$, which means that $k_1$ remains the lowest integer for which $P_{k_1}(D)$ contains non-zero entries. We can therefore run this iterative process repeatedly, resulting in $P_{k_1}(D')$ becoming a 0-matrix and on the next iteration, $k_1$ will increase to the new lowest integer for which plane $P_{k_1}(D')$ contains non-zero entries until $D'$ is a 0-hypermatrix. Therefore $A-B$ can be expressed as a sum of pairs of T-blocks with opposite depth occupying the same vertical lines.
Now, assume that $A-B$ can be expressed as a sum of pairs of T-blocks with opposite depth occupying the same vertical lines. This means that, in any vertical line $V$ of $A-B$, $$\sum_{T \in T_V} d(T) = \sum_T d(T)-d(T) = 0\text{.}$$ Which, by Theorem \[ASHL\_difference\], means that $L(A) = L(B)$.
So, for any ASHM $A$ with ASHL $L = L(A)$, we have that $A_n(L)$ is the set of all ASHMs $B$ for which $A-B$ can be expressed as a sum of pairs of T-blocks with opposite depth occupying the same vertical lines. The following theorem tells us the smallest dimension an ASHL $L$ can have if $A_n(L)$ contains more than one element.
\[size\_thm\] The minimum $n$ for which two distinct $n \times n \times n$ ASHMs $A$ and $B$ can satisfy $L(A) = L(B)$ is 4.
Two ASHMs $A$ and $B$ can satisfy $L(A) = L(B)$ only if both ASHMs contain at least one negative entry [@ashmbib].
The following are the only two $3 \times 3$ ASHMs containing negative entries, and these do not satisfy $L(A) = L(B)$.
$$A = {{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& & + \\
& + & \\
+ & & \end{pmatrix}}}}
\hspace{-0.1cm}\nearrow\hspace{-0.1cm}{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& + & \\
+ & - & + \\
& + & \end{pmatrix}}}}
\hspace{-0.1cm}\nearrow\hspace{-0.1cm}{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
+ & & \\
& + & \\
& & +\end{pmatrix}}}}$$ $$B = {{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
+ & & \\
& + & \\
& & +\end{pmatrix}}}}
\hspace{-0.1cm}\nearrow\hspace{-0.1cm}{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& + & \\
+ & - & + \\
& + & \end{pmatrix}}}}
\hspace{-0.1cm}\nearrow\hspace{-0.1cm}{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& & + \\
& + & \\
+ & & \end{pmatrix}}}}$$
$$L(A) = {{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
3 & 2 & 1 \\
2 & 2 & 2 \\
1 & 2 & 3\end{pmatrix}}}}
\hspace{1.5cm}
L(B) ={{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
1 & 2 & 3 \\
2 & 2 & 2 \\
3 & 2 & 1\end{pmatrix}}}}$$
The following example is a pair of $4 \times 4 \times 4$ ASHMs with the same corresponding ASHL.
$$A =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& & & + \\
& & + & \\
& + & & \\
+ & & & \end{pmatrix}}}}
\nearrow{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& & + & \\
& + & - & + \\
+ & - & + & \\
& + & & \end{pmatrix}}}}
\nearrow{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& + & & \\
+ & & & \\
& & & + \\
& & + & \end{pmatrix}}}}
\nearrow{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
+ & & & \\
& & + & \\
& + & & \\
& & & +\end{pmatrix}}}}$$
$$B =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& & & + \\
& + & & \\
& & + & \\
+ & & & \end{pmatrix}}}}
\nearrow{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& & + & \\
& & & + \\
+ & & & \\
& + & & \end{pmatrix}}}}
\nearrow{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
& + & & \\
+ & - & + & \\
& + & - & + \\
& & + & \end{pmatrix}}}}
\nearrow{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
+ & & & \\
& + & & \\
& & + & \\
& & & +\end{pmatrix}}}}$$
$$L(A) = L(B) =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
4 & 3 & 2 & 1 \\
3 & 2 & 3 & 2 \\
2 & 3 & 2 & 3 \\
1 & 2 & 3 & 4\end{pmatrix}}}}$$
Therefore the minimum dimension for which two ASHMs $A$ and $B$ can satisfy $L(A) = L(B)$ is 4.
Note that $A_n(L)$ can contain ASHMs with different numbers of non-zero elements. In the following example, the number of non-zero entries in $A$ is 68, while the number of non-zero entries in $B$ is 76.
\[2ashm\]
$$A =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & \textbf{+} & & \\
& & & & \textbf{+} & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & \textbf{+} & & & & \\
& & \textbf{+} & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & \textbf{+} & & & & \\
& & \textbf{+} & & & & & \\
& & & & & & \textbf{+} & \\
& & & & \textbf{+} & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & \textbf{+} & \\
& & & & & & & \textbf{+} \\
& & & & \textbf{+} & & & \\
& & & & & \textbf{+} & & \\
& & & \textbf{+} & & & & \\
& & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& \textbf{+} & & & & & & \\
& & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & \\
& & & & & \textbf{+} & & \\
& & & & & & \textbf{+} & \\
& & & & & & & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & \textbf{+} & & & \\
& & & & & \textbf{+} & & \\
& & & \textbf{+} & & & & \\
& \textbf{+} & & & \textbf{-} & & & \textbf{+} \\
\textbf{+} & & & \textbf{-} & & & \textbf{+} & \\
& & & & \textbf{+} & & & \\
& & \textbf{+} & & & & & \\
& & & \textbf{+} & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & & & & & & & \\
& \textbf{+} & & & & & & \\
& & \textbf{+} & & & & & \\
& & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
& & & & \textbf{+} & & & \\
& & & & & & \textbf{+} & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & & & \textbf{+} & & \\
& & & & \textbf{+} & & & \\
& & \textbf{+} & & & & & \\
& & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & \\
& \textbf{+} & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & \textbf{+} & & & & \\
& \textbf{+} & & & & & & \\
& & & & & \textbf{+} & & \\
& & & & \textbf{+} & & & \end{pmatrix}}}}$$
$$B =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & \textbf{+} & & \\
& & & & \textbf{+} & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & \textbf{+} & & & & \\
& & \textbf{+} & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & \textbf{+} & & & & \\
& & \textbf{+} & & & & & \\
& & & & & & \textbf{+} & \\
& & & & \textbf{+} & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & \textbf{+} & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & \textbf{+} & \\
& & & & & & & \textbf{+} \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & \textbf{-} & \textbf{+} & & \\
& & & & \textbf{+} & & & \\
& & \textbf{+} & & & & & \\
& \textbf{+} & & & & & & \\
\textbf{+} & & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& \textbf{+} & & & & & & \\
& & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & \\
& & \textbf{+} & \textbf{-} & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
& & & & & \textbf{+} & & \\
& & & & & & \textbf{+} & \\
& & & & & & & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & \textbf{+} & & & \\
& & & & & \textbf{+} & & \\
& & & \textbf{+} & & & & \\
& \textbf{+} & & & \textbf{-} & & & \textbf{+} \\
\textbf{+} & & & \textbf{-} & & & \textbf{+} & \\
& & & & \textbf{+} & & & \\
& & \textbf{+} & & & & & \\
& & & \textbf{+} & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & & & & & & & \\
& \textbf{+} & & & & & & \\
& & \textbf{+} & & & & & \\
& & & & \textbf{+} & & & \\
& & & \textbf{+} & \textbf{-} & \textbf{+} & & \\
& & & & & & & \textbf{+} \\
& & & & \textbf{+} & & & \\
& & & & & & \textbf{+} & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & & & \textbf{+} & & \\
& & & \textbf{+} & & & & \\
& & \textbf{+} & \textbf{-} & \textbf{+} & & & \\
& & & \textbf{+} & & & & \\
\textbf{+} & & & & & & & \\
& \textbf{+} & & & & & & \end{pmatrix}}}}
\hspace{-0.2cm}\nearrow
\hspace{-0.16cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
& & \textbf{+} & & & & & \\
\textbf{+} & & & & & & & \\
& & & & & & & \textbf{+} \\
& & & & & & \textbf{+} & \\
& & & \textbf{+} & & & & \\
& \textbf{+} & & & & & & \\
& & & & & \textbf{+} & & \\
& & & & \textbf{+} & & & \end{pmatrix}}}}$$
$$L(A) = L(B) =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
6 & 4 & 8 & 2 & 5 & 1 & 3 & 7 \\
8 & 6 & 2 & 4 & 1 & 5 & 7 & 3 \\
4 & 1 & 6 & 5 & 3 & 7 & 2 & 8 \\
1 & 5 & 4 & 6 & 4 & 3 & 8 & 5 \\
5 & 2 & 7 & 6 & 4 & 6 & 5 & 1 \\
2 & 8 & 3 & 7 & 5 & 4 & 1 & 6 \\
7 & 3 & 5 & 1 & 6 & 8 & 4 & 2 \\
3 & 7 & 1 & 5 & 8 & 2 & 6 & 4\end{pmatrix}}}}$$
The Maximum Number of Equal Entries of an ASHL
==============================================
The following question is also posed in Brualdi and Dahl’s paper [@ashmbib].
What is the maximum number of times an integer can occur as an entry of an $n \times n$ ASHL?
It is shown in their paper that an integer can occur $2n$ times in an $n \times n$ ASHL, and it is asked if the maximum is equal to $2n$. The following example exceeds this bound.
\[max\_entries7\] Here, 4 occurs as an entry in this $7 \times 7$ ASHL 29 times. $$A =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$
$$L(A) =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
6 & 3 & 1 & 4 & 7 & 5 & 2 \\
3 & 4 & 4 & 4 & 4 & 4 & 5 \\
1 & 4 & 4 & 4 & 4 & 4 & 7 \\
4 & 4 & 4 & 4 & 4 & 4 & 4 \\
7 & 4 & 4 & 4 & 4 & 4 & 1 \\
5 & 4 & 4 & 4 & 4 & 4 & 3 \\
2 & 5 & 7 & 4 & 1 & 3 & 6\end{pmatrix}}}}$$
This is the highest possible number of times an entry can be repeated in a $7 \times 7$ ASHL, as each number $1, 2, \dots, n$ must appear exactly once in the first and last rows and columns of an $n \times n$ ASHL. This means that the upper bound for such a construction is $(n-2)^2 + 4$, which is $(7-2)^2+4=29$, in the $n=7$ case.
We define the *diamond positions* of an $n \times n$ ASM $A$ to be the positions of $A$ corresponding to non-zero entries of the diamond ASM $D_n$.
Example \[max\_entries7\] can be generalised in the following way.
\[max\_entries\] For a given $n$, there exists an $n \times n$ ASHL such that
- $\frac{n+1}{2}$ occurs as an entry $\frac{n^2+4n-19}{2}$ times, if $n$ is odd;
- $\frac{n}{2}$ occurs as an entry $\frac{n^2+4n-20}{2}$ times, if $n$ is even.
Let $p = \lfloor \frac{n+1}{2} \rfloor$, $m = \lceil\frac{n+1}{2}\rceil$, and note that $p=m$ for odd $n$. We construct an ASHM $A$ with the required properties as follows.
- $P_p(A) = D_n$, and for $k = 1, 2, \dots, p-1$, plane $P_{p \pm k}(A)$ contains the diamond ASM $D_{n-2k}$ such that there is a $+$ entry in every position where there is a $-$ entry in the diamond ASM contained in the plane $P_{p \pm (k-1)}$.
- The other non-zero entries of $P_{p-1}(A)$ are a diagonal of $+$ entries from $A_{1,m+2,p-1}$ to $A_{p-2,n,p-1}$, a diagonal of $-$ entries from $A_{2,m+2,p-1}$ to $A_{p-2,n-1,p-1}$, a diagonal of $+$ entries from $A_{m+2,1,p-1}$ to $A_{n,p-2,p-1}$, and a diagonal of $-$ entries from $A_{m+2,2,p-1}$ to $A_{n-1,p-2,p-1}$.
- The other non-zero entries of $P_1(A)$ are a diagonal of $+$ entries from $A_{1,m+1,1}$ to $A_{p-1,n,1}$ and a diagonal of $+$ entries from $A_{m+1,1,1}$ to $A_{n,p-1,1}$.
- The other non-zero entries of $P_2(A)$ are an anti-diagonal of $+$ entries from $A_{2,p-2,2}$ to $A_{p-2,2,2}$, an anti-diagonal of $+$ entries from $A_{m+2,n-1,2}$ to $A_{n-1,m+2,2}$, and $+$ entries in $A_{1,1,2}$ and $A_{n,n,2}$.
- For $k=2, \dots, p-3$, the other non-zero entries of $P_{p-k}(A)$ are an anti-diagonal of $+$ entries from $A_{1,k,p-k}$ to $A_{k,1,p-k}$, and an anti-diagonal of $+$ entries from $A_{n-k+1,n,p-k}$ to $A_{n,n-k+1,p-k}$.
- For $k = 1, 2, \dots, p-1$, the entries of $P_{p+k}(A)$ not containing $D_{n-2k}$ (as outlined in the first step) satisfy $A_{i,j,p+k} = A_{n-i, j, p-k}$.
- If $n$ is even, the non-zero entries of $P_n(A)$ are an anti-diagonal from $A_{p,1,n}$ to $A_{1,p,n}$ and an anti-diagonal from $A_{n,m,n}$ to $A_{m,n,n}$.
We see that $p$ occurs in all diamond positions of $L = L(A)$ because $$(p-k+1) - (p-k+2) + \dots -(p+k-2) + (p+k-1) = p\text{.}$$
The other occurances of $p$ as entries of $L$ occur along diagonals from $L_{2,m+2}$ to $L_{p-2,n-1}$ and from $L_{m+2,2}$ to $L_{n-1,p-2}$ by $$1 - (p-1) + (n+p-m-1) = n-m+1 = p\text(,)$$ and along antidiagonals from $L_{p-2,2}$ to $L_{2,p-2}$ and from $L_{n-1, m+2}$ to $L_{m+2,n-1}$ by $$2 - (p+1) + (n+p-m) = n-m+1 = p\text{.}$$
: $${{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+}\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow
\hdots
\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \textbf{-} & \textbf{+} & \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \hdots & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \textbf{-} & \textbf{+} & \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \ddots & \ddots & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \iddots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \ddots & \ddots & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \iddots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \hdots & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} \\
\vspace{-0.1cm} \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$ $$\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \iddots & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \iddots & \iddots & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \iddots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \iddots & \iddots & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \textbf{-} & \textbf{+} & \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \hdots & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \textbf{-} & \textbf{+} & \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow
\hdots
\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\vspace{-0.1cm} \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$
It can be easily seen that each plane of this hypermatrix is an ASM. All vertical lines of $A$ corresponding to diamond positions of $L$ clearly have the alternating property. All vertical lines corresponding to the diagonal from $(2,p+2)$ to $(p-2,n-1)$, the diagonal from $(p+2,2)$ to $(n-1,p-2)$, the anti-diagonal from $(2,p-2)$ to $(p-2,2)$, and the anti-diagonal from $(p+2,n-1)$ to $(n-1, p+2)$ have exactly three non-zero entries, which alternate $+, - , +$. All other vertical lines contain exactly one non-zero entry, namely one $+$ entry. Therefore this is an ASHM.
The $p$ entries occur in the following positions of the corresponding ASHL. $${{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\color{white} 0 & p & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 00 \\
\vspace{-0.1cm}\color{white} 0 & \color{white} 0 & \color{white} 0 & p & p & p & p & p & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white} 0 & \iddots & \iddots & p & \vdots & p & \ddots & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0& p & \iddots & \iddots &\color{white} p &\color{white} p & \color{white} p &\ddots & \ddots & p & \color{white} 0 \\
\color{white} 0\color{white} 0 & p & p &\color{white} p & \color{white} p &\color{white} p &\color{white} p &\color{white} p & p & p & \color{white} 0 \\
p & p & \hdots & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \hdots & p & p \\
\vspace{-0.1cm}\color{white} 0 & p & p & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \color{white} p & p & p & \color{white} 0\color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & p & \ddots & \ddots & \color{white} p & \color{white} p & \color{white} p & \iddots & \iddots & p & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \ddots & p & \vdots & p & \iddots & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & p & p & p & p & p & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 00 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & p & \color{white} 0\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$
As outlined above, $p$ occurs as an entry in the diamond positions of $L$, and also occurs as every entry in the diagonal from $L_{2,p+2}$ to $L_{p-2,n-1}$, the diagonal from $L_{p+2,2}$ to $L_{n-1,p-2}$, the anti-diagonal from $L_{2,p-2}$ to $L_{p-2,2}$, and the anti-diagonal from $L_{p+2,n-1}$ to $L_{n-1, p+2}$.
Therefore $p$ occurs as an entry of $L$ a total of $\frac{n^2+4n-19}{2}$ times: $$(1 + 3 + \dots + n-2 + n + n-2 + \dots + 3 +1) + 4\Big(\frac{n+1}{2}-3\Big) = \Big(\frac{n+1}{2}\Big)^2 + \Big(\frac{n-1}{2}\Big)^2 + (2n -10) = \frac{n^2+4n-19}{2}$$
$$\hspace{-0.4cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 &\color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+}\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow
\hdots
\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \hdots & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \hdots & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \ddots & \ddots & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \hdots & \color{white} \textbf{-} & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \hdots & \color{white} \textbf{-} & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \color{white} \textbf{-} & \color{white} \textbf{+} &\color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \ddots & \ddots & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \hdots &\color{white} \textbf{-} &\color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \hdots & \color{white} \textbf{+} &\color{white} \textbf{-} &\color{white} \textbf{+} & \textbf{-} & \textbf{+} \\
\textbf{+} & \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \hdots & \color{white} \textbf{+} &\color{white} \textbf{-} &\color{white} \textbf{+} & \textbf{-} & \textbf{+}& \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \color{white} \textbf{+} &\color{white} \textbf{-} & \color{white} \textbf{+} &\color{white} \textbf{-} & \color{white} \hdots &\color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots &\color{white} \textbf{-} & \color{white} \textbf{+} &\color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$ $$\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \iddots & \textbf{-} & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \iddots & \iddots & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \hdots & \color{white} \textbf{-} & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \hdots & \color{white} \textbf{-} & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \color{white} \textbf{-} & \color{white} \textbf{+} &\color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \hdots & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} \\
\vspace{-0.1cm}\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \iddots & \iddots & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \hdots & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} \textbf{+} & \hdots & \color{white} \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} \textbf{-} & \color{white} \textbf{+} & \color{white} \textbf{-} & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow
\hdots
\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\textbf{+} & \color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white}0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white}0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\vspace{-0.1cm}\color{white}0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 &\color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$
It can be easily seen that each plane of this hypermatrix is an ASM. All vertical lines of $A$ corresponding to diamond positions of $L$ clearly have the alternating property. All vertical lines corresponding to the diagonal from $(2,m+2)$ to $(p-2,n-1)$, the diagonal from $(m+2,2)$ to $(n-1,p-2)$, the anti-diagonal from $(2,p-2)$ to $(p-2,2)$, and the anti-diagonal from $(m+2,n-1)$ to $(n-1, m+2)$ have exactly three non-zero entries, which alternate $+, - , +$. All other vertical lines contain exactly one non-zero entry, namely one $+$ entry. Therefore this is an ASHM.
The $p$ entries occur in the following positions of the corresponding ASHL. $${{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\color{white} 0 & p & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 00 \\
\vspace{-0.1cm}\color{white} 0 & \color{white} 0 & \color{white} 0 & p & \color{white} 0 & p & p & p & p & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & \color{white} 0 & \iddots & \color{white} 0 & \iddots & p & \vdots & p & \ddots & \ddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0& p & \color{white} 0 & \iddots & \iddots &\color{white} p &\color{white} p & \color{white} p &\ddots & \ddots & p & \color{white} 0 \\
\color{white} 0\color{white} 0 & \color{white} 0 & p & p &\color{white} p & \color{white} p &\color{white} p &\color{white} p &\color{white} p & p & p & \color{white} 0 \\
\color{white} 0 & p & p & \hdots & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \hdots & p & p \\
p & p & \hdots & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \hdots & p & p & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & p & p & \color{white} p & \color{white} p & \color{white} p & \color{white} p & \color{white} p & p & p & \color{white} 0 & \color{white} 0 \\
\vspace{-0.1cm}\color{white} 0 & p & \ddots & \ddots & \color{white} p & \color{white} p & \color{white} p & \iddots & \iddots & \color{white} 0 & p & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \ddots & \ddots & p & \vdots & p & \iddots &\color{white} 0 & \iddots & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & p & p & p & p & \color{white} 0 & p & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 00 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & p & \color{white} 0\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$
As outlined above, $p$ occurs as an entry in the diamond positions of $L$, and also occurs as every entry in the diagonal from $L_{2,m+2}$ to $L_{p-2,n-1}$, the diagonal from $L_{m+2,2}$ to $L_{n-1,p-2}$, the anti-diagonal from $L_{2,p-2}$ to $L_{p-2,2}$, and the anti-diagonal from $L_{m+2,n-1}$ to $L_{n-1, m+2}$.
Therefore $p$ occurs as an entry of $L$ a total of $\frac{n^2+4n-20}{2}$ times: $$2(1 + 3 + \dots + n-1) + 4\Big(\frac{n+1}{2}-3\Big) = 2\Big(\frac{n}{2}\Big)^2 + (2n -10) = \frac{n^2+4n-20}{2}$$
Note that this bound is not tight. This is currently our best general construction, but we have constructed specific examples narrowly exceeding this bound.
In the $n=11$ case, the construction outlined in the proof of Theorem \[max\_entries\] gives an $n \times n \times n$ ASHM $A$ with ASHL $L(A)$ containing the same entry $73$ times. The ASHM $B$, with $L(B)$ containing the same entry $77$ times, is obtained by the addition of T-blocks to $A$ as follows. $$B = A + T_{2,1,3:\,3,2,4} + T_{9,10,3:\,10,11,4} - T_{2,2,4:\,11,10,8} + T_{9,1,8:\,10,2,9} + T_{2,10,8:\,3,11,9}$$ Explicitly, $B$ is the following ASHM. $$B = {{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+}\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} \\ \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$ $$\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.12cm}\nearrow\hspace{-0.12cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$
Which has the following corresponding ASHL. $$L(B) =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
2 & 4 & 3 & 7 & 11 & 6 & 1 & 5 & 9 & 8 & 10 \\
3 & 8 & 7 & 6 & 6 & 6 & 6 & 6 & 5 & 4 & 9 \\
4 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 8 \\
7 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 5 \\
11 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 1 \\
6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\
1 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 11 \\
5 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 7 \\
8 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 4 \\
9 & 8 & 5 & 6 & 6 & 6 & 6 & 6 & 7 & 4 & 3 \\
10 & 4 & 9 & 5 & 1 & 6 & 11 & 7 & 3 & 8 & 2\end{pmatrix}}}}$$
In the $n=13$ case, the construction outlined in the proof of Theorem \[max\_entries\] gives an $n \times n \times n$ ASHM $A$ with ASHL $L(A)$ containing the same entry $101$ times. The following ASHM $B$ exceeds this, with $L(B)$ containing the same entry $103$ times. $$B =
{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+}\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow$$ $$\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \textbf{-} & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \textbf{-} & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \textbf{-} & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{-} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}
\hspace{-0.2cm}\nearrow\hspace{-0.2cm}{{ \tiny \arraycolsep=0.08\arraycolsep\ensuremath{\begin{pmatrix}
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 \\
\color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \textbf{+} & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0 & \color{white} 0\end{pmatrix}}}}$$
$$L(B) =
{{ \small \arraycolsep=0.8\arraycolsep\ensuremath{\begin{pmatrix}
4 & 11 & 5 & 10 & 6 & 1 & 7 & 13 & 12 & 2 & 3 & 9 & 8 \\
11 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 3 & 4 & 10 & 5 & 9 \\
5 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 13 & 4 & 10 & 3 \\
10 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 6 & 13 & 4 & 2 \\
6 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 3 & 12 \\
1 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 13 \\
7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 \\
13 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 1 \\
12 & 3 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 6 \\
2 & 4 & 13 & 6 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 10 \\
3 & 10 & 4 & 13 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 5 \\
9 & 5 & 10 & 4 & 3 & 7 & 7 & 7 & 7 & 7 & 7 & 7 & 11 \\
8 & 9 & 3 & 2 & 12 & 13 & 7 & 1 & 6 & 10 & 5 & 11 & 4\end{pmatrix}}}}$$
We conclude by posing the following problem.
Is it possible to construct an $n \times n \times n$ ASHM $A$ for which $(n-2)^2 +4$ entries of $L(A)$ are equal, for $n > 7$?
Acknowledgements {#acknowledgements .unnumbered}
================
I would firstly like to thank Dr Trent Marbach, who first brought [@ashmbib] to my attention. I would also like to thank my PhD supervisors, Dr Rachel Quinlan and Dr Kevin Jennings, for supplying guidance on the presentation of this paper. This research was funded by a Hardiman Research Scholarship.
[9]{} R. Brualdi, G. Dahl. *Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Squares*. Advances in Applied Mathematics, **95**(10): 1016, 2018.
A. Lascoux, M.-P. Schützenberger *Treillis et bases des groupes de Coxeter* Electronic Journal of Combinatorics, **3**(2 R): 1-35, 1996.
R. Brualdi, K. Kiernan, S. Meyer, M. Schroeder. *Patterns of Alternating Sign Matrices*. Linear Algebra and its Applications, **438**(10): 3967-3990, 2013.
|
---
abstract: 'Double neutron star mergers are strong sources of gravitational waves. The upcoming advanced gravitational wave detectors are expected to make the first detection of gravitational wave bursts (GWBs) associated with these sources. Proposed electromagnetic counterparts of a GWB include a short GRB, an optical macronova, and a long-lasting radio afterglow. Here we suggest that at least some GWBs could be followed by an early afterglow lasting for thousands of seconds, if the post-merger product is a highly magnetized, rapidly rotating, massive neutron star rather than a black hole. This afterglow is powered by dissipation of a proto-magnetar wind. The X-ray flux is estimated to be as bright as $(10^{-8}-10^{-7})~{\rm erg~s^{-1}~cm^{-2}}$. The optical flux is subject to large uncertainties but could be as bright as 17th magnitude in R-band. We provide observational hints of such a scenario, and discuss the challenge and strategy to detect these signals.'
author:
- Bing Zhang
title: 'Early X-ray and optical afterglow of gravitational wave bursts from mergers of binary neutron stars'
---
Introduction
============
Mergers of neutron-star/neutron-star (NS-NS) binaries are strong sources of gravitational waves [e.g. @kramer06]. The upcoming advanced gravitational wave detectors such as Advanced LIGO [@ligo] and Advanced VIRGO [@virgo] are expected to expand the detection horizon to a few hundred Mpc for NS-NS mergers as early as 2015. Theoretical motivation [@eichler89; @narayan92; @rosswog12] and observational progress [e.g. @gehrels05; @barthelmy05a; @berger05] suggest that at least some short gamma-ray bursts (SGRBs) may be related to NS-NS mergers. This hypothesis can be proved when both a SGRB and a gravitational wave burst (GWB) are detected in coincidence with each other in trigger time and direction. On the other hand, observations of SGRBs suggest that at least some of them are collimated [e.g. @burrows06; @depasquale10]. Since the strength of the gravitational wave signals does not sensitively depend on the orientation of the NS-NS merger orbital plane with respect to the line of sight, most GWBs would not be associated with SGRBs even if the SGRB-GWB association is established. Searching for electromagnetic counterparts of SGRB-less GWBs is essential to confirm the astrophysical origin of the GWBs, and to advance our understanding of the compact star merger physics. In the literature, an optical “macronova” [@lipaczynski98; @kulkarni05; @metzger10] due to decay of the ejecta launched during the merger[^1] and a long-lasting radio afterglow due to interaction between the ejecta and the ambient medium [@nakar11; @metzger12; @piran12] have been predicted. Both are challenging to detect [@metzger12]. Here we suggest another possible electromagnetic counterpart of GWBs. We argue that if the post-merger product is a short-lived massive neutron star rather than a black hole, a SGRB-less GWB could be followed by an early X-ray and optical afterglow extending for thousands of seconds. We provide observational hints of such a possibility in §2. In §3, we estimate the duration and brightness of the X-ray and optical afterglows, and discuss their detectability. A brief summary is given in §4.
Massive neutron star as the post-merger object
==============================================
There are two lines of reasoning to suspect that NS-NS mergers can produce a massive NS rather than a black hole, which may survive for an extended period of time of the observational interest. The first is along the line of the observations of NSs and NS-NS binaries in the Galaxy (e.g. [@lattimer12] for a review). A secure lower limit of the maximum NS mass is set by PSR J1614-2230 (in a NS-WD binary) to $1.97\pm 0.04 M_\odot$ through a precise measurement of the Shapiro delay [@demorest10]. NSs with possibly even higher masses, albeit with large uncertainties, are also suggested. For example, the NS candidate in the X-ray binary 4U 1700-377 has a mass $2.44 \pm 0.27 M_\odot$ [@rawls11], and the NS in the NS-WD binary PSR B1516+02B has a mass $2.08 \pm 0.19 M_\odot$ [@freire08]. A stiff equation of state (EOS) of neutron matter is demanded by the data. Although current data do not allow to differentiate among various stiff EOS models, most of these stiff-EOS NS models predict a maximum NS mass close to or higher than $2.5 M_\odot$ for a non-rotating NS [@lattimer12]. For rapidly spinning NSs which are likely relevant for the post-merger products, the maximum mass can be even higher due to a centrifugal support. On the other hand, the observations of the Galactic NS-NS systems suggest that the NS mass in these systems peak at $1.35 M_\odot$, and the sum of the two NS masses for a significant fraction of the population is around $2.6 M_\odot$ [@lattimer12]. Numerical simulations suggest that NS-NS mergers typically eject several percent solar masses [@rosswog12]. As a result, the post-merger products of at lease a fraction (e.g. $f_{\rm NS} \sim 0.5$) of NS-NS merger events should have a total mass below the maximum NS mass of a rapidly spinning NS. This NS would not collapse until loosing a significant amount of angular momentum within the characteristic spin-down time scale. Such a possibility was suggested by [@dai06] and [@gao06] to interpret X-ray flares and plateaus following SGRBs, and is now strengthened by additional data.
The second line of reasoning is based on the observations of the SGRB X-ray afterglows. The most direct evidence of a spinning-down object at the SGRB central engine is in GRB 090515 detected by Swift [@rowlinson10]. After a short prompt emission phase lasting for $T_{90} = 0.036 \pm 0.016$ s, the burst showed an X-ray plateau that lasted for $\sim 240$ s, after which the flux declines rapidly, and became undetectable by XRT at $\sim 500$ s after the trigger [@rowlinson10]. Such a steady plateau with rapid decline would be a signature of a magnetar at the central engine [@zhangmeszaros01a; @troja07]. Even though no redshift measurement was made for this burst, an analysis suggests that the presumed heavy NS has parameters consistent with a magnetar for a reasonable redshift range [@rowlinson10]. A later systematic analysis of Swift SGRB X-ray lightcurves suggests that a significant fraction of SGRBs have evidence of an X-ray plateau followed by a steep drop in flux, which is consistent with a magnetar central engine [@rowlinson12]. If SGRBs are associated with NS-NS mergers, it is likely that a millisecond magnetar survived in these SGRBs.
Another indirect piece of evidence is X-ray flares following some SGRBs [@barthelmy05a]. A possible interpretation is the magnetic activity of a differentially rotating massive NS after a NS-NS merger [@dai06]. If the magnetic field strength of this post-merger massive NS is not too high (similar to that of normal pulsars), the magnetic activity of the NS has the right time scale and luminosity to account for X-ray flares.
Early X-ray and optical afterglow of NS-NS merger-induced GWBs
==============================================================
At least some SGRBs are collimated [@burrows06; @depasquale10]. For the standard X-ray afterglow component (that originates from the external shock of the SGRB jet), the afterglow jet opening angle is believed to be comparable to the prompt emission jet opening angle, so that a GWB without a SGRB association would have a very faint “orphan” afterglow peaking at a time when the jet is decelerated enough so that the $1/\Gamma$ cone enters line-of-sight. The prospects of detecting such a SGRB orphan afterglow are poor. Here we suggest that the afterglow powered by a rapidly spinning massive NS has a much wider solid angle than the solid angle of the SGRB jet, so that [*SGRB-less GWBs can also have a bright afterglow from a dissipating proto-magnetar wind with a large solid angle*]{}. At the base of the central engine (light cylinder), the wind launched from the millisecond magnetar is essentially isotropic. Numerical simulations suggest that this proto-magnetar wind from a NS-NS merger progenitor would be collimated by the ejecta launched during the merger process, but with a much larger angle, $30^\circ-40^\circ$, than the case of a massive-star core-collapse progenitor [@bucciantini12]. This is much larger than the jet opening angle inferred from the afterglow modeling of some SGRBs [@burrows06; @depasquale10]. A wider solid angle of proto-magnetar wind than the GRB jet angle was also inferred from an analysis of the magnetar engine candidates for long GRBs [@lyons10].
In the following, we adopt the [*ansatz*]{} that some NS-NS mergers produce a massive magnetar. The proto-magnetar wind is essentially isotropic at the base, with a wide solid angle $\theta_{w,1} \sim 40^{\circ}$ for a free wind (with a beaming factor $f_{b,w,1}=\Delta \Omega_{w,1} / 4\pi \sim 0.2$) and an even larger solid angle $\Delta \Omega_{w,2}$ in the equatorial direction for a confined wind that pushes the heavy ejecta launched during the merger phase (with a beaming factor $f_{b,w,2}=\Delta \Omega_{w,2} / 4\pi
\sim 0.8$, so that the total beaming factor is $f_{b,w}=f_{b,w,1}+f_{b,w,2} \sim 1$). This hypothesis applies regardless of whether the GWB is associated with a SGRB. If there is a GWB/SGRB association, we expect that SGRB jets have a much smaller solid angle. For example, if the typical SGRB jet opening angle is $\theta_{j} \sim 10^\circ$, one has the jet beaming factor $f_{b,j} = \Delta \Omega_j/4\pi \sim 0.015$, so that $\Delta \Omega_j \ll \Delta \Omega_{w,1} < \Delta \Omega_{w,2}$.
The NS-NS merger event rate is very uncertain. The rate inferred from the Galactic NS-NS systems has a wide range $2 - 2\times 10^{4}
~{\rm Gpc^{-3}~yr^{-1}}$ [@phinney91; @kalogera04; @abadie10]. This is consistent with the upper limit $2\times 10^5~{\rm Gpc^{-3}
~yr^{-1}}$ set by the current non-detection with the last LIGO and VIRGO run [@ligo]. Within the advanced LIGO horizon $\sim 300$ Mpc, the NS-NS merger rate (and therefore GWB rate) would be $R_{\rm GWB}\sim (0.2 - 2000) ~{\rm yr^{-1}}$. Among these, $R_{\rm GWB-ag}\sim (0.1 - 1000) (f_{\rm NS}/0.5) (f_{b,w})
~{\rm yr^{-1}}$ would have strong afterglow emission associated with the proto-magnetar wind, most of which would not have a SGRB association, since the line of sight is outside the SGRB cone even if there is a SGRB/GWB association.
After the merger, the proto-NS is initially very hot and cools via neutrino emission. After about 10 seconds, the NS is cooled enough so that a Poynting-flux-dominated outflow can be launched [@usov92; @metzger11]. It will be spun down by magnetic dipole radiation and by the torque of a strong electron-positron pair wind flowing out from the magnetosphere. Since before the merger the two NSs are in the Keplerian orbits, the post-merger product should be near the break-up limit. We take $P_0=1~{\rm ms}~P_{0,-3}$ as a typical value of the initial spin period of the proto-magnetar. An uncertain parameter is the polar-cap magnetic field of the dipole magnetic field component, $B_{p}$, which depends on whether the $\alpha-\Omega$ dynamo is efficiently operating, and on the magnetic field strength of the parent NSs if the dynamo mechanism is not efficient. Given nearly the same amount of the total rotation energy $E_{\rm rot} = (1/2) I \Omega_0^2 \sim 2\times 10^{52}~{\rm erg}$ ($\Omega_0 = 2\pi/P$), the luminosity, and hence, the afterglow flux critically depend on $B_p$. As a rough estimate, we apply the dipole spindown formula. Correcting for the beaming factor $f_w$ and the efficiency factor $\eta_x$ to convert the spin down luminosity to the observed X-ray luminosity in the detector band, one gets $$\begin{aligned}
F_x & = & \frac{\eta_x L_{sd}}{4 \pi f_{b,w} D_L^2}
\simeq 2\times 10^{-8}~{\rm erg~s^{-1}~cm^{-2}} \nonumber \\
& \times & \eta_{x,-2} f_{b,w}^{-1} \left(\frac{D_L}{300~{\rm Mpc}}
\right)^{-2} I_{45} P_{0,3}^{-2} T_{sd,3}^{-1},
\label{Fx}\end{aligned}$$ where $L_{sd} = I \Omega_0^2 / (2 T_{sd})$ is the characteristic spindown luminosity, and $$T_{sd} \simeq 2\times 10^3~{\rm s}~ I_{45} B_{p,15}^{-2} P_{0,-3}^2
R_6^{-6}
\label{Tsd}$$ is the characteristic spindown time scale. Here $I = 10^{45} I_{45}$ is the moment of inertia (typical value $I_{45} = 1.5$ for a massive NS), $R = 10^6 R_6$ is the radius of the NS, and the convention $Q_x = Q/10^x$ has been adopted. Here we have assumed that a good fraction ($\eta_x \sim 0.01$) of spin down energy is released in the X-ray band. This is based on the following two considerations: First, some SGRBs indeed have a bright X-ray plateau that is likely due to the magnetar spindown origin [@rowlinson10; @rowlinson12], which suggests that the main energy channel of releasing the magnetic dissipation energy is in the X-ray band. Second, a rough theoretical estimate shows that the typical energy band of a dissipating magnetized wind with a photon luminosity $L_\gamma \sim 10^{49}~{\rm erg~s^{-1}}$ could be in X-rays.
We consider two mechanisms to dissipate the magnetar wind energy to radiation. (1) In the free wind zone with solid angle $\Delta \Omega_{w,1}$, one may consider a magnetized wind with a luminosity $L_w$ and magnetization parameter $\sigma(R)$ dissipated at a radius $R$ from the central engine. Assuming that the magnetic energy is abruptly converted to the internal energy of power-law distributed electrons (such as in the scenario of the ICMART model), one can generally estimate the typical synchrotron energy as [@zhangyan11] $E_p \simeq 80~{\rm keV}~ L_{\gamma,48}^{1/2} R_{15}^{-1} \eta_{x,-2}^{3/2}
\sigma_4^2$. A cooled-down proto-magnetar typically has $\sigma_0
\sim 10^9$ at the central engine [@metzger11]. A magnetized flow can be quickly accelerated to $\Gamma \sim \sigma_0^{1/3}
\sim 10^3$ at $R_0 \sim 10^7$ cm, where $\sigma \sim \sigma_0^{2/3}
\sim 10^6$ [@komissarov09]. After this phase, the flow may still accelerate as $\Gamma \propto R^{1/3}$, with $\sigma$ falling as $\propto R^{-1/3}$ [@drenkhahn02]. At $R \sim 10^{15}$ cm, one has $\sigma \sim 2\times 10^3$, so that $E_p \sim 3.7$ keV, which is in the X-ray band. (2) One can also consider the confined magnetar wind zone with solid angle $\Delta \Omega_{w,2}$ where the magnetar wind is expanding into a heavy ejecta launched during the merger process[^2]. The magnetic energy may be rapidly discharged upon interaction between the wind and the ejecta, which occurs at a radius $R \sim v t_{\rm delay} = 3\times 10^{10}~{\rm cm}
(v/0.1 c) t_{\rm delay,1}$, where $v \sim 0.1 c$ is the speed of ejecta, and $t_{\rm delay} \sim 10$ s is the delay time between the merger and the launch of a high-$\sigma$ magnetar wind. The Thomson optical depth for a photon to pass through the ejecta shell is $\tau_{th} \sim
\sigma_{\rm T} M_{\rm ej} / (4\pi R^2 m_p) \sim 7\times 10^8
(M_{ej}/(0.01M_\odot)) \gg 1$. So the spectrum of the dissipated wind is thermal-like. One can estimate the typical energy $\sim k (L_w/4\pi R^2 \sigma)^{1/4}
\sim 5~{\rm keV} L_{w,49}^{1/4} (R/(3\times 10^{10}~{\rm cm}))^{-1/2}$, which is also in the X-ray band.
One can see that the X-ray band flux of the early afterglow (Eq.\[Fx\]) is very high, well above the sensitivity threshold of Swift XRT [@moretti07] $$F_{x,th} = 6 \times 10^{-12}~{\rm erg~s^{-1}~cm^{-2}}~ T_{\rm obs,2}^{-1},$$ where $T_{\rm obs}$ is the observation time. The light curve is expected to be flat (a plateau) lasting for a duration $T_{sd}$ followed by a $t^{-2}$ decay. However, since the NS spins down quickly in the $t^{-2}$ regime, it is likely that it would lose centrifugal support and collapse to a black hole shortly after the end of the plateau. In this case, essentially all the materials collapse into the BH, without substantial accretion afterwards. The light curve then shows a very sharp drop in flux at the end of the plateau, similar to what is seen in GRB 090515 [@rowlinson10].
The challenge to detect such a bright X-ray afterglow following a GWB is its short duration (Eq.\[Tsd\]) and the large error box of a GWB trigger. This requires a Swift-like space detector for quick slew, but the error box of the GWB trigger, typically a few tens to a hundred square degrees [@abadie12], is much larger than the XRT field of view (0.16 square degree). How to efficiently search for the bright X-ray source within $T_{sd}$ in such a large sky area is challenging. Even though some strategies using Swift have been proposed [@kanner12], the current searches for the GWB afterglow typically happen about half-day after the GWB trigger [@evans12]. The problem can be alleviated if $B_p$ of the proto-NS is weaker. For example, even for a typical pulsar field $B_p \sim 10^{12}$ G [@dailu98b; @dai06], the X-ray luminosity can be still as high as $5\times 10^{-11} ~{\rm erg~s^{-1}~cm^{-2}}$ for detection, while the duration of the plateau extends to $T_{sd} \sim 2
\times 10^7$ s. This would give enough time to search for the X-ray afterglow. However, strong magnetic fields are likely generated during the merger events [@price06]. Very likely one has to face the large-error-box, short-duration problem. An ideal strategy to observe this early afterglow is to design a large field-of-view imaging X-ray telescope, preferably with fast-slewing capability. Such a telescope, even with a moderate sensitivity, can catch the bright early X-ray afterglows of SGRB-less GWBs. The new mission concept ISS-Lobster [@gehrels12] invokes an X-ray wide-field imager with a 0.5-sr field of view that covers $\sim 50\%$ of the sky every 3 hours, which is ideal to detect this bright X-ray afterglow.
The optical flux of the proto-magnetar wind is subject to uncertainties. In the free wind zone (solid angle $\Delta \Omega_{w,1}$), the emission spectral shape is synchrotron. If one has the standard $F_\nu
\propto \nu^{1/3}$ synchrotron spectrum below $E_p$, the specific X-ray flux at 1 keV $F_\nu({\rm X}) \sim 4$ mJy would correspond to a R-band magnitude 17. This would be an optimistic estimate of the optical brightness. For the confined wind zone (solid angle $\Delta \Omega_{w,2}$), the spectrum of a dissipating wind is quasi-thermal, and the optical flux is greatly suppressed. Indeed, no bright optical emission was detected during the plateau phase of GRB 090515 [@rowlinson10], suggesting that the optical emission of a dissipative proto-magnetar wind is suppressed. Nonetheless, the interaction between the magnetar wind and the ejecta in the confined wind zone can give very interesting radiation signatures in the optical band (H. Gao et al. 2013, in preparation). Wide-field optical telescopes are essential to search for such optical GWB afterglows in the large GWB error box.
The gravitational wave signals from these GWBs have an interesting signature: after the standard chirp signal during the in-spiral and merger phases [@flanagan98; @kobayashimeszaros03], there should be an extended GW emission episode afterwards due to a secular bar-mode instability of the newly formed proto-magnetar [@corsi09]. The signature is in the advanced LIGO frequency band, and can in principle be detected. Jointly detecting such a GW signal along with the X-ray afterglow would give an unambiguous identification of the proto-magnetar nature of the central engine.
Some SGRBs are followed by an extended emission, which sometimes can be very bright [e.g. @gehrels06]. It is unclear whether the extended emission shares the same solid angle with the short hard spikes. If it has a wider solid angle than the short hard spike emission, as expected in the magnetar engine scenario [e.g. @metzger08], then such a bright extended emission (lasting $\sim 100$ s) can be also associated with SGRB-less GWBs. This emission is brighter than the X-ray afterglow emission discussed above, and can be readily detected by wide-field imagers such as ISS-Lobster.
Summary
=======
We have proposed another electromagnetic counterpart of GWBs from NS-NS mergers. It applies to the cases when the two NSs are not very massive (as observed in Galactic double NS systems), so that the post-merger product has a mass below the maximum mass of a rapidly spinning NS. We show that such a scenario is plausible in view of the observations of Galactic NSs, NS-NS systems, and SGRB afterglows. The proto-magnetar would eject a wide-beam wind, whose dissipation would power an X-ray afterglow as bright as $\sim (10^{-8}-10^{-7})
~{\rm erg~s^{-1}~cm^{-2}}$. The duration is typically $10^3 - 10^4$ s, depending on the strength of the dipolar magnetic fields. It is challenging to detect the X-ray afterglow with the current facilities such as Swift, but a wide-field X-ray imager (such as ISS-Lobster) would be ideal to catch this bright X-ray signal. The optical afterglow flux is subject to large uncertainties, but could be as bright as 17th magnitude in R band. Prompt, deep optical follow-up observations of GWBs are desirable. The detection of these signals would confirm the astrophysical origin of GWBs, and shed light into the physics of NS-NS mergers and the NS equation of state.
I thank a Cheung Kong scholarship of China, the hospitality of the KIAA and Department of Astronomy of Peking University, and the sabbatical committee of the UNLV faculty senate, to provide me an ideal working environment to conduct research efficiently. I thank stimulative discussion with Xue-Feng Wu, He Gao, Zi-Gao Dai, and Yi-Zhong Fan, and helpful comments from Kunihito Ioka, Elenora Troja and an anonymous referee. This work is partially supported by NSF AST-0908362.
[50]{} natexlab\#1[\#1]{}
, J., [Abbott]{}, B. P., [Abbott]{}, R., [et al.]{} 2010, Classical and Quantum Gravity, 27, 173001
—. 2012, , 541, A155
, B. P., [Abbott]{}, R., [Adhikari]{}, R., [et al.]{} 2009, Reports on Progress in Physics, 72, 076901
, F., [Alshourbagy]{}, M., [Amico]{}, P., [et al.]{} 2008, Classical and Quantum Gravity, 25, 114045
, S. D., [Chincarini]{}, G., [Burrows]{}, D. N., [et al.]{} 2005, , 438, 994
, E., [Kulkarni]{}, S. R., [Fox]{}, D. B., [et al.]{} 2005, , 634, 501
, N., [Metzger]{}, B. D., [Thompson]{}, T. A., & [Quataert]{}, E. 2012, , 419, 1537
, D. N., [Grupe]{}, D., [Capalbi]{}, M., [et al.]{} 2006, , 653, 468
, A., & [M[é]{}sz[á]{}ros]{}, P. 2009, , 702, 1171
, Z. G., & [Lu]{}, T. 1998, , 333, L87
, Z. G., [Wang]{}, X. Y., [Wu]{}, X. F., & [Zhang]{}, B. 2006, Science, 311, 1127
, M., [Schady]{}, P., [Kuin]{}, N. P. M., [et al.]{} 2010, , 709, L146
, P. B., [Pennucci]{}, T., [Ransom]{}, S. M., [Roberts]{}, M. S. E., & [Hessels]{}, J. W. T. 2010, , 467, 1081
, G., & [Spruit]{}, H. C. 2002, , 391, 1141
, D., [Livio]{}, M., [Piran]{}, T., & [Schramm]{}, D. N. 1989, , 340, 126
, P. A., [Fridriksson]{}, J. K., [Gehrels]{}, N., [et al.]{} 2012, , 203, 28
, [É]{}. [É]{}., & [Hughes]{}, S. A. 1998, , 57, 4535
, P. C. C., [Ransom]{}, S. M., [B[é]{}gin]{}, S., [et al.]{} 2008, , 675, 670
, W.-H., & [Fan]{}, Y.-Z. 2006, [Chin. J. Astron. Astrophys.]{}, 6, 513
, N., [Barthelmy]{}, S. D., & [Cannizzo]{}, J. K. 2012, in IAU Symposium, Vol. 285, IAU Symposium, ed. E. [Griffin]{}, R. [Hanisch]{}, & R. [Seaman]{}, 41–46
, N., [Sarazin]{}, C. L., [O’Brien]{}, P. T., [et al.]{} 2005, , 437, 851
, N., [Norris]{}, J. P., [Barthelmy]{}, S. D., [et al.]{} 2006, , 444, 1044
, V., [Kim]{}, C., [Lorimer]{}, D. R., [et al.]{} 2004, , 601, L179
, J., [Camp]{}, J., [Racusin]{}, J., [Gehrels]{}, N., & [White]{}, D. 2012, , 759, 22
, S., & [M[é]{}sz[á]{}ros]{}, P. 2003, , 589, 861
, S. S., [Vlahakis]{}, N., [K[ö]{}nigl]{}, A., & [Barkov]{}, M. V. 2009, , 394, 1182
, M., [Stairs]{}, I. H., [Manchester]{}, R. N., [et al.]{} 2006, Science, 314, 97
, S. R. 2005, ArXiv Astrophysics e-prints (arXiv:astro-ph/0510256)
, K., [Ioka]{}, K., & [Shibata]{}, M. 2012, ArXiv e-prints (arXiv:1209.5747)
, J. M. 2012, Annual Review of Nuclear and Particle Science, 62, 485
, L.-X., & [Paczy[ń]{}ski]{}, B. 1998, , 507, L59
, N., [O’Brien]{}, P. T., [Zhang]{}, B., [et al.]{} 2010, , 402, 705
, B. D., & [Berger]{}, E. 2012, , 746, 48
, B. D., [Giannios]{}, D., [Thompson]{}, T. A., [Bucciantini]{}, N., & [Quataert]{}, E. 2011, , 413, 2031
, B. D., [Quataert]{}, E., & [Thompson]{}, T. A. 2008, , 385, 1455
, B. D., [Mart[í]{}nez-Pinedo]{}, G., [Darbha]{}, S., [et al.]{} 2010, , 406, 2650
, A., [Perri]{}, M., [Capalbi]{}, M., [et al.]{} 2007, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6688, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
, E., & [Piran]{}, T. 2011, , 478, 82
, R., [Paczynski]{}, B., & [Piran]{}, T. 1992, , 395, L83
, E. S. 1991, , 380, L17
, T., [Nakar]{}, E., & [Rosswog]{}, S. 2012, ArXiv e-prints (arXiv:1204.6242)
, D. J., & [Rosswog]{}, S. 2006, Science, 312, 719
, M. L., [Orosz]{}, J. A., [McClintock]{}, J. E., [et al.]{} 2011, , 730, 25
, S., [Piran]{}, T., & [Nakar]{}, E. 2012, ArXiv e-prints (arXiv:1204.6240)
, A., & [O’Brien]{}, P. 2012, in -Ray Bursts 2012 Conference (GRB 2012)
, A., [O’Brien]{}, P. T., [Tanvir]{}, N. R., [et al.]{} 2010, , 409, 531
, E., [Cusumano]{}, G., [O’Brien]{}, P. T., [et al.]{} 2007, , 665, 599
, V. V. 1992, , 357, 472
, B., & [M[é]{}sz[á]{}ros]{}, P. 2001, , 552, L35
, B., & [Yan]{}, H. 2011, , 726, 90
[^1]: [@kyutoku12] conjectured that a tip of such an ejecta can reach relativistic speed and give broad-band afterglow in a wide solid angle.
[^2]: I thank Xue-Feng Wu for pointing out this possibility.
|
---
abstract: 'Some issues in inclusive and exclusive diffractive processes are discussed.'
address: ' Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel'
author:
- 'Aharon Levy [^1]'
title: 'Inclusive and exclusive diffraction: overview of the experimental presentations at Diffraction2004'
---
Introduction
============
There were 20 experimental presentations at this conference, all by expert people in the field. All these talks were plenary, so there is no point of me summarizing again the contents of each talk. Instead, I would like to present a personal view and touch upon few selected issues which were presented here.
What is diffraction?
======================
In spite of the fact that diffractive processes have a long history, it is still not easy to give a precise and concise definition of what one calls a diffractive reaction. It is clearly a process where, in an exchange picture, no color is exchanged. This however includes all the colorless particles. A further requirement is that the exchanged messenger has the quantum number of the vacuum, which, in the Regge picture is named the Pomeron. Elastic scattering is usually presented as an example of a diffractive process. However, at low energies it can proceed also through an exchange with quantum numbers different from the vacuum, called a Reggeon. At high energies, where the kinematics allows for a large rapidity range, it is more useful to talk about rapidity gaps. Due to the fact that no color is exchanged, there is a suppression of gluon radiation and therefore a rapidity gap is produced between the two vertices of the interaction. Of course, both the Pomeron and the Reggeon are colorless, however the rapidity gap produced by a Reggeon is exponentially suppressed, while that of the Pomeron remains constant. This prompted Bjorken [@bj] to define diffraction as processes in which the large rapidity gap is not exponentially suppressed.
How practical is this definition? This we shall see in the following sections.
Diffraction in inclusive processes
==================================
Kinematical variables
---------------------
I will try to define here all the variables needed in the following sections. Let us first look at a diffractive reaction $ep\to epX$ at HERA, depicted in figure \[fig:ddis\].
![Schematic representation of the deep inelastic diffractive scattering. []{data-label="fig:ddis"}](diff-kin-bw.eps){width="0.5\hsize"}
A virtual photon $\gamma^*$ ($Q^2\equiv-q^2$) interacts with the proton through a colorless exchange with vacuum quantum numbers. A mass $M_X$ of the hadronic system recoils against the proton. The square of the momentum transfer at the proton vertex is $t$. The fraction of the proton momentum carried by the exchange is denoted by ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$. The quark struck by the virtual photon carries a fraction $\beta$ of the momentum of the colorless exchange. (Note that sometimes $z$ is used instead of $\beta$). The last two variables are connected to Bjorken $x$ as follows: $x=\beta{x_{{{\scriptscriptstyle {I\!\!P}}}}}$.
At the Tevatron, where one studies the diffractive reaction $\bar{p} p
\to \bar{p} X$, the equivalent variable to ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ is denoted by $\xi$.
Selecting diffractive processes
-------------------------------
There are three methods used at HERA to select diffractive events. One [@lps] uses the Leading Proton Spectrometer (LPS) to detect the scattered proton and by choosing the kinematic region where the scattered proton looses very little of its initial longitudinal energy, it ensures that the event was diffractive. A second method [@lrg] simply request a large rapidity gap (LRG) in the event and fits the data to contributions coming from Pomeron and Reggeon exchange. The third method [@mx] uses the distribution of the mass of the hadronic system seen in the detector, $M_X$, to isolate diffractive events. We will refer to these three as LPS, LRG and $M_X$ methods. At the Fermilab Tevatron [@dino] diffractive processes are being studied by tagging events with either a rapidity gap or a leading hadron.
The LPS method has the advantage of detecting the scattered proton and thus excluding proton dissociative processes. However, in order to ensure that the scattered proton resulted from a diffractive process, one requires ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.01, where ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ is the amount of longitudinal momentum lost by the scattered proton. This cut removes contributions coming from Reggeon exchanges [@xpcut].
The LRG method selects events which also include some proton dissociative processes and Reggeon contributions. The latter can be removed by the same ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.01 cut as above. The proton dissociative processes can be removed provided their mass is large enough to produce signals in some forward tagging devices. The contribution of low mass proton dissociation can be estimated. In the analysis of the H1 collaboration, processes with proton dissociation into masses below 1.6 GeV amount to about 10% [@sebastian].
The $M_X$ method which subtracts the exponentially suppressed large rapidity gap events, in principle subtracts also the Reggeon contribution and is left only with the proton dissociative background. These can not be removed for masses below 2.4 GeV, which constitutes about 30% of the selected diffractive events [@capua].
At the Tevatron, single diffractive events, $\bar{p} p \to \bar{p} X$, are selected by tagging the scattered $\bar{p}$.
Diffractive structure function
------------------------------
In figure \[fig:allsf\] the diffractive structure function measurements with all three HERA methods are presented. The $M_X$ data have been multiplied by a factor of 0.69 to correct for the proton dissociation background. No correction was done to the H1 data. All methods seem in general to agree with each other in their overlapping kinematic region.
![Comparison of ${x_{{{\scriptscriptstyle {I\!\!P}}}}}{F_2^{D(3)}}$ measured by H1 and ZEUS, as a function of ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ in overlapping bins of $\beta$ and $Q^2$. []{data-label="fig:allsf"}](lps-mx-h1.eps){width="1.0\hsize"}
$Q^2$ dependence of $\lambda$
-----------------------------
The $x$ behaviour of the inclusive structure function $F_2$ at a given $Q^2$ is well described by an $x^{-\lambda}$ form. The value of $\lambda$ is connected to the Pomeron intercept, $\lambda={\alpha_{{{\scriptscriptstyle {I\!\!P}}}}}(0)-1$. The value of $\lambda$ is approximately constant till $Q^2\approx$ 1 GeV$^2$, and then rises almost linearly with $\ln Q^2$.
It is of interest to see whether the ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ behaviour of ${x_{{{\scriptscriptstyle {I\!\!P}}}}}{F_2^{D(3)}}$ shows a similar pattern. To this end, a fit of the form ${x_{{{\scriptscriptstyle {I\!\!P}}}}}{F_2^{D(3)}}\propto {x_{{{\scriptscriptstyle {I\!\!P}}}}}^{-\lambda}$ was performed, for different $Q^2$ intervals.
Figure \[fig:lama\] shows the value of $\lambda$ as function of $Q^2$ from fits to the $x$ behaviour of $F_2$ and from the ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ behaviour of ${x_{{{\scriptscriptstyle {I\!\!P}}}}}{F_2^{D(3)}}$. The precision data of $F_2$ makes it possible to see a very significant rise with $Q^2$. The ${x_{{{\scriptscriptstyle {I\!\!P}}}}}{F_2^{D(3)}}$ data does not have the precision needed for a clear $Q^2$ dependence. There is a trend similar to that of $F_2$, but given the large errors of the data, the behaviour is also consistent with no $Q^2$ dependence.
![The $Q^2$ dependence of $\lambda$ obtained from fits to $F_2\propto x^{-\lambda}$ and ${x_{{{\scriptscriptstyle {I\!\!P}}}}}{F_2^{D(3)}}\propto {x_{{{\scriptscriptstyle {I\!\!P}}}}}^{-\lambda}$. The band indicates the value corresponding to the soft Pomeron. []{data-label="fig:lama"}](lama.eps){width="1.0\hsize"}
NLO QCD fits to ${x_{{{\scriptscriptstyle {I\!\!P}}}}}{F_2^{D(3)}}$
-------------------------------------------------------------------
It has been proven [@QCD-fact] that QCD factorization works for diffractive processes at HERA. This allows to use the DGLAP [@dglap] evolution equations to get diffractive parton distribution functions. Given the fact that for describing diffractive processes one needs more variables, $t$, ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$, $\beta$, $Q^2$, one would actually like to evolve in $\beta$ and $Q^2$ for fixed $t$ and ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$. $t$ is usually hard to measure and one integrates over it. Thus, ideally one would like to evolve for fixed ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ values. The statistics of the presently available data is not sufficient for carrying this out.
Ingelman and Schlein [@ingelman-schlein] suggested to consider the exchanged Pomeron as a particle having internal structure. Under this assumption, the diffractive process is described as a multistep event: the proton ’radiates’ a Pomeron having a fraction ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ of the proton momentum. The virtual photon interacts in a deep inelastic process off the Pomeron, scattering of a parton in the Pomeron carrying a fraction $\beta$ of the Pomeron momentum. There is thus a flux factor at the proton vertex, dependent only on ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ ($t$ is integrated out), and a Pomeron structure function $F_2^{D(2)}$. This picture assumes Regge factorization, an assumption which has to be checked by the data.
Using the proven QCD factorization together with the assumed Regge factorization, one gets diffractive parton distributions.
Regge factorization
-------------------
The assumption of Regge factorization clearly does not hold in the inclusive case. The value of $\lambda$ is clearly $Q^2$ dependent. However both the diffractive H1 data and the LPS data can be described by an NLO QCD fit with one fixed value of ${\alpha_{{{\scriptscriptstyle {I\!\!P}}}}}$(0), as shown by Schätzel [@sebastian] and by Capua [@capua] at this meeting. In case of the LPS data, a cut on ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.01 has to be made. For the H1 analysis, one needs to add the Reggeon contribution. The statistics of the LPS data were not sufficient to repeat the fit in different $Q^2$ bins. The H1 analysis, shows some indication of a rise of $\lambda$ with $Q^2$ (see figure \[fig:lama\]), though with quite large error bars. The uncertainty comes not only from the statistics but also from the way one needs to isolate the Pomeron contribution from the Pomeron + Reggeon fit.
We can conclude that in inclusive diffractive processes, for ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.01, the data can be consistent with Regge factorization. There is a clear need for more precise data.
Diffractive parton distribution functions
-----------------------------------------
The H1 data, which have a wide kinematical coverage in $Q^2$ and in $\beta$, have been used to extract diffractive parton distributions (dpdfs), shown in figure \[fig:dpdfs\].
![The resulting parton density distributions in the Pomeron, using a NLO QCD fit (shaded line) compared to a LO fit (solid line).[]{data-label="fig:dpdfs"}](pdfs.ps){width="1.0\hsize"}
One sees a dominance of the gluon distribution, which is the outcome of the fact that the data show positive scaling violation up to quite high $\beta$ values. This is quantified in figure \[fig:gluon-fract\], which shows that for the region $0.01 < \beta < 1$, the gluons carry 80% of the Pomeron momentum.
![The gluon momentum fraction from a NLO QCD fit, as a function of $Q^2$. []{data-label="fig:gluon-fract"}](gluonfrac.eps){width="0.6\hsize"}
The same conclusion is reached by the LPS analysis. Note that the validity of the diffractive parton distribution functions (dpdfs) is in the following kinematic region: $Q^2 >$ 3 GeV$^2$, $M_X >$ 2 GeV and ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.05. Are these dpdfs portable just like in the inclusive pdfs case? In other words, does QCD factorization work?
QCD factorization test
----------------------
The diffractive parton distributions, obtained by H1 from the NLO QCD fit, were used to calculate expectations of other diffractive processes. The agreement with data is very good for diffractive $D^*$ as well as for diffractive dijet production [@vinokurova]. However, the expectations for the Tevatron results are by one order of magnitude too high [@mesropian]. This is not surprising as QCD factorization should not hold for diffractive hadron-hadron reactions. Furthermore, the Tevatron data is measured in the kinematic region $0.035 < \xi <0.095$, where the Reggeon exchange dominates and thus would not be called diffraction. As mentioned above, the validity of the H1 fit is for ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.05 (${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ at HERA is $\xi$ at the Tevatron). The fact that QCD factorization seems to fail for hadron-hadron data is also explained by introducing the notion of survival probability of the rapidity gap [@survival]. Taking it at face value, this would mean that for the Tevatron processes, the survival probability is about 0.1. Since the photon has a hadronic part (’resolved photon’), this notion can be tested in diffractive photoproduction of dijets [@renner; @vinokurova]. Indeed it seems that using a survival probability of 0.34 [@kaidalov-survival], one can describe the resolved photon data. There seems to be some uncertainty about the conclusion concerning the direct part: while the H1 measurement is below the expectations, the ZEUS result is consistent with it.
An interesting attempt to fit the combined data of inclusive and inclusive diffraction data was carried out by Martin, Ryskin and Watt [@mrw], who included absorption correction to the QCD analysis. While the quark distributions they get are not very different from those of H1, their gluon distribution is significantly lower than the H1 one. This results in expectations which are by almost factor of 3 lower than the H1 ones for the comparison with the Tevatron data (see figure \[fig:watt\]).
![Effective diffractive structure function for dijet production in $p\bar{p}$ interactions as a function of $\beta$, compared to expectations of different sets of diffractive parton distribution functions. []{data-label="fig:watt"}](mrw.epsi){width="0.9\hsize"}
In this case the survival probability would be closer to that of the resolved photon case.
Ratio of $\sigma^D$ to $\sigma_{tot}$
-------------------------------------
Diffractive processes were said to constitute about 10% of the total inclusive DIS processes. However, the ratio of $\sigma^D/\sigma^{tot}$ is $Q^2$ dependent. It can be as high as 20% at $Q^2 \approx$ 3 GeV$^2$, going down to about 10% at $Q^2 \approx$ 30 GeV$^2$, as can be seen in fig \[fig:sigratio\], where this ratio is shown as a function of $Q^2$, for the kinematic region $ 200 < W <
245$ GeV, $M_X < 35$ GeV, and $M_N < 2.3$ GeV.
![The ratio of diffractive to total cross section as a function of $Q^2$, for a selected kinematic region. []{data-label="fig:sigratio"}](sigratio.eps){width="0.8\hsize"}
One should however keep in mind that the $Q^2$ dependence might be an outcome of the fact that in this presentation of the ratio, different regions of $\beta$ are covered for different values of $Q^2$. Note that this ratio at $Q^2$ = 100 GeV$^2$ goes down to $\approx$ 5% for ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.03 [@sebastian].
This ratio has been measured for the first time for Charged Current diffractive processes [@sebastian; @rautenberg]. It is in the range of 2-3 % for ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$ and $x <$ 0.05.
One can calculate the ratio of diffractive to total cross section for specific processes and check whether the parton distributions obtained from the NLO QCD analysis fulfill the Pumplin bound [@pumplin]. This was done in [@kaidalov-survival] for diffractive to inclusive dijet production induced by gluons and is displayed in figure \[fig:unitarity\].
![The ratio of diffractive to inclusive dijet production cross section as a function of $x$ of the gluon for different scales of the hard scattering, for the recent H1 diffractive parton distribution functions. Also shown is the unitarity limit, called Pumplin bound. []{data-label="fig:unitarity"}](pumplin.eps){width="0.8\hsize"}
As seen, the bound is clearly violated for relatively low scales at low $x$. This might indicate that unitarity effects are already present in the gluon sector.
Summary on inclusive diffraction
--------------------------------
This subsection is more a presentation of some questions than a real summary. Is Regge factorization broken? Also for ${x_{{{\scriptscriptstyle {I\!\!P}}}}}<$ 0.01? There is a need for more precise measurements to come to a clear conclusion. Ideally, so as not to be dependent on the Regge factorization assumption in diffraction, one would like to do a QCD analysis for fixed values of ${x_{{{\scriptscriptstyle {I\!\!P}}}}}$. This will need much higher statistics than presently available.
There is a breaking of QCD factorization when using the parton distribution densities to compare to hadron-hadron data. This is interpreted by the introduction of the large rapidity gap survival probability. The value of the survival probability seems to be in the range of 0.1-0.3.
The presently obtained gluon momentum densities seem to give results which are violating the Pumplin bound, in certain kinematical regions. This could be the indication of the presence of unitarity effects.
There is a large ratio of diffractive to total cross section which decreases with $Q^2$.
Exclusive diffractive processes
===============================
Introduction
------------
This section describes exclusive processes like electroproduction of vector mesons or Deeply Virtual Compton Scattering (DVCS). The situation in this cases is much simpler as these processes are clearly diffractive processes at the high energies where they are measured. There still exist the problem of isolating the ’elastic’ process from the proton dissociative one. By measuring the cross section for a limited low $t$ range, the contribution of the latter is minimized.
Soft to hard transition
-----------------------
One of the nice features seen in these data is the transition from soft to hard processes as one increases the scale.
![A compilation of elastic photoproduction of vector mesons, as a function of $W$. The total $\gamma p$ cross section is plotted for comparison. []{data-label="fig:xsect"}](xsect.eps){width="0.95\hsize"}
This transition is seen also for the photoproduction of vector mesons, where the mass serves as the scale. Figure \[fig:xsect\] shows the photoproduction cross section as a function of the $\gamma p$ center of mass energy, $W$, for different vector mesons. The light vector mesons, $\rho$, $\omega$ and $\Phi$ show an energy dependence which is characteristic of a soft process (the total $\gamma p$ cross section is also shown for comparison). For the heavier vector mesons, the energy dependence becomes much steeper, as expected from hard processes. Note also that the real part of the amplitude increases with the hardness of the process and is a further reason for the sharp energy dependence.
The soft to hard transition can also be seen for a given vector meson, by changing the $Q^2$ of the process. The cross section is parameterized as $W^\delta$ and $\delta$ is seen to increase with $Q^2$. To compare all the vector mesons on one plot [@ciesielski] one shows $\delta$ as function of $Q^2+M_V^2$, with $M_V$ being the mass of the vector meson. As seen in figure \[fig:delvm\], one gets an increase of $\delta$ from values of about 0.2 (soft) at the low scale end to a value of about 1 (hard) at high scales.
![The parameter $\delta$ from a fit of the form $W^\delta$ to the cross section data of exclusive production of VMs, as a function of $Q^2+M_V^2$. []{data-label="fig:delvm"}](deltavm.eps){width="0.72\hsize"}
Effective Pomeron trajectory
----------------------------
Using the energy dependence of vector meson electroproduction at fixed $t$ values, one can obtain the parameters of the effective trajectory exchanged in the process. This way the effective trajectory of the Pomeron was determined for the $\rho$, $\phi$ and $J/\psi$ electroproduction. A summary plot of the intercepts and slopes for all three VMs, as function of $Q^2+M_V^2$, is presented in figure \[fig:apom\].
![Compilation of ${\alpha_{{{\scriptscriptstyle {I\!\!P}}}}}$(0) and ${\alpha^\prime_{{\scriptscriptstyle {I\!\!P}}}}$ values, extracted in exclusive VM production, as a function of $Q^2+M_V^2$.[]{data-label="fig:apom"}](apom.eps){width="1.0\hsize"}
The intercept of the low mass VMs are consistent with that of the soft Pomeron. This is not the case for the $J/\psi$ which has a significantly higher intercept. As for the slope, all values are lower that that of the soft Pomeron, as expected from hard processes.
Sizes of vector mesons
----------------------
The $t$ distribution of the VMs can be well described by an exponentially falling cross section, with a slope $b$. This slope is connected to the size of the VM. At low $Q^2$ the size of the light VMs is large, decreasing with $Q^2$ from a value of $\approx$ 10 GeV$^{-2}$ to about 5 GeV$^{-2}$. For the $J/\psi$ the size is small already at low $Q^2$. This can be seen in figure \[fig:bvm\], where the slope $b$ is plotted as function of $Q^2$.
![Exponential slope of the $t$ distribution measured for exclusive VM production as a function of $Q^2$. The lines are to guide the eye. []{data-label="fig:bvm"}](bvm.eps){width="1.0\hsize"}
The dotted lines are just to guide the eye. The data seem to converge at high $Q^2$ on a value of $b \approx$ 5 GeV$^{-2}$.
$R=\sigma_L/\sigma_T$
---------------------
The ratio $R$ of the longitudinal photon cross section, $\sigma_L$, to that of the transverse photon, $\sigma_T$, can be obtained by studying the decay distribution of the VM and assuming s-channel helicity conservation. This has been done for $\rho$ [@sandacz; @grebenyuk; @ciesielski], $\phi$ [@ciesielski] and $J/\psi$ [@ciesielski].
![The $Q^2$ dependence of $R$ for exclusive $\rho^0$ electroproduction. []{data-label="fig:r-compass"}](compas.eps){width="0.8\hsize"}
Figure \[fig:r-compass\] shows as an example the very impressive preliminary measurements of $R$ as a function of $Q^2$ for the $\rho$ VM by the COMPASS collaboration [@sandacz]. For all VMs, $R$ is rising with $Q^2$.
![A compilation of the values of $R$ for $\rho^0$, $\phi$ and $J/\psi$, as a function of $Q^2$.The lines are a fit to the data of the form $R=a(Q^2/M_V^2)^b$. []{data-label="fig:rvm"}](rvm.eps){width="1.0\hsize"}
A compilation of $R$ for all three VMs is shown in figure \[fig:rvm\], as function of $Q^2$. The lines are a fit to the data of the form $R=a(Q^2/M_V^2)^b$, which indicate that $R$ scales with $Q^2/M_V^2$.
Configurations of the photon
----------------------------
The photon is described as fluctuating into a $q\bar{q}$ pair. When the relative $k_T$ between the pair is small we speak of a large spatial configuration, while if the relative $k_T$ is large, the photon fluctuates into a small spatial configuration. Longitudinal photons have small configurations, while transverse photons consist of both large and small configuration. A small spatial configuration leads to hard processes and thus to a steep energy dependence of the cross section. A large configuration has a shallower energy dependence, as expected in soft processes.
It came therefore as a surprise that the ratio $R$ for the $\rho$ electroproduction process is $W$ independent for $W$ values up to 120 GeV and $Q^2$ up to 19 GeV$^2$ [@halina]. The ratio $R$ is $W$ independent also for $J/\psi$ [@ciesielski].
This means that for some reason in case of VM electroproduction the large configurations of the transverse photon are suppressed.
Another process which seems to send the same message is DVCS. The energy dependence of this reaction indicates a hard process. The $W^\delta$ dependence yields [@hiller] $\delta$=0.98$\pm$0.44 for the H1 measurement and $\delta$=0.75$\pm$0.15$^{+0.08}_{-0.06}$ for ZEUS. Such a steep energy dependence would be expected from a dominant longitudinal photon. However, in DVCS one goes from a virtual photon to a real one, $\gamma^*\to\gamma$. Assuming s-channel helicity conservation, this means that, since the real photon is transverse, also the initial virtual photon has to be transverse. Thus, the steep energy dependence of the DVCS cross section means that the large configuration in the transverse photon is suppressed.
Summary on exclusive diffraction
--------------------------------
Exclusive diffractive processes become hard once the scale gets large. The properties of the effective Pomeron exchange at the larger scales are consistent with that of a hard process. The large configurations of the transverse photon seem to be suppressed in exclusive VM, including real photon, production.
Outlook
-------
Diffractive processes are expected to be measured with improved machines and detectors at HERA, the Tevatron and at RHIC. Whether we want it or not, a large portion of the interactions measured at LHC will be of diffractive nature [@white]. In fact, the exclusive diffractive production of the Higgs boson has been proposed as a potential background free method to search for the light Higgs at LHC. One can also dream about a future $ep$ collider which will allow to reach kinematic region where phase transitions can be observed. Clearly diffraction is a subject which will occupy us for quite some time to come.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to acknowledge and thank the organizers for the excellent organization of a very pleasant conference in a most beautiful location.
[99]{}
James D. Bjorken, [*Hard diffraction and deep inelastic scattering*]{}, Proceedings of DIS94, Eilat, Israel,(ed. A. Levy) p.151, 1994.
ZEUS Collab., S. Chekanov et al., DESY-04-131, hep-ex/0408009 (2004).
H1 Collab., C. Adloff et al., Nucl. Phys. [**B619**]{}, 3 (2001).
ZEUS Collab., J. Breitweg et al., Eur. Phys. J. [**C6**]{}, 43 (1999).
See eg K. Goulianos, [*Hadronic diffraction: where do we stand?*]{}, hep-ph/0407035 (2004).
K. Golec-Biernat, J. Kwiecinski and A. Szczurek, Phys. Rev. [**D56**]{}, 3955 (1997).
S. Schätzel (for the H1 Collaboration), [*H1 diffractive structure function measurements and QCD fits*]{}, these proceedings.
M. Capua (for the ZEUS Collaboration), [*ZEUS results on inclusive diffraction*]{}, these proceedings.
J.C. Collins, Phys. Rev. [**D 557**]{}, 3051 (1998); erratum Phys. Rev. [**D 61**]{}, 2000 (1998); A. Berera and D.E. Soper, Phys. Rev. [**D 53**]{}, 6162 (1996); L. Trentadue and G. Veneziano, Phys. Lett. [**B 323**]{}, 201 (1994).
Y.L. Dokshitzer, Sov. Phys. JETP [**46**]{}, 641 (1977) \[Zh. Eksp. Teor. Fiz. [**73**]{}, 1216 (1977)\]; V.N. Gribov and L.N. Lipatov, Yad. Fiz. [**15**]{}, 1218 (1972) \[Sov. J. Nucl. Phys. [ **15**]{}, 675 (1972)\]; V.N. Gribov and L.N. Lipatov, Yad. Fiz. [ **15**]{}, 781 (1972) \[Sov. J. Nucl. Phys. [**15**]{}, 438 (1972)\]; G. Altarelli and G. Parisi, Nucl. Phys. [**B 126**]{}, 298 (1977).
G. Ingelman and P.E. Schlein, Phys. Lett. [**B 152**]{}, 256 (1985).
S. Vinokurova (for the H1 Collaboration), [*The diffractive production of charm and jets*]{}, these proceedings.
C. Mesropian (for the CDF Collaboration), [ *Diffraction at CDF in run II*]{}, these proceedings.
See eg E. Gotsman, E. Levin and U. Maor, Phys. Lett. [**B 438**]{}, 229 (1998).
R. Renner (for the ZEUS Collaboration), [*ZEUS results on diffractive dijets*]{}, these proceedings.
A.B. Kaidalov et al., Phys. Lett. [**B 567**]{}, 61 (2003).
A.D. Martin, M.G. Ryskin and G. Watt, Phys. Rev. [**D 70**]{}, 091502 (2004).
J. Rautenberg (for the ZEUS Collaboration), [ *Diffractive charged current (ZEUS)*]{}, these proceedings.
J. Pumplin, Phys. Rev. [**D 8**]{}, 2899 (1973).
R. Ciecielski (for the ZEUS Collaboration), [ *Recent ZEUS results on vector mesons*]{}, these proceedings.
A. Sandacz (for the COMPASS Collaboration), [ *Diffractive $\rho^0$ production at COMPASS*]{}, these proceedings.
O. Grebenyuk (for the HERMES Collaboration), [ *Exclusive electroproduction of pions and vector meson at HERMES*]{}, these proceedings.
See eg H. Abramowicz,[*Diffractive scattering*]{}, hep-ex/0410002 (2004).
K. Hiller (for the H1 Collaboration), [*High $|t|$ diffraction and DVCS from H1*]{}, these proceedings.
S. White, these proceedings.
[^1]: Supported in part by the Israel Science Foundation (ISF).
|
---
abstract: 'We report the results of *(a)* extensive follow-up observations of the gamma-ray pulsar J1732$-$3131 that has been recently detected at decameter wavelengths, and *(b)* deep searches for counterparts of 9 other *radio-quiet* gamma-ray pulsars at 34 MHz, using the Gauribidanur radio telescope. No periodic signal from J1732$-$3131 could be detected above a detection threshold of $8\sigma$, even with an effective integration time of more than 40 hours. However, the average profile obtained by combining data from several epochs, at a dispersion measure of 15.44 pc cm$^{-3}$, is found to be consistent with that from the earlier detection of this pulsar at a confidence level of $99.2\%$. We present this consistency between the two profiles as an evidence that J1732$-$3131 is a faint radio pulsar with an average flux density of 200–400 mJy at 34 MHz. Detection sensitivity of our deep searches, despite the extremely bright sky background at such low frequencies, is generally comparable to that of higher frequency searches for these pulsars, when scaled using reasonable assumptions about the underlying pulsar spectrum. We provide details of our deep searches, and put stringent upper limits on the decameter wavelength flux densities of several *radio-quiet* gamma-ray pulsars.'
author:
- |
Yogesh Maan$^{1,2}$[^1] and H. A. Aswathappa$^{1}$\
$^{1}$Raman Research Institute, Bangalore 560080, India\
$^{2}$Joint Astronomy Programme, Indian Institute of Science, Bangalore 560012, India
title: 'Deep searches for decameter wavelength pulsed emission from *radio-quiet* gamma-ray pulsars'
---
\[firstpage\]
pulsars: general – pulsars: individual: J1732$-$3131
Introduction
============
The Large Area Telescope (LAT) on board the *Fermi* gamma-ray satellite, with its unprecedented sensitivity, has revolutionized the study of gamma-ray emitting pulsars, increasing the known population from less than 10 to 121 pulsars[^2] [@fermi_catalog13; @Pletsch13]. About one-third (40) of these pulsars were discovered in blind searches of the LAT data [@Abdo09; @Saz10; @Pletsch12a; @Pletsch12b; @Pletsch12c; @Pletsch13]. Despite deep searches at frequencies $\gtrsim500$ MHz [@Saz10; @Ray11; @Pletsch12a], confirmed radio counterparts of only 4 of these have been detected so far [@Camilo09; @Abdo10; @Pletsch12a], suggesting a large fraction of gamma-ray pulsar population to be *radio-quiet*[^3].
A likely explanation for the apparent absence of radio emission from the majority of the LAT-discovered pulsars is that their narrow radio beams miss the line of sight towards earth [@BJ99; @WR11], and hence appear as *radio-quiet*. However, the radio emission beam is expected to become wider at low frequencies [radius-to-frequency mapping in radio pulsars; @Cordes78], increasing the probability of our line of sight passing through the beam. With this in mind, we used the archival data of the pulsar/transient survey at 34.5 MHz, carried out using the Gauribidanur radio telescope during 2002–2006, to search for decameter-wavelength pulsed emission from several of the LAT-discovered pulsars. A possible detection of radio counterpart of the LAT-discovered pulsar J1732$-$3131, resulting from the above search, was reported earlier [@MAD12 hereafter Paper I]. Weak (and periodic) pulsed emission from J1732$-$3131 was detected in only one of the several observing sessions. Although scintillation may explain the detection in only one session, another likely possibility is that the radio emission from LAT-discovered pulsars might not be persistent, i.e., they might appear in *radio-bright* mode only once in a while. Two categories of radio pulsars — intermittent pulsars [@Kramer06] and rotating radio transients [RRATs; @McLaughlin06] — are well known for such emission behavior.
Deep search program: motivation
-------------------------------
Motivated by the intriguing detection of J1732$-$3131, we embarked on an observing program of deep searches for the decameter-wavelength counterparts of the so-called radio-quiet gamma-ray pulsars, using the Gauribidanur radio telescope at 34 MHz. In the first phase of this *deep search program*, each of the target sources in the selected sample of 10 gamma-ray pulsars[^4] was observed in multiple ($>20$) sessions. *Deep* searches for persistent periodic signals were realized by time-aligning and co-adding the data from these multiple sessions, as described in Section 3. While the significant enhancement in sensitivity achieved this ways is important, the deep search program was motivated by two more crucial factors:
1. Even for the handful of pulsars which are detectable at such low frequencies, the received periodic signals are very weak. Especially at decameter wavelengths, interstellar and ionospheric scintillation, and contamination from radio frequency interference (RFI), can hinder detection of such weak signals. Hence, a weak source, even if intrinsically persistent, may not be detected in all the observing sessions. In addition, the source may also be intrinsically variable. Hence, it is important to observe the same field multiple times.
2. Assuming that our noise statistics are Gaussian, a detection even at $5\sigma$ might appear quite significant (chance probability of such a detection is less than $0.6\times10^{-6}$). But the measured statistics generally deviate from the expected Gaussian nature due to RFI contamination and/or systematics contributed by the receiver, and hence the possibility that a $5\sigma$ detection from a single observing session is due to some weak RFI can not be ruled out. However, detection of even a relatively weak periodic signal, but in more than one observing sessions on different days, consistent in pulse-shape and at the same phase of the period, is highly unlikely to be a manifestation of noise (i.e., a chance occurrence) or some RFI. Such consistency across observing sessions, is therefore crucial to raise the level of confidence in establishing the astrophysical origin of an otherwise weak signal.
All the LAT-discovered pulsars which we have searched for, are isolated pulsars with periods in the range 48–444 ms, and only J1813$-$1246 and J1954$+$2836 have periods below 100 ms. Among the pulsars for which deep searches have been carried out, J1732$-$3131 is followed-up most extensively (125 observing sessions). We present here results of our sensitive searches using these follow-ups of J1732$-$3131 and 9 other pulsars, as well as those using the archival data, and provide useful constraints on the decameter-wavelength flux densities of several radio-quiet gamma-ray pulsars. Section 2 describes details of the archival data and our new observations. In section 3, we explain the search methodologies. Section 4 presents results of follow-up searches of J1732$-$3131 and several other gamma-ray pulsars, and the upper limits obtained on flux densities of these targets, followed by conclusions in section 5.
Observations and pre-search data processing
===========================================
The archival as well as the new observations were carried out using the Gauribidanur radio telescope. The telescope originally consisted of an array of 640 dipoles ($160\times4$ rows) in the east-west direction (hereafter EW array) and an array of 360 dipoles extending southwards from the center of the EW array [@DSS89]. Presently, *only* the EW arm of this telescope is maintained, and the survey as well as the new observations were carried out using this array in coherent phased-array mode. The beam widths of the EW array are 21 arcmin and $25{\ensuremath{^{\circ}}}\times\sec{\rm (zenith~angle)}$ in right ascension (RA) and declination (Dec), respectively, with an effective collective area of about $12000~{\rm m}^2$ at the instrumental zenith ($+14{\ensuremath{^{\circ}}}.1$ Dec). The target source is tracked during the observation by steering the phased-array beam electronically. In both sets of observations, data were acquired using the portable pulsar receiver[^5] (hereafter PPR; Deshpande, Ramkumar, Chandrasekaran and Vinutha, in preparation) as described in Section 2.3.
Survey observations
-------------------
The pulsar/transient survey was carried out in the years 2002–2006 using the EW array at 34.5 MHz, with a bandwidth of 1.05 MHz. The full accessible declination range ($-45{\ensuremath{^{\circ}}}$ to $+75{\ensuremath{^{\circ}}}$) could be covered with 5 discrete pointings in declination: $-30{\ensuremath{^{\circ}}}$, $-05{\ensuremath{^{\circ}}}$, $+14{\ensuremath{^{\circ}}}$, $+35{\ensuremath{^{\circ}}}$ and $+55{\ensuremath{^{\circ}}}$. Appropriate pointings were made to cover a large range in right ascension. Apart from J1732$-$3131, data towards 16 other[^6] gamma-ray pulsars are available from single/multiple observing sessions of this survey. Other details of the survey observations towards these sources are given in Table \[ulimits\_archival\].
New observations
----------------
Under the deep search observing program, new observations of 10 radio-quiet gamma-ray pulsars were carried out in multiple sessions spread over several months in 2012. For these observations, a bandwidth of 1.53 MHz centered at 34 MHz was used. Further, these observations could use only $80\%$ of the potential collecting area, since $20\%$ of the EW array dipoles ($10\%$ at each of the two far ends) were not available. However, a slightly larger bandwidth and longer session duration, as compared to those of the survey observations, together provided about $18\%$ improvement in sensitivity, despite the $20\%$ loss in the collecting area. Further relevant details of these observations can be found in Table \[ulimits\_new\]. Two radio pulsars, B0834$+$06 and B1919$+$21, were also observed regularly as “control pulsars”. The position coordinates of the pulsars J0633$+$0632 and J0633$+$1746 (\[RA,Dec\]$=$\[06:34:26, 6.5[$^{\circ}$]{}\] and \[06:34:38, 17.8[$^{\circ}$]{}\] respectively, precessed to the epoch of observations) lie close to each other. We observed both of these pulsars simultaneously by pointing towards the direction \[06:34:26, 10[$^{\circ}$]{}.0\] (since both the pulsars fall in the same beam, and well above the half power points).
Data acquisition and pre-search processing
------------------------------------------
In each of the observing sessions, PPR was used to directly record the raw signal voltage sequence at the Nyquist rate (with 2-bit, 4-level quantization), while tracking the source. In the off-line processing, the voltage time sequence is Fourier transformed in blocks of lengths appropriate for a chosen spectral resolution in the resultant dynamic spectrum, and successive raw power-spectra are averaged to achieve desired temporal resolution. For the archival data, appropriate parameters are chosen to achieve 256 spectral channels across 1.05 MHz bandwidth centered around 34.5 MHz, and a temporal resolution of $\sim1.95$ ms. For the new observations, the resultant dynamic spectrum consists of 1024 channels across 1.53 MHz bandwidth centered around 34 MHz, with a temporal resolution of $\sim2$ ms.
To identify RFI contaminated parts of the data, *robust*[^7] mean and standard deviation are computed, and an appropriate threshold in signal-to-noise ratio (S/N) is used separately in the frequency and time domains. First the RFI contaminated frequency channels are identified, and data from these channels are excluded while identifying the time samples contaminated with RFI. The RFI contaminated frequency channels as well as time samples are excluded from any further processing. Most of the observations were conducted in the night time, and typically only a few percent ($<5\%$) of the data were found to be RFI contaminated. From the new observations, time-intervals cumulating to about one observing session duration were rejected for J0633$+$0632/J0633$+$1746 and J1809$-$2332. Several of the observing sessions towards J1732$-$3131 happened to be in the day time, and only 85 sessions worth of effective integration time could be used out of a total of 125 observing sessions.
Search methods and sensitivity
==============================
The individual observing session data were searched for presence of single bright pulses as well as for pulsed signals at the expected periods of the respective gamma-ray pulsars. While the detailed methodologies of these two kinds of searches can be found in @ythesis and Section 2 of Paper I, a brief overview is provided below.
Single pulse search
-------------------
Searching for bright single pulses involves dedispersing the data at a number of trial dispersion measures (DM), and subjecting the individual time series, corresponding to each of the trial DMs, to a common detection criterion, i.e. an appropriate S/N threshold. For optimum detections, the individual time series are systematically smoothed with a template of varying width, effectively carrying out a search across the pulse-width as well. We sample the template width range in a logarithmic manner, with a step of 2 (i.e., we use $2^n$ time-samples wide templates, where $n$ varies from 0 to a maximum chosen value, in steps of 1)[^8]. We carried out the single pulse search in two different ranges of DMs: 0–20 pc cm$^{-3}$ and 20–50 pc cm$^{-3}$, with the consecutive trial DMs in the two ranges differing by 0.01 pc cm$^{-3}$ and 0.05 pc cm$^{-3}$, respectively. The maximum match filter widths used for the two ranges are 128 ms and 256 ms, respectively. The S/N threshold is chosen based on how many “false alarms” can be tolerated in the final candidate list. For $N_{tot}$ number of points in a time series, the expected number of “false-alarms”, $N_f$, crossing a threshold of $\eta$ (in units of the rms noise) solely due to noise, are given by: $$erf(\eta/1.414) = 1 - 2 \times N_f / N_{tot}
\label{eq_thresh}$$ where $erf()$ is the *error function*. Allowing 5 false alarms from each of the trial DMs[^9], implies a S/N threshold less than 5. Note that scaling-up of the denominator on the right-hand side of Eq. \[eq\_thresh\] appropriately, so as to account for the number of trial widths as well, does not make the implied threshold significantly different from 5. So, we have used a detection threshold of 5 in our single pulse searches. However, detections marginally above this threshold can be confirmed only when reasonable number of single pulses are detected at the same DM. In case of detection of a single bright pulse, we need to insist on larger S/N ($\geq8$), so that consistency as well as the dispersive nature of the signal can be checked across the bandwidth.
Search for dispersed periodic pulses
------------------------------------
The periodicity search using data from individual observing sessions involves folding the time series corresponding to each of the frequency channels over the expected period of the respective gamma-ray pulsars. The *folded dynamic spectrum* is then used to search for a dispersed signal, in a way similar to that used in the deep searches for dispersed periodic pulses described below (Section \[sect\_dsearch\]). We also search over a narrow range of period offsets around the expected period. Extending the search in the period-domain is particularly important for the archival data, since the observation epoch is well before the launch of the Fermi mission and the validity of the back-projected gamma-ray ephemeris can not be ensured. For the parameters of our search, the optimum S/N threshold, as suggested by @Handbook04, is about 5. However, we set a slightly higher S/N threshold of 8 to account for any low level RFI, as well as to be able to check for consistency of a signal across the observation bandwidth.
The multiple observing sessions towards each of the target sources allowed us to explore any transient or non-persistent periodic emission from these pulsars. The multiple session data from the new observations were used to carry out deep searches, details of which are given below.
Deep search for dispersed periodic pulses {#sect_dsearch}
-----------------------------------------
Since the rotation ephemerides for the gamma-ray pulsars are known from timing of the LAT data[^10], multiple session data from the new observations could be used advantageously to enhance our sensitivity for detecting a periodic signal. For each of our target gamma-ray pulsars, we use the pulsar timing software <span style="font-variant:small-caps;">Tempo</span>[^11] along with the corresponding timing model, to predict the pulsar period and the pulse phase. The dynamic spectrum for each of the observing sessions is folded over the predicted pulse period (the time-ranges identified as RFI-contaminated in the pre-search processing are excluded). The *folded dynamic spectra* from all the observing sessions of a particular source are then phase-aligned and co-added. While co-adding, the average band-shape modulation is removed, and the frequency channels identified as RFI contaminated in individual observing sessions are excluded. Also, to account for possible differences in the effective integration time of individual sessions (i.e., the RFI-free observation duration), a suitably weighted average of the folded dynamic spectra is computed.
To search for a dispersed signal, the final co-added (or more precisely, *averaged*) folded dynamic spectrum is dedispersed for a number of trial dispersion measures, and the significance of the resultant average profiles is assessed. To enhance the signal-to-noise ratio (S/N), the profiles are smoothed to a resolution of about $20{\ensuremath{^{\circ}}}$ to $30{\ensuremath{^{\circ}}}$ in pulse-longitude, and sum-of-squares (Paper I) or $\chi^2$ [@Leahy83] is used as the figure of merit to assess the profile significance. An in-house developed software pipeline was used to perform the above search. The pipeline was successfully verified using observations of our control pulsars.
----------------------- -------------- ---------------------------------- ------------------------------------ ----------------------------------- --------------- ------------------------ ------------------------
Sr. Target PSR Pointing Pointing t$_{obs}$ (s) $T_{\rm sky}$ $S_{\rm min}^{\rm SP}$ $S_{\rm min}^{\rm P0}$
No. Dec. (${\ensuremath{^{\circ}}}$) offset (${\ensuremath{^{\circ}}}$) \[$N_{\rm sessions} \times\tau$\] (K) (Jy) (mJy)
1 J0357$+$3205 $+35$ 3 2$\times$1200 18100 72 218
2 J0633$+$0632 $+14$ 7 1$\times$1200 19900 91 278
3 J0633$+$1746 $+14$ 4 1$\times$1200 19900 77 234
4 J1741$-$2054 $-30$ 9 1$\times$1200 62600 386 1174
5 J1809$-$2332 $-30$ 6 3$\times$1200 62400 340 1036
6 J1813$-$1246 $-05$ 8 4$\times$1200 76000 387 1179
7 J1826$-$1256 $-05$ 8 3$\times$1200 80100 408 1242
8 J1846$+$0919 $+14$ 5 1$\times$1200 62900 254 774
9 J1907$+$0602 $+14$ 8 1$\times$1200 71700 357 1087
10 J1954$+$2836 $+35$ 6 2$\times$1200 45000 202 614
11 J1957$+$5033 $+55$ 4 1$\times$1200 35300 173 527
12 J1958$+$2846 $+35$ 6 2$\times$1200 47100 211 642
13 J2021$+$4026 $+35$ 5 4$\times$1200 43200 184 560
14 J2032$+$4127 $+35$ 6 2$\times$1200 38200 171 521
15 J2055$+$2539 $+35$ 9 2$\times$1200 35900 199 605
16 J2238$+$5903 $+55$ 4 1$\times$1200 28100 138 420
\[ulimits\_archival\]
----------------------- -------------- ---------------------------------- ------------------------------------ ----------------------------------- --------------- ------------------------ ------------------------
Notes. — (1) “Pointing offset” is the difference between the pointing declination and the true declination of the target pulsar. (2) Since the computation of sensitivity limits do not take into account any possible offset in RA, the limits in some cases might be underestimated, at most (i.e., in the worst case) by a factor of 2. (3) $\tau$ is the individual observing session duration, and t$_{obs}$ is the total observation duration of all the sessions towards a particular source.
------------------ -------------- ----------------------------------- --------------- ------------------------ ------------------------ -------------------- ----------------- ------ --------------------------------- -------------------------------
Sr. Target PSR t$_{obs}$ (s) $T_{\rm sky}$ $S_{\rm min}^{\rm SP}$ $S_{\rm min}^{\rm P0}$
No. \[$N_{\rm sessions} \times\tau$\] (K) (Jy) (mJy) $S_{\rm previous}$ $\nu_{\rm obs}$ Ref. $S_{\rm previous}^{\rm Scaled}$ $S_{\rm min}^{\rm P0,Scaled}$
(mJy) (MHz) ($\mu$Jy) ($\mu$Jy)
1 J0357$+$3205 24$\times$1800 17200 67 34 0.043 327 c 2 20
2 J0633$+$0632 45$\times$1800 19900 77 28 0.075 327 c 4 17
3 J0633$+$1746 45$\times$1800 19900 102 38 150.0 35 a 94 22
4 J1732$-$3131 85$\times$1800 51400 271 73 0.059 1374 c 57 43
5 J1809$-$2332 20$\times$1800 74900 348 193 0.026 1352 c 24 114
6 J1836$+$5925 33$\times$1800 24700 128 55 0.070 350 c 4 32
7 J2021$+$4026 24$\times$1800 44100 181 92 0.051 820 c 17 54
8 J2055$+$2539 22$\times$1800 30400 114 60 0.085 327 b 5 35
9 J2139$+$4716 23$\times$1800 32500 143 74 0.171 350 d 11 44
10 J2238$+$5903 22$\times$1800 29700 154 82 0.027 820 c 9 48
\[ulimits\_new\]
------------------ -------------- ----------------------------------- --------------- ------------------------ ------------------------ -------------------- ----------------- ------ --------------------------------- -------------------------------
[$N_{\rm sessions}$ is modified (lowered) so that t$_{obs}$ provides the *effective* integration time (i.e., the integration time after excluding the RFI contaminated time intervals).]{}
[ The upper flux density limits presented for these pulsars are modified by the correction factors for the respective offsets from the pointing declination.]{}
[Using a pulse duty cycle of $50\%$ (instead of $10\%$) for J1732$-$3131, as indicated by its average profile, would increase the corresponding $S_{\rm min}^{\rm P0}$ by a factor of 3.]{}
[References — (a) Ramachandran et al. 1998; (b) Saz Parkinson et al. 2010; (c) Ray et al. 2011; (d) Pletsch et al. 2012b.]{}
Single pulse search sensitivity
-------------------------------
In our single-pulse searches, the peak flux density of a temporally resolved pulse [@CM03], is given by: $$S_{\rm peak}^{\rm SP} = (S/N)_{\rm peak} \times
\frac{2 k_B T_{sys}}{A_e(z)\sqrt{n_p\, W\,\Delta\nu}}
\label{sps_speak}$$ where, $T_{\rm sys}$ is the system temperature, $A_e(z)$ is the effective collecting area as a function of zenith-angle ($z$), $\Delta\nu$ is the observation bandwidth, $n_p$ is the number of polarizations (1 for Gauribidanur telescope), and $(S/N)_{\rm peak}$ is peak signal-to-noise ratio of the pulse corresponding to a smoothing optimum for its observed width of $W$.
Note that the observed pulse width is contributed to by various pulse broadening effects, viz. intrinsic pulse width, interstellar scattering, receiver filter response time, and residual dispersion smearing across individual frequency channels. However, the scatter broadening at such low frequencies dominates over other pulse broadening effects even for moderate values of DM. Hence, at moderately high DMs, sensitivity of our single pulse search, in terms of pulse energy (i.e., $S_{\rm peak}^{\rm SP}\times W$), becomes independent of the *intrinsic* pulse width. This is clearly seen in Figure \[fig\_sps\_smin\] which shows the minimum detectable pulse energy for intrinsic pulse widths of 1, 10 and 50 ms, as a function of DM. For instance, beyond a DM of about 25 pc cm$^{-3}$, the minimum detectable pulse energy for all the pulses with intrinsic widths $\leq 10$ ms is same. We have followed the scatter broadening dependence on DM as modelled by @Bhat04. Also, we have used the collecting area corresponding to a pointing declination at or near the instrumental zenith of $14{\ensuremath{^{\circ}}}.1$. For a declination away from zenith, the sensitivity will decrease by a factor of $\sec(z)$.
Periodic signal search sensitivity
----------------------------------
For periodicity searches, the minimum detectable flux density $S_{\rm min}^{\rm P0}$, i.e., at the threshold signal-to-noise ratio $(S/N)_{\rm min}$, is given by [@Vivek82]: $$S_{\rm min}^{\rm P0} = (S/N)_{\rm min} \times
\frac{2 k_B T_{\rm sys}}{A_e(z)\sqrt{n_p\, t_{\rm obs}\, \Delta\nu}}
\sqrt{\frac{W}{P-W}}$$ where, $W$ is the pulse width, $P$ is the pulse period and $t_{\rm obs}$ is the total integration time. For archival data, $t_{\rm obs}$ is equal to the total observation duration of a single session (i.e., about 1200 $s$). For new observations, $t_{\rm obs}$ equals the cumulative observation duration of all the sessions.
Results and Discussion
======================
Searches using the archival data
--------------------------------
Our searches for bright single pulses as well as for periodic signals using the archival data did not result in any further detection of decameter-wavelength counterparts of radio-quiet gamma-ray pulsars. For the archival data, the upper flux density limits for periodic as well as single pulse emission are presented in Table \[ulimits\_archival\]. To enable easy comparison with the flux density limits at higher radio frequencies available in literature, generally computed for a detection limit of $5\,\sigma$, the upper limits presented in Table \[ulimits\_archival\] are also computed for a (S/N)$_{\rm min}$ of 5. For the archival observations, our target sources were generally offset from the pointing center of the beam. To calculate the factor by which the gain reduces at the target source declination, relative to the beam-center declination, we assume a theoretical beam-gain pattern: $P(\theta)=[\sin{(\pi D \sin{\theta}/\lambda)}/(\pi D \sin{\theta}/\lambda)]^2$, where $D=20$ m and $\lambda=8.8$ m. The flux density limits estimated at the beam center are then scaled-up using the above correction factors computed for respective source position offsets[^12]. Note that we have carried out the above correction only for the offsets in declination. Possible offsets in RA are less than 1 minute (i.e., above the half-power points in the beam-gain pattern). Whenever archival data are available from multiple sessions, the offsets in RA are different for different sessions, and generally the RA offset is negligible at least for one of the sessions. Hence, Table \[ulimits\_archival\] presents the sensitivity limits for the best case when there is no offset in RA, and the limits in some cases might be underestimated, at most (i.e., in the worst case) by a factor of 2. The sensitivity limits for the single pulse search ($S_{\rm min}^{\rm SP}$) are computed for a nominal pulse width of $100$ ms, while those for the periodicity search ($S_{\rm min}^{\rm P0}$) are computed for a pulse duty cycle of $10\%$ and observation-duration of a single observing session, i.e., $1200$ s.
Deep follow-up observations of J1732$-$3131
-------------------------------------------
We carried out extensive follow-up observations of J1732$-$3131, distributed in 125 sessions, amounting to a total of 62.5 hours of observation time. In our deep search using an effective integration time of about 42.5 hours (after rejecting the RFI contaminated time sections), we could not (re-)detect any readily apparent (i.e., above a detection threshold of $8\sigma$) periodic signal from J1732$-$3131. Our searches for single bright pulses as well as for periodic signal using the individual session data also did not result in any significant candidate above our detection threshold of $8\sigma$.
Although we did not have any significant detection, the possibility of a signal weaker than our detection threshold can not be ruled out. Since we have an estimate of the DM from our candidate detection of this pulsar ($15.44\pm0.32$ pc cm$^{-3}$; Paper I), we can look for weak periodic signals at this DM that are consistent over multiple observing sessions. Furthermore, allowing for the possibility that the periodic signal might be very weak, if at all present, we carefully chose the observing sessions that are virtually free from RFI contamination (assessed by visual inspection of the dynamic spectrum), and where the dedispersed folded profiles were found to have full-swing S/N (i.e., peak-to-peak S/N) more than 4. Such average profiles, corresponding to 21 sessions, are phase-aligned and presented in Figure \[j1732\_mep\]. For comparison, we have overlaid the average profile from the original detection (dotted line; hereafter *the old profile*) on the net average profile of all the 21 sessions (solid line; hereafter *the new profile*) in the upper panel. The two profiles are manually aligned, since accuracy of the time-stamp in the archival data is not adequate enough. The two profiles, observed 10 years apart, exhibit striking similarity, and both are consistent with each other within the noise uncertainties. As a quantitative measure of the similarity, the Pearson (normalized) correlation coefficient between the two profiles is found to be 0.85.
To further assess statistical significance of the apparent similarity between the two profiles, we performed Monte-Carlo simulation. An individual realization in our simulation involves generating a random noise profile and finding its cross-correlation with the old profile. To be compatible with the smoothed profiles shown in Figure \[j1732\_mep\], the random noise profile is also smoothed with a $45{\ensuremath{^{\circ}}}$ wide window. The resultant noise profile is cross-correlated with the old profile at all possible phase-shifts, and the maximum (normalized) correlation coefficient is noted down. We simulated 10 million such independent realizations. The maximum correlation coefficient was found to be $\geq0.85$ (i.e., equal to or greater than the correlation found between the old and the new profile) only in $0.8\%$ of these realizations. Hence, the probability of the old and the new profiles having the same origin is estimated to be 0.992. In other words, the two profiles are consistent with each other at a confidence level of $99.2\%$.
The observed consistency between the average profile shape obtained by combining data from multiple epochs and that from the original detection 10 years ago, compels us to infer that (a) our candidate detection (Paper I) was not a mere manifestation of noise or RFI, and hence (b) the LAT-pulsar J1732$-$3131 is not radio-quiet. If true, the dispersion measure of this pulsar is $15.44\pm0.32$ pc cm$^{-3}$ (Paper I). Also, our earlier estimate of the average flux density (i.e., pulse-energy/period) of this pulsar in Paper I ($\sim4$ Jy; at 34.5 MHz) was most probably affected by scintillation. The new average profile provides a better estimate (since the scintillation effects are expected to average out), and suggests the average flux density to be 200–400 mJy at 34 MHz. With this new estimate, non-detection of this pulsar at higher radio frequencies could be explained with a spectral index $\lesssim -2.3$, assuming no turn-over [@Izvekova81] in the spectrum. This upper limit on the spectral index lies on the steeper edge of the range of spectral indices for normal pulsars [$-1.4\pm1.0$; @Bates13].
New observations towards other target sources
---------------------------------------------
In a couple of observing sessions towards the telescope pointing direction of RA=06:34:26, Dec=10[$^{\circ}$]{}, we detected a few ultra-bright pulses at two different DMs of about $2$ pc cm$^{-3}$ and $3.3$ pc cm$^{-3}$, respectively. However, when dedispersed at the DMs suggested by the bright single pulses, no significant signal was found at the expected periodicities of our target pulsars J0633$+$0632 and J0633$+$1746, which would have been in the telescope beam centered at above coordinates. Energies of these strong pulses in the two observing sessions are comparable to typical energies of giant pulses from the Crab pulsar at decameter wavelengths [@Popov06]. More detailed investigations of these single pulses will be reported elsewhere.
No significant pulsed (periodic or transient) signal, above a detection threshold of $8\sigma$, was found towards the directions of other selected gamma-ray pulsars. The upper limits on corresponding flux densities, for a detection limit[^13] of $5\sigma$, are presented in Table \[ulimits\_new\]. For computing the periodic signal search sensitivity ($S_{\rm min}^{\rm P0}$), we have excluded the time-intervals rejected as RFI contaminated from the total integration time. To compare with the earlier searches at higher frequencies, we have also compiled the flux density limits ($S_{\rm previous}$) from literature, along with their corresponding observation frequencies ($\nu_{\rm obs}$), in Table \[ulimits\_new\]. In case of the limits being available at several frequencies, the one at the lowest frequency (i.e., closest to 34 MHz) has been used. Wherever needed, these limits were scaled to $5\sigma$-level, before compiling into the table. For comparison, our limits at decameter wavelengths and those from literature are scaled to 1.4 GHz using a spectral index of $-2.0$, and presented as $S_{\rm min}^{\rm P0,Scaled}
\left(=S_{\rm min}^{\rm P0}\times\left[\frac{1400}{34}\right]^{-2}\right)$ and $S_{\rm previous}^{\rm Scaled}
\left(=S_{\rm previous}\times\left[\frac{1400}{\nu_{\rm obs}}\right]^{-2}\right)$, respectively. We have assumed that there is no spectral turn-over above our observation frequency (i.e., 34 MHz). Note that @Bates13 and @Maron00 have estimated the average spectral index for normal pulsars to be $-1.4\pm1.0$ and $-1.8\pm0.2$, respectively. Our assumed spectral index (i.e., $-2.0$), although lying on the steeper side, is well consistent with both these estimates. Despite the large background sky-temperature at our observing frequency, for a couple of pulsars our flux density limits are better than those from deep searches at higher radio frequencies, and in other cases they are only within a factor of few of the limits from shorter wavelength searches (provided the spectral index of these sources is equal to or steeper than $-2.0$).
The above comparison of flux density limits may appear to be optimistic, since we have not assumed any turn-over in the spectrum. However, even with a turn-over around 80–100 MHz, our flux density limits scale to typically a few hundreds of $\mu$Jy at 1.4 GHz. Further, if the lack of radio emission from the LAT-discovered pulsars is indeed due to unfavorable viewing geometries, then the pulsars which could possibly be detected at decameter wavelengths can be expected to have steep spectra. If we assume a fairly steep spectrum with an index of $-3.0$ [for comparison, the spectral index of B0943+10 is $-3.7\pm0.36$; @Maron00], most of our flux limits scale to less than $100\mu$Jy at 1.4 GHz, and some of them are still comparable to those reported at higher frequencies.
The possibility that some of our target sources are “radio-loud”, but have flux densities below our detection limits, can not be ruled out. The very faint radio emission from J1732$-$3131 which could be assessed only by making use of its DM estimated from earlier detection (Paper I), indicates the possibility of very faint emission from a few more of the (so far) *radio-quiet* gamma-ray pulsars. However, lack of radio detection from most of our target sources indicates that a large fraction of our sample may indeed be radio-quiet. Consequently, the high fraction of gamma-ray pulsars being radio-quiet is consistent with the predictions of “narrow polar-cap” models [e.g., @Sturrock71; @RS75] for radio beams and “fan-beam outer magnetosphere” models [e.g., @Romani96] for gamma-ray emission.
Conclusions
===========
The following points summarize the results of our deep searches for decameter wavelength counterparts of several *radio-quiet* gamma-ray pulsars:
1. We have shown that the 34 MHz average profile of the LAT-discovered pulsar J1732$-$3131 obtained by effectively integrating over more than 10 hours of new observations carried out at different epochs (Figure \[j1732\_mep\]) is consistent with that from the first radio detection of this pulsar [@MAD12] at a confidence level of $99.2\%$. We present this consistency as an evidence that J1732$-$3131 is a faint radio pulsar *(and not radio-quiet)* at decameter wavelengths.
2. We have put stringent upper limits on pulsed (transient as well as periodic signal) radio emission from several of the radio-quiet gamma-ray pulsars at decameter wavelengths (Table \[ulimits\_new\]). Despite the extremely bright sky background at decameter wavelengths, the flux density limits obtained from our deep searches are comparable to those from higher frequency searches of these pulsars, when scaled to 1.4 GHz assuming a spectral index of $-2.0$ and no turn-over in the spectrum.
We would also like to emphasize that in the process of carrying out the deep searches, the Gauribidanur radio telescope is now appropriately equipped with a sensitive setup to detect and study known periodic signals with average flux densities as low as a few mJy, even at such low frequencies.
Acknowledgments {#acknowledgments .unnumbered}
===============
We gratefully acknowledge the support from the observatory staff. We thank the anonymous referee for a critical review of our manuscript, as well as for the comments and suggestions which helped in improving the manuscript. YM is thankful to Avinash Deshpande for useful discussions and comments on the manuscript. YM is grateful to Paul Ray and other members of the LAT-team for providing the up-to-date timing models of several gamma-ray pulsars. We gratefully thank Indrajit V. Barve, Hariharan K., Rajalingam M., and several other colleagues, for their help with the observations at several occasions. The Gauribidanur radio telescope is jointly operated by the Raman Research Institute and the Indian Institute of Astrophysics.
\[lastpage\]
[^1]: E-mail: [email protected]; Current Address: National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Pune 411007, India.
[^2]: Additional 28 pulsars detected in gamma-rays are reported to have publications in preparation [@fermi_catalog13], further increasing the total number of gamma-ray pulsars to 149.
[^3]: Setting a new convention, the 2$^{\rm nd}$ *Fermi* LAT catalog of gamma-ray pulsars labels all the pulsars with 1.4 GHz flux density $<30\,\mu$Jy as “radio-quiet”. However, as in the usual convention, we use the term radio-quiet only for those pulsars which have no detectable radio flux for an observer at earth.
[^4]: Our sample also includes J1732$-$3131, with the aim of making its confirmatory (re-)detection.
[^5]: <http://www.rri.res.in/~dsp_ral/ppr/ppr_main.html>
[^6]: Radio counterparts of 3 of these 16 gamma-ray pulsars are known. The radio counterpart of J1907$+$0602 was reported while our searches were ongoing [@Abdo10], while those of J1741$-$2054 and J2032$+$4127 were already known [@Camilo09].
[^7]: It is possible that the computed mean and standard deviation get biased by a few very strong pulses. To get an unbiased (or robust) estimate, mean and standard deviation are recalculated by using the previous estimates to detect and exclude the strong pulses above a given S/N threshold. This process is continued iteratively till the computed mean and standard deviation no more differ from their respective values in the previous iteration.
[^8]: As evident from Eq. \[sps\_speak\], for a given peak flux density, the highest achievable S/N of a pulse is directly proportional to square-root of its width. Hence, for an optimum width-search, we sample the trial pulse width range in a logarithmic manner.
[^9]: Our choice of tolerable number of false alarms is admittedly large, to increase the probability of detecting the faint pulses.
[^10]: [The up-to-date timing models of several gamma-ray pulsars are provided by the LAT team at <https://confluence.slac.stanford.edu/display/GLAMCOG/LAT+Gamma-ray+Pulsar+Timing+Models>]{}
[^11]: For more information about <span style="font-variant:small-caps;">Tempo</span>, please refer to the website: <http://www.atnf.csiro.au/research/pulsar/tempo/> .
[^12]: As explained in Paper-I, system temperature at the beam center is estimated by computing a weighted average of sky temperature estimates [@DU90] at several points across the large beam using a theoretical beam-gain pattern.
[^13]: As mentioned earlier, the flux density limits are computed for a (S/N)$_{\rm min}$ of 5, to enable easy comparison with the flux density limits at higher radio frequencies available in literature.
|
---
abstract: 'In this article we consider a Fokker-Planck equation on ${\mathbb{R}}^d$ with a non-local, mass preserving perturbation. We first give a spectral analysis of the unperturbed Fokker-Planck operator in an exponentially weighted $L^2$-space. In this space the perturbed Fokker-Planck operator is an isospectral deformation of the Fokker-Planck operator, i.e. the spectrum of the Fokker-Planck operator is not changed by the perturbation. In particular, there still exists a unique (normalized) stationary solution of the perturbed evolution equation. Moreover, the perturbed Fokker-Planck operator generates a strongly continuous semigroup of bounded operators. Any solution of the perturbed equation converges towards the stationary state with exponential rate $-1$, the same rate as for the unperturbed Fokker-Planck equation. Moreover, for any $k\in{\mathbb{N}}$ there exists an invariant subspace with codimension $k$ (if $d=1$) in which the exponential decay rate of the semigroup equals $-k$.'
address:
- 'Institute for Analysis and Scientific Computing, Technical University Vienna, Wiedner Hauptstraße 8, A-1040 Vienna'
- 'Institute for Analysis and Scientific Computing, Technical University Vienna, Wiedner Hauptstraße 8, A-1040 Vienna'
author:
- Dominik Stürzer
- Anton Arnold
bibliography:
- 'referenz.bib'
title: 'Spectral Analysis and Long-Time Behaviour of a Fokker-Planck Equation with a Non-Local Perturbation'
---
Introduction {#sec1}
============
This work deals with the analysis of the following class of perturbed Fokker-Planck equations:
\[pert\_fp\] $$\begin{aligned}
{\partial}_t f&=&\nabla\cdot(\nabla f+{\mathbf{x}}f)+\Theta f=:Lf+\Theta f\label{pert_fp:1}\\
f|_{t=0}&=&{\varphi}({\mathbf{x}}),\label{pert_fp:2}
\end{aligned}$$
where $t\ge 0,\,{\mathbf{x}}\in{\mathbb{R}}^d$ with $d\in{\mathbb{N}}$, and $f=f(t,{\mathbf{x}})$. Here, ${\partial}_t f$ denotes the time derivative. The linear, non-local operator $\Theta$ is given by a convolution $\Theta f={\vartheta}*f$ with respect to ${\mathbf{x}}$, where its kernel ${\vartheta}$ is assumed to be time-independent and with zero mean, i.e. $\int_{{\mathbb{R}}^d}{\vartheta}({\mathbf{x}}){\,\mathrm{d}}{\mathbf{x}}=0$. Also, it is assumed to satisfy certain regularity conditions, which will be specified in the Sections \[sec3\] and \[sec35\].
The above equation is mainly motivated by the quantum-kinetic Wigner-Fokker-Planck equation, describing so-called open quantum systems, see [@Arnold2010; @alms]. It is of the form $$\begin{aligned}
{\partial}_t u&= \nabla_{{\mathbf{x}},\mathbf v}\cdot(\nabla_{{\mathbf{x}},\mathbf v} u+ (\nabla_{{\mathbf{x}},\mathbf v} A+\mathbf{F})u)+\Xi[V]u\\
u|_{t=0}&=u_0,\end{aligned}$$ where $u=u(t,{\mathbf{x}},\mathbf v)$ is the phase-space quasi-density, with ${\mathbf{x}},\mathbf v\in{\mathbb{R}}^d$ denoting position and momentum. The given coefficient function $\nabla_{{\mathbf{x}},\mathbf v} A+\mathbf{F}$ is affine in $({\mathbf{x}},\mathbf v)$ and models the confinement and friction of the system. $\Xi[V]$ is a non-local operator (convolution in $\mathbf v$) determined by an external potential $V({\mathbf{x}})$. One question of interest in this problem is to show the existence of a unique normalized stationary state, and to prove uniform exponential convergence of the solution to the stationary state. In the case of a quadratic confinement potential with a small perturbation these questions have been answered positively in [@Arnold2010], see also [@afn2008] for an operator-theoretic approach. However, from the physical point of view, the restriction to nearly quadratic potentials seems quite artificial. This raises the question if the results can be extended to a more general family of (confining) potentials. In order to gain insight into what can be expected and what mechanisms are responsible for the actual behaviour, we shall consider here (\[pert\_fp\]) as a similar, yet simplified model, which still preserves the essential structure. The non-local operator $\Xi[V]$, which is a convolution in $\mathbf v$, is replaced by a convolution with kernel ${\vartheta}$. This represents a first step towards the full analysis.
Other examples of non-local perturbations in Fokker-Planck equations appear e.g. in the linearized vorticity formulation of the 2D Navier-Stokes equations (cf. (12)-(14) in [@gw]) or in electronic transport models (cf. the linearization of equations (1), (6), (7) in [@lk]).
For the unperturbed equation (\[pert\_fp\]), i.e. the case ${\vartheta}=0$, the natural functional setting is the space $L^2(\mu^{-1})$, with the weight function $\mu({\mathbf{x}})=\exp(-|{\mathbf{x}}|^2/2)$. Here, $\mu/(2\pi)^{d/2}$ is the unique steady state with normalized mass, i.e. $\int_{{\mathbb{R}}^d}\mu/(2\pi)^{d/2}{\,\mathrm{d}}{\mathbf{x}}=1$, and all solutions to initial conditions with mass one decay towards this state with exponential rate of at least $-1$, see e.g. [@bakry]. However, if $\Theta$ is added, the situation often becomes more complicated. One reason is that many non-local (convolution) operators are unbounded in the space $L^2(\mu^{-1})$. This can be illustrated for the simple example with the convolution kernel ${\vartheta}=\delta_{-\alpha}-\delta_{\alpha}, \,\alpha\in{\mathbb{R}}$, in one dimension. It corresponds to the operator $(\Theta f)(x)=f(x+\alpha)-f(x-\alpha),\,x\in{\mathbb{R}}$, which is unbounded in $L^2(\mu^{-1})$. In this case one can show (with an eigenfunction expansion) that every (non-trivial) stationary state of (\[pert\_fp\]) is [*not*]{} even an element of $L^2(\mu^{-1})$. Thus, this space is not suitable for our intended large-time analysis, since it is “too small”. This motivates to consider (\[pert\_fp\]) in some larger space $L^2(\omega)$, with a weight $\omega$ growing slower than $\mu^{-1}$. Due to the previous discussion we shall choose $\omega$ such that a large class of non-local operators becomes bounded. But the new space should not be “too large” either, since we would risk to loose many convenient properties (like the spectral gap) of the unperturbed Fokker-Planck operator. In $L^2({\mathbb{R}}^d)$, e.g., the spectrum of $L$ is the left half plane $\{\lambda\in{\mathbb{C}}:{\operatorname{Re}}\lambda\le d/2\}$, cf. [@Metafune2001]. It will turn out that $\omega({\mathbf{x}}):=\cosh \beta |{\mathbf{x}}|,\,\beta>0$, is a convenient choice. Moreover, there is a useful characterization of the functions of $L^2(\omega)$ in terms of their Fourier transform, see Lemma \[analyticity\].
Here we focus on the Fokker-Planck operator in exponentially weighted spaces. For $L^2$-spaces with polynomial weights, the spectrum of $L$ was studied in [@Gallay2002]. Furthermore, our results complement the analysis of Metafune [@Metafune2001], where a larger class of Ornstein-Uhlenbeck operators is investigated in unweighted $L^p$-spaces with $p\ge 1$.
This paper is organized as follows. Since the analysis in the $d$-dimensional case is very similar to the one-dimensional case, we first discuss (in Sections \[sec2\] and \[sec3\]) the one-dimensional problem in great detail, to keep the notation and arguments more concise. In Section \[sec35\], we generalize the proofs to higher dimensions.
In Section \[sec2\] we investigate the one-dimensional Fokker-Planck operator in $L^2(\omega)$ (denoted by ${\mathcal L}$), and show that its spectrum is $-{\mathbb{N}}_0$, and consists entirely of eigenvalues. All eigenspaces are one-dimensional, in particular the stationary state is unique up to normalization. Moreover, the operator ${\mathcal L}$ generates a $C_0$-semigroup of uniformly bounded operators on $L^2(\omega)$, and any solution of (\[pert\_fp\]) for $\Theta=0$ converges towards the (appropriately scaled) stationary solution with exponential rate of at least $-1$. More generally, for any $k\in{\mathbb{N}}_0$ there exists an ${\mathcal L}$-invariant subspace of $L^2(\omega)$ with codimension $k$ in which the associated semigroup has an exponential decay rate of $-k$. Section \[sec3\] is dedicated to the perturbed Fokker-Planck operator ${\mathcal L}+\Theta$ in one dimension. Using the compactness of the resolvent of ${\mathcal L}$ and ladder operators we show that ${\mathcal L}+\Theta$ is an isospectral deformation of the unperturbed operator ${\mathcal L}$, i.e. $\sigma({\mathcal L}+\Theta)=\sigma({\mathcal L})=-{\mathbb{N}}_0$. The spectrum still consists only of eigenvalues with one-dimensional eigenspaces, which ensures the existence of a unique normalized steady state of (\[pert\_fp\]) in $L^2(\omega)$. Finally we show that the semigroup generated by ${\mathcal L}+\Theta$ still has the same decay properties as the one generated by ${\mathcal L}$. In particular the solutions of (\[pert\_fp\]) with normalized mass decay to the stationary state with exponential rate of at least $-1$. In Section \[sec4\] we present simulation results, which illustrate the decay rates obtained before.
The Fokker-Planck Operator in Weighted $L^2$-Spaces {#sec2}
===================================================
Here and in Section \[sec3\] we shall consider the one-dimensional Fokker-Planck equation, i.e. $d=1$. For the Fourier transform we use the convention $${\mathcal F}_{x\to\xi}f\equiv\hat f(\xi):=\int_{\mathbb{R}}f(x){\mathrm{e}}^{-{{\mathrm{i}}}x\xi}{\,\mathrm{d}}x.$$ With this scaling we may identify $\hat f(0)$ with the [*mass*]{} of $f$.
For an analytic function $f$ on a simply connected domain $\Omega$ we denote the line integral of $f$ along a path from $a$ to $b$ inside of $\Omega$ by $$\int_{a\to b} f(\zeta){\,\mathrm{d}}\zeta.$$ In order to properly define complex powers, we specify a branch of the logarithm. For $\xi\in{\mathbb{C}}\setminus\{0\}$ we set $\ln\xi:=\log|\xi|+{{\mathrm{i}}}\arg\xi$, with $\arg\xi\in[-\frac\pi2,\frac{3\pi}2)$, and $\log(\cdot)$ is the natural logarithm on ${\mathbb{R}}^+$. For $\zeta\in{\mathbb{C}}$ we may then define $\xi^{-\zeta}:=\exp(-\zeta\ln(\xi))$.\[log\]
On a domain $\Omega\subseteq{\mathbb{R}}$ we call a real-valued function $w\in L^1_{\text{loc}}(\Omega)$ a [*weight function*]{} if it is bounded from below by a positive constant a.e. on every compact subset of $\Omega$. We denote the corresponding weighted $L^p$-space by $L^p(\Omega;w)\equiv L^p(\Omega;w(x){\,\mathrm{d}}x)$, where $1\le p\le \infty$. The space $L^2(\Omega;w)$ is equipped with the inner product $${\langle}f,g{\rangle}_{\Omega,w}=\int_\Omega f\bar g w{\,\mathrm{d}}x,$$ and the norm $\|\cdot\|_{\Omega,w}$.
Also, we introduce weighted Sobolev spaces. For two weight functions $w_0$ and $w_1$ and $1\le p\le\infty$, the space $W^{1,p}(\Omega;w_0,w_1)$ consists of all functions $f\in L^p(\Omega;w_0)$, whose distributional derivative satisfies $f'\in L^p(\Omega;w_1)$. We equip the space $W^{1,2}(\Omega;w_0,w_1)$ with the norm $$\|f\|_{\Omega,w_0,w_1}:=\big(\|f\|_{\Omega,w_0}^2+\|f'\|_{\Omega,w_1}^2 \big)^{\frac12},$$ see [@Kufner1984]. If $\Omega={\mathbb{R}}$ we shall omit the symbol $\Omega$ in these notations.
Furthermore, we present some definitions and properties concerning unbounded operators and their spectrum. Let $X,{\mathcal X}$ be Hilbert spaces. If $X$ is continuously and densely embedded in ${\mathcal X}$, we write $X{\hookrightarrow}{\mathcal X}$, and $X{\hookrightarrow}{\hookrightarrow}{\mathcal X}$ indicates that the embedding is compact. ${\mathscr C}(X)$ denotes the set of all closed operators $A$ in $X$ with dense domain $D(A)$. The set of all bounded operators $A:X\to {\mathcal X}$ is ${\mathscr B}(X,{\mathcal X})$; if $X={\mathcal X}$ we just write ${\mathscr B}(X)$. A closed, linear subspace $Y\subset X$ is said to be [*invariant*]{} under $A\in{\mathscr C}(X)$ (or [*$A$-invariant*]{}) iff $D(A)\cap Y$ is dense in $Y$ and ${\operatorname{ran}}A|_{Y}\subset Y$, see e.g. [@Albrecht2003]. For an operator $A\in{\mathscr C}(X)$ its range is ${\operatorname{ran}}A$, its null space is $\ker A$, and its algebraic null space is $M(A):=\bigcup_{k\ge 0}\ker A^k$. For any $\zeta\in{\mathbb{C}}$ lying in the resolvent set $\rho(A)$, we denote the resolvent by $R_A(\zeta):=(\zeta-A)^{-1}$. The complement of $\rho(A)$ is the spectrum $\sigma(A)$, and $\sigma_p(A)$ is the point spectrum. For an isolated subset $\sigma'\subset\sigma(A)$ the corresponding [*spectral projection*]{} ${\mathrm{P}\!}_{A, \sigma'}$ is defined via the line integral $$\label{def:spec_proj}
{\mathrm{P}\!}_{A, \sigma'}:=\frac 1{2\pi{{\mathrm{i}}}}\oint_\Gamma R_A(\zeta){\,\mathrm{d}}\zeta,$$ where $\Gamma$ is a closed Jordan curve with counter-clockwise orientation, strictly separating $\sigma'$ from $\sigma(A)\setminus\sigma'$, with $\sigma'$ in the inside of $\Gamma$ and $\sigma(A)\setminus\sigma'$ on the outside. The following results can be found in [@kato Section III.6.4] and [@taylay Section V.9]: The spectral projection is a bounded projection operator, decomposing $X$ into two $A$-invariant subspaces, namely ${\operatorname{ran}}{\mathrm{P}\!}_{A,\sigma'}$ and $\ker {\mathrm{P}\!}_{A,\sigma'}$. This property is referred to as the [*reduction of $A$ by ${\mathrm{P}\!}_{A, \sigma'}$.*]{} A remarkable property of this decomposition is the fact that $\sigma(A|_{{\operatorname{ran}}{\mathrm{P}\!}_{A,\sigma'}})= \sigma'$ and $\sigma(A|_{\ker {\mathrm{P}\!}_{A, \sigma'}})=\sigma(A)\backslash \sigma'$. Most of the time we will be concerned with the situation where $\sigma'=\{\lambda\}$, i.e. an isolated point of the spectrum. For further results see the Appendix \[app:space\_enlarg\].
A final remark concerns constants occurring in estimates: Throughout this article, $C$ denotes some positive constant, not necessarily always the same. Dependence on certain parameters will be indicated in brackets, e.g. $C(t)$ for dependence on $t$.
We begin our analysis by investigating the unperturbed one-dimensional Fokker-Planck operator $Lf:=f''+xf'+f$ in various weighted spaces. The natural space to consider $L$ in is $E:=L^2(1/\mu)$ with $\mu(x):=\exp(-x^2/2)$. We use the notation $\|\cdot\|_E$ for the norm and ${\langle}\cdot,\cdot{\rangle}_E$ for the inner product. Writing the operator in the form $$Lf=\bigg(\bigg(\frac f\mu\bigg)'\mu\bigg)'$$ shows that $L|_{C_0^\infty}$ is symmetric and dissipative in $E$. Then, the proper definition of $L$ is obtained by the closure of $L|_{C_0^\infty}$, and this procedure yields its domain $D(L)\subset E$. In the subsequent theorem we summarize some important properties of $L$ in $E$, see [@Metafune2001; @bakry; @helffernier]. Since $L$ in $E$ is isometrically equivalent to the (dimensionless) quantum harmonic oscillator Hamiltonian $H=-\Delta-1/2+x^2/4$ in $L^2({\mathbb{R}})$, we transfer many results of $H$ (see [@parmegg] and [@resi Theorem XIII.67]) to $L$. For the properties of the spectral projections, see also [@kato Section V.3.5].
\[prop:fp\_in\_H\] The Fokker-Planck operator $L$ in $E$ has the following properties:
1. \[st\_fp:1\] $L$ with $D(L)=\{f\in E:f''+xf'+f\in E\}$ is self-adjoint and has a compact resolvent.
2. \[st\_fp:2\] The spectrum is $\sigma(L)=-{\mathbb{N}}_0$, and it consists only of eigenvalues.
3. \[st\_fp:3\] For each eigenvalue $-k\in\sigma(L)$ the corresponding eigenspace is one-dimensional, spanned by $\mu_k:= \frac 1{\sqrt{2\pi}}H_k\mu$, where $$H_k(x)= \mu(x)^{-1}{\frac{\mathrm{d}^{k}{}}{\mathrm{d}{x}^{k}}}\mu(x)$$ is the $k$-th Hermite polynomial.
4. \[st\_fp:4\] The eigenvectors $(\mu_k)_{k\in{\mathbb{N}}_0}$ form an orthogonal basis of $E$.
5. \[st\_fp:45\] There holds the spectral representation $$\label{spec_proj_in_E}L=\sum_{k\in{\mathbb{N}}_0}-k\Pi_{L,k},\quad\text{where}\quad \Pi_{L,k}:=\frac{\sqrt{2\pi}}{k!}\mu_k{\langle}\cdot,\mu_k{\rangle}_E$$ is the spectral projection onto the $k$-th eigenspace.
6. \[st\_fp:5\] The operator $L$ generates a $C_0$-semigroup of contractions on $E_k$ for all $k\in{\mathbb{N}}_0$, where $E_k:=\ker(\Pi_{L,0}+\cdots+\Pi_{L,k-1}),\,k\ge 1$, and $E_0:=E$ are $L$-invariant subspaces of $E$. The semigroup satisfies the estimate $$\big\|{\mathrm{e}}^{tL}|_{E_k}\big\|_{{\mathscr B}(E_k)}\le{\mathrm{e}}^{-kt},\quad\forall k\in{\mathbb{N}}_0.$$
Hence, the Fokker-Planck equation ${\partial}_t f=Lf$ has a unique stationary solution with normalized mass, given by $\mu_0$. Its orthogonal complement $E_1$ consists of all elements of $E$ with zero mass. And according to Result (\[st\_fp:5\]) for $k=1$, any solution of ${\partial}_t f=Lf$ with unit mass converges towards $\mu_0$ with exponential rate of at least $-1$ in the $E$-norm.
In order to analyze the perturbed equation , we quickly find that $E$ is not appropriate. For example, for the simple (unbounded) perturbation $\Theta f(x):=f(x+\alpha)-f(x-\alpha),\,\alpha\in{\mathbb{R}}$, we can explicitly compute the stationary solution $f_0$ of and expand it with respect to the orthogonal basis $(\mu_k)_{k\in{\mathbb{N}}}$ of $E$. The obtained Fourier coefficients form a divergent sequence, and so $f_0\notin E$. Therefore we consider some larger space $L^2(\omega)$ instead of $E$, with a weight function $\omega$ growing more slowly than $\mu^{-1}$. Thereby we choose $\omega$ such that $\Theta$ becomes a bounded operator in $L^2(\omega)$ for a large family of convolution kernels. E.g., one can easily verify that $\Theta f(x)=f(x+\alpha)-f(x-\alpha)$ is bounded in $L^2(\exp(\beta|x|^\gamma))$ iff $\gamma\in[0,1]$ (for $\beta>0$). At the same time, $\omega$ should grow fast enough such that $L$ still has a spectral gap in $L^2(\omega)$, i.e. there exists some $a<0$ such that $\{\zeta\in{\mathbb{C}}:{\operatorname{Re}}\zeta>a\}\cap\sigma(L)=\{0\}$. These requirements suggest that exponentially growing weights would be good candidates, growing as fast as permissible while still admitting a large class of non-local operators. So, for the rest of this paper, we choose the weight function $\omega(x)=\cosh\beta x$ for some fixed $\beta>0$, and use the corresponding space ${\mathcal E}:=L^2(\cosh\beta x)$. As we will see in the following, the space ${\mathcal E}$ is very convenient also for technical purposes, since it can easily be characterized using the Fourier transform.
\[analyticity\] For $f\in{\mathcal E}$ we have the following properties:
1. \[analyt:i\] There holds $f\in {\mathcal E}$ iff its Fourier transform $\hat f$ possesses an analytic continuation (still denoted by $\hat f$) to the open strip $\Omega_{\beta/2}:=\{z\in {\mathbb{C}}:|{\operatorname{Im}}z|<\beta/2\}$, which satisfies $$\label{seid}
\sup_{\substack{|b|<\beta/2\\b\in{\mathbb{R}}}}\|\hat f(\cdot+{{\mathrm{i}}}b)\|_{L^2({\mathbb{R}})}<\infty.$$
2. \[analyt:ii\] For $\xi\in{\mathbb{R}}$ and $|b|<\beta/2$, $\hat f$ is explicitly given by $\hat f(\xi+{{\mathrm{i}}}b)={\mathcal F}_{x\to\xi}({\mathrm{e}}^{bx}f(x))$.
3. \[analyt:iii\] The following function lies in $L^2({\mathbb{R}})$: $$\label{hatf}
\xi\mapsto\hat f\Big(\xi\pm{{\mathrm{i}}}\frac\beta 2 \Big):={\mathcal F}_{x\to\xi}({\mathrm{e}}^{\pm\frac\beta 2x}f(x)),\quad\text{for a.e.~}\xi\in{\mathbb{R}}.$$ Moreover, $b\mapsto\hat f(\cdot+{{\mathrm{i}}}b)$ lies in $C([-\beta/2,\beta/2];L^2({\mathbb{R}}))$. In particular is a natural continuation of $\hat f$ from $\Omega_{\beta/2}$ to the closure $\overline{\Omega_{\beta/2}}$.
The proof is deferred to the Appendix \[app:proof\]. In the following, $\hat f$ always denotes the extension of the Fourier transform of $f\in{\mathcal E}$ according to Lemma \[analyticity\] (\[analyt:ii\])-(\[analyt:iii\]). Using this convention, we introduce an alternative norm on the space ${\mathcal E}$: $$\label{norm_w}
{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega^2:=\|\hat f(\cdot+{{\mathrm{i}}}\beta/2)\|_{L^2({\mathbb{R}})}^2+\|\hat f(\cdot-{{\mathrm{i}}}\beta/2)\|_{L^2({\mathbb{R}})}^2,$$ which is equal to $4\pi\|f\|^2_\omega$.
Furthermore, we notice that there holds a Poincaré-type inequality in ${\mathcal E}$:
The inequality $$\label{poincare}
\| f\|_\omega\le C_\beta\| f'\|_\omega$$ holds for all $f\in W^{1,2}(\omega,\omega)$, where $C_\beta>0$ is a constant only depending on $\beta$.
Use $|\widehat {f'\,}\!(\xi)|=|\xi\hat f(\xi)|$, and $|\xi|\ge \beta/2$ on $|{\operatorname{Im}}\xi|=\beta/2$. Then apply the norm ${|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}\cdot{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega$.
Our next step is to properly define the Fokker-Planck operator in ${\mathcal E}$. To this end we first define the distributional Fokker-Planck operator $\mathfrak Lf:=f''+xf'+f$ for $f\in{\mathscr S}'$.
\[reso\_estim\] Let $\zeta\in{\mathbb{C}}$ with ${\operatorname{Re}}\zeta\ge 1+\beta^2/2$, and consider the resolvent equation $(\zeta-\mathfrak L)f=g$ for $f,g\in{\mathcal E}$. Then there exists a constant $C>0$ independent of $f,g$, such that $$\label{comp_est}
\|f\|_\varpi+\|f'\|_\omega \le C\|g\|_\omega,$$ where $\varpi(x)=(1+|x|)\omega(x)$.
Let us fix $\zeta\in{\mathbb{C}}$ with ${\operatorname{Re}}\zeta\ge 1+\beta^2/2$. Now we consider the resolvent equation $(\zeta-\mathfrak L)f=g$ for $f,g\in {\mathcal E}\subset{\mathscr S}'$. Applying ${\langle}\cdot,f{\rangle}_\omega$ to both sides yields: $$\begin{aligned}
\int_{\mathbb{R}}\bar f g\omega{\,\mathrm{d}}x&= \int_{\mathbb{R}}\zeta |f|^2\omega-(f'+xf)'\bar f\omega{\,\mathrm{d}}x\\
&= \int_{\mathbb{R}}|f'|^2\omega+|f|^2(x\omega'+\zeta\omega)+f'\bar f\omega' +f\bar f' x\omega{\,\mathrm{d}}x. \end{aligned}$$ Next we take the real part: $$\begin{aligned}
{\operatorname{Re}}\int_{\mathbb{R}}\bar f g\omega{\,\mathrm{d}}x&= \int_{\mathbb{R}}|f'|^2\omega+|f|^2(x\omega'+{\operatorname{Re}}(\zeta)\omega)+\frac 12|f^2|'(\omega' +x\omega){\,\mathrm{d}}x\nonumber\\
&= \int_{\mathbb{R}}|f'|^2\omega+\frac 12|f|^2\tilde\omega{\,\mathrm{d}}x,\label{4star}\end{aligned}$$ with $\tilde\omega:=-\omega''+x\omega'+(2{\operatorname{Re}}\zeta-1)\omega$. For our choice $\omega(x)=\cosh \beta x$ we obtain $\tilde\omega(x)=(2{\operatorname{Re}}\zeta-1-\beta^2)\omega(x)+x\beta \sinh\beta x$. For ${\operatorname{Re}}\zeta\ge 1+\beta^2/2$, $\tilde\omega$ is strictly positive. Thus, $\tilde\omega$ is a weight function, and it has the asymptotic behaviour $\tilde\omega(x)\sim\beta|x|\omega(x)$ as $x\to\pm\infty$. Applying the Cauchy-Schwarz inequality to the left hand side of (\[4star\]) yields $$\frac12\|f\|_{\tilde\omega}^2+ \|f'\|^2_\omega\le \|f\|_\omega\|g\|_\omega.$$ For the left hand side we use $\omega(x)\le \tilde\omega(x)$ and the Poincaré inequality (\[poincare\]) to obtain $$\frac 12\|f\|_{\tilde\omega}+\frac 1{C_\beta}\|f'\|_\omega \le \|g\|_\omega.$$ The result follows, since the weight functions $\tilde\omega$ and $\varpi$ define equivalent norms.
\[cor\_dissip\] The operator $(L-1-\beta^2/2)|_{C_0^\infty({\mathbb{R}})}$ is dissipative in ${\mathcal E}$.
We use the result for $\zeta= 1+\beta^2/2$. We then estimate the right hand side for $f\in C_0^\infty({\mathbb{R}})$: $${\operatorname{Re}}\int_{\mathbb{R}}\bar f(L-\zeta)f{\,\mathrm{d}}x \le -\Big(C_\beta+\frac 12\Big)\|f\|_\omega^2\le 0,$$ where we used the Poincaré inequality and $\tilde\omega\ge\omega$.
The above results can be used to establish the proper definition of the Fokker-Planck operator in ${\mathcal E}$:
\[abschluss\_LL\] The operator $L|_{C_0^\infty({\mathbb{R}})}$ is closable in ${\mathcal E}$, and its closure ${\mathcal L}:={\operatorname{cl}}_{\mathcal E}L|_{C_0^\infty({\mathbb{R}})}$ has the domain of definition $D({\mathcal L})=\{f\in{\mathcal E}: \mathfrak Lf\in{\mathcal E}\}$. For $f\in D({\mathcal L})$ we have ${\mathcal L}f=\mathfrak Lf$.
The proof is deferred to the Appendix \[app:proof\]. It also yields the following result:
\[wob\] The resolvent set $\rho({\mathcal L})$ is non-empty. It contains the half-plane $\{\zeta\in{\mathbb{C}}:{\operatorname{Re}}\zeta\ge 1+\beta^2/2\}$.
As it turns out, the resolvent estimate is strong enough to prove compactness of the resolvent. To this end we shall use the following simplified version of [@Opic1989 Theorem 2.4]:
Let $w,w_0,w_1$ be weight functions, and $(\Omega_n)_{n\in{\mathbb{N}}}$ a monotonically increasing sequence of subsets of ${\mathbb{R}}$ that converges to ${\mathbb{R}}$. Assume that for all $n\in{\mathbb{N}}$ there holds the compact embedding $W^{1,2}(\Omega_n;w_0,w_1){\hookrightarrow}{\hookrightarrow}L^2(\Omega_n; w)$. Then $$W^{1,2}(w_0,w_1){\hookrightarrow}{\hookrightarrow}L^2(w)\quad{\Leftrightarrow}\quad \lim_{n\to\infty}\sup_{\|f\|_{w_0,w_1}\le 1}\|f\|_{{\mathbb{R}}\backslash\Omega_n;w}=0.$$
From this we deduce immediately the following lemma:
\[comp\_embed\] Let $w,w_0,w_1$ be weight functions. If $\lim_{|x|\to\infty}w(x)/w_0(x)=0$, then the compact embedding holds: $$W^{1,2}(w_0,w_1){\hookrightarrow}{\hookrightarrow}L^2(w).$$
This compact embedding allows to prove that $R_{\mathcal L}(\zeta)$ is compact:
\[trm:compactness\] For any $\zeta \in\rho({\mathcal L})$ the resolvent operator $R_{\mathcal L}(\zeta):{\mathcal E}\to{\mathcal E}$ is compact. In particular $\sigma({\mathcal L})=\sigma_p({\mathcal L})$, i.e. the spectrum of ${\mathcal L}$ consists entirely of eigenvalues.
To begin with, we fix some $\zeta\in{\mathbb{C}}$ with ${\operatorname{Re}}\zeta\ge 1+\beta^2/2$. According to Lemma \[reso\_estim\] we have the estimate , which we can reformulate: There exists a constant $C>0$ such that $$\|R_{\mathcal L}(\zeta)g\|_{\varpi,\omega}\le C\|g\|_\omega,\quad \forall g\in{\mathcal E}.$$ Hence $R_{\mathcal L}(\zeta)\in{\mathscr B}({\mathcal E}, W^{1,2}(\varpi,\omega))$. Now we have the asymptotic behaviour $\omega(x)/\varpi(x)\sim 1/|x|\to 0$ as $x\to\pm\infty.$ Therefore we may apply Lemma \[comp\_embed\] for $w=w_1=\omega$ and $w_0=\varpi$, which yields the compact embedding $W^{1,2}(\varpi,\omega){\hookrightarrow}{\hookrightarrow}{\mathcal E}$. Thus, the resolvent $R_{\mathcal L}(\zeta)\in{\mathscr B}({\mathcal E})$ is compact for ${\operatorname{Re}}\zeta\ge 1+\beta^2/2$. But this already implies the compactness of $R_{\mathcal L}(\zeta)$ for all $\zeta\in\rho({\mathcal L})$, cf. [@kato Theorem III.6.29]. The same reference confirms that $\sigma({\mathcal L})=\sigma_p({\mathcal L})$.
With these preparations we can now characterize the spectrum of ${\mathcal L}$:
\[prop\_spec\] We have $\sigma({\mathcal L})=-{\mathbb{N}}_0$. Each eigenspace is one-dimensional, and for $k\in{\mathbb{N}}_0$ we have $\ker(k+{\mathcal L})={\operatorname{span}}\{\mu_k\}$.
We consider the Fourier transform of the eigenvalue equation $(\zeta-\mathfrak L)f=0$ for $f\in {\mathcal E}$. The general solution of the Fourier-transformed equation on the real line reads: $$\label{branchis}
\hat f(\xi)=C_\pm\mu(\xi)\xi^{-\zeta},\quad \xi\in{\mathbb{R}}^\pm.$$ For details see the computation in the beginning of the Appendix \[app:b\] for $g={\vartheta}=0$. Since $f\in{\mathcal E}$, $\hat f$ has to be analytic in $\Omega_{\beta/2}$, see Lemma \[analyticity\]. With the specification of the complex logarithm in Section \[sec2\] we may extend both parts of $\hat f$ from analytically to the complex half-planes $\{{\operatorname{Re}}\xi>0\}$ and $\{{\operatorname{Re}}\xi<0\}$ respectively. However, if $\zeta\in{\mathbb{C}}\setminus{\mathbb{Z}}$, the two extensions do not meet continuously at the imaginary axis, thus $\hat f$ is not analytic in $\Omega_{\beta/2}$ (except for the trivial case $C_\pm=0$). If $\zeta\in{\mathbb{Z}}$, we obtain continuity of $\hat f$ at the imaginary axis (without $\xi=0$) iff $C_-=C_+$. But for $\zeta\in{\mathbb{N}}$, $\hat f$ still has a pole at $\xi=0$, thus it is not analytic. In the remaining case $\zeta\in -{\mathbb{N}}_0$ the function $\hat f$ from has an analytic extension to ${\mathbb{C}}$, when we choose $C_-=C_+$. So $f\in{\mathcal E}$ solves the eigenvalue equation for $\zeta$ iff $\zeta\in-{\mathbb{N}}_0$. And according to the eigenspaces are still spanned by the $\mu_k,\,k\in{\mathbb{N}}_0$, since $\hat\mu_k(\xi)=({{\mathrm{i}}}\xi)^k\mu(\xi)$.
The main difference to $L$ in $E$ is that the eigenfunctions do not form an orthogonal basis any more. However, we are still able to transfer the concept of the $L$-invariant subspaces $E_k\subset E$ to ${\mathcal E}$.
\[spec\_pr\] For every $k\in{\mathbb{N}}$ we have the following facts:
1. \[ho:i\] The subspace ${\mathcal E}_k:={\operatorname{cl}}_{\mathcal E}E_k$ is ${\mathcal L}$-invariant, and $\sigma({\mathcal L}|_{{\mathcal E}_k})=\{-k,-k-1,\ldots\}$
2. \[ho:ii\] The spectral projection $\Pi_{{\mathcal L},k}$ of ${\mathcal L}$ associated to the eigenvalue $-k$ satisfies $$\ker\Pi_{{\mathcal L},k}={\mathcal E}_{k+1}\oplus{\operatorname{span}}\{\mu_{k-1},\ldots,\mu_0\},\quad{\operatorname{ran}}\Pi_{{\mathcal L},k}= {\operatorname{span}}\{\mu_k\}.$$ Moreover, $\ker\Pi_{{\mathcal L},0}={\mathcal E}_1$ and ${\operatorname{ran}}\Pi_{{\mathcal L},0}={\operatorname{span}}\{\mu_0\}$.
3. \[ho:iii\] There holds ${\mathcal E}={\mathcal E}_k\oplus {\operatorname{span}}\{\mu_{k-1},\ldots,\mu_0\}$.
Since $\sigma(L)=\sigma({\mathcal L})$, and $R_L(\zeta)\subset R_{\mathcal L}(\zeta)$ for all $\zeta\in{\mathbb{C}}\setminus(-{\mathbb{N}}_0)$, we conclude from that for any $\sigma'\subset\sigma({\mathcal L})$ there holds $\Pi_{L,\sigma'}\subset \Pi_{{\mathcal L},\sigma'}$, and they are bounded projections in $E$ and ${\mathcal E}$, respectively. For $\sigma':=\{0,\ldots,-k+1\},\, k\in{\mathbb{N}},$ we apply Lemma \[lem:proj\] from the appendix: ${\operatorname{ran}}\Pi_{{\mathcal L},\sigma'}={\operatorname{cl}}_{\mathcal E}{\operatorname{ran}}\Pi_{L,\sigma'}={\operatorname{cl}}_{\mathcal E}{\operatorname{span}}\{\mu_0,\ldots,\mu_{k-1}\}={\operatorname{span}}\{\mu_0,\ldots,\mu_{k-1}\}$ and $\ker\Pi_{{\mathcal L},\sigma'}={\operatorname{cl}}_{\mathcal E}\ker\Pi_{L,\sigma'}={\operatorname{cl}}_{\mathcal E}E_k=:{\mathcal E}_k$. This shows . Since the projection $\Pi_{{\mathcal L},\sigma'}$ is bounded, the range and kernel indeed represent a decomposition of ${\mathcal E}$, thus we also obtain Result .
For we use the same arguments as before, with $\sigma'=\{-k\}$ instead.
Next we characterize the subspaces ${\mathcal E}_k$.
\[char:e\_k\] For $k\in-{\mathbb{N}}$ the subspace ${\mathcal E}_k$ is explicitly given by $$\label{e_k} {\mathcal E}_k=\left\{f\in{\mathcal E}:\int_{\mathbb{R}}f(x)x^j{\,\mathrm{d}}x=0,\, 0\le j\le k-1\right\}.$$ Furthermore, there holds $$\label{f_char_e_k}{\mathcal E}_k=\left\{f\in{\mathcal E}:\hat f^{(j)}(0)=0,\,0\le j\le k-1\right\},$$ where $\hat f^{(j)}$ denotes the $j$-th derivative of the Fourier transform of $f$.
The functionals $\psi_j:f\mapsto\int_{\mathbb{R}}f(x)x^j{\,\mathrm{d}}x,\,j\in{\mathbb{N}}$, are continuous in ${\mathcal E}$. We define $\tilde \psi_j:=\psi_j|_{E}$. Let $f\in E_k=\{\mu_0,\ldots,\mu_{k-1}\}^{\perp_E}$. The orthogonality condition then reads $$0={\langle}f,\mu_j{\rangle}_E=\int_{\mathbb{R}}f(x)\mu_j(x)\mu(x)^{-1}{\,\mathrm{d}}x=\frac 1{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)H_j(x){\,\mathrm{d}}x,\quad\forall 0\le j\le k-1,$$ which is equivalent to $\tilde\psi_0(f)=\ldots=\tilde\psi_{k-1}(f)=0$. Applying Lemma \[lem:funct\] from the appendix with ${\mathcal X}={\mathcal E}$ and $X=E$ yields ${\operatorname{cl}}_{\mathcal E}E_k=\{f\in{\mathcal E}:\psi_j(f)=0,\,0\le j\le k-1\}$, which is equal to ${\mathcal E}_k$ by definition. This proves (\[e\_k\]).
The second equality (\[f\_char\_e\_k\]) immediately follows from $$\int_{\mathbb{R}}f(x)x^j{\,\mathrm{d}}x={\mathcal F}_{x\to\xi}[f(x)x^j](0)={{\mathrm{i}}}^j\hat f^{(j)}(0),\quad\forall j\in{\mathbb{N}}_0.$$
The representation (\[e\_k\]) of the ${\mathcal E}_k$ also holds in polynomially weighted spaces, which is shown in [@Gallay2002 Appendix A].
The final result of this section deals with the analysis of the semigroup ${\mathrm{e}}^{t{\mathcal L}}$ generated by ${\mathcal L}$ in ${\mathcal E}$. We already know that $L$ generates a $C_0$-semigroup $({\mathrm{e}}^{tL})_{t\ge 0}$ of bounded operators in $E$, and from [@Gallay2002 Appendix A] we get its representation (for $f\in E$): $$\label{expl_semi}
{\mathcal F}_{x\to\xi}\big[{\mathrm{e}}^{tL}f\big]=\exp\Big(-\frac{\xi^2}2(1-{\mathrm{e}}^{-2t})\Big)\hat f\big(\xi{\mathrm{e}}^{-t}\big),\quad t\ge0.$$ This formula can be extended to $f\in {\mathcal E}$, yielding a family $(S(t))_{t\ge 0}$ of operators in ${\mathcal E}$.
\[cx\_0\] The family of operators $(S(t))_{t\ge0}$ given by is a family of bounded operators in ${\mathcal E}$.
In order to show that the operators $S(t)$ are bounded, we use the norm ${|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}\cdot{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega$. So we estimate $\|{\mathcal F}[ S(t)f](\xi+{{\mathrm{i}}}\beta/2)\|$, the estimate for the other term in ${|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}\cdot{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega$ is analogous: $$\begin{aligned}
\big\|{\mathcal F}[ S(t)f](\cdot+{{\mathrm{i}}}\beta/2)\big\|^2_{L^2({\mathbb{R}})} & = \int_{\mathbb{R}}\exp\Big(\Big[-\xi^2+\frac{\beta^2}4\Big](1-{\mathrm{e}}^{-2t})\Big)\Big|\hat f\Big(\Big[\xi+{{\mathrm{i}}}\frac\beta 2\Big]{\mathrm{e}}^{-t}\Big)\Big|^2{\,\mathrm{d}}\xi\label{zw_nn}\\
&\le \exp\Big(\frac{\beta^2}4\Big)\int_{\mathbb{R}}\Big|\hat f\Big(\Big[\xi+{{\mathrm{i}}}\frac\beta 2\Big]{\mathrm{e}}^{-t}\Big)\Big|^2{\,\mathrm{d}}\xi\nonumber\\
&=\exp\Big(\frac{\beta^2}4+t\Big)\int_{\mathbb{R}}\Big|\hat f\Big(\xi+{{\mathrm{i}}}{\mathrm{e}}^{-t}\frac\beta 2\Big)\Big|^2{\,\mathrm{d}}\xi\nonumber\\
&\le \exp\Big(\frac{\beta^2}4+t\Big){|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_{\cosh({\mathrm{e}}^{-t}\beta x)}^2 \le \exp\Big(\frac{\beta^2}4+t\Big){|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}^2_\omega\nonumber\end{aligned}$$ So $( S(t))_{t\ge0}$ is a family of bounded operators in ${\mathcal E}$, and there exists a constant $M>0$ with $$\| S(t)\|_{{\mathscr B}({\mathcal E})}\le M{\mathrm{e}}^{t/2},\quad t\ge0.$$
The operator ${\mathcal L}$ is the infinitesimal generator of the $C_0$-semigroup $(S(t))_{t\ge0}$ in ${\mathcal E}$.
According to [@pazy Theorem 1.4.5], Corollary \[cor\_dissip\] implies that ${\mathcal L}-1-\beta^2/2={\operatorname{cl}}_{\mathcal E}(L|_{C_0^\infty}-1-\beta^2/2)$ is dissipative in ${\mathcal E}$. From Proposition \[prop\_spec\] we also know that any $\zeta\in{\mathbb{C}}$ with ${\operatorname{Re}}\zeta>0$ lies in $\rho({\mathcal L})$. So we can apply the Lumer-Phillips Theorem [@pazy Theorem 1.4.3] and find that ${\mathcal L}$ generates a $C_0$-semigroup $({\mathrm{e}}^{t{\mathcal L}})_{t\ge 0}$ of bounded operators. Since ${\mathrm{e}}^{t{\mathcal L}}$ and $S(t)$ are both bounded in ${\mathcal E}$ and coincide on the dense subspace $D(L)\subset{\mathcal E}$, we get ${\mathrm{e}}^{t{\mathcal L}}=S(t)$ in ${\mathcal E}$ for all $t\ge 0$.
As a consequence we write ${\mathrm{e}}^{t{\mathcal L}}:=S(t)$ for the semigroup generated by ${\mathcal L}$, and the representation holds for all $f\in{\mathcal E}$.
\[unpert\_decay\] For every $k\in{\mathbb{N}}_0$ we have:
1. \[1i\] The space ${\mathcal E}_k$ is invariant under the family $({\mathrm{e}}^{t{\mathcal L}})_{t\ge0}$.
2. \[2ii\] There exists some $C_k>0$ such that $$\|{\mathrm{e}}^{t{\mathcal L}}|_{{\mathcal E}_k}\|_{{\mathscr B}({\mathcal E}_k)}\le C_k{\mathrm{e}}^{-kt},\quad t\ge 0.$$
The closed subspaces ${\mathcal E}_k$ are ${\mathcal L}$-invariant, so they are also invariant under $({\mathrm{e}}^{t{\mathcal L}})_{t\ge0 }$.
In order to show , we use the first line of and make the additional assumption $t\ge1$: $$\begin{aligned}
\big\|{\mathcal F}[{\mathrm{e}}^{t{\mathcal L}}f](\xi+{{\mathrm{i}}}\beta/2)\big\|^2_{L^2({\mathbb{R}}_\xi)} & \le {\mathrm{e}}^{\frac{\beta^2}4} \int_{\mathbb{R}}{\mathrm{e}}^{-\frac{\xi^2}2}\Big|\Big[\xi+{{\mathrm{i}}}\frac\beta 2\Big]{\mathrm{e}}^{-t}\Big|^{2k}\left|\frac{\hat f\Big(\Big[\xi+{{\mathrm{i}}}\frac\beta 2\Big]{\mathrm{e}}^{-t}\Big)}{\Big(\Big[\xi+{{\mathrm{i}}}\frac\beta 2\Big]{\mathrm{e}}^{-t}\Big)^k}\right|^2{\,\mathrm{d}}\xi\label{hello}\end{aligned}$$ Here we used the inequality $\frac 12<1-{\mathrm{e}}^{-2t}<1$ for $t\ge1$. In the following we use the Poincaré inequality : $$\begin{aligned}
\left\|\frac{\hat f\Big(\Big[\xi+{{\mathrm{i}}}\frac\beta 2\Big]{\mathrm{e}}^{-t}\Big)}{\Big(\Big[\xi+{{\mathrm{i}}}\frac\beta 2\Big]{\mathrm{e}}^{-t}\Big)^k}\right\|_{L^\infty({\mathbb{R}}_\xi)}&=\left\|{\mathcal F}_{x\to\xi}\left(\exp\Big(\frac\beta2{\mathrm{e}}^{-t}x\Big){\mathcal F}^{-1}_{\xi\to x}\left[\frac{\hat f(\xi)}{\xi^k}\right]\right)\right\|_{L^\infty({\mathbb{R}}_\xi)} \\
&\le\left\|\exp\Big(\frac\beta2{\mathrm{e}}^{-t}x\Big){\mathcal F}^{-1}_{\xi\to x}\left[\frac{\hat f(\xi)}{\xi^k}\right]\right\|_{L^1({\mathbb{R}}_x)}\\
&\le \tilde C(t)\left\|{\mathcal F}^{-1}_{\xi\to x}\left[\frac{\hat f(\xi)}{\xi^k}\right]\right\|_\omega\\
&\le C(t)\left\|({{\mathrm{i}}}{\partial}_x)^k{\mathcal F}^{-1}_{\xi\to x}\left[\frac{\hat f(\xi)}{\xi^k}\right]\right\|_\omega = C(t)\|f\|_\omega.\end{aligned}$$ Thereby, the constant $\tilde C(t)$ is given by $$\tilde C(t)=\int_{\mathbb{R}}\frac{\exp(\beta{\mathrm{e}}^{-t}x)}{\cosh\beta x}{\,\mathrm{d}}x,$$ which is uniformly bounded for $t\ge 1$. Inserting this result in yields for $t\ge1$ $$\begin{aligned}
\big\|{\mathcal F}[{\mathrm{e}}^{t{\mathcal L}}f](\xi+{{\mathrm{i}}}\beta/2)\big\|^2_{L^2({\mathbb{R}}_\xi)}&\le C{\mathrm{e}}^{\frac{\beta^2}4}{\mathrm{e}}^{-2kt}\|f\|^2_\omega \int_{\mathbb{R}}{\mathrm{e}}^{-\frac{\xi^2}2}\Big|\xi+{{\mathrm{i}}}\frac\beta 2\Big|^{2k}{\,\mathrm{d}}\xi\\
&= C {\mathrm{e}}^{-2kt}\|f\|^2_\omega.\end{aligned}$$ Thus there exists a constant $C>0$ such that ${|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}{\mathrm{e}}^{t{\mathcal L}}f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega\le C{\mathrm{e}}^{-kt}{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega$ for all $t\ge 1$. From Lemma \[cx\_0\] we also know that the semigroup is uniformly bounded for $t\in[0,1]$, so altogether we get the desired decay estimate for the semigroup in ${\mathcal E}_k$.
Before we turn to the perturbed Fokker-Planck equation, we summarize our results so far:
\[extension\_theorem\] Let $\omega(x):=\cosh\beta x$ for some $\beta>0$. Then the Fokker-Planck operator $L|_{C_0^\infty({\mathbb{R}})}$ is closable in ${\mathcal E}=L^2(\omega)$, and its closure ${\mathcal L}={\operatorname{cl}}_{\mathcal E}L|_{C_0^\infty({\mathbb{R}})}$ has the following properties:
1. \[fp:res:i\] The spectrum satisfies $\sigma({\mathcal L})=-{\mathbb{N}}_0$, and $\ker({\mathcal L}+k)={\operatorname{span}}\{\mu_k\}$ for any $k\in{\mathbb{N}}_0$. The eigenfunctions satisfy the relation $\mu_k=\mu_0^{(k)}$, the $k$-th derivative of $\mu_0$.
2. The resolvent $R_{\mathcal L}(\zeta)$ is compact in ${\mathcal E}$ for all $\zeta\notin -{\mathbb{N}}_0$.
3. \[fp:res:ii\] For any $k\in{\mathbb{N}}_0$ the closed subspace ${\mathcal E}_k:={\operatorname{cl}}_{\mathcal E}{\operatorname{span}}\{\mu_k,\mu_{k+1},\ldots\}$ is an ${\mathcal L}$-invariant subspace of ${\mathcal E}$, and ${\operatorname{span}}\{\mu_0,\ldots,\mu_{k-1}\}$ is a complement. In particular ${\mathcal E}_0={\mathcal E}$.
4. \[fp:res:iii\] The spectral projection $\Pi_{{\mathcal L},k}$ to the eigenvalue $-k\in-{\mathbb{N}}_0$ fulfills ${\operatorname{ran}}\Pi_{{\mathcal L},k}={\operatorname{span}}\{\mu_k\}$ and $\ker\Pi_{{\mathcal L},k}={\mathcal E}_{k+1}\oplus {\operatorname{span}}\{\mu_{k-1},\ldots,\mu_0\}$ for $k\in{\mathbb{N}}_0$.
5. \[fp:res:iv\] For any $k\in{\mathbb{N}}_0$ the operator ${\mathcal L}$ generates a $C_0$-semigroup on ${\mathcal E}_k$, and there exists a constant $C_k\ge 1$ such that we have the estimate $$\left\|{\mathrm{e}}^{t{\mathcal L}}|_{{\mathcal E}_k}\right\|_{{\mathscr B}({\mathcal E}_k)}\le C_k {\mathrm{e}}^{-kt},\quad\forall t\ge 0.$$
More generally, the results of Theorem \[extension\_theorem\] hold for all weight functions $\omega(x)=\exp(\beta|x|^\gamma)$ with either $\gamma\in(0,2)$ and $\beta>0$ or $\gamma=2$ and $\beta\in(0,\frac 12]$. This can be shown by using the results from [@gmm], where an operator decomposition method is used to transfer spectral properties of operators from a Banach space to a larger Banach space. For a detailed discussion of the application of [@gmm], see [@meidiss].
The sequence of eigenfunctions $(\mu_k)_{k\in{\mathbb{N}}_0}$ is an orthogonal basis of $E$. In the larger space ${\mathcal E}$, the linear hull ${\operatorname{span}}\{\mu_k:k\in{\mathbb{N}}_0\}$ is still dense, due to the continuous embedding $E{\hookrightarrow}{\mathcal E}$.
Also, each $f\in{\mathcal E}$ can (formally) uniquely be decomposed according to the sequence of spectral projections $(\Pi_{{\mathcal L},k})_{k\in{\mathbb{N}}_0}$, see the proof of Proposition \[pert:spec\]. But the obtained series may diverge in ${\mathcal E}$. As an example we consider $f(x):=\exp(-|x|)\in L^2(\cosh x)$. Since $f$ is symmetric, we have $\Pi_{{\mathcal L},k}f=0$ if $k$ is odd. For $k=2n,\,n\in{\mathbb{N}}_0$, one can show the asymptotic behaviour for $n\to\infty$: $$\|\Pi_{{\mathcal L},2n}f\|_\omega=\mathcal{O}\Big(\frac{\sqrt{(2n)!}}{n^{1/4}}\Big),$$ where we use the explicit representation for the Hermite polynomials $H_{2n}$ from (5.5.4) in [@szego], and the asymptotic expansions for $H_{2n}$ given in [@szego Theorem 8.22.9]. Therefore, the formal series $\sum_{n\in{\mathbb{N}}_0}\Pi_{{\mathcal L},2n} f$ is divergent in ${\mathcal E}$. So the sequence $(\mu_k)_{k\in{\mathbb{N}}_0}$ is neither a Schauder basis nor a representation system of ${\mathcal E}$. However, the sequence $(\mu_k/\|\mu_k\|_E)_{k\in{\mathbb{N}}_0}$ is still a Bessel system, see [@christensen; @bilgus] for the definitions.
Analysis of the Perturbed Operator {#sec3}
==================================
So far we have discussed the one-dimensional Fokker-Planck operator ${\mathcal L}$ in ${\mathcal E}=L^2(\omega)$, with $\omega(x)=\cosh\beta x$. In this section we investigate the properties of the perturbed (one-dimensional) operator ${\mathcal L}+\Theta$ in ${\mathcal E}$, and we shall summarize the results in Theorem \[final\_pert\_trm\]. We begin by specifying the assumptions we make on the perturbation $\Theta$.
\[pagge\] [**(C) Conditions on $\boldsymbol\Theta$:**]{} We assume that $\Theta f={\vartheta}*f$, for $f\in{\mathcal E}$, where ${\vartheta}$ is a tempered distribution that fulfills the following properties in $\Omega_{\beta/2}$ for some $\beta>0$:
1. The Fourier transform $\hat {\vartheta}$ can be extended to an analytic function in $\Omega_{\beta/2}$ (also denoted by $\hat {\vartheta}$), and $\hat{\vartheta}\in L^\infty(\Omega_{\beta/2})$.
2. It holds $\hat{\vartheta}(0)=0$, i.e. ${\vartheta}$ has zero mean.
3. The mapping $\xi\mapsto{\operatorname{Re}}\int_0^1\hat{\vartheta}(\xi s)/s{\,\mathrm{d}}s$ is essentially bounded in $\Omega_{\beta/2}$.
\[rem31\] If the conditions [**(C)(i)-(ii)**]{} hold for ${\vartheta}$, then the mapping $\xi\mapsto\int_0^1\hat{\vartheta}(\xi s)/s{\,\mathrm{d}}s$ is analytic in $\Omega_{\beta/2}$. This becomes clear when writing $\hat{\vartheta}(\xi s)/s= \xi\hat {\vartheta}(\xi s)/(\xi s)$, which is analytic for all $s\in(0,1]$ and can be continuously extended to $\hat{\vartheta}'(0)\xi$ for $s=0$. The analyticity of $\xi\mapsto\int_0^1\hat{\vartheta}(\xi s)/s{\,\mathrm{d}}s$ on $\Omega_{\beta/2}$ then follows from [@dettman Theorem 4.9.1].
\[thetaeine\] There holds $\Theta f\in{\mathcal E}$ for all $f\in{\mathcal E}$ iff the condition [**(C)(i)**]{} holds.
Clearly, $\widehat{\Theta f}=\hat{\vartheta}\hat f$ is analytic in $\Omega_{\beta/2}$ for $f\in{\mathcal E}$. According to Lemma \[analyticity\] there holds $\Theta f\in{\mathcal E}$ iff $$\label{theta_in_e}
\sup_{|b|<\beta/2}\|(\hat{\vartheta}\hat f) (\cdot+{{\mathrm{i}}}b)\|_{L^2({\mathbb{R}})}<\infty,$$ where we use $\widehat {\Theta f}=\hat{\vartheta}\hat f$. Now we apply Hölder’s inequality and find that (\[theta\_in\_e\]) holds for all $f\in{\mathcal E}$ iff ${\vartheta}$ satisfies [**(C)(i)**]{}.
As a consequence of the above lemma and , the product $\hat{\vartheta}\hat f$ itself is the Fourier transform of an element of ${\mathcal E}$. So we may define $(\hat{\vartheta}\hat f)(\cdot\pm{{\mathrm{i}}}\beta/2)\in L^2({\mathbb{R}})$ for $f\in{\mathcal E}$ according to (\[hatf\]) whenever ${\vartheta}$ satisfies [**(C)(i)**]{}. With this we obtain according to Lemma \[analyticity\] (\[analyt:iii\]): $$\label{fthet}
b\mapsto (\hat{\vartheta}\hat f)(\cdot+{{\mathrm{i}}}b)\in C([-\beta/2,\beta/2];L^2({\mathbb{R}})).$$
\[lemma:bdd\_theta\] The convolution $\Theta$ is bounded in ${\mathcal E}$ if the condition [**(C)(i)**]{} holds.
We apply the norm (\[norm\_w\]) to $\Theta f$. The Fourier transform turns the convolution into a multiplication, so we get according to and [**(C)(i)**]{}
$$\begin{aligned}
{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}\Theta f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega^2&=\int_{\mathbb{R}}|\hat{\vartheta}\hat f(\xi-{{\mathrm{i}}}\beta/2)|^2{\,\mathrm{d}}\xi+\int_{\mathbb{R}}|\hat{\vartheta}\hat f(\xi+{{\mathrm{i}}}\beta/2)|^2{\,\mathrm{d}}\xi\\
&=\lim_{b\nearrow\beta/2}\Big[\int_{\mathbb{R}}|\hat{\vartheta}\hat f(\xi-{{\mathrm{i}}}b)|^2{\,\mathrm{d}}\xi+\int_{\mathbb{R}}|\hat{\vartheta}\hat f(\xi+{{\mathrm{i}}}b)|^2{\,\mathrm{d}}\xi\Big]\\
&\le \|\hat{\vartheta}\|^2_{L^\infty(\Omega_{\beta/2})}\lim_{b\nearrow\beta/2}\Big[\int_{\mathbb{R}}|\hat f(\xi-{{\mathrm{i}}}b)|^2{\,\mathrm{d}}\xi+\int_{\mathbb{R}}|\hat f(\xi+{{\mathrm{i}}}b)|^2{\,\mathrm{d}}\xi\Big]\\
&=\|\hat{\vartheta}\|^2_{L^\infty(\Omega_{\beta/2})}{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega^2.\end{aligned}$$
\[remark\_theta\_inv\] Under the assumption [**(C)**]{} there holds $\Theta:{\mathcal E}_k\to{\mathcal E}_{k+1}\subset{\mathcal E}_k$ for every $k\in{\mathbb{N}}$.
According to Proposition \[char:e\_k\], $f\in{\mathcal E}_k$ iff $\xi=0$ is a zero of $\hat f(\xi)$ of order greater or equal to $k$. Because of the assumption $\hat {\vartheta}(0)=0$ the Fourier transform $\widehat{\Theta f}=\hat {\vartheta}\hat f$ has a zero at least of order $k+1$ for $f\in{\mathcal E}_k$, so $\Theta f \in{\mathcal E}_{k+1}$.
\[cor:inv\_subsp\] Let [**(C)**]{} hold, and $k\in{\mathbb{N}}_0$. Then the space ${\mathcal E}_k$ is an $({\mathcal L}+\Theta)$-invariant subspace of ${\mathcal E}$.
Since the conditions [**(C)**]{} are not very handy for direct applications, the following lemma gives some criteria that are simpler to verify and sufficient for [**(C)**]{}.
\[hinr\_theta\] Let $\beta>0$ and $\omega(x)=\cosh\beta x$, and assume that ${\vartheta}\in{\mathscr S}'$ fulfills
1. \[zeromas\] $\hat {\vartheta}(0)=0$,
2. ${\vartheta}={\vartheta}_W+{\vartheta}_D$ with ${\vartheta}_W\in W^{1,1}(\omega^{\frac12},\omega^{\frac12})$ and ${\vartheta}_D\in D:=\{\sum_{j=1}^n a_j\delta_{x_j}:a_j\in{\mathbb{C}},\,x_j\in{\mathbb{R}},\,n\in{\mathbb{N}}\}$, where $\delta_{x_j}$ denotes the delta distribution located at $x_j$.
Then $\Theta f={\vartheta}*f$ satisfies [**(C)**]{} for this $\beta>0$.
In general $\hat{\vartheta}_W(0)$ and $\hat{\vartheta}_D(0)$ are not zero, so it is convenient to define ${\vartheta}_W^*:={\vartheta}_W+M\mu$ and ${\vartheta}_D^*:={\vartheta}_D-M\mu$, where $M:=\hat{\vartheta}_D(0)/\sqrt{2\pi}$. Then $\hat{\vartheta}_W^*$ and ${\vartheta}_D^*$ have zero mass, and we still have $\hat{\vartheta}_W^*\in W^{1,1}(\omega^{\frac12},\omega^{\frac12}).$ Since ${\mathcal F}_{x\to\xi} \delta_{x_j}={\mathrm{e}}^{-{{\mathrm{i}}}\xi x_j}$ and $\hat\mu(\xi)=\sqrt{2\pi}\mu(\xi)$, it is immediate that ${\vartheta}^*_D$ satisfies [**(C)(i)**]{}. In order to see [**(C)(iii)**]{} for ${\vartheta}^*_D$, we note that the integral occurring in this condition can be rewritten as the line integral from $0$ to $\xi$: $$\int_{0\to\xi}\frac{\hat{\vartheta}_D^*(z)}{z}{\,\mathrm{d}}z$$ which is path-independent in ${\mathbb{C}}$ (and thus in $\Omega_{\beta/2}$), since $\hat{\vartheta}_D^*$ is an entire function and has a zero at $0$. Therefore the integral itself is analytic, and thus uniformly bounded on every compact subset of ${\mathbb{C}}$. Because of this, it is sufficient to show uniform boundedness of this integral as $|\xi|\to\infty$ in $\Omega_{\beta/2}$. We outline this for the map $\xi\mapsto{\mathrm{e}}^{-{{\mathrm{i}}}x_j\xi}$ for any fixed $x_j\in{\mathbb{R}}$ and ${\operatorname{Re}}\xi>1$, the case ${\operatorname{Re}}\xi<-1$ is analogous. Thereby we choose the following integration path (note that we may start from $z=1$, since the integral from $0$ to $1$ is a constant) $$\begin{aligned}
\left|\int_{1\to\xi}\frac{{\mathrm{e}}^{-{{\mathrm{i}}}x_jz}}{z}{\,\mathrm{d}}z\right|& \le\left| \int_1^{{\operatorname{Re}}(\xi)}\frac{{\mathrm{e}}^{-{{\mathrm{i}}}x_jz}}{z}{\,\mathrm{d}}z\right|+\left|\int_{{\operatorname{Re}}(\xi)\to{\operatorname{Re}}(\xi)+{{\mathrm{i}}}{\operatorname{Im}}(\xi)}\frac{{\mathrm{e}}^{-{{\mathrm{i}}}x_jz}}{z}{\,\mathrm{d}}z\right|\\
&\le\left| \int_{x_j}^{{\operatorname{Re}}(\xi)x_j}\frac{{\mathrm{e}}^{-{{\mathrm{i}}}z}}{z}{\,\mathrm{d}}z\right|+\frac\beta 2{\mathrm{e}}^{|x_j|\beta/2}.\end{aligned}$$ The first integral is known to remain uniformly bounded as ${\operatorname{Re}}(\xi)\to+\infty$. For estimating the second integral we used $\xi\in\Omega_{\beta/2}$ and ${\operatorname{Re}}\xi\ge1$. Since $\hat\mu=\sqrt{2\pi}\mu$ decays sufficiently fast in $\Omega_{\beta/2}$, it is clear that the integral of $\hat\mu(z)/z$ from $1$ to $\xi$ also remains uniformly bounded as $\xi\to+\infty$. Altogether, we conclude that $\hat{\vartheta}_D^*$ satisfies [**(C)(iii)**]{}.
Now we verify the same properties for ${\vartheta}_W^*$. Since ${\vartheta}_W^*\in L^1(\omega^{\frac 12})$, we may extend $\hat {\vartheta}_W^*$ to an analytic function in $\Omega_{\beta/2}$, and there holds (\[hatf\]), cf. [@dautli5 Proposition XVI.1.3]. The Fourier transform is a continuous map from $L^1({\mathbb{R}})$ to $B_0({\mathbb{R}})$, i.e. the continuous functions decaying at infinity, equipped with the uniform norm. Therefore, ${\vartheta}_W^*\in L^1(\omega^{\frac 12})$ implies $$\|\hat {\vartheta}_W^*\|_{L^\infty(\Omega_{\beta/2})}=\sup_{|b|<\frac\beta2}\sup_{\xi\in{\mathbb{R}}}|\hat{\vartheta}_W^*(\xi+{{\mathrm{i}}}b)|
\le \!\!\sup_{|b|<\frac\beta2}\|{\vartheta}_W^*(x){\mathrm{e}}^{bx}\|_{L^1({\mathbb{R}})}
\le \|{\vartheta}_W^*(x){\mathrm{e}}^{\frac\beta 2|x|}\|_{L^1({\mathbb{R}})}<\infty,$$ So [**(C)(i)**]{} is satisfied. For [**(C)(iii)**]{} it is sufficient to show that for some $c>0$ and all $\xi\in\Omega_{\beta/2}$ with $|\xi|\ge 1$ there holds $|\hat{\vartheta}_W^*(\xi)|\le c/|\xi|$, which is fulfilled if ${\mathcal F}({{\vartheta}_W^*}')\in\ L^\infty(\Omega_{\beta/2})$. Analogously to the previous part of the proof we obtain that this is satisfied if ${{\vartheta}_W^*}'\in L^1(\omega^{\frac 12})$. We conclude that ${\vartheta}_W^*$ fulfills [**(C)(i)**]{} and [**(C)(iii)**]{} if ${\vartheta}_W^*\in W^{1,1}(\omega^{\frac12},\omega^{\frac12})$.
Finally, ${\vartheta}$ satisfies the condition [**(C)(ii)**]{} due to the assumption (\[zeromas\]).
For the rest of the article, we shall always assume that $\Theta$ satisfies the condition [**(C)**]{} for some fixed $\beta>0$ , and we choose the weight function $\omega(x)=\cosh \beta x$ with this particular $\beta$. The first result about the perturbed Fokker-Planck operator is the following lemma:
\[sigma\_perturb\] The operator ${\mathcal L}+\Theta$ has compact resolvent in ${\mathcal E}$.
A bounded perturbation of an infinitesimal generator with compact resolvent has compact resolvent again, see [@engel Proposition III.1.12]. Then the result follows by combining the results of Theorems \[trm:compactness\] and \[extension\_theorem\] for ${\mathcal L}$, and Corollary \[lemma:bdd\_theta\] for $\Theta$.
As a consequence, the spectrum of ${\mathcal L}+\Theta$ in ${\mathcal E}$ is non-empty and consists only of eigenvalues. In order to characterize the entire spectrum, we introduce the following ladder operators[^1], namely the [*annihilation operator*]{} $$\alpha^-:{\mathcal E}_1\to{\mathcal E}:f\mapsto \int_{-\infty}^x f(y){\,\mathrm{d}}y,$$ and its formal inverse $\alpha^+:f\mapsto f'$, the [*creation operator*]{}.
\[prop\_annil\] The annihilation operator $\alpha^-$ has the following properties:
1. \[anihi:i\] For any $k\in{\mathbb{N}}$ there holds $\alpha^-\in{\mathscr B}({\mathcal E}_k,{\mathcal E}_{k-1})$.
2. \[anihi:ii\] In ${\mathcal E}_1$ the operators $\Theta$ and $\alpha^-$ commute.
3. \[anihi:iii\] Let $f\in{\mathcal E}_1,\,\zeta\in{\mathbb{C}}$ such that $({\mathcal L}+\Theta)f=\zeta f$. Then $$({\mathcal L}+\Theta)(\alpha^-f)=(\zeta+1) (\alpha^-f).$$
First we show (\[anihi:i\]). The property $\alpha^-:{\mathcal E}_k\to{\mathcal E}_{k-1}$ can be verified by using the explicit representation (\[e\_k\]) of the ${\mathcal E}_k$, and integration by parts (first for $f\in C_0^\infty({\mathbb{R}})$). The boundedness of $\alpha^-$ follows immediately from the Poincaré inequality (\[poincare\]). Property (\[anihi:ii\]) holds true since $\Theta$ is a convolution. For Result (\[anihi:iii\]) one applies $\alpha^-$ to the equation $({\mathcal L}+\Theta)f=\zeta f$, and uses the identity $\alpha^-({\mathcal L}f)={\mathcal L}(\alpha^-f)-\alpha^-f$ and the Property (\[anihi:ii\]).
By using the annihilation operator, we are able to prove:
\[pert:spec\] We have the following spectral properties of ${\mathcal L}+\Theta$ in ${\mathcal E}$:
1. \[ps:i\] $\sigma({\mathcal L}+\Theta)=-{\mathbb{N}}_0$.
2. \[ps:ii\] For each $k\in{\mathbb{N}}_0$, the eigenspace $\ker({\mathcal L}+\Theta+k)$ is one-dimensional.
3. \[ps:iii\] The eigenfunction $f_k$ to the eigenvalue $-k\in\-{\mathbb{N}}_0$ is explicitly given by (up to a normalization constant) $$\label{rec_f_k}
f_k=(\alpha^+)^kf_0=f_0^{(k)},\quad\text{and}\quad \hat f_0(\xi)=\exp\Big(-\frac{\xi^2}2+\int_0^1\frac{\hat{\vartheta}(\xi s)}s{\,\mathrm{d}}s\Big),\quad\xi\in\Omega_{\beta/2}.$$
In particular, $f_0$ is the unique stationary solution with unit mass of the perturbed Fokker-Planck equation (\[pert\_fp\]) in one dimension.
In order to show (\[ps:i\]) we first prove that $\bigcap_{k\in{\mathbb{N}}}{\mathcal E}_k=\{0\}$. According to (\[f\_char\_e\_k\]) there holds $$\bigcap_{k\in{\mathbb{N}}}{\mathcal E}_k=\left\{f\in{\mathcal E}:\hat f^{(k)}(0)=0,\, k\in{\mathbb{N}}_0\right\}.$$ But for $f\in{\mathcal E}$, $\hat f$ is analytic, and the only analytic function with a zero of infinite order is the zero function, which proves the statement.
Thus, for any eigenfunction $f$, there exists a unique $k\in{\mathbb{N}}_0$ such that $f\in{\mathcal E}_k\backslash{\mathcal E}_{k+1}$, which is the minimal $k\in{\mathbb{N}}_0$ with the property $\Pi_{{\mathcal L},k} f\neq 0$. Applying this projection to the eigenvalue equation yields $$\Pi_{{\mathcal L},k}({\mathcal L}+\Theta)f=-k\Pi_{{\mathcal L},k}f=\zeta\Pi_{{\mathcal L},k}f,$$ where we used $\Theta f\in{\mathcal E}_{k+1}$ (cf. Lemma \[remark\_theta\_inv\]). Hence, the eigenvalue corresponding to $f$ satisfies $\zeta=-k$. Thus $\sigma({\mathcal L}+\Theta)\subseteq-{\mathbb{N}}_0$. If now $f_k$ is an eigenfunction with eigenvalue $-k$, we can apply $k$ times the continuous operator $\alpha^-$ to $f_k$, and create eigenfunctions to all eigenvalues $\{-k+1,\ldots,0\}$. So either $\sigma({\mathcal L}+\Theta)=-{\mathbb{N}}_0$ or $\sigma({\mathcal L}+\Theta)=\{-k_0,\ldots,0\}$, i.e. there exists some minimal eigenvalue $-k_0$. But the latter scenario is actually not possible, because then the operator $({\mathcal L}+\Theta)|_{{\mathcal E}_{k_0+1}}$ would have empty spectrum in ${\mathcal E}_{k_0+1}$, which contradicts the fact that it still has a compact resolvent in ${\mathcal E}_{k_0+1}$.
In order to verify (\[ps:ii\]) we recall from the first part of the proof that if $f$ is an eigenfunction of ${\mathcal L}+\Theta$ to the eigenvalue $-k$, then $k={\operatorname{argmin}}\{\Pi_{{\mathcal L},j}f\neq 0:j\in{\mathbb{N}}_0\}$. In particular, $$\label{3.6}
\Pi_{{\mathcal L},k}f\neq 0$$ for such an eigenfunction. Assume that $\dim\ker({\mathcal L}+\Theta+k)>1$ for some $k\in{\mathbb{N}}_0.$ Thus we may choose two linearly independent eigenfunctions to the eigenvalue $-k$. Since $\dim{\operatorname{ran}}\Pi_{{\mathcal L},k}=1$, we can find a linear combination of these two eigenfunctions, yielding an eigenfunction $f$ which satisfies $\Pi_{{\mathcal L},k}f=0$. But this contradicts and hence $\dim\ker({\mathcal L}+\Theta+k)=1$.
For the third result (\[ps:iii\]) we consider the Fourier transform of the eigenvalue equation $({\mathcal L}+\Theta)f_k=-k f_k$ for $k\in{\mathbb{N}}_0$. This yields the following differential equation for $\hat f_k$: $$\xi\hat f_k'(\xi)=\big(\hat {\vartheta}(\xi)+k-\xi^2\big)\hat f_k(\xi).$$ Its general solution reads $$\hat f_k(\xi)=c_k\xi^k q(\xi),\quad\text{with}\quad q(\xi):=\exp\Big(-\frac{\xi^2}2+\int_0^1\frac{\hat{\vartheta}(\xi s)}s{\,\mathrm{d}}s\Big),$$ for all $k\in{\mathbb{N}}_0$, with $c_k\in{\mathbb{C}}$. We may now fix $c_k:={{\mathrm{i}}}^k$, which completes the proof.
\[def:spec:proj:pert\] The spectral projection $\mathcal P_k$ of ${\mathcal L}+\Theta$ corresponding to the eigenvalue $-k\in-{\mathbb{N}}_0$ fulfills $${\operatorname{ran}}\mathcal P_k={\operatorname{span}}\{f_k\},\quad\ker\mathcal P_k={\mathcal E}_{k+1}\oplus{\operatorname{span}}\{f_{k-1},\ldots,f_0\},$$ with the eigenfunctions $f_k,\ldots,f_0$ given in (\[rec\_f\_k\]). Therefore, all singularities of the resolvent are of order one, and for all $k\in{\mathbb{N}}_0$ there holds $M({\mathcal L}+\Theta+k)=\ker({\mathcal L}+\Theta+k)$.
The set $\mathcal{K}_k:={\mathcal E}_{k+1}\oplus{\operatorname{span}}\{f_{k-1},\ldots,f_0\}$ is invariant under ${\mathcal L}+\Theta$, cf. Corollary \[cor:inv\_subsp\]. Therefore the algebraic eigenspace satisfies $M({\mathcal L}+\Theta+k)=\ker({\mathcal L}+\Theta+k)={\operatorname{span}}\{f_k\}$, being the complement of $\mathcal K_k$. In particular we obtain the $({\mathcal L}+\Theta)$-invariant decomposition ${\mathcal E}=\mathcal K_k\oplus M({\mathcal L}+\Theta+k)$, and $\sigma(({\mathcal L}+\Theta)|_{\mathcal K_k})=-{\mathbb{N}}_0\backslash\{-k\}$. So we can apply Lemma \[uniqueness\] from the appendix, which yields the properties of the spectral projections.
Since $\dim \mathcal P_k=1$ and $M({\mathcal L}+\Theta+k)=\ker({\mathcal L}+\Theta+k)$, the singularity of $R_{{\mathcal L}+\Theta}(\zeta)$ at $\zeta=-k$ is a pole of order one, see Proposition \[spec\_prop\] (\[itemiii\])-(\[itemiv\]).
Having explicitly determined the spectrum of the perturbed Fokker-Planck operator, we now turn to the generated semigroup and the corresponding decay rates. We start with the fact that ${\mathcal L}+\Theta$ generates a $C_0$-semigroup:
\[prop\_3\_11\] For each $k\in{\mathbb{N}}_0$ the operator $({\mathcal L}+\Theta)|_{{\mathcal E}_k}$ is the infinitesimal generator of a $C_0$-semigroup on ${\mathcal E}_k$. The semigroup on ${\mathcal E}$ preserves mass, i.e. $$\int_{\mathbb{R}}f(x){\,\mathrm{d}}x=\int_{\mathbb{R}}[{\mathrm{e}}^{t({\mathcal L}+\Theta)}f](x){\,\mathrm{d}}x,\quad\forall t\ge 0.$$
According to Theorem \[extension\_theorem\] the operator ${\mathcal L}$ generates a $C_0$-semigroup on ${\mathcal E}_k$ for every $k\in{\mathbb{N}}_0$, and due to Lemma \[remark\_theta\_inv\] and Corollary \[lemma:bdd\_theta\] we have $\Theta|_{{\mathcal E}_k}\in{\mathscr B}({\mathcal E}_k)$. Now a bounded perturbation of the infinitesimal generator of a $C_0$-semigroup is again infinitesimal generator, see [@engel Theorem III.1.3], and so the first result follows.
To show the conservation of mass we use the decomposition of $({\mathrm{e}}^{t({\mathcal L}+\Theta)})_{t\ge 0}$ by $\mathcal P_0$ corresponding to ${\mathcal E}={\mathcal E}_1\oplus{\operatorname{span}}\{f_0\}$. The space ${\mathcal E}_1$ consists of all massless functions, so the part $\mathcal P_0f$ alone determines the mass of any $f\in{\mathcal E}$. Since ${\mathcal E}_1$ and ${\operatorname{span}}\{ f_0\}$ are both invariant under the semigroup, $\mathcal P_0$ and $({\mathrm{e}}^{t({\mathcal L}+\Theta)})_{t\ge 0}$ commute. Furthermore we have $\mathcal P_0 f\in\ker({\mathcal L}+\Theta)$, and hence ${\mathrm{e}}^{t({\mathcal L}+\Theta)}\mathcal P_0 f=\mathcal P_0 f$ for all $t\ge 0$. Altogether we obtain $\mathcal P_0 {\mathrm{e}}^{t({\mathcal L}+\Theta)}f=\mathcal P_0 f$ for all $f\in{\mathcal E},\,t\ge 0$, i.e. the semigroup preserves mass.
Next we investigate the decay rate of $({\mathrm{e}}^{t({\mathcal L}+\Theta)})_{t\ge 0}$ on the subspaces ${\mathcal E}_k$. To this end we define: $$\hat\psi(\xi):=\exp\Big(\int_0^1\frac{\hat{\vartheta}(\xi s)}s{\,\mathrm{d}}s\Big),\quad\xi\in\Omega_{\beta/2},$$ which is analytic in $\Omega_{\beta/2}$ according to Remark \[rem31\].
\[eigs:Psi\] The map $\Psi:f\mapsto f*\psi$ has the properties:
1. \[klam\_i\] For each $k\in{\mathbb{N}}_0$, $\Psi:{\mathcal E}_k\to{\mathcal E}_k$ is a bijection, with inverse $\Psi^{-1}:f\mapsto f*{\mathcal F}^{-1}[1/\hat\psi]$.
2. $\Psi,\Psi^{-1}\in{\mathscr B}({\mathcal E})$.
We define $\bar\Psi:f\mapsto f*{\mathcal F}^{-1}[1/\hat\psi]$. Due to the condition [**(C)(iii)**]{} there holds $\Psi f,\bar\Psi f\in {\mathcal E}$ for all $f\in{\mathcal E}$, which is shown analogously to Lemma \[thetaeine\]. Let now $f\in{\mathcal E}_k$ for some $k\in{\mathbb{N}}_0$. Then $\hat f(\xi)$ has a zero of order greater or equal to $k$ at $\xi=0$, cf. Proposition \[char:e\_k\]. Since $\hat\psi$ and $1/\hat\psi$ are analytic in $\Omega_{\beta/2}$, the zero at $\xi=0$ of ${\mathcal F}_{x\to\xi}\Psi f=\hat f(\xi)\hat\psi(\xi)$ and of ${\mathcal F}_{x\to\xi}\bar\Psi f=\hat f(\xi)/\hat\psi(\xi)$ is of the same order as of $\hat f$. So $\Psi,\bar\Psi:{\mathcal E}_k\to{\mathcal E}_k$ for all $k\in{\mathbb{N}}_0$.
By applying the Fourier transform, we see that $\Psi\circ\bar\Psi f=\bar\Psi\circ\Psi f=f$ for all $f\in{\mathcal E}$, i.e. $\bar\Psi=\Psi^{-1}$, and $\Psi,\Psi^{-1}:{\mathcal E}_k\to{\mathcal E}_k$ are bijections for all $k\in{\mathbb{N}}_0$.
Finally, as in Corollary \[lemma:bdd\_theta\] one proves the boundedness of $\Psi$ and $\Psi^{-1}$ by using the assumption [**(C)(iii)**]{}.
The map $\Psi$ plays a crucial role in the analysis of the perturbed Fokker-Planck operator ${\mathcal L}+\Theta$, because it relates the eigenspaces of ${\mathcal L}$ to the eigenspaces of ${\mathcal L}+\Theta$: According to Proposition \[pert:spec\] we have: $$\label{psi}
f_k=\Psi\mu_k,\quad k\in{\mathbb{N}}_0.$$ By using this property of $\Psi$ we obtain the following result:
\[aquiv:reso\] Let $k\in{\mathbb{N}}_0$ and $\zeta\in{\mathbb{C}}\backslash\{-k,-k-1,\ldots\}$. Then there holds $$\label{equiv:resolvents}
R_{{\mathcal L}+\Theta}(\zeta)|_{{\mathcal E}_k} =\Psi\circ R_{\mathcal L}(\zeta)\circ\Psi^{-1}|_{{\mathcal E}_k}.$$ In particular there exists a constant $\tilde C_k>0$ such that $$\label{reso:est}
\big\|\big(R_{{\mathcal L}+\Theta}(\zeta)|_{{\mathcal E}_k}\big)^n\big\|_{{\mathscr B}({\mathcal E}_k)}\le \frac{\tilde C_k}{({\operatorname{Re}}\zeta+k)^n},\quad {\operatorname{Re}}\zeta>{-k},\, n\in{\mathbb{N}}.$$
We fix $k\in{\mathbb{N}}_0$. Then for all $j\ge k$ and $\zeta\in{\mathbb{C}}\backslash\{-k,-k-1,\ldots\}$ there holds due to (\[psi\]): $$R_{\mathcal L}(\zeta)\mu_j=\frac{\mu_j}{\zeta+j}=\Psi^{-1}\circ R_{{\mathcal L}+\Theta}(\zeta)f_j= \Psi^{-1}\circ R_{{\mathcal L}+\Theta}(\zeta)\circ\Psi\mu_j.$$ So we have $ R_{\mathcal L}(\zeta)=\Psi^{-1}\circ R_{{\mathcal L}+\Theta}(\zeta)\circ\Psi$ in the space ${\operatorname{span}}\{\mu_j:j\ge k\}\subset E_k$, which is dense in ${\mathcal E}_k$. Then this identity extends to ${\mathcal E}_k$ due to the continuity of the occurring operators.
In order to prove the resolvent estimate (\[reso:est\]) we use $$\big(R_{{\mathcal L}+\Theta}(\zeta)|_{{\mathcal E}_k}\big)^n=R_{{\mathcal L}+\Theta}(\zeta)^n|_{{\mathcal E}_k}=\Psi\circ R_{{\mathcal L}}(\zeta)^n\circ \Psi^{-1}|_{{\mathcal E}_k},$$ which follows from (\[equiv:resolvents\]) and Lemma \[eigs:Psi\] (\[klam\_i\]). Because of $\Psi,\Psi^{-1}\in{\mathscr B}({\mathcal E}_k)$ we conclude $$\label{circ_estim}
\big\|\big(R_{{\mathcal L}+\Theta}(\zeta)|_{{\mathcal E}_k}\big)^n\big\|_{{\mathscr B}({\mathcal E}_k)}\le \|\Psi\|_{{\mathscr B}({\mathcal E}_k)} \big\|\big(R_{{\mathcal L}}(\zeta)|_{{\mathcal E}_k}\big)^n\big\|_{{\mathscr B}({\mathcal E}_k)}\|\Psi^{-1}\|_{{\mathscr B}({\mathcal E}_k)}.$$ Due to the semigroup estimate in Theorem \[extension\_theorem\] (\[fp:res:iv\]) there holds $$\big\|\big(R_{{\mathcal L}}(\zeta)|_{{\mathcal E}_k}\big)^n\big\|_{{\mathscr B}({\mathcal E}_k)}\le \frac{C_k}{({\operatorname{Re}}\zeta+k)^n},\quad {\operatorname{Re}}\zeta>-k,\,n\in{\mathbb{N}},$$ according to the Hille-Yosida theorem. Inserting this estimate in (\[circ\_estim\]) shows (\[reso:est\]).
Since $\Psi,\Psi^{-1}\in{\mathscr B}({\mathcal E})$, the norm $\|\Psi(\cdot)\|_\omega$ is equivalent to $\|\cdot\|_\omega$ on ${\mathcal E}$. Therefore, the map $\Psi:({\mathcal E},\|\Psi(\cdot)\|_\omega)\to ({\mathcal E},\|\cdot\|_\omega)$ is an isometric isomorphism. Thus, according to the operator ${\mathcal L}$ in $({\mathcal E},\|\Psi(\cdot)\|_\omega)$ is isometrically equivalent to ${\mathcal L}+\Theta$ in $({\mathcal E},\|\cdot\|)$.
Let $k\in{\mathbb{N}}_0$. Then there exists a constant $\tilde C_k>0$ such that $$\label{pert_dec_rate}
\big\|{\mathrm{e}}^{t({\mathcal L}+\Theta)}|_{{\mathcal E}_k}\big\|_{{\mathscr B}({\mathcal E}_k)} \le \tilde C_k{\mathrm{e}}^{-kt},\quad t\ge 0.$$
The result immediately follows from (\[reso:est\]) by application of the Hille-Yosida theorem.
\[conv\_to\_f\_0\] The above result implies the exponential convergence of any solution of (\[pert\_fp\]) towards the (appropriately scaled) stationary state: Choose any $f\in{\mathcal E}$. Then there exists a unique constant $m\in{\mathbb{C}}$ (the “mass” of $f$) such that $\mathcal P_0 f=m f_0$. So $f-mf_0=(1-\mathcal P_0)f\in{\mathcal E}_1$, cf. Lemma \[def:spec:proj:pert\], which implies ${\mathrm{e}}^{t({\mathcal L}+\Theta)}f-mf_0={\mathrm{e}}^{t({\mathcal L}+\Theta)}(f-mf_0)\in{\mathcal E}_1$ for all $t\ge 0$, due to Proposition \[prop\_3\_11\]. With (\[pert\_dec\_rate\]) and $k=1$ this implies $$\|{\mathrm{e}}^{t({\mathcal L}+\Theta)}f-mf_0\|_\omega\le \tilde C_1\|f-mf_0\|_\omega{\mathrm{e}}^{-t},\quad t\ge0.$$
In the one dimensional case we can explicitly compute the Fourier transform of $R_{{\mathcal L}+\Theta}(\zeta)g$, see Proposition \[f\_trafo\_reso\]: For any $k\in{\mathbb{N}}_0$, ${\operatorname{Re}}\zeta>-k$, and $g\in {\mathcal E}_k$, the unique solution $f\in{\mathcal E}_k$ of $(\zeta-{\mathcal L}-\Theta)f=g$ satisfies $$\hat f(\xi)={\mathcal F}_{x\to\xi} [R_{{\mathcal L}+\Theta}(\zeta)g]=\hat f_0(\xi)\int_0^1\frac{\hat g(s\xi)}{\hat f_0(s\xi)}s^{\zeta-1}{\,\mathrm{d}}s,\quad \xi\in\Omega_{\beta/2},$$ where $s^{\zeta}={\mathrm{e}}^{\zeta\log s}$ and $\log$ is the natural logarithm on ${\mathbb{R}}^+.$ One can use this representation for an alternative proof of the resolvent estimate (\[reso:est\]). However, this becomes less convenient in higher dimensions, since it is then not clear how to properly compute the explicit Fourier transform of $R_{{\mathcal L}+\Theta}(\zeta)$.
Now we summarize our results in the final theorem:
\[final\_pert\_trm\] Let ${\mathcal E}=L^2(\omega)$, where $\omega(x)=\cosh\beta x$, for some $\beta>0$, and let $\Theta$ fulfill the condition [**(C)**]{} for this $\beta>0$. Then the perturbed operator ${\mathcal L}+\Theta$ has the following properties in ${\mathcal E}$:
1. It has compact resolvent, and $\sigma({\mathcal L}+\Theta)=\sigma_p({\mathcal L}+\Theta)=-{\mathbb{N}}_0$.
2. There holds $M({\mathcal L}+\Theta+k)=\ker({\mathcal L}+\Theta+k)={\operatorname{span}}\{f_k\}$, where $f_k$ is the eigenfunction to the eigenvalue $-k$ given by (\[rec\_f\_k\]). The eigenfunctions are related by $f_k=f_0^{(k)}$.
3. The spectral projection $\mathcal P_k$ corresponding to the eigenvalue $-k\in-{\mathbb{N}}$ fulfills $${\operatorname{ran}}\mathcal P_k={\operatorname{span}}\{f_k\},\quad\ker\mathcal P_k={\mathcal E}_{k+1}\oplus{\operatorname{span}}\{ f_{k-1},\ldots, f_0\},$$ where the $({\mathcal L}+\Theta)$-invariant spaces ${\mathcal E}_k$ are explicitly given in (\[e\_k\]). Moreover, ${\operatorname{ran}}\mathcal P_0={\operatorname{span}}\{f_0\}$ and $\ker\mathcal P_0={\mathcal E}_1$.
4. For every $k\in{\mathbb{N}}_0$, the operator $({\mathcal L}+\Theta)|_{{\mathcal E}_k}$ generates a $C_0$-semigroup $({\mathrm{e}}^{t({\mathcal L}+\Theta)}|_{{\mathcal E}_k})_{t\ge 0}$ in ${\mathcal E}_k$, which satisfies the estimate $$\|{\mathrm{e}}^{t({\mathcal L}+\Theta)}|_{{\mathcal E}_k}\|_{{\mathscr B}({\mathcal E}_k)}\le\tilde C_k{\mathrm{e}}^{-k t},\quad t\ge0,$$ where the constant $\tilde C_k>0$ is independent of $t$.
Apparently, the particular choice of $\beta>0$ has no influence on the above results, except possibly for the constants $\tilde C_k$. In practice, the constant $\beta$ may therefore be chosen arbitrarily small, such that $\Theta$ satisfies [**(C)**]{} for this $\beta$.
The Higher-Dimensional Case {#sec35}
===========================
As already mentioned in the introduction, the preceding results can be generalized to higher dimensions without much additional effort. Most proofs are analogous to the ones in the one-dimensional case. Therefore we give here only an outline of the steps leading to the extension of Theorem \[final\_pert\_trm\] to higher dimensions.
In this section we consider the perturbed Fokker-Planck equation (\[pert\_fp\]) on ${\mathbb{R}}^d$, where $d\in{\mathbb{N}}$ is the spatial dimension. Elements of ${\mathbb{R}}^d$ resp. ${\mathbb{C}}^d$ are represented by bold letters, e.g. ${\mathbf{x}}\in{\mathbb{R}}^d,{\boldsymbol{\xi}}\in{\mathbb{C}}^d$, and we write ${\mathbf{x}}=(x_1,\ldots,x_d)$. For a multi-index ${\mathbf{k}}\in{\mathbb{N}}_0^d$ we define $|{\mathbf{k}}|:=k_1+\cdots+k_d$, ${\mathbf{x}}^{\mathbf{k}}:=x_1^{k_1}\cdots x_d^{k_d}$ and ${\mathbf{k}}!:=k_1!\cdots k_d!$. Furthermore $$D^{\mathbf{k}}:=\frac{{\partial}^{|{\mathbf{k}}|}}{{\partial}x_1^{k_1}\cdots {\partial}x_d^{k_d}}.$$ We adopt the notation for weighted Sobolev spaces on ${\mathbb{R}}^d$ from Section \[sec2\], as well as the normalization of the Fourier transform.
We consider the Fokker-Planck operator on ${\mathbb{R}}^d$ given by $$L f:=\nabla\cdot\bigg(\mu\nabla\bigg(\frac{f}{\mu}\bigg)\bigg)=\Delta f+{\mathbf{x}}\cdot\nabla f+df,$$ where $\mu({\mathbf{x}}):=\exp(-{\mathbf{x}}\cdot{\mathbf{x}}/2)$. The natural space to consider $L$ in is $E:=L^2(1/\mu)$. Since it is isometrically equivalent to the harmonic oscillator $H:=-\Delta- d/2+|{\mathbf{x}}|^2/4$ in $L^2({\mathbb{R}}^d)$, we transfer many results of $H$ (see [@parmegg] and [@resi Theorem XIII.67]) to $L$. In the following we summarize some properties of $L$ in $E$ (see also [@Metafune2001; @bakry; @helffernier]):
\[prop:fp\_in\_H:d\] The Fokker-Planck operator $L$ in $E$ has the following properties:
1. \[st\_fp:1:d\] $L$ with $D(L)=\{f\in E:Lf\in E\}$ is self-adjoint and has a compact resolvent.
2. \[st\_fp:2:d\] The spectrum is $\sigma(L)=-{\mathbb{N}}_0$, and it consists only of eigenvalues.
3. \[st\_fp:3:d\] For each eigenvalue $-k\in\sigma(L)$ the corresponding eigenspace has the dimension $\binom{k+d-1}{k}$, and it is spanned by the eigenfunctions $$\mu_{\mathbf{k}}({\mathbf{x}}):=\prod_{\ell=1}^d \mu_{k_\ell}(x_\ell),\quad |{\mathbf{k}}|=k,$$ where the $\mu_j$ are defined in Theorem \[prop:fp\_in\_H\].
4. \[st\_fp:4:d\] The eigenfunctions $(\mu_{\mathbf{k}})_{{\mathbf{k}}\in{\mathbb{N}}^d_0}$ form an orthogonal basis of $E$.
5. \[st\_fp:45:d\] The spectral projection $\Pi_{L,k}$ onto the $k$-th eigenspace is given by $$\Pi_{L,k}=\sum_{|{\mathbf{k}}|=k}\Pi_{L,{\mathbf{k}}},\quad\text{where}\quad \Pi_{L,{\mathbf{k}}}:=\frac{(2\pi)^{d/2}}{{\mathbf{k}}!}\mu_{\mathbf{k}}{\langle}\cdot,\mu_{\mathbf{k}}{\rangle}_{E}.$$ There holds the spectral representation $L=\sum_{k\in{\mathbb{N}}_0}-k\Pi_{L,k}.$
6. \[st\_fp:5:d\] The operator $L$ generates a $C_0$-semigroup of contractions on $E_k$ for all $k\in{\mathbb{N}}_0$, where $E_k:=\ker(\Pi_{L,0}+\cdots+\Pi_{L,k-1}),\,k\ge 1$, and $E_0:=E$. The semigroup satisfies the estimate $$\big\|{\mathrm{e}}^{tL}|_{E_k}\big\|_{{\mathscr B}(E_k)}\le{\mathrm{e}}^{-kt},\quad\forall k\in{\mathbb{N}}_0.$$
The next step is to properly define $L$ in ${\mathcal E}:=L^2(\omega)$ with a weight $\omega({\mathbf{x}})=\cosh\beta|{\mathbf{x}}|$ with $\beta>0$. As in the one-dimensional case we have a characterization of ${\mathcal E}$ by the Fourier transform. Due to (a small variant of) [@resi2 Theorem IX.13] we have: There holds $f\in{\mathcal E}$ iff $\hat f$ has an analytic extension (denoted by $\hat f$ as well) to the set $\Omega_{\beta/2}:=\{\mathbf z\in{\mathbb{C}}^d:|{\operatorname{Im}}\mathbf z|<\beta/2\}$ and $$\label{f_in_E:d}
\sup_{\substack{|\mathbf b|<\beta/2\\ \mathbf b\in{\mathbb{R}}^d}}\|\hat f(\cdot+{{\mathrm{i}}}\mathbf b)\|_{L^2({\mathbb{R}}^d)}<\infty.$$
For any $\mathbf b\in{\mathbb{R}}^d$ with $|\mathbf b|<\beta /2$ we have $\hat f({\boldsymbol{\xi}}+{{\mathrm{i}}}\mathbf b)={\mathcal F}_{{\mathbf{x}}\to{\boldsymbol{\xi}}}\big({\mathrm{e}}^{\mathbf b\cdot{\mathbf{x}}}f({\mathbf{x}})\big).$ The right hand side still makes sense for $|\mathbf b|=\beta/2$ as an $L^2({\mathbb{R}}^d)$-function. And according to this identity and Plancherel’s formula there holds $\mathbf b\mapsto \hat f(\cdot+{{\mathrm{i}}}\mathbf b)\in C(\overline{B(\beta/2,0)};L^2({\mathbb{R}}^d))$, where $B(\beta/2,0):=\{\mathbf b\in{\mathbb{R}}^d:|\mathbf b|<\beta/2\}$. We can use this fact to define the norm $$\label{f_norm:d}
{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}f{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}^2_\omega:=\sum_{\ell=1}^d \Big\|\hat f\Big(\cdot+{{\mathrm{i}}}\frac\beta2\boldsymbol{\delta}_\ell\Big)\Big\|^2_{L^2({\mathbb{R}}^d)}+\Big\|\hat f\Big(\cdot-{{\mathrm{i}}}\frac\beta2\boldsymbol{\delta}_\ell\Big)\Big\|^2_{L^2({\mathbb{R}}^d)},$$ where $\boldsymbol{\delta}_\ell\in{\mathbb{R}}^d$ is the vector whose $\ell$-th component is one, and all others are zero. The norm ${|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}\cdot{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega$ is equivalent to $\|\cdot\|_\omega$.
In ${\mathcal E}$ there hold Poincaré-type inequalities:
\[pr\_pc:d\] For every ${\mathbf{k}}\in{\mathbb{N}}_0^d$ there exists a constant $C_{\mathbf{k}}>0$ such that for all $f\in C_0^\infty({\mathbb{R}}^d)$: $$\label{poincare:d}
\|f\|_\omega\le C_{\mathbf{k}}\|D^{\mathbf{k}}f\|_\omega.$$
For the proof see Appendix \[app:proof\]. A similar statement is given in [@GOIII Theorem 14.5]. By using this Poincaré inequality we can generalize Lemma \[reso\_estim\]: Let again $\mathfrak L =\Delta+{\mathbf{x}}\cdot\nabla+d$ be the distributional Fokker-Planck operator. For $f,g\in{\mathcal E}\subset{\mathscr S}'$ with $(\zeta-\mathfrak L)f=g$ we have the estimate $$\label{djan}
\|f\|_\varpi+\|\nabla f\|_\omega\le C\|g\|_\omega,$$ where $\varpi({\mathbf{x}})=(2{\operatorname{Re}}\zeta-d)\omega({\mathbf{x}})+{\mathbf{x}}\cdot\nabla\omega({\mathbf{x}})-\Delta\omega({\mathbf{x}})$, which is a weight function for ${\operatorname{Re}}\zeta$ sufficiently large. Now we may proceed analogously to the proof of Lemma \[abschluss\_LL\] and show that $\mathfrak L|_{C_0^\infty({\mathbb{R}}^d)}$ is closable in ${\mathcal E}$, and its closure ${\mathcal L}$ has the domain $D({\mathcal L})=\{f\in{\mathcal E}:\mathfrak L f\in{\mathcal E}\}$. From [@Opic1989 Theorem 2.4] we get the compact embedding $W^{1,2}(\varpi,\omega){\hookrightarrow}{\hookrightarrow}{\mathcal E}$, and together with the estimate this implies the compactness of the resolvent of ${\mathcal L}$, analogously to Theorem \[trm:compactness\]. Hence, the spectrum of ${\mathcal L}$ consists only of eigenvalues, and there holds
In ${\mathcal E}$ we have $\sigma({\mathcal L})=-{\mathbb{N}}_0$. The eigenspaces are still spanned by the $\mu_\mathbf k$.
We consider the Fourier transform of the eigenvalue equation ${\mathcal L}f=\zeta f$, and by setting $\tilde f({\boldsymbol{\xi}}):=\hat f({\boldsymbol{\xi}})/\hat \mu({\boldsymbol{\xi}})$ we get analogously to the calculation in the Appendix \[app:b\] the equation $$\label{simpl_eveq}
{\boldsymbol{\xi}}\cdot\nabla\tilde f({\boldsymbol{\xi}})=-\zeta\tilde f({\boldsymbol{\xi}}).$$ For each $j\in\{1,\ldots,d\}$ the function $\tilde f(0,\ldots,0,\xi_j,0,\ldots, 0)$ needs to be analytic in $\Omega_{\beta/2}$, and satisfies for $\tilde g=0$. So, as in the Appendix \[app:b\] we find that it is necessary that $\zeta\in -{\mathbb{N}}_0$.
For $k:=-\zeta\in{\mathbb{N}}_0$ and ${\boldsymbol{\xi}}\in{\mathbb{R}}^d$ we obtain by differentiating with respect to $\xi_j$: $${\boldsymbol{\xi}}\cdot\nabla\Big({\frac{\partial^{}{\tilde f({\boldsymbol{\xi}})}}{\partial{\xi_j}^{}}}\Big)=(k-1)\Big({\frac{\partial^{}{\tilde f({\boldsymbol{\xi}})}}{\partial{\xi_j}^{}}}\Big).$$ Thus, for any ${\mathbf{k}}\in{\mathbb{N}}_0^d$ with $|{\mathbf{k}}|=k$ we get $${\boldsymbol{\xi}}\cdot\nabla\big(D^{{\mathbf{k}}}\tilde f({\boldsymbol{\xi}})\big)=0,$$ and all characteristics meet at ${\boldsymbol{\xi}}=0$. $\hat f$ is analytic on ${\mathbb{R}}^d$. Hence, the continuity of $D^{\mathbf{k}}\tilde f({\boldsymbol{\xi}})$ at ${\boldsymbol{\xi}}=0$ implies $D^{{\mathbf{k}}}\tilde f({\boldsymbol{\xi}})=C$ for some constant $C\in{\mathbb{C}}$. This holds for any $|{\mathbf{k}}|=k$, so the general solution of is a linear combination of all ${\boldsymbol{\xi}}^{{\mathbf{k}}}$ with $|{\mathbf{k}}|=-\zeta=k$. Therefore, the Fourier transform of an eigenfunction $f$ with $({\mathcal L}+k) f=0$ is a linear combination of the ${\boldsymbol{\xi}}^{{\mathbf{k}}}\mu({\boldsymbol{\xi}})$ with $|{\mathbf{k}}|=k$ (and, equivalently, $f({\mathbf{x}})$ is a linear combination of the $D^{{\mathbf{k}}}\mu({\mathbf{x}})$). Then, according to Theorem \[prop:fp\_in\_H:d\] and Theorem \[prop:fp\_in\_H\] , the eigenspace for $\zeta=-k$ is spanned by the $\mu_{\mathbf{k}}$.
As in Proposition \[spec\_pr\] we can define the ${\mathcal L}$-invariant subspaces ${\mathcal E}_k:={\operatorname{cl}}_{\mathcal E}E_k={\operatorname{cl}}_{\mathcal E}{\operatorname{span}}\{\mu_{\mathbf{k}}:|{\mathbf{k}}|\ge k\}$ for all $k\in{\mathbb{N}}_0$, and $\sigma({\mathcal L}|_{{\mathcal E}_k})=\{-k,-k-1,\ldots\}$. By applying Lemma \[lem:funct\] we get by induction $$\label{e_k:d}
{\mathcal E}_k=\Big\{f\in{\mathcal E}:\int_{{\mathbb{R}}^d} f({\mathbf{x}}){\mathbf{x}}^{\mathbf{k}}{\,\mathrm{d}}{\mathbf{x}}=0, \, |{\mathbf{k}}|\le k-1\Big\}=\big\{f\in{\mathcal E}:D^{\mathbf{k}}\hat f(0)=0, \, |{\mathbf{k}}|\le k-1\big\}.$$ Analogously to Proposition \[spec\_prop\] we can also characterize the spectral projections corresponding to the eigenvalues $-k\in-{\mathbb{N}}_0$, see the result of Theorem \[extension\_theorem:d\] below. Finally, as in the one-dimensional case, one shows that ${\mathcal L}$ generates a $C_0$-semigroup of bounded operators $({\mathrm{e}}^{t{\mathcal L}})_{t\ge 0}$, which is given by the formula (cf. [@Gallay2002 Appendix A]) $${\mathcal F}_{{\mathbf{x}}\to{\boldsymbol{\xi}}}\big[{\mathrm{e}}^{t{\mathcal L}}f\big]=\exp\Big(-\frac{{\boldsymbol{\xi}}\cdot{\boldsymbol{\xi}}}2(1-{\mathrm{e}}^{-2t})\Big)\hat f\big({\boldsymbol{\xi}}{\mathrm{e}}^{-t}\big),\quad t\ge0.$$ The corresponding decay estimates on the subspaces ${\mathcal E}_k$ can be shown as in the proof of Proposition \[unpert\_decay\]. Thereby one uses the norm and the Poincaré inequality .
\[extension\_theorem:d\] In ${\mathcal E}:=L^2(\omega)$, with $\omega({\mathbf{x}})=\cosh\beta|{\mathbf{x}}|$ and $\beta>0$, the operator $L$ is closable, and ${\mathcal L}:={\operatorname{cl}}_{\mathcal E}L$ has the following properties:
1. \[fp:res:i:d\] The spectrum satisfies $\sigma({\mathcal L})=-{\mathbb{N}}_0$, and $M({\mathcal L}+k)=\ker({\mathcal L}+k)={\operatorname{span}}\{\mu_{\mathbf{k}}:|{\mathbf{k}}|=k\}$ for any $k\in{\mathbb{N}}_0$. The eigenfunctions satisfy $\mu_{\mathbf{k}}=D^{\mathbf{k}}\mu_{\mathbf 0}$.
2. \[fp:res:ii:d\] For any $k\in{\mathbb{N}}_0$ the closed subspace ${\mathcal E}_k:={\operatorname{cl}}_{\mathcal E}{\operatorname{span}}\{\mu_{\mathbf{k}}:|{\mathbf{k}}|\ge k\}$ is an ${\mathcal L}$-invariant subspace of ${\mathcal E}$, and ${\operatorname{span}}\{\mu_{\mathbf{k}}:|{\mathbf{k}}|\le k-1\}$ is a complement. In particular ${\mathcal E}_0={\mathcal E}$.
3. \[fp:res:iii:d\] The spectral projection $\Pi_{{\mathcal L},k}$ to the eigenvalue $-k\in-{\mathbb{N}}_0$ fulfills ${\operatorname{ran}}\Pi_{{\mathcal L},k}={\operatorname{span}}\{\mu_{\mathbf{k}}:|{\mathbf{k}}|=k\}$ and $\ker\Pi_{{\mathcal L},k}={\mathcal E}_{k+1}\oplus {\operatorname{span}}\{\mu_{\mathbf{k}}:|{\mathbf{k}}|<k\}$.
4. \[fp:res:iv:d\] For any $k\in{\mathbb{N}}_0$ the operator ${\mathcal L}$ generates a $C_0$-semigroup on ${\mathcal E}_k$, and there exists a constant $C_k\ge 1$ such that we have the estimate $$\left\|{\mathrm{e}}^{t{\mathcal L}}|_{{\mathcal E}_k}\right\|_{{\mathscr B}({\mathcal E}_k)}\le C_k {\mathrm{e}}^{-kt},\quad\forall t\ge 0.$$
Next we specify the conditions on the perturbation $\Theta$.
[**($\mathbf{C_d}$) Conditions on $\boldsymbol\Theta$:**]{} We assume that $\Theta f={\vartheta}*f$, for $f\in{\mathcal E}$, where ${\vartheta}$ is a tempered distribution that fulfills the following properties in $\Omega_{\beta/2}$ for some $\beta>0$:
1. The Fourier transform $\hat {\vartheta}$ can be extended to an analytic function in $\Omega_{\beta/2}$ (also denoted by $\hat {\vartheta}$), and $\hat{\vartheta}\in L^\infty(\Omega_{\beta/2})$.
2. It holds $\hat{\vartheta}(\mathbf 0)=0$, i.e. ${\vartheta}$ has zero mean.
3. The mapping ${\boldsymbol{\xi}}\mapsto{\operatorname{Re}}\int_0^1\hat{\vartheta}({\boldsymbol{\xi}}s)/s{\,\mathrm{d}}s$ is essentially bounded in $\Omega_{\beta/2}$.
Condition [**($\mathbf{C_d}$)(i)**]{} ensures that $\Theta\in{\mathscr B}({\mathcal E})$, which is seen by using the norm ${|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}\cdot{|{}\hspace{-0.5mm}{}|{}\hspace{-0.5mm}{}|}_\omega$. And due to [**($\mathbf{C_d}$)(ii)**]{} we have $\Theta:{\mathcal E}_k\to{\mathcal E}_{k+1}$ for all $k\in{\mathbb{N}}_0$. In the following we always assume that [**($\mathbf{C_d}$)**]{} holds.
\[pert:spec:d\] We have the following spectral properties of ${\mathcal L}+\Theta$ in ${\mathcal E}$:
1. \[eins\] $\sigma({\mathcal L}+\Theta)=-{\mathbb{N}}_0$.
2. \[zwei\] For each $k\in{\mathbb{N}}_0$, the eigenspace $\ker({\mathcal L}+\Theta+k)$ has the dimension $\binom{k+d-1}{k}$.
3. Under appropriate scaling, the eigenfunctions $f_{\mathbf{k}}$ to the eigenvalue $-k\in\-{\mathbb{N}}_0$ are explicitly given by $$\label{rec_f_k:d}
f_{\mathbf{k}}=D^{\mathbf{k}}f_{\mathbf 0},\quad|{\mathbf{k}}|=k,$$ where $$\label{f_k:d}
\hat f_{\mathbf 0}({\boldsymbol{\xi}}):=\exp\Big(-\frac{{\boldsymbol{\xi}}\cdot{\boldsymbol{\xi}}}2+\int_0^1\frac{\hat{\vartheta}({\boldsymbol{\xi}}s)}s{\,\mathrm{d}}s\Big),\quad {\boldsymbol{\xi}}\in\Omega_{\beta/2}\subset{\mathbb{C}}^d.$$
Thereby $f_{\mathbf 0}$ is the unique stationary solution of the perturbed Fokker-Planck equation (\[pert\_fp\]) with unit mass.
Since the resolvent is compact (see the discussion above), the spectrum consists only of eigenvalues. As in the one-dimensional case one shows $\sigma({\mathcal L}+\Theta)\subset-{\mathbb{N}}_0$ by applying $\Pi_{{\mathcal L},k}$ to the eigenvalue equation. This also implies $\dim\ker(k+{\mathcal L}+\Theta)\le\dim{\operatorname{ran}}\Pi_{{\mathcal L},k}=\binom{k+d-1}{k}$. Then one verifies that the functions $f_{\mathbf{k}}$ given in (\[rec\_f\_k:d\]) are eigenfunctions, and lie in ${\mathcal E}$, according to the condition (\[f\_in\_E:d\]). Since $\dim{\operatorname{span}}\{f_{\mathbf{k}}:|{\mathbf{k}}|=k\}=\binom{k+d-1}{k}$, there are no further eigenfunctions, due to the previous estimate on the dimension of the eigenspaces. So $\ker(k+{\mathcal L}+\Theta)={\operatorname{span}}\{f_{\mathbf{k}}:|{\mathbf{k}}|=k\}$ for all $k\in{\mathbb{N}}_0$.
Now we introduce $$\hat \psi({\boldsymbol{\xi}}):=\exp\Big(\int_0^1\frac{\hat{\vartheta}({\boldsymbol{\xi}}s)}s{\,\mathrm{d}}s\Big),\quad {\boldsymbol{\xi}}\in\Omega_{\beta/2},$$ and the mapping $\Psi:f\mapsto f*\psi$. The results of Lemma \[eigs:Psi\] for $\Psi$ still hold, and due to (\[f\_k:d\]) we have for all ${\mathbf{k}}\in{\mathbb{N}}_0$: $$f_{\mathbf{k}}=\Psi\mu_{\mathbf{k}}.$$ As in Proposition \[aquiv:reso\] we obtain $R_{{\mathcal L}+\Theta}(\zeta)|_{{\mathcal E}_k} =\Psi\circ R_{\mathcal L}(\zeta)\circ\Psi^{-1}|_{{\mathcal E}_k},$ for all $k\in{\mathbb{N}}_0$ and $\zeta\in{\mathbb{C}}\backslash\{-k,-k-1,\ldots\}.$ The estimates (\[reso:est\]) and (\[pert\_dec\_rate\]) also hold here, and for the convergence of $f(t)={\mathrm{e}}^{t({\mathcal L}+\Theta)}f$ to the stationary solution see Remark \[conv\_to\_f\_0\]. As in Section \[sec3\] we finally have:
\[final\_pert\_trm:d\] Let ${\mathcal E}=L^2(\omega({\mathbf{x}}){\,\mathrm{d}}{\mathbf{x}})$, where $\omega({\mathbf{x}})=\cosh\beta |{\mathbf{x}}|$, for some $\beta>0$ and ${\mathbf{x}}\in{\mathbb{R}}^d$, and let $\Theta$ fulfill the condition [**($\mathbf{C_d}$)**]{} for this $\beta>0$. Then the perturbed operator ${\mathcal L}+\Theta$ has the following properties in ${\mathcal E}$:
1. It has compact resolvent, and $\sigma({\mathcal L}+\Theta)=\sigma_p({\mathcal L}+\Theta)=-{\mathbb{N}}_0$.
2. There holds $M({\mathcal L}+\Theta+k)=\ker({\mathcal L}+\Theta+k)={\operatorname{span}}\{f_{\mathbf{k}}:|{\mathbf{k}}|=k\}$, where the $f_{\mathbf{k}}$ are the eigenfunctions given by (\[rec\_f\_k:d\]). They are related by $f_{\mathbf{k}}=D^{\mathbf{k}}f_{\mathbf 0}$.
3. The spectral projection ${\mathcal P}_{k}$ to the eigenvalue $-k\in-{\mathbb{N}}_0$ fulfills ${\operatorname{ran}}{\mathcal P}_{k}={\operatorname{span}}\{f_{\mathbf{k}}:|{\mathbf{k}}|=k\}$ and $\ker{\mathcal P}_{k}={\mathcal E}_{k+1}\oplus {\operatorname{span}}\{f_{\mathbf{k}}:|{\mathbf{k}}|<k\}$, where the $({\mathcal L}+\Theta)$-invariant spaces ${\mathcal E}_k$ are explicitly given in (\[e\_k:d\]).
4. For every $k\in{\mathbb{N}}_0$, the operator $({\mathcal L}+\Theta)|_{{\mathcal E}_k}$ generates a $C_0$-semigroup $({\mathrm{e}}^{t({\mathcal L}+\Theta)}|_{{\mathcal E}_k})_{t\ge 0}$ in ${\mathcal E}_k$, which satisfies the estimate $$\|{\mathrm{e}}^{t({\mathcal L}+\Theta)}|_{{\mathcal E}_k}\|_{{\mathscr B}({\mathcal E}_k)}\le\tilde C_k{\mathrm{e}}^{-k t},\quad t\ge0,$$ where the constant $\tilde C_k>0$ is independent of $t$.
Simulation Results {#sec4}
==================
In this section we shall illustrate numerically the exponential convergence for the one-dimensional perturbed Fokker-Planck equation (\[pert\_fp\]), with ${\vartheta}:={\varepsilon}(\delta_{-\alpha}-\delta_{\alpha})$, i.e. $\Theta f(x)={\varepsilon}(f(x+\alpha)-f(x-\alpha))$, for some ${\varepsilon},\alpha\in{\mathbb{R}}$. The eigenfunctions $f_k$ of the evolution operator ${\mathcal L}+\Theta$ can be obtained by an inverse Fourier transform, with $\hat f_k$ explicitly given in (\[rec\_f\_k\]). If the initial condition ${\varphi}$ is a (finite) linear combination of the $f_k$, the solution to (\[pert\_fp\]) reads explicitly $$f(t,x)={\mathrm{e}}^{t({\mathcal L}+\Theta)}\Big[\sum_{j=1}^n a_jf_{k_j}\Big]=\sum_{j=1}^n a_j{\mathrm{e}}^{-k_j t}f_{k_j},\quad \forall t\ge 0.$$
In the simulation we use a mass conserving Crank-Nicolson finite difference scheme for (\[pert\_fp\]). It is employed on the spatial interval $[-25,25]$ (with $1500$ gridpoints) along with zero-flux boundary conditions. Moreover, we choose $\alpha={\varepsilon}=2$ and $\beta=1$, i.e. ${\mathcal E}=L^2(\cosh x)$.
The following numerical results verify the decaying behaviour of solutions to (\[pert\_fp\]), and yield an estimate to the constants $\tilde C_k$ from Theorem \[final\_pert\_trm\]. First we consider the initial condition ${\varphi}_1=(f_1-1.32f_2)/\|f_1-1.32f_2\|_\omega$. For the corresponding solution we plot $\|f(t,\cdot)\|_\omega$ in Figure \[fig\_a\]. Since the sequence $(f_k)_{k\in{\mathbb{N}}}$ is not orthogonal in ${\mathcal E}$, the initial decay rate is here smaller than the individual decay rate of $f_1$ (i.e. $-1$). But after some time, the $f_1$-term becomes dominant, and the decay rate approaches $-1$. For large times, the norm behaves approximately like $1.73 \,{\mathrm{e}}^{-t},$ so we have the lower bound $\tilde C_1\ge 1.73$.
As a second example we choose the initial condition ${\varphi}_2=(\chi_{[-4,0]}-\chi_{[0,4]})/\|\chi_{[-4,0]}-\chi_{[0,4]}\|_\omega$. It lies in ${\mathcal E}_1$ since it is massless. The evolution of $\|f(t,\cdot)\|_\omega$ is displayed in Figure \[fig\_b\]. Here, the norm even increases initially. Only after some time, the norm begins to decay with a rate tending to $-1$. For large times $t$, the norm behaves approximately like $22.53\,{\mathrm{e}}^{-t}$, which shows $\tilde C_1\ge 22.53$.
Spectral Projections {#app:space_enlarg}
====================
In this section we review some properties of spectral projections and resolvents, cf. [@taylay Chapters V.9-10], [@yosida Chapter VIII.8] and [@kato Sections III.6.4-5].
Here, $X$ is a Hilbert space, $A\in{\mathscr C}(X)$, and we assume $\lambda\in\sigma(A)$ to be an isolated point of the spectrum. Then the corresponding spectral projection ${\mathrm{P}\!}_{A,\lambda}$ is defined by (\[def:spec\_proj\]), and $\lambda$ is an isolated singularity of the resolvent $R_A(\zeta)$.
\[fds\] For every $n\in{\mathbb{N}}$ we have $$\begin{aligned}
{\operatorname{ran}}(\lambda-A)^n\supseteq & \ker {\mathrm{P}\!}_{A,\lambda},\\
\ker(\lambda-A)^n\subseteq & {\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}.\end{aligned}$$ There exists some $n\in{\mathbb{N}}$ such that both inclusion relations become equalities iff $\lambda$ is a pole of $R_A(\zeta)$. In this case $\lambda\in\sigma_p(A)$, i.e. an eigenvalue.
\[spec\_prop\] For the reduction of $A$ by a fixed spectral projection ${\mathrm{P}\!}_{A,\lambda}$ we have:
1. There holds ${\mathrm{P}\!}_{A,\lambda}D(A)\subset D(A)$, and $\ker {\mathrm{P}\!}_{A,\lambda}$ and ${\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}$ are $A$-invariant subspaces of $X$.
2. $A|_{{\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}}\in{\mathscr C}({{\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}})$ and $A|_{\ker {\mathrm{P}\!}_{A,\lambda}}\in{\mathscr C}({\ker {\mathrm{P}\!}_{A,\lambda}})$.
3. There holds $\sigma(A|_{{\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}})=\{\lambda\}$ and $\sigma(A|_{\ker {\mathrm{P}\!}_{A,\lambda}})=\sigma(A)\backslash\{\lambda\}$. Furthermore $A|_{{\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}}\in{\mathscr B}({\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda})$.
4. \[itemiii\] If $\dim {\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}<\infty$, then $\lambda-A|_{{\operatorname{ran}}{\mathrm{P}\!}_{A,\lambda}}$ is nilpotent, $\lambda$ is a pole of $R_A(\zeta)$, and $\lambda\in\sigma_p(A)$.
5. \[itemiv\] If $\lambda$ is a pole of $R_A(\zeta)$, then $M(\lambda-A)=\ker(\lambda-A)$ iff the pole has order one.
For a finite number of isolated points of the spectrum we have:
\[uniqueness\] For $N\in{\mathbb{N}}_0$, let $A$ have isolated points of the spectrum $\zeta_0,\ldots, \zeta_{N-1}$, which are eigenvalues with $\dim
M(\zeta_k-A)<\infty$ for all $0\le k\le N-1$. Assume there exists a closed subspace $Y\subset X$, such that
1. \[1:i\] $Y$ is $A$-invariant, and $\sigma(A|_{Y})\cap \{\zeta_0,\ldots,\zeta_{N-1}\}=\emptyset$.
2. \[1:ii\] $X$ can be decomposed as $X=Y\oplus M(\zeta_0-A)\oplus\ldots\oplus M(\zeta_{N-1}-A)$.
Then $Y=\ker \Pi_A$, where $\Pi_A:=\Pi_{A,0}+\cdots+\Pi_{A,N-1}$ is the sum of the spectral projections $\Pi_{A,k}$ corresponding to the $\zeta_k$, and $M(\zeta_k-A)={\operatorname{ran}}\Pi_{A,k}$ for all $0\le k\le N-1$.
According to the assumptions there holds $\sigma(A|_Y)=\sigma(A)\backslash\{\zeta_0,\ldots,\zeta_{N-1}\}$, and therefore the map $\zeta\mapsto R_A(\zeta)|_{Y}$ is analytic in $\rho(A)\cup\{\zeta_0,\ldots,\zeta_{N-1}\}$. Due to the definition (\[def:spec\_proj\]) of spectral projections this implies that $\Pi_{A,k}Y\equiv0$ for every $\Pi_{A,k}$, and therefore $Y\subseteq \ker \Pi_A$. On the other hand we have $M(\zeta_k-A)\subseteq{\operatorname{ran}}\Pi_{A,k}$ for all $0\le k\le N-1$, according to Proposition \[fds\]. From (\[1:ii\]) we conclude that the inclusions have to be equalities, otherwise $\ker\Pi_A\cap{\operatorname{ran}}\Pi_A\neq\{0\}$, which is impossible.
Fourier Transform of the Resolvent {#app:b}
==================================
This section deals with the explicit computation of the Fourier transform of the resolvent $R_{{\mathcal L}+\Theta}(\zeta)$ of the (one-dimensional) perturbed Fokker-Planck operator ${\mathcal L}+\Theta$ in ${\mathcal E}$, where $\Theta$ fulfills the condition [**(C)**]{}. We begin by considering the resolvent equation $$(\zeta-{\mathcal L}-\Theta)f=g$$ on ${\mathbb{R}}$, where we assume ${\operatorname{Re}}\zeta>{-k}$ and $f,g\in{\mathcal E}_k$ for some $k\in{\mathbb{N}}_0$. We apply the Fourier transform, which yields the following differential equation: $$\xi\Big[\hat f'(\xi)+\Big(\xi+\frac{\zeta-\hat{\vartheta}(\xi)}{\xi}\Big)\hat f(\xi)\Big]=\hat g(\xi).$$ By defining $\tilde f:=\hat f/\hat f_0$ and $\tilde g:=\hat g/\hat f_0$ we obtain the equivalent equation $$\label{ode:tilde_f}
\xi\tilde f'(\xi)+\zeta\tilde f(\xi)=\tilde g(\xi).$$ The general solution for $\xi\in{\mathbb{R}}^\pm$ reads $$\label{tf}
\tilde f(\xi)=\int_0^1\tilde g(\xi s)s^{\zeta-1}{\,\mathrm{d}}s+C_\pm\xi^{-\zeta}=:I(\xi)+C_\pm\xi^{-\zeta},$$ where the $C_\pm\in{\mathbb{C}}$ are integration constants to be determined.
First we shall show that the integral $I(\xi)$ is an analytic function on $\Omega_{\beta/2}$: If $g\in{\mathcal E}_k$, then $\tilde g$ is analytic in $\Omega_{\beta/2}$ and has a zero at $\xi=0$ of order not less than $k$, see (\[f\_char\_e\_k\]). Therefore, for any fixed $\zeta\in{\mathbb{C}}$ with ${\operatorname{Re}}\zeta>-k$, $$\tilde g(\xi s)s^{\zeta-1}=\frac{\tilde g(\xi s)}{s^k}s^{\zeta+k-1},\quad s \in (0,1],$$ is locally integrable at $s=0$, and $I(\xi)$ is well defined for all $\xi\in\Omega_{\beta/2}$. To see that it is actually analytic, we define $I_{\varepsilon}(\xi):=\int_{\varepsilon}^1 G_k(\xi,s)s^{\zeta+k-1}{\,\mathrm{d}}s$ for ${\varepsilon}\in[0,1)$, where $$G_k(\xi, s):=\begin{cases}
\displaystyle \frac{\tilde g(\xi s)}{ s^k},\quad &s\in(0,1],\\
\displaystyle\frac{\tilde g^{(k)}(0)\xi^k}{k!},\quad& s=0,
\end{cases}$$ for $\xi\in\Omega_{\beta/2}$. The function $G_k(\cdot,s)$ is analytic in $\Omega_{\beta/2}$ for all (fixed) $s\in[0,1]$, and $G_k$ is continuous in $\Omega_{\beta/2}\times [0,1]$. According to [@dettman Theorem 4.9.1], the functions $I_{\varepsilon}(\xi)$ are analytic in $\Omega_{\beta/2}$ for all ${\varepsilon}\in(0,1)$. Now we show that $(I_{\varepsilon})_{{\varepsilon}\in(0,1)}$ converges normally to $I$ in $\Omega_{\beta/2}$ as ${\varepsilon}\to 0$: Let $K\subset\Omega_{\beta/2}$ be compact. Then we have $$\begin{aligned}
\sup_{\substack{\xi\in K\\s\in[0,1]}}|G_k(\xi,s)|&\le\sup_{\substack{\xi\in K_0\\s\in[0,1]}}|G_k(\xi,s)|=\sup_{\substack{\xi\in K_0\backslash\{0\}\\s\in(0,1]}}\Big|\frac{\tilde g(\xi s)}{(\xi s)^k}\xi^k\Big|\nonumber\\
& \le \sup_{\xi\in K_0\backslash\{0\}}\Big|\frac{\tilde g(\xi )}{\xi ^k}\Big|\cdot\sup_{\xi\in K_0}|\xi^k|=:C_K<\infty,\label{ce_ka}\end{aligned}$$ since $\tilde g(\xi)/\xi^k$ is analytic in $\Omega_{\beta/2}$ (the singularity at $\xi=0$ is removable). Thereby, $ K_0$ is an appropriate convex, compact set with $\{0\}\cup K\subseteq K_0\subset \Omega_{\beta/2}$, and $C_K>0$ is a constant. With (\[ce\_ka\]) we obtain the following estimate for $\xi\in K$ and $0<{\varepsilon}\le 1$: $$|I(\xi)-I_{\varepsilon}(\xi)|=\Big|\int_0^{\varepsilon}G_k(\xi,s)s^{\zeta+k-1}{\,\mathrm{d}}s\Big|\le C_K \frac{{\varepsilon}^{{\operatorname{Re}}\zeta+k}}{{\operatorname{Re}}\zeta+k}.$$ Since ${\operatorname{Re}}\zeta+k>0$, this shows the normal convergence of the analytic functions $I_{\varepsilon}$ towards $I$. According to [@dettman Theorem 4.2.3] this implies that $I(\xi)$ is analytic in $\Omega_{\beta/2}$.
Now it remains to determine the constants $C_\pm$ in (\[tf\]). If we require $f\in{\mathcal E}_k$, it is necessary that $\tilde f$ is analytic in $\Omega_{\beta/2}$ and has a zero of order not less than $k$ at $\xi=0$. As already shown, $I(\xi)$ is analytic in $\Omega_{\beta/2}$. Furthermore, for $g\in{\mathcal E}_k$ and all (fixed) $s\in[0,1]$, $\xi\mapsto G_k(\xi,s)$ has a zero of order not less than $k$ at $\xi=0$. Therefore $I(\xi)=\int_0^1G_k(\xi,s)s^{\zeta+k-1}{\,\mathrm{d}}s$ has the same property, so ${\mathcal F}^{-1}I\in{\mathcal E}_k$. Thus, it is sufficient to consider the term $C_\pm\xi^{-\zeta}$. If $\zeta\notin-{\mathbb{N}}_0$, then $\xi^{-\zeta}$ is not analytic in $\Omega_{\beta/2}$ anyway, hence $C_+=C_-=0$. If $\zeta\in\{-k+1,\ldots,-1\}$ for $g\in{\mathcal E}_k$, $\xi^{-\zeta}$ is analytic, and we obtain $C_+=C_-$ because we require continuity of the solution. But the order of the zero of $\xi^{-\zeta}$ is at most $k-1$. Since we need a zero of at least order $k$, we again obtain $C_+=C_-=0$. The conclusion of the above analysis is summarized in the following proposition:
\[f\_trafo\_reso\] Let $g\in{\mathcal E}_k$ for some $k\in{\mathbb{N}}_0$, and ${\operatorname{Re}}\zeta>{-k}$. Then the unique $f\in{\mathcal E}_k$ with $f=R_{{\mathcal L}+\Theta}(\zeta)g$ satisfies $$\hat f(\xi)=\hat f_0(\xi)\int_0^1\frac{\hat g(s\xi)}{\hat f_0(s\xi)}s^{\zeta-1}{\,\mathrm{d}}s,\quad\xi\in\Omega_{\beta/2}.$$
Deferred Proofs and Lemmata {#app:proof}
===========================
For $f\in{\mathcal E}$ there holds $f(x){\mathrm{e}}^{bx}\in L^2({\mathbb{R}})$ for all $b\in[-\frac\beta 2,\frac\beta2]$. Therefore $\hat f$ is analytic in $\Omega_{\beta/2}$ according to [@resi2 Theorem IX.13]. Due to part (b) of the proof of that theorem (see page 132 in [@resi2]), Result (\[analyt:ii\]) follows. We proceed to the proof of (\[analyt:i\]). If $f\in {\mathcal E}$ and $b\in [-\frac\beta 2,\frac\beta2]$, we clearly have $$\|f(x){\mathrm{e}}^{bx}\|_{L^2({\mathbb{R}})} \le \|f(x){\mathrm{e}}^{\frac\beta 2|x|}\|_{L^2({\mathbb{R}})} \le \sqrt 2\|f\|_\omega.$$ On the left hand side we insert the identity from and use Plancherel’s identity, which shows . Conversely, let us now assume that $\hat f$ is analytic in $\Omega_{\beta/2}$ and that holds. We shall now show that $f\in{\mathcal E}$. Due to these assumptions we conclude from [@resi2 Theorem IX.13] that $f(x){\mathrm{e}}^{bx}\in\ L^2({\mathbb{R}})$ for all $b\in (-\frac\beta 2,\frac\beta2)$. For these values of $b$ we may therefore use the representation from (\[analyt:ii\]). We insert it in and after applying Plancherel’s identity we get $$\label{sup_f_hat_e}
\sup_{\substack{|b|<\beta/2\\b\in{\mathbb{R}}}}\|f(x){\mathrm{e}}^{b|x|}\|_{L^2({\mathbb{R}})}<\infty.$$ But this is only possible if $f\in{\mathcal E}$, otherwise the supremum in would not be finite.
Finally we show (\[analyt:iii\]). For $f\in{\mathcal E}$ there holds $f(x){\mathrm{e}}^{\pm\frac \beta 2x}\in L^2({\mathbb{R}})$, and therefore $\xi\mapsto\hat f(\xi\pm{{\mathrm{i}}}\beta/2)$, as defined in (\[hatf\]), is again an element of $L^2({\mathbb{R}})$. With this definition we now show $b\mapsto\hat f(\cdot+{{\mathrm{i}}}b)\in C([-\beta/ 2,\beta /2];L^2({\mathbb{R}}))$. Due to Plancherel’s identity we may show equivalently that $b\mapsto f(x){\mathrm{e}}^{bx}$ is continuous in $L^2({\mathbb{R}})$. To this end we fix $b,b_0\in[-\beta/2,\beta/2]$, and we split the integral for any $R>0$: $$\begin{aligned}
\|f(x){\mathrm{e}}^{bx}-f(x){\mathrm{e}}^{b_0x}\|_{L^2({\mathbb{R}})}^2 = \int\displaylimits_{{\mathbb{R}}\backslash[-R,R]}\!\!|f(x)|^2({\mathrm{e}}^{b_0x}-{\mathrm{e}}^{bx})^2{\,\mathrm{d}}x+\int_{-R}^R|f(x)|^2({\mathrm{e}}^{b_0x}-{\mathrm{e}}^{bx})^2{\,\mathrm{d}}x\label{1_2}\end{aligned}$$ Now, for any ${\varepsilon}>0$ we can find some $R=R({\varepsilon})>0$ such that $\int_{{\mathbb{R}}\backslash[-R,R]}|f(x)|^2{\mathrm{e}}^{\beta|x|}{\,\mathrm{d}}x<{\varepsilon}$. So we get for the first integral (independent of $b,b_0$) $$\begin{aligned}
\int\displaylimits_{{\mathbb{R}}\backslash[-R,R]}\!\!|f(x)|^2({\mathrm{e}}^{b_0x}-{\mathrm{e}}^{bx})^2{\,\mathrm{d}}x &\le \int\displaylimits_{{\mathbb{R}}\backslash[-R,R]}\!\! |f(x)|^2{\mathrm{e}}^{2|x|\max\{|b|,|b_0|\}}{\,\mathrm{d}}x\\
&\le \int\displaylimits_{{\mathbb{R}}\backslash[-R,R]}\!\! |f(x)|^2{\mathrm{e}}^{\beta|x|}{\,\mathrm{d}}x<{\varepsilon}.\end{aligned}$$ The second integral in converges to zero, for any fixed $R>0$, as $b\to b_0$. Altogether $$\lim_{b\to b_0} \|f(x){\mathrm{e}}^{bx}-f(x){\mathrm{e}}^{b_0x}\|_{L^2({\mathbb{R}})}^2<{\varepsilon}.$$
According to Corollary \[cor\_dissip\] the operator $(L-1-\beta^2/2)|_{C_0^\infty({\mathbb{R}})}$ is dissipative, so it is closable (cf. [@pazy Theorem 1.4.5 (c)]), and so is $L|_{C_0^\infty({\mathbb{R}})}$. We define ${\mathcal L}:={\operatorname{cl}}_{\mathcal E}L|_{C_0^\infty({\mathbb{R}})}$, and the domain $D({\mathcal L})$ consists of all $f\in{\mathcal E}$ such that there exists some $h\in{\mathcal E}$ such that (for some $(f_n)_{n\in{\mathbb{N}}}\subset C_0^\infty({\mathbb{R}})$) $$\begin{cases}
\lim_{n\to\infty}\|f_n-f\|_\omega=0,\\
\lim_{n\to\infty}\|Lf_n-h\|_\omega=0.
\end{cases}$$ For such $f$ we have ${\mathcal L}f:= h=\mathfrak L f$. Therefore $D({\mathcal L})\subseteq\{f\in {\mathcal E}:\mathfrak Lf\in{\mathcal E}\}$. Since $\|\cdot\|_E$ is stronger than $\|\cdot\|_\omega$ we also have $D(L)\subset D({\mathcal L})$.
Finally we need to show that the above inclusion for the domain indeed is an equality. We take $\zeta\in{\mathbb{C}}$ with ${\operatorname{Re}}\zeta\ge 1+\beta^2/2$. From Theorem \[prop:fp\_in\_H\] and the dissipativity of $\zeta-{\mathcal L}$ we know that $(\zeta-{\mathcal L})^{-1}|_E=(\zeta-L)^{-1}$ is a well-defined operator on $E$. And from we conclude that this is even a bounded operator in ${\mathcal E}$ with dense domain $E$. Therefore, also its closure ${\operatorname{cl}}_{\mathcal E}( (\zeta-{\mathcal L})^{-1}|_E)=(\zeta-{\mathcal L})^{-1}$ is bounded in ${\mathcal E}$, and therefore $\zeta\in\rho({\mathcal L})$. Now assume that there is some $f\in{\mathcal E}\backslash D({\mathcal L})$ such that $\mathfrak Lf\in{\mathcal E}$. Because $\zeta\in\rho({\mathcal L})$, $\zeta-{\mathcal L}:D({\mathcal L})\to{\mathcal E}$ is a bijection, and therefore there exists a unique $\mathfrak f\in D({\mathcal L})$ with $(\zeta-{\mathcal L})\mathfrak f=(\zeta-\mathfrak L)f$, which is equivalent to the existence of $\mathfrak f^\star\in{\mathcal E}$ with $\mathfrak f^\star\neq 0$ such that $(\zeta-\mathfrak L)\mathfrak f^\star=0$. But according to this is impossible.
\[lem:proj\] Consider two Hilbert spaces $X{\hookrightarrow}{\mathcal X}$, and a projection ${\mathrm{P}\!}_{\mathcal X}\in{\mathscr B}({\mathcal X})$, such that ${\mathrm{P}\!}_X:={\mathrm{P}\!}_{\mathcal X}|_X\in{\mathscr B}(X)$. Then ${\operatorname{ran}}{\mathrm{P}\!}_{\mathcal X}={\operatorname{cl}}_{\mathcal X}{\operatorname{ran}}{\mathrm{P}\!}_X$ and $\ker{\mathrm{P}\!}_{\mathcal X}={\operatorname{cl}}_{\mathcal X}\ker{\mathrm{P}\!}_X$.
We give here the proof of the equality of the ranges, the other identity can be shown analogously, using the complementary projections instead. On the one hand we have ${\operatorname{ran}}{\mathrm{P}\!}_X\subseteq{\operatorname{ran}}{\mathrm{P}\!}_{\mathcal X}$, and so ${\operatorname{cl}}_{\mathcal X}{\operatorname{ran}}{\mathrm{P}\!}_X\subseteq {\operatorname{ran}}{\mathrm{P}\!}_{\mathcal X}$, since ${\operatorname{ran}}{\mathrm{P}\!}_{\mathcal X}$ is closed in ${\mathcal X}$ due to the boundedness of ${\mathrm{P}\!}_{\mathcal X}$. On the other hand ${\mathrm{P}\!}_{\mathcal X}={\operatorname{cl}}_{\mathcal X}{\mathrm{P}\!}_X$, which implies ${\operatorname{ran}}{\mathrm{P}\!}_{\mathcal X}\subseteq{\operatorname{cl}}_{\mathcal X}{\operatorname{ran}}{\mathrm{P}\!}_X$.
\[lem:funct\] Let $X{\hookrightarrow}{\mathcal X}$ be Hilbert spaces, and $\psi_0,\ldots,\psi_{k-1}\in{\mathscr B}({\mathcal X},{\mathbb{C}}),\,k\in{\mathbb{N}}$, be linearly independent functionals. Then $\tilde \psi_j:=\psi_j|_X\in{\mathscr B}(X,{\mathbb{C}})$ for all $0\le j\le k-1$, and $$\bigcap_{j=0}^{k-1}\ker\psi_j={\operatorname{cl}}_{\mathcal X}\bigcap_{j=0}^{k-1}\ker\tilde\psi_j.$$
The boundedness of the $\tilde \psi_j$ is an immediate consequence of $X{\hookrightarrow}{\mathcal X}$. In order to show the second statement, we notice that according to the Riesz representation theorem there exists a unique $x_j\in X$ such that $\tilde\psi_j(\cdot)={\langle}\cdot,x_j{\rangle}_X$ for every $0\le j\le k-1$, where ${\langle}\cdot,\cdot{\rangle}_X$ denotes the inner product in $X$. The set $\{x_0,\ldots,x_{k-1}\}$ is linearly independent, because the corresponding functionals are. We now apply the Gram-Schmidt process to $\{x_0,\ldots,x_{k-1}\}$ to obtain the orthonormal family $\{\hat x_0,\ldots,\hat x_{k-1}\}$ with same linear hull. As a consequence, there exists a regular matrix $\Lambda:=(\lambda_\ell^j)_{\ell,j}\in{\mathbb{C}}^{k\times k}$ such that $\hat x_\ell=\sum_{j=0}^{k-1}\lambda_\ell^jx_j$. With this we get $$\hat x_\ell{\langle}\cdot,\hat x_\ell{\rangle}_X=\sum_{i,j=0}^{k-1}\lambda_\ell^i\bar\lambda_\ell^j x_i{\langle}\cdot,x_j{\rangle}_X=\sum_{i,j=0}^{k-1}\lambda_\ell^i\bar\lambda_\ell^j x_i\tilde\psi_j(\cdot),\quad 0\le \ell\le k-1.$$ We may now define the orthogonal projection $$\label{wasweis}
{\mathrm{P}\!}_X:=\sum_{\ell=0}^{k-1}\hat x_\ell{\langle}\cdot,\hat x_\ell{\rangle}_X=\sum_{i,j,\ell=0}^{k-1}\lambda_\ell^i\bar\lambda_\ell^j x_i\tilde\psi_j(\cdot).$$ It can naturally be extended to a projection ${\mathrm{P}\!}_{\mathcal X}$ in ${\mathcal X}$ by replacing the $\tilde \psi_j$ by $\psi_j$. Since $\psi_j\in{\mathscr B}({\mathcal X},{\mathbb{C}})$ for all $0\le j\le k-1$, there follows ${\mathrm{P}\!}_{\mathcal X}\in{\mathscr B}({\mathcal X})$ from (\[wasweis\]). Now we apply Lemma \[lem:proj\] to ${\mathrm{P}\!}_X\subset{\mathrm{P}\!}_{\mathcal X}$ to obtain $\ker{\mathrm{P}\!}_{\mathcal X}={\operatorname{cl}}_{\mathcal X}\ker{\mathrm{P}\!}_X$.
Now it remains to characterize the kernels of the projections. Due to (\[wasweis\]) we have ${\mathrm{P}\!}_X f=0$ in $X$ iff $$\label{sys_eq:psi}
\sum_{j=0}^{k-1}\tilde\psi_j(f)\sum_{\ell=0}^{k-1}\lambda_\ell^i\bar\lambda_\ell^j=0,\quad 0\le i\le k-1,$$ since the vectors $x_i$ are linearly independent. We note that the sums $\sum_{\ell=0}^{k-1}\lambda_\ell^i\bar\lambda_\ell^j$ for $0\le i,j\le k-1$ are the elements of the matrix $\Lambda_2:=\Lambda\Lambda^*,$ where $\Lambda^*$ is the Hermitian conjugate of $\Lambda$. Since $\Lambda_2$ is regular, it follows that (\[sys\_eq:psi\]) holds iff $\tilde\psi_j(f)=0$ for all $0\le j\le k-1.$ The proof of ${\mathrm{P}\!}_{\mathcal X}f=0$ iff $\psi_j(f)=0$ for all $0\le j\le k-1$ is analogous.
We only consider the situation $|{\mathbf{k}}|=1$, the estimate for higher derivatives follows by repeated application of that result. Without loss of generality we assume ${\mathbf{k}}=(1,0,\ldots,0)$ for the proof. For the norm we use the equivalent weight $\omega_\star({\mathbf{x}})=\cosh\beta x_1 \cosh\beta x_2 \allowbreak\cdots\allowbreak \cosh\beta x_d$. In this context we write $\hat {\mathbf{x}}_1:=(x_2,\ldots,x_d)$ and $\omega_\star(\hat {\mathbf{x}}_1):=\omega_\star({\mathbf{x}})/\cosh\beta x_1$. By applying the one-dimensional Poincaré inequality we obtain for $f\in C_0^\infty({\mathbb{R}}^d)$: $$\begin{aligned}
\|f\|^2_{\omega_\star} &= \int_{{\mathbb{R}}^d}|f({\mathbf{x}})|^2 \omega_\star({\mathbf{x}}){\,\mathrm{d}}{\mathbf{x}}\nonumber\\
&=\int_{\mathbb{R}}\Big[\Big(\int_{{\mathbb{R}}^{d-1}} |f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12}\Big]^2\cosh\beta x_1 {\,\mathrm{d}}x_1\nonumber\\
&\le C_\beta\int_{\mathbb{R}}\Big[{\frac{\partial^{}{}}{\partial{x_1}^{}}}\Big(\int_{{\mathbb{R}}^{d-1}} |f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12}\Big]^2\cosh\beta x_1 {\,\mathrm{d}}x_1.\label{duck}\end{aligned}$$ For the inner integral we compute $$\begin{aligned}
{\frac{\partial^{}{}}{\partial{x_1}^{}}}\Big(\int_{{\mathbb{R}}^{d-1}} |f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12} & = \frac12\cdot\frac{\int_{{\mathbb{R}}^{d-1}} {\frac{\partial^{}{}}{\partial{x_1}^{}}}|f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1}{\Big(\int_{{\mathbb{R}}^{d-1}} |f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12}}\\
&\le \frac{\Big(\int_{{\mathbb{R}}^{d-1}} |f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12}\Big(\int_{{\mathbb{R}}^{d-1}} |{\frac{\partial^{}{}}{\partial{x_1}^{}}}f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12}}{\Big(\int_{{\mathbb{R}}^{d-1}} |f({\mathbf{x}})|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12}}\\
&= \Big(\int_{{\mathbb{R}}^{d-1}} \Big|{\frac{\partial^{}{}}{\partial{x_1}^{}}}f({\mathbf{x}})\Big|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)^{\frac 12}.\end{aligned}$$ Inserting this in we conclude $$\|f\|^2_{\omega_\star} \le C_\beta \int_{\mathbb{R}}\Big(\int_{{\mathbb{R}}^{d-1}} \Big|{\frac{\partial^{}{}}{\partial{x_1}^{}}}f({\mathbf{x}})\Big|^2 \omega_\star(\hat {\mathbf{x}}_1){\,\mathrm{d}}\hat{\mathbf{x}}_1\Big)\cosh\beta x_1{\,\mathrm{d}}x_1=C_\beta\Big\|{\frac{\partial^{}{f}}{\partial{x_1}^{}}}\Big\|_{\omega_\star}^2.$$
Acknowledgement {#acknowledgement .unnumbered}
---------------
The authors were supported by the FWF (project I 395-N16 and the doctoral school “Dissipation and dispersion in non-linear partial differential equations”) and the ÖAD-project “Long-time asymptotics for evolution equations in chemistry and biology”.
[^1]: One of the best-known applications of ladder operators occurs in the spectral analysis of the quantum harmonic oscillator, see e.g. [@helffer2002].
|
---
abstract: 'We introduce and study coordinate-wise powers of subvarieties of ${\mathbb{P}}^n$, i.e. varieties arising from raising all points in a given subvariety of ${\mathbb{P}}^n$ to the $r$-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of ${\mathbb{P}}^n$ under the quotient of ${\mathbb{P}}^n$ by the action of the finite group ${\mathbb{Z}}_r^{n+1}$. We determine the degree of coordinate-wise powers and study their defining equations, in particular for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.'
address:
- 'Indian Statistical Institute (ISI) Kolkata'
- 'Max-Planck Institute *Mathematics in the Sciences* Leipzig'
- 'Department of Mathematical Sciences, University of Copenhagen'
author:
- Papri Dey
- Paul Görlach
- Nidhi Kaihnsa
title: 'Coordinate-wise Powers of Algebraic Varieties'
---
Introduction
============
Recently, Hadamard products of algebraic varieties have been attracting attention of geometers. These are subvarieties $X \star Y$ of projective space ${\mathbb{P}}^n$ that arise from multiplying coordinate-by-coordinate any two points $x \in X$, $y \in Y$ in given subvarieties $X,Y$ of ${\mathbb{P}}^n$. In applications, they first appeared in [@CMS], where the variety associated to the restricted Boltzmann machine was described as a repeated Hadamard product of the secant variety of ${\mathbb{P}}^1 \times \ldots \times {\mathbb{P}}^1
\subset {\mathbb{P}}^{2^n-1}$ with itself. Further study in [@BCK], [@FOW], [@BCFL], [@CCFL] made progress towards understanding Hadamard products. Of particular interest are the $r$-th Hadamard powers $X^{\star r} := X \star \ldots \star X$ of an algebraic variety $X \subset {\mathbb{P}}^n$. They are the multiplicative analogue of secant varieties that play a central role in classical projective geometry: The $r$-th secant variety $\sigma_r(X)$ is the closure of the set of coordinate-wise sums of $r$ points in $X$. Its subvariety corresponding to sums of $r$ equal points is the original variety $X$. In the multiplicative setting, the Hadamard power $X^{\star r}$ replaces $\sigma_r(X)$, but it does not typically contain $X$ if $[1:\ldots:1] \notin X$. As a multiplicative substitute for the inclusion $X \subset \sigma_r(X)$, it is natural to study the subvariety of $X^{\star r}$ given by coordinate-wise products of $r$ equal points in $X$. Formally, for a projective variety $X \subset {\mathbb{P}}^n$ and an integer $r\in{\mathbb{Z}}$ (possibly negative), we are interested in studying its image under the rational map $$\varphi_r \colon {\mathbb{P}}^n \dashrightarrow {\mathbb{P}}^n, \qquad [x_0:\ldots:x_n] \mapsto
[x_0^r:\ldots:x_n^r].$$ We call the image, $X^{\circ r}$, of $X$ under $\varphi_r$ the $r$-th [coordinate-wise power of $X \subset {\mathbb{P}}^n$]{}.
In this article, we investigate these coordinate-wise powers $X^{\circ r}$ with a main focus on the case $r>0$. These varieties show up naturally in many applications. For the Grassmannian variety $\operatorname{Gr}(k,{\mathbb{P}}^n)$ in its Plücker embedding, the intersection with its $r$-th coordinate-wise power $\operatorname{Gr}(k,{\mathbb{P}}^n) \cap
\operatorname{Gr}(k,{\mathbb{P}}^n)^{\circ r}$ was described combinatorially in terms of matroids in [@Len] for even $r$. In [@Bon], highly singular surfaces in ${\mathbb{P}}^3$ have been constructed as preimages of a specific singular surface under the morphism $\varphi_r$ for $r>0$. In the case $r=-1$, the map $\varphi_r$ is a classical Cremona transformation and images of varieties under this transformation are called [reciprocal varieties]{} whose study has received particular attention in the case of linear spaces, see [@LSV], [@KV] and [@FSW]. For $r > 0$, the coordinate-wise powers $X^{\circ r}$ of a variety $X \subset
{\mathbb{P}}^n$ have the following natural interpretation: The quotient of ${\mathbb{P}}^n$ by the finite subgroup ${\mathbb{Z}}_r^{n+1}$ of the torus $({\mathbb{C}}^*)^{n+1}$ is again a projective space. The image of a variety $X \subset {\mathbb{P}}^n$ in ${\mathbb{P}}^n/{\mathbb{Z}}_r^{n+1} \cong {\mathbb{P}}^n$ is the variety $X^{\circ r}$, since $\varphi_r
\colon {\mathbb{P}}^n \to {\mathbb{P}}^n$ is the geometric quotient of ${\mathbb{P}}^n$ by ${\mathbb{Z}}_r^{n+1}$. In other words, coordinate-wise powers of algebraic varieties are images of subvarieties of ${\mathbb{P}}^n$ under the quotient by a certain finite group. The case $r=2$ has the special geometric significance of quotienting by the group generated by reflections at the coordinate hyperplanes of ${\mathbb{P}}^n$. We are, therefore, especially interested in [coordinate-wise squares]{} of varieties.
A particular application of interest is the variety of orthostochastic matrices. An orthostochastic matrix is a matrix arising by squaring each entry of an orthogonal matrix. In other words, they are points in the coordinate-wise square of the variety of orthogonal matrices. Orthostochastic matrices play a central role in the theory of majorization [@MOA] and are closely linked to finding real symmetric matrices with prescribed eigenvalues and diagonal entries, see [@Hor] and [@Mir]. Recently, it has also been shown that studying the variety of orthostochastic matrices is central to the existence of determinantal representations of bivariate polynomials and their computation, see [@Dey].
As a further application, we show that coordinate-wise squares of linear spaces show up naturally in the study of symmetric matrices with a degenerate spectrum of eigenvalues.
The article is structured as follows: As customary when studying any variety, first and foremost, we compute the degree of $X^{\circ r}$. We use this to derive the degree of the variety of orthostochastic matrices. In Section \[sec:Hypersurfaces\], we dig a little deeper and find explicitly the defining equations of the coordinate-wise powers of hypersurfaces. We define [*generalised power sum hypersurfaces*]{} and give relations between their dual and reciprocal varieties. We study in more detail coordinate-wise powers of linear spaces in the final section. We show the dependence of the degree of the coordinate-wise powers of a linear space on the combinatorial information captured by the corresponding linear matroid. Particular attention is drawn to the case of coordinate-wise squares of linear spaces. For low-dimensional linear spaces we give a complete classification. We also describe the defining ideal for the coordinate-wise square of general linear spaces of arbitrary dimension in a high-dimensional ambient space, and we link this question to the study of symmetric matrices with a codimension 1 eigenspace.
#### **Acknowledgements**
The authors would like to thank Mateusz Michałek and Bernd Sturmfels for their guidance and suggestions. This work was initiated while the first author was visiting Max Planck Institute MiS Leipzig. The financial support by MPI Leipzig which made this visit possible is gratefully acknowledged. The second and third author were funded by the International Max Planck Research School [*Mathematics in the Sciences*]{} (IMPRS) during this project.
Degree formula {#sec:Degree}
==============
Throughout this article, we work over ${\mathbb{C}}$. We denote the homogeneous coordinate ring of ${\mathbb{P}}^n$ by ${\ensuremath{{\mathbb{C}}[\mathbf{x}]}}:=
{\mathbb{C}}[x_0,\ldots,x_n]$. For any integer $r \in {\mathbb{Z}}$, we consider the rational map $$\varphi_r \colon {\mathbb{P}}^n \dashrightarrow {\mathbb{P}}^n, \qquad [x_0:\ldots:x_n]
\mapsto [x_0^r:\ldots:x_n^r].$$
For $r \geq 0$, the rational map $\varphi_r$ is a morphism. Throughout, let $X \subset {\mathbb{P}}^n$ be a projective variety, not necessarily irreducible. We denote by $X^{\circ r} \subset {\mathbb{P}}^n$ the image of $X$ under the rational map $\varphi_r$. More explicitly, $$X^{\circ r} := \begin{cases}
\overline{\varphi_r(X \setminus V(x_0 x_1 \ldots x_n))} &\text{if } r <
0, \\
\varphi_r(X) &\text{if } r\geq 0. \\
\end{cases}$$ For $r < 0$, we will only consider the case that no irreducible component of $X$ is contained in any coordinate hyperplane of ${\mathbb{P}}^n$. We call the image $X^{\circ r} \subset {\mathbb{P}}^n$ the [*$r$-th coordinate-wise power*]{} of $X$. In the case $r=-1$, the variety $X^{\circ (-1)}$ is called the [*reciprocal variety*]{} of $X$. We primarily focus on positive coordinate-wise powers in this article, and therefore we will from now on always assume $r > 0$ unless explicitly stated otherwise.
Observe that $\varphi_r \colon {\mathbb{P}}^n \to {\mathbb{P}}^n$ is a finite morphism, and hence, the image $X^{\circ r}$ of $X$ under $\varphi_r$ has same dimension as $X$.
The cyclic group ${\mathbb{Z}}_r$ of order $r$ is identified with the group of $r$-th roots of unity $\{\xi \in {\mathbb{C}}\mid \xi^r = 1\}$. We consider the action of the $(n+1)$-fold product ${\mathbb{Z}}_r^{n+1} := {\mathbb{Z}}_r\times \ldots \times {\mathbb{Z}}_r$ on ${\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ given by rescaling the variables $x_0, \ldots, x_n$ with $r$-th roots of unity. We denote the quotient of ${\ensuremath{{\mathbb{Z}}_r^{n+1}}}$ by the subgroup $\{(\xi,\xi,\ldots,\xi) \in
{\mathbb{C}}^r \mid \xi^r = 1\} \subset {\ensuremath{{\mathbb{Z}}_r^{n+1}}}$ as ${\ensuremath{\mathcal{G}_{r}}}:= {\ensuremath{{\mathbb{Z}}_r^{n+1}}}/{\mathbb{Z}}_r$. The group action of ${\ensuremath{{\mathbb{Z}}_r^{n+1}}}$ on ${\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ determines a linear action of ${\ensuremath{\mathcal{G}_{r}}}$ on ${\mathbb{P}}^n$. In this way, we can also view ${\ensuremath{\mathcal{G}_{r}}}$ as a subgroup of $\operatorname{Aut}({\mathbb{P}}^n)$. For $r=2$, this has the geometric interpretation of being the linear group action generated by reflections at coordinate hyperplanes. Note that ${\ensuremath{\mathcal{G}_{r}}}$ does not act on the vector space ${\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d$ of homogeneous polynomials of degree $d$, instead it acts on ${\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$.
Given a projective variety, the following proposition describes set-theoretically the preimage under $\varphi_r$ of its coordinate-wise $r$-th power.
\[prop:preimage\] Let $X \subset {\mathbb{P}}^n$ be a variety and let $X^{\circ r} \subset {\mathbb{P}}^n$ be its coordinate-wise $r$-th power. The preimage $\varphi_r^{-1}(X^{\circ r})$ is given by $\bigcup_{\tau\in {\ensuremath{\mathcal{G}_{r}}}}\tau\cdot X$.
This follows from $X^{\circ r} = \varphi_r(X)$ and the fact that $\varphi_r^{-1}(\varphi_r(p))=\{\tau\cdot p \mid \tau\in {\ensuremath{\mathcal{G}_{r}}}\}$ for all $p\in
X$.
In particular, for $r=2$, we obtain the following geometric description.
The preimage of $X^{\circ 2}$ under $\varphi_2 \colon {\mathbb{P}}^n \to {\mathbb{P}}^n$ is the union over the orbit of $X$ under the subgroup of $\operatorname{Aut}({\mathbb{P}}^n)$ generated by the reflections in the coordinate hyperplanes.
In the following theorem, we give a degree formula for the coordinate-wise powers of an irreducible variety.
\[prop:DegreeFormula\] Let $X \subset {\mathbb{P}}^n$ be an irreducible projective variety. Let $\operatorname{Stab}_r(X) := \{\tau \in {\ensuremath{\mathcal{G}_{r}}}\mid \tau \cdot X = X\}$ and $\operatorname{Fix}_r(X) := \{\tau \in {\ensuremath{\mathcal{G}_{r}}}\mid {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\tau
{\aftergroup\egroup\originalright}|_{X}
}} = \operatorname{id}_X\}$. Then the degree of the $r$-th coordinate-wise power of $X$ is $$\deg X^{\circ r} = \frac{|\operatorname{Fix}_r(X)|}{|\operatorname{Stab}_r(X)|} \: r^{\dim X} \deg
X.$$
Let $H_1, \ldots, H_k \subset {\mathbb{P}}^n$ for $k := \dim X^{\circ r} = \dim X$ be general hyperplanes whose common intersection with $X^{\circ r}$ consists of finitely many reduced points. We want to determine $|X^{\circ r} \cap
\bigcap_{i=1}^k H_i|$. By \[prop:preimage\], we have $$\varphi_r^{-1}{\mathopen{}\mathclose\bgroup\originalleft}(X^{\circ r} \cap \bigcap_{i=1}^k H_i{\aftergroup\egroup\originalright}) =
\bigcup_{\tau \in {\ensuremath{\mathcal{G}_{r}}}} \tau \cdot {\mathopen{}\mathclose\bgroup\originalleft}(X \cap \bigcap_{i=1}^k
\varphi_r^{-1} H_i{\aftergroup\egroup\originalright}).$$ Note that each $\varphi_r^{-1} H_i$ is a hypersurface of degree $r$ fixed under the ${\ensuremath{\mathcal{G}_{r}}}$-action, and their common intersection with $X$ consists of finitely many reduced points by Bertini’s theorem (as in [@FOV99 3.4.8]). By Bézout’s theorem, ${\mathopen{}\mathclose\bgroup\originalleft}|X \cap \bigcap_{i=1}^k \varphi_r^{-1} H_i{\aftergroup\egroup\originalright}| = r^k \deg X$. We note that $Z := X \cap {\mathopen{}\mathclose\bgroup\originalleft}( \bigcup_{\tau \in {\ensuremath{\mathcal{G}_{r}}}\setminus \operatorname{Stab}_r(X)}
\tau \cdot X {\aftergroup\egroup\originalright})$ is of dimension $<k$ by irreducibility of $X$. Therefore, the common intersection of $k$ general hyperplanes $H_i$ with $\varphi_r(Z)$ is empty. This implies that the intersection of $\tau \cdot X$ and $\tau' \cdot X$ does not meet $\bigcap_{i=1}^k \varphi_r^{-1} H_i$ for all $\tau, \tau' \in {\ensuremath{\mathcal{G}_{r}}}$ with $\tau' \cdot \tau^{-1} \notin \operatorname{Stab}_r(X)$. Hence, the above can be written as a disjoint union $$\bigcup_{\tau \in {\ensuremath{\mathcal{G}_{r}}}} \tau \cdot {\mathopen{}\mathclose\bgroup\originalleft}(X \cap \bigcap_{i=1}^k
\varphi_r^{-1} H_i{\aftergroup\egroup\originalright}) = \bigsqcup_{j=1}^s
\tau_j \cdot {\mathopen{}\mathclose\bgroup\originalleft}(X \cap \bigcap_{i=1}^k \varphi_r^{-1} H_i{\aftergroup\egroup\originalright}),$$ where $\tau_1, \ldots, \tau_s \in {\ensuremath{\mathcal{G}_{r}}}$ for $s = |{\ensuremath{\mathcal{G}_{r}}}|/|\operatorname{Stab}_r(X)|$ represent the cosets of $\operatorname{Stab}_r(X)$ in ${\ensuremath{\mathcal{G}_{r}}}$.
In particular, $${\mathopen{}\mathclose\bgroup\originalleft}|\varphi_r^{-1}{\mathopen{}\mathclose\bgroup\originalleft}(X^{\circ r} \cap \bigcap_{i=1}^k
H_i{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright}| =
\frac{|{\ensuremath{\mathcal{G}_{r}}}|}{|\operatorname{Stab}_r(X)|} \: r^k \deg X.$$ For a general point $p \in X$, we have $\{\tau \in {\ensuremath{\mathcal{G}_{r}}}\mid \tau \cdot p = p\} = \operatorname{Fix}_r(X)$. Then \[prop:preimage\] shows that a general point of $X^{\circ r} = \varphi_r(X)$ has $|{\ensuremath{\mathcal{G}_{r}}}|/|\operatorname{Fix}_r(X)|$ preimages under $\varphi_r$, so for general hyperplanes $H_i$ we conclude $$\deg X^{\circ r} = {\mathopen{}\mathclose\bgroup\originalleft}|X^{\circ r} \cap \bigcap_{i=1}^k H_i{\aftergroup\egroup\originalright}| =
\frac{|\operatorname{Fix}_r(X)|}{|{\ensuremath{\mathcal{G}_{r}}}|}
\:
{\mathopen{}\mathclose\bgroup\originalleft}|\varphi_r^{-1}{\mathopen{}\mathclose\bgroup\originalleft}(X^{\circ r} \cap \bigcap_{i=1}^k
H_i{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright}| =
\frac{|\operatorname{Fix}_r(X)|}{|\operatorname{Stab}_r(X)|} \: r^k \deg X.$$
Orthostochastic matrices {#ssec:Orthostochastic}
------------------------
We use \[prop:DegreeFormula\] to compute the degree of the variety of orthostochastic matrices. By ${\ensuremath{\mathbb{O}(m)}} \subset {\mathbb{P}}^{m^2}$ (resp. ${\ensuremath{\mathbb{SO}(m)}}
\subset {\mathbb{P}}^{m^2}$) we mean the projective closure of the affine variety of orthogonal (resp. special orthogonal) matrices in ${\mathbb{A}}^{m^2}$. It was shown in [@Dey] that the problem of deciding whether a bivariate polynomial can be expressed as the determinant of a definite/monic symmetric linear matrix polynomial (a [determinantal representation]{}) is closely linked to the problem of finding the defining equations of the variety ${\ensuremath{\mathbb{O}(m)}}^{\circ 2}$. In the case $m=3$, the defining equations of ${\ensuremath{\mathbb{O}(3)}}^{\circ 2}$ are known [@CD Proposition 3.1] and based on this knowledge, it was shown in [@DeyPhD Section 4.2] how to compute a determinantal representation for a cubic bivariate polynomial or decide that none exists. For arbitrary $m$, the ideal of defining equations may be very complicated, but we are still able to compute its degree:
\[prop:DegreeOrthIm\] We have ${\ensuremath{\mathbb{O}(m)}}^{\circ 2} = {\ensuremath{\mathbb{SO}(m)}}^{\circ 2}$ and its degree is $$\deg {\ensuremath{\mathbb{O}(m)}}^{\circ 2} = 2^{(m-1)^2} \;
\frac{\deg{\ensuremath{\mathbb{O}(m)}}}{2^{\binom{m+1}{2}}} \leq 2^{(m-1)^2}.$$
The variety ${\ensuremath{\mathbb{O}(m)}}$ consists of two connected components that are isomorphic to ${\ensuremath{\mathbb{SO}(m)}}.$ The images of these components under $\varphi_2
\colon {\mathbb{P}}^{m^2} \to {\mathbb{P}}^{m^2}$ coincide. In particular, ${\ensuremath{\mathbb{O}(m)}}^{\circ 2} =
{\ensuremath{\mathbb{SO}(m)}}^{\circ 2}$ and $\deg {\ensuremath{\mathbb{O}(m)}} = 2 \deg {\ensuremath{\mathbb{SO}(m)}}$. We determine $\operatorname{Fix}_2({\ensuremath{\mathbb{SO}(m)}})$ and $\operatorname{Stab}_2({\ensuremath{\mathbb{SO}(m)}})$.
Identify elements of ${\ensuremath{\mathcal{G}_{2}}}$ with $m\times m$-matrices whose entries are $\pm
1$. Then a group element $S \in {\ensuremath{\mathcal{G}_{2}}} = \{\pm 1\}^{m \times m}$ acts on the affine open subset ${\mathbb{A}}^{m^2} \subset {\mathbb{P}}^{m^2}$ corresponding to $m \times
m$-matrices $M \in {\mathbb{C}}^{m\times m}$ as $S \circ M$, where $S \circ M$ denotes the Hadamard product (i.e. entry-wise product) of matrices. Clearly, $\operatorname{Fix}_2({\ensuremath{\mathbb{SO}(m)}})$ is trivial, or else every special orthogonal matrix would need to have a zero entry at a certain position.
We claim that $\operatorname{Stab}_2({\ensuremath{\mathbb{SO}(m)}}) \subset \{S \in \{\pm 1\}^{m \times m} \mid
\operatorname{rk}S = 1\}$. Indeed, assume that $S \in \{\pm 1\}^{m \times m}$ lies in $\operatorname{Stab}_2({\ensuremath{\mathbb{SO}(m)}})$, but is not of rank $1$. Then $m \geq 2$ and we may assume that the first two columns of $S$ are linearly independent. Consider the vectors $u,v \in {\mathbb{C}}^m$ given by $$u_i := \begin{cases} 1 &\text{if } i < m, \\ -1 &\text{if } i=m \end{cases}
\qquad \text{and} \qquad
v_i := \begin{cases} 2^{i-1} &\text{if } i < m, \\ 2^{m-1}-1 &\text{if } i=m \end{cases} \qquad \text{for all $i \in \{1, \ldots,m\}.$}$$ Since $u$ and $v$ are orthogonal, we can find a special orthogonal matrix $M
\in {\mathbb{C}}^{m \times m}$ whose first two columns are $M_{\bullet 1}
= u/\|u \|_2$ and $M_{\bullet 2} = v/\| v \|_2$. But $S \in
\operatorname{Stab}_2({\ensuremath{\mathbb{SO}(m)}})$, so the matrix $S \circ M$ must be a special orthogonal matrix. In particular, the first two columns of $S \circ M$ must be orthogonal, i.e. $$\label{eq:powersOf2}
0 = \sum_{i=1}^m (S_{i1}u_i) (S_{i2} v_i) = -(S_{m1} S_{m2}) (2^{m-1}-1) +
\sum_{i=1}^{m-1} (S_{i1} S_{i2}) 2^{i-1}.$$ Since $S_{i1} S_{i2} = \pm 1$ for all $i$, we have $|\sum_{i=1}^{m-1} (S_{i1}
S_{i2}) 2^{i-1}| \leq 2^{m-1}-1$, and equality in holds if and only if $S_{i1} S_{i2} = S_{j1} S_{j2}$ for all $i,j \in \{1,\ldots,m\}$. However, this contradicts the linear independence of the first two columns of $S$. Hence, the claim follows.
Any rank 1 matrix in $\{\pm 1\}^{m\times m}$ can be uniquely written as $u
v^T$ with $u,v \in \{\pm 1\}^m$ and $u_1=1$. Such a rank 1 matrix $S = uv^T$ lies in $\operatorname{Stab}_2({\ensuremath{\mathbb{SO}(m)}})$ if and only if for each special orthogonal matrix $M \in {\mathbb{C}}^{m \times m}$ the matrix $$S \circ M = (uv^T) \circ M = \operatorname{diag}(u_1,\ldots,u_m) \: M \:
\operatorname{diag}(v_1,\ldots,v_m)$$ is again a special orthogonal matrix. This is true if and only if $\prod_{i=1}^m u_i = \prod_{i=1}^m v_i$. Therefore, $$\begin{aligned}
\operatorname{Stab}_2({\ensuremath{\mathbb{SO}(m)}}) &= \{u v^T \mid u,v \in \{\pm 1\}^m, \: u_1=1, \:
{\textstyle \prod_i u_i = \prod_i v_i}\},
\end{aligned}$$ and, thus, $|\operatorname{Stab}_2({\ensuremath{\mathbb{SO}(m)}})| = 2^{2m-2}$.
Since ${\ensuremath{\mathbb{SO}(m)}} \subset {\mathbb{P}}^{m^2}$ is irreducible, applying \[prop:DegreeFormula\] gives $$\deg {\ensuremath{\mathbb{SO}(m)}}^{\circ 2} = \frac{1}{2^{2m-2}} \: 2^{\binom{m}{2}} \deg
{\ensuremath{\mathbb{SO}(m)}} = 2^{\binom{m}{2}-2m+1} \deg {\ensuremath{\mathbb{O}(m)}}
= 2^{(m-1)^2} \; \frac{\deg{\ensuremath{\mathbb{O}(m)}}}{2^{\binom{m+1}{2}}}.$$
Finally, we observe that the affine variety of orthogonal matrices in ${\mathbb{A}}^{m^2}$ is an intersection of $\smash{\binom{m+1}{2}}$ quadrics which correspond to the polynomials given by the equation $M^T M =
\operatorname{id}$ satisfied by orthogonal matrices $M \in {\mathbb{C}}^{m \times m}$. Therefore, its projective closure ${\ensuremath{\mathbb{O}(m)}}
\subset {\mathbb{P}}^{m^2}$ must satisfy $\smash{\deg {\ensuremath{\mathbb{O}(m)}} \leq
2^{\binom{m+1}{2}}}$. This shows $\deg {\ensuremath{\mathbb{O}(m)}}^{\circ 2} \leq 2^{(m-1)^2}$.
\[rem:DegreeOfOrth\] The degree of ${\ensuremath{\mathbb{O}(m)}}$ (resp. ${\ensuremath{\mathbb{SO}(m)}}$) is known for all $m$ by [@BBBKR], namely $$\deg {\ensuremath{\mathbb{O}(m)}} = 2^m \det {\mathopen{}\mathclose\bgroup\originalleft}(\: \binom{2m-2i-2j}{m-2i} \:{\aftergroup\egroup\originalright})_{1\leq
i,j\leq \lfloor m/2 \rfloor}.$$ Table \[tbl:DegreesOrthIm\] shows the resulting degrees of ${\ensuremath{\mathbb{O}(m)}}^{\circ 2} = {\ensuremath{\mathbb{SO}(m)}}^{\circ 2}$ for some values of $m$.
Linear spaces
-------------
We now determine the degree of coordinate-wise powers $L^{\circ r}$ for a linear space $L \subset {\mathbb{P}}^n$, based on \[prop:DegreeFormula\]. It can be expressed in terms of the combinatorics captured by the matroid of $L \subset
{\mathbb{P}}^n$. We briefly recall some basic definitions for matroids associated to linear spaces in ${\mathbb{P}}^n$. We refer to [@Oxl] for a detailed introduction to matroid theory.
Let $L \subset {\mathbb{P}}^n$ be a linear space. The combinatorial information about the intersection of $L$ with the linear coordinate spaces in ${\mathbb{P}}^n$ is captured in the [linear matroid]{} $\mathcal M_L$. It is the collection of index sets $I \subset \{0,1,\ldots,n\}$ such that $L$ does not intersect $V(\{x_i \mid i \notin I\})$. Formally, $$\mathcal M_L := \{I \subset \{0,1,\ldots,n\} \; \mid \; L \cap V(\{x_i \mid
i \notin I\}) = \emptyset\}.$$ Different conventions about linear matroids exist in the literature, and some authors take a dual definition for the linear matroid of $L$.
The set $\{0,1,\ldots,n\}$ is the [ground set]{} of the matroid. Index sets $I \in \mathcal M_L$ are called [independent]{}, while index sets $I \in \operatorname{Pow}(\{0,1,\ldots,n\})
\setminus \mathcal M_L$ are called dependent. An index $i \in \{0,1,\ldots,n\}$ is called a [coloop]{} of $\mathcal M_L$ if, for all $I \subset
\{0,1,\ldots,n\}$, the condition $I \in \mathcal M_L$ holds if and only if $I
\cup \{i\} \in \mathcal M_L$ holds. Geometrically, an index $i \in
\{0,1,\ldots,n\}$ is a coloop of $\mathcal M_L$ if and only if $L \subset
V(x_i)$.
A subset $E \subset \{0,1,\ldots,n\}$ is called [irreducible]{} if there is no non-trivial partition $E = E_1
\sqcup E_2$ with $$I \in \mathcal M_L \quad \Leftrightarrow \quad I\cap E_1 \in \mathcal M_L
\text{ and }
I\cap E_2 \in \mathcal M_L \qquad \forall I \subset E.$$ The maximal irreducible subsets of $\{0,1,\ldots,n\}$ are called [components]{} of $\mathcal M_L$ and they form a partition of $\{0,1,\ldots,n\}$. Geometrically, a component of $\mathcal M_L$ is a minimal non-empty subset of $\{0,1,\ldots,n\}$ with the property that $L \cap V(x_i \mid
i \in I)$ and $L \cap V(x_i \mid i \notin I)$ together span the linear space $L$. In the following result, we determine the degree of $L^{\circ r} \subset {\mathbb{P}}^n$ as an invariant of the linear matroid $\mathcal M_L$.
\[prop:LinearDegreeFormula\] Let $L \subset {\mathbb{P}}^n$ be a linear space of dimension $k$. Let $s$ be the number of coloops and $t$ the number of components of the associated linear matroid $\mathcal M_L$. Then $$\deg L^{\circ r} = r^{k + s - t + 1}.$$
By \[prop:DegreeFormula\], we need to determine the cardinality of the groups $$\operatorname{Stab}_r(L) = \{\tau \in {\ensuremath{\mathcal{G}_{r}}}\mid \tau \cdot L = L\} \qquad \text{and} \qquad
\operatorname{Fix}_r(L) =
\{\tau \in {\ensuremath{\mathcal{G}_{r}}}\mid {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\tau
{\aftergroup\egroup\originalright}|_{L}
}} = \operatorname{id}_L\}.$$ Consider the affine cone over $L$, which is a $(k+1)$-dimensional subspace $W \subset {\mathbb{C}}^{n+1}$. We denote the canonical basis of ${\mathbb{C}}^{n+1}$ by $e_0,
\ldots, e_n$.
We observe that $|\operatorname{Fix}_r(L)| = |\{\tau \in {\ensuremath{{\mathbb{Z}}_r^{n+1}}}\mid {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\tau
{\aftergroup\egroup\originalright}|_{W}
}} =
\operatorname{id}\}|$. For $\tau \in {\ensuremath{{\mathbb{Z}}_r^{n+1}}}$, we have $$\begin{aligned}
{{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\tau
{\aftergroup\egroup\originalright}|_{W}
}} = \operatorname{id}\quad
&\Leftrightarrow \quad W \subset \langle e_i \mid i \in \{0,1,\ldots,n\}
\text{ s.t.\ } \tau_i = 1 \rangle \\
&\Leftrightarrow \quad L \subset V(x_i) \quad \forall i \in
\{0,1,\ldots,n\}
\text{ s.t.\ } \tau_i \neq 1 \\
&\Leftrightarrow \quad \tau_i = 1 \text{ for all $i \in \{0,1,\ldots,n\}$
which
are not a coloop of $\mathcal M_L$}.
\end{aligned}$$ From this, we see that $|\operatorname{Fix}_r(L)| = r^s$.
For the stabiliser of $L$, we have $|\operatorname{Stab}_r(L)| = \frac{1}{r}\: |\{\tau \in
{\ensuremath{{\mathbb{Z}}_r^{n+1}}}\mid \tau \cdot W = W\}|$. If $\tau \in {\ensuremath{{\mathbb{Z}}_r^{n+1}}}$, then $$\begin{aligned}
\tau \cdot W = W \quad
&\Leftrightarrow \quad W = \bigoplus_{\xi \in {\mathbb{Z}}_r} W \cap \langle e_i \mid i
\in \{0,1,\ldots,n\} \text{ s.t.\ } \tau_i = \xi \rangle \\
&\Leftrightarrow \quad \text{For each $\xi \in {\mathbb{Z}}_r$, the set $\{i \in
\{0,1,\ldots,n\} \mid \tau_i = \xi\}$ is a union of} \\
&\phantom{{}\Leftrightarrow{} {}\quad{}} \text{components of $\mathcal M_L$.}
\\
&\Leftrightarrow \quad \forall C \subset \{0,1,\ldots,n\} \text{ component of
}
\mathcal M_L, \; \exists \xi \in {\mathbb{Z}}_r \text{ s.t. } \tau_i = \xi \text{ for
all }
i
\in C.
\end{aligned}$$ In particular, there are precisely $r^t$ elements $\tau \in {\ensuremath{{\mathbb{Z}}_r^{n+1}}}$ with $\tau
\cdot W = W$. We deduce that $|\operatorname{Stab}_r(L)| = r^{t-1}$, which concludes the proof by \[prop:DegreeFormula\].
\[cor:DegreeMatroidInvariant\] The degree of the coordinate-wise $r$-th power of a linear space only depends on the associated linear matroid. If $L_1, L_2 \subset {\mathbb{P}}^n$ are linear spaces such that the linear matroids $\mathcal M_{L_1}$ and $\mathcal
M_{L_2}$ are isomorphic (i.e. they only differ by a permutation of $\{0,1,\ldots,n\}$), then $L_1^{\circ r} \subset {\mathbb{P}}^n$ and $L_2^{\circ r}
\subset {\mathbb{P}}^n$ have the same degree.
\[cor:generalLinearDegree\] Let $L \subset {\mathbb{P}}^n$ be a linear space of dimension $k$. Then $\deg
L^{\circ r} \leq r^k$. For general k-dimensional linear spaces in ${\mathbb{P}}^n$, equality holds.
Every coloop of $\mathcal M_L$ forms a component of $\mathcal M_L$ and the set $\{0,1,\ldots,n\}\setminus \{\text{coloops}\}$ is a union of components, hence $t\leq s+1$. Therefore, by Proposition \[prop:LinearDegreeFormula\], $\deg L^{\circ r}
\leq r^k$. A *general* linear space $L \in \operatorname{Gr}(k,{\mathbb{P}}^n)$ intersects only those linear coordinate space in ${\mathbb{P}}^n$ of dimension at least $n-k$. Therefore, the linear matroid of a general linear space is the uniform matroid: $\mathcal M_L = \{I \subset \{0,1,\ldots,n\}\, \mid\, |I| \leq n-k-1\}$. It is easily checked from the definitions that this matroid has no coloops and only one component.
\[ex:DegreeHyperplane\] We illustrate \[prop:LinearDegreeFormula\] for hyperplanes. Up to permuting and rescaling the coordinates of ${\mathbb{P}}^n$, each hyperplane is given by $L = V(f)$ with $f = x_0+\ldots+x_m$ for some $m \in \{0,1,\ldots,n\}$. Its linear matroid is $$\mathcal M_L = \{\emptyset, \{0\}, \{1\}, \ldots, \{m\}\}.$$ The components of this matroid are the set $\{0,1,\ldots,m\}$ and the singletons $\{i\}$ for $i \geq m+1$. The matroid $\mathcal M_L$ has no coloops for $m \geq 1$ and the unique coloop $0$ if $m = 0$. Then \[prop:LinearDegreeFormula\] shows $\deg
L^{\circ r} = r^{m-1}$ for $m \geq 1$, and $\deg L^{\circ r} = 1$ for $m =
0$. For $m = 3$, $n = 3$ and $r=2$, we obtain a quartic surface which we illustrate in Figure \[fig:QuarticSurfaceFromPlane\].
Hypersurfaces {#sec:Hypersurfaces}
=============
In this section, we study the coordinate-wise powers of hypersurfaces. Here, by a hypersurface, we mean a pure codimension 1 variety. In particular, hypersurfaces are assumed to be reduced, but are allowed to have multiple irreducible components. We describe a way to find the explicit equation describing the image of the given hypersurface under the morphism $\varphi_r$. We define generalised power sum symmetric polynomials and we give a relation between duality and reciprocity of hypersurfaces defined by them. Finally, we raise the question whether and how the explicit description of coordinate-wise powers of hypersurfaces may lead to results on the coordinate-wise powers for arbitrary varieties.
The defining equation
---------------------
The defining equation of a degree $d$ hypersurface is a square-free (i.e. reduced) polynomial unique up to scaling, corresponding to a unique $f \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$. We work with points in ${\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$, i.e. polynomials up to scaling. We do not always make explicit which degree $d$ we are talking about if it is irrelevant to the discussion. The product of $f \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$ and $g \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_{d'})$ is well-defined up to scaling, i.e. as an element $fg \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_{d+d'})$. Equally, we talk about irreducible factors etc. of elements of ${\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$.
Since the finite morphism $\varphi_r $ preserves dimensions, the coordinate-wise $r$-th power of a hypersurface is again a hypersurface, leading to the following definition.
\[def:definingEquation\] Let $f\in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$ be square-free and $V(f) \subset {\mathbb{P}}^n$ be the corresponding hypersurface. We denote by $f^{\circ r} \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_{d'})$ the defining equation of the hypersurface $V(f)^{\circ r}$, i.e. $$V(f^{\circ r}) = V(f)^{\circ r}.$$
For a given square-free polynomial $f$, we want to compute $f^{\circ r}$. To this end, we introduce the following auxiliary notion.
\[def:sym\] Let $f \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$ be square-free. We define $\operatorname{\mathfrak{s}}_r(f) \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_{d'})$ as follows:
(i) If $f$ is irreducible and $f \neq x_i \ \forall i \in
\{0,1,\ldots,n\}$, then we define $\operatorname{\mathfrak{s}}_r(f) \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_{d'})$ to be the product over the orbit ${\ensuremath{\mathcal{G}_{r}}}\cdot f \subset {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$. For $f=x_i$, we define $\operatorname{\mathfrak{s}}_r(f):=x_i^r.$
(ii) If $f=f_1 f_2 \ldots f_m$ where $f_i \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$ are irreducible, then we define $$\operatorname{\mathfrak{s}}_r(f):=\operatorname{lcm}\{\operatorname{\mathfrak{s}}_r(f_1), \operatorname{\mathfrak{s}}_r(f_2), \ldots, \operatorname{\mathfrak{s}}_r(f_m)\}.$$
Observe that in case (ii), determining $\operatorname{\mathfrak{s}}_r(f)=\operatorname{lcm}\{\operatorname{\mathfrak{s}}_r(f_1),
\operatorname{\mathfrak{s}}_r(f_2), \ldots, \operatorname{\mathfrak{s}}_r(f_m)\}$ is straightforward, assuming the decomposition of $f$ into irreducible factors $f_1, \ldots, f_m$ is known. Indeed, the irreducible factors of each $\operatorname{\mathfrak{s}}_r(f_i)$ are immediate from case (i) of the definition, so determining the least common multiple does not require any additional factorization.
\[lem:symInvolvesOnlyPowers\] Let $f\in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$ be square-free. Then $\operatorname{\mathfrak{s}}_r(f)\in
{\mathbb{P}}({\mathbb{C}}[x_0^r,\ldots,x_n^r]_{d'})$, and the principal ideal generated by $\operatorname{\mathfrak{s}}_r(f)$ in the subring ${\mathbb{C}}[x_0^r,\ldots,x_n^r] \subset {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ is $(f) \cap {\mathbb{C}}[x_0^r,\ldots,x_n^r]$.
It is enough to show the claim for $f$ irreducible because we can deduce the general case in the following manner. If $f$ factors into irreducible factors as $f = f_1
f_2 \ldots f_m$, then $$\begin{aligned}
(f) \cap {\mathbb{C}}[x_0^r,\ldots,x_n^r]
&= (f_1)\cap \ldots \cap (f_m) \cap {\mathbb{C}}[x_0^r,\ldots,x_n^r]
= \bigcap_{i=1}^m ((f_i)\cap {\mathbb{C}}[x_0^r,\ldots,x_n^r]) \\
&= \bigcap_{i=1}^m \, (\operatorname{\mathfrak{s}}_r(f_i))
= (\operatorname{lcm}\{\operatorname{\mathfrak{s}}_r(f_1), \operatorname{\mathfrak{s}}_r(f_2), \ldots, \operatorname{\mathfrak{s}}_r(f_m)\}) = (\operatorname{\mathfrak{s}}_r(f)).
\end{aligned}$$
We now assume that $f$ is irreducible. If $f = x_i$ for some $i \in
\{0,1,\ldots,n\}$, then the claim holds trivially by the definition of $\operatorname{\mathfrak{s}}_r(f)$. Let $f \neq x_i$ for all $i$ and $g$ be a polynomial representing $\operatorname{\mathfrak{s}}_r(f) \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_{md})$. By definition, $\operatorname{\mathfrak{s}}_r(f)$ is fixed under the action of ${\ensuremath{\mathcal{G}_{r}}}$, hence $\tau \cdot g$ is a multiple of $g$ for all $\tau \in {\ensuremath{{\mathbb{Z}}_r^{n+1}}}$. Since $g$ is not divisible by $x_i$, it must contain a monomial not divisible by $x_i$. This shows that $g$ is fixed by $\tau^{{\smash[t]{(i)}}} = (1, \ldots, 1, \zeta, 1, \ldots, 1) \in {\ensuremath{{\mathbb{Z}}_r^{n+1}}}$, where the $i$-th position of $\tau^{{\smash[t]{(i)}}}$ is a primitive $r$-th root of unity. Since $\tau^{{\smash[t]{(0)}}}, \tau^{{\smash[t]{(1)}}}, \ldots, \tau^{{\smash[t]{(n)}}}$ generate the group ${\ensuremath{{\mathbb{Z}}_r^{n+1}}}$, we have $\tau \cdot g = g$ for all $\tau \in {\ensuremath{{\mathbb{Z}}_r^{n+1}}}$. Hence, $g$ lies in the invariant ring ${\ensuremath{{\mathbb{C}}[\mathbf{x}]}}^{{\ensuremath{{\mathbb{Z}}_r^{n+1}}}} =
{\mathbb{C}}[x_0^r,\ldots,x_n^r]$, i.e. $\operatorname{\mathfrak{s}}_r(f) \in
{\mathbb{P}}({\mathbb{C}}[x_0^r,\ldots,x_n^r]_{d^{\prime}})$.
If $h \in (f)$ is a polynomial in ${\mathbb{C}}[x_0^r,\ldots,x_n^r]$, then $h$ is invariant under the action of ${\ensuremath{{\mathbb{Z}}_r^{n+1}}}$ on ${\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$, so $h \in (\tau \cdot f)$ for all $\tau \in {\ensuremath{\mathcal{G}_{r}}}$. By the definition of $\operatorname{\mathfrak{s}}_r(f)$ and irreducibility of $\tau \cdot f$, this shows $h \in \operatorname{\mathfrak{s}}_r(f)$. We conclude $(f) \cap
{\mathbb{C}}[x_0^r,\ldots,x_n^r] = (\operatorname{\mathfrak{s}}_r(f))$.
Based on \[def:sym\] and \[lem:symInvolvesOnlyPowers\], the following proposition gives a method to find the equation of the coordinate-wise power of a hypersurface.
\[prop:hypersurfaces\] Let $V(f) \subset {\mathbb{P}}^n$ be a hypersurface. The defining equation $f^{\circ
r}$ of its coordinate-wise $r$-th power is given by replacing each occurrence of $x_i^r$ in $\operatorname{\mathfrak{s}}_r(f)$ by $x_i$ for all $i \in \{0,1,\ldots,n\}$.
Since $V(f)^{\circ r} \subset {\mathbb{P}}^n$ is the image of $V(f)$ under $\varphi_r
\colon {\mathbb{P}}^n \to {\mathbb{P}}^n$, its ideal $(f^{\circ r}) \subset {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ is the preimage under the ring homomorphism $\psi \colon {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}\to {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$, $x_i \mapsto x_i^r$ of the ideal $(f) \subset {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$. The claim is therefore an immediate consequence of \[lem:symInvolvesOnlyPowers\].
For clarity, we illustrate the above results for a hyperplane in ${\mathbb{P}}^3$.
\[ex:quarticSingularSurface\] For $n=3$ and $f := x_0+x_1+x_2+x_3 \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_1)$, we have $$\begin{aligned}
\begin{multlined} \operatorname{\mathfrak{s}}_2(f) = (x_0+x_1+x_2+x_3) (x_0+x_1+x_2-x_3)
(x_0+x_1-x_2+x_3) (x_0+x_1-x_2-x_3) \\ (x_0-x_1+x_2+x_3)
(x_0-x_1+x_2-x_3) (x_0-x_1-x_2+x_3) (x_0-x_1-x_2-x_3). \end{multlined}
\end{aligned}$$ Expanding this expression, we obtain a polynomial in ${\mathbb{C}}[x_0^2,x_1^2,x_2^2,x_3^2]$ and, substituting $x_i^2$ by $x_i$, we obtain by Proposition \[prop:hypersurfaces\] that the coordinate-wise square $V(f)^{\circ 2} \subset {\mathbb{P}}^3$ is the vanishing set of $$\begin{aligned}
f^{\circ 2} &= x_0^4 - 4x_0^3x_1 + 6x_0^2x_1^2 - 4x_0x_1^3 + x_1^4 -
4x_0^3x_2 + 4x_0^2x_1x_2 + 4x_0x_1^2x_2 - 4x_1^3x_2 + 6x_0^2x_2^2 +
4x_0x_1x_2^2 + 6x_1^2x_2^2 \\
&- 4x_0x_2^3 - 4x_1x_2^3 + x_2^4 - 4x_0^3x_3 + 4x_0^2x_1x_3 +
4x_0x_1^2x_3 - 4x_1^3x_3 + 4x_0^2x_2x_3 - 40x_0x_1x_2x_3 + 4x_1^2x_2x_3
+ 4x_0x_2^2x_3 \\
&+ 4x_1x_2^2x_3 - 4x_2^3x_3 + 6x_0^2x_3^2 + 4x_0x_1x_3^2 + 6x_1^2x_3^2
+
4x_0x_2x_3^2 + 4x_1x_2x_3^2 + 6x_2^2x_3^2 - 4x_0x_3^3 - 4x_1x_3^3 -
4x_2x_3^3 + x_3^4.
\end{aligned}$$
![The coordinate-wise square of the plane $V(x_0+x_1+x_2+x_3)
\subset {\mathbb{P}}^3$.[]{data-label="fig:QuarticSurfaceFromPlane"}](QuarticSurfaceFromPlaneInP3New.png){width="19.50000%"}
This rational quartic surface is illustrated in Figure \[fig:QuarticSurfaceFromPlane\]. It is a Steiner surface with three singular lines forming the ramification locus of ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\varphi_2
{\aftergroup\egroup\originalright}|_{V(f)}
}} \colon V(f) \to V(f)^{\circ 2}$.
\[ex:SquaringTheCircle\] Consider the plane conic $C = V(f) \subset {\mathbb{P}}^2$ given by $f := (x_1-ax_0)^2+(x_2-bx_0)^2-(c x_0)^2$ for some $a,b,c \in {\mathbb{R}}$ with $c > 0$. In the affine chart $x_0=1$, this corresponds over the real numbers to the circle with center $(a,b)$ and radius $c$. From \[prop:hypersurfaces\], we show that the coordinate-wise square of the circle $C \subset {\mathbb{P}}^2$ can be a line, a parabola or a singular quartic curve. See Figure \[fig:SquaringTheCircle\] for an illustration of the following three cases:
(i) If the circle $C$ is centered at the origin (i.e. $a=b=0$), then $\operatorname{\mathfrak{s}}_r(f) = f$ and $C^{\circ 2} \subset {\mathbb{P}}^2$ is the line defined by the equation $f^{\circ 2} = x_1+x_2-c^2x_0.$
(ii) If the center of the circle lies on a coordinate-axis and is not the origin (i.e. $ab=0$, but $(a,b) \neq (0,0)$), then $C^{\circ 2} \subset
{\mathbb{P}}^2$ is a conic. Say $a=0$, then $C^{\circ 2}$ is defined by the equation $f^{\circ 2} = (x_1+x_2)^2 + 2(b^2-c^2)x_0x_1-2(b^2+c^2)x_0x_2 +
(b^2-c^2)^2x_0^2.$ In the affine chart $x_0 = 1$, $C$ is a circle and $C^{\circ 2}$ is a parabola.
(iii) If the center of the circle does not lie on a coordinate-axis, then $|{\ensuremath{\mathcal{G}_{r}}}\cdot f| = 4$. Therefore, $C^{\circ 2}$ is a quartic plane curve. Its equation can be computed explicitly using \[prop:hypersurfaces\]. Being the image of a conic, the quartic curve $C^{\circ 2}$ is rational, hence it cannot be smooth. In fact, its singularities are the two points $[0:1:-1]$ and $[a^2+b^2:b^2(c^2-a^2-b^2):a^2(c^2-a^2-b^2)]$ in ${\mathbb{P}}^2$. They form the branch locus of ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\varphi_2
{\aftergroup\egroup\originalright}|_{C}
}} \colon C \to C^{\circ 2}$. The point $[0:1:-1] \in {\mathbb{P}}^2$ is the image of the two complex points $[0:1:\pm i]$ at infinity lying on all of the four conics $\tau \cdot C$ for $\tau \in
{\ensuremath{\mathcal{G}_{2}}}$. The other singular point of $C^{\circ 2}$ is the image under $\varphi_2$ of the two intersection points of the two circles $C$ and $\tau \cdot C$ for $\tau = [1:-1:-1] \in {\ensuremath{\mathcal{G}_{2}}}$ inside the affine chart $x_0=1.$
${{\vcenter{\hbox{\includegraphics[height=4cm]{squaringcircles4.png}}}}}\qquad
\longrightarrow
\qquad
{{\vcenter{\hbox{\includegraphics[height=4cm]{CirclesSquared.png}}}}}$
Let $f \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$ be irreducible and $f \neq x_i$. Then the Newton polytope of $f^{\circ
r} $ arises from the Newton polytope of $f$ by rescaling according to the cardinality of the orbit ${\ensuremath{\mathcal{G}_{r}}}\cdot f \subset {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$: $$\operatorname{Newt}(f^{\circ r}) = \frac{|{\ensuremath{\mathcal{G}_{r}}}\cdot f|}{r} \cdot \operatorname{Newt}(f) \subset
{\mathbb{R}}^{n+1}.$$ Indeed, we have $\operatorname{Newt}(\tau \cdot f) = \operatorname{Newt}(f)$ for all $\tau \in {\ensuremath{\mathcal{G}_{r}}}$, and since $\operatorname{Newt}(g h) = \operatorname{Newt}(g) + \operatorname{Newt}(h)$ holds for all polynomials $g,h$, we have $\operatorname{Newt}(\operatorname{\mathfrak{s}}_r(f)) = |{\ensuremath{\mathcal{G}_{r}}}\cdot f| \cdot \operatorname{Newt}(f)$ by \[def:sym\]. Replacing $x_i^r$ by $x_i$ rescales the Newton polytope with the factor $\frac{1}{r}$, so the claim follows.
Duals and reciprocals of power sum hypersurfaces
------------------------------------------------
We now highlight the interactions between coordinate-wise powers, dual and reciprocal varieties for the case of [power sum hypersurfaces]{} $V(x_0^p+\ldots+x_n^p) \subset {\mathbb{P}}^n$. Specifically, we determine explicitly all hypersurfaces that arise from power sum hypersurfaces by repeatedly taking duals and reciprocals as the coordinate-wise $r$-th power of some hypersurface. In this subsection, we also allow $r$ to take negative integer values.
Recall that the [reciprocal variety]{} $V(f)^{\circ (-1)}$ of a hypersurface $V(f) \subset {\mathbb{P}}^n$ not containing any coordinate hyperplane of ${\mathbb{P}}^n$ is defined as the closure of $\varphi_{-1}(V(f)\setminus V(x_0 x_1 \ldots x_n))$ in ${\mathbb{P}}^n$. We denote it also by $\operatorname{\mathcal{R}}V(f)$. For linear spaces the reciprocal variety and its Chow form has been studied in detail in [@KV].
We also recall the definition of the [dual variety]{} of $V(f)
\subset {\mathbb{P}}^n$. Consider the set of hyperplanes in ${\mathbb{P}}^n$ that arise as the projective tangent space at a smooth point of $V(f)$. This is a subset of the dual projective space $({\mathbb{P}}^n)^*$ and its Zariski closure is the [dual variety]{} of $V(f)$, which we denote by $V(f)^*$ or $\operatorname{\mathcal{D}}V(f)$. We identify $({\mathbb{C}}^{n+1})^*$ with ${\mathbb{C}}^{n+1}$ via the standard bilinear form and therefore identify $({\mathbb{P}}^n)^*$ with ${\mathbb{P}}^n$.
Consider the [power sum polynomial]{} ${\ensuremath{\mathfrak{f}}_{p}} :=
x_0^p+\ldots+x_n^p \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_p)$ for $p \in {\mathbb{N}}$. As before, we regard polynomials only up to scaling. For power sums with negative exponents we consider the numerator of the rational function as $${\ensuremath{\mathfrak{f}}_{-p}} := (x_1 x_2 x_3 \ldots x_n)^p+(x_0 x_2 x_3 \ldots
x_n)^p+\ldots+(x_0 x_1 x_2 \ldots x_{n-1})^p \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_{np}) \quad
\text{for } p \in {\mathbb{N}}.$$ In particular, ${\ensuremath{\mathfrak{f}}_{-1}} \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_n)$ is the elementary symmetric polynomial of degree $n$. Recall that the morphism $\varphi_r \colon {\mathbb{P}}^n \to {\mathbb{P}}^n$ for $r > 0$ is finite, hence preserves dimension. Since $\varphi_{-1} \colon {\mathbb{P}}^n
\dashrightarrow {\mathbb{P}}^n$ is a birational map, the rational map $\varphi_{-r} = \varphi_{-1} \circ \varphi_r$ also preserves dimensions: $\dim
V({\ensuremath{\mathfrak{f}}_{p}})^{\circ (-r)} = \dim V({\ensuremath{\mathfrak{f}}_{p}})$. We therefore extend \[def:definingEquation\] to include the defining equation of $V({\ensuremath{\mathfrak{f}}_{p}})^{\circ r}$ by ${\ensuremath{\mathfrak{f}}_{p}}^{\circ r}$ for all $p,r \in {\mathbb{Z}}\setminus \{0\}$. For the constant polynomial ${\ensuremath{\mathfrak{f}}_{0}} = 1 \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_0)
$, we define ${\ensuremath{\mathfrak{f}}_{0}}^{\circ r} := 1 $ for all $r
\in {\mathbb{Z}}\setminus \{0\}$.
\[lem:powSumRescaling\] For all $s \in {\mathbb{Z}}$ and $r, \lambda \in {\mathbb{Z}}\setminus \{0\}$, we have ${\ensuremath{\mathfrak{f}}_{\lambda s}}^{\circ (\lambda r)} = {\ensuremath{\mathfrak{f}}_{s}}^{\circ r}$.
For $\lambda > 0$, we have $\varphi_{\lambda}^{-1} (V({\ensuremath{\mathfrak{f}}_{s}})) =
V({\ensuremath{\mathfrak{f}}_{\lambda s}})$, hence $$V({\ensuremath{\mathfrak{f}}_{\lambda s}}^{\circ (\lambda r)}) = \varphi_r(\varphi_\lambda(
V({\ensuremath{\mathfrak{f}}_{\lambda s}}))) = \varphi_r (V({\ensuremath{\mathfrak{f}}_{s}})) = V({\ensuremath{\mathfrak{f}}_{s}}^{\circ
r}),$$ where we have used the surjectivity of $\varphi_\lambda \colon {\mathbb{P}}^n \to {\mathbb{P}}^n$. For $\lambda < 0$, we use the above to see $$V({\ensuremath{\mathfrak{f}}_{\lambda s}}^{\circ (\lambda r)}) =
(V({\ensuremath{\mathfrak{f}}_{\lambda s}})^{\circ (-\lambda)})^{\circ (-r)} =
V({\ensuremath{\mathfrak{f}}_{-s}})^{\circ (-r)} =
(\operatorname{\mathcal{R}}V({\ensuremath{\mathfrak{f}}_{-s}}))^{\circ r}.$$ The reciprocal variety of $V({\ensuremath{\mathfrak{f}}_{-s}})$ is $V({\ensuremath{\mathfrak{f}}_{s}})$ for all $s \in
{\mathbb{Z}}$. Hence, $V({\ensuremath{\mathfrak{f}}_{\lambda s}}^{\circ (\lambda r)}) = V({\ensuremath{\mathfrak{f}}_{s}})^{\circ
r}$.
This naturally leads us to the our next definition.
For any rational number $p = \frac{s}{r} \in {\mathbb{Q}}$ ($r,s \in {\mathbb{Z}}$, $r \neq
0$), we define the [generalised power sum polynomial]{} ${\ensuremath{\mathfrak{f}}_{p}} := {\ensuremath{\mathfrak{f}}_{s}}^{\circ r} \in {\mathbb{P}}({\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d)$.
By Lemma \[lem:powSumRescaling\], the generalised power sum polynomial ${\ensuremath{\mathfrak{f}}_{p}}$ is well-defined. With this definition, we get the following duality result for hypersurfaces generalising Example 4.16 in [@GKZ].
\[thm:dualPowSum\] Let $p, q \in {\mathbb{Q}}\setminus \{0\}$ be such that $\frac{1}{p}+\frac{1}{q} = 1$. Then $V({\ensuremath{\mathfrak{f}}_{p}})^* = V({\ensuremath{\mathfrak{f}}_{q}})$.
Write $p = \frac{s}{r}$ with $r \in {\mathbb{Z}}\setminus \{0\}$, $s \in
{\mathbb{Z}}_{>0}$. Let $b \in V({\ensuremath{\mathfrak{f}}_{p}}) = \varphi_r(V({\ensuremath{\mathfrak{f}}_{s}}))$ be a smooth point of $V({\ensuremath{\mathfrak{f}}_{p}}) \setminus V(x_0 x_1 \ldots x_n)$, and let $a \in
V({\ensuremath{\mathfrak{f}}_{s}}) \setminus V(x_0 x_1 \ldots x_n)$ be such that $b =
\varphi_r(a)$. The morphism $\varphi_r \colon {\mathbb{P}}^n \setminus V(x_0 x_1 \ldots
x_n) \to {\mathbb{P}}^n \setminus V(x_0 x_1 \ldots x_n)$ induces a linear isomorphism on projective tangent spaces $\mathbb{T}_a {\mathbb{P}}^n = {\mathbb{P}}^n \to {\mathbb{P}}^n =
\mathbb{T}_{b} {\mathbb{P}}^n$ given by $\operatorname{diag}(ra_0^{r-1}, ra_1^{r-1},\ldots,
ra_n^{r-1})$. This maps $$\mathbb{T}_a V({\ensuremath{\mathfrak{f}}_{s}}) = V{\mathopen{}\mathclose\bgroup\originalleft}(\sum_{i=0}^n (\partial_i
{\ensuremath{\mathfrak{f}}_{s}})(a) \: x_i{\aftergroup\egroup\originalright}) \subset {\mathbb{P}}^n \quad \text{onto} \quad
\mathbb{T}_b V({\ensuremath{\mathfrak{f}}_{p}}) = V{\mathopen{}\mathclose\bgroup\originalleft}(\sum_{i=0}^n \frac{(\partial_i
{\ensuremath{\mathfrak{f}}_{s}})(a)}{ra_i^{r-1}} \: x_i{\aftergroup\egroup\originalright}) \subset {\mathbb{P}}^n.$$ In particular, $V({\ensuremath{\mathfrak{f}}_{p}})^* \subset {\mathbb{P}}^n$ is the image of the rational map $$V({\ensuremath{\mathfrak{f}}_{s}}) \dashrightarrow {\mathbb{P}}^n, \qquad x \mapsto
{\mathopen{}\mathclose\bgroup\originalleft}[\frac{\partial_0
{\ensuremath{\mathfrak{f}}_{s}}}{rx_0^{r-1}}: \frac{\partial_1
{\ensuremath{\mathfrak{f}}_{s}}}{rx_1^{r-1}}:\ldots:\frac{\partial_n
{\ensuremath{\mathfrak{f}}_{s}}}{rx_n^{r-1}}{\aftergroup\egroup\originalright}].$$ From $\partial_i {\ensuremath{\mathfrak{f}}_{s}} = s x_i^{s-1}$ we conclude that $V({\ensuremath{\mathfrak{f}}_{p}})^* = \varphi_{s-r}(V({\ensuremath{\mathfrak{f}}_{s}})) = V({\ensuremath{\mathfrak{f}}_{s/(s-r)}}) =
V({\ensuremath{\mathfrak{f}}_{q}})$.
This statement can be understood as an algebraic analogue of the duality theory for $\ell^p$-spaces $({\mathbb{R}}^n, |\cdot|_p)$. Indeed, let $p,q \geq 1$ be rational with $\frac{1}{p}+\frac{1}{q} = 1$. The unit ball in $({\mathbb{R}}^n,|\cdot|_p)$ is $U_p := \{v \in {\mathbb{R}}^n \mid \sum_{i} v_i^p = 1\}$ and, by $\ell_p$-duality, hyperplanes tangent to $U_p$ correspond to the points on the unit ball $U_q$ of the dual normed vector space $({\mathbb{R}}^n, |\cdot|_q)$. The complex projective analogues of the unit balls $U_p \subset {\mathbb{R}}^n$ are the generalised power sum hypersurfaces $V({\ensuremath{\mathfrak{f}}_{p}}) \subset {\mathbb{P}}^n$ and \[thm:dualPowSum\] shows the previous statement in this setting.
Using \[prop:hypersurfaces\] we can compute ${\ensuremath{\mathfrak{f}}_{p}}$ for any $p
\in {\mathbb{Q}}$ explicitly. In particular, we make the following observation:
\[lem:genPowSumSubst\] Let $s \in {\mathbb{N}}$ and $r \in {\mathbb{Z}}\setminus \{0\}$ be relatively prime. Then ${\ensuremath{\mathfrak{f}}_{s/r}}$ arises from ${\ensuremath{\mathfrak{f}}_{1/r}}$ by substituting $x_i \mapsto x_i^s$ for all $i \in
\{0,1,\ldots,n\}$.
This follows from the explicit description of the polynomials ${\ensuremath{\mathfrak{f}}_{s/r}} =
{\ensuremath{\mathfrak{f}}_{s}}^{\circ r}$ and ${\ensuremath{\mathfrak{f}}_{1/r}} = {\ensuremath{\mathfrak{f}}_{1}}^{\circ r}$ given by \[prop:hypersurfaces\].
By Lemma \[lem:genPowSumSubst\], in order to determine the generalised power sum polynomials ${\ensuremath{\mathfrak{f}}_{p}}$, we may restrict our attention to ${\ensuremath{\mathfrak{f}}_{1/r}}$. These have a particular geometric interpretation as repeated dual-reciprocals of the linear space $V(x_0+x_1+\ldots+x_n) \subset {\mathbb{P}}^n$ as in \[repeatedDR\].
\[cor:dualReciprocals\] The repeated dual-reciprocals of generalised power sum hypersurfaces $V({\ensuremath{\mathfrak{f}}_{p}})$ are given by $$\begin{aligned}
(\operatorname{\mathcal{D}}\operatorname{\mathcal{R}})^k \: V({\ensuremath{\mathfrak{f}}_{p}}) &= V({\ensuremath{\mathfrak{f}}_{p/(1+kp)}}) \qquad \forall k
\in {\mathbb{N}}, \; p \in {\mathbb{Q}}\setminus \{0, -\frac{1}{k}, -\frac{1}{k-1},\ldots,-1\}
\quad \text{and} \\
(\operatorname{\mathcal{R}}\operatorname{\mathcal{D}})^k \: V({\ensuremath{\mathfrak{f}}_{p}}) &= V({\ensuremath{\mathfrak{f}}_{p/(1-kp)}}) \qquad \forall k
\in {\mathbb{N}}, \; p \in {\mathbb{Q}}\setminus \{0, \frac{1}{k}, \frac{1}{k-1}, \ldots, 1\}.
\end{aligned}$$
We show the claim for $V({\ensuremath{\mathfrak{f}}_{p}})$ by induction on $k$. For $k=0$, the claim is trivial. For $k > 0$, we get by induction hypothesis: $$\begin{aligned}
(\operatorname{\mathcal{D}}\operatorname{\mathcal{R}})^k \: V({\ensuremath{\mathfrak{f}}_{p}}) &= \operatorname{\mathcal{D}}\operatorname{\mathcal{R}}V({\ensuremath{\mathfrak{f}}_{p/(1+(k-1)p)}})
= (V({\ensuremath{\mathfrak{f}}_{p/(1+(k-1)p)}})^{\circ (-1)})^* \\ &\stackrel{(*)}{=}
V({\ensuremath{\mathfrak{f}}_{-p/(1+(k-1)p)}})^* \stackrel{(**)}{=} V({\ensuremath{\mathfrak{f}}_{p/(1+kp)}}),
\end{aligned}$$ where $(*)$ holds by \[lem:powSumRescaling\] and $(**)$ by \[thm:dualPowSum\]. From this, we also see $$(\operatorname{\mathcal{R}}\operatorname{\mathcal{D}})^k \: V({\ensuremath{\mathfrak{f}}_{p}}) = \operatorname{\mathcal{R}}(\operatorname{\mathcal{D}}\operatorname{\mathcal{R}})^k \operatorname{\mathcal{R}}V({\ensuremath{\mathfrak{f}}_{p}}) = \operatorname{\mathcal{R}}(\operatorname{\mathcal{D}}\operatorname{\mathcal{R}})^k V({\ensuremath{\mathfrak{f}}_{-p}}) = \operatorname{\mathcal{R}}V({\ensuremath{\mathfrak{f}}_{-p/(1-kp)}}) = V({\ensuremath{\mathfrak{f}}_{p/(1-kp)}}),$$ concluding the proof.
\[repeatedDR\] For $r > 0$, the repeated alternating reciprocals and duals of the linear space $V({\ensuremath{\mathfrak{f}}_{1}}) \subset {\mathbb{P}}^n$ are the coordinate-wise powers of $V({\ensuremath{\mathfrak{f}}_{1}})$ given as $$\begin{aligned}
{\underbrace{\operatorname{\mathcal{D}}\operatorname{\mathcal{R}}\operatorname{\mathcal{D}}\operatorname{\mathcal{R}}\ldots \operatorname{\mathcal{D}}\operatorname{\mathcal{R}}}_{2r-2}} \:
V({\ensuremath{\mathfrak{f}}_{1}}) &= V({\ensuremath{\mathfrak{f}}_{1}})^{\circ r} \qquad
\text{and} \qquad
{\underbrace{\operatorname{\mathcal{R}}\operatorname{\mathcal{D}}\operatorname{\mathcal{R}}\ldots \operatorname{\mathcal{D}}\operatorname{\mathcal{R}}}_{2r-1}} \:
V({\ensuremath{\mathfrak{f}}_{1}}) = V({\ensuremath{\mathfrak{f}}_{1}})^{\circ (-r)}.
\end{aligned}$$
Let $n = 3$ and $f:=x_0+x_1+x_2+x_3$. The reciprocal variety of the plane $V(f)\subset{\mathbb{P}}^3$ is given by ${\ensuremath{\mathfrak{f}}_{-1}} = x_1 x_2
x_3+x_0x_2x_3+x_0x_1x_3+x_0x_1x_2$. Its dual is $V({\ensuremath{\mathfrak{f}}_{1/2}}) = V({\ensuremath{\mathfrak{f}}_{1}})^{\circ 2} \subset {\mathbb{P}}^3$ by \[thm:dualPowSum\]. This is the quartic surface from Example \[ex:quarticSingularSurface\]. Higher iterated dual-reciprocal varieties of $V(f)$ can be explicitly computed analogous to Example \[ex:quarticSingularSurface\] via \[cor:dualReciprocals\]. For instance, the surface $\operatorname{\mathcal{D}}\operatorname{\mathcal{R}}\operatorname{\mathcal{D}}\operatorname{\mathcal{R}}V(f) \subset
{\mathbb{P}}^3$ is the coordinate-wise cube of $V(f)$ which is the degree 9 surface illustrated in Figure \[fig:cubeOfPlane\].
![The iterated dual-reciprocal $\operatorname{\mathcal{D}}\operatorname{\mathcal{R}}\operatorname{\mathcal{D}}\operatorname{\mathcal{R}}V(f) \subset {\mathbb{P}}^3$[]{data-label="fig:cubeOfPlane"}](CubeOfPlane4.pdf){width="20.00000%"}
The construction of the generalised power sum hypersurfaces $V({\ensuremath{\mathfrak{f}}_{p}})$ may be understood in a broader context of coordinate-wise powers with rational exponents: For a subvariety $X \subset {\mathbb{P}}^n$, and a rational number $p = r/s$ with $r \in {\mathbb{Z}}$ and $s \in {\mathbb{Z}}_{>0}$ relatively prime, we may define the coordinate-wise $p$-th power $X^{\circ p} := \varphi_s^{-1}(X^{\circ r}) = (\varphi_s^{-1}(X))^{\circ r}$. This is a natural generalisation of the coordinate-wise integer powers $X^{\circ r}$. With this definition, the generalised power sum hypersurface $V({\ensuremath{\mathfrak{f}}_{p}})$ is the $1/p$-th coordinate-wise power of $V({\ensuremath{\mathfrak{f}}_{1}})$. While we focus on coordinate-wise powers to integral exponents in this article, many results easily transfer to the case of rational exponents. For instance, the defining ideal of $X^{\circ (r/s)}$ is obtained by substituting $x_i \mapsto x_i^s$ in each of the generators of the vanishing ideal of $X^{\circ r}$ – in particular, the number of minimal generators for these two ideals agree.
From hypersurfaces to arbitrary varieties?
------------------------------------------
We briefly discuss to what extent \[prop:hypersurfaces\] can be used to determine coordinate-wise powers of arbitrary varieties, and mention the difficulties involved in this approach.
Let $r > 0$ and let $f_1, \ldots, f_m$ be homogeneous polynomials vanishing on a variety $X
\subset {\mathbb{P}}^n$. Their $r-$th coordinate-wise powers give rise to the inclusion $X^{\circ r} \subset V(f_1^{\circ r}, \ldots, f_m^{\circ r})$. We may ask when equality holds, which leads us to the following definition, reminiscent of the notion of *tropical bases* in Tropical Geometry [@MS Section 2.6].
\[def:powerBasis\] A set of homogeneous polynomials $f_1, \ldots, f_m \subset {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ is an $r$-th power basis of the ideal $I = (f_1, \ldots, f_m)$ if the following equality of sets holds: $$V(f_1, \ldots, f_m)^{\circ r} = V(f_1^{\circ r}, \ldots, f_m^{\circ r}).$$
We show the existence of such power bases for a given ideal in the following proposition.
\[prop:higherCodimension\] Let $I \subset {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ be a homogeneous ideal. Then for each $r$, there exists an $r$-th power basis of $I$.
Let $J$ denote the defining ideal of $V(I)^{\circ r} \subset
{\mathbb{P}}^n$. If $J$ is generated by homogeneous polynomials $g_1, \ldots, g_m \in
{\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$, we define $f_1, \ldots, f_m \in {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ to be their images under the ring homomorphism ${\ensuremath{{\mathbb{C}}[\mathbf{x}]}}\to {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$, $x_i \mapsto x_i^r$. Then $f_i\in I$, since $$V(f_i) = \varphi_r^{-1}(V(g_i)) \supset
\varphi_r^{-1}(V(I)^{\circ r}) \supset
V(I).$$ On the other hand, we have $f_i^{\circ r} = g_i$, since $V(f_i)^{\circ r} = \varphi_r(\varphi_r^{-1}(V(g_i))) = V(g_i)$ by surjectivity of $\varphi_r$. Therefore, $f_1^{\circ r}, \ldots,
f_m^{\circ r}$ generate $J$. Enlarging $f_1, \ldots, f_m$ to a generating set of $I$ gives an $r$-th power basis of $I$.
\[prop:higherCodimension\] shows the existence of $r$-th power bases, but explicitly determining one a priori is nontrivial. For the variety of orthostochastic matrices ${\ensuremath{\mathbb{O}(m)}}^{\circ 2}$ as in \[ssec:Orthostochastic\], it is natural to suspect that the quadratic equations defining the variety of orthogonal matrices would form a power basis for $r = 2$. This is the question discussed in [@CD Section 3], where it was shown that this is true for $m = 3$, but not for $m \geq 6$. The cases $m = 4,5$ are an open problem [@CD Problem 6.2]. Our results on the degree of ${\ensuremath{\mathbb{O}(m)}}^{\circ 2}$ reduce this open problem to the computation whether explicitly given polynomials $f_1^{\circ 2}, \ldots, f_k^{\circ 2}$ describe an irreducible variety of the correct dimension and degree. Straightforward implementations of this computation seem to be beyond current computer algebra software.
In the following two examples, we will see that even in the case of squaring codimension 2 linear spaces, obvious candidates for $f_1, \ldots, f_m$ do not form a power basis.
\[ex:Codim2GensDontWork\] Let $I := (f_1, f_2) \subset {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ be the ideal defining the line in ${\mathbb{P}}^3$ that is given by $f_1 := x_0+x_1+x_2+x_3$ and $f_2 := x_1+2x_2+3x_3$. The polynomials $f_1^{\circ 2}$ and $f_2^{\circ 2}$ have degrees 4 and 2, respectively, by \[prop:hypersurfaces\]. Note that the polynomial $f_3 := 3x_0^2-x_1^2+x_2^2-3x_3^2 = 3(x_0-x_1-x_2-x_3)f_1+2(x_1+x_2)f_2$ also lies in $I$, so the ideal of $V(I)^{\circ 2}$ contains the linear form $f_3^{\circ 2} = 3x_0-x_1+x_2-3x_3$. The polynomials $f_1,f_2$ do not form a power basis of $I$. In fact, one can check that $V(f_1^{\circ 2}, f_2^{\circ 2})
\subset {\mathbb{P}}^3$ is the union of four rational quadratic curves, one of which is $V(I)^{\circ 2}$, see Figure \[fig:4Conics\] for an illustration. A power basis of $I$ is given by $f_1,f_2,f_3$.
![Distinction between $V(f_1^{\circ 2}, f_2^{\circ 2})$ and $V(f_1,f_2)^{\circ 2}$[]{data-label="fig:4Conics"}](4QuadraticCurves.pdf){width="20.00000%"}
\[ex:Codim2CircuitsDontWork\] Another natural choice for polynomials $f_1, \ldots, f_m$ in the ideal of a linear space $X \subset {\mathbb{P}}^n$ consists of the *circuit forms*, i.e. linear forms vanishing on $X$ that are minimal with respect to the set of occurring variables. However, for $$X := V(x_0+x_1+x_2+x_3+x_4, \; x_1+2x_2+3x_3+4x_4) \subset {\mathbb{P}}^4,$$ these circuit forms are $$\begin{gathered}
f_1 = x_1+2x_2+3x_3+4x_4, \quad f_2 = x_0-x_2-2x_3-3x_4, \quad f_3 =
2x_0+x_1-x_3-2x_4, \\
f_4 = 3x_0+2x_1+x_2-x_4, \quad f_5 =
4x_0+3x_1+2x_2+x_3,
\end{gathered}$$ and one can check that the point $[16:16:1:36:9] \in {\mathbb{P}}^4$ lies in $V(f_1^{\circ 2}, \ldots, f_5^{\circ 2})$, but not in $X^{\circ 2}$. In particular, $f_1, \ldots, f_5$ is not an $r$-th power basis for $r = 2$.
We have seen in \[ex:Codim2GensDontWork\] and \[ex:Codim2CircuitsDontWork\] that even for the case of linear spaces of codimension 2 it is not an easy task to a priori identify an $r$-th power basis.
The following proposition shows how one can straightforwardly find a very large $r$-th power basis of an ideal $I$, without first computing the ideal of $V(I)^{\circ r}$.
\[prop:stupidPowerBasis\] If $g_1, \ldots, g_k \in {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}_d$ are forms of degree $d$, then taking $(k-1)r^n+1$ general linear combinations of $g_1, \ldots, g_k$ produces an $r$-th power basis of $(g_1, \ldots, g_k)$.
We assume that $g_1, \ldots, g_k$ are linearly independent, or else we can replace them with a linearly independent subset. For $m := (k-1)r^n+1$, let $f_1, \ldots, f_m \in
\langle g_1, \ldots, g_k \rangle$ be such that no $k$ of them are linearly dependent. For $X := V(g_1, \ldots, g_k)$, we will show that $V(f_1^{\circ r}, \ldots, f_m^{\circ r}) = X^{\circ r}$ by comparing the preimages of both sides under $\varphi_r \colon {\mathbb{P}}^n \to {\mathbb{P}}^n$.
By Proposition \[prop:preimage\], we have $\varphi_r^{-1}(X^{\circ r}) =
\bigcup_{\tau \in {\ensuremath{\mathcal{G}_{r}}}} \tau \cdot X$ and $$\varphi_r^{-1}(V(f_1^{\circ r}, \ldots, f_m^{\circ r})) =
\bigcap_{i=1}^m \varphi_r^{-1} (\varphi_r(V(f_i))) =
\bigcap_{i=1}^m \bigcup_{\tau \in {\ensuremath{\mathcal{G}_{r}}}} \tau \cdot V(f_i).$$ Let $p \in \varphi_r^{-1} (V(f_1^{\circ r}, \ldots, f_m^{\circ r})) \subset
{\mathbb{P}}^n$. Then for each $i \in \{1,\ldots,m\}$ there exists some $\tau \in {\ensuremath{\mathcal{G}_{r}}}$ with $p \in \tau \cdot V(f_i)$ using the last equality above. Since $m > (k-1)|{\ensuremath{\mathcal{G}_{r}}}|$, by pigeonhole principle there must exist $\tau
\in {\ensuremath{\mathcal{G}_{r}}}$ and $i_1, i_2,\ldots, i_k \in \{1,\ldots,m\}$ distinct with $p \in
\bigcap_{j=1}^k \tau \cdot V(f_{i_j}) = \tau \cdot
V(f_{i_1},\ldots,f_{i_k})$. Since, by assumption, no $k$ of them are linearly dependent $f_{i_1}, \ldots, f_{i_k}$ span $\langle g_1,
\ldots, g_k\rangle$. Therefore, $V(f_{i_1},\ldots,f_{i_k}) = X$, and hence, $p\in\tau \cdot
V(f_{i_1},\ldots,f_{i_k})$ implies that $p \in \tau \cdot X \subset \varphi_r^{-1}(X^{\circ r})$. This shows $\varphi_r^{-1} (V(f_1^{\circ r}, \ldots, f_m^{\circ r})) \subset
\varphi_r^{-1}(X)$. The reverse inclusion is trivial.
In particular, \[prop:stupidPowerBasis\] shows that for a subvariety of ${\mathbb{P}}^n$ defined by $k$ forms of degree $d$, its coordinate-wise $r$-th power can be described set-theoretically by the vanishing of $(k-1)r^n+1$ forms of degree $\leq d r^{n-1}$. However, we will see in \[sec:LinSpaces\] that for linear spaces this bound is rather weak in many cases and should be expected to allow dramatic refinement in general. We raise the following as a broad open question:
When does a set of homogeneous polynomials form an $r$-th power basis? For a given ideal $I$, do there exist polynomials $f_1, \ldots, f_m \in I$ that simultaneously form an $r$-th power basis for all $r$?
Linear spaces {#sec:LinSpaces}
=============
In this section, we specialise to linear spaces $L \subset {\mathbb{P}}^n$ and investigate their coordinate-wise powers $L^{\circ r}$. First, we highlight the dependence of $L^{\circ r}$ on the geometry of a finite point configuration associated to $L \subset {\mathbb{P}}^n$. For $r=2$, we point out its relation to symmetric matrices with degenerate eigenvalues. Based on this, we classify the coordinate-wise squares of lines and planes. Finally, we turn to the case of squaring linear spaces in high-dimensional ambient space.
Point configurations
--------------------
We study the defining ideal of $L^{\circ r}$ for a $k$-dimensional linear space $L \subset {\mathbb{P}}^n$. The degrees of its minimal generators do not change under rescaling and permuting coordinates of ${\mathbb{P}}^n$, i.e. under the actions of the algebraic torus $\mathbb{G}_m^{n+1} = ({\mathbb{C}}^*)^{n+1}$ and the symmetric group $\mathfrak{S}_{n+1}$. Fixing a $(k+1)$-dimensional vector space $W$, we have the identification $$\begin{aligned}
\{\text{\small orbits of $\operatorname{Gr}(k,{\mathbb{P}}^n)$ under $\mathbb{G}_m^{n+1}
\rtimes
\mathfrak{S}_{n+1}$}\} &\leftrightarrow {\mathopen{}\mathclose\bgroup\originalleft}\{\substack{\text{\small
finite multi-sets $Z \subset {\mathbb{P}}W^*$ with $\langle Z \rangle = {\mathbb{P}}W^*$} \\
\text{\small of cardinality $\leq n+1$ up to
$\operatorname{Aut}({\mathbb{P}}W^*)$}}{\aftergroup\egroup\originalright}\} \\
L = \operatorname{im}({\mathbb{P}}W \xhookrightarrow{[\ell_0:\ell_1:\ldots:\ell_s:0:\ldots:0]}
{\mathbb{P}}^n) \qquad & \substack{\mapsfrom \\ \mapsto} \qquad Z = \{[\ell_0],
[\ell_1], \ldots, [\ell_s]\} \subset {\mathbb{P}}W^*, s \leq n.
\end{aligned}$$ Hence, we may express coordinate-wise powers of a linear space $L$ in terms of the corresponding finite multi-set $Z \subset {\mathbb{P}}W^*$. In fact, it is easy to check that the degrees of the minimal generators of the defining ideal only depend on the underlying set $Z$, forgetting repetitions in the multi-set. We study coordinate-wise powers of a linear space in terms of the corresponding non-degenerate finite point configuration.
For the entirety of Section \[sec:LinSpaces\], we establish the following notation: Let $L\subset {\mathbb{P}}^n$ be a linear space of dimension $k$. We understand $L$ as the image of a chosen linear embedding $\iota\colon {\mathbb{P}}W \xhookrightarrow{[\ell_0:\ldots:\ell_n]} {\mathbb{P}}^n$, where $W$ is a $(k+1)$-dimensional vector space and $\ell_0, \ldots, \ell_n \in W^*$ are linear forms defining $\iota$. Consider the finite set of points $Z
\subset {\mathbb{P}}W^*$ given by $$Z := \{[\ell_i] \in {\mathbb{P}}W^* \mid 0 \leq i \leq n \text{ such that } \ell_i
\neq 0\}.$$
Since $\ell_0, \ell_1, \ldots, \ell_n \in W^*$ define the linear embedding $\iota$, they cannot have a common zero in $W$. Hence, the linear span of $Z$ is the whole space ${\mathbb{P}}W^*$. We denote by $I(Z) \subset \operatorname{Sym}^\bullet W$ the defining ideal of $Z \subset {\mathbb{P}}W^*$. The subspace of degree $r$ forms vanishing on $Z$ is written as $I(Z)_r \subset \operatorname{Sym}^r W$.
The main technical tool is the following observation that $L^{\circ r} \subset {\mathbb{P}}^n$ equals (up to a linear re-embedding) the image of the $r$-th Veronese variety $\nu_r({\mathbb{P}}W)\subset {\mathbb{P}}\operatorname{Sym}^r W$ under the projection from the linear space ${\mathbb{P}}(I(Z)_r) \subset {\mathbb{P}}\operatorname{Sym}^r W$.
\[lem:intrinsicDescription\] The diagram $$\begin{tikzcd}
& {\mathbb{P}}W \arrow[hookrightarrow]{r}{\nu_r}
\arrow{dr}{\psi} \arrow{d}{\varphi_r \circ \iota}
& {\mathbb{P}}\operatorname{Sym}^r W \arrow[dashed]{d}{\pi} \\
{\mathbb{P}}^n \arrow[hookleftarrow]{ur}{\iota} \arrow{r}{\varphi_r} & {\mathbb{P}}^n
\arrow[hookleftarrow]{r}{\vartheta} &
{\mathbb{P}}(\operatorname{Sym}^r W/I(Z)_r)
\end{tikzcd}$$ commutes, where $\nu_r$ is the $r$-th Veronese embedding, $\pi$ is the linear projection of ${\mathbb{P}}\operatorname{Sym}^r W$ from the linear space ${\mathbb{P}}(I(Z)_r)$, $\psi$ is a morphism and $\vartheta$ is a linear embedding.
We observe that the morphism $\varphi_r \circ \iota$ is given by $$\varphi_r \circ \iota \colon {\mathbb{P}}W \to {\mathbb{P}}^n, \qquad [v] \mapsto [\ell_0^r(v)
: \ell_1^r(v) \ldots : \ell_n^r(v)].$$ The $n+1$ elements $\ell_i^r \in \operatorname{Sym}^r W^*$ correspond to a linear map $\chi \colon \operatorname{Sym}^r W \to {\mathbb{C}}^{n+1}$ via the natural identification $(\operatorname{Sym}^r W^*)^{n+1} = \operatorname{Hom}_{\mathbb{C}}(\operatorname{Sym}^r W, {\mathbb{C}}^{n+1})$.
The rational map $\bar{\chi}$ between projective spaces corresponding to the linear map $\chi$ gives the following commuting diagram: $$\begin{tikzcd}
{\mathbb{P}}W \arrow[hookrightarrow]{r}{\nu_r} \arrow{d}{\varphi_r\, \circ\,
\iota} &
{\mathbb{P}}\operatorname{Sym}^r W \arrow[dashed]{d}{\pi} \\
{\mathbb{P}}^n \arrow[hookleftarrow]{r}{\vartheta}
\arrow[leftarrow, dashed]{ur}{\bar{\chi}} &
{\mathbb{P}}(\operatorname{Sym}^r W/\ker \chi),
\end{tikzcd}$$ where $\vartheta$ is the linear embedding of projective spaces induced by factoring $\chi$ over $\operatorname{Sym}^r W/\ker \chi$. In particular, $\nu_r({\mathbb{P}}W) \cap
{\mathbb{P}}(\ker \chi) = \emptyset$, since $\varphi_r \circ \iota$ is defined everywhere on ${\mathbb{P}}W$. Hence, ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\pi
{\aftergroup\egroup\originalright}|_{\nu_r({\mathbb{P}}W)}
}} \colon \nu_r({\mathbb{P}}W) \to {\mathbb{P}}(\operatorname{Sym}^r
W/\ker \chi)$ is a morphism.
Finally, we claim that $\ker \chi = I(Z)_r$. Once we know this, defining $\psi := {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
\pi
{\aftergroup\egroup\originalright}|_{\nu_r({\mathbb{P}}W)}
}} \circ \nu_r$ completes the claimed diagram.
Let $f \in \operatorname{Sym}^r W$ such that $f\in I(Z)_r$. Naturally identifying $W$ and $W^{**}$, we may view $f$ as a form of degree $r$ on $W^*$. Then, the condition that $f \in I(Z)_r$ translates to $f(\ell_i) = 0 \ \forall i$. Viewing $f$ as a symmetric $r$-linear form $W^*
\times \ldots \times W^* \to {\mathbb{C}}$, we have $f(\ell_i,
\ldots, \ell_i) = 0 \ \forall i$. Also, when $f$ is considered as a linear form on $\operatorname{Sym}^r W^*$, $f(\ell_i^r) = 0 \ \forall i$. The latter expression is equivalent to $f \in \ker \chi$, via the identification of $W$ and $W^{**}$. We conclude $I(Z)_r = \ker \chi$.
In particular, we deduce the following:
\[prop:NoQuadrics\] Let $L$ be a linear space such that the finite set of points $Z$ does not lie on a degree $r$ hypersurface. Then the ideal of $L^{\circ r}$ is generated by linear and quadratic forms.
Since $I(Z)_r = 0$, we deduce from \[lem:intrinsicDescription\] that $L^{\circ r} = \varphi_r(L)$ is a linear re-embedding of the $k$-dimensional $r$-th Veronese variety $\nu_r({\mathbb{P}}W) \subset {\mathbb{P}}\operatorname{Sym}^r W$. The ideal of this Veronese variety is generated by quadrics. Since $\dim \operatorname{Sym}^r W = \tbinom{k+r}{r}$, the linear re-embedding $\vartheta \colon {\mathbb{P}}\operatorname{Sym}^r W \hookrightarrow {\mathbb{P}}^n$ adds $n-\smash{\binom{k+r}{r}}+1$ linear forms to the ideal.
Degenerate eigenvalues and squaring
-----------------------------------
We now specialise to the case of coordinate-wise squaring, i.e. $r = 2$. This case has special geometric importance, since it corresponds to computing the image of a linear space under the quotient of ${\mathbb{P}}^n$ by the reflection group generated by the coordinate hyperplanes. In this section through \[prop:eigenvaluesEarly\] we point out that the case of coordinate-wise square of a linear space is closely related to studying symmetric matrices with a degenerate spectrum of eigenvalues. Here, we interpret ${\mathbb{P}}\operatorname{Sym}^2 {\mathbb{F}}^{k+1}$ (for ${\mathbb{F}}= {\mathbb{R}}$ or ${\mathbb{C}}$) as the projective space consisting of symmetric $(k+1) \times (k+1)$-matrices up to scaling with entries in ${\mathbb{F}}$.
\[prop:eigenvaluesEarly\] Let $X \subset {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{R}}^{k+1}$ be the set of real symmetric $(k+1)\times
(k+1)$-matrices with an eigenvalue of multiplicity $\geq k$. Then the Zariski closure of $X$ in ${\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1}$ is projectively equivalent to the projective cone over the coordinate-wise square $L^{\circ 2}$ of any $k$-dimensional linear space $L$ whose point configuration $Z \subseteq {\mathbb{P}}W^*$ lies on a unique and smooth quadric.
Let $L \subset {\mathbb{P}}^n$ be a $k$-dimensional linear space such that $I(Z)_2$ is spanned by a smooth quadric $q \in {\mathbb{P}}\operatorname{Sym}^2 W$. Choosing coordinates of $W \cong {\mathbb{C}}^{k+1}$, we identify points in ${\mathbb{P}}\operatorname{Sym}^2 W$ with complex symmetric $(k+1) \times (k+1)$-matrices up to scaling and we can assume $q = \operatorname{id}\in {\mathbb{P}}\operatorname{Sym}^2 W$. The second Veronese variety $\nu_2({\mathbb{P}}W) \subset {\mathbb{P}}\operatorname{Sym}^2 W$ consists of rank $1$ matrices. Let $X_0
\subset {\mathbb{P}}(\operatorname{Sym}^2 W/\langle q \rangle)$ be the image of $\nu_2({\mathbb{P}}W)$ under the natural projection. By \[lem:intrinsicDescription\], $X_0$ is the coordinate-wise square $L^{\circ 2}$ up to a linear re-embedding.
The projective cone over $X_0 \cong L^{\circ 2}$ is the subvariety $X_1
\subset {\mathbb{P}}\operatorname{Sym}^2 W$ consisting of complex symmetric matrices $M$ such that the set $M+\langle \operatorname{id}\rangle$ contains a matrix of rank $\leq 1$. We observe that the rank of $M-\lambda \operatorname{id}$ is the codimension of the eigenspace of $M$ with respect to $\lambda \in {\mathbb{C}}$. Hence, $$X_1 = \{M \in {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1} \mid M \text{ has an eigenspace of
codimension~$\leq 1$}\}.$$
We are left to show that $X_1$ is the Zariski closure in ${\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1}$ of $X \subset {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{R}}^{k+1}$. Since real symmetric matrices are diagonalizable, the multiplicity of an eigenvalue is the dimension of the corresponding eigenspace. Hence, $X_1 \cap {\mathbb{P}}\operatorname{Sym}^2
{\mathbb{R}}^{k+1}= X$. The set $X$ is the orbit of the line $V := \{\operatorname{diag}(\lambda,
\ldots, \lambda, \mu) \mid [\lambda:\mu] \in {\mathbb{P}}_{\mathbb{R}}^1\}$ under the action of $O(k+1).$ The action is given by conjugation with orthogonal matrices and the stabiliser is $O(k) \times \{\pm 1\}$. Therefore, $X$ has real dimension $\dim V + \dim O(k+1) - \dim O(k) = k+1$. Also, $X_1$ is the projective cone over $X_0 \cong L^{\circ 2}$, so it is a ${(k+1)}$-dimensional irreducible complex variety. We conclude that $X_1$ is the Zariski closure of $X$ in ${\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1}$.
We illustrate \[prop:eigenvaluesEarly\] in the case of $3 \times 3$-matrices:
Consider the set of real symmetric $3 \times 3$-matrices with a repeated eigenvalue. We denote its Zariski closure in ${\mathbb{P}}\operatorname{Sym}^3 {\mathbb{C}}^2$ by $Y$. By \[prop:eigenvaluesEarly\], it can be understood in terms of the coordinate-wise square $L^{\circ 2}$ for some plane $L$. We make this explicit as follows: Consider the planar point configuration $$Z = \{[1:i:0], [1:-i:0], [1:0:i], [1:0:-i], [0:1:i]\} \subseteq {\mathbb{P}}^2,$$ lying only on the conic $V(x^2+y^2+z^2)$. Let $L$ be the corresponding plane in ${\mathbb{P}}^4$, given as the image of $$\iota \colon {\mathbb{P}}^2 \hookrightarrow {\mathbb{P}}^4, \qquad [x:y:z] \mapsto [x+iy:x-iy:x+iz:x-iz:y+iz].$$ Under the linear embedding $$\begin{aligned}
\psi \colon {\mathbb{P}}^4 &\hookrightarrow {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^3, \\
{\scriptsize [a:b:c:d:e]} &\mapsto {\scriptsize
\begin{bmatrix}
2(a+b+c+d) & 3i(-a+b) & 3i(-c+d) \\
3i(-a+b) & 6(-2a-2b+c+d) & 3i(-a-b+c+d-2e) \\
3i(-c+d) & 3i(-a-b+c+d-2e) & 6(a+b-2c-2d),
\end{bmatrix}},
\end{aligned}$$ the plane $L$ gets mapped into $Y$. Indeed, it is easily checked that a point $[x:y:z]$ gets mapped to the matrix $-4(x^2+y^2+z^2)\operatorname{id}+12(x,y,z)^T(x,y,z)$ under the composition $\psi \circ \iota \colon {\mathbb{P}}^2 \to {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^3$; note that this matrix has a repeated eigenvalue. More precisely, \[prop:eigenvaluesEarly\] shows that $Y$ is the projective cone over $\psi(L^{\circ 2})$ with the vertex $\operatorname{id}$.
In \[ssec:squaringHighDim\] we give an explicit set-theoretic description of the coordinate-wise square of a linear space in high-dimensional ambient space. We will show the following result as a special case of \[thm:uniqueQuadricCase\]. Given a matrix $A \in {\mathbb{C}}^{s \times s}$, we denote a $2\times 2$ minor of $A$ by $A_{ij|k\ell}$ where $i,j$ are the rows and $k,\ell$ are the columns of the minor.
\[cor:eigenvalues\] Let $s \geq 4$. A symmetric matrix $A \in {\mathbb{C}}^{s \times s}$ has an eigenspace of codimension $\leq 1$ if and only if its $2\times 2$-minors satisfy the following for $i,j,k,\ell \leq s$ distinct: $$A_{ij|k\ell} = 0, \qquad A_{ik|i\ell} = A_{jk|j\ell} \qquad \text{and}
\qquad A_{ik|ik} - A_{i\ell|i\ell} = A_{jk|jk} - A_{j \ell|j \ell}.$$ These equations describe the Zariski closure in the complex vector space $\operatorname{Sym}^2 {\mathbb{C}}^s$ of the set of real symmetric matrices with an eigenvalue of multiplicity $\geq s-1$.
Squaring lines and planes
-------------------------
In this subsection we consider the low-dimensional cases and classify the coordinate-wise squares of lines and planes in arbitrary ambient spaces.
\[thm:squaringLines\] Let $L$ be a line in ${\mathbb{P}}^n$.
(i) If $|Z| = 2$, then $L^{\circ 2}$ is a line in ${\mathbb{P}}^n$.
(ii) If $|Z| > 2$, then $L^{\circ 2}$ is a smooth conic in ${\mathbb{P}}^n$.
Since $Z \subset {\mathbb{P}}W^*$ spans the projective line ${\mathbb{P}}W^*$, we must have $|Z| \geq 2$.
If $|Z| > 2$, then $I(Z)_2 = 0$, since no non-zero quadratic form on the projective line ${\mathbb{P}}W^*$ vanishes on all points of $Z$. Then \[lem:intrinsicDescription\] implies that $L^{\circ 2} = (\varphi_2 \circ
\iota)({\mathbb{P}}W) $ is a linear re-embedding of $\nu_2({\mathbb{P}}W) $, which is a smooth conic in the plane ${\mathbb{P}}\operatorname{Sym}^2 W \cong {\mathbb{P}}^2$.
If $|Z| = 2$, then $\dim I(Z)_2 = 1$, since up to scaling there is a unique quadric vanishing on the points $Z $. By \[lem:intrinsicDescription\], the image $\varphi_2(L)$ lies in a projective line ${\mathbb{P}}^1 \cong \vartheta({\mathbb{P}}(\operatorname{Sym}^2 W/I(Z)_2)) \subset {\mathbb{P}}^n$. On the other hand $\dim L^{\circ 2} = \dim L = 1$. Hence, $L^{\circ 2} =
\varphi_2(L)$ is a line in ${\mathbb{P}}^n$.
\[rem:lineMatroidInvariant\] We observe that the two possibilities in \[thm:squaringLines\] for the coordinate-wise square of a line $L$ differ in degree. In particular, \[cor:DegreeMatroidInvariant\] shows that it only depends on the linear matroid $\mathcal M_L$ whether $L^{\circ 2} $ is a line or a (re-embedded) plane conic.
In the Grassmannian of lines $\operatorname{Gr}(1, {\mathbb{P}}^n)$, consider the locus $\Gamma
\subset \operatorname{Gr}(1, {\mathbb{P}}^n)$ of those lines $L $ whose coordinate-wise square $L^{\circ 2}$ is a line. Considering Plücker coordinates $p_{ij}$ on the Grassmannian $\operatorname{Gr}(1, {\mathbb{P}}^n)$, we observe that $\Gamma$ is the subvariety of $\operatorname{Gr}(1, {\mathbb{P}}^n)$ given by the vanishing of $p_{ij} p_{jk} p_{ki}$ for all $i,j,k \in \{0,1, \ldots, n\}$ distinct: $$\Gamma = V(p_{ij} p_{jk} p_{ki} \mid i,j,k \in \{0,1, \ldots, n\} \text{
distinct}) \subset \operatorname{Gr}(1, {\mathbb{P}}^n).$$ Indeed, if $L$ is the image of an embedding ${\mathbb{P}}^1 \xhookrightarrow{B}
{\mathbb{P}}^n$ given by a chosen rank $2$ matrix $B \in {\mathbb{C}}^{(n+1)\times 2}$, then $Z
\subset ({\mathbb{P}}^1)^*$ is the set of points corresponding to the non-zero rows of $B$. Then $|Z|=2$ if and only if among any three distinct rows of $B$ there always exist two linearly dependent rows. In terms of the Plücker coordinates, which are given by the $2 \times 2$-minors of $B$, this translates into the vanishing condition above.
\[thm:squaringPlanes\] Let $L$ be a plane in ${\mathbb{P}}^n$. The defining ideal $I \subset {\ensuremath{{\mathbb{C}}[\mathbf{x}]}}$ of $L^{\circ
2}$ depends on the geometry of the planar configuration of $Z\subset {\mathbb{P}}W^*$ as follows (see Figure \[fig:PointConfiguration\]):
(i) If $Z$ is not contained in any conic, then $I$ is minimally generated by $n-5$ linear forms and 6 quadratic forms.
(ii) If $Z$ is contained in a unique conic $Q \subset {\mathbb{P}}W^*$, we distinguish two cases:
(a) If $Q$ is irreducible, then $I$ is minimally generated by $n-4$ linear forms and 7 cubic forms.
(b) If $Q$ is reducible, then $L^{\circ 2}$ is the complete intersection of $n-4$ hyperplanes and 2 quadrics.
(iii) If $Z$ is contained in several conics, we distinguish three cases:
(a) If $|Z| = 3$, then $I$ is minimally generated by $n-2$ linear forms.
(b) If $|Z| = 4$ and no three points of $Z$ are collinear, then $I$ is minimally generated by $n-3$ linear forms and one quartic form.
(c) If $|Z| \geq 3$ and all but one of the points of $Z$ lie on a line, then $I$ is minimally generated by $n-3$ linear forms and one quadratic form.
---------------- ---------- ---------- ----------- ----------- -----------
\[-0.2cm\] (i) (ii).(a) (ii).(b) (iii).(a) (iii).(b) (iii).(c)
---------------- ---------- ---------- ----------- ----------- -----------
Notice that $k=2$, so $\dim W = 3$.
(i) If $I(Z)_2 = 0$, then $L^{\circ 2} \subset {\mathbb{P}}^n$ is by \[lem:intrinsicDescription\] a linear re-embedding of the Veronese surface $\nu_2({\mathbb{P}}W) \subset {\mathbb{P}}\operatorname{Sym}^2 W$. The ideal of the $\nu_2({\mathbb{P}}W)$ is minimally generated by six quadrics. Indeed, choosing a basis for $W$, we may understand points in ${\mathbb{P}}\operatorname{Sym}^2 W$ as symmetric $3\times 3$-matrices up to scaling. Then $\nu_2({\mathbb{P}}W)$ is the subvariety corresponding to symmetric rank 1 matrices, which is the vanishing set of the six quadratic polynomials corresponding to the $2 \times 2$-minors. Since $\dim {\mathbb{P}}\operatorname{Sym}^2 W = 5$, the linear re-embedding ${\mathbb{P}}\operatorname{Sym}^2 W
\hookrightarrow {\mathbb{P}}^n$ adds $n-5$ linear forms to $I$.
(ii) We can choose a basis $\{z_0,z_1,z_2\}$ of $W$ such that the unique reduced plane conic through $Z \subset {\mathbb{P}}W^*$ is with respect to these coordinates given by the vanishing of either $q_1 := z_0^2-2z_1 z_2 \in \operatorname{Sym}^2 W$ or $q_2 := z_1 z_2 \in \operatorname{Sym}^2 W$.
We consider the basis $\{z_1^2, z_2^2, 2z_0 z_1, 2z_0 z_2, 2z_1 z_2\}$ of $\operatorname{Sym}^2 W/\langle q_1\rangle$ and the basis $\{z_0^2, z_1^2, z_2^2, 2z_0
z_1, 2z_0 z_2\}$ of $\operatorname{Sym}^2 W/\langle q_2\rangle$. With respect to these choices of bases, the morphism $\psi\colon {\mathbb{P}}W \to
{\mathbb{P}}(\operatorname{Sym}^2 W/I(Z)_2)$ is given as $$\begin{aligned}
&\psi\colon {\mathbb{P}}^2 \to {\mathbb{P}}^4, \quad [a_0:a_1:a_2] \mapsto [a_1^2 : a_2^2 :
a_0 a_1 : a_0 a_2 : a_0^2+a_1a_2] \\
\text{or} \qquad &\psi\colon {\mathbb{P}}^2 \to {\mathbb{P}}^4, \quad [a_0:a_1:a_2] \mapsto
[a_0^2 : a_1^2 : a_2^2 : a_0 a_1 : a_0 a_2].
\end{aligned}$$ In the first case, we checked computationally with `Macaulay2` [@M2] that the ideal is minimally generated by seven cubics. A structural description of these quadrics and cubics will be given in the proof of \[thm:uniqueQuadricCase\]. The image of the second morphism is a complete intersection of two binomial quadrics. By \[lem:intrinsicDescription\], the coordinate-wise square $L^{\circ 2}$ arises from the image of $\psi$ via a linear re-embedding ${\mathbb{P}}^4
\hookrightarrow {\mathbb{P}}^n$, producing additional $n-4$ linear forms in $I$.
(iii) In case (a), the set $Z$ consists of three points spanning the projective plane ${\mathbb{P}}W^*$, so $\dim \operatorname{Sym}^2 W/I(Z)_2 = 3$. Then by \[lem:intrinsicDescription\], the coordinate-wise square $L^{\circ 2}$ is contained in a plane ${\mathbb{P}}^2 \cong \vartheta({\mathbb{P}}(\operatorname{Sym}^2
W/I(Z)_2))
\subset {\mathbb{P}}^n$. On the other hand, $\dim L^{\circ 2} = \dim L = 2$, so $L^{\circ 2} \subset {\mathbb{P}}^n$ must be a plane in ${\mathbb{P}}^n$.
For case (b), we may assume that $$Z = \{[1:0:0],[0:1:0],[0:0:1],[-1:-1:-1]\}$$ for a suitably chosen basis $\{\ell_0,\ell_1,\ell_2\}$ of $W^*$. By \[lem:intrinsicDescription\], $L^{\circ 2} \subset {\mathbb{P}}^n$ is a linear re-embedding of the image of $\psi \colon {\mathbb{P}}W \to {\mathbb{P}}(\operatorname{Sym}^2
W/I(Z)_2)$. On the other hand, the plane $L' := V(x_0+x_1+x_2+x_3) \subset {\mathbb{P}}^3$ is the image of ${\mathbb{P}}W \xhookrightarrow{[\ell_0:\ell_1:\ell_2:-\ell_0-\ell_1-\ell_2]} {\mathbb{P}}^3$, so $Z$ can also be viewed as the finite set of points associated to $L'$. Applying \[lem:intrinsicDescription\] to $L' \subset {\mathbb{P}}^3$ shows that the image of $\psi \colon {\mathbb{P}}W \to {\mathbb{P}}(\operatorname{Sym}^2 W/I(Z)_2)$ is the coordinate-wise square ${L'}^{\circ 2} \subset {\mathbb{P}}^3$. Hence, $L^{\circ 2} \subset {\mathbb{P}}^n$ is a linear re-embedding of the quartic surface from \[ex:quarticSingularSurface\] into higher dimension.
Finally, we consider case (c). Consider three points $p_1,p_2,p_3 \in
Z$ lying on a line $T \subset {\mathbb{P}}W^*$. Then $T$ must be an irreducible component of each conic through $Z$. Since $Z$ spans the projective plane ${\mathbb{P}}W^*$, there must also be a point $p_0 \in Z$ outside of $T$. All points in $Z \setminus \{p_0\}$ must lie on the line $T$, as otherwise there could be at most one conic passing through $Z$. If $Z' := \{p_0,p_1,p_2,p_3\}
\subset Z$, then each conic passing through $Z'$ also passes through $Z$, i.e. $I(Z)_2 = I(Z')_2$.
We may choose a basis $z_0,z_1,z_2$ of $W$ such that $Z' \subset {\mathbb{P}}W^*$ with respect to these coordinates is given by $$Z' = \{[1:0:0],[0:1:0],[0:0:1],[0:1:1]\}.$$ The plane $L' := V(x_1+x_2-x_3) \subset {\mathbb{P}}^3$ is the image of ${\mathbb{P}}^2 \xhookrightarrow{[z_0:z_1:z_2:z_1+z_2]} {\mathbb{P}}^3$, so $Z'$ can be viewed as the finite set of points associated to $L'$. \[lem:intrinsicDescription\] shows that ${L'}^{\circ 2} \subset {\mathbb{P}}^3$ coincides with the image of the morphism $\psi \colon {\mathbb{P}}W \to {\mathbb{P}}(\operatorname{Sym}^2 W/I(Z')_2)$. On the other hand, \[lem:intrinsicDescription\] shows that $L^{\circ 2} \subset {\mathbb{P}}^n$ is a linear re-embedding of ${\mathbb{P}}W \to {\mathbb{P}}(\operatorname{Sym}^2
W/I(Z)_2)$. From $I(Z)_2 = I(Z')_2$, we deduce that $L^{\circ
2} \subset {\mathbb{P}}^n$ is a linear re-embedding of the quadratic surface $${L'}^{\circ 2} = V(x_1+x_2-x_3)^{\circ 2} = V(x_1^2+x_2^2+x_3^2-2x_1
x_2-2x_2 x_3
-2x_3x_1) \subset {\mathbb{P}}^3,$$ as we compute from \[prop:hypersurfaces\].
\[rem:planeNoMatroidInvariant\] Opposed to \[rem:lineMatroidInvariant\], the structure of the coordinate-wise square of a plane $L \subset {\mathbb{P}}^n$ does *not* only depend on the linear matroid of $L$: For $n=5$, it can happen both in case (i) and case (ii).(a) of \[thm:squaringPlanes\] that $\mathcal M_L = \{I \subset \{0,1,\ldots,5\} \mid |I| \leq 3\}$.
Squaring in high ambient dimensions {#ssec:squaringHighDim}
-----------------------------------
Consider the case of $k$-dimensional linear spaces in ${\mathbb{P}}^n$ for $n \gg k$. For a *general* linear space $L \in \operatorname{Gr}(k,{\mathbb{P}}^n)$, the finite set of points $Z$ does not lie on a quadric. We know from \[prop:NoQuadrics\] that the coordinate-wise square $L^{\circ 2}$ is a linear re-embedding of the $k$-dimensional second Veronese variety. In this subsection, we investigate the first degenerate case where the point configuration $Z$ is a unique quadric.
The following theorem gives the structure of coordinate-wise squares as the one appearing in \[prop:eigenvaluesEarly\]. We will also prove \[cor:eigenvalues\] by deriving the polynomials vanishing on the set of symmetric matrices with a comultiplicity $1$ eigenvalue. \[prop:eigenvaluesEarly\] shows that \[cor:eigenvalues\] is a special case of the theorem stated below.
\[thm:uniqueQuadricCase\] Let $L \subset {\mathbb{P}}^n$ be linear space of dimension $k$. If the point configuration $Z$ lies on a unique quadric of rank $s$, then $L^{\circ 2}$ can set-theoretically be described as the vanishing set of $n-\binom{k+2}{2}+2$ linear forms and $$\begin{cases}
(k+3)(k+2)(k+1)(k-2)/12 \text{ quadratic forms}, &\text{if } s \geq 4, \\
(k+3)(k+2)(k+1)(k-2)/12 \text{ quadratic and 7 cubic forms}, &\text{if }
s = 3, \\
(k+3)(k+2)(k+1)(k-2)/12 + 2 \text{ quadratic forms}, &\text{if } s = 2.
\end{cases}$$
In fact, for $s \geq 3$, we show that the claim holds *scheme-theoretically*, see Remark \[rem:schemeTheoretic\]. We believe that in fact for arbitrary $s$ the claim is even true *ideal-theoretically*.
The remainder of this subsection is dedicated to the proof of \[thm:uniqueQuadricCase\]. It reduces to the following elimination problem. Let $k \geq 1$ and $s \geq 2$. Consider a symmetric $(k+1)\times (k+1)$-matrix of variables $Y := (y_{ij})_{1\leq i,j \leq k+1}$ and the corresponding polynomial ring ${\ensuremath{{\mathbb{C}}[\mathbf{y}]}}:= {\mathbb{C}}[y_{ij} ]/(y_{ij}-y_{ji}).$ Over the polynomial ring ${\ensuremath{{\mathbb{C}}[\mathbf{y},t]}}$, we consider the matrix $M := Y+tI_s$, where we define the matrix $$I_s := \operatorname{diag}(\underbrace{1,\ldots,1}_{s}, \underbrace{0,\ldots,0}_{k+1-s})
\in {\mathbb{C}}^{(k+1)\times(k+1)}.$$
Henceforth, we denote the $2\times 2$-minors of $Y$ with rows $i\neq j$ and columns $\ell \neq m$ by $Y_{ij|\ell m} := y_{i\ell} y_{j m} - y_{im} y_{j \ell} \in {\ensuremath{{\mathbb{C}}[\mathbf{y}]}},$ and correspondingly $M_{ij|\ell m} \in {\ensuremath{{\mathbb{C}}[\mathbf{y},t]}}$ for the $2 \times 2$-minors of $M$. Let $J_0 \subset {\ensuremath{{\mathbb{C}}[\mathbf{y},t]}}$ denote the ideal generated by the $2
\times 2$-minors of $M$. By $J := J_0 \cap {\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$ we denote the ideal in ${\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$ obtained by eliminating $t$ from $J_0$. We explicitly describe the elimination ideal $J$ for all values of $k$ and $s$.
\[prop:matrixCompletion\] The vanishing set $V(J) \subset {\mathbb{P}}^{\binom{k+2}{2}-1}$ can set-theoretically be described as the zero set of $$\begin{cases}
(k+3)(k+2)(k+1)(k-2)/12 \text{ quadratic forms}, &\text{if } s \geq 4, \\
(k+3)(k+2)(k+1)(k-2)/12 \text{ quadratic and 7 cubic forms}, &\text{if }
s = 3, \\
(k+3)(k+2)(k+1)(k-2)/12 + 2 \text{ quadratic forms}, &\text{if } s = 2.
\end{cases}$$
First, we observe that \[thm:uniqueQuadricCase\] follows directly from \[prop:matrixCompletion\].
Analogous to the proof of \[prop:eigenvaluesEarly\], we identify ${\mathbb{P}}\operatorname{Sym}^2 W$ with ${\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1}$ such that $q = I_s$. By \[lem:intrinsicDescription\], the coordinate-wise square $L^{\circ 2} $ is a linear re-embedding of the variety obtained by the projection of $\nu_2({\mathbb{P}}W)$ from the point $q=I_s
\in {\mathbb{P}}\operatorname{Sym}^2 W$. Note that $V(J)$ describes the set of points $Y\in {\mathbb{P}}\operatorname{Sym}^2 W$ lying on the line joining $q $ with some point in $\nu_2({\mathbb{P}}W)$. Hence, the projection from $q$ is given by intersecting $V(J)$ with a hyperplane $H \subset {\mathbb{P}}\operatorname{Sym}^2 W$ not containing $q = I_s$.
From \[prop:matrixCompletion\], we know that $V(J) \cap H$ has a set-theoretic description inside $H \cong {\mathbb{P}}^{\binom{k+2}{2}-2}$ as the zero set of the indicated number of quadric and cubic forms. The coordinate-wise square $L^{\circ
2}$ is by \[lem:intrinsicDescription\] the image of $V(J)
\cap H$ under a linear embedding $\vartheta\colon H \hookrightarrow {\mathbb{P}}^n$, leading to additional $n-\binom{k+2}{2}+2$ linear forms vanishing on $L^{\circ 2}$.
We prove \[prop:matrixCompletion\] in several steps. First, we describe a set $\mathcal{X}$ of certain low-degree polynomials in the ideal $J$. Secondly, we show that $V(\mathcal{X}) = V(J)$. Finally, we identify a subset of $\mathcal{X}$ providing minimal generators of the ideal $(\mathcal
X) \subset {\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$, consisting of the claimed number of quadratic and cubic forms.
The following sets of polynomials in ${\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$ are contained in the ideal $J$: $$\begin{aligned}
\mathcal{E} &:= \{Y_{ij|\ell m} \:\mid \:\{i,j\} \cap \{\ell,m\} \subset
\{s+1, \ldots,k+1\}\}, \\
\mathcal{F} &:= \{Y_{i\ell|i m} - Y_{j\ell|jm} \:\mid\: i, j \leq s,\:
\{\ell\}
\cap \{m\} \subset \{s+1, \ldots,k+1\}\}, \\
\mathcal{G} &:=
\{Y_{ij|ij} - Y_{j\ell|j\ell} + Y_{\ell m|\ell m} - Y_{m i|m i} \:\mid\:
i, j, \ell, m \leq s \text{ distinct}\},\\
\mathcal{H}_1 &:=
\{y_{i\ell}(Y_{ij|ij}-Y_{i\ell|i\ell})-(y_{\ell \ell}-y_{jj})Y_{ij|j\ell}
\mid
i,j,\ell \leq s\}, \\
\mathcal{H}_2 &:=
\{(y_{ii}-y_{jj})Y_{ij|ij}+(y_{jj}-y_{\ell \ell})Y_{j \ell|j \ell}+(y_{\ell
\ell}-y_{ii})Y_{\ell i|\ell i}
\mid i,j, \ell \leq s\}.\end{aligned}$$
Using that $$\label{eq:minorsOfMY}
\begin{aligned}
Y_{ij|ij} &= M_{ij|ij}-(y_{ii}+y_{jj})t-t^2 \qquad \text{for all $i,j
\leq s$
distinct and} \\
Y_{i\ell|j\ell} &= M_{i \ell|j \ell} - ty_{ij} \qquad \hspace{2.11cm}
\text{for all $\ell
\leq s$, $\{i\} \cap \{j\} \subset \{s+1,\ldots,k+1\}$},
\end{aligned}$$ we can check that $$\begin{aligned}
Y_{ij|\ell m} &= M_{ij|\ell m}, \\
Y_{i\ell|i m} - Y_{j\ell|jm} &= M_{i\ell|i m} - M_{j\ell|jm}, \\
Y_{ij|ij} - Y_{j\ell|j\ell} + Y_{\ell m|\ell m} - Y_{m i|m i} &= M_{ij|ij}
- M_{j\ell|j\ell} + M_{\ell m|\ell m} - M_{m i|m
i}, \\
y_{i\ell}(Y_{ij|ij}-Y_{i\ell|i\ell})-(y_{\ell \ell}-y_{jj})Y_{ij|j\ell} &=
y_{i\ell}(M_{ij|ij}-M_{i\ell|i\ell})-(y_{\ell \ell}-y_{jj})M_{ij|j\ell}, \\
(y_{ii}-y_{jj})Y_{ij|ij}+(y_{jj}-y_{\ell \ell})Y_{j \ell|j \ell}+(y_{\ell
\ell}-y_{ii})Y_{\ell i|\ell i} &=
(y_{ii}-y_{jj})M_{ij|ij}+(y_{jj}-y_{\ell \ell})M_{j \ell|j
\ell}+(y_{\ell \ell}-y_{ii})M_{\ell i|\ell i}
\end{aligned}$$ holds for respective indices $i,j,\ell,m$. From this, we conclude that these polynomials are contained in $J_0 \cap {\ensuremath{{\mathbb{C}}[\mathbf{y}]}}= J$.
From now on, we denote $\mathcal{X} := \mathcal{E} \cup \mathcal{F} \cup \mathcal{G} \cup
\mathcal{H}_1 \cup \mathcal{H}_2$. These polynomials describe $V(J)$:
\[lem:setTheoreticEquality\] Inside ${\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1} = {\mathbb{P}}^{\binom{k+2}{2}-1}$, we consider the open sets $$U_1 := {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1} \setminus \{I_s\} \qquad \text{and} \qquad U_2
:=
{\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1} \setminus
{\mathopen{}\mathclose\bgroup\originalleft}\{{\mathopen{}\mathclose\bgroup\originalleft}(\begin{smallmatrix} \begin{smallmatrix}* & * \\ * & *
\end{smallmatrix} & 0 \\ 0 & 0\end{smallmatrix}{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright}\}.$$
(i) If $s \geq 3$, then $V(\mathcal X)$ and $V(J)$ agree scheme-theoretically on $U_1$.
(ii) If $s = 2$, then $V(\mathcal X)$ and $V(J)$ agree scheme-theoretically on $U_2$.
(iii) For $s$ arbitrary, $V(\mathcal X)$ and $V(J)$ agree set-theoretically.
For $k \leq 5$, we have checked computationally with a straightforward implementation in `Macaulay2` [@M2] that even the ideal-theoretic equality $(\mathcal X) = J$ holds. We now argue that from this we can conclude the claim for arbitrary $k$.
(i) Let $s \geq 3$. We need to show that the ideal generated by $\mathcal
X
\subset {\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$ coincides with $J \subset
{\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$ after localisation at any element in the set $$\{y_{ij} \mid \{i\} \cap
\{j\} \subset \{s+1, \ldots,k+1\}\} \cup \{y_{ii}-y_{jj} \mid i,j \leq
s\},$$ since the union of the corresponding non-vanishing sets $D(y_{ij}),
D(y_{ii}-y_{jj})$ is $U_1$.
In order to show that $(\mathcal X)$ and $J$ agree after localisation at $y_{i_0j_0}$ for $\{i_0\} \cap \{j_0\} \subset \{s+1,\ldots,k+1\}$, we may substitute $y_{i_0 j_0} = 1$ in both ideals. For a fixed $\ell_0
\leq s$ distinct from $i_0$ and $j_0$, we note that $t+Y_{i_0\ell_0|j_0\ell_0} = M_{i_0 \ell_0| j_0 \ell_0} \in
{{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J_0
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0}=1}
}}.$ Hence, eliminating $t$ from ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J_0
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0}=1}
}}$ just amounts to replacing $t = -Y_{i_0 \ell_0|j_0 \ell_0}$ in each occurrence of $t$ in the minors $M_{ij|\ell m}$ (for $i \neq j$, $\ell \neq m$) generating the ideal $J_0$.
According to , this leads to the following generators of ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0} = 1}
}}$:
- [>p[0.4]{}>p[0.4]{}]{} $Y_{i_0
\ell_0|j_0\ell_0}^2-(y_{ii}+y_{jj})Y_{i_0
\ell_0|j_0\ell_0}+Y_{ij|ij}$ & for $i\neq j \leq s$,
- [>p[0.3]{}>p[0.5]{}]{} $-y_{ij}Y_{i_0 \ell_0|j_0\ell_0}+Y_{i \ell|j\ell}$ & for $\ell \leq s, \{i\} \cap \{j\} \subset \{s+1,\ldots,k+1\}$,
- [>p[0.1]{}>p[0.7]{}]{} $Y_{ij|\ell m}$ & for $\{i,j\}\cap \{\ell,m\}
\subset\{s+1,\ldots,k+1\}$.
To check that ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0} = 1}
}} = {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
(\mathcal X)
{\aftergroup\egroup\originalright}|_{y_{i_0
j_0} = 1}
}}$, we need to check that each of these polynomials belong to ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
(\mathcal X)
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0} =
1}
}}$. For this, it is enough to see that they can be expressed in terms of those polynomials in $\mathcal
X$ that only involve variables with indices among $\{i_0,j_0,\ell_0,i,j,\ell\}$. This corresponds to showing the claim for a corresponding symmetric submatrix of $M$ of size at most $6 \times 6$. We conclude that it is enough to check ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0} = 1}
}} = {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
(\mathcal X)
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0} = 1}
}}$ for $k
\leq 5$.
Similarly, in order to show that ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J
{\aftergroup\egroup\originalright}|_{y_{i_0 i_0}-y_{j_0 j_0}=1}
}} = {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
(\mathcal X)
{\aftergroup\egroup\originalright}|_{y_{i_0
i_0}-y_{j_0 j_0}=1}
}}$ holds for $i_0,j_0 \leq s$ distinct, we realise that $t+Y_{i_0 \ell_0|i_0 \ell_0}-Y_{j_0 \ell_0|j_0 \ell_0} = M_{i_0 \ell_0|i_0
\ell_0} - M_{j_0 \ell_0|j_0 \ell_0} \in {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J_0
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0}=1}
}}$ holds for fixed $\ell_0 \leq s$ distinct from $i_0$ and $j_0$. Therefore, replacing $t = Y_{j_0 \ell_0|j_0 \ell_0}-Y_{i_0
\ell_0|i_0 \ell_0}$ in the expressions for the $2 \times 2$-minors of $M$ describes generators of ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J
{\aftergroup\egroup\originalright}|_{y_{i_0 i_0}-y_{j_0 j_0}=1}
}}$. As before, these polynomials involve variables with at most six distinct indices, so it is enough to verify the claim for $k \leq 5$ by the same argument as above.
(ii) For $s = 2$, the argument from (i) still shows ${{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
J_0
{\aftergroup\egroup\originalright}|_{y_{i_0
j_0} = 1}
}} = {{
{\mathopen{}\mathclose\bgroup\originalleft}.\kern-\nulldelimiterspace
(\mathcal X)
{\aftergroup\egroup\originalright}|_{y_{i_0 j_0} = 1}
}}$ for $\{i_0, j_0\}
\cap \{3,\ldots,k+1\} \neq \emptyset$. For the localisation at $y_{12}$ and at $y_{11}-y_{22}$, the argument does not apply since we cannot choose $\ell_0$ distinct from $\{i_0,j_0\} = \{1,2\}$ as before. Hence, we have shown the equality of $V(\mathcal X)$ and $V(J)$ only on $U_2$.
(iii) We observe that the polynomials in $\mathcal X$ vanish on the point $I_s \in {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1}$, and that $I_s \in V(J)$ by definition of $J$. Together with (i), this proves the claim for $s \geq 3$.
For $s = 2$, the polynomials in $\mathcal X$ vanish on all symmetric matrices of the form $A = {\mathopen{}\mathclose\bgroup\originalleft}(\begin{smallmatrix} \begin{smallmatrix}a & c \\ c & b
\end{smallmatrix} & 0 \\ 0 & 0\end{smallmatrix}{\aftergroup\egroup\originalright}) \in \operatorname{Sym}^2
{\mathbb{C}}^{k+1}$. On the other hand, each such matrix is a point in $V(J)$, since $A+t_0 I_2$ is a matrix of rank $\leq 1$ for $t_0 \in {\mathbb{C}}$ such that $t_0^2+(a+b)t_0+(ab-c^2)=0$. Together with (ii), we conclude that $V(\mathcal X) = V(J)$ holds set-theoretically.
\[lem:linearIndependenceAmongSets\] The vector spaces spanned by the polynomials in $\mathcal X$ satisfy:
(i) $\langle \mathcal E \cup \mathcal F \cup \mathcal G
\rangle = \langle \mathcal E \rangle \oplus \langle \mathcal F \rangle
\oplus
\langle \mathcal G \rangle,$
(ii) $\langle \mathcal H_1 \cup
\mathcal H_2 \rangle \cap (\mathcal E, \mathcal F, \mathcal G) = \emptyset$ for $s = 3$,
(iii) $\mathcal H_1 \cup \mathcal H_2 \subset (\mathcal E, \mathcal F,
\mathcal G)$ for $s \neq 3$.
Let $\mathcal M_{\mathcal E} \subset {\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$ denote the set of monomials occurring in one of the polynomials of $\mathcal E$, and analogously for $\mathcal F$, $\mathcal G$, $\mathcal H_1$ and $\mathcal H_2$.
(i) This follows from the observation that $\mathcal{M}_{\mathcal E}$, $\mathcal{M}_{\mathcal
F}$ and $\mathcal{M}_{\mathcal G}$ are disjoint sets.
(ii) For $s = 3$, note that $\mathcal G = \emptyset$ and none of the monomials in $\mathcal M_{\mathcal E} \cup \mathcal M_{\mathcal F}$ is of the form $y_{ij}y_{\ell m}$ with $i,j,\ell,m \leq 3$. On the other hand, the monomials in $\mathcal M_{\mathcal H_1} \cup \mathcal M_{\mathcal H_2}$ are of the form $y_{i_1 j_1} y_{i_2 j_2} y_{i_3 j_3}$ with $i_1, i_2,
i_3, j_1, j_2, j_3 \leq 3$. Hence, no monomial in $\mathcal M_{\mathcal H_1}
\cup \mathcal M_{\mathcal H_2}$ is a multiple of any of the monomials in $\mathcal M_{\mathcal E} \cup \mathcal M_{\mathcal F}$, so $\langle \mathcal
H_1 \cup
\mathcal H_2 \rangle \cap (\mathcal E, \mathcal F, \mathcal G) = \emptyset$.
(iii) If $s = 2$ we have $\mathcal H_1 \cup \mathcal H_2=\emptyset$, so the claim is trivial. Let $s \geq 4$. Then for all $i,j,\ell,m \leq s$ distinct, we have $$\begin{aligned}
&\hspace{0.75cm}y_{i\ell}(Y_{ij|ij}-Y_{i\ell|i\ell})-(y_{\ell
\ell}-y_{jj})Y_{ij|j\ell} \\
&\hspace{0.75cm}= -2y_{j m}Y_{i j| \ell m}-y_{j m}Y_{i m| j \ell} -y_{i
\ell}(Y_{i
\ell|
i
\ell}-Y_{\ell j| \ell j}+Y_{j m| j
m}-Y_{m i| m i})+y_{i m}(Y_{j \ell| j m}-Y_{i \ell| i m})\\
&\hspace{0.75cm}\phantom{{}=} +y_{i j}(Y_{i j| i
\ell}-Y_{m j|
m \ell})+(y_{i i}-y_{j j})(Y_{i j| \ell j}-Y_{i m| \ell m})-y_{j
\ell}(Y_{i
\ell| j \ell}-Y_{i m| j m}) \in (\mathcal E, \mathcal F, \mathcal G),
\\[0.3em]
&\hspace{0.75cm}(y_{ii}-y_{jj})Y_{ij|ij}+(y_{jj}-y_{\ell \ell})Y_{j \ell|j
\ell}+(y_{\ell
\ell}-y_{ii})Y_{\ell i|\ell i} \\
&\hspace{0.75cm}= (y_{i i}-y_{j j})(Y_{i j| i
j}-Y_{j \ell| j \ell}+Y_{\ell m| \ell m}-Y_{m i| m i})+(y_{\ell
\ell}-y_{i i})(Y_{i \ell| i \ell}-Y_{\ell j| \ell j}+Y_{j m| j m}-Y_{m i|
m i})\\
&\hspace{0.75cm}\phantom{{}=}+y_{\ell m}(Y_{i \ell| i m}-Y_{j \ell| j
m})-y_{j m}(Y_{i j|
i m}-Y_{\ell j| \ell m})+y_{i m}(Y_{j i| j m}-Y_{\ell i| \ell m}) \in
(\mathcal E, \mathcal F, \mathcal G),
\end{aligned}$$ so $\mathcal H_1$ and $\mathcal H_2$ lie in the ideal generated by $\mathcal
E$, $\mathcal F$ and $\mathcal G$.
Next, we identify maximal linearly independent subsets of $\mathcal E$, $\mathcal F$, $\mathcal G$.
\[lem:basesEFG\] The following sets are bases for the vector spaces $\langle \mathcal E
\rangle$, $\langle \mathcal F \rangle$ and $\langle \mathcal G \rangle$: $$\begin{aligned}
\mathcal B_{\mathcal E} &:= \{Y_{ij|\ell m} \mid i < j, \ell < m, i
\leq
\ell \leq j \text{ s.t.\ } \{i,j\} \cap \{\ell,m\} \subset \{s+1, \ldots,
k+1\} \text{ and } j \leq m \text{ if } i=\ell\}, \\
\mathcal B_{\mathcal F} &:= \{Y_{i\ell|i m} - Y_{1\ell|1m} \mid 2
\leq i
\leq s, \: 2 \leq \ell \leq m \text{ s.t. } i \notin
\{\ell, m\}, \: \{\ell\} \cap \{m\} \subset \{s+1, \ldots,k+1\}\} \\
&\phantom{{}:={}}\cup \{Y_{i1|im} - Y_{21|2m} \mid 3 \leq i \leq s,\:
3 \leq m \leq k+1,\: i \neq m\}
\; \cup\; \{Y_{i1|i2} - Y_{31|32} \mid i \in
\{4,\ldots,s\}\}, \\
\mathcal B_{\mathcal G} &:= \{Y_{12|12} - Y_{2\ell|2\ell} + Y_{\ell m|\ell
m} - Y_{m1|m1} \:\mid\: 3 \leq m \leq s-1,\: \ell \in \{3,4,\ldots,m-1\}
\cup
\{s\}\} \\
&\phantom{{}:={}}\cup \{Y_{1s|1s} - Y_{s2|s2} + Y_{2m|2m} - Y_{m1|m1}
\:\mid\: 3 \leq m \leq s-1\}.
\end{aligned}$$
The polynomials in $\mathcal E$ not contained in $\mathcal
B_{\mathcal E} \cup (- \mathcal B_{\mathcal E})$ are the polynomials $Y_{ij|\ell m}$ for $i < j <
\ell < m$. However, these can be expressed as $Y_{ij|\ell m} = Y_{i \ell|j
m}-Y_{im|j \ell} \in \langle \mathcal B_{\mathcal E}\rangle$. Hence $\mathcal B_{\mathcal E}$ spans $\langle \mathcal E \rangle$. For $i <
j$, $\ell < m$ with $i \leq \ell \leq j$ such that $\{i,j\} \cap
\{\ell,m\} \subset \{s+1, \ldots, k+1\}$, we note that $Y_{ij|\ell m} \in
{\ensuremath{{\mathbb{C}}[\mathbf{y}]}}$ is the unique polynomial in $\mathcal B_{\mathcal
E}$ containing the monomial $y_{im} y_{\ell j}$. In particular, the polynomials in $\mathcal B_{\mathcal E}$ are linearly independent, so $\mathcal B_{\mathcal E}$ forms a basis of $\langle \mathcal E \rangle$.
If $i,j,\ell,m \in \{1,\ldots,k+1\}$ with $\ell < m$ are such that $Y_{i\ell|i m} - Y_{j\ell|jm} \in \mathcal F \setminus (\mathcal B_{\mathcal
F} \cup -\mathcal B_{\mathcal F})$, then $$Y_{i\ell|i m} - Y_{j\ell|jm} = \begin{cases}
(Y_{i\ell|i m} - Y_{1\ell|1m})-(Y_{j\ell|j m} - Y_{1\ell|1m}) &\text{if }
\ell, m \neq 1, \\
(Y_{i1|im} - Y_{21|2m})-(Y_{j1|jm} - Y_{21|2m}) &\text{if }
\ell = 1, m \neq 2, \\
(Y_{i1|i2} - Y_{31|32})-(Y_{j1|j2} - Y_{31|32}) &\text{if }
\ell = 1, m = 2,
\end{cases}$$ so $\mathcal B_{\mathcal F}$ spans $\langle \mathcal F \rangle$. Each of the polynomials $Y_{i\ell|i m} - Y_{j\ell|jm}$ in $\mathcal B_{\mathcal F}$ contains a monomial not occurring in any of the other polynomials of $\mathcal B_{\mathcal F}$, namely $y_{ii} y_{\ell m}$. Therefore, the polynomials in $\mathcal B_{\mathcal F}$ are linearly independent.
For $3 \leq m \leq s-1$ and $\ell \in \{3,\ldots,m-1\} \cup \{s\}$, the polynomial $Y_{12|12} - Y_{2\ell|2 \ell} + Y_{\ell m|\ell m} - Y_{m1|m1}$ is the unique polynomial in $\mathcal B_{\mathcal G}$ containing the monomial $y_{\ell \ell} y_{m m}$. In particular, if a linear combination of polynomials in $\mathcal B_{\mathcal G}$ is zero, none of the above polynomials can occur in this linear combination. The remaining polynomials in $\mathcal B_{\mathcal G}$ are of the form $Y_{1s|1s} - Y_{s2|s2} +
Y_{2m|2m} - Y_{m1|m1}$ for $3 \leq m \leq s-1$. Among these, the polynomial containing the monomial $y_{22} y_{mm}$ is unique. We conclude that the polynomials in $\mathcal B_{\mathcal G}$ are linearly independent.
We observe that $$\mathcal G \subset \big\{\sum_{i,j=1}^s a_{ij} Y_{ij|ij} \mid A=(a_{ij})
\in
{\mathbb{C}}^{s \times s} \text{ symmetric with } a_{ii} = 0 \text{ and }
(1,\ldots,1) A
=0\big\}.$$ The vector space of symmetric $s \times s$-matrices with zero diagonal and whose columns all sum to zero is of dimension $\binom{s}{2}-s$, so $\dim_{\mathbb{C}}\langle \mathcal G\rangle \leq \binom{s}{2}-s$. On the other hand, we can count that $|\mathcal B_{\mathcal G}| = \binom{s-3}{2}+2(s-3) =
\binom{s}{2}-s$, so $\mathcal B_{\mathcal G}$ is a basis of $\langle \mathcal G \rangle$.
By \[lem:setTheoreticEquality\], $V(J) = V(\mathcal X)$ holds set-theoretically. For $s = 3$, we observe that $\mathcal H_1 \cup \mathcal H_2$ consists up to sign of seven linearly independent cubics, so by \[lem:linearIndependenceAmongSets\], the ideal $(\mathcal X)$ is in this case minimally generated by those seven cubics and the polynomials in $\mathcal B_{\mathcal E}$, $\mathcal B_{\mathcal F}$ and $\mathcal
B_{\mathcal G}$ from \[lem:basesEFG\].
For $s \neq 3$, \[lem:linearIndependenceAmongSets\] and \[lem:basesEFG\] show that $(\mathcal X)$ is minimally generated just by the polynomials $\mathcal B_{\mathcal E} \cup \mathcal B_{\mathcal F} \cup
\mathcal B_{\mathcal G}$. Straightforward counting gives: $$\begin{aligned}
|\mathcal B_{\mathcal E}| &= {\textstyle 2 \binom{k+1}{4} + (k-s+1)
\binom{k}{2} + \binom{k-s+1}{2}} \\
&=(k^4-6sk^2+4k^3+6s^2-6sk+5k^2-6s+2k)/12,
\\
|\mathcal B_{\mathcal F}| &= {\textstyle (s-1)
{\mathopen{}\mathclose\bgroup\originalleft}(\binom{k-1}{2}+(k-s+1){\aftergroup\egroup\originalright}) +
(s-2)(k-2) + \binom{s-3}{1}} \\
&=
\begin{cases}
(sk^2-k^2+sk-2s^2-3k+4s-2)/2 &\text{if } s \geq 3, \\
(sk^2-k^2+sk-2s^2-3k+4s)/2 &\text{if } s = 2,
\end{cases} \\
|\mathcal B_{\mathcal G}| &=
\begin{cases}
{\textstyle \binom{s}{2}-s = (s^2-3s)/2} &\text{if }s \geq 3, \\
0 &\text{if } s=2.
\end{cases}
\end{aligned}$$ Adding up these cardinalities gives the claimed number of quadratic forms.
\[rem:schemeTheoretic\] In fact, for $s \geq 3$, our proof shows that $V(\mathcal X)$ is the same scheme as $V(J)$ away from the point $I_s \in {\mathbb{P}}\operatorname{Sym}^2 {\mathbb{C}}^{k+1}$. In the proof of \[thm:uniqueQuadricCase\], we considered $V(J) \cap H$, where $H$ is a hyperplane not containing $I_s$. Since $V(J) \cap H = V(\mathcal X) \cap H$ scheme-theoretically, we conclude that our equations for $L^{\circ 2}$ in \[thm:uniqueQuadricCase\] describe not only the correct set, but even the correct scheme. In fact, we believe that we have ideal-theoretic equality for the specified set of polynomials, but our proof stops short of verifying this.
We now prove the result about eigenspaces of symmetric matrices stated as \[cor:eigenvalues\]. It follows directly from the proof of \[prop:matrixCompletion\].
A complex symmetric matrix $A \in {\mathbb{C}}^{s \times s}$ has an eigenspace of codimension 1 with respect to an eigenvalue $\lambda \in {\mathbb{C}}$ if and only if the matrix $A-\lambda \operatorname{id}$ is of rank 1, which means that $A \in
V(J)$ for the case $s = k+1$. By \[lem:setTheoreticEquality\] and \[lem:linearIndependenceAmongSets\], this is equivalent to the vanishing of the equations ${\mathcal E\cup \mathcal F \cup \mathcal G}$, which are the above relations among $2\times 2$-minors for $s = k+1 \geq 4$. The second claim was proved in \[prop:eigenvaluesEarly\].
The proof of \[thm:uniqueQuadricCase\] was based on relating the coordinate-wise square $L^{\circ 2}$ in the case $\dim_{\mathbb{C}}I(Z)_2 = 1$ to the question when a symmetric matrix can by completed to a rank 1 matrix by adding a multiple of $I_s$. In the same spirit, for *arbitrary* linear spaces $L $ (no restrictions on the set of quadrics containing $Z $), determining the ideal of the coordinate-wise square $L^{\circ 2}$ boils down to the following problem in symmetric rank 1 matrix completion:
\[prob:MatrixProblem\] For a fixed matrix $B \in {\mathbb{C}}^{(n+1) \times (k+1)}$ of rank $k+1$, find the defining equations of the set $${\mathopen{}\mathclose\bgroup\originalleft}\{M \in {\mathbb{C}}^{(k+1)\times(k+1)} \text{ symmetric } \mid
\begingroup \footnotesize \begin{array}{c}
\exists P \in {\mathbb{C}}^{(k+1)\times(k+1)} \text{ symmetric such that }
BPB^T \\ \text{ has a zero diagonal and } \operatorname{rk}(M+P) = 1
\end{array} \endgroup {\aftergroup\egroup\originalright}\}.$$
Indeed, let $L $ be an arbitrary linear space of dimension $k$ and let $B \in {\mathbb{C}}^{(n+1) \times (k+1)}$ be a chosen matrix of full rank describing $L$ as the image of the linear embedding ${\mathbb{P}}^k \hookrightarrow {\mathbb{P}}^n$ given by $B$. Then the rows of $B$ form the finite set of points $Z \subset ({\mathbb{P}}^k)^*$. Identifying quadratic forms on ${\mathbb{P}}^k$ with symmetric $(k+1)\times
(k+1)$-matrices, the subspace $I(Z)_2 \subset \operatorname{Sym}^2 ({\mathbb{C}}^{k+1})^*$ corresponds to $$I(Z)_2 = \{P \in {\mathbb{C}}^{(k+1)\times(k+1)} \text{ symmetric such that }
BPB^T \\ \text{ has a zero diagonal}\}.$$ By \[lem:intrinsicDescription\], the coordinate-wise square $L^{\circ 2}
$ is a linear re-embedding of projecting the second Veronese variety $$\nu_2({\mathbb{P}}^k) = \{\text{rank~$1$ symmetric $(k+1)\times (k+1)$-matrices up to
scaling}\}$$ from ${\mathbb{P}}(I(Z)_2)$, so describing the ideal of $L^{\circ 2}$ corresponds to solving \[prob:MatrixProblem\] for the given matrix $B$. Similarly, describing the coordinate-wise $r$-th power of a linear space corresponds to the analogous problem in symmetric rank $1$ tensor completion.
By \[lem:intrinsicDescription\], determining the coordinate-wise $r$-th power of a linear space corresponds to describing the projection of the $r$-th Veronese variety from a linear space of the form ${\mathbb{P}}(I(Z)_r)$ for a non-degenerate finite set of points $Z$. We may ask how general this problem is, and pose the question which linear subspaces of ${\mathbb{P}}\operatorname{Sym}^r W$ are of the form ${\mathbb{P}}(I(Z)_r)$:
\[qu:whichProjections\] Which linear subspaces of ${\mathbb{C}}[z_0,\ldots,z_k]_r$ can be realised as the set of degree $r$ polynomials vanishing on some non-degenerate finite set of points in ${\mathbb{P}}^k$ of cardinality $\leq n+1$?
We envision that an answer to this question may lead to insights into describing which varieties can occur as the coordinate-wise $r$-th power of some linear space in ${\mathbb{P}}^n$.
[BBBKR17]{}
M. Brandt, J. Bruce, T. Brysiewicz, R. Krone, and E. Robeva: [*The degree of $\mathrm{SO}(n,\mathbb{C})$*]{}, Combinatorial Algebraic Geometry: Selected Papers From the 2016 Apprenticeship Program, Ed. by G. G. Smith and B. Sturmfels. Springer New York, New York, NY, 229–246 (2017).
C. Bocci, E. Carlini, and J. Kileel: [*Hadamard products of linear spaces.*]{} [ J. Algebra]{}, [**448**]{}, 595–617, (2016).
C. [Bonnaf[é]{}]{}: [*A surface of degree $24$ with $1440$ singularities of type $D_4$*]{}, arXiv: 1804.08388.
C. [Bocci]{}, G. [Calussi]{}, G. [Fatabbi]{}, and A. [Lorenzini]{}: [*The Hilbert function of some Hadamard products*]{}, [Collectanea Mathematica]{}, [**69**]{}, 205–220, (2018).
G. [Calussi]{}, E. [Carlini]{}, G. [Fatabbi]{}, and A. [Lorenzini]{}: [*On the Hadamard product of degenerate subvarieties*]{}, arXiv: 1804.01388.
O. Chterental and D. Ž. oković: [*On orthostochastic, unistochastic and qustochastic matrices*]{}, [Linear Algebra Appl.]{}, [**428(4)**]{}, 1178–1201, (2008).
M. A. Cueto, J. Morton, and B. Sturmfels: [*Geometry of the restricted [B]{}oltzmann machine*]{}, [Algebraic methods in statistics and probability [II]{}, volume 516 of Contemp. Math]{}. Amer. Math. Soc., Providence, RI, 135–153 (2010).
P. [Dey]{}: [*Characterization of determinantal bivariate polynomials*]{}, arXiv: 1708.09559.
P. [Dey]{}: [*Monic symmetric/hermitian determinantal representations of multi-variate polynomials*]{}. PhD thesis. Department of Electrical Engineering, Indian Institute of Technology Bombay, 2017.
J. A. De Loera, B. Sturmfels, and C. Vinzant: [*The central curve in linear programming*]{}, [Found. Comput. Math.]{}, [**12(4)**]{}, 509–540, (2012).
H. Flenner, L. O’Carroll, and W. Vogel: [*Joins and intersections*]{}, [Springer Monographs in Mathematics]{}, (1999).
N. Friedenberg, A. Oneto, and R. L. Williams: [*Minkowski sums and Hadamard products of algebraic varieties*]{}, Combinatorial Algebraic Geometry: Selected Papers From the 2016 Apprenticeship Program, Ed. by G. G. Smith and B. Sturmfels. Springer New York, New York, NY, 133–157, (2017).
A. Fink, D. E. Speyer, and A. Woo: [*A Gröbner basis for the graph of the reciprocal plane*]{}, [J. Commut. Algebra]{}, Advance publication, (2018).
I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky: [Discriminants, resultants, and multidimensional determinants]{}, Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, (1994).
D. R. Grayson and M. E. Stillman: Macaulay2, a software system for research in algebraic geometry. Available at <http://www.math.uiuc.edu/Macaulay2/>.
A. Horn: [*Doubly stochastic matrices and the diagonal of a rotation matrix*]{}, [Amer. J. Math.]{}, [**76**]{}, 620–630, (1954).
M. [Kummer]{} and C. [Vinzant]{}: [*The Chow form of a reciprocal linear space*]{}, arXiv: 1610.04584.
M. [Lenz]{}: [*On powers of Plücker coordinates and representability of arithmetic matroids*]{}, [Adv. in Appl. Math.]{}, [**112**]{}, 101911, (2020).
L. Mirsky: [*Results and problems in the theory of doubly-stochastic matrices*]{}, [Z. Wahrscheinlichkeitstheorie und Verw. Gebiete]{}, [**1**]{}, 319–334, (1963).
A. W. Marshall, I. Olkin, and B. C. Arnold: [*Inequalities: theory of majorization and its applications*]{}, Springer Series in Statistics. Springer, New York, second edition, (2011).
D. Maclagan and B. Sturmfels: [Introduction to tropical geometry]{}, [Graduate Studies in Mathematics]{}, American Mathematical Society, Providence, RI, (2015).
James Oxley: [Matroid theory]{}, [Oxford Graduate Texts in Mathematics]{}, Oxford University Press, Oxford, second edition, (2011).
|
TIFR/TH/97-15\
NORDITA–97/31 P\
hep-ph/9705273\
\
[Paul Hoyer$^1$ and D.P. Roy$^2$]{}\
$^1$ Nordita, Blegdamsvej 17, DK-2100 Copenhagen, Denmark\
\[2mm\] $^2$ T.I.F.R., Homi Bhabha Road, Mumbai 400 005, India.
Using an intrinsic parton model we estimate the rough shape and size of the intrinsic gluon component of the nucleon, corresponding to an energy scale $Q$ of the order ${\Lambda_{QCD}}$. It is nearly as hard in shape as the valence quark, while its size accounts for a quarter of the nucleon momentum. Both are in qualitative agreement with the input leading twist gluon distribution assumed by Glück, Reya and Vogt at this scale in order to reproduce the observed distributions at $Q^2 \gsim 1$ GeV$^2$ via perturbative QCD evolution.
While the $Q^2$ dependence of the parton distribution functions are successfully explained by perturbative QCD, one cannot predict the shape and size of these distribution functions at the starting scale of $Q^2$. The standard practice is to use phenomenological parametrisations of the quark and gluon distributions in the Bjorken $x$ variable \[1\] at an input scale of Q\^2\_0 = 1 - 2 [[GeV]{}]{}\^2 . \[highqodef\] Roughly speaking, in this $Q^2$ region the nucleon momentum seems to be shared about equally between the three valence quarks and an infinite number of soft gluons, with the gluon distribution function g(x) x\^[-1]{} [at low]{} x . \[softgluedist\] Thus to a first approximation [\_g]{}&& \^1\_0 x g (x) dx , \[softegfract\]\
\[3mm\] |x\_g && = 0 , \[softxgfract\] where ${\varepsilon_g}$ is the total fraction of the nucleon momentum which is carried by gluons and $\bar x_g$ is the average momentum fraction carried per gluon. There is so far no theoretical understanding of the size or the shape of the input gluon distribution represented by Eqs. (\[softegfract\]), (\[softxgfract\]) even at a qualitative level.
A bold step along this direction was taken by Glück, Reya and Vogt (GRV) \[2, 3, 4\] by extending the perturbative QCD evolution of the leading twist parton distribution functions down to Q\^2\_0 0.2 [[GeV]{}]{}\^2 , \[lowqodef\] [[*ie*]{}]{}, to a $Q_0$ of the same order as ${\Lambda_{QCD}}$. This corresponds to the regime of long distance and time scales, where the input parton distributions can be regarded as intrinsic to the nucleon \[5\]. An initial attempt to generate the canonical gluon distribution represented by (\[highqodef\]) - (\[softxgfract\]) by starting only with the three valence quarks at the low input scale of (\[lowqodef\]) did not succeed \[2\]. However, GRV did reproduce the measured gluon distribution by adding a valence-like gluon component to the valence quarks at the low input scale. While the shape of the input gluon distribution is roughly similar to that of the valence quarks, the gluons carry a smaller momentum fraction [\_g]{}= 0.25 [and]{} [\_q]{}= 0.60 [at]{} Q\^2\_0 0.2 [[GeV]{}]{}\^2 , \[hardfract\] in the case of leading order QCD evolution \[4\]. The remaining 15% of the nucleon momentum is attributed to a sea quark component whose shape is also assumed to be valence like.
In this note we shall provide rough estimates of the shape and size of the intrinsic gluon component, corresponding to the long time scale of [Eq. (\[lowqodef\])]{} above. For this purpose we shall use the model of Brodsky [[*et al.*]{}]{} \[5\] for the long time scale structure of the nucleon wavefunction. As we shall see below, the predicted intrinsic gluon component has a roughly similar shape and size as the above mentioned inputs of GRV \[2-4\].
According to the model of \[5\], the partons in the nucleon Fock states |qqq, |qqqg, |qqq|q q, \[fockdecomp\] have similar velocities in order to stay together over a long time scale. More quantitatively, the probability $P_n$ of an $n$-parton Fock state is given by P\_n (E)\^[-2]{} (m\_N\^2-\^n\_[i=1]{} [[m\_[i]{}]{}\^2 x\_i]{})\^[-2]{} . \[fockprob\] Here $\Delta E$ is the energy difference, in the infinite momentum frame, between the nucleon and the Fock state, and ${m_{\perp i}}^2 = m^2_i + k^2_{\perp i}$ is the squared transverse mass of parton $i$. The distribution (\[fockprob\]), motivated by old fashioned perturbation theory \[5\], is relevant for the long time-scale $(\propto 1/\Delta E)$ structure of the nucleon. The leading twist $Q^2$ evolution, on the other hand, reflects the increasing resolution of short-lived ‘extrinsic’ states created through single parton splitting. In the spirit of GRV we thus propose using [Eq. (\[fockprob\])]{} to determine the input, low $Q^2$ ‘valence’ distribution to which leading twist evolution is applied.
According to [Eq. (\[fockprob\])]{} the probability distribution of a given Fock state is peaked at x\_i = . \[peakprob\] In particular, if there are heavy partons in a Fock state then they will carry a large fraction of the nucleon momentum. This led to the suggestion of an intrinsic charm component $|qqq\bar c c\rangle$ of the nucleon, where the charm quark pair carries the bulk of the nucleon momentum \[5\]. The EMC data on muon induced dimuons \[6\] seems to indicate the presence of such a hard intrinsic charm component in the nucleon \[7\], but there is no definitive experimental evidence for it so far. For fixed ${m_{\perp i}}$, the probability distribution of [Eq. (\[fockprob\])]{} implies a power-law fall-off $(1-x)^n$ for the parton distributions, with $n=3$ and 4 for the valence quark and gluon respectively \[5, 8\]. A similar model was used in \[9\] to predict a hard fragmentation function for the charm quark into a charmed hadron, which is in good agreement with experimental data.
In the present case we are interested in the long time scale structure of the nucleon wavefunction in terms of the multi-gluon Fock states |qqq, |qqqg, |qqqgg, . Consider the $n$-parton Fock state consisting of the 3 valence quarks and ($n-3$) gluons. We take the parton momenta to be distributed according to [Eq. (\[fockprob\])]{} with a common transverse mass [m\_[i]{}]{}k\^2\_\^[12]{} 0.3 - 0.4 [[GeV]{}]{}, \[kvalue\] [[*ie*]{}]{}, with a typical intrinsic momentum corresponding to a hadronic scale of [${\cal O}(1\ {{\rm fm}})$]{}. This implies an equipartition of the nucleon momentum among the $n$ partons, [[*ie*]{}]{}, |x\_q = |x\_g = 1/n , [\_q]{}= 3/n , [\_g]{}= (n-3)/n . \[fockfrac\] Thus if a single Fock state were dominant the shape of the gluon distribution would be identical to that of the valence quark. However, in general we have to consider the contribution of all nucleon Fock states |N= A\_3 |qqq+ A\_4 |qqqg+ A\_5 |qqqgg+ . \[focksum\] Here C\_n = |A\_n|\^2 = P\_n (x\_1 , …, x\_n) \^n\_[i=1]{} dx\_i \[cndef\] represents the net probability for the $n$-parton Fock state, and $\sum_{n=3} C_n = 1$. Thus [\_q]{}&=& \_[n=3]{} [3 C\_n n]{} = 3 = 3|x\_q\
\[3mm\] [\_g]{}&=& \_[n=3]{} [(n-3) C\_n n]{} = ,\
|x\_g &=& = / , \[fracexpr\] [[*ie*]{}]{}, in general $\bar x_g < \bar x_q$. As we shall see below, the intrinsic gluon distribution can be soft $(\bar x_g \simeq 0)$ or nearly as hard as the valence quark $(\bar x_g \sim \bar x_q)$ depending on the nature of the coefficients $C_n$.
In order to proceed further we need to know the $n$-dependence of the probability factors $C_n$ of Eqs. (\[fockprob\]) and (\[cndef\]). We shall assume that the $n$-dependence is mainly determined by the energy denominators of [Eq. (\[fockprob\])]{} evaluated at their most likely configuration (\[peakprob\]), with ${m_{\perp i}}$ given by [Eq. (\[kvalue\])]{}. This gives (neglecting the nucleon mass term in [Eq. (\[fockprob\])]{}), C\_n 1/n\^4 . \[cn4\] Consequently, |x\_q = \^\_3 [dnn\^5]{} / \^\_3 [dn n\^4]{} . = [1 4]{} , [\_q]{}= [3 4]{} , \[qfrac4\] [\_g]{}= [1 4]{} , = [32]{} , |x\_g = = [16]{} . \[gfrac4\] Thus the average momentum fractions of gluons and quarks are similar $(\bar x_g \simeq 0.7 \bar x_q)$, while their total momentum fractions are in the ratio ${\varepsilon_g}\colon {\varepsilon_q}= 1 \colon 3$. Both features are in qualitative agreement with the input gluon distribution of GRV at $Q^2_0 \simeq 0.2\ {{\rm GeV}}^2$ \[4\], as discussed above.
It is instructive to see how sensitive the results are to the assumed $n$-dependence of the probability factors $C_n$. Let us consider the three other cases C\_n 1/n , 1/n\^2 , [and]{} 1/n\^3 . The resulting average and total momentum fractions as well as ${\langle{n_g}\rangle}$ are shown in Table I along with those of Eqs. (\[cn4\]) - (\[gfrac4\]). We see that the shape and size of the intrinsic gluon distribution depend sensitively on the distribution of Fock states. While both the quark and the gluon distributions are hard for $C_n \propto 1/n^4$, they are both soft for $C_n \propto 1/n$. Simultaneously the total momentum fraction carried by the gluons $({\varepsilon_g})$ increases from 1/4 to 1.
It may also be noticed from Table I that $C_n \propto 1/n^2$ corresponds to hard quark and soft gluon distributions, each carrying half the nucleon momentum fraction, as in the case of the canonical parametrisation at higher $Q^2$ given by Eqs. (\[highqodef\]) - (\[softxgfract\]). However, in this perturbative regime the virtual photon scatters from Fock states having a short life-time of [${\cal O}(1/Q)$]{}. Hence the dynamics is not determined by the Fock states of lowest energy, as in our intrinsic model based on [Eq. (\[fockprob\])]{}.
Given that the parton distributions depend sensitively on the Fock state probabilities, we find it significant that the probability distribution (\[cn4\]) of intrinsic states gives roughly the right shape and size of the valence parton distributions, as required for the GRV input at the appropriate scale of $Q_0^2 \simeq 0.2\
{{\rm GeV}}^2$ \[2-4\]. It would be interesting to compare the Fock state distribution of [Eq. (\[cn4\])]{} with that of solvable field theory models, such as QCD$_{1+1}$ \[10\] and dimensionally reduced QCD \[11\].
So far we have neglected the (presumably small) intrinsic sea quark component. Regardless of our specific model, there are at least two reasons to expect that the sea quark component should, like the gluon, have a hard distribution at the low momentum scale of [Eq. (\[lowqodef\])]{}, as is indeed the case in the GRV input \[4\]. (i) The presence of soft sea quarks $(x_i \rightarrow 0)$ in any Fock state will imply the corresponding $\Delta E \rightarrow \infty$, making such states irrelevant for the long time scale structure. (ii) The perturbative evolution of sea quarks in the small $x$ region is driven by the small $x$ behavior of gluons. Thus a soft sea component cannot develop as long as the gluon component remains hard.
It should be emphasized that the hard quark and gluon distributions at the low scale of [Eq. (\[lowqodef\])]{} represent leading twist parton distributions which are not directly measurable in electron scattering. At such low values of $Q^2$ the physical cross section is in fact dominated by higher twist contributions – the photon scatters coherently from several quarks. Being of leading twist, the $Q^2$ evolution of the parton distributions are known, however, and they can thus be compared with data at a higher scale, such as that given by (\[highqodef\]). This was, of course, how GRV arrived at their parametrization of the parton distributions at low $Q^2$.
In summary, we have estimated the rough shape and size of the intrinsic valence quark and gluon components of the nucleon. At the low scale $Q^2 \simeq 0.2\ {{\rm GeV}}^2$, [[*ie*]{}]{}, for $Q=
{{\cal O}({\Lambda_{QCD}})}$, we assume the nucleon Fock state probabilities to be proportional to $(\Delta E)^{-2}$, where $\Delta E$ is the excitation energy of the state. In this approach the average momentum of an intrinsic gluon turns out to be similar to that of a valence quark. The total momentum fractions carried by gluons and quarks are in the ratio 1 : 3. Both features are in qualitative agreement with the shape and size of the input parton distributions found by GRV \[2–4\] at $Q^2 \simeq 0.2\ {{\rm GeV}}^2$. When evolved to $Q^2 \gsim 1\
{{\rm GeV}}^2$ these distributions reproduce the experimental data. Thus our model, taken together with the GRV analysis, provides a theoretical basis for understanding the shape and size of the observed gluon distribution at $Q^2
= 1-2\ {{\rm GeV}}^2$.
It is a pleasure to thank Profs. S. Brodsky, R.M. Godbole, J. Kwiecinski, E. Reya, R.G. Roberts and G.G. Ross for illuminating discussions.
3em
1. See e.g. A.D. Martin, R.G. Roberts and W.J. Stirling, [*Phys. Lett.*]{} [**B 387**]{} (1996) 419, hep-ph/9606345.
2. M. Glück, E. Reya and A. Vogt, [*Z. Phys.*]{} [**C 48**]{} (1990) 471.
3. M. Glück, E. Reya and A. Vogt, [*Z. Phys.*]{} [**C 53**]{} (1992) 127; [*Phys. Lett.*]{} [**B 306**]{} (1993) 391.
4. M. Glück, E. Reya and A. Vogt, [*Z. Phys*]{} [**C 67**]{} (1995) 433.
5. S.J. Brodsky, P. Hoyer, C. Peterson and N. Sakai, [*Phys. Lett.*]{} [**B 93**]{} (1980) 451; S.J. Brodsky, C. Peterson and N. Sakai, [*Phys. Rev.*]{} [**D 23**]{} (1981) 2745.
6. EMC Collaboration: J.J. Aubert et. al, [*Nucl. Phys.*]{} [**B 213**]{} (1983) 31.
7. D.P. Roy, On the Indication of Hard Charm in the Latest EMC Dimuon Data, TIFR/TH/83-1 (1983); see also B. Harris, J. Smith and R. Vogt, [*Nucl. Phys.*]{} [**B 461**]{} (1996) 181, hep-ph/9508403.
8. S.J. Brodsky and I. Schmidt, [*Phys. Lett.*]{} [**B 234**]{} (1990) 144; S.J. Brodsky, M. Burkhardt and I. Schmidt, [*Nucl. Phys.*]{} [**B 441**]{} (1995) 197, hep-ph/9401328.
9. M. Suzuki, [*Phys. Lett.*]{} [**B 71**]{} (1977) 139; see also J.D. Bjorken, [*Phys. Rev.*]{} [**D 17**]{} (1978) 171.
10. K. Hornbostel, S. J. Brodsky and H. C. Pauli, [*Phys. Rev.*]{} [**D 41**]{} (1990) 3814.
11. F. Antonuccio and S. Dalley, [*Phys. Lett.*]{} [**B376**]{} (1996) 154, hep-th/9512106.
[**Table I**]{}: The average and total momentum fractions carried by the valence quarks and the intrinsic gluons are shown along with the average number of gluons for four different types of Fock state distributions.
--------------- --------------------------- -------------------- --------------------------- -------------------- -----------------
$C_n$ ${\varepsilon_q}$ $\bar x_q$ ${\varepsilon_g}$ $\bar x_g$ $<n_g>$
$1/n$ 0 0 1 0 $\infty$
$1/n^2$ 1/2 1/6 1/2 0 $\infty$
$1/n^3$ 2/3 2/9 1/3 1/9 3
$1/n^4$ 3/4 1/4 1/4 1/6 3/2
--------------- --------------------------- -------------------- --------------------------- -------------------- -----------------
|
---
address: |
${}^a$University of California at San Diego, La Jolla, CA 92093-0319\
${}^b$University of Kentucky, Lexington, KY 40506-0055
author:
- 'COLIN MORNINGSTAR$^{a}$ and MIKE PEARDON$^{b}$'
title: GLUEBALLS FROM IMPROVED LATTICE ACTIONS
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Glueballs and hybrid mesons are presently of great interest theoretically and experimentally. The lattice formulation of QCD provides an ideal setting in which to carry out theoretical studies of such systems from first principles using sophisticated numerical simulations. In order to extract the physical properties of glueballs and hybrid mesons from such simulations, systematic errors from the finite lattice spacing $a$ must be removed or made acceptably small. There are two approaches to accomplishing this: (1) using finer grids or (2) using improved actions on coarse grids. The first approach is much simpler and has been used in almost all previous glueball and hybrid meson studies. However, this approach requires vast computational power. As the grid is made finer, many more lattice sites are needed to maintain the physical volume of the system. The simulation costs rise typically as $1/a^6$ as $a$ is decreased. Because of this, lattice studies of glueballs have in the past been dominated by large collaborations using some of the world’s fastest supercomputers.
Here, we show that the second approach, the use of improved actions, can be used to study glueballs much more efficiently. Improved actions have smaller lattice spacing errors, and hence, permit the use of much coarser lattices which can be simulated using modern computer workstations. The key to the success of the improvement program is the reliable determination of the couplings of the interactions terms in the action. Much effort over the past decade has been directed towards this problem. Recently, two competing methods have emerged, one which uses block-renormalization group transformations, another which advocates a judicious combination of mean field theory and perturbation theory. In this work, we use the latter approach.
A novel feature of our calculations is the use of anisotropic lattices in which the temporal spacing $a_t$ is much smaller than the spacing $a_s$ in the spatial directions. This allows much more efficient glueball mass measurements by exploiting the enhanced signal-to-noise of the glueball correlation functions at smaller temporal separations. Mean-field link renormalization[@TI] is crucial for maintaining the proper renormalized anisotropy $a_t/a_s$.
Our results are shown in Fig. \[fig:scaling\]. The scalar glueball mass from the improved action exhibits dramatically reduced cutoff contamination compared to the Wilson action. Finite-$a_s$ errors are seen to be small for the tensor and pseudo-vector glueballs, although differences between the $E^{++}$ and $T_2^{++}$ representations indicate small violations of rotational invariance, especially for large $a_s$. These results clearly show that glueballs can be studied without the use of supercomputers, provided that simulations are carried out using improved actions on anisotropic lattices.
References {#references .unnumbered}
==========
[99]{}
J. Sexton, [*et al.*]{}, ; P. De Forcrand [*et al.*]{}, ; C. Michael and M. Teper, ; UKQCD Collaboration, . G.P. Lepage and P.B. Mackenzie, .
|
---
abstract: 'We compute the submodule of finite support of the tensor product of two modules $M$ and $N$ and estimate its length in terms of $ M$ and $N$. Also, we compute some higher local cohomology modules of tensor products.'
author:
- Mohsen Asgharzadeh
title: finite support of tensor products
---
Introduction
============
In this note $(R,{\frak{m}})$ is a commutative, noetherian and local ring of dimension $d$. Also, all modules are finitely generated. By ${\operatorname{H}}^0_{{\frak{m}}}(M)$ we mean the elements of $M$ that are annihilated by some power of ${\frak{m}}$. We consider to ${\operatorname{H}}^0_{{\frak{m}}}(M\otimes_RN)$ and denote its length by ${h}^0(M\otimes_RN)$.
(See [@Wolmer Page 704]) Can one estimate ${h}^0(M\otimes_RN)$ in terms of $M$ and $N$?
Under various assumptions on the ring and on the modules, Vasconcelos proved several bounds on ${h}^0(M\otimes_RN)$. For example, when $R$ is regular $N$ is locally free and ${\operatorname{pd}}(M)<\dim(R)$. He asked for a similar extension when the ring is Gorenstein with isolated singularity, see [@2010 Question 8.2]. In §2 we slightly extend Vasconcelos’ bounds. Also, we present results in the singular case, see Proposition \[v5\] and \[buchs\].
In the case the ring is Gorenstein of dimension $d\geq1$ and $M$ has a presentation $0\to R^n\stackrel{\varphi}{\longrightarrow}R^{n+d-1} \to M\to 0$ where $I_n(\varphi)$ is ${\frak{m}}$-primary, Vasconcelos proved ${h}^0 (M\otimes_RM)\leq d\left((d - 1)\deg(M) + \ell(\frac{R}{I_n(\varphi)})\right)^2$, here $\ell(-)$ is the length function. In [@2010 Question 8.1], he asked how good is the estimate compared to ${h}^0 (M\otimes_RM)$? In §3 we present some explicit computations. For example, there is a situation for which $ d((d - 1)\deg(M) + \ell(\frac{R}{I_n(\varphi)}))^2>{h}^0 (M\otimes_RM)^2$, see Proposition \[squar\].
In §4 we partially answer Vasconcelos’ question on the torsion part of tensor products. For example over a $3$-dimensional Cohen-Macaulay local domain and a reflexive module $M$ such that ${\operatorname{pd}}(M)<\infty$, we show $M$ is free provided $M^{\otimes 3}$ is torsion-free.
Suppose $M$ and $N$ are vector bundles over a Cohen-Macaulay ring. This in turn implies that the length of ${\operatorname{H}}^{<d}_{{\frak{m}}}(M\otimes_R N)$ is finite. In §5 we try to understand $\ell({\operatorname{H}}^{+}_{{\frak{m}}}(M\otimes_R N))$. We do this in three subsections. §5.A deals with the low-dimensional cases. Also, Remark \[sph\] slightly extends a result of Auslander. By a result of Auslander (see [@au Theorem 3.7]) and Huneke-Wiegand (see [@tensor2 §4]) vanishing of ${\operatorname{H}}^{i}_{{\frak{m}}}(M\otimes_R M^\ast)$ for some $i$ implies freeness of $M$ if we impose some conditions on $i$, $d$ and $M$. Here, the notation $M^\ast$ stands for ${\operatorname{Hom}}_R(M,R)$. What can say in a more general setting? This is subject of §5.B. In subsection §5.C we compute ${\operatorname{H}}^{+}_{{\frak{m}}}(M\otimes_R N)$ in some singular cases.
Bounds on ${h}^0(-\otimes \sim)$: after Vasconcelos
====================================================
By $\mu(-)$ we mean the minimal number of elements that need to generate $(-)$.
\[0\] Let $M$ be of finite length. Then ${h}^0(M\otimes_RN)\leq\ell(M)\mu(N)$.
The proof is by induction on $\ell(M)$. Suppose $\ell(M)=1$. Then $M=R/ {\frak{m}}$. By definition, ${\operatorname{H}}^0_{{\frak{m}}}(M\otimes_RN)=M\otimes_RN=\frac{N}{{\frak{m}}N}$ and so ${h}^0(M\otimes_RN)=\mu(N)=\ell(M)\mu(N)$. We look at the exact sequence $0\to R/ {\frak{m}}\to M\to \overline{M}\to 0$ where $\ell(\overline{M})=\ell(M)-1$. By induction, $\ell(\overline{M}\otimes_R N)\leq\ell(\overline{M})\mu(N)$. The sequence induces $ R/ {\frak{m}}\otimes_RN\stackrel{g}{\longrightarrow}M\otimes_RN\stackrel{f}{\longrightarrow}\overline{M}\otimes_RN\to 0$. Since $R/ {\frak{m}}\otimes_RN\twoheadrightarrow {\operatorname{im}}(g)\to 0$ is surjective, $\ell(\ker(f))=\ell({\operatorname{im}}(g))\leq\mu(N)$. We have $$\ell(M\otimes_R N)=\ell(\overline{M}\otimes_R N)+\ell(\ker(f))\leq\ell(\overline{M}\otimes_RN)+\ell(N/ {\frak{m}}N)\leq\ell(\overline{M})\mu(N)+\mu(N).$$ So, $\ell({\operatorname{H}}^0_{{\frak{m}}}(M\otimes _RN))= \ell(M\otimes _RN)\leq(\ell(M)-1)\mu(N)+\mu(N)=\mu(N)\ell(M)$.
The particular case in the next result stated in [@2010 Proposition 2.1] without proof:
\[reduction\] One has ${h}^0(M\otimes_RN)\leq{h}^0(M)\mu(N)+{h}^0(\frac{M}{{\operatorname{H}}^0_{{\frak{m}}}(M)}\otimes_RN)$. In particular, $${h}^0(M\otimes_RN)\leq{h}^0(M)\mu(N)+{h}^0(N)\mu(N)+{h}^0(M/ {\operatorname{H}}^0_{{\frak{m}}}(M) \otimes_R\ N/ {\operatorname{H}}^0_{{\frak{m}}}(N)).$$
We may assume neither $M$ nor $N$ are of finite length (see Lemma \[0\]). We look at $0\to {\operatorname{H}}^0_{{\frak{m}}}(M)\to M\to \widetilde{M}:=\frac{M}{{\operatorname{H}}^0_{{\frak{m}}}(M)}\to 0$. Apply $-\otimes_RN$ to it and look at the induced long exact sequence $${\operatorname{Tor}}_1^R(\widetilde{M},N)\to {\operatorname{H}}^0_{{\frak{m}}}(M)\otimes_RN\stackrel{f}{\longrightarrow}M\otimes_RN\to \widetilde{M}\otimes_RN\to 0.$$ The sequences $0\to \ker(f)\to M\otimes_RN\to \widetilde{M}\otimes_RN\to 0 $ and ${\operatorname{Tor}}_1^R(\widetilde{M},N)\to {\operatorname{H}}^0_{{\frak{m}}}(M)\otimes_RN\to \ker(f)\to 0 $ are exact. From the second, $\ell(\ker(f))\leq\ell({\operatorname{H}}^0_{{\frak{m}}}(M)\otimes_RN)\leq{h}^0(M)\mu(N)$, see Lemma \[0\]. The first one deduces the exact sequence $0\to {\operatorname{H}}^0_{{\frak{m}}}(\ker(f))\to {\operatorname{H}}^0_{{\frak{m}}}( M\otimes_RN)\to {\operatorname{H}}^0_{{\frak{m}}}(\widetilde{M}\otimes_RN)\to {\operatorname{H}}^1_{{\frak{m}}}(\ker(f)).$ So, ${h}^0(M\otimes_R N)\leq {h}^0(\ker(f))+ {h}^0(\widetilde{M}\otimes _RN) = \ell(\ker(f))+ {h}^0(\widetilde{M}\otimes _RN) \leq{h}^0(M)\mu(N)+{h}^0(\widetilde{M}\otimes_R N).$ Repeat this for $N$, we have $$\begin{array}{ll}
{h}^0(M\otimes_RN)&\leq{h}^0(M)\mu(N)+{h}^0(\widetilde{M}\otimes_RN)\\
&\leq{h}^0(M)\mu(N)+{h}^0(N)\mu(\widetilde{N})+{h}^0(\widetilde{M}\otimes_R\widetilde{N})\\
&\stackrel{(\ast)}\leq{h}^0(M)\mu(N)+{h}^0(N)\mu(N)+{h}^0(\widetilde{M}\otimes_R\widetilde{N}),
\end{array}$$where $(\ast)$ follows by applying $(-)\otimes_R R/{\frak{m}}$ to $N\twoheadrightarrow \widetilde{N}\to 0$ to see that $N/ {\frak{m}}N\twoheadrightarrow \widetilde{N}/ {\frak{m}}\widetilde{N}\to 0$. In particular, $\dim(\widetilde{N}/ {\frak{m}}\widetilde{N})\leq \dim(N/ {\frak{m}}N)$. This completes the proof.
By ${h}^i(-)$ we mean $\ell({\operatorname{H}}^i_{{\frak{m}}}(-))$ provided it is finite. By ${\operatorname{pd}}(-)$ we mean the projective dimension. We look at the minimal free resolution of $M$: $\cdots \to R^{\beta_{i}(M)}\stackrel{f_{i}}{\longrightarrow}R^{\beta_{i-1}(M)}\to\cdots\to R^{\beta_{0}(M)}\to M\to 0.$ The $i^{th}$ *syzygy* module of $M$ is ${\operatorname{Syz}}_i(M) := \ker(f_{i-1})$ for all $i>0$. The following is in [@2010 Theorem 4.1] under the additional assumption that $R$ is Gorenstein.
\[vector\] Let $R$ be an equi-dimensional and generalized Cohen-Macaulay local ring, and $N$ be locally free and of constant rank over the punctured spectrum. If ${\operatorname{pd}}(M)<{\operatorname{depth}}(R)$, then ${h}^0(M\otimes_RN)\leq
\sum_{i=0}^{{\operatorname{pd}}(M)}\beta_i(M){h}^i(N)$.
Let $p:={\operatorname{pd}}(M)$. We may assume $N$ is not of finite length (see Lemma \[0\]). The assumptions implies that $N$ is generalized Cohen-Macaulay and of dimension equal to $\dim (R)$. We look at $0\to {\operatorname{Syz}}_1(M)\to R^{\beta_{0}(M)}\to M\to 0$. Apply $-\otimes_RN$ to it and look at the induced long exact sequence $$0\to{\operatorname{Tor}}_1^R(M,N)\to {\operatorname{Syz}}_1(M)\otimes_RN\stackrel{f}{\longrightarrow}R^{\beta_{0}(M)}\otimes_RN\to M\otimes_RN\to 0.$$ We have $0\to \ker(f)\to R^{\beta_{0}}\otimes_R N\to M\otimes_R N\to 0 $ and $0\to{\operatorname{Tor}}_1^R(M,N)\to {\operatorname{Syz}}_1(M)\otimes_R N\to \ker(f)\to 0.$ Since $N$ is locally free, ${\operatorname{Tor}}_1^R(M,N)$ is of finite length. Thus, ${\operatorname{H}}^0_{{\frak{m}}}({\operatorname{Tor}}_1^R(M,N))={\operatorname{Tor}}_1^R(M,N)$ and ${\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Tor}}_1^R(M,N))=0$. We apply $\Gamma_{{\frak{m}}}$ to these sequences to deduce the long exact sequences:
1. $0\to{\operatorname{H}}^0_{{\frak{m}}}({\operatorname{Tor}}_1^R(M,N))\to {\operatorname{H}}^0_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_R N)\to{\operatorname{H}}^0_{{\frak{m}}}(\ker(f))\to{\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Tor}}_1^R(M,N))=0, $
2. $0\to {\operatorname{H}}^0_{{\frak{m}}}(\ker(f))\to {\operatorname{H}}^0_{{\frak{m}}}(R^{\beta_{0}(M)}\otimes_RN)\to {\operatorname{H}}^0_{{\frak{m}}}(M\otimes_RN)\to {\operatorname{H}}^1_{{\frak{m}}}(\ker(f)).$
Also, ${\operatorname{H}}^+_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN)\simeq{\operatorname{H}}^+_{{\frak{m}}}(\ker(f))$. We use these to conclude that: $${h}^0(M\otimes_RN)\leq \ell({\operatorname{H}}^1_{{\frak{m}}}(\ker(f)))+\beta_{0}(M){h}^0(N)=\ell({\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN))+\beta_{0}(M){h}^0(N).$$ In the same vein, $\ell({\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN))\leq\ell({\operatorname{H}}^2_{{\frak{m}}}({\operatorname{Syz}}_2(M)\otimes_RN))+\beta_{1}(M){h}^1(N).$ Thus $$\begin{array}{ll}
{h}^0(M\otimes_RN)&\leq \ell({\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN))+\beta_{0}(M){h}^0(N)\\
&\leq\ell({\operatorname{H}}^2_{{\frak{m}}}({\operatorname{Syz}}_2(M)\otimes_RN))+\beta_{1}(M){h}^1(N)+\beta_{0}(M){h}^0(N).
\end{array}$$ Repeating this, $
{h}^0(M\otimes_RN)
\leq\ell({\operatorname{H}}^p_{{\frak{m}}}({\operatorname{Syz}}_{p}(M)\otimes_RN))+\sum_{i=0}^{p-1}\beta_i(M){h}^i(N)=\sum_{i=0}^{p}\beta_i(M){h}^i(N).$
By ${\operatorname{hdeg}}(M)$ we mean the *cohomological degree*, see [@Wolmer] for its definition. The following contains more data than [@2010 Theorem 4.2] via dealing with ${\operatorname{pd}}(A)=\dim(R)$.
\[cvector\] Let $R$ be a $d$-dimensional regular local ring, $M$ a module and $N$ be locally free over the punctured spectrum. Then$${h}^0(M\otimes_RN)\leq \left\{
\begin{array}{rl}
d{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N) & \ \ \ \ \ \ \ \ \ \ \ \ \text{if } {\operatorname{pd}}(M)< d\\
(d+1){\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)-1 & \ \ \ \ \ \ \ \ \ \ \ \ \text{if } {\operatorname{pd}}(M)= d
\end{array} \right.$$
Due to Lemma \[0\] we can assume that neither $M$ nor $N$ are artinian. The claim in the case ${\operatorname{pd}}(M)< d$ is in [@2010 Theorem 4.2]. Suppose ${\operatorname{pd}}(M)=d$. Since $M$ is not artinian, $M\neq \Gamma_{{\frak{m}}}(M)$. We denote $M/ \Gamma_{{\frak{m}}}(M)$ by $\widetilde{M}$. Note that ${\operatorname{depth}}(\widetilde{M})>0$. Due to Auslander-Buchsbaum formula, ${\operatorname{pd}}(\widetilde{M})< d$. We combine Lemma \[reduction\] with the first part to see $${h}^0(M\otimes_RN)\leq{h}^0(M)\mu(N)+{h}^0(\widetilde{M}\otimes_RN)\leq{h}^0(M)\mu(N)+d{\operatorname{hdeg}}(\widetilde{M}){\operatorname{hdeg}}(N).$$ Recall from definition that ${h}^0(M)\leq{\operatorname{hdeg}}(M)$. By [@Wolmer Theorem 1.10], $\beta_i(N)\leq\beta_i(k){\operatorname{hdeg}}(N)$. We use this for $i=0$ to see $\mu(N)\leq{\operatorname{hdeg}}(N)$. In view of [@Wolmer Proposition 2.8(a)] we have ${\operatorname{hdeg}}(\widetilde{M})={\operatorname{hdeg}}(M)-\ell(\Gamma_{{\frak{m}}}(M) )<{\operatorname{hdeg}}(M)$. We putt all of these together to see $${h}^0(M\otimes_RN)\leq{h}^0(M)\mu(N)+d{\operatorname{hdeg}}(\widetilde{M}){\operatorname{hdeg}}(N)<{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)+d {\operatorname{hdeg}}(M) {\operatorname{hdeg}}(N).$$ The claim is now clear.
\[1\] Let $R$ be a $d$-dimensional regular local ring. Assume one of the following items hold: i) $d=1$, ii) $d=2$ and $M$ is torsion-free, iii) $d=3$ and $M$ is reflexive. Then ${h}^0(M\otimes_RN)<(d+1){\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)$ for any finitely generated module $N$.
It follows that $M$ is locally free. In view of Proposition \[cvector\] we get the desired claim.
The next result slightly extends [@2010 Proposition 3.4]:
\[1g\] Let $(R,{\frak{m}})$ be a $1$-dimensional complete local integral domain containing a field, $M$ and $N$ be finitely generated. Let $J$ be the Jacobian ideal. Then $${h}^0(M\otimes_RN)\leq{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)(2+\deg(R)\ell(\frac{R}{J}))-{\operatorname{rank}}(M){\operatorname{rank}}(N)\deg(R)\ell(\frac{R}{J}).$$In particular, ${h}^0(M\otimes_RN)\leq(2+\deg(R)\ell(\frac{R}{J})){\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)$.
Due to Lemma \[0\], neither $M$ nor $N$ are artinian. Let $\widetilde{M}:=\frac{M}{{\operatorname{H}}^0_{{\frak{m}}}(M)}$. This is nonzero and of positive depth. Thus, $\widetilde{M}$ is maximal Cohen-Macaulay. Over any 1-dimensional reduced local ring, the category of maximal Cohen-Macaulay modules coincides with the category of torsion free modules. Hence $\widetilde{M}$ and $\widetilde{N}$ are torsion free. In view of [@i], we see $J{\operatorname{Ext}}^1_R(-,\sim)=0$. We combine this with the proof of [@2010 Proposition 3.4] to see ${h}^0(\widetilde{M}\otimes_R\widetilde{N})\leq\left(\mu(\widetilde{M})\mu(\widetilde{N})-{\operatorname{rank}}(\widetilde{M}){\operatorname{rank}}(\widetilde{N})\right)\deg(R)\ell(\frac{R}{J}).$ Recall that $\mu(\widetilde{M})\leq \mu(M)$. Denote the fraction field of $R$ by $Q(R)$. Recall that ${\operatorname{H}}^0_{{\frak{m}}}(M)\otimes_R Q(R)=0$. We apply the exact functor $-\otimes_RQ(R)$ to $0\to{\operatorname{H}}^0_{{\frak{m}}}(M)\to M \to \widetilde{M}\to 0$ to see the sequence $0={\operatorname{H}}^0_{{\frak{m}}}(M)\otimes_R Q(R)\to M\otimes_R Q(R)\to \widetilde{M}\otimes_R Q(R)\to 0$ is exact. From this ${\operatorname{rank}}(M)={\operatorname{rank}}(\widetilde{M})$. Therefore, ${h}^0(\widetilde{M}\otimes_R\widetilde{N})
\leq\left(\mu(M)\mu(N)-{\operatorname{rank}}(M){\operatorname{rank}}(N)\right)\deg(R)\ell(\frac{R}{J}).$ In view of Lemma \[reduction\] we have $$\begin{array}{ll}
{h}^0(M\otimes_RN)&\leq{h}^0(M)\mu(N)+{h}^0(N)\mu(N)+{h}^0(\widetilde{M}\otimes_R\widetilde{N})\\
&\leq {h}^0(M)\mu(N)+{h}^0(N)\mu(N)+\left(\mu(M)\mu(N)-{\operatorname{rank}}(M){\operatorname{rank}}(N)\right)\deg(R) \ell(\frac{R}{J})\\
&\leq{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)(2+\deg(R) \ell(\frac{R}{J}))-{\operatorname{rank}}(M){\operatorname{rank}}(N)\deg(R) \ell(\frac{R}{J}).
\end{array}$$
\[v5\] Let $R$ be a Gorenstein ring with isolated singularity and $M$ be maximal Cohen-Macaulay. Then ${h}^0(M\otimes_RN)$ can estimate in terms of $M$ and $N$.
Maximal Cohen-Macaulay modules over Gorenstein rings are reflexive, e.g., $M$ is reflexive. We may assume $N$ is not of finite length (see Lemma \[0\]). In view of Lemma \[reduction\], we may replace $N$ with $N/ \Gamma_{{\frak{m}}}(N)$ and assume in addition that ${\operatorname{depth}}(N)>0$. This implies that ${\operatorname{Hom}}_R(-,N)$ has positive depth provided ${\operatorname{Hom}}_R(-,N)\neq 0$. Let $D(-)$ be the Auslander’s transpose. We look at the exact sequence $${\operatorname{Tor}}^R_2 (D(M^{\ast}),N)\stackrel{f}{\longrightarrow}M^{\ast\ast}\otimes _RN\stackrel{g}{\longrightarrow}{\operatorname{Hom}}_R(M^{\ast},N)\stackrel{h}{\longrightarrow}{\operatorname{Tor}}^R_1 (D(M^{\ast}),N)\to 0.$$ Without loss of the generality we can assume that ${\operatorname{Hom}}_R(-,N)\neq 0$. Note that $M^{\ast}$ is maximal Cohen-Macaulay and so locally free over punctured spectrum. Since $D(-)$ behaves nicely with respect to localization, we see that $D(M^{\ast})$ is of finite length. Hence ${\operatorname{Tor}}^R_2 (D(M^{\ast}),N)$ is of finite length. Due to ${\operatorname{Tor}}^R_2 (D(M^{\ast}),N)\twoheadrightarrow {\operatorname{im}}(f)\to 0$ we see ${\operatorname{im}}(f)$ is of finite length. We have the following exact sequences $0\to\ker(h)\to{\operatorname{Hom}}_R(M^{\ast},N)\to{\operatorname{Tor}}^R_1 (D(M^{\ast}),N)\to 0 $ and $0\to\ker(g)\to M^{\ast\ast}\otimes_R N\to\ker(h)\to 0.$ Also, ${\operatorname{Tor}}^R_2 (D(M^{\ast}),N)\twoheadrightarrow{\operatorname{im}}(f)=\ker(g).$ Since ${\operatorname{depth}}({\operatorname{Hom}}(M^{\ast},N))>0$ the first sequence says that ${\operatorname{depth}}(\ker(h))>0$. From the second sequence we have ${h}^0(M\otimes_RN)={h}^0(\ker(g)).$ From the third, we have ${h}^0(\ker(g))=\ell({\operatorname{im}}(f))\leq\ell({\operatorname{Tor}}^R_2 (D(M^{\ast}),N))$. In sum, ${h}^0(M\otimes_RN)\leq\ell({\operatorname{Tor}}^R_2 (D(M^{\ast}),N))\leq \beta_2(N)\ell(D(M^{\ast})).$
\[buchs\] Let $R$ be a Cohen-Macaulay local ring of dimension $d>1$, $M$ be perfect of projective dimension one and $N$ be Buchsbaum of dimension $d$. Then ${h}^0(M\otimes_RN)<
3{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)$. Suppose in addition that ${\operatorname{depth}}(N)>0$. Then ${h}^0(M\otimes_RN)\leq
2{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)$.
Let $\widetilde{N}:=\frac{N}{{\operatorname{H}}^0_{{\frak{m}}}(N)}$. In view of [@bus Proposition I.2.22], $\widetilde{N}$ is Buchsbaum. Since $\dim(N)=d>0$, we deuce that $\widetilde{N}\neq 0$. It follows by definition that ${\operatorname{depth}}(\widetilde{N})>0$, ${\operatorname{H}}^+_{{\frak{m}}}(\widetilde{N})\simeq{\operatorname{H}}^+_{{\frak{m}}}(N)$ and that $\dim(N)=\dim(\widetilde{N})$. Recall from [@yosh Proposition 2.7]:
1. Let $A$ be a Cohen-Macaulay local ring of dimension $d>1$ and $P$ be perfect of depth one. If $Q$ is Buchsbaum of positive depth and maximal dimension, then ${h}^0(P\otimes_A Q)= \mu(P)({h}^0(Q) + {h}^1(Q))$.
Recall that ${\operatorname{hdeg}}(\widetilde{N})={\operatorname{hdeg}}(N)-\ell(\Gamma_{{\frak{m}}}(N) )$, $\mu(-)\leq{\operatorname{hdeg}}(-)$ and that ${h}^{<d}(-)\leq{\operatorname{hdeg}}(-)$. In view of Lemma \[reduction\] we have $$\begin{array}{ll}
{h}^0(M\otimes_RN)&\leq{h}^0(N)\mu(M)+{h}^0(M\otimes_R\widetilde{N})\\
&= {h}^0(N)\mu(M)+\mu(M)({h}^0(\widetilde{N}) + {h}^1(\widetilde{N}))\\
&\leq{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)+2{\operatorname{hdeg}}(M){\operatorname{hdeg}}(\widetilde{N})\\
&={\operatorname{hdeg}}(M){\operatorname{hdeg}}(N)+2{\operatorname{hdeg}}(M)({\operatorname{hdeg}}(N)-\Gamma_{{\frak{m}}}(N) )\\
&\leq3{\operatorname{hdeg}}(M){\operatorname{hdeg}}(N),
\end{array}$$and we remark that if $\Gamma_{{\frak{m}}}(N)\neq 0$, then the last inequality is strict. This completes the proof.
Toward sharping the bound on ${h}^0(M\otimes_R M)$
==================================================
We look at $M$ with a presentation of the form $0\to R^n\stackrel{\varphi}{\longrightarrow}R^{n+d-1} \to M\to 0$ where $d=\dim R$. Question 1.2 deals with the sharpness of ${h}^0 (M\otimes_RM)\leq d\left((d - 1)\deg(M) +
\ell(\frac{R}{I_n(\varphi)})\right)^2$. Suppose $d=2$ and $n=1$. Let us repeat the assumption: $M$ has a presentation of the form $0\to R\stackrel{\varphi}{\longrightarrow}R^{2} \to M\to 0$ where the ideal $I_1(\varphi)$ is ${\frak{m}}$-primary. The bound translates to ${h}^0 (M\otimes_RM)\leq 2(\deg(M) + \ell(\frac{R}{I_1(\varphi)}))^2$.
\[m20\] Let $(R,{\frak{m}},k)$ be a $2$-dimensional regular local ring. Then ${h}^0({\frak{m}}\otimes_R{\frak{m}})=1$.
Let $x$ and $y$ be a generating set of ${\frak{m}}$ and look at $\zeta:=x\otimes y-y\otimes x$. We have $$x\zeta=x(x\otimes y-y\otimes x)=x^2\otimes y-xy\otimes x=xy\otimes x-xy\otimes x=0.$$ Similarly, $y\zeta=0$, so that ${\frak{m}}\zeta=0$. By definition, $\zeta\in{\operatorname{H}}^0_{{\frak{m}}}({\frak{m}}\otimes_R{\frak{m}})$. Again due to definition, ${\operatorname{H}}^0_{{\frak{m}}}({\frak{m}}\otimes_R{\frak{m}})$ is submodule of the torsion part of $
{\frak{m}}\otimes_R{\frak{m}}$. On the other hand, the torsion part of $ {\frak{m}}\otimes_R{\frak{m}}$ is ${\operatorname{Tor}}^R_2(k,k)$ (see [@tensor Lemma 1.4]) which is a vector space of dimension equal to $\beta_2(k)=1$. From these, ${\operatorname{H}}^0_{{\frak{m}}}({\frak{m}}\otimes_R{\frak{m}})=\zeta R\simeq k$. In particular, ${h}^0({\frak{m}}\otimes_R{\frak{m}})=\ell({\operatorname{H}}^0_{{\frak{m}}}({\frak{m}}\otimes_R{\frak{m}}))=1$.
The difference $2(\deg(M) + \ell(\frac{R}{I_n(\varphi)}))^2-{h}^0 (M\otimes_RM)$ may be large:
\[squar\] Let $(R,{\frak{m}},k)$ be a $2$-dimensional Cohen-Macaulay local domain and $I$ be an ideal generated by a full parameter sequence. Then ${h}^0(I\otimes_RI)={\operatorname{hdeg}}(R/I)$. In particular, $${h}^0 (I\otimes_RI)=\ell(R/I)\lneqq
2\left(\deg(I) + \ell(R/I)\right)^2.$$
Let $x$ and $y$ be a generating set of $I$. The notation $\mathbb{K}(I; R)$ stands for the Koszul complex of $R$ with respect to $I$. That is $$\mathbb{K}(I; R):=0{\longrightarrow}R\stackrel{(^{+y}_{-x})}{\longrightarrow}R^2\stackrel{(x,y)}{\longrightarrow}R{\longrightarrow}R/I{\longrightarrow}0.$$ This is a minimal free resolution of $R/I$. In view of definition, $$\mathbb{K}(I; R)\otimes_R R/I\simeq0{\longrightarrow}R/I\stackrel{0}{\longrightarrow}R/I\oplus R/I\stackrel{0}{\longrightarrow}R/I{\longrightarrow}R/I\otimes
R/I{\longrightarrow}0.$$ By definition, ${\operatorname{tor}}(I\otimes _RI)\simeq{\operatorname{Tor}}^R_2(R/I,R/I)\simeq{\operatorname{H}}_2(\mathbb{K}(I; R)\otimes_R \frac{R}{I})\simeq\frac{R}{I}.$ We look at the exact sequence $0\to {\operatorname{tor}}(I\otimes _RI)\to I\otimes_R I\to \frac{I\otimes _RI}{{\operatorname{tor}}(I\otimes_R I)}\to 0.$ Since $\frac{I\otimes_R I}{{\operatorname{tor}}(I\otimes _RI)}$ is torsion-free, ${\operatorname{H}}^0_{{\frak{m}}}(\frac{I\otimes_R I}{{\operatorname{tor}}(I\otimes_R I)})=0$. We put this in $0\to {\operatorname{H}}^0_{{\frak{m}}}( {\operatorname{tor}}(I\otimes _RI) )\to{\operatorname{H}}^0_{{\frak{m}}}( I\otimes _RI )\to {\operatorname{H}}^0_{{\frak{m}}}(\frac{I\otimes _RI}{{\operatorname{tor}}(I\otimes_R I)})$ to see that ${\operatorname{H}}^0_{{\frak{m}}}( {\operatorname{tor}}(I\otimes _RI) )\simeq{\operatorname{H}}^0_{{\frak{m}}}( I\otimes_R I )$. Since $\ell(\frac{R}{I})<\infty$, ${\operatorname{H}}^0_{{\frak{m}}}( I\otimes _RI )\simeq{\operatorname{H}}^0_{{\frak{m}}}( {\operatorname{tor}}(I\otimes_R I) )\simeq{\operatorname{H}}^0_{{\frak{m}}}( R/I )\simeq R/I.$ Thus, ${h}^0(I\otimes_RI)=\ell(R/I)$.
\[2au\] Let $(R,{\frak{m}},k)$ be a $2$-dimensional regular local ring and $0\neq M$ be torsion-free. Then ${h}^0(M\otimes_RM)=0$ if and only if $M$ is free.
The if part is trivial. Suppose $M$ is not free. Since $M$ is $({\operatorname{S}}_1)$ it follows that ${\operatorname{pd}}(M)=1$. We claim that ${\operatorname{Tor}}_1^R(M,M)=0$. Suppose on the contradiction that ${\operatorname{Tor}}_1^R(M,M)\neq0$. Let ${\frak{p}}$ be any height one prime ideal. Since $R_{{\frak{p}}}$ is a discreet valuation ring and $M_{{\frak{p}}}$ is torsion-free, it follows that $M_{{\frak{p}}}$ is free over $R_{{\frak{p}}}$. From this, ${\operatorname{Tor}}_1^R(M,M)$ is of finite length. Thus, ${\operatorname{depth}}({\operatorname{Tor}}_1^R(M,M))=0$. We recall the following result of Auslander (see [@au Theorem 1.2]):
1. Let $S$ be a local ring, ${\operatorname{pd}}(A)<\infty$. Let $q$ be the largest number such that ${\operatorname{Tor}}_q^S(A, B)\neq0$. If ${\operatorname{depth}}({\operatorname{Tor}}_q^S(A, B))\leq1$, then ${\operatorname{depth}}(B)={\operatorname{depth}}({\operatorname{Tor}}_q^S(A, B))+{\operatorname{pd}}(A)-q.$
We use this for $A=B=M$ and $q=1$, to see $1={\operatorname{depth}}(M)={\operatorname{depth}}({\operatorname{Tor}}_1^R(M, M))+{\operatorname{pd}}(M)-q=0+1-1=0,$ a contradiction. Thus, ${\operatorname{Tor}}_1^R(M,M)=0$. This vanishing result allow us to use:
1. (see [@au Corollary 1.3]) Let $S$ be a local ring, $A$ and $B$ be of finite projective dimension. If ${\operatorname{Tor}}_+^S(A, B)=0$, then ${\operatorname{pd}}(A)+{\operatorname{pd}}(B)=
{\operatorname{pd}}(A\otimes_S B)$.
From this, ${\operatorname{pd}}(M\otimes_RM)=2$. By Auslander-Buchsbaum, ${\operatorname{depth}}(M\otimes_RM)=0$. Consequently, ${h}^0(M\otimes_RM)\neq0$.
It may be natural to extend the above result to the $3$-dimensional case by replacing torsion-free with reflexive. This is not the case:
\[3\] Let $(R,{\frak{m}},k)$ be a $3$-dimensional regular local ring. Then ${h}^0(M\otimes_RN)=0$ for any reflexive modules $M$ and $N$.
We put $L:=M\oplus N$. It is enough to show ${h}^0(L\otimes_RL)=0$. Without loss of the generality, $L$ is not free. This implies that ${\operatorname{pd}}(L)=1$. By Auslander-Buchsbaum, ${\operatorname{depth}}(L)=2$. This implies that ${\operatorname{H}}^0_{{\frak{m}}}(L)={\operatorname{H}}^1_{{\frak{m}}}(L)=0$. Let $0\to R^n\to R^m\to L\to 0$ be a free resolution of $L$. It follows that $0\to{\operatorname{Tor}}^R_1(L,L)\to L^n\to L^m\to L\otimes_R L\to 0$. Denote the fraction field of $R$ by $Q(R)$. Recall that ${\operatorname{Tor}}^R_1(L,L)\otimes_R Q(R)=0$, i.e., ${\operatorname{Tor}}^R_1(L,L)$ is torsion. Since ${\operatorname{Tor}}^R_1(L,L)$ is torsion, ${\operatorname{Tor}}^R_1(L,L)\subset L^n$ and $ L^n$ is torsion-free, we get that ${\operatorname{Tor}}^R_1(L,L)=0$. We put this in the above sequence to see $0\to L^n\to L^m\to L\otimes_R L\to 0$ is exact. This sequence induces $0={\operatorname{H}}^0_{{\frak{m}}}(L^m)\to {\operatorname{H}}^0_{{\frak{m}}}(L\otimes_R L)\to{\operatorname{H}}^1_{{\frak{m}}}(L^m) =0$. So, ${h}^0(L\otimes_RL)=0$.
Let us consider to another situation:
Let $(R,{\frak{m}},k)$ be a $d$-dimensional regular local ring with $d>2$ and $I$ be a Gorenstein ideal of height two. Then ${h}^0(I\otimes_RI)=0$.
Due to a result of Serre, $I$ generated by a regular sequence $x$ and $y$. Since ${\operatorname{H}}^0_{{\frak{m}}}(I\otimes_R I)\subset {\operatorname{tor}}(I\otimes_R I)$, we deduce that ${\operatorname{H}}^0_{{\frak{m}}}(I\otimes_R I)\subset {\operatorname{H}}^0_{{\frak{m}}}({\operatorname{tor}}(I\otimes_R I))$. The Koszul complex of $R$ with respect to $x$ and $y$ is a free resolution of $R/I$. Then, ${\operatorname{tor}}(I\otimes _RI)={\operatorname{Tor}}^R_2(R/I,R/I)\simeq {\operatorname{H}}_2(\mathbb{K}(I; R)\otimes _RR/I)=R/I.$ Recall that depth of $R/I$ is positive. By the cohomological characterization of depth, ${\operatorname{H}}^0_{{\frak{m}}}(R/I)=0$. We put all things together to deduce that ${\operatorname{H}}^0_{{\frak{m}}}(I\otimes_R I)\simeq {\operatorname{H}}^0_{{\frak{m}}}({\operatorname{tor}}(I\otimes _RI))={\operatorname{H}}^0_{{\frak{m}}}(R/I)=0.$ So, ${h}^0(I\otimes_RI)=0$.
torsion in tensor products
==========================
In [@2010 Question 8.4] Vasconcelos posed some questions. For example, let $R$ be a one-dimensional domain and $M$ a torsion-free module such that $M\otimes_R M$ is torsion-free. Is $M$ free?
(See [@tensor 4.7]) Let $R$ be a one-dimensional local domain with a canonical module which is not Gorenstein. Then there is a non-free and torsion-free module $M$ such that $M\otimes_R M$ is torsion-free.
Also, he asked the following:
Let $R$ be a local domain and $M$ be torsion-free. Is there an integer $e$ guaranteeing that if $M$ is not free, then the tensor power $M^{\otimes e}$ has nontrivial torsion?
\[\] Let $R$ be a $3$-dimensional Cohen-Macaulay local domain and $M$ be a reflexive module such that ${\operatorname{pd}}(M)<\infty$. If $M^{\otimes 3}$ is torsion-free, then $M$ is free.
Since $M$ is torsion-free it is a submodule of a free module $F$. Let $C:=\frac{F}{M}$. There is nothing to prove if $C=0$. Without loss of the generality we assume that $C\neq0$. Note that ${\operatorname{pd}}(M)\leq 1$. Suppose on the contradiction that ${\operatorname{pd}}(M)\neq 0$, i.e., ${\operatorname{pd}}(M)= 1$. We look at the exact sequence $0\to M\to F\to C\to 0\quad(\ast)$. The induced long exact sequence, presents the natural isomorphisms ${\operatorname{Tor}}^R_{i+1}(C,M)\simeq{\operatorname{Tor}}^R_{i}(M,M)$ for all $i>0$. Since ${\operatorname{pd}}(M)=1$, ${\operatorname{Tor}}^R_{\geq 2}(C,M)=0$ and so ${\operatorname{Tor}}^R_{+}(M,M)=0$. This vanishing result allow us to compute ${\operatorname{pd}}(M \otimes_R M) $, see Fact \[2au\].B). By Auslander-Buchsbaum formula, $
{\operatorname{depth}}(M) + {\operatorname{depth}}(M)
= {\operatorname{depth}}(R) + {\operatorname{depth}}(M \otimes_R M). $ From ${\operatorname{depth}}(M)=2$ we see ${\operatorname{depth}}(M \otimes_R M)=1$. Again, $(\ast)$ yields the following exact sequence $$0{\longrightarrow}{\operatorname{Tor}}^R_1(C,M^{\otimes 2}){\longrightarrow}M^{\otimes 3}{\longrightarrow}M^{\otimes 2}\otimes_R F{\longrightarrow}M^{\otimes 2}\otimes_R C{\longrightarrow}0$$and the natural isomorphisms ${\operatorname{Tor}}^R_{i+1}(C,M^{\otimes 2})\simeq{\operatorname{Tor}}^R_{i}(M,M^{\otimes 2})$ for all $i>0$. Recall that ${\operatorname{Tor}}^R_1(C,M^{\otimes 2})$ is torsion. Since ${\operatorname{Tor}}^R_1(C,M^{\otimes 2})$ is torsion, ${\operatorname{Tor}}^R_1(C,M^{\otimes 2})\subset M^{\otimes 3}$ and $M^{\otimes 3}$ is torsion-free, we get that ${\operatorname{Tor}}^R_1(C,M^{\otimes 2})=0$. In order to show ${\operatorname{Tor}}^R_2(C,M^{\otimes 2})=0$ we use a trick of Peskine-Szpiro. Since the assumptions are not the same, we present the details. First, we show ${\operatorname{Tor}}^R_2(C,M^{\otimes 2})$ is of finite length. Indeed, let ${\frak{p}}\neq {\frak{m}}$ be in support of $M$. Since $M_{{\frak{p}}}$ is reflexive and of finite projective dimension, it is $({\operatorname{S}}_2)$. Since ${\operatorname{depth}}(R_{{\frak{p}}})=\dim R_{{\frak{p}}}<3$ it follows that ${\operatorname{pd}}(M_{{\frak{p}}})={\operatorname{depth}}(R_{{\frak{p}}})-{\operatorname{depth}}(M_{{\frak{p}}})= 0$. That is $M$ is locally free over the punctured spectrum. From this, ${\operatorname{Tor}}^R_{1}(M,M^{\otimes 2})$ is of finite length. By the natural isomorphism, ${\operatorname{Tor}}^R_2(C,M^{\otimes 2})\simeq {\operatorname{Tor}}^R_{1}(M,M^{\otimes 2})$. Hence $\ell({\operatorname{Tor}}^R_2(C,M^{\otimes 2}))<\infty$. By $(\ast)$, we have ${\operatorname{pd}}(C)=2$. Let $0\to F_2\to F_1\to F_0\to C\to 0$ be a free resolution of $C$. Apply $-\otimes_RM^{\otimes 2}$ to it we have $${\operatorname{Tor}}^R_2(C,M^{\otimes 2})=\ker\left(F_2\otimes_R M^{\otimes 2}\to F_1\otimes_R M^{\otimes 2}\right)\subset \bigoplus_{{\operatorname{rank}}(F_2)} M^{\otimes 2}.$$Note that $M^{\otimes 2}$ is of positive depth. Any non-zero submodule of a module of positive depth is a module of positive depth. We apply this for the pair ${\operatorname{Tor}}^R_2(C,M^{\otimes 2})\subset \bigoplus_{{\operatorname{rank}}(F_2)} M^{\otimes 2}$ to deduce that ${\operatorname{Tor}}^R_{2}(C,M^{\otimes 2})=0$. Since ${\operatorname{pd}}(C)=2$, ${\operatorname{Tor}}^R_{+}(C,M^{\otimes 2})=0$. This allow us to apply Fact \[2au\].B) to see $
{\operatorname{depth}}(C) + {\operatorname{depth}}(M^{\otimes 2})
\stackrel{(+)}= {\operatorname{depth}}(R) + {\operatorname{depth}}(M^{\otimes 2}\otimes_R C)$. By Auslander-Buchsbaum formula, ${\operatorname{depth}}(C)=1$. Recall that ${\operatorname{depth}}(M^{\otimes 2})=1$. We see left side of $(+)$ is $2$ and the right hand side is at least $3$. This is a contradiction. In sum, $M$ is free.
\[t2\] Let $R$ be a local ring of depth $2$ and $M$ be torsion-free such that ${\operatorname{pd}}(M)<\infty$. If $M^{\otimes 2}$ is torsion-free, then $M$ is free.
Suppose on the contradiction that $M$ is not free. Since $M$ is torsion-free it is a submodule of a free module $F$. Let $C:=\frac{F}{M}$. Without loss of the generality we assume that $C\neq0$. We look at the exact sequence $0\to M\to F\to C\to 0$. The induced long exact sequence, presents the natural isomorphisms ${\operatorname{Tor}}^R_{i+1}(C,M)\simeq{\operatorname{Tor}}^R_{i}(M,M)$ for all $i>0$. It follows by Auslander-Buchsbaum that ${\operatorname{pd}}(M)= 1$. We conclude that ${\operatorname{Tor}}^R_{\geq 2}(C,M)=0$. Thus ${\operatorname{Tor}}^R_{+}(M,M)=0$. We recall from Fact \[2au\].B) that $
{\operatorname{depth}}(M) + {\operatorname{depth}}(M)
\stackrel{(+)}= {\operatorname{depth}}(R) + {\operatorname{depth}}(M \otimes_R M) $. Also, ${\operatorname{depth}}(M \otimes_R M)>0$ because it is torsion-free. The left hand side of $(+)$ is $2$ and the right hand side is at least $3$. This contradiction says that $M$ is free.
higher cohomology of tensor products
=====================================
. We need the following result:
\[hw\](See [@tensor2 Theorem 2.4]) Let $R$ be such that its completion is a quotient of equicharacteristic regular local ring by a nonzero element. Let $r$ be such that $0\leq r< \dim R$. Assume $M\otimes N$ is $({\operatorname{S}}_{r+1})$ over the punctured spectrum and at least one them is of constant rank and ${\operatorname{pd}}(M)<\infty$. Then ${\operatorname{H}}^{r}_{{\frak{m}}}( N\otimes_R M)=0$ and both of $M$ and $N$ has depth at least $r$ if and only if ${\operatorname{depth}}(N)+{\operatorname{depth}}(M)\geq \dim R+r +1$.
It may be nice to determine the case for which ${\operatorname{depth}}(M)+{\operatorname{depth}}(N)$ is minimum. Recall that $M$ is called $p$-*spherical* if ${\operatorname{pd}}(M) = p$ and ${\operatorname{Ext}}^i_R(M,R) = 0$ for $i\neq0$ and $i \neq p$.
\[sph\] Let $R$ be such that its completion is a quotient of equicharacteristic regular local ring by a nonzero element and $M$ be torsion-free of constant rank, of projective dimension $p\in \mathbb{N}$ and locally free. Then ${\operatorname{depth}}(M)+{\operatorname{depth}}(M^\ast)= \dim R+1$ if and only if $M$ is $p$-spherical.
Suppose ${\operatorname{depth}}(M)+{\operatorname{depth}}(M^\ast)= \dim R+1$. In view of Fact \[hw\] $M\otimes_R M^\ast$ is torsion-free. Let $j$ be the smallest positive integer such that ${\operatorname{Ext}}^j_R(M,R) \neq0$. Such a thing exists. Set $f:R^{\beta_{j}(M)}\to R^{\beta_{j-1}(M)}$. We look at $L:={\operatorname{coker}}(f^\ast)$ and the inclusion ${\operatorname{Ext}}^j_R(M,R)\subset L $. Since ${\operatorname{pd}}(M)<\infty$, $M$ is generically free. Thus, ${\operatorname{Tor}}_1^R(M,-)$ is torsion. We put this along with the proof of [@au 3.8(e)] to see that ${\operatorname{Tor}}^R_{j+1}(L,M)=0$. By the rigidity theorem of Lichtenbaum [@l Theorem 3], ${\operatorname{Tor}}^R_{i}(L,M)=0$ for all $i>j$. Since ${\operatorname{depth}}(L)=0$ this says that ${\operatorname{pd}}(M)\leq j$. By definition, $M$ is $p$-spherical. Conversely, assume that $M$ is $p$-spherical. There is an exact sequence $0\to M^\ast\to ( R^{\beta_{0}(M)})^\ast\to\ldots \to (R^{\beta_{p}(M)})^\ast\to L\to 0$. Since ${\operatorname{Ext}}^p_R(M,R)\subset L $ and $\ell({\operatorname{Ext}}^j_R(M,R))<\infty $ we deduce that ${\operatorname{depth}}(L)=0$. It turns out that ${\operatorname{depth}}(M^\ast)=p+1$. Due to Auslander-Buchsbaum formula, ${\operatorname{depth}}(M)+{\operatorname{depth}}(M^\ast)= \dim R+1$.
Let $R$ be a regular local ring of dimension $3$ and $M$ a reflexive module.
1. Always ${\operatorname{H}}^{0}_{{\frak{m}}}( M\otimes _RM)=0$.
2. If ${\operatorname{H}}^{i}_{{\frak{m}}}( M\otimes _RM)=0$ for some $0<i<3$, then $M$ is free.
The first item is in Observation \[3\]. We may assume that $i>0$ and that $M\neq 0$. Reflexive modules over 2-dimensional regular local rings are free. From this, $M$ is locally free over the punctured spectrum. We apply Fact \[hw\] for $r=i$, to see that $2{\operatorname{depth}}(M)\geq \dim R+i+1\geq 5$. That is $2<\frac{5}{2}\leq{\operatorname{depth}}(M)\leq \dim(M)\leq 3$. Thus, ${\operatorname{depth}}(M)= 3$. Due to Auslander-Buchsbaum, $M$ is free.
In view of [@tensor2 Example 1.8] there is a non-free ideal $I$ of $R:=\frac{K[[x,y,z,w]]}{(xy-uv)}$ such that $I\otimes I^\ast$ is torsion-free.
Let $(R,{\frak{m}},k)$ be a local ring of depth at least $3$. Then i) ${\frak{m}}\otimes_R {\frak{m}}^\ast$ is torsion-free, ii) ${\frak{m}}$ is locally free, and iii) ${\operatorname{H}}^{2}_{{\frak{m}}}({\frak{m}}\otimes_R {\frak{m}}^\ast)=0$.
Clearly ${\frak{m}}$ is non-free and locally free, and that ${\operatorname{Ext}}^{<3}_R(k,R)={\operatorname{H}}^{<3}_{{\frak{m}}}(R)=0$. We look at $0\to {\frak{m}}\to R\to k\to 0 \quad(\ast)$. It yields that $0=k^\ast\to {\frak{m}}^\ast\to R^\ast\to{\operatorname{Ext}}^1_R(k,R)=0$, i.e., ${\frak{m}}^\ast\simeq R$. Also, $(\ast)$ implies that $0={\operatorname{H}}^1_{{\frak{m}}}(k)\to{\operatorname{H}}^2_{{\frak{m}}}({\frak{m}})\to {\operatorname{H}}^2_{{\frak{m}}}(R)=0$. So, ${\operatorname{H}}^{2}_{{\frak{m}}}({\frak{m}}\otimes_R {\frak{m}}^\ast)\simeq{\operatorname{H}}^2_{{\frak{m}}}({\frak{m}})=0$.
.
\[main5b\] Let $(R,{\frak{m}},k)$ be a regular local ring and $M$ be an indecomposable Buchsbaum module of dimension $d:=\dim(R)$ which is not Cohen-Macaulay.
1. If ${\operatorname{depth}}(M)=1$, then $${h}^i(M\otimes _RM)=\left\{
\begin{array}{rl}
{d}\choose{2}& \ \ \ \ \ \ \ \ \ \ \ \ \text{if }\ \ i=0\\
d+1 & \ \ \ \ \ \ \ \ \ \ \ \ \text{if } \ \ i=1\\
0& \ \ \ \ \ \ \ \ \ \ \ \ \text{if }\ \ 2\leq i< d
\end{array} \right.$$In particular, $M\otimes_R M$ is not Buchsbaum.
2. If $d>3$ and $M$ is almost Cohen-Macaulay, then $${h}^i(M\otimes_R M^\ast)=
\left\{
\begin{array}{rl}
0& \ \ \ \ \text{ if } \ \ i\in\{0\}\cup[3,d-2]\\
1& \ \ \ \ \text{ if } \ \ i=1\\
d& \ \ \ \ \text{ if }\ \ i=2\ \ or \ \ i=d-1
\end{array}\right.$$ In particular, $M\otimes_R M^\ast$ is quasi-Buchsbaum. Against to $M$ and $M^\ast$, $M\otimes_R M^\ast$ is not Buchsbaum.
i\) First, we state a more general claim:
1. Let $(A,{\frak{n}},k)$ be a Cohen-Macaulay local ring of dimension at least two and $I\lhd A$ be ${\frak{n}}$-primary. Then$${h}^i(I\otimes_A {\frak{n}})=\left\{
\begin{array}{rl}
\beta_2(A/I)& \ \ \ \ \ \ \ \ \ \ \ \ \text{if } \ \ i=0\\
\mu(I)+\ell(A/I) & \ \ \ \ \ \ \ \ \ \ \ \ \text{if }\ \ i=1\\
0& \ \ \ \ \ \ \ \ \ \ \ \ \text{if } \ \ 2\leq i< \dim A\\
\end{array} \right.$$
Indeed, let $d:=\dim A$. We look at $0\to {\frak{n}}\to A\to k\to 0$ and we drive the following exact sequence$$0{\longrightarrow}{\operatorname{Tor}}_1^A(k,I){\longrightarrow}I\otimes_A {\frak{n}}{\longrightarrow}I{\longrightarrow}I\otimes_A k{\longrightarrow}0\quad(\ast)$$ Recall that $I\otimes_A k\simeq\frac{I}{I{\frak{n}}}\simeq k^{\mu(I)}$ and ${\operatorname{Tor}}_1^A(k,I)\simeq{\operatorname{Tor}}_2^A(k,A/I)\simeq k^{\beta_2(A/I)}$. We break down $(\ast)$ into
1. $0\to k^{\beta_2(A/I)}\to I\otimes_A {\frak{n}}\to L\to 0$ and
2. $0\to L\to I\to k^{\mu(I)}\to 0$.
We conclude from a) the exact sequence $0\to {\operatorname{H}}^0_{{\frak{n}}}(k^{\beta_2(A/I)})\to {\operatorname{H}}^0_{{\frak{n}}}(I\otimes_A {\frak{n}})\to {\operatorname{H}}^0_{{\frak{n}}}(L)$. It follows from b) that the sequence $0\to {\operatorname{H}}^0_{{\frak{n}}}(L)\to {\operatorname{H}}^0_{{\frak{n}}}(I)=0$ is exact. We combine these to see $\ell({\operatorname{H}}^0_{{\frak{n}}}(I\otimes_R {\frak{n}}))=\ell({\operatorname{H}}^0_{{\frak{n}}}(k^{\beta_2(A/I)}))={\beta_2(A/I)}$. From a) we have ${\operatorname{H}}^1_{{\frak{n}}}(I\otimes_R {\frak{n}})\simeq {\operatorname{H}}^1_{{\frak{n}}}(L)$. From b), $$0={\operatorname{H}}^0_{{\frak{n}}}(I){\longrightarrow}{\operatorname{H}}^0_{{\frak{n}}}(k^{\mu(I)}){\longrightarrow}{\operatorname{H}}^1_{{\frak{n}}}(L)\simeq{\operatorname{H}}^1_{{\frak{n}}}(I\otimes_R {\frak{n}}){\longrightarrow}{\operatorname{H}}^1_{{\frak{n}}}(I){\longrightarrow}{\operatorname{H}}^1_{{\frak{n}}}(k^{\mu(I)})=0.$$ In order to compute ${\operatorname{H}}^1_{{\frak{n}}}(I)$, we look at $0\to I\to A\to A/I\to 0$. This induces $0={\operatorname{H}}^0_{{\frak{n}}}(A)\to{\operatorname{H}}^0_{{\frak{n}}}(A/I)\to {\operatorname{H}}^1_{{\frak{n}}}(I) \to {\operatorname{H}}^1_{{\frak{n}}}(A)= 0 .$ Thus, ${\operatorname{H}}^1_{{\frak{n}}}(I)\simeq{\operatorname{H}}^0_{{\frak{n}}}(A/I)=A/I$. We put all of these together to see $0\to k^{\mu(I)}\to {\operatorname{H}}^1_{{\frak{n}}}(I\otimes_A {\frak{n}})\to A/I\to0.$ We conclude that ${h}^1(I\otimes_A {\frak{n}})=\mu(I)+\ell(A/I)$. Let $2\leq i< d$. Recall that ${\operatorname{H}}^i_{{\frak{n}}}(I\otimes_A {\frak{n}})\simeq{\operatorname{H}}^i_{{\frak{n}}}(L)\simeq{\operatorname{H}}^i_{{\frak{n}}}(I)$. We look at $0={\operatorname{H}}^{i-1}_{{\frak{n}}}(A/I)\to {\operatorname{H}}^i_{{\frak{n}}}(I) \to {\operatorname{H}}^i_{{\frak{n}}}(A)= 0 $ to deduce that ${\operatorname{H}}^i_{{\frak{n}}}(I\otimes_A {\frak{n}})\simeq{\operatorname{H}}^i_{{\frak{n}}}(I)=0$. This completes the proof of Claim A). Recall from [@goto Corollary (3.7)] that:
1. Let $(A,{\frak{n}})$ be a regular local ring and $P$ be an indecomposable Buchsbaum module of maximal dimension. Then $P\simeq {\operatorname{Syz}}_{i}(\frac{A}{{\frak{n}}})$ where $i={\operatorname{depth}}(P)$.
In the light of Fact A) we see $M={\operatorname{Syz}}_{1}(k)={\frak{m}}$. Note that $\beta_2(k)$ is equal to ${d}\choose{2}$ and $\mu({\frak{m}})=d$. It follows by the assumptions that $\dim(R)\geq2$. Claim A) yields that: $${h}^i(M\otimes _RM)=\left\{
\begin{array}{rl}
{d}\choose{2}& \ \ \ \ \ \ \ \ \ \ \ \ \text{if }\ \ i=0\\
d+1 & \ \ \ \ \ \ \ \ \ \ \ \ \text{if } \ \ i=1\\
0& \ \ \ \ \ \ \ \ \ \ \ \ \text{if }\ \ 2\leq i< d
\end{array} \right.$$ To see the particular case, we recall from [@goto Theorem (1.1)] that:
1. Let $(A,{\frak{n}})$ be a regular local ring and $P$ be Buchsbaum. Then $P\simeq\bigoplus_{0\leq i\leq \dim (A)}{\operatorname{Syz}}_i(\frac{A}{{\frak{n}}})^{{h}^i}$ where ${h}^i:={h}^i(P)$ for all $0\leq i< \dim A$.
Suppose on the contradiction that $M\otimes_R M$ is Buchsbaum. Due to Fact B), $M\otimes_R M\simeq\bigoplus_{0\leq i\leq d}{\operatorname{Syz}}_i(k)^{{h}^i}$ where ${h}^i:={h}^i(M\otimes_R M)$ for $i\neq d$. It turns out that $M\otimes_R M\stackrel{(\natural)}\simeq k^{{d}\choose{2}}\bigoplus{\operatorname{Syz}}_{1}(k)^{\oplus (d+1)}\bigoplus R^{n}$ for some $n\geq0$. Since $M\simeq{\frak{m}}$, we see the rank of left hand side of $(\natural)$ is one. The rank of right hand side is $0+(d+1)+n$. Since $n\geq 0$, we get to a contradiction. So, $M\otimes_R M$ is not Buchsbaum.
ii\) We recall that $M$ is called almost Cohen-Macaulay if ${\operatorname{depth}}(M)\geq\dim( M)-1$. Since $M$ is not Cohen-Macaulay, ${\operatorname{depth}}(M)=\dim (M)-1=d-1$. In the light of Fact A), $M:={\operatorname{Syz}}_{d-1}(k)$. Since $M$ is locally free, ${\operatorname{Tor}}_1^R(M,M^\ast)$ is of finite length. We look at $0\to R \to R^d\to M\to 0$ and we drive the following exact sequence$$0{\longrightarrow}{\operatorname{Tor}}_1^R(M,M^\ast){\longrightarrow}M^\ast {\longrightarrow}(M^\ast)^d{\longrightarrow}M\otimes _RM^\ast{\longrightarrow}0.$$ We break down it into $0\to {\operatorname{Tor}}_1^R(M,M^\ast)\to M^\ast\to L\to 0$ and $0\to L\to (M^\ast)^d\to M\otimes_R M^\ast\to 0$. It follows from the first sequence that $0={\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Tor}}_1^R(M,M^\ast))\to {\operatorname{H}}^1_{{\frak{m}}}(M^\ast)\to {\operatorname{H}}^1_{{\frak{m}}}(L)\to{\operatorname{H}}^2_{{\frak{m}}}({\operatorname{Tor}}_1^R(M,M^\ast))=0$. Similarly, ${\operatorname{H}}^{+}_{{\frak{m}}}(M^\ast)\simeq{\operatorname{H}}^{+}_{{\frak{m}}}(L)$. Recall that $M^\ast$ is reflexive. In particular it is $({\operatorname{S}}_2)$. So, $ {\operatorname{H}}^{1}_{{\frak{m}}}(L)\simeq{\operatorname{H}}^{1}_{{\frak{m}}}(M^\ast)=0$. It follows from the second short exact sequence that $0={\operatorname{H}}^0_{{\frak{m}}}((M^\ast)^d)\to {\operatorname{H}}^0_{{\frak{m}}}(M\otimes_R M^\ast)\to {\operatorname{H}}^{1}_{{\frak{m}}}(L)=0$. From this, ${h}^0(M\otimes _RM^\ast)=0$.
1. (See [@ag Proposition A.1]) Let $A$ be a ring, a necessarily and sufficient condition for which $P$ be projective is that $\varphi_P:P\otimes_AP^\ast\to {\operatorname{Hom}}_A(P,P)$ is (surjective) an isomorphism.
Since $M$ is locally free, it follows from Fact A) that $K:=\ker(\varphi_{M})$ and $C:={\operatorname{coker}}(\varphi_{M})$ are of finite length and that $C\neq 0$. From this, ${\operatorname{H}}^0_{{\frak{m}}}(C)=C\neq 0$, ${\operatorname{H}}^+_{{\frak{m}}}(C)= {\operatorname{H}}^+_{{\frak{m}}}(K)= 0$. We look at $0\to K\to M\otimes_R M^\ast\to{\operatorname{im}}(\varphi_{M})\to 0 $ and $0\to{\operatorname{im}}(\varphi_{M})\to {\operatorname{Hom}}_R(M,M) \to C\to 0 $. Since ${\operatorname{depth}}(M)>1$ another result of Auslander-Goldman ([@ag Proposition 4.7]) says that ${\operatorname{depth}}({\operatorname{Hom}}_R(M,M) )>1$, i.e., ${\operatorname{H}}^0_{{\frak{m}}}({\operatorname{Hom}}_R(M,M))={\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Hom}}_R(M,M))=0$. We apply this along with the long exact sequences of local cohomology modules to see
1. $0={\operatorname{H}}^1_{{\frak{m}}}(K)\to {\operatorname{H}}^1_{{\frak{m}}}(M\otimes _RM^\ast)\to{\operatorname{H}}^1_{{\frak{m}}}({\operatorname{im}}(\varphi_M))\to {\operatorname{H}}^2_{{\frak{m}}}(K)=0$
2. $0= {\operatorname{H}}^0_{{\frak{m}}}({\operatorname{Hom}}_R(M,M)){\longrightarrow}{\operatorname{H}}^0_{{\frak{m}}}(C){\longrightarrow}{\operatorname{H}}^1_{{\frak{m}}}({\operatorname{im}}(\varphi_M)){\longrightarrow}{\operatorname{H}}^1_{{\frak{m}}}({\operatorname{Hom}}_R(M,M))=0,$
e.g., ${\operatorname{H}}^1_{{\frak{m}}}(M\otimes _RM^\ast)\simeq{\operatorname{H}}^1_{{\frak{m}}}({\operatorname{im}}(\varphi_M))\simeq{\operatorname{H}}^0_{{\frak{m}}}(C) \simeq C\simeq {\operatorname{Tor}}^R_1(D(M),M)$, because ${\operatorname{coker}}(\varphi _M)={\operatorname{Tor}}^R_1 (D(M),M).$ Let ${\frak{m}}=(x_1,\ldots, x_d)$. In view of $0\to R \stackrel{(x_1,\ldots, x_d)}{\longrightarrow}R^d\to M\to 0$ we see $D(M)= {\operatorname{coker}}\left(R^d\stackrel{(x_1,\ldots, x_d)}{\longrightarrow}R\right)=\frac{R}{{\frak{m}}}.$ Also, ${\operatorname{Tor}}^R_1(D(M),M)\simeq {\operatorname{Tor}}^R_1(k,{\operatorname{Syz}}_{d-1}(k))={\operatorname{Tor}}_d^R(k,k)=k.$ Combining these, ${h}^1(M\otimes_R M^\ast)=\ell({\operatorname{Tor}}^R_1(D(M),M))=1$. Also, ${\frak{m}}{\operatorname{H}}^1_{{\frak{m}}}(M\otimes _RM^\ast)=0$.
1. (See [@bv Proposition 4.1]) Let $(A,{\frak{n}})$ be a local ring, $L$ be locally free and $N$ be of depth at least $3$. Then ${\operatorname{Ext}}^i_A (L, N)\simeq {\operatorname{H}}^{i+1}_{{\frak{m}}}( N\otimes_A L^{\ast})$ for all $1\leq i\leq {\operatorname{depth}}(N)-2$.
By this ${\operatorname{H}}^2_{{\frak{m}}}(M\otimes_R M^\ast)\simeq{\operatorname{Ext}}^1_R(M,M)$, because ${\operatorname{depth}}(M)=d-1\geq3$. Apply ${\operatorname{Hom}}_R(-,M)$ to $0\to R \to R^d\to M\to 0$ to see $0\to {\operatorname{Hom}}_R(M,M) \to {\operatorname{Hom}}_R(R^d,M)\to {\operatorname{Hom}}_R(R,M)\to {\operatorname{Ext}}^1_R(M,M)\to 0$. Thus, ${\operatorname{H}}^2_{{\frak{m}}}(M\otimes_R M^\ast)\simeq{\operatorname{Ext}}^1_R(M,M)={\operatorname{coker}}\left(M^d\stackrel{(x_1,\ldots, x_d)}{\longrightarrow}M\right)=\frac{M}{{\frak{m}}M}.$ Hence, ${h}^2(M\otimes_R M^\ast)=\ell(\frac{M}{{\frak{m}}M})=\mu(M)= \beta_{d-1}(k)=d$. Also, ${\frak{m}}{\operatorname{H}}^2_{{\frak{m}}}(M\otimes _RM^\ast)=0$.
Let $3\leq i\leq d-2$. Due to Fact D) we know that ${\operatorname{H}}^{i}_{{\frak{m}}}(M\otimes_R M^\ast)\simeq {\operatorname{Ext}}^{i-1}_R(M,M)=0$, because ${\operatorname{pd}}(M)=1$. Thus, ${h}^i(M\otimes_R M^\ast)=0$.
Here, we compute ${h}^{d-1}(M\otimes_R M^\ast)$. To this end, we recall from [@tensor2 Proposition 4.1] that:
1. Let $A$ and $B$ be locally free over a regular local ring $(S,{\frak{n}})$ of dimension $d\geq 3$ and let $2\leq j\leq d-1$. Then ${\operatorname{H}}^j_{{\frak{n}}}(A\otimes_S B)^v\simeq {\operatorname{H}}^{d+1-j}_{{\frak{n}}}(A^{\ast}\otimes_S B^{\ast})$, where $(-)^v$ is the Matlis duality.
Since $d-1\geq2$, ${\operatorname{Syz}}_{d-1}(k)$ is a second syzygy, it is reflexive. Also, $\ell((-)^v)=\ell(-)$. We use these to see $${h}^{d-1}(M\otimes_R M^\ast)=
\ell({\operatorname{H}}^{d-1}_{{\frak{m}}}(M\otimes_R M^\ast)^v)=\ell({\operatorname{H}}^{2}_{{\frak{m}}}(M^\ast\otimes_R M^{\ast\ast}))=\ell({\operatorname{H}}^{2}_{{\frak{m}}}(M^\ast\otimes_R M))=d.$$Since Matlis duality preserves the annihilator we deduce that ${\frak{m}}{\operatorname{H}}^{d-1}_{{\frak{m}}}(M^\ast\otimes_R M)=0$.
We proved that ${\frak{m}}{\operatorname{H}}^{<d}_{{\frak{m}}}(M\otimes_R M^\ast)=0$. By definition, $M\otimes_R M^\ast$ is quasi-Buchsbaum. In view of $0\to R \to R^d\to M\to 0$ we see $0\to M^\ast\to R^d\to R$ is exact. Thus, $M^\ast={\operatorname{Syz}}_2(R/ {\frak{m}})$ is Buchsbaum. Note that ${\operatorname{rank}}(M)={\operatorname{rank}}(M^\ast)=d-1$, because $0\to M^\ast\to R^d\to {\frak{m}}\to 0$. Thus, ${\operatorname{rank}}(M\otimes_R M^\ast)=(d-1)^2.$ Also, ${\operatorname{rank}}({\operatorname{Syz}}_1(k))=1$, because ${\operatorname{Syz}}_1(k)={\frak{m}}$. Suppose on the contradiction that $M\otimes_R M^\ast$ is Buchsbaum. Due to Fact B) there is an $n\geq 0$ such that $$M\otimes_R M^\ast={\operatorname{Syz}}_1(k)\bigoplus{\operatorname{Syz}}_{2}(k)^{\oplus d}\bigoplus {\operatorname{Syz}}_{d-1}(k)^{\oplus d}\bigoplus R^{n}.$$ The left hand side is a vector bundle of rank $(d-1)^2$. The right hand side is a vector bundle of rank $1+d (d-1)+d (d-1)+n$. Since $n\geq 0$, we get to a contradiction.
Over a regular local ring $(R,{\frak{m}},k)$ of dimension $d>1$, Auslander was looking for a vector bundle $M$ without free summand of dimension $d$ such that ${\operatorname{pd}}(M)={\operatorname{pd}}(M^\ast)$ and ${\operatorname{H}}^{0}_{{\frak{m}}}( M\otimes _RM^\ast)=0$. He proved the existence of $M$ is equivalent to the oddness of $d$.
\[ausbun\]Let $(R,{\frak{m}},k)$ be a regular local ring of odd dimension $d$ and $M$ be as above. If $M$ is Buchsbaum, then $M\simeq {\operatorname{Syz}}_{\frac{d+1}{2}}(k)^{\oplus m} $ for some $m$.
Suppose first that $M$ is indecomposable. By Fact \[main5b\].A) $M\simeq{\operatorname{Syz}}_i(k)$ where $i:={\operatorname{depth}}(M)$. Since $M$ has no free direct summand, $i<d$. This allow us to use [@goto Lemma 3.2] to see $M^\ast={\operatorname{Syz}}_{d-i+1}(k)$. We deduce from $d-i={\operatorname{pd}}(M)={\operatorname{pd}}(M^\ast)={\operatorname{pd}}({\operatorname{Syz}}_{d-i+1}(k))=d-(d-i+1)$ that $i=\frac{d+1}{2}$. In particular, $M= {\operatorname{Syz}}_{\frac{d+1}{2}}(k)$. As a second case, suppose $M$ is decomposable and has a direct summand other than ${\operatorname{Syz}}_{\frac{d+1}{2}}(k)$. In view of Fact \[main5b\].B) there is an $I\subset[1,d-1]$ such that $M\simeq\bigoplus_{i\in I}{\operatorname{Syz}}_i(k)^{{h}^i}$. Note that ${\operatorname{pd}}(M)=\sup_{i\in I}\{{\operatorname{pd}}({\operatorname{Syz}}_i(k))\}=\sup_{i\in I}\{d-i\}=d-\inf\{i:i\in I\}.$ Let $j$ be such that $j=d-\inf\{i:i\in I\}$. Recall that ${\operatorname{Syz}}_i(k)^\ast={\operatorname{Syz}}_{d-i+1}(k)$. Since ${\operatorname{pd}}(M)={\operatorname{pd}}(M^\ast)$ it follows that ${\operatorname{Syz}}_{d-j+1}(k)$ is a direct summand of $M$. One of $j$ and $d-j$ is smaller than $\frac{d+1}{2}$. Without loss of the generality, we assume that $j<\frac{d+1}{2}$ (one may use [@tensor2 Theorem 2.4] to get a contradiction. Here, we follow our simple reasoning:) We look at $0\to{\operatorname{Syz}}_j(k)\to R^{\beta_{j-1}(k)}\to{\operatorname{Syz}}_{j-1}(k)\to 0$. This induces $$0\to{\operatorname{Tor}}^R_1({\operatorname{Syz}}_j(k),{\operatorname{Syz}}_{j-1}(k))\to{\operatorname{Syz}}_j(k)\otimes_R {\operatorname{Syz}}_{j}(k)\to R^{\beta_{j-1}(k)}\otimes_R {\operatorname{Syz}}_{j}(k)\to {\operatorname{Syz}}_{j}(k)\otimes_R {\operatorname{Syz}}_{j-1}(k)\to 0.$$Note that ${\operatorname{Tor}}^R_1({\operatorname{Syz}}_j(k),{\operatorname{Syz}}_{j-1}(k))\simeq{\operatorname{Tor}}^R_j({\operatorname{Syz}}_j(k), k )\simeq{\operatorname{Tor}}^R_{j+j}(k , k)\simeq k ^{\oplus\beta_{2j}(k)}.$ Since $j<\frac{d-1}{2}$ we have $2j\leq d$. We conclude from this that ${\operatorname{Tor}}^R_1({\operatorname{Syz}}_j(k),{\operatorname{Syz}}_{j-1}(k))$ is nonzero and of finite length. Since $$k\subset{\operatorname{Tor}}^R_1({\operatorname{Syz}}_j(k),{\operatorname{Syz}}_{j-1}(k))\subset {\operatorname{Syz}}_j(k)\otimes_R {\operatorname{Syz}}_{j}(k)\subset M\otimes_R M^\ast,$$ we see that ${\operatorname{H}}^0_{{\frak{m}}}(M\otimes _RM^\ast)\neq 0$, a contradiction.
. Recall that vanishing of ${\operatorname{H}}^{2}_{{\frak{m}}}(M\otimes_R M^{\ast})$ over regular local rings implies freeness of $M^\ast$. This can’t be extended into hypersurfaces: Let $R:=\frac{K[[x,y,z,w]]}{(xy-uv)}$ and $I:=(x,y)$. Then ${\operatorname{H}}^{2}_{{\frak{m}}}(I\otimes_R I^\ast)=0$ but $I^\ast$ is not free. In fact, the following stated implicitly in [@tensor2]:
Let $R$ be a hyper-surface of dimension $d\geq2$ and $M$ be torsion-free, locally free and of constant rank. Assume one the following holds: i) ${\operatorname{H}}^1_{{\frak{m}}}(M\otimes _R M^\ast)= {\operatorname{H}}^2_{{\frak{m}}}(M\otimes _R M^\ast)=0$, or ii) ${\operatorname{H}}^0_{{\frak{m}}}(M\otimes _R M^\ast)= {\operatorname{H}}^1_{{\frak{m}}}(M\otimes _R M^\ast)=0$. Then $M^\ast$ is free.
\[kan\] Let $R$ be a Cohen-Macaulay local ring with isolated Gorenstein singularity and possessing a canonical module. Let $i>0$. Then ${\operatorname{H}}^{i}_{{\frak{m}}}(\omega_R\otimes _R \omega_R^\ast)\neq0$ if and only if $i=1$ or $i=\dim( R)$.
By isolated Gorenstein singularity we mean a non Gorenstein ring which is Gorenstein over the punctured spectrum. From this, $d:=\dim R\neq 0$. Since $(\omega_R)_{{\frak{p}}}\simeq \omega_{R_{{\frak{p}}}}\neq 0$, ${\operatorname{Supp}}(\omega_R)={\operatorname{Spec}}(R)$. Also, ${\operatorname{Ass}}({\operatorname{Hom}}_R(\omega_R,R))={\operatorname{Supp}}(\omega_R)\cap{\operatorname{Ass}}(R)={\operatorname{Spec}}(R)\cap{\operatorname{Ass}}(R)={\operatorname{Ass}}(R)$. From this, ${\operatorname{Supp}}(\omega_R^\ast)={\operatorname{Spec}}(R)$. It follows that ${\operatorname{Supp}}(\omega_R\otimes \omega_R^\ast)={\operatorname{Spec}}(R)$. Thus, $\dim(\omega_R\otimes_R \omega_R^\ast)=d$. By Gorthendieck’s non-vanishing theorem, ${\operatorname{H}}^{d}_{{\frak{m}}}(\omega_R\otimes_R \omega_R^\ast)\neq0$. This completes the proof in the case $d=1$. We assume that $d\geq 2$. Let $\varphi_{\omega_R}:\omega_R\otimes_R\omega_R^\ast\to {\operatorname{Hom}}_R(\omega_R,\omega_R)$. Recall that ${\operatorname{Hom}}_R (\omega_R, \omega_R)\simeq R$ and that ${\operatorname{H}}^0_{{\frak{m}}}(R)={\operatorname{H}}^1_{{\frak{m}}}(R)=0$. Since $\omega_R$ is locally free, it follows from Fact \[main5b\].C) that $K:=\ker(\varphi_{\omega_R})$ and $C:={\operatorname{coker}}(\varphi_{\omega_R})$ are of finite length and that $C\neq 0$. From this, ${\operatorname{H}}^0_{{\frak{m}}}(C)=C\neq 0$, ${\operatorname{H}}^+_{{\frak{m}}}(C)= {\operatorname{H}}^+_{{\frak{m}}}(K)= 0$. We look at $0\to K\to \omega_R\otimes_R \omega_R^\ast\to{\operatorname{im}}(\varphi_{\omega_R})\to 0 $ and $0\to{\operatorname{im}}(\varphi_{\omega_R})\to R \to C\to 0 $. It follows that ${\operatorname{H}}^1_{{\frak{m}}}(\omega_R\otimes _R \omega_R^\ast)\simeq{\operatorname{H}}^1_{{\frak{m}}}({\operatorname{im}}(\varphi_{\omega_R}))\simeq{\operatorname{H}}^0_{{\frak{m}}}(C)\simeq C\neq 0.$ This completes the proof in the case $d=2$. Assume that $d>2$. We proved that ${\operatorname{H}}^{1}_{{\frak{m}}}(\omega_R\otimes_R \omega_R^\ast)\neq0$ and ${\operatorname{H}}^{d}_{{\frak{m}}}(\omega_R\otimes _R \omega_R^\ast)\neq0$. Let $2\leq i\leq d-1$. Then ${\operatorname{H}}^{i}_{{\frak{m}}}(\omega_R\otimes _R \omega_R^\ast)\simeq {\operatorname{H}}^i_{{\frak{m}}}({\operatorname{im}}(\varphi_{\omega_R}))\simeq{\operatorname{H}}^{i-1}_{{\frak{m}}}(C)=0$.
(Part of [@yosh Conjecture 3.4]) Let $R$ be a Cohen-Macaulay local ring, $M$ be perfect and $N$ be Buchsbaum and of maximal dimension. If ${\operatorname{pd}}(M)\leq{\operatorname{depth}}(N)$, then ${h}^i(M\otimes_RN)=
\sum_{j=0}^{{\operatorname{pd}}(M)}\beta_j(M){h}^{j+i}(N)$ for all $i< \dim(M).$
\[vector2\] Let $R$ be a Cohen-Macaulay local ring, $M$ be perfect and $N$ be locally free and of constant rank. Then ${h}^i(M\otimes_RN)\leq
\sum_{j=0}^{{\operatorname{pd}}(M)}\beta_j(M){h}^{j+i}(N)$ for all $i< \dim(M).$
For every module $L$ of finite projective dimension, we have ${\operatorname{grade}}(L) + \dim (L) = \dim( R)$. In particular, if $L$ is perfect then $\dim (L)=\dim( R)-{\operatorname{pd}}(L)$. Therefore, things reduced to show ${h}^i(M\otimes_RN)\leq
\sum_{j=0}^{{\operatorname{pd}}(M)}\beta_j(M){h}^{j+i}(N)$ for all $i< \dim (R)-{\operatorname{pd}}(M).$ We may assume that ${\operatorname{pd}}( M)>0$. There is nothing to prove if $\dim (R)-{\operatorname{pd}}( M)=0$. Without loss of the generality, ${\operatorname{pd}}(M)<\dim( R)={\operatorname{depth}}(R)$. Now, the case $i=0$ is in Proposition \[vector\]. We may assume that $i>0$. Let $f:{\operatorname{Syz}}_1(M)\otimes_RN\to R^{\beta_{0}(M)}\otimes_RN$ and recall from Proposition \[vector\] that ${\operatorname{H}}^i_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN)\simeq{\operatorname{H}}^i_{{\frak{m}}}(\ker(f))$ and there is an exact sequence $ {\operatorname{H}}^i_{{\frak{m}}}(R^{\beta_{0}(M)}\otimes_RN)\to {\operatorname{H}}^i_{{\frak{m}}}(M\otimes_RN)\to {\operatorname{H}}^{i+1}_{{\frak{m}}}(\ker(f)).$ Hence $${h}^i(M\otimes_RN)\leq \ell({\operatorname{H}}^{i+1}_{{\frak{m}}}(\ker(f)))+\beta_{0}(M){h}^i(N)=\ell({\operatorname{H}}^{i+1}_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN))+\beta_{0}(M){h}^i(N).$$ In the same vein, $\ell({\operatorname{H}}^{i+1}_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN))\leq\ell({\operatorname{H}}^{i+2}_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN))+\beta_{1}(M){h}^{i+1}(N).$ Therefore, $$\begin{array}{ll}
{h}^i(M\otimes_RN)&\leq \ell({\operatorname{H}}^{i+1}_{{\frak{m}}}({\operatorname{Syz}}_1(M)\otimes_RN))+\beta_{0}(M){h}^i(N)\\
&\leq\ell({\operatorname{H}}^{i+2}_{{\frak{m}}}({\operatorname{Syz}}_2(M)\otimes_RN))+\beta_{1}(M){h}^{i+1}(N)+\beta_{0}(M){h}^i(N).
\end{array}$$ Repeating this, $
{h}^i(M\otimes_RN)
\leq\ell({\operatorname{H}}^{i+\ell}_{{\frak{m}}}({\operatorname{Syz}}_{\ell}(M)\otimes_RN))+\sum_{j=0}^{\ell-1}\beta_j(M){h}^{j+i}(N).$ Putting $\ell:={\operatorname{pd}}(M)-i$, $
{h}^i(M\otimes _R N)
\leq\ell({\operatorname{H}}^{{\operatorname{pd}}(M)}_{{\frak{m}}}({\operatorname{Syz}}_{{\operatorname{pd}}( M)}(M)\otimes _RN))+\sum_{j=0}^{\ell-1}\beta_j(M){h}^{j+i}(N)=\sum_{j=0}^{{\operatorname{pd}}( M)}\beta_j(M){h}^{j+i}(N).$
The same proof shows that: Let $R$ be equi-dimensional and generalized Cohen-Macaulay local ring and $N$ be locally free and of constant rank. If ${\operatorname{pd}}(M)<{\operatorname{depth}}(R)$, then ${h}^i(M\otimes_RN)\leq
\sum_{j=0}^{{\operatorname{pd}}(M)}\beta_j(M){h}^{j+i}(N)$ for all $i< {\operatorname{depth}}(R)-{\operatorname{pd}}(M).$
[99]{} M. Auslander and O. Goldman, *Maximal orders*, Trans. AMS **97** (1960), 1–24.
M. Auslander, *Modules over unramified regular local rings*, Illinois J. Math. [**[5]{}**]{} (1961) 631–-647.
W. Bruns and U. Vetter, *Length formulas for the local cohomology of exterior powers*, Math. Z. [**[191]{}**]{} (1986), 145–158.
S. Goto, *Maximal Buchsbaum modules over regular local rings and a structure theorem for generalized Cohen-Macaulay modules*, Commutative algebra and combinatorics, Adv. Stud. Pure Math. [**[11]{}**]{}, Kinokuniya, Tokyo, North-Holland, Amsterdam (1987), 39–64.
C. Huneke and R. Wiegand, *Tensor products of modules and the rigidity of Tor*, Math. Ann. [**[299]{}**]{} (1994), 449–476.
C. Huneke and R. Wiegand, *Tensor products of modules, rigidity and local cohomology*, Math. Scand. [**[81]{}**]{} (1997), 161–-183.
S.B. Iyengar and R. Takahashi, *The Jacobian ideal of a commutative ring and annihilators of cohomology*, J. algebra, to appear.
S. Lichtenbaum, *On the vanishing of Tor in regular local rings*, Ill. J. Math. [**[10]{}**]{} (1966), 220–226.
C. Peskine and L. Szpiro, *Dimension projective finie et cohomologie locale*, Publ. Math. IHES. [**42**]{} (1973), 47-–119.
J. Stückrad, and W. Vogel, *Buchsbaum rings and applications. An interaction between algebra, geometry, and topology*, Mathematische Monographien [**[21]{}**]{}, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986.
Wolmer V. Vasconcelos, *Cohomological degrees and applications*, Commutative algebra, 667–-707, Springer, New York, 2013.
Wolmer V. Vasconcelos, *Length complexity of tensor products*, Comm. Algebra [**[38]{}**]{} (2010), no. 5, 1743–-1760.
K.I. Yoshida, *A note on multiplicity of perfect modules of codimension one*, Comm. Algebra [**25**]{} (1997), no. 9, 2807-–2816.
|
---
abstract: 'This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a bridge graph, generated by paths started in a fixed state at any time. The bridge graph is made into a unimodular network by marking it and selecting a root in a specified fashion. The unimodularity of this network is leveraged to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e., recurrence both forwards and backwards in time, is introduced and shown to be a key property in uniquely distinguishing paths in the Doeblin graph, and also a decisive property for Markov chains indexed by ${\mathbb{Z}}$. Properties related to simulating the bridge graph are also studied.'
author:
- |
François Baccelli[^1], Mir-Omid Haji-Mirsadeghi[^2], and\
James T. Murphy III[^3]
title: '<span style="font-variant:small-caps;">Doeblin Trees</span>'
---
**MSC2010:** 05C80, 60J10, 60G10, 60D05. **Keywords:** Doeblin graph, bridge graph, bi-recurrent path, unimodular network, eternal family tree, coupling from the past, Markov chain.
Introduction
============
The first incarnation of the Propp and Wilson [@propp1996exact] coupling from the past (CFTP) algorithm was designed to build a perfect sample from the stationary distribution $\pi$ of an irreducible, aperiodic, and positive recurrent Markov chain on a finite state space $S$. It uses a Doeblin-type coupling of a family of copies of the Markov chain started in all possible states at all possible times, whereby when two chains meet, they merge. This coupling is represented with a random directed graph on ${\mathbb{Z}}\times S$ depicting the trajectories of these Markov chains. Below, this random graph will be referred to as the [**Doeblin graph**]{} of the chain.
Prior to this research, the study of this random graph has been mostly a by-product of research on perfect simulation. In 1992–1993, Borovkov and Foss [@borovkov1992stochastically; @borovkov1994two] laid out the framework of stochastically recursive sequences (SRS), of which Markov chains are a special case, and they proved the main results on the existence of a stationary version of an SRS to which non-stationary versions converge in a certain sense. The CFTP algorithm itself was introduced by Propp and Wilson in 1996 in [@propp1996exact] for obtaining samples from the stationary distribution of a Markov chain. The CFTP algorithm can be seen as a specialization of the general ideas of [@borovkov1992stochastically] for SRS to the Markov case aiming at perfect simulation. Foss and Tweedie [@foss1998perfect] then gave a necessary and sufficient condition for the CFTP algorithm to converge a.s. From 1996 to 2000, many papers [@fill1997interruptible; @propp1998coupling; @murdoch1998exact; @propp1998get; @kendall1998perfect; @haggstrom1999exact; @haggstrom1999characterization; @moller1999perfect; @kendall2000perfect; @wilson2000couple; @meng2000towards] investigated how to improve CFTP implementations or how to apply CFTP or a CFTP-inspired algorithm to obtain a perfect sample from a particular Markov chain’s stationary distribution. Of particular importance is Wilson’s read-once CFTP algorithm [@wilson2000couple], which allows CFTP to be done by only simulating forwards in time. A review of perfect simulation in stochastic geometry up to that point is provided in [@moller2001review]. Since then, [@kendall2004geometric; @connor2007perfect] showed that (possibly impractical) generalizations of the CFTP algorithm can be applied under weaker conditions, and [@foss2003extended] gives a CFTP-like algorithm that applies even in the non-Markovian setting.
In this paper, focus is shifted away from finding an individual sample from the stationary distribution of a Markov chain, and instead properties of the Doeblin graph as a whole are studied. The SRS framework will be used, but, because the Markov case is a fundamental special case, most sections will spell out what can be said in the Markov case. The main tool of study is the theory of unimodular random (rooted) networks in the sense of Aldous and Lyons [@aldous2007processes]. Unimodular networks are rooted networks where, heuristically, the root is picked uniformly at random. In order to generalize this concept for infinite networks, instead of picking the root uniformly at random, the network is required to satisfy a mass transport principle. The primary new object of study is the subgraph of bridges between a fixed recurrent state, which is referred to as the [**bridge graph**]{} and is roughly inspired by the population process in [@baccelli2018renewal]. The subgraph is defined by looking at processes started at any time from this fixed state. General setup and definitions of the Doeblin graph and the bridge graph are given in . Random networks and how to view subgraphs like the bridge graph as random networks are handled in . The main theorem is then proved in .
proves the main theorem, identifying the unimodular structure in the bridge graph. studies properties of the bridge graph that are inherited due to its I/F component structure as a unimodular network. Here I/F refers to the class of a component in the sense of the foil classification theorem in unimodular networks in [@baccellieternal], which is reviewed in . The most interesting case is when $S$ is infinite and the Doeblin graph is connected. In this case (see ), although there may be infinitely many bi-infinite paths in the Doeblin graph, there exists a unique [**bi-recurrent**]{} path, a bi-infinite path that visits every state infinitely often in the past, as well as in the future. This unique path also has the property that the states in $S$ that the path traverses form a stationary version of the original Markov chain (or SRS), and hence give samples from its stationary distribution. Indeed, the original CFTP algorithm ultimately computes the time zero point on the bi-recurrent path. By embedding Markov chains inside Doeblin graphs, bi-recurrence is also shown to be a decisive property for Markov chains indexed by ${\mathbb{Z}}$. shows that if a Markov chain ${{({X_t})}}_{t \in {\mathbb{Z}}}$ has an irreducible, aperiodic, and positive recurrent transition matrix, then ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is stationary if and only if it is bi-recurrent for any (and hence every) state. The I/F structure of a component leads to further useful qualitative properties discussed in . In reversed time, the bridge tree can be seen as a multi-type branching-like process where the types are the elements of $S$, and for which there is at most one child of each type per generation. The nodes in this branching process are either mortal (i.e., with finitely many descendants) or immortal (resp. infinitely many). The mortal descendants of the nodes on the bi-infinite path form a stationary sequence of finite trees. Mean values in these trees are linked to coupling times by mass-transport relations. Finally, gives results that are relevant to simulating the bridge graph, such as approximating the bridge graph by finite networks, and viewing the process of vertical slices of the bridge graph as a Markov chain in its own right. The final section gives several bibliographical comments, which make connections of the present research to other works.
The Doeblin Graph {#sec:doeblin-graphs}
=================
Definition {#sec:doeblin-graph-denf}
----------
In this section, the Doeblin graph is constructed. Fix a probability space $(\Omega, {\mathcal{F}}, {\mathbf{P}})$, a countable state space $S$, and a complete separable metric space $\Xi$ for the remainder of the document. The first ingredient needed is a [**pathwise transition generator**]{}, a function $h_{\mathrm{gen}}: S\times \Xi \to S$ that will be used for determining transitions between states of $S$. Such an ${h_{\mathrm{gen}}}$, combined with a [**driving sequence**]{} ${{({\xi_t})}}_{t \in {\mathbb{N}}}$, is used to give a pathwise representation of a stochastic process ${{({X_t})}}_{t \in {\mathbb{N}}}$ satisfying $$\label{eqn:srs}
X_{t+1} := {h_{\mathrm{gen}}}(X_t, \xi_t),\qquad t {\geqslant}0.$$ is the defining property of a stochastically recursive sequence (SRS) in the sense of Borovkov and Foss [@borovkov1992stochastically]. If the driving sequence is taken to be i.i.d. and independent of $X_0$, then ${{({X_t})}}_{t \in {\mathbb{N}}}$ is a (discrete time) Markov chain with transition matrix $P = {{({p_{x,y}})}}_{x,y \in S}$ determined by $p_{x,y} := {\mathbf{P}}({h_{\mathrm{gen}}}(x,\xi_0) = y)$ for each $x,y \in S$. It is a classical result that, when $\Xi := [0,1]$, all possible transition matrices $P$ can be achieved by choosing ${h_{\mathrm{gen}}}$ and the distribution of $\xi_0$ accordingly (c.f. Chapter 17 in [@borovkov2013stochastic]). Many processes in this paper will be indexed by ${\mathbb{Z}}$ or an interval of ${\mathbb{Z}}$ instead of just ${\mathbb{N}}$. The pathwise transition generator ${h_{\mathrm{gen}}}$ and a stationary and ergodic *bi-infinite* driving sequence $\xi:= {{({\xi_{t}})}}_{t\in {\mathbb{Z}}}$, are fixed for the remainder of the document. The notation for the transition matrix $P = {{({p_{x,y}})}}_{x,y \in S}$ is also fixed for the remainder of the document, even when $\xi$ is not assumed to be i.i.d.
The space ${\mathbb{Z}}\times S$ should be thought of as time and space coordinates, with $(t,x) \in {\mathbb{Z}}\times S$ being in state $x$ at time $t$. The vertices and edges of a graph ${\Gamma}$ will be written $V({\Gamma})$ and $E({\Gamma})$, and if $V({\Gamma}) \subseteq {\mathbb{Z}}\times S$, the vertices of ${\Gamma}$ sitting at a particular time $t$ or in a particular state $x$ will be denoted, respectively, as $$\begin{aligned}
V_t({\Gamma}) := {\left\{{(s,y) \in V({\Gamma}) : s=t}\right\}},\quad
V^x({\Gamma}) := {\left\{{(s,y) \in V({\Gamma}) : y=x}\right\}}.\end{aligned}$$ Note that $V_t({\Gamma})$ and $V^x({\Gamma})$ are subsets ${\mathbb{Z}}\times S$, i.e. their elements have both a time component and a space component. If instead just states (elements of $S$) or just times (elements of ${\mathbb{Z}}$) are desired, then the following are used instead $$\begin{aligned}
\label{defn:subscriptt}
{\Gamma}_t := {\left\{{x \in S: (t,x) \in V_t({\Gamma})}\right\}},\quad
{\Gamma}^x := {\left\{{t \in {\mathbb{Z}}: (t,x) \in V^x({\Gamma})}\right\}}.\end{aligned}$$
Then the [**Doeblin graph**]{} ${\mathbf{G}}= {\mathbf{G}}({h_{\mathrm{gen}}}, \xi)$ is constructed as follows. It has vertices $V({\mathbf{G}}) := {\mathbb{Z}}\times S$. The edges of ${\mathbf{G}}$ are determined by the [**follow map**]{} ${f_+}: V({\mathbf{G}}) \to V({\mathbf{G}})$, which is a random map giving directions of where each vertex should move to in the next time step. It is defined by $${f_+}(t,x) := (t+1,{h_{\mathrm{gen}}}(x,\xi_{t})),\qquad (t,x) \in {\mathbb{Z}}\times S.$$ That is, let the edges of ${\mathbf{G}}$ be drawn from each $(t,x) \in {\mathbb{Z}}\times S$ to ${f_+}(t,x)$. By saying a function $f:A \to B$ is a random map, it is meant that $f:A\times \Omega \to B$ is measurable and the second argument will be omitted. Iterates of ${f_+}$ are denoted by ${f_+}^n$ for $n {\geqslant}0$. Thinking of each vertex in ${\mathbf{G}}$ as an individual, one may also interpret the follow map as mapping each vertex to its parent vertex.
Modeling {#sec:doeblin-graph-modeling}
--------
When dealing with paths in ${\mathbf{G}}$, it will often be convenient to ignore the time coordinate and focus only on the space coordinate. If ${{({X_t})}}_{t \in I}$ is a stochastic process defined on $\Omega$ which takes values in $S$ and is such that ${{({t,X_t})}}_{t \in I}$ is a.s. a path in ${\mathbf{G}}$ over some fixed time interval $I\subseteq {\mathbb{Z}}$, then ${{({X_t})}}_{t \in I}$ is called the [**state path (in ${\mathbf{G}}$)**]{} corresponding to the path ${{({t,X_t})}}_{t \in I}$. That is, there are two ways of looking at every route through ${\mathbf{G}}$: as a path ${{({t,X_t})}}_{t \in I} \subseteq V({\mathbf{G}})$, or as a state path ${{({X_t})}}_{t \in I} \subseteq S$.
\[lem:embed-srs-into-doeblin-graph\] Let $I\neq\emptyset$ be an interval in ${\mathbb{Z}}$. Suppose that ${{({X_t})}}_{t \in I}$ is a stochastic process taking values in $S$ a.s. satisfying the recurrence relation $X_{t+1} = {h_{\mathrm{gen}}}(X_t, \xi_t)$ for each $\inf I {\leqslant}t < \sup I$, where ${h_{\mathrm{gen}}}$ and ${{({\xi_t})}}_{t \in I}$ are the same as are used to define ${\mathbf{G}}$. Then ${{({X_t})}}_{t \in I}$ is a state path in ${\mathbf{G}}$.
One must check that ${(t, X_t)}_{t \in I}$ is a.s. a path in ${\mathbf{G}}$. Fix $t \in I$. Since $V({\mathbf{G}}) = {\mathbb{Z}}\times S$, $(t, X_t)$ is certainly a vertex of ${\mathbf{G}}$. If $t+1 \in I$ as well, one must check the edge $e$ from $(t, X_t)$ to $(t+1, X_{t+1})$ is a.s. an edge in ${\mathbf{G}}$. The edges of ${\mathbf{G}}$ are defined to be from each $(t,x) \in {\mathbb{Z}}\times S$ to $(t+1, {h_{\mathrm{gen}}}(x,\xi_t))$, so the relation $X_{t+1} = {h_{\mathrm{gen}}}(X_t, \xi_t)$ holding a.s. implies the edge $e$ is a.s. an edge of ${\mathbf{G}}$.
In particular, says that any SRS whose driving sequence is defined for all times in ${\mathbb{Z}}$ can be seen as living inside a Doeblin graph, namely the one generated by its driving sequence and choosing ${h_{\mathrm{gen}}}$ to be the same as in the definition of the SRS.
State paths started at a deterministic vertex will also be used heavily. For the remainder of the document, let $F^{(t,x)}:= {{({F^{(t,x)}_s})}}_{s {\geqslant}t}$ be the [**state path in ${\mathbf{G}}$ started at time $t$ in state $x$**]{}, i.e., $F^{(t,x)}$ is a re-indexing of the states traversed by $f_+$ defined by $$\begin{aligned}
(s, F^{(t,x)}_s) = {f_+}^{s-t}(t,x),\qquad (t,x) \in {\mathbb{Z}}\times S,\quad s {\geqslant}t.\end{aligned}$$ One has that $F^{(t,x)}$ is a version of the SRS or Markov chain started in state $x$ with initial condition given at time $t$. Generally speaking, throughout the paper, a parenthesized superscript, as in $F^{(t,x)}$, refers to a starting location. For every $x \in S$, the distribution of ${{({F^{(t,x)}_{s+t}})}}_{s {\geqslant}0}$ does not depend on $t$ because $\xi$ is stationary. An example of a Doeblin graph and path of $F^{(t,x)}$ are drawn in .
![An example of a Doeblin graph with the path corresponding to the state path $F^{(t,x)}$ distinguished. All edges are directed from left to right.[]{data-label="fig:meft-example"}](meft-fig.pdf){width="\linewidth"}
It has already been noted (see [@borovkov2013stochastic]) that a Markov chain ${{({X_t})}}_{t \in {\mathbb{N}}}$ with any given desired transition matrix can be constructed as an SRS with i.i.d. driving sequence. The following is an analogous result saying that any Markov chain ${{({X_t})}}_{t \in {\mathbb{Z}}}$ may be realized as a state path in a Doeblin graph with i.i.d. driving sequence. Note here that the time index set is all of ${\mathbb{Z}}$, not just ${\mathbb{N}}$.
\[thm:embed-into-general-meff\] Suppose that ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a Markov chain with transition matrix $P$ on some probability space, where $P$ is the same as was defined for the Doeblin graph ${\mathbf{G}}$. Also suppose the driving sequence $\xi$ is i.i.d. Then there is a probability space $(\Omega',{\mathcal{F}}',{\mathbf{P}}')$ and ${{({X_t'})}}_{t \in {\mathbb{Z}}}\sim {{({X_t})}}_{t \in {\mathbb{Z}}}$ on $\Omega'$ such that ${{({X_t'})}}_{t \in {\mathbb{Z}}}$ is state path in ${\mathbf{G}}'$, where ${\mathbf{G}}'$ is the Doeblin graph generated by some i.i.d. driving sequence $\xi'={{({\xi_{t}'})}}_{t\in {\mathbb{Z}}} \sim \xi$ in $\Omega'$ with pathwise transition generator ${h_{\mathrm{gen}}}$. Moreover, for each $t \in {\mathbb{Z}}$, $X'_t$ is independent of ${{({\xi_s'})}}_{s {\geqslant}t}$.
Consider a probability space housing independent copies of ${{({X_t})}}_{t \in {\mathbb{Z}}}$ and ${\mathbf{G}}$. Then consider for each $t \in {\mathbb{Z}}$ the state path in ${\mathbf{G}}$ started at $X_t$. The distributions of these state paths determine a consistent set of finite dimensional distributions for the desired pair of processes $({{({X'_t})}}_{t \in {\mathbb{Z}}}, {{({\xi'_t})}}_{t \in {\mathbb{Z}}})$. By the Kolmogorov extension theorem, the result follows. The full proof of is given in the appendix.
Basic Properties
----------------
Plainly, ${\mathbf{G}}$ is acyclic as an undirected graph because all outgoing edges point forward one unit in time and each vertex has only one outgoing edge. When ${\mathbf{G}}$ is a.s. connected, it is called a [**Doeblin Eternal Family Tree**]{} or a [**[Doeblin [[EFT]{}]{}]{}**]{} for short. More generally, ${\mathbf{G}}$ may have up to countably many components and is referred to as a [**Doeblin Eternal Family Forest**]{} or [**[Doeblin [[EFF]{}]{}]{}**]{}. The [[EFT]{}]{} and [[EFF]{}]{} terminology is inspired by [@baccellieternal] and the word eternal refers to the fact that every vertex of ${\mathbf{G}}$ has a unique outgoing edge. That is, there is no individual that is an ancestor of all other individuals. An [[EFF]{}]{} is a more general object than an [[EFT]{}]{}, i.e. an [[EFF]{}]{} may also be an [[EFT]{}]{}.
If the driving sequence $\xi$ is i.i.d., so that the state paths $F^{(t,x)}$ for each $(t,x) \in {\mathbb{Z}}\times S$ are Markov chains, then say that ${\mathbf{G}}$ is [**Markovian**]{}. If $\xi$ is such that for each $t \in {\mathbb{Z}}$, ${{({{f_+}(t,x)})}}_{x \in S}$ is an independent family, then ${\mathbf{G}}$ is said to have [**vertical independence**]{}. If ${\mathbf{G}}$ is Markovian and has vertical independence, then say that ${\mathbf{G}}$ has [**fully independent transitions**]{}.
Some later results are only valid for [[EFTs]{}]{}, so the following result gives an easy case when ${\mathbf{G}}$ can be shown to be connected.
\[prop:teaser-indep-transitions-components\] Suppose ${\mathbf{G}}$ has fully independent transitions, and $P$ is irreducible and positive recurrent with period $d$. Then a.s. ${\mathbf{G}}$ has $d$ components. In particular, if $P$ is irreducible, aperiodic, and positive recurrent, then ${\mathbf{G}}$ is an [[EFT]{}]{}.
The case of a general $d$ is reduced to $d=1$ by viewing the chain only every $d$ steps and with state space restricted to one of the $d$ classes appearing in a cyclic decomposition of the state space. Consider the state paths in ${\mathbf{G}}$ started at $(0,x)$ and $(0,y)$ for any two $x,y$. Strictly before hitting the diagonal, the pair of state paths has the same distribution as a product chain, i.e. two independent copies of the chain with one started at $x$ and the other at $y$. The product chain is irreducible, aperiodic, and positive recurrent, and therefore a.s. hits the diagonal, showing the state paths started at $(0,x)$ and $(0,y)$ eventually merge. The full proof of is given in the appendix.
A $\xi$-measurable subgraph ${\Gamma}= {\Gamma}({{({\xi_{t}})}}_{t \in {\mathbb{Z}}})$ of ${\mathbf{G}}$ is called [**shift-covariant**]{} if, for all $s \in {\mathbb{Z}}$, ${\Gamma}({{({\xi_{t+s}})}}_{t\in {\mathbb{Z}}})$ is a.s. the time-translation of ${\Gamma}$ by $-s$. Say a state path ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is [**shift-covariant**]{} if the corresponding path in ${\mathbf{G}}$ is shift-covariant. In other words, if the driving sequence $\xi$ is translated by some amount $s$ in time, then shift-covariant objects are also translated in time by the same amount. Let $E \in {\mathcal{F}}$ be $\xi$-measurable, say $1_E = g({{({\xi_{t}})}}_{t\in {\mathbb{Z}}})$. Say that $E$ is [**shift-invariant**]{} if $g({{({\xi_{t}})}}_{t\in {\mathbb{Z}}})= g({{({\xi_{t+1}})}}_{t\in {\mathbb{Z}}})$ a.s. That is, shift-invariant events are those events whose occurence is unaffected by time translations of the driving sequence $\xi$. One has that ${\mathbf{P}}(E) \in {\left\{{0,1}\right\}}$ for all shift-invariant events $E$ due to the ergodicity of $\xi$. All of the following are shift-invariant and hence happen with probability zero or one: ${\mathbf{G}}$ is locally finite, ${\mathbf{G}}$ contains no cycles, ${\mathbf{G}}$ is connected, ${\mathbf{G}}$ has exactly $n \in {\mathbb{N}}\cup{\left\{{\infty}\right\}}$ components, ${\mathbf{G}}$ contains exactly $n \in {\mathbb{N}}\cup{\left\{{\infty}\right\}}$ bi-infinite paths. Generally it will be obvious whether an event is shift-invariant.
When ${\mathbf{G}}$ is a Markovian, one needs to be cautious that not all state paths in ${\mathbf{G}}$ are Markov chains with transition matrix $P$.
\[ex:nonproperstatepath\] Let $S:={\mathbb{Z}}$ and suppose ${\mathbf{G}}$ has fully independent transitions with $p_{x,x-1} = p_{x,x} = p_{x,x+1} = \frac{1}{3}$ for all $x \in S$. Choose $X_0$ to be the smallest element of ${\mathbb{Z}}$ (in some well-ordering of ${\mathbb{Z}}$) such that $F^{(0,X_0)}_1 = F^{(0,X_0)}_2$. In this case, a.s. $X_1 = X_2$, so ${{({X_t})}}_{t \in {\mathbb{N}}}$ is not even Markovian.
The problem with the path in the previous example is that it looks into the future. Namely, the value of $X_0$ depends on information at time $1$ and time $2$. To exclude state paths like those in , the notion of properness is introduced. For a nonempty interval $I$ of ${\mathbb{Z}}$, if for each $t \in I$, $X_t$ is independent of ${{({\xi_{s}})}}_{s {\geqslant}t}$, then ${{({X_t})}}_{t \in I}$ is called a [**proper**]{} state path. In the Markovian case, if $I$ has a minimum element $t_0$, then to show that a state path ${{({X_t})}}_{t \in I}$ is proper it is sufficient that $X_{t_0}$ is independent of ${{({\xi_{s}})}}_{s {\geqslant}t_0}$ because for any $s \in {\mathbb{N}}$, $X_{t_0+s}$ is measurable with respect to the $\sigma$-algebra generated by $X_{t_0}$ and $\xi_{t_0},\ldots,\xi_{t_0 + s-1}$. Unlike general state paths in ${\mathbf{G}}$, proper state paths inherit a Markov transition structure.
\[lem:proper-state-path-is-markov-chain\] Suppose ${\mathbf{G}}$ is Markovian. If ${{({X_t})}}_{t \in I}$ is a proper state path in ${\mathbf{G}}$ over a nonempty interval $I\subseteq {\mathbb{Z}}$, then ${{({X_t})}}_{t \in I}$ is a Markov chain with transition matrix $P$.
Fix $t < \sup I$. Let $E:={\left\{{X_{t}=x_t,\ldots, X_{t-k}=x_{t-k}}\right\}}$ be given with $k \in {\mathbb{N}}$ such that $t-k {\geqslant}\inf I$, and $x_t,\ldots,x_{t-k} \in S$. Note that whether $E$ occurs is a function of $X_{t-k}$ and $\xi_{t-k},\ldots, \xi_{t-1}$, so the fact that $X_{t-k}$ is independent of ${{({\xi_s})}}_{s {\geqslant}t-k}$ and the fact that $\xi$ is i.i.d. imply that $E$ is independent of ${{({\xi_{s}})}}_{s {\geqslant}t}$. Then for any $x \in S$, $$\begin{aligned}
{\mathbf{E}}[1_{{\left\{{X_{t+1}=x}\right\}}} 1_{E}]
&={\mathbf{E}}[1_{{\left\{{h(x_t, \xi_{t})=x}\right\}}} 1_{E}]\\
&={\mathbf{P}}(h(x_t, \xi_{t})=x) {\mathbf{P}}(E)\\
&= p_{x_t,x} {\mathbf{P}}(E)\\
&= {\mathbf{E}}[p_{X_t,x} 1_E].
\end{aligned}$$ Since $E$ was an arbitrary cylinder set, it follows that for all $x\in S$, $${\mathbf{P}}(X_{t+1} = x \mid {{({X_s})}}_{s \in I, s{\leqslant}t}) = p_{X_{t},x}.$$ Thus ${{({X_s})}}_{s \in I}$ is a Markov chain with transition matrix $P$.
Connections with CFTP
---------------------
Consider the following structural result that will be expanded upon in . It is a special case of and , which will be proved later.
\[prop:mainthm-teaser\] Suppose ${\mathbf{G}}$ is Markovian, and that $P$ is irreducible, aperiodic, and positive recurrent. Then a.s. in every component of ${\mathbf{G}}$ there exists a unique bi-infinite path that visits every state in $S$ infinitely often in the past. All other bi-infinite paths in ${\mathbf{G}}$ do not visit any state infinitely often in the past. If ${\mathbf{G}}$ is an [[EFT]{}]{}, then with $\beta_t$ denoting the state at time $t$ of the unique bi-infinite path visiting every state infinitely often in the past, one has that ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ is a stationary Markov chain with transition matrix $P$, so that $\beta_t \sim \pi$ for all $t \in {\mathbb{Z}}$, where $\pi$ is the invariant distribution for $P$.
The main result of the original Propp and Wilson paper can be translated into the language of [Doeblin [[EFFs]{}]{}]{} and summarized as follows. The reader is encouraged to ponder what it says about the structure of ${\mathbf{G}}$, and in doing so one sees that is has much the same spirit as .
\[prop:PW-CFTP\] If $S$ is finite and ${\mathbf{G}}$ is Markovian and an [[EFT]{}]{} (which, since $S$ is finite, necessitates that $P$ is irreducible and aperiodic), then there is an a.s. finite time $\tau$ such that all paths in ${\mathbf{G}}$ started at any time $t {\leqslant}-\tau$ have merged by time $0$, all reaching a common vertex $(0, \beta_0)$. Moreover, $\beta_0 \sim \pi$, where $\pi$ is the stationary distribution of $P$, and there is an algorithm $A$ that a.s. terminates in finite time returning $\beta_0$.
In fact, the $\beta_0$ appearing in and the $\beta_0$ appearing in are the same. That is, the perfect sampling algorithm $A$ is ultimately computing the point in ${\mathbf{G}}$ on the unique bi-infinite path and returning its state. This can be seen by the fact that, since all paths started at time $-\tau$ reach the common vertex $(0,\beta_0)$, any bi-infinite path in ${\mathbf{G}}$ must also pass through $(0,\beta_0)$. However, what is notably absent in is any mention of an algorithm to compute $\beta_0$. Whether such an algorithm exists in general is not studied in the present research.
Bridge Graphs {#sec:MBEFFs}
-------------
The primary tool used in this document will be the theory of unimodular networks in the sense of [@aldous2007processes]. Local finiteness is essential in the theory of unimodular networks, but the Doeblin graph ${\mathbf{G}}$ may not be locally finite, as the following result shows.
If $\sum_{x\in S} p_{x,y} < \infty$ for all $y \in S$, then ${\mathbf{G}}$ is a.s. locally finite. If ${\mathbf{G}}$ has fully independent transitions and for some $y\in S$, $\sum_{x \in S} p_{x,y} = \infty$, then ${\mathbf{G}}$ is a.s. not locally finite.
Both statements follow from the Borel-Cantelli lemmas. That is, for any fixed $(t,y) \in {\mathbb{Z}}\times S$, if $\sum_{x \in S}p_{x,y} < \infty$, then a.s. one has that only finitely many of the events ${\left\{{{f_+}(t-1,x) = (t,y)}\right\}}_{x \in S}$ occur, showing $(t,y)$ has finite in-degree, and hence finite degree, in ${\mathbf{G}}$. On the other hand, if ${\mathbf{G}}$ has fully independent transitions and for some fixed $(t,y) \in {\mathbb{Z}}\times S$ one has $\sum_{x \in S}p_{x,y} = \infty$, then a.s. infinitely many of the events ${\left\{{{f_+}(t-1,x) = (t,y)}\right\}}_{x \in S}$ occur, so that $(t,y)$ has infinite degree.
The remedy taken here is to instead concentrate on particular subgraphs of ${\mathbf{G}}$. In this section, subgraphs are introduced that are locally finite under a positive recurrence assumption and turn out to have nice properties when considered as random networks.
For each $(t,x) \in {\mathbb{Z}}\times S$, and each $y \in S$, let $$\begin{aligned}
\tau^{(t,x)}(y) :=\inf{\left\{{s > t: F^{(t,x)}_s = y}\right\}}, \qquad \sigma^{(t,x)}(y) := \tau^{(t,x)}(y) - t\end{aligned}$$ be, respectively, the [**return time**]{} and [**time until return**]{} of $F^{(t,x)}$ to $y$. The word return is used even when $y \neq x$, in which case it may be that $F^{(t,x)}$ is not part of a state path that has visited $y$ before time $t$. Note that the distribution of $\sigma^{(t,x)}(y)$ does not depend on $t$ because $\xi$ is stationary. Call a state $x \in S$ [**positive recurrent**]{} if ${\mathbf{E}}[\sigma^{(0,x)}(x)] < \infty$ or [**recurrent**]{} if $\sigma^{(0,x)}(x) < \infty$ a.s. In the Markovian case these are the usual definitions. If a state $x \in S$ is recurrent, then indeed for every $t \in {\mathbb{Z}}$, $F^{(t,x)}$ visits $x$ infinitely often.
For each fixed $x \in S$, consider the subgraph ${\mathbf{B}}(x)$ of ${\mathbf{G}}$ of all paths starting from state $x$ at any time. That is, ${\mathbf{B}}(x)$ is the subgraph of ${\mathbf{G}}$ with $$\begin{aligned}
\label{eq:Bxvertices}
V({\mathbf{B}}(x))
:= \bigcup_{t \in {\mathbb{Z}}}{\left\{{(s,F^{(t,x)}_s) : s {\geqslant}t}\right\}}
= \bigcup_{t \in {\mathbb{Z}}}{\left\{{{f_+}^n(t,x) : n {\geqslant}0}\right\}}.\end{aligned}$$ Call ${\mathbf{B}}(x)$ the [**bridge graph for state $x$**]{} and refer to it as either a [**[bridge [[EFF]{}]{}]{}**]{} or [**[bridge [[EFT]{}]{}]{}**]{} depending on whether it is a forest or a tree. Note that one of these possibilities happens with probability $1$ because the number of components in ${\mathbf{B}}(x)$ is shift-invariant.
For the remainder of the document, assume there exists a positive recurrent state $x^* \in S$, which is fixed, and the notation ${\mathbf{B}}:= {\mathbf{B}}(x^*)$ refers to the bridge graph for state $x^*$.
An example bridge graph appears in .
![An example bridge graph, in this case for state $x^* = 0$, sitting inside the Doeblin graph.[]{data-label="fig:meft-and-bridge-example"}](meft-and-bridge-fig.pdf){width="\linewidth"}
Equivalently, ${\mathbf{B}}$ can be described in terms of descendants of vertices, viewing directed edges in ${\mathbf{G}}$ as pointing from a vertex to its parent. For each $(t,y) \in {\mathbb{Z}}\times S$, define the [**descendants**]{} of $(t,y)$ in ${\mathbf{G}}$ to be $$\begin{aligned}
D^{(t,y)} &:= {\left\{{(s,x) \in {\mathbb{Z}}\times S : F^{(s,x)}_t = y}\right\}}.\end{aligned}$$ Then ${\mathbf{B}}$ is also the subgraph of ${\mathbf{G}}$ with $$\begin{aligned}
V({\mathbf{B}}) = {\left\{{(t,y) \in {\mathbb{Z}}\times S : \exists s,\, (s,x^*) \in D^{(t,y)}}\right\}}.\end{aligned}$$ That is, ${\mathbf{B}}$ is the subgraph of ${\mathbf{G}}$ generated by vertices that have some descendant in state $x^*$. In particular, recalling ,
$$\begin{aligned}
y \in {\mathbf{B}}_t \iff \exists s,\, x^* \in D^{(t,y)}_s,\qquad (t,y) \in {\mathbb{Z}}\times S.\end{aligned}$$
shows that if ${\mathbf{G}}$ is a.s. connected, then ${\mathbf{B}}$ is too.
\[lem:G-eft-implies-B-eft\] If $u,v \in V({\mathbf{B}})$ are in the same component of ${\mathbf{G}}$, then they are in the same component of ${\mathbf{B}}$. In particular, if ${\mathbf{G}}$ is an [[EFT]{}]{}, then ${\mathbf{B}}$ is an [[EFT]{}]{}.
Consider times $s,t \in {\mathbb{Z}}$. Suppose $(s,x^*)$ and $(t,x^*)$ are in the same component of ${\mathbf{G}}$. Then $F^{(s,x^*)}$ and $F^{(t,x^*)}$ meet at some point. But, by definition, the paths of $F^{(s,x^*)}$ and $F^{(t,x^*)}$ are included in ${\mathbf{B}}$. Hence $(s,x^*)$ and $(t,x^*)$ are in the same component of ${\mathbf{B}}$. Now if $u,v \in V({\mathbf{B}})$ are in the same component of ${\mathbf{G}}$, $u$ is in the same component in ${\mathbf{G}}$ as some $(s,x^*)$ and $v$ is in the same component of ${\mathbf{G}}$ as some $(t, x^*)$, and $(s,x^*)$ and $(t, x^*)$ are in the same component of ${\mathbf{B}}$ by the previous part. Hence $u,v$ are in the same component of ${\mathbf{B}}$.
The condition that ${\mathbf{B}}$ is an [[EFT]{}]{} is equivalent to strong coupling convergence (defined and studied in [@borovkov1992stochastically; @borovkov1994two; @foss2003extended]) of $F^{(0,x^*)}$ to a stationary version of the SRS. However, simple conditions for ${\mathbf{B}}$ to be an [[EFT]{}]{} are not known outside of the Markovian case, where showed that if $P$ is irreducible, aperiodic, and positive recurrent, then ${\mathbf{G}}$ is an [[EFT]{}]{}. Another (not neccessarily easy to check) condition for ${\mathbf{B}}$ to be an [[EFT]{}]{} will be given in .
The main tool used in this paper is unimodularity of random networks. The first form of unimodularity used is stationarity, i.e., the unimodularity of the deterministic network ${\mathbb{Z}}$ rooted at $0$ and with neighboring integers connected. Unimodularity of ${\mathbb{Z}}$ gives a helpful way to reorganize proofs based on stationarity in terms of transporting mass between different times. Recall that a (measurable) group action $\theta:{\mathbb{Z}}\times \Omega \to \Omega$ of ${\mathbb{Z}}$ on $\Omega$ is called ${\mathbf{P}}$-invariant if ${\mathbf{P}}(\theta_t \in \cdot) = {\mathbf{P}}$ for all $t \in {\mathbb{Z}}$. The shift operator on $\Xi ^{\mathbb{Z}}$ is an example of such an action.
\[lem:mtpforZ\] Suppose $w:{\mathbb{Z}}\times {\mathbb{Z}}\to {\mathbb{R}}_{{\geqslant}0}$ is a random map. Also suppose $\theta:{\mathbb{Z}}\times \Omega \to \Omega$ is a ${\mathbf{P}}$-invariant ${\mathbb{Z}}$-action on $\Omega$, and that the two are compatible in the sense that $w(s,t)\circ \theta_r = w(s+r,t+r)$ almost surely for each $s,t,r \in {\mathbb{Z}}$. Then with $w^+:= \sum_{t \in {\mathbb{Z}}} w(0,t)$ and $w^- := \sum_{s \in {\mathbb{Z}}} w(s,0)$, one has $$\begin{aligned}
{\mathbf{E}}[w^+] = {\mathbf{E}}[w^-].
\end{aligned}$$
One calculates $${\mathbf{E}}[w^+] = \sum_{t \in {\mathbb{Z}}} {\mathbf{E}}[w(0,t)]
= \sum_{t \in {\mathbb{Z}}} {\mathbf{E}}[w(0,t) \circ \theta_{-t}]
= \sum_{t \in {\mathbb{Z}}} {\mathbf{E}}[w(-t,0)]
= {\mathbf{E}}[w^-]$$ as desired.
The mass transport principle for ${\mathbb{Z}}$ immediately gives the following.
\[prop:mbeff-is-locally-finite\] For all $t \in {\mathbb{Z}}$, ${\mathbf{E}}[\#{\mathbf{B}}_t] {\leqslant}{\mathbf{E}}[\sigma^{(0,x^*)}(x^*)]$. In particular, ${\mathbf{B}}$ is a.s. locally finite, even if ${\mathbf{G}}$ itself is not.
Without loss of generality, $\Omega$ is the canonical space $\Xi^{{\mathbb{Z}}}$, with the driving sequence ${{({\xi_{t}})}}_{t \in {\mathbb{Z}}}$ being coordinate maps. Then $\theta:{\mathbb{Z}}\times \Omega \to \Omega$ defined by $\theta_s({{({\xi_{t}})}}_{t\in {\mathbb{Z}}}) := {{({\xi_{s+t}})}}_{t\in {\mathbb{Z}}}$ is a ${\mathbf{P}}$-invariant measurable ${\mathbb{Z}}$-action on $\Omega$. Choose the mass transport $w(s,t) := 1_{{\left\{{\sigma^{(s,x^*)}(x^*) > t-s > 0}\right\}}}$. The fact that one has $\sigma^{(s,x^*)}(x^*) \circ \theta_r = \sigma^{(s+r,x^*)}(x^*)$ for all $s,t,r \in {\mathbb{Z}}$ implies $w$ is compatible with $\theta$. Then $w^+ = \sigma^{(0,x^*)}(x^*) -1$, and $w^- = \#{\left\{{s<0: \sigma^{(s,x^*)}(x^*) > |s|}\right\}} {\geqslant}\#{\mathbf{B}}_0-1$, where this inequality follows from the fact that for every $y \in {\mathbf{B}}_0 \setminus {\left\{{x^*}\right\}}$, there is $s<0$ such that $\sigma^{(s,x^*)}(x^*) > |s|$ and $F^{(s,x^*)}_0 = y$. Thus the mass transport principle for ${\mathbb{Z}}$ gives ${\mathbf{E}}[\sigma^{(0,x^*)}(x^*)-1] {\geqslant}{\mathbf{E}}[\#{\mathbf{B}}_0-1]$, from which the result follows.
The proof style of may be repeated in many different ways and the boilerplate setup of the proof can be mostly omitted once one understands the flow of the proof. The shortened version of the proof of is given to exemplify how much can be omitted without losing the main idea.
Let the mass transport $w(s,t)$ send mass $1$ from $s$ to all times $t$ strictly after $s$ and strictly before $F^{(s,x^*)}$ returns to $x^*$. Then $w^+ = \sigma^{(0,x^*)}(x^*)-1$ and $w^- = \#{\left\{{s < 0: \sigma^{(s,x^*)}(x^*) > |s|}\right\}} {\geqslant}\#{\mathbf{B}}_0 - 1$, where this inequality follows from the fact that for every $y \in {\mathbf{B}}_0 \setminus {\left\{{x^*}\right\}}$, there is $s<0$ such that $\sigma^{(s,x^*)}(x^*) > |s|$ and $F^{(s,x^*)}_0 = y$. The mass transport principle finishes the claim.
One now sees the versatility of using even the simplest form of unimodularity. A list of mass transports and the results they give, all by following the same proof style, appears in in the appendix. Some of the mass transports give new results, and others recover well-known results, such as fact that $\pi(y)/\pi(x^*)$ is the expected number of visits of a Markov chain started at $x^*$ to $y$ before returning to $x^*$, and $1/\pi(x^*)$ is the expected return time of a Markov chain started at $x^*$ to return to $x^*$, where $\pi$ is the invariant distribution for the Markov chain. The next section reviews the more general theory of random networks and unimodularity, then shows how to embed subgraphs of ${\mathbf{G}}$ as random networks, so that eventually one may find a unimodular structure inside ${\mathbf{G}}$.
Random Networks {#sec:randomnetworks}
===============
Definition and Basic Properties
-------------------------------
See [@aldous2007processes; @khezeli2017shift] for a more thorough review of random networks than what is provided here. A [**network**]{} is a graph ${\Gamma}=(V({\Gamma}),E({\Gamma}))$ equipped with a complete separable metric space $(\Xi_{\Gamma}, d_{\Xi_{\Gamma}})$ called the [**mark space**]{} and two maps from $V({\Gamma})$ and ${\left\{{(v,e): v \in V({\Gamma}), e \in E({\Gamma}), v \sim e}\right\}}$ to $\Xi_{\Gamma}$, where $\sim$ is used for adjacency of vertices or edges. The image of $v$ (resp. $(v,e)$) in $\Xi_{\Gamma}$ is called its [**mark**]{}, which is extra information associated to the vertex (resp. edge). The mark of $(v,e)$ may also be thought of as the mark of $e$ considering it to be a directed edge with initial vertex $v$. The graph distance between $v$ and $w$ is denoted $d_{\Gamma}(v,w)$. Unless explicitly mentioned otherwise, networks are assumed to be nonempty, locally finite, and connected.
An [**isomorphism**]{} between two networks with the same mark space is a graph isomorphism that also preserves the marks. A [**rooted network**]{} is a pair $({\Gamma},o)$ in which ${\Gamma}$ is a network and $o$ is a distinguished vertex of ${\Gamma}$ called the [**root**]{}. An [**isomorphism**]{} of rooted networks is a network isomorphism that takes the root of one network to the root of the other. Similar definitions apply to doubly rooted networks $({\Gamma},o,v)$. For convenience, from now on consider only networks with mark space $({\Xi_{\mathrm{univ}}}, d_{{\Xi_{\mathrm{univ}}}})$, where ${\Xi_{\mathrm{univ}}}$ is some fixed uncountable complete separable metric space, such as ${\mathbb{N}}^{\mathbb{N}}$ or the Hilbert cube, since all possible mark spaces are homeomorphic to a subset of such a ${\Xi_{\mathrm{univ}}}$. Let $\mathcal{G}$ denote the set of isomorphism classes of nonempty, locally finite, connected networks, and let $\mathcal{G}_{*}$ (resp. $\mathcal{G}_{**}$) be the set of isomorphism classes of singly (resp. doubly) rooted networks of the same kind. The isomorphism class of a network ${\Gamma}$ (resp. $({\Gamma},o)$, or $({\Gamma},o,v)$) is denoted by $[{\Gamma}]$ (resp. $[{\Gamma},o]$ or $[{\Gamma},o,v]$).
The sets $\mathcal{G}_{*}$ and $\mathcal{G}_{**}$ are equipped with natural metrics making them complete separable metric spaces (cf. [@aldous2007processes]). The distance $d_{\mathcal{G}_*}([{\Gamma}_1, o_1], [{\Gamma}_2, o_2])$ between the isomorphism classes of $({\Gamma}_1,o_1)$ and $({\Gamma}_2,o_2)$ is $1/(1+\alpha)$, where $\alpha$ is the supremum of those $r>0$ such that there is a rooted isomorphism of the balls of graph-distance $\lfloor r \rfloor$ around the roots of ${\Gamma}_1, {\Gamma}_2$ such that each pair of corresponding marks has distance less than $1/r$. The distance on $\mathcal{G}_{**}$ is defined similarly and the projections $[{\Gamma},o,v] \mapsto [{\Gamma},o]$ and $[{\Gamma},o,v] \mapsto [{\Gamma},v]$ are continuous.
A [**random (rooted) network**]{} is a random element in $\mathcal{G}_{*}$ equipped with its Borel $\sigma$-algebra $\mathcal{B}(\mathcal{G}_{*})$. A random network $[{\boldsymbol{{\Gamma}}},{\boldsymbol{o}}]$ is called [**unimodular**]{} if for all measurable $g:\mathcal{G}_{**} \to {\mathbb{R}}_{{\geqslant}0}$, the following [**mass transport principle**]{} is satisfied: $$\begin{aligned}
\label{eq:mtp}
{\mathbf{E}}\sum_{v \in V({\boldsymbol{{\Gamma}}})} g[{\boldsymbol{{\Gamma}}},{\boldsymbol{o}},v] = {\mathbf{E}}\sum_{v \in V({\boldsymbol{{\Gamma}}})} g[{\boldsymbol{{\Gamma}}},v,{\boldsymbol{o}}].\end{aligned}$$ Heuristically, the root of a unimodular network is picked uniformly at random from its vertices. However, since there is no uniform distribution on an infinite set of vertices, the mass transport principle is used in lieu of requiring the root to be picked uniformly at random. One should take care to note that the sums in the previous equation depend only on the isomorphism class $[{\boldsymbol{{\Gamma}}},{\boldsymbol{o}}]$ and not which representative is used.
Next, the notions of covariant vertex-shifts, foils, connected components, and the cardinality classification of components of a unimodular network are reviewed. See [@baccellieternal] for a reference on these concepts. A [**(covariant) vertex-shift**]{} is a map $\Phi$ which associates to each network ${\Gamma}$ a function $\Phi_{\Gamma}: V({\Gamma}) \to V({\Gamma})$ such that $\Phi$ commutes with network isomorphisms and the function $[{\Gamma},o,v] \to 1_{{\left\{{\Phi_{\Gamma}(o)=v}\right\}}}$ is measurable on $\mathcal{G}_{**}$. For a vertex-shift $\Phi$, define two equivalence relations on each network ${\Gamma}$ by saying $u,v \in V({\Gamma})$ are in the same [**$\Phi$-foil**]{} if $\Phi_{\Gamma}^n(u) = \Phi_{\Gamma}^n(v)$ for some $n \in {\mathbb{N}}$, or in the same [**$\Phi$-component**]{} if $\Phi_{\Gamma}^n(u) = \Phi_{\Gamma}^m(v)$ for some $n,m \in {\mathbb{N}}$. Two vertices are in the same $\Phi$-component if their forward orbits under $\Phi$ intersect, whereas they are in the same $\Phi$-foil if, after some finite number of applications of $\Phi$, the vertices meet. The [**$\Phi$-graph of ${\Gamma}$**]{} is the graph drawn on ${\Gamma}$ with vertices $V({\Gamma})$ and edges from each $v \in V({\Gamma})$ to $\Phi_{\Gamma}(v)$. The following is a special case of the classification theorem appearing in [@baccellieternal].
\[thm:foil-classification-theorem\] Let $[{\boldsymbol{{\Gamma}}},{\boldsymbol{o}}]$ be a unimodular network and $\Phi$ a vertex-shift. Almost surely, every vertex has finite degree in the $\Phi$-graph of ${\boldsymbol{{\Gamma}}}$. In addition, each component $C$ of the $\Phi$-graph of ${\boldsymbol{{\Gamma}}}$ falls in one of the following three classes:
1. Class F/F: $C$ and all its foils are finite, and there is a unique cycle in $C$.
2. Class I/F: $C$ is infinite but all its foils are finite, there are no cycles in $C$, and there is a unique bi-infinite path in $C$.
3. Class I/I: $C$ is infinite and all its foils are infinite, and there are no cycles or bi-infinite paths in $C$.
The last tool needed from [@baccellieternal] is the so-called no infinite/finite inclusion lemma, which is used heavily in the proof of . To state it, the following definitions are needed. A [**covariant subset (of the set of vertices)**]{} is a map $C$ which associates to each network ${\Gamma}$ a set $C_{\Gamma}\subseteq V({\Gamma})$ such that $C$ commutes with network isomorphisms, and such that $[{\Gamma},o]\mapsto 1_{{\left\{{o \in C_{\Gamma}}\right\}}}$ is measurable. A [**covariant (vertex) partition**]{} is a map $\Pi$ which associates to all networks ${\Gamma}$ a partition $\Pi_{\Gamma}$ of $V({\Gamma})$ such that $\Pi$ commutes with network isomorphisms, and such that the (well-defined) subset ${\left\{{[G,o,v]: v \in \Pi_G(o)}\right\}} \subseteq \mathcal{G}_{**}$ is measurable, where $\Pi_{\Gamma}(o)$ denotes the partition element in $\Pi_{\Gamma}$ containing $o$. Then one has the following.
\[lem:no-infinite-finite-inclusion\] Let $[{\boldsymbol{{\Gamma}}},{\boldsymbol{o}}]$ be a unimodular network, $\Pi$ a covariant partition, and $C$ a covariant subset. Almost surely, there is no infinite element $E$ of $\Pi_{{\boldsymbol{{\Gamma}}}}$ such that $E \cap C_{{\boldsymbol{G}}}$ is finite and nonempty.
Embedding Subgraphs of the Doeblin Graph as Random Networks
-----------------------------------------------------------
In order to view a subgraph of ${\mathbf{G}}$ as a random network, one must ensure the subgraph is nonempty, locally finite, connected, and a root ${\boldsymbol{o}}$ has been suitably chosen. Since the vertices of ${\mathbf{G}}$ come from the fixed countable space ${\mathbb{Z}}\times S$, the following setup will help to verify all the technicalities.
Let $V:= {\mathbb{Z}}\times S$. Suppose that $${\boldsymbol{{\Gamma}}}:\Omega \to {\left\{{0,1}\right\}}^V \times \Xi^V \times {\left\{{0,1}\right\}}^{V\times V} \times \Xi^{V \times V}
=: (f_V, \xi_V, f_E, \xi_E)$$ is measurable (where the codomain is given its product topology and corresponding Borel $\sigma$-algebra). Then ${\boldsymbol{{\Gamma}}}(\omega)$ can be considered for each $\omega \in \Omega$ as a (possibly empty, possibly not locally finite, possibly disconnected) network in the following way. For each $u,v \in V$, interpret
1. \[it:d1\] $f_V(v)$ as the indicator that $v \in V({\boldsymbol{{\Gamma}}})$,
2. \[it:d2\] $f_E(u,v)$ as the indicator that the edge ${\left\{{u,v}\right\}} \in E({\boldsymbol{{\Gamma}}})$,
3. \[it:d3\] $\xi_V(u)$ as the mark of $u$, and
4. \[it:d4\] $\xi_E(u,v)$ as the mark of the vertex-edge pair $(u, {\left\{{u,v}\right\}})$.
That is, use \[it:d1,it:d2,it:d3,it:d4\] to *define* $V({\boldsymbol{{\Gamma}}}), E({\boldsymbol{{\Gamma}}})$, and the marks of vertices and edges in ${\boldsymbol{{\Gamma}}}$. Note that $f_E$ must be symmetric because edges are not directed, but $\xi_E$ may not be, since each edge is associated with two marks, one per vertex. If one wants to consider directed edges, one instead uses undirected edges and uses the marks on edges to specify which direction the edge should point. The definition of $\xi_V(u)$ when $u \notin V({\boldsymbol{{\Gamma}}})$ is irrelevant, and similarly for the definition of $f_E(u,v)$ and $\xi_E(u,v)$ if either of $u$ or $v$ is not in $V({\boldsymbol{{\Gamma}}})$. All statements about the network defined by ${\boldsymbol{{\Gamma}}}$ are then translated into statements about the maps $(f_V, \xi_V, f_E, \xi_E)$. For instance, $$\begin{aligned}
\text{\{${\boldsymbol{{\Gamma}}}$ is not empty\}} =
{\left\{{\sum_{v \in V} f_V(v) > 0}\right\}}.\end{aligned}$$ This is exactly the kind of construction used to define the Doeblin graph ${\mathbf{G}}$. In the case of ${\mathbf{G}}$,
1. $f_V=1$ on ${\mathbb{Z}}\times S$,
2. $f_E\left((t,x), (t+1, h(x, \xi_{t}))\right)
=f_E\left((t+1, h(x, \xi_{t})),(t,x)\right) =1$ for all $(t,x) \in {\mathbb{Z}}\times S$ and $f_E = 0$ otherwise,
3. $\xi_V(t,x) = \xi_{t}$ for all $(t,x) \in {\mathbb{Z}}\times S$, and
4. $\xi_E\left((t,x), (t+1, h(x, \xi_{t,x}))\right)=1$ for all $t \in {\mathbb{Z}}, x \in S$ to indicate the edge is directed forwards in time.
This construction also works for the bridge graph ${\mathbf{B}}$ as well. When a ${\boldsymbol{{\Gamma}}}$ has been constructed as in this section, one can see ${\boldsymbol{{\Gamma}}}$ as a random network after any measurable choice of root, given that it is nonempty and locally finite.
\[lem:class-of-tangible-network-is-measurable\] Suppose ${\boldsymbol{{\Gamma}}}=(f_V,\xi_V,f_E,\xi_E)$ is as above and a.s. ${\boldsymbol{{\Gamma}}}$ is nonempty, locally finite, and connected. Then for any measurable choice of root ${\boldsymbol{o}} \in V({\boldsymbol{{\Gamma}}})$, $[{\boldsymbol{{\Gamma}}}, {\boldsymbol{o}}]$ is a random network.
Write the event that $[{\boldsymbol{{\Gamma}}},{\boldsymbol{o}}]$ is within $\epsilon>0$ of some fixed network $[{\Gamma}, o]$ as a countable union over rooted isomorphic copies $({\Gamma}', o')$ of $({\Gamma},o)$ with vertices in $V$ of the event that ${\boldsymbol{o}}=o'$, the neighborhood of radius $\lceil\frac{1}{\epsilon}\rceil$ around ${\boldsymbol{o}}$ is exactly ${\Gamma}'$, and the marks $\xi_V(u)$ for $u \in V({\Gamma}')$ and $\xi_E(v,w)$ for ${\left\{{v,w}\right\}} \in E({\Gamma}')$ are within $\epsilon$ of the corresponding vertex and edge marks of $({\Gamma}', o')$. Each of these conditions individually are written in terms of events using the maps $f_V,\xi_V, f_E, \xi_E$, showing the desired measurability of $\omega \mapsto [{\boldsymbol{{\Gamma}}}(\omega), {\boldsymbol{o}}(\omega)]$. The full proof of given in the appendix.
Thus indeed ${\mathbf{G}}$ may be seen as a random network when rooted and marked, assuming it is locally finite and connected. But the question remains whether this may be done in such a way as to make ${\mathbf{G}}$ unimodular. The first approach one might take is to investigate whether ${\mathbf{G}}$, rooted at $(0,X_0)$ for some (random) choice of $X_0\in S$, is unimodular. Two natural choices, at least in the standard CFTP setup, are to take $X_0$ to be the output of the CFTP algorithm, or to take $X_0$ to be independent of ${\mathbf{G}}$. For simplicity, the standard CFTP setup refers to the case where ${\mathbf{G}}$ has fully independent transitions, $S$ is finite, and the CFTP algorithm succeeds a.s. The following proposition determines when ${\mathbf{G}}$ can be unimodular under the previous choices of $X_0$.
\[prop:naiive-unimodularity-in-G\] Suppose ${\mathbf{G}}$ is an [[EFT]{}]{}, that ${\mathbf{G}}$ has each $(t,x) \in V({\mathbf{G}})$ marked by $(x, \xi_{t})$, and that $X_0$ is a random choice in $S$. Then
- if $[{\mathbf{G}},(0,X_0)]$ is unimodular, then $S$ is finite and $X_0$ is uniformly distributed on $S$,
- if $X_0$ is independent of ${\mathbf{G}}$ and uniformly distributed on a finite $S$, then $[{\mathbf{G}}, (0,X_0)]$ is unimodular, and
- if $X_0$ is the output of the CFTP algorithm in the standard CFTP setup, then $[{\mathbf{G}}, (0,X_0)]$ is unimodular if and only if $S$ has a single element.
The first point follows by constructing for each $x,y \in S$ a mass transport that, when applied to ${\mathbf{G}}$, sends mass 1 within vertical slices of ${\mathbf{G}}$ from the vertex in state $x$ to the vertex in state $y$. Unimodularity then gives ${\mathbf{P}}(X_0 = x) = {\mathbf{P}}(X_0 = y)$. The second point follows from the definition of unimodularity. The third point follows by noting that the output of the CFTP algorithm has at least one child, but unimodularity implies that it must have one on average, so a.s. it has one child. A nonempty tree where every vertex has one incoming and one outgoing edge is isomorphic to ${\mathbb{Z}}$, so $S$ can only have one state. The full proof of is given in the appendix.
While choosing $X_0$ uniformly distributed on $S$ and independent of ${\mathbf{G}}$ works when $S$ is finite, unimodularity of the whole ${\mathbf{G}}$ is doomed in the general case, as there is no uniform distribution on an infinite $S$. This is the reason for introducing the bridge graph ${\mathbf{B}}$, which is locally finite. However, the bridge graph may still not be connected, so a spine is added to it to make it connected.
\[cor:bridge-graphs-embedding\] Let $\overline{{\mathbf{B}}}$ be ${\mathbf{B}}$ [**with spine added**]{}, i.e. with edges from each $(t,x^*)$ to $(t+1,x^*)$ for all $t \in {\mathbb{Z}}$ added. Then for any measurable marks and any measurable choice of root ${\boldsymbol{o}} \in V({\mathbf{B}})$, $[\overline{{\mathbf{B}}}, {\boldsymbol{o}}]$ is a random network.
One has that $(0,x^*) \in V(\overline{{\mathbf{B}}})$, so $\overline{{\mathbf{B}}}$ is nonempty. Also $\overline{{\mathbf{B}}}$ is locally finite by and the fact that adding the spine has increased the degree of each vertex by at most two. Finally, since each $v \in V(\overline{{\mathbf{B}}})$ is connected to some $(t, x^*)$, and the spine in $\overline{{\mathbf{B}}}$ connects all such vertices, $\overline{{\mathbf{B}}}$ is connected. finishes the claim.
Everything is in place to see the unimodular structure hidden in ${\mathbf{G}}$, which is handled in the next section.
Unimodularizability and its Consequences {#sec:main-thm}
========================================
Unimodularizability of the Bridge Graph {#sec:unimodularizability-of-bridge}
---------------------------------------
The following result identifies the unimodular structure inside ${\mathbf{G}}$. For the rest of the document, each $(t,y)\in V({\mathbf{B}})$ is marked by $(y,\xi_{t})$ whenever considered as a vertex in a rooted network.
\[thm:main-thm-mbeft-is-unimodularizable\] Any random network with distribution $$\label{eq:unimodular-measure}
{{\mathbf{P}}^{\mathsmaller{\square}}}(A) := \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]} {\mathbf{E}}\left[ \sum_{w \in V_0({\mathbf{B}})} 1_{{\left\{{[\overline{{\mathbf{B}}}, w] \in A}\right\}}}\right],
\qquad A \in \mathcal{B}(\mathcal{G}_*).$$ is unimodular. The spine need not be added and ${\mathbf{B}}$ may also be used instead of $\overline{{\mathbf{B}}}$ if ${\mathbf{B}}$ is already connected.
One may interpret the distribution ${{\mathbf{P}}^{\mathsmaller{\square}}}$ as a size-biased version of the network obtained by starting with ${\mathbf{B}}$ and selecting the root uniformly from ${\mathbf{B}}_0$.
By , $\overline{{\mathbf{B}}}$ with marks as specified and any choice of root is a random network. Therefore, all the quantities in the following calculation are measurable. Let $g:\mathcal{G}_{**} \to {\mathbb{R}}_{{\geqslant}0}$ be given. One has $$\begin{aligned}
&\int_{\mathcal{G}_{*}} \sum_{v \in V({\Gamma})}
g[{\Gamma},o,v]\,{{\mathbf{P}}^{\mathsmaller{\square}}}(d[{\Gamma},o])\\
&= \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]} {\mathbf{E}}\sum_{y \in
{\mathbf{B}}_0}\sum_{v \in V({\mathbf{B}})} g[\overline{{\mathbf{B}}},(0,y), v]\\
&= \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]} \sum_{y,y' \in S, t \in {\mathbb{Z}}}{\mathbf{E}}\left[
1_{{\left\{{(0,y), (t,y') \in V({\mathbf{B}})}\right\}}}g[\overline{{\mathbf{B}}},(0,y),
(t,y')]\right].
\end{aligned}$$ Stationarity on ${\mathbb{Z}}$ implies the right hand side is equal to $$\begin{aligned}
& \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]} \sum_{y,y' \in S, t \in {\mathbb{Z}}}{\mathbf{E}}\left[
1_{{\left\{{(-t,y), (0,y') \in V({\mathbf{B}})}\right\}}}g[\overline{{\mathbf{B}}},(-t,y),
(0,y')]\right]\\
= &\frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]} \sum_{y,y' \in S, t \in {\mathbb{Z}}}{\mathbf{E}}\left[
1_{{\left\{{(t,y), (0,y') \in V({\mathbf{B}})}\right\}}}g[\overline{{\mathbf{B}}},(t,y),
(0,y')]\right]\\
= &\frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]} {\mathbf{E}}\sum_{y' \in
{\mathbf{B}}_0}\sum_{v \in V({\mathbf{B}})} g[\overline{{\mathbf{B}}},v, (0,y')]\\
=&\int_{\mathcal{G}_{*}} \sum_{v \in V({\Gamma})}
g[{\Gamma},v,o]\,{{\mathbf{P}}^{\mathsmaller{\square}}}(d[{\Gamma},o]).
\end{aligned}$$ Thus ${{\mathbf{P}}^{\mathsmaller{\square}}}$ is the distribution of a unimodular network.
The view of ${{\mathbf{P}}^{\mathsmaller{\square}}}$ as a size-biased version of a network is formalized in the following.
\[prop:size-biased-dist\] Let ${\boldsymbol{o}}$ be, conditionally on $V_0({\mathbf{B}})$, uniformly distributed on $V_0({\mathbf{B}})$ and independent of ${\mathbf{B}}$. Then under the size-biased measure $\hat {\mathbf{P}}(E) := \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]} {\mathbf{E}}[\#{\mathbf{B}}_0 1_E]$ for each $E \in {\mathcal{F}}$, the random network $[\overline{{\mathbf{B}}}, {\boldsymbol{o}}]$ has the distribution ${{\mathbf{P}}^{\mathsmaller{\square}}}$.
In what follows, $V$ ranges over the sets for which ${\mathbf{P}}(V_0({\mathbf{B}}) = V) > 0$, of which there are at most countably many because ${\mathbf{B}}_0$ is a.s. a finite subset of the countable $S$. For any $A \in \mathcal{B}(\mathcal{G}_*)$ and with $C := {\mathbf{E}}[\#{\mathbf{B}}_0]$, $$\begin{aligned}
&\hat {\mathbf{P}}([\overline{{\mathbf{B}}}, {\boldsymbol{o}}] \in A) \\
&= \frac{1}{C} {\mathbf{E}}[\# {\mathbf{B}}_0 1_{{\left\{{[\overline{{\mathbf{B}}}, {\boldsymbol{o}}] \in A}\right\}}}]\\
&= \frac{1}{C}\sum_{V} \left\vert V\right\vert {\mathbf{P}}(V_0({\mathbf{B}})=V) {\mathbf{P}}([\overline{{\mathbf{B}}}, {\boldsymbol{o}}] \in A \mid V_0({\mathbf{B}}) = V)\\
&= \frac{1}{C}\sum_{V} \sum_{v\in V} \left\vert V\right\vert {\mathbf{P}}(V_0({\mathbf{B}})=V)
{\mathbf{P}}({\boldsymbol{o}} = v, [\overline{{\mathbf{B}}}, v] \in A \mid V_0({\mathbf{B}}) = V)
\end{aligned}$$ which, by the conditional independence of ${\boldsymbol{o}}$ and ${\mathbf{B}}$, is $$\begin{aligned}
&= \frac{1}{C}\sum_{V} \sum_{v \in V}\left\vert V\right\vert {\mathbf{P}}(V_0({\mathbf{B}})=V)
\frac{1}{\left\vert V\right\vert}{\mathbf{P}}([\overline{{\mathbf{B}}}, v] \in A \mid V_0({\mathbf{B}}) = V)\\
&= \frac{1}{C}{\mathbf{E}}\left[\sum_{V} \sum_{v \in V}
1_{{\left\{{[\overline{{\mathbf{B}}}, v] \in A, V_0({\mathbf{B}}) = V}\right\}}}\right]\\
&= {{\mathbf{P}}^{\mathsmaller{\square}}}(A)
\end{aligned}$$ as claimed.
I/F Component Properties {#sec:i-f-component-properties}
------------------------
For any measurable event $A\subseteq \mathcal{G}_*$ in the $\sigma$-algebra of [**root-invariant**]{} events, i.e., such that if $[{\Gamma},o] \in A$ then $[{\Gamma},v] \in A$ for all $v \in V({\Gamma})$, one has $$\begin{aligned}
{{\mathbf{P}}^{\mathsmaller{\square}}}(A)
= \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]}{\mathbf{E}}\left[\sum_{w \in
V_0({\mathbf{B}})} 1_{{\left\{{[\overline{{\mathbf{B}}}, w] \in A}\right\}}} \right]
= \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]}{\mathbf{E}}\left[\#{\mathbf{B}}_0
1_{{\left\{{[\overline{{\mathbf{B}}} , (0,x^*)] \in A}\right\}}} \right].\end{aligned}$$ This immediately gives the following.
\[lem:same-root-inv-events-measure-zero\] One has that ${{\mathbf{P}}^{\mathsmaller{\square}}}$ and ${\mathbf{P}}([\overline{{\mathbf{B}}},(0,x^*)] \in \cdot)$ have the same root-invariant sets of measure $0$ or $1$.
Next, a vertex-shift that is designed to follow the arrows in ${\mathbf{B}}$ is defined. It plays the same role as ${f_+}$ but is defined for all networks. From now on, let $\Phi$ denote the [**follow vertex-shift**]{} defined on any network ${\Gamma}$ for each $u \in V({\Gamma})$ by $\Phi_{{\Gamma}}(u):=v$ if either:
1. there is a unique outgoing edge from $u$ and this edge terminates at $v$, or
2. $u$ is in state $x^*$ and there is a unique outgoing edge from $u$ that does not terminate at a vertex in state $x^*$, and this edge terminates at $v$.
If neither of the two conditions above is met for any $v \in V({\Gamma})$, define $\Phi_{{\Gamma}}(u):=u$ for concreteness. Here a vertex is considered to be in a state $y \in S$ when the first component of its mark is $y$ (recall that a vertex $(t,y) \in V({\mathbf{B}})$ is marked by $(y, \xi_t)$). The second clause in the definition of $\Phi$ is there because of the presence of the spine in $\overline{{\mathbf{B}}}$, so that if the root is in state $x^*$ the vertex-shift will choose to follow the arrow in ${\mathbf{B}}$ instead of following the arrow to the next element of the spine, unless the two coincide. By construction, $\Phi_{\overline{{\mathbf{B}}}}(t,x) = {f_+}(t,x)$ for all $(t,x) \in V(\overline{{\mathbf{B}}})$.
The event that all $\Phi$-components of a network are of class I/F is root-invariant, and moreover it has ${\mathbf{P}}([\overline{{\mathbf{B}}}, (0,x^*)] \in \cdot)$-probability one because the $\Phi$-graph of $\overline{{\mathbf{B}}}$ is ${\mathbf{B}}$ itself, the $\Phi$-components of $\overline{{\mathbf{B}}}$ are the components of ${\mathbf{B}}$, and the $\Phi$-foils of $\overline{{\mathbf{B}}}$ are subsets of the sets ${{({V_t({\mathbf{B}})})}}_{t \in {\mathbb{Z}}}$, which are finite. Hence ${{\mathbf{P}}^{\mathsmaller{\square}}}$ is concentrated on the set of networks having only $\Phi$-components of I/F class. It follows that any a.s. root-invariant properties that follow from ${{\mathbf{P}}^{\mathsmaller{\square}}}$ being unimodular and having I/F components automatically apply to ${\mathbf{P}}([\overline{{\mathbf{B}}}, (0,x^*)] \in \cdot)$ as well. Such properties will be referred to as [**I/F component properties**]{} and are explored in .
### Bi-recurrent Paths {#sec:bi-recurrence}
This section studies bi-infinite paths in ${\mathbf{G}}$ and identifies special bi-infinite paths that have a certain recurrence property backwards in time. Firstly, it is possible to have multiple bi-infinite paths in ${\mathbf{G}}$ because ${\mathbf{G}}$ is disconnected.
Consider the case where $S := {\left\{{1,2}\right\}}$ and ${h_{\mathrm{gen}}}$ and ${{({\xi_{t}})}}_{t \in {\mathbb{Z}}}$ are chosen so that the transition $(t,1) \to (t+1,2)$ occurs if and only if $(t,2) \to (t+1,1)$ occurs. In this case ${\mathbf{G}}$ has two components a.s. Each component is itself a bi-infinite path.
Moreover, even when ${\mathbf{G}}$ is connected, it it still possible to have multiple bi-infinite paths in ${\mathbf{G}}$.
\[ex:deterministic-falling\] Consider the case of $S:={\mathbb{N}}$ with fully independent transitions. Let the transition matrix $P$ be determined as follows. In state $0$, transition to a $\mathrm{Geom}(1/2)$ random variable, and from any other $n \neq 0$, deterministically transition from $n$ to $n-1$. In this case, from every vertex $(s,x) \in {\mathbb{Z}}\times S$, there is a bi-infinite path ${{({t,X_t})}}_{t \in {\mathbb{Z}}}$ in ${\mathbf{G}}$ for which $X_{s-k} = k+x$ for all $k {\geqslant}0$. Thus there are infinitely many bi-infinite paths, despite the fact that in this case ${\mathbf{G}}$ is an [[EFT]{}]{}, which follows from .
In , even though ${\mathbf{G}}$ is connected, ${\mathbf{G}}$ has infinitely many bi-infinite paths. However, amongst the bi-infinite paths, there is one special bi-infinite path. The special path is the unique bi-infinite path that visits every state infinitely often *in the past*. It turns out that this is the correct kind of path to look for in general. A bi-infinite sequence ${{({x_t})}}_{t \in {\mathbb{Z}}}$ in $S$ is called [**bi-recurrent for state $x$**]{} if ${\left\{{t\in {\mathbb{Z}}: x_t = x}\right\}}$ is unbounded above and below. If ${{({x_t})}}_{t \in {\mathbb{Z}}}$ is bi-recurrent for every $x\in S$, it is simply called [**bi-recurrent**]{}. A state path ${{({X_t})}}_{t \in I}$ in ${\mathbf{G}}$ is called [**bi-reccurent (for state $x$)**]{} if a.s. its trajectory is bi-recurrent (for state $x$). Recall that $\Phi$ denotes the follow vertex-shift. The existence of bi-infinite paths in $\Phi$-components of a network is an I/F property, and hence one has the following.
\[prop:existence-of-bi-rec-paths\] It holds that ${\mathbf{B}}$ has a unique bi-infinite path in each component a.s. The corresponding state paths are bi-recurrent for $x^*$ and these are the only state paths in all of ${\mathbf{G}}$ that are bi-recurrent for $x^*$. Moreover, for each $y \in S$, these state paths either a.s. never visit $y$, or are bi-recurrent for $y$.
By , ${{\mathbf{P}}^{\mathsmaller{\square}}}$-a.e. network has a unique bi-infinite path in each $\Phi$-component, where $\Phi$ is the follow vertex-shift. But having a unique bi-infinite path in each $\Phi$-component is a root-invariant event, and hence ${\mathbf{P}}$-a.s. $\overline{{\mathbf{B}}}$ has a unique bi-infinite path in each $\Phi$-component. Since the $\Phi$-components of $\overline{{\mathbf{B}}}$ are the components of ${\mathbf{B}}$, ${\mathbf{P}}$-a.s. every component of ${\mathbf{B}}$ contains a unique bi-infinite path.
Let $\Pi$ be the covariant partition of $\Phi$-components. Define the covariant subset $C$ on a network ${\Gamma}$ by letting $C_{\Gamma}$ be the subset of vertices of ${\Gamma}$ that are either the first or last visit to a given state $y \in S$, if they exist, on the unique bi-infinite path in their $\Phi$-component of ${\Gamma}$, if such a path exists. The no infinite/finite inclusion lemma, , implies that ${{\mathbf{P}}^{\mathsmaller{\square}}}$ is concentrated on the set of networks ${\Gamma}$ with no first or last visit to $y$ on the unique bi-infinite paths in each $\Phi$-component of ${\Gamma}$. This property is root-invariant and hence a.s. the state paths corresponding to the unique bi-infinite paths in each component of ${\mathbf{B}}$ either do not visit state $y$ or are bi-recurrent for $y$. Taking a countable union over $y \in S$ shows this property holds simultaneously for all $y \in S$. Since the unique bi-infinite path in each component of ${\mathbf{B}}$ at least hits $x^*$, one may at least conclude the paths are bi-recurrent for $x^*$. Finally, there cannot be any other bi-recurrent state paths for $x^*$ in ${\mathbf{G}}$ because, by definition, a bi-recurrent state path in ${\mathbf{G}}$ will lie in ${\mathbf{B}}$ since it visits $x^*$ at arbitrarily large negative times.
The next result applies to the nicest case, where ${\mathbf{G}}$ is a tree.
\[cor:ergodic-meft-unique-bi-recurrent\] Suppose that ${\mathbf{G}}$ is an [[EFT]{}]{}. Then ${\mathbf{G}}$ contains a unique (up to measure zero modifications) state path ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ that is bi-recurrent for $x^*$. Moreover, ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ is shift-covariant, stationary, and for each $t \in {\mathbb{Z}}$ one has that $\beta_t$ is measurable with respect to $\sigma(\xi_s : s < t)$. Additionally, ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ is bi-recurrent for every $x \in S$ that is positive recurrent.
shows that a.s. there is a unique bi-infinite path in each component of ${\mathbf{B}}$, and the corresponding state paths are bi-recurrent for $x^*$. Since ${\mathbf{G}}$ a.s. has only one component, ${\mathbf{B}}$ does too. The second part of then implies the bi-recurrent state path for $x^*$ in ${\mathbf{B}}$ is the only bi-recurrent state path for $x^*$ in ${\mathbf{G}}$. One would like to define ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ to be the unique bi-recurrent state path for $x^*$ in ${\mathbf{G}}$. However, in that case, ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ would only be defined a.s. For concreteness, define $\beta_t$ for each $t \in {\mathbb{Z}}$ by letting $\beta_t := \lim_{s \to -\infty} F^{(s,x^*)}_t$ on the event that the limit exists, and $\beta_t := x^*$ otherwise. On the a.s. event $E$ that ${\mathbf{B}}$ is connected, $\#{\mathbf{B}}_t < \infty$ for all $t \in {\mathbb{Z}}$, and there is a unique bi-infinite path in ${\mathbf{B}}$, one has that ${{({t,\beta_t})}}_{t \in {\mathbb{Z}}}$ coincides with the unique bi-infinite path in ${\mathbf{B}}$. This is because if, for some $t \in {\mathbb{Z}}$, $\lim_{s \to -\infty} F_t^{(s,x^*)}$ does not exist, then either $\#{\mathbf{B}}_t = \infty$, or there exist two states $x,y \in S$ such that $(t,x)$ and $(t,y)$ have (necessarily disjoint) locally finite infinite trees of descendants in ${\mathbf{B}}$. The former case is forbidden on $E$, and, in the latter case, König’s lemma would imply the existence of two distinct bi-infinite paths in ${\mathbf{B}}$, which is also forbidden on $E$. Thus $\lim_{s \to -\infty} F_t^{(s,x^*)}$ exists for all $t \in {\mathbb{Z}}$ on the event $E$, and, on this event, the unique bi-infinite path in ${\mathbf{B}}$ must therefore be ${{({t,\beta_t})}}_{t \in {\mathbb{Z}}}$. The shift-covariance and hence stationarity of ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ follows from its definition in terms of $F^{(s,x^*)}$ for each $s \in {\mathbb{Z}}$. For each $t \in {\mathbb{Z}}$, measurability of $\beta_t$ with respect to $\sigma(\xi_s : s < t)$ also follows from its definition, since each $F^{(r,x^*)}_t$ with $r {\leqslant}t$ is $\sigma(\xi_s : s < t)$-measurable.
Now let ${{({Y_t})}}_{t \in {\mathbb{Z}}}$ be the unique bi-recurrent state path for some other $y \in S$ that is positive recurrent. Since ${\mathbf{G}}$ is a.s. connected, ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ and ${{({Y_t})}}_{t \in {\mathbb{Z}}}$ eventually merge, a.s. However, stationarity forbids that there is a first time such that $\beta_t = Y_t$, so it must be that $\beta_t = Y_t$ for all $t \in {\mathbb{Z}}$. Thus ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ is bi-recurrent for every $y \in S$ that is positive recurrent.
shows that, like in the standard CFTP setup, there is a $\beta_0$ living at time $0$ in ${\mathbf{G}}$ that is a perfect sample from the stationary distribution of the Markov chain or SRS. However, unlike in the standard CFTP setup, it is not known whether there is an algorithm that can find $\beta_0$ in finite time.
Another consequence of the existence of bi-recurrent paths in ${\mathbf{B}}$ is that one can bound the number of components of ${\mathbf{B}}$.
\[cor:B-has-finitely-many-components\] The a.s. constant number $n$ of components of ${\mathbf{B}}$ is no larger than $\min{\left\{{ k : {\mathbf{P}}(\#{\mathbf{B}}_0 = k)>0}\right\}} < \infty$. In particular, ${\mathbf{B}}$ has finitely many connected components, even if ${\mathbf{G}}$ has infinitely many components, and if ${\mathbf{P}}(\#{\mathbf{B}}_0 = 1) > 0$, then ${\mathbf{B}}$ is an [[EFT]{}]{}.
The number of components of ${\mathbf{B}}$ is shift-invariant and hence a.s. constant. Each component of ${\mathbf{B}}$ contains a bi-recurrent path by . Each bi-recurrent path intersects $V_0({\mathbf{B}})$ in a different element since they are in different components of ${\mathbf{B}}$. It follows that $n {\leqslant}\#{\mathbf{B}}_0$ a.s. If ${\mathbf{P}}(\# {\mathbf{B}}_0 = k) > 0$ for some $k$, then it follows that $n {\leqslant}k$.
The deterministic cycle on $n$ states shows that the bound in can be achieved for each $n$. In general, any bi-infinite stationary process on $S$ (or any countable set) must be bi-recurrent.
\[prop:stationary-implies-birec\] Suppose that ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a stationary process taking values in $S$. Then a.s. ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is bi-recurrent for every $x \in {\left\{{X_t}\right\}}_{t \in {\mathbb{Z}}}$.
For each $x \in S$, stationarity forbids that there is a first or last visit of ${{({X_t})}}_{t \in {\mathbb{Z}}}$ to $x$ since such an occurrence would have to be equally likely to happen at all times $t \in {\mathbb{Z}}$. Thus, a.s. either $x \notin {\left\{{X_t}\right\}}_{t \in {\mathbb{Z}}}$ or ${\left\{{t \in {\mathbb{Z}}: X_t = x}\right\}}$ must be unbounded both above and below. The countability of $S$ finishes the claim.
The remainder of the section specializes to the Markovian setting again. In the Markovian setting, bi-recurrence is actually equivalent to stationarity in the irreducible, aperiodic, positive recurrent case.
\[thm:stationary-iff-birecurrent\] Suppose that $P$ is irreducible, aperiodic, and positive recurrent, and that ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a Markov chain with transition matrix $P$. Then ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is stationary if and only if it is bi-recurrent for any (and hence every) state.
By , it is possible to assume without loss of generality that ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a state path in the Doeblin graph ${\mathbf{G}}$ with fully independent transitions. By , ${\mathbf{G}}$ is an [[EFT]{}]{} and therefore implies that ${\mathbf{G}}$ contains a bi-recurrent state path ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ that is, for all $y \in S$, the a.s. unique bi-recurrent state path for state $y$ in ${\mathbf{G}}$. Moreover, $\beta_t \sim \pi$ for all $t \in {\mathbb{Z}}$, where $\pi$ is the stationary distribution for $P$. If ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is bi-recurrent for some $y\in S$, then, by uniqueness, $X_t = \beta_t$ for all $t \in {\mathbb{Z}}$, a.s. In particular, ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is stationary. The converse follows from and irreducibility.
A bi-infinite path in ${\mathbf{G}}$ whose state path is not bi-recurrent for any state $x \in S$ will be called [**spurious**]{}. Observe the difference between spurious bi-infinite paths and the unique bi-recurrent path in . Viewed in reverse time, a spurious path must run off to $\infty$ in the sense that for every finite set $F\subseteq S$, the reversed path eventually leaves $F$ forever. It is possible for ${\mathbf{G}}$ to contain spurious bi-infinite paths, as was seen in .
![The Doeblin graph from with the bi-recurrent path and a spurious path distinguished. \[fig:meft-falling-example\]](meft-falling-fig.pdf){width="\linewidth"}
Say that $P^n$ [**converges uniformly (to $\pi$ as $n \to \infty$)**]{} if $P$ is irreducible, aperiodic, and positive recurrent with stationary distribution $\pi$, and $\sup_{x \in S}\left\Vert P^n(x,\cdot) - \pi\right\Vert \to 0$ as $n\to \infty$. For example, this is automatic if $P$ is irreducible, aperiodic, and $S$ is finite. Some authors call $P$ uniformly ergodic, but the term ergodic is not used here to avoid a terminology collision with ergodic theory. Uniform convergence to $\pi$ is also equivalent (cf. [@meyn2009markov] Theorem 16.0.2 (v)) to the statement that there is $m$ such that $P^m(x,\cdot) {\geqslant}\varphi(\cdot)$ for all $x \in S$, for a measure $\varphi$ which is not the zero measure. It is also equivalent (cf. [@foss1998perfect] Theorem 4.2) to the fact that the CFTP algorithm succeeds in the case of fully independent transitions, i.e. the backwards vertical coupling time $\inf {\left\{{t>0 : F^{(-t,x)}_{0} = F^{(-t,y)}_{0}, \forall x,y \in S}\right\}}$ is a.s. finite.
Together, the following two results say that when a Markov chain that mixes uniformly is started in the infinite past, it has converged to its stationary distribution by any finite time.
\[prop:unif-ergodicity-no-spurious-paths\] Suppose $P^n$ converges uniformly to $\pi$ as $n \to \infty$ and ${\mathbf{G}}$ has fully independent transitions. Then ${\mathbf{G}}$ contains no spurious bi-infinite paths.
For every $s<t$ let $C_{s,t}$ be the event that $F^{(s,x)}_t = F^{(s,y)}_t$ for all $x,y \in S$. That is, $C_{s,t}$ is the event that starting at time $s$, all paths in ${\mathbf{G}}$ collapse to a single state by time $t$. Note that ${\mathbf{P}}(C_{s,t})$ depends only on $t-s$. Since ${\mathbf{G}}$ has fully independent transitions and $P^n$ converges to $\pi$ uniformly as $n \to \infty$, by e.g. Theorem 5.2 in [@foss1998perfect], there exists some $k \in {\mathbb{N}}$ such that ${\mathbf{P}}(C_{s,t}) > 0$ when $t-s{\geqslant}k$. Consider $E_n := C_{-k(n+1), -kn}$ for each $n \in {\mathbb{N}}$. One has ${\mathbf{P}}(E_n) = {\mathbf{P}}(E_0)>0$ for all $n$ and the $E_n$ are independent. It follows that a.s. infinitely many of them occur. On an $\omega$ for which infinitely many $E_n$ occur, there is at most one bi-infinite path in ${\mathbf{G}}$, and thus any bi-infinite path in ${\mathbf{G}}$ must coincide with the unique bi-recurrent path guaranteed to exist by .
It is a classical result that it is possible to find a bi-infinite stationary version ${{({X_t})}}_{t \in {\mathbb{Z}}}$ of a Markov chain that has a stationary distribution. The following shows that, in the case of uniform convergence to $\pi$, this is the only way to extend a Markov chain to have time index set all of ${\mathbb{Z}}$. That is, if ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a Markov chain that conveges uniformly to its stationary distribution, then it must be that $X_t
\sim \pi$ for all $t \in {\mathbb{Z}}$.
\[prop:bi-infinite-stationary-unif-ergodic\] Suppose $P^n$ converges uniformly to $\pi$ as $n \to \infty$. Then every Markov chain ${{({X_t})}}_{t \in {\mathbb{Z}}}$ with transition matrix $P$ is stationary and bi-recurrent. The subtle assumption here is that the time index set is all of ${\mathbb{Z}}$.
By , one may assume ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a state path in ${\mathbf{G}}$ with fully independent transitions, which is then an [[EFT]{}]{} by . Since $P^n$ converges uniformly to $\pi$ as $n \to \infty$, ${\mathbf{G}}$ contains no spurious bi-infinite paths by , and hence ${{({X_t})}}_{t \in {\mathbb{Z}}}$ must be the bi-recurrent state path. then implies ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is stationary.
may fail for an irreducible, aperiodic, and positive recurrent $P$ if $P$ does not converge uniformly to its stationary distribution. Indeed, it was already shown, e.g., in , that it is possible for ${\mathbf{G}}$ to admit spurious bi-infinite paths. If ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a proper state path in ${\mathbf{G}}$ that corresponds to a spurious bi-infinite path, then ${{({X_t})}}_{t \in {\mathbb{Z}}}$ is a Markov chain with transition matrix $P$, but it is not stationary since it is not bi-recurrent. Recall that ${\mathbf{B}}(x)$ denotes the bridge graph in ${\mathbf{G}}$ using $x$ as the base point instead of $x^*$.
\[prop:intersection-is-bi-inf-path\] Suppose $P$ is irreducible, aperiodic, and positive recurrent, and that ${\mathbf{G}}$ has fully independent transitions. If
1. $S$ is infinite,
2. ${\mathbf{G}}$ is locally finite, and
3. ${\mathbf{G}}$ contains no spurious bi-infinite paths,
then $$\begin{aligned}
\bigcap_{x \in S} V({\mathbf{B}}(x)) = {\left\{{(t,\beta_t): t \in {\mathbb{Z}}}\right\}},
\end{aligned}$$ where ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ is the unique bi-recurrent state path in ${\mathbf{G}}$. That is, the bi-recurrent path in ${\mathbf{G}}$ is the only thing common to all of the [bridge [[EFTs]{}]{}]{}. Alternatively, if $S$ is finite and has at least 2 states, then a.s. $$\begin{aligned}
\bigcap_{x \in S} V({\mathbf{B}}(x)) \supsetneq {\left\{{(t,\beta_t): t \in {\mathbb{Z}}}\right\}}.
\end{aligned}$$
For each $x \in S$, the bi-recurrent path is in ${\mathbf{B}}(x)$ because it is bi-recurrent for $x$. Suppose $S$ is infinite, ${\mathbf{G}}$ is locally finite, and that ${\mathbf{G}}$ contains no spurious bi-infinite paths. Consider a vertex $v \in V({\mathbf{G}})$ not on the bi-recurrent path. The tree of all descendants of $v$ in ${\mathbf{G}}$ must be finite, else König’s lemma would give a bi-infinite path in ${\mathbf{G}}$ that is distinct from the unique bi-recurrent path since $v$ is not on the bi-recurrent path. Since ${\mathbf{G}}$ contains no spurious bi-infinite paths, this is impossible. Since the tree of descendants of $v$ is finite but $S$ is infinite, there is some state $x \in S$ such that $v$ has no descendant in state $x$. In particular, $v \notin V({\mathbf{B}}(x))$, showing that nothing off the bi-recurrent path can be common to all the [bridge [[EFTs]{}]{}]{}.
Next suppose that $2 {\leqslant}\# S < \infty$. It suffices to give a finite deterministic graph ${\Gamma}$ that is a subgraph of ${\mathbf{G}}$ with positive probability such that when some time-translate of ${\Gamma}$ is a subgraph of ${\mathbf{G}}$, $\bigcap_{x \in S} V({\mathbf{B}}(x))$ contains a vertex not on the unique bi-infinite path in ${\mathbf{G}}$. Firstly, since $S$ is finite, choose a tree $T$ on ${\mathbb{Z}}\times S$ that occurs with positive probability and is an example witnesses of the a.s. finiteness of the backwards vertical coupling time $\inf {\left\{{t>0 : F^{(-t,x)}_{0} = F^{(-t,y)}_{0}, \forall x,y \in S}\right\}}$. Suppose $T$ is rooted at $(0,x_0)$. In particular, $V(T) \subseteq (-\infty,0] \times S$. By irreducibility of $P$ and the fact that $\#S {\geqslant}2$, choose $L=(x_0,x_1,\ldots,x_n)$ a finite path in $S$ using only positive probability transitions from $x_0$ back to $x_0=x_n$ that passes through all states of $S$ and has the property that $x_i \neq x_{i+1}$ for any $i$. Note that $$\begin{aligned}
L_0 := {\left\{{(t, x_t) : t=0,\ldots, n}\right\}},\qquad
L_1 := {\left\{{(t+1, x_t) : t=0,\ldots, n}\right\}}
\end{aligned}$$ do not intersect. Moreover, $L_0$ and $T$ intersect only at the vertex $(0,x_0)$, and $L_1$ and $T$ do not intersect. Let ${\Gamma}$ be the union of $T$, $L_0$, and $L_1$. The edges of $T$, $L_0$, and $L_1$ all occur with positive probability in ${\mathbf{G}}$, and none of them have the same initial vertex, so that in fact they are comprised of independent edges in ${\mathbf{G}}$. Since ${\Gamma}$ has only a finite number of edges, it follows that ${\Gamma}\subseteq {\mathbf{G}}$ occurs with positive probability. Moreover, when ${\Gamma}\subseteq {\mathbf{G}}$ occurs, the vertex $(n,x_{n-1}) \in V({\mathbf{B}}(x))$ for all $x \in S$, but it is not on the bi-infinite path. This is because, by construction, $(0,x_0)$ is on the bi-infinite path in ${\mathbf{G}}$ and therefore $L_0$ makes up a segment of the bi-infinite path in ${\mathbf{G}}$. But, $L_1$ includes a representative for every state, so for every $x \in S$ there is an $s \in {\mathbb{Z}}$ such that $x \in
D^{(n,x_{n-1})}_s$. Finally, $V(L_0) \cap V(L_1) = \emptyset$ so $(n,x_{n-1})$ is not on the bi-infinite path in ${\mathbf{G}}$.
### Other I/F Component Properties {#sec:other-i-f-properties}
The existence and uniqueness of a bi-infinite path in each $\Phi$-component of a network is one I/F property that was studied at length in , which centered around bi-recurrent paths in ${\mathbf{B}}$. However, there are many other potential things to say about ${\mathbf{B}}$ following from its I/F structure. A few of them are discussed in this brief section.
The first is the general structure of a network with only I/F components. Each component of ${\mathbf{B}}$ contains a unique bi-infinite path. Points on a bi-infinite path are sometimes referred to as [**immortals**]{} due to the fact that they do not disappear after an infinite number of applications of the follow vertex-shift $\Phi$. A component [**evaporates**]{} if each point disappears after a finite number (depending on the point) of applications of $\Phi$. Thus, in the case of ${\mathbf{B}}$, none of the components evaporate. [**Mortals**]{} are those points in $V({\mathbf{B}})$ that do disappear after a finite number of applications of $\Phi$, i.e. those that have only finitely many descendants. Each component of ${\mathbf{B}}$ contains a bi-infinite path of immortals, and each immortal has exactly one child who is immortal. Thus the immortals within a component are ordered like ${\mathbb{Z}}$ in a shift-covariant way. Hanging off of each immortal is then a (possibly empty) tree of mortals, the descendants of the immortal who are not themselves immortal and whose closest immortal ancestor is the given immortal. With this viewpoint, each component of ${\mathbf{B}}$ can be seen as a shift-covariant bi-infinite sequence of finite rooted trees, where each immortal is the root of its tree. If there is only one component of ${\mathbf{B}}$, then it has already been noted that there is a unique bi-infinite path in ${\mathbf{B}}$ whose state path ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ is stationary. However, more can be said in this case. If there is only one component of ${\mathbf{B}}$, then in fact the whole sequence ${{({[Q_t, (t,\beta_t)]})}}_{t \in {\mathbb{Z}}}$ is stationary, where $Q_t$ is the tree hanging from the immortal $(t,\beta_t)$. It is important here that the isomorphism class of $Q_t$ is used and each vertex $(t,y) \in V({\mathbf{B}})$ is marked with $(y,\xi_t)$, otherwise the sequence would not be stationary due to the strictly increasing time coordinate. This view of ${\mathbf{B}}$ as a joining of trees gives an alternative way of looking at ${\mathbf{B}}$ compared to the view of ${\mathbf{B}}$ as a union of bridges between $x^*$ at different times. Yet another viewpoint is that of ${\mathbf{B}}$ as a sequence of vertical slices. This idea has already been explored slightly in that the way the root was chosen in the definition of the unimodular measure ${{\mathbf{P}}^{\mathsmaller{\square}}}$ is by choosing a root from one of these vertical slices. The view of ${\mathbf{B}}$ as a sequence of vertical slices is explored more in and is the main topic of .
Additionally, the list of mass transports given in the appendix gives some integrability results relating these three viewpoints. In particular, in each way of viewing ${\mathbf{B}}$ there is a natural way to split ${\mathbf{B}}$ into pieces. In the view of ${\mathbf{B}}$ as a joining of a sequence of trees of mortals hanging off an immortal, the vertices are partitioned by which tree they are in. In the view of ${\mathbf{B}}$ as a sequence of vertical slices, the vertices are partitioned by which slice they are in. In the view of ${\mathbf{B}}$ as paths started from state $x^*$, vertices are partitioned by the time they first return to $x^*$. In fact, the mass transport arguments given in the appendix show that the mean number of vertices in a partition element is the same for all three viewpoints. See the list of mass transports in the appendix for a more detailed description of these results and other finer-grained results.
Applications to Simulating the Bridge Graph {#sec:simulation-applications}
-------------------------------------------
### Local Weak Convergence to the Bridge Graph {#sec:local-weak-convergence}
It was shown in that the measure ${{\mathbf{P}}^{\mathsmaller{\square}}}$ may be thought of as an appropriately size-biased version of a network with the root picked uniformly at random from individuals at time $0$. A common reason for size-biasing to show up is when picking uniformly at random across a population and asking the size of the group an individual is in. Picking uniformly at random is what unimodularity models, so one might expect that a unimodular network can be approximated by picking the root uniformly at random from a very large but finite sub-network. At present, whether all unimodular networks can be approximated in this way is an open problem [@aldous2007processes]. In the case of the unimodular [bridge [[EFF]{}]{}]{}, it will be shown directly that indeed it can be approximated by finite sub-networks with a root picked uniformly at random.
In this section, different ways of approximating the unimodular version of ${\mathbf{B}}$ by finite subgraphs are considered. Recall that $\overline{{\mathbf{B}}}$ denotes ${\mathbf{B}}$ with spine added, i.e. ${\mathbf{B}}$ with edges connecting each $(t,x^*)$ to $(t+1,x^*)$. For a finite interval $I \subseteq {\mathbb{Z}}$ define $V_{I}({\mathbf{B}}) := \cup_{t \in I} V_t({\mathbf{B}})$ and let $\overline{{\mathbf{B}}} \cap I$ denote the subgraph of $\overline{{\mathbf{B}}}$ induced by $V_{I}({\mathbf{B}})$. Also define $V'_{I}({\mathbf{B}}) := \cup_{t \in I} {\left\{{(s,F^{(t,x^*)}_s): t {\leqslant}s {\leqslant}\sup I}\right\}}$ to be the vertices of ${\mathbf{B}}$ obtained by simulating paths starting from $x^*$ within the time window $I$, and let $\overline{{\mathbf{B}}} \sqcap I$ denote the graph it induces in $\overline{{\mathbf{B}}}$. Two ways of approximating ${\mathbf{B}}$ are then as follows:
1. Restrict to $[-n,0]$ and pick a uniform root in $V_{[-n,0]}({\mathbf{B}})$.
2. Simulate paths starting from $x^*$ in the window $[-n, n]$, which gives the vertices of $\overline{{\mathbf{B}}} \sqcap [-n,n] \subseteq \overline{{\mathbf{B}}}$, then pick a uniform root in $V'_{[0,n]}({\mathbf{B}})$.
After choosing a large viewing window $I$, a vertex picked at random will not likely be near the edge of this window, so the effects of throwing away all but this finite window can be controlled. However, the first method involves perfect knowledge of some finite window of ${\mathbf{B}}$. Practically speaking, when $S$ is infinite, one does not have a way to be sure that one has computed all of ${\mathbf{B}}$ in a finite window, as the only tool available is to simulate sample paths starting from different locations. This is the motivation for the second method of picking a root. For, even if the edge effects caused by only viewing simulations of paths in ${\mathbf{B}}$ from $-n$ to $n$ cannot be controlled, the edge effects from $0$ to $n$ can be controlled using the information from simulating from $-n$ to $n$. It will be shown shortly that both of these methods enjoy convergence in the local weak sense to the measure ${{\mathbf{P}}^{\mathsmaller{\square}}}$.
\[lem:ergodicEFTavg\] For any strictly increasing sequence of finite intervals ${{({I_n})}}_{n \in {\mathbb{N}}}$ in ${\mathbb{Z}}$, and any function $g \in L^1({{\mathbf{P}}^{\mathsmaller{\square}}})$, one has $$\begin{aligned}
\frac{1}{\#I_n{\mathbf{E}}[\#{\mathbf{B}}_0] } \sum_{v \in V_{I_n}({\mathbf{B}})} g[\overline{{\mathbf{B}}}, v] \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g]
\end{aligned}$$ and $$\begin{aligned}
\frac{1}{\#V_{I_n}({\mathbf{B}})} \sum_{v \in V_{I_n}({\mathbf{B}})} g[\overline{{\mathbf{B}}}, v] \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g],
\end{aligned}$$ where both convergences happen ${\mathbf{P}}$-a.s. as $n \to \infty$. In particular $$\begin{aligned}
\frac{\#V_{I_n}({\mathbf{B}})}{{\mathbf{E}}[\#V_{I_n}({\mathbf{B}})]}
=\frac{\#V_{I_n}({\mathbf{B}})}{\#I_n{\mathbf{E}}[\#{\mathbf{B}}_0]} \to 1.
\end{aligned}$$
Assume without loss that $\Omega = \Xi^{\mathbb{Z}}$ is the canonical space and ${{({\theta_t})}}_{t \in {\mathbb{Z}}}$ is the family of shift operators defined by $\theta_t({{({\xi_s})}}_{s \in {\mathbb{Z}}}) = {{({\xi_{t+s}})}}_{s \in {\mathbb{Z}}}$. Both statements follow from rewriting $$\begin{aligned}
\sum_{v \in V_{I_n}({\mathbf{B}})} g[\overline{{\mathbf{B}}},v]
= \sum_{t \in I_n} \left(\sum_{x \in {\mathbf{B}}_t} g[\overline{{\mathbf{B}}}, (t,x)]\right)
= \sum_{t \in I_n} g_0 \circ \theta_t,
\end{aligned}$$ where $g_0 := \sum_{x \in {\mathbf{B}}_0} g[\overline{{\mathbf{B}}}, (0,x)]$. The pointwise ergodic theorem for amenable groups (cf. [@lindenstrauss2001pointwise]) then proves the claim.
\[prop:localweakconvergence\] Fix any strictly increasing sequence of finite intervals ${{({I_n})}}_{n \in {\mathbb{N}}}$ in ${\mathbb{Z}}$, and for each $n \in {\mathbb{N}}$, let ${\boldsymbol{o}}_n$ be, conditionally on $V_{I_n}({\mathbf{B}})$, uniformly distributed on $V_{I_n}({\mathbf{B}})$ and independent of $\overline{{\mathbf{B}}} \cap I_n$ (including its marks). Then for all bounded measurable $g:\mathcal{G}_*\to {\mathbb{R}}_{{\geqslant}0}$ depending only on vertices at some bounded distance to the root, one has $$\begin{aligned}
\frac{1}{\# V_{I_n}({\mathbf{B}})}\sum_{v \in V_{I_n}({\mathbf{B}})}g[\overline{{\mathbf{B}}}\cap{I_n}, v] \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g],\qquad {\mathbf{P}}\text{-a.s.}
\end{aligned}$$ as $n \to \infty$. In particular, $$\begin{aligned}
{\mathbf{P}}([\overline{{\mathbf{B}}} \cap I_n, {\boldsymbol{o}}_n] \in \cdot) \to {{\mathbf{P}}^{\mathsmaller{\square}}}, \qquad n \to \infty
\end{aligned}$$ in the sense of local weak convergence.
Fix $N\in {\mathbb{N}}$ and let $g:\mathcal{G}_* \to {\mathbb{R}}_{{\geqslant}0}$ measurable, bounded, and such that $g$ depends only on vertices at graph distance at most $N$ from the root. One has $$\begin{aligned}
&{\mathbf{E}}[g[\overline{{\mathbf{B}}}\cap{I_n}, {\boldsymbol{o}}_n]]\\
&= {\mathbf{E}}[{\mathbf{E}}[g[\overline{{\mathbf{B}}}\cap{I_n}, {\boldsymbol{o}}_n] \mid V_{I_n}({\mathbf{B}})]] \\
&= {\mathbf{E}}\left[\frac{1}{\# V_{I_n}({\mathbf{B}})}\sum_{v \in V_{I_n}({\mathbf{B}})}g[\overline{{\mathbf{B}}}\cap{I_n}, v]\right] \\
&= {\mathbf{E}}\left[\left(\frac{{\mathbf{E}}[\# V_{I_n}({\mathbf{B}})]}{\# V_{I_n}({\mathbf{B}})}\right) \left(\frac{1}{{\mathbf{E}}[\#V_{I_n}({\mathbf{B}})]}\sum_{v \in V_{I_n}({\mathbf{B}})}g[\overline{{\mathbf{B}}}\cap{I_n}, v]\right)\right].
\end{aligned}$$ Call the two parenthesized expressions in the previous expectation $a_n$ and $b_n$ respectively, then it will be shown that $a_n b_n \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g]$ a.s., from which it also follows that ${\mathbf{E}}[a_n b_n] \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g]$ by dominated convergence. This will prove the claims. By stationarity and linearity of expectation, for each $n \in {\mathbb{N}}$, $$\begin{aligned}
{{\mathbf{E}}^{\mathsmaller{\square}}}[g]
=\frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_{0}]}{\mathbf{E}}\left[\sum_{v \in {\mathbf{B}}_{0}}g[\overline{{\mathbf{B}}}, v]\right]
={\mathbf{E}}\left[\frac{1}{{\mathbf{E}}[\#V_{I_n}({\mathbf{B}})]}\sum_{v \in V_{I_n}({\mathbf{B}})}g[\overline{{\mathbf{B}}}, v]\right].
\end{aligned}$$ Call the inside of the last expectation $c_n$. Letting ${[{\Gamma}, o]}_N$ denote the neighborhood of size $N$ around $o$ in a network ${\Gamma}$, for all $n > N$ $$\begin{aligned}
&\left\vert b_n - c_n \right\vert \\
&{\leqslant}\frac{1}{\#I_n{\mathbf{E}}[\#{\mathbf{B}}_{0}]}\sum_{v \in V_{I_n}({\mathbf{B}})}\left\vert g[\overline{{\mathbf{B}}}\cap{I_n}, v]-g[\overline{{\mathbf{B}}}, v]\right\vert\\
&{\leqslant}\frac{2\Vert g\Vert_\infty}{\#I_n{\mathbf{E}}[\#{\mathbf{B}}_{0}]}\#{\left\{{v \in V_{I_n}({\mathbf{B}}) : {[\overline{{\mathbf{B}}}\cap{I_n},v]}_N \neq {[\overline{{\mathbf{B}}},v]}_N}\right\}}\\
&{\leqslant}\frac{2\Vert g\Vert_\infty}{\#I_n{\mathbf{E}}[\#{\mathbf{B}}_{0}]}\left(\sum_{k = \min I_n}^{\min I_n +N} \#{\mathbf{B}}_{k} + \sum_{k=\max I_n-N}^{\max I_n} \#{\mathbf{B}}_{k}\right)\\
&{\leqslant}\frac{2\Vert g\Vert_\infty}{\#I_n{\mathbf{E}}[\#{\mathbf{B}}_{0}]}\left(\sum_{k \in I_n} \#{\mathbf{B}}_{k} - \sum_{k=\min I_n+ N}^{\max I_n - N} \#{\mathbf{B}}_{k}\right)\\
&\to 2\Vert g \Vert_\infty - 2\Vert g\Vert_\infty\\
&= 0
\end{aligned}$$ as $n\to \infty$, ${\mathbf{P}}$-a.s., by . But also $c_n \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g]$ and $a_n \to 1$, ${\mathbf{P}}$-a.s., also by . Hence $a_n b_n \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g]$, ${\mathbf{P}}$-a.s., as claimed.
\[prop:localweakconvergenceofsimulationeasy\] Fix any increasing sequence of finite intervals ${{({I_n})}}_{n \in {\mathbb{N}}} = {{({[-a_n, b_n]})}}_{n \in {\mathbb{N}}}$ in ${\mathbb{Z}}$ containing $0$ with $a_n\to \infty$ and $b_n$ strictly increasing. For each $n \in {\mathbb{N}}$, let ${\boldsymbol{o}}'_n$ be, conditionally on $V'_{I_n}({\mathbf{B}})$, uniformly distributed on $V'_{[0,b_n]}({\mathbf{B}})$ and independent of $\overline{{\mathbf{B}}} \sqcap I_n$ (including its marks). Then for all bounded measurable $g:\mathcal{G}_*\to {\mathbb{R}}_{{\geqslant}0}$ depending only on vertices at some bounded distance to the root, one has $$\begin{aligned}
\frac{1}{\# (V'_{I_n}({\mathbf{B}}) \cap{[0,b_n]})}\sum_{v \in V'_{I_n}({\mathbf{B}}) \cap{[0,b_n]}}g[\overline{{\mathbf{B}}}\sqcap I_n, v] \to {{\mathbf{E}}^{\mathsmaller{\square}}}[g],\qquad {\mathbf{P}}\text{-a.s.}
\end{aligned}$$ In particular, $$\begin{aligned}
{\mathbf{P}}([\overline{{\mathbf{B}}} \sqcap I_n, {\boldsymbol{o}}_n'] \in \cdot) \to {{\mathbf{P}}^{\mathsmaller{\square}}}, \qquad n \to \infty
\end{aligned}$$ in the sense of local weak convergence.
Fix $N\in {\mathbb{N}}$ and let $g:\mathcal{G}_* \to {\mathbb{R}}_{{\geqslant}0}$ measurable, bounded, and such that $g$ depends only on vertices at graph distance at most $N$ from the root. The finiteness of ${\mathbf{B}}_0$ implies that one has that ${[\overline{{\mathbf{B}}} \sqcap I_n,v]}_N = {[\overline{{\mathbf{B}}} \cap I_n, v]}_N = {[\overline{{\mathbf{B}}}, v]}_N$ eventually as $n \to \infty$ for all $v \in V_{0}({\mathbf{B}})$, and hence for all $v \in V_{I_n}({\mathbf{B}}) \cap {[0,b_n-N]}$ eventually as $n\to \infty$ as well. For the same reason $V'_{I_n}({\mathbf{B}}) \cap [0,b_n] = V_{[0,b_n]}({\mathbf{B}})$ eventually as $n \to \infty$ as well. It follows that eventually $$\begin{aligned}
&\frac{1}{\#( V'_{I_n}({\mathbf{B}}) \cap [0,b_n])}\sum_{v \in V'_{I_n}({\mathbf{B}})\cap[0,b_n]}g[\overline{{\mathbf{B}}}\sqcap I_n, v]\\
&=\frac{1}{\#V_{[0,b_n]}({\mathbf{B}})}\sum_{v \in V_{[0,b_n]}({\mathbf{B}})}g[\overline{{\mathbf{B}}}\cap I_n, v] \\
&+ \frac{1}{\#V_{[0,b_n]}({\mathbf{B}})}\sum_{v \in V_{[b_n-N+1, b_n]}({\mathbf{B}})}(g[\overline{{\mathbf{B}}} \sqcap I_n, v] - g[\overline{{\mathbf{B}}} \cap I_n, v]).
\end{aligned}$$ Of the last two terms, $\frac{1}{\#V_{[0,b_n]}({\mathbf{B}})}\sum_{v \in V_{[0,b_n]}({\mathbf{B}})}g[\overline{{\mathbf{B}}}\cap I_n, v]\to {{\mathbf{E}}^{\mathsmaller{\square}}}[g]$ by , so it suffices to show that the last term goes to $0$. Indeed, $$\begin{aligned}
&\left|\frac{1}{\#V_{[0,b_n]}({\mathbf{B}})}\sum_{v \in V_{[b_n-N+1, b_n]}({\mathbf{B}})}(g[\overline{{\mathbf{B}}} \sqcap I_n, v] - g[\overline{{\mathbf{B}}} \cap I_n, v])\right|\\
&{\leqslant}\frac{2\Vert g \Vert_\infty \# V_{[b_n-N+1,b_n]}({\mathbf{B}}) }{\#V_{[0,b_n]}({\mathbf{B}})}\\
&= \frac{2\Vert g \Vert_\infty (\# V_{[0,b_n]}({\mathbf{B}}) - \#V_{[0,b_n-N]}({\mathbf{B}})) }{\#V_{[0,b_n]}({\mathbf{B}})}\\
&\to 2\Vert g \Vert_\infty(1 - 1)\\
&= 0
\end{aligned}$$ as desired.
### Renewal Structure of the Bridge Graph {#sec:renewal-structure-of-mbefts}
In this section, the driving sequence $\xi$ is assumed to be i.i.d., i.e. ${\mathbf{G}}$ is Markovian. One may ask whether the bridge graph ${\mathbf{B}}$ admits any kind of renewal structure. Is it possible that ${\mathbf{B}}_t$ contains only one state? This is not necessarily possible. Indeed, if $p_{x,x} = 0 $, then ${\mathbf{B}}_t$ contains at least two states for every $t \in {\mathbb{Z}}$. It is true, though, that ${\mathbf{B}}_t$ is infinitely often equal to any set that it has positive probability of being equal to. Let $S_{{\mathbf{B}}}$ denote the [**possible configurations**]{} of ${\mathbf{B}}_0$, i.e. $S_{{\mathbf{B}}} := {\left\{{E \subseteq S : {\mathbf{P}}({\mathbf{B}}_0=E)>0}\right\}}$. By , $S_{{\mathbf{B}}}$ consists only of finite subsets of $S$ and is therefore countable.
\[lem:general-regeneration-points\] For any subset $E \in S_{{\mathbf{B}}}$, the set of $t$ for which ${\mathbf{B}}_t=E$ forms a simple stationary point process $\Psi_E$ on ${\mathbb{Z}}$ with ${\mathbf{P}}(\Psi_E({\mathbb{Z}})=\infty)=1$ and intensity $\lambda_E ={\mathbf{P}}({\mathbf{B}}_0=E)$. In particular, ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ is bi-recurrent for each $E \in S_{\mathbf{B}}$.
For $E \in S_{\mathbf{B}}$, the event that there is a $t$ such that ${\mathbf{B}}_t=E$ is shift-invariant and has positive probability. Therefore it happens almost surely. The set of such $t$ is shift-covariant and therefore determines a simple stationary point process $\Psi_E$. The previous line implies that $\Psi_E$ contains at least one point, and therefore infinitely many a.s. One calculates $\lambda_E = {\mathbf{E}}[\Psi_E({\left\{{0}\right\}})] = {\mathbf{E}}[1_{{\left\{{{\mathbf{B}}_0=E}\right\}}}]$, completing the proof.
Moreover, ruling out obvious hurdles to ${\mathbf{B}}_t$ being a singleton is sufficient.
\[lem:regeneration-points\] Suppose ${\mathbf{G}}$ is an [[EFT]{}]{} and has fully independent transitions. Assume that $p_{x^*,x^*}>0$. Then ${\left\{{x^*}\right\}} \in S_{{\mathbf{B}}}$.
By , $\#{\mathbf{B}}_t$ is a.s. finite for each $t \in {\mathbb{Z}}$, and thus it is possible to choose $x_1,\ldots,x_n \in S$ such that ${\mathbf{P}}({\mathbf{B}}_0 = {\left\{{x_1,\ldots,x_n}\right\}})>0$. Since ${\mathbf{G}}$ is an [[EFT]{}]{}, choose a tree $T\subseteq {\mathbb{Z}}\times S$ with leaves $(0,x_1),\ldots,(0,x_n)$ and root $(t,x^*)$ for some $t>0$ such that ${\mathbf{P}}(T \subseteq {\mathbf{G}})>0$. With $[t] := {\left\{{0,\ldots,t}\right\}}$, let $I:={\left\{{s \in [t]: x^* \notin T_s}\right\}}$. Then $$\begin{aligned}
{\mathbf{P}}({\mathbf{B}}_t = {\left\{{x^*}\right\}})
&{\geqslant}{\mathbf{P}}({\mathbf{B}}_0 = {\left\{{x_1,\ldots,x_n}\right\}}, T \subseteq {\mathbf{G}}, F^{(s,x^*)}_{s+1}=x^*, \forall s \in I)\\
&{\geqslant}{\mathbf{P}}({\mathbf{B}}_0 = {\left\{{x_1,\ldots,x_n}\right\}}){\mathbf{P}}(T \subseteq {\mathbf{G}}) {\mathbf{P}}(F^{(s,x^*)}_{s+1}=x^*, \forall s \in I)\\
&= {\mathbf{P}}({\mathbf{B}}_0 = {\left\{{x_1,\ldots,x_n}\right\}}){\mathbf{P}}(T \subseteq {\mathbf{G}}) {(p_{x,x})}^{\#I}\\
&> 0.
\end{aligned}$$ To justify the use of independence in the previous calculation, note that ${\mathbf{B}}_0$ is ${{({\xi_{s}})}}_{s<0}$-measurable, whereas the events ${\left\{{T\subseteq {\mathbf{G}}}\right\}}$ and ${\left\{{F^{(s,x^*)}_{s+1}=x^*, \forall s \in I}\right\}}$ are ${{({\xi_{s}})}}_{s {\geqslant}0}$-measurable, so the first is independent of the second two. Then the second is independent of the third because, by construction, they involve disjoint sets of edges in ${\mathbf{G}}$.
Now it is possible to see the renewal structure in ${\mathbf{B}}$. Namely, ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ is itself an irreducible, aperiodic, and positive recurrent Markov chain under certain conditions.
\[prop:Bx-renewal-structure\] One has that ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ is a Markov chain on $S_{{\mathbf{B}}}$. Additionally, ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ is stationary and bi-recurrent for every $E \in S_{{\mathbf{B}}}$. Its transition matrix $P_{{\mathbf{B}}}$ is irreducible and positive recurrent. If ${\mathbf{G}}$ is an ${{EFT}}{}$ with fully independent transitions and $p_{x^*,x^*}>0$, then ${\left\{{x^*}\right\}} \in S_{\mathbf{B}}$ and $P_{{\mathbf{B}}}({\left\{{x^*}\right\}},{\left\{{x^*}\right\}})>0$ so $P_{{\mathbf{B}}}$ is aperiodic as well.
By , $\#{\mathbf{B}}_t$ is a.s. finite for each $t \in {\mathbb{Z}}$. Moreover, ${\mathbf{B}}_{t+1} = {\left\{{x^*}\right\}} \cup {\left\{{F^{(t,y)}_{t+1}:y\in {\mathbf{B}}_t}\right\}}$, so indeed ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ is a Markov chain on the finite subsets of $S$ since, for each $t \in {\mathbb{Z}}$, ${\mathbf{B}}_{t+1}$ is a function of ${\mathbf{B}}_t$ and $\xi_t$. Here the running assumption that $\xi$ is i.i.d. is used. By , ${{({{\mathbf{B}}_t})}}_{t\in{\mathbb{Z}}}$ is bi-recurrent for every state $E\subseteq S$ such that ${\mathbf{P}}({\mathbf{B}}_0=E)>0$. In particular, the chain must be irreducible on $S_{{\mathbf{B}}}$, else a return to some state $E_1$ could not occur after a return to another state $E_2$ for some $E_1,E_2$ that do not communicate. Since ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ is shift-covariant it is stationary. The existence of a positive stationary distribution (the law of ${\mathbf{B}}_0$) for the irreducible $P_{{\mathbf{B}}}$ implies $P_{{\mathbf{B}}}$ is positive recurrent. If ${\mathbf{G}}$ is an [[EFT]{}]{} with fully independent transitions, then shows that ${\left\{{x^*}\right\}} \in S_{{\mathbf{B}}}$. Then $p_{x^*,x^*}>0$ implies $P_{{\mathbf{B}}}({\left\{{x^*}\right\}},{\left\{{x^*}\right\}})>0$ as well, so $P_{{\mathbf{B}}}$ is also aperiodic in that case.
It is possible that the $S_{{\mathbf{B}}}$ is strictly smaller than the set of all finite subsets of $S$ containing $x^*$.
Consider $S := {\left\{{0,1,2}\right\}}$ and $x^*:= 0$ with $p_{0,0} = p_{0,1} = p_{0,2} = \frac13$ and $p_{1,0} = p_{2,0} = 1$. That is, from $0$ make a uniform choice of where to jump, and from $1$ and $2$ deterministically return to $0$. Fix $t \in {\mathbb{Z}}$. In this case, if $1 \in {\mathbf{B}}_t$, it must be that $F^{(t-1,0)}_t = 1$. Similarly, if $2 \in {\mathbf{B}}_t$, it must be that $F^{(t-1,0)}_t = 2$. Thus it cannot be that both $1,2 \in {\mathbf{B}}_t$, and hence ${\left\{{0,1,2}\right\}} \notin S_{{\mathbf{B}}}$.
However, if *every* state has a chance to be lazy, then $S_{{\mathbf{B}}}$ does turn out to be the set of all finite subsets of $S$ containing $x$.
\[prop:lazy-S-B-is-everything\] Suppose ${\mathbf{G}}$ has fully independent transitions, $P$ is irreducible, and $p_{y,y}>0$ for all $y \in S$. Then ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ is an irreducible, aperiodic, positive recurrent, and stationary Markov chain on the set of all finite subsets of $S$ containing $x^*$.
The assumptions imply that, in fact, $P$ is irreducible, aperiodic, and positive recurrent (since $x^*$ is always assumed positive recurrent), so implies that the only item left to show is that $S_{{\mathbf{B}}}$ contains all finite subsets of $S$ containing $x^*$. Let a finite set $E$ containing $x^*$ be given. Call $(y_1,\ldots,y_n)$ with each $y_i \in S$ a [**possible path**]{} if $\prod_{i=1}^{n-1} p_{y_i,y_{i+1}} > 0$. For the rest of the proof, all paths considered are possible paths. One would like to simply draw a path from $x^*$ to each $y \in S$ where after a path reaches its destination it becomes constant while it waits for the other paths to finish. This approach is slightly flawed because it may be that, for instance, every path from $x^*$ to $z$ passes through $y$. In this case, one must draw the path from $x^*$ to $y$ before the path from $x^*$ to $z$, otherwise the resulting graph would have a vertex with multiple outgoing edges, which is an impossibility in ${\mathbf{G}}$. However, the approach will work as long as it is possible to draw the paths in an order such that no interference occurs.
{width="\linewidth"}
Define a partial order $\prec_0$ on $E$ by saying $y \prec_0 z$ if all paths from $x^*$ to $z$ pass through $y$ with the convention that the trivial path $(x^*)$ does not pass through $x^*$ (to prohibit $x^* \prec_0 x^*$). Since $E$ is finite, there is a $\prec_0$-maximal element $x_0 \in E$. That is, for all $y \in E$ there is a path from $x^*$ to $y$ that does not hit $x_0$. Choose a path $L_0$ from $x^*$ to $x_0$. With $\prec_n$, $x_0,\ldots,x_n$, and $L_0,\ldots,L_{n}$ defined, as long as $E\setminus {\left\{{x_0,\ldots,x_n}\right\}} \neq \emptyset$, recursively define $\prec_{n+1}$, $x_{n+1}$, and $L_{n+1}$ as follows. By construction, for all $y \in E \setminus {\left\{{x_0,\ldots,x_n}\right\}}$, there is a path from $x^*$ to $y$ that avoids $x_0,\ldots,x_n$. Define $\prec_{n+1}$ on $E\setminus{\left\{{x_0,\ldots,x_n}\right\}}$ by saying $y \prec z$ if all paths from $x^*$ to $z$ avoiding $x_0,\ldots,x_n$ pass through $y$. Then it is possible to choose a $\prec_{n+1}$-maximal element $x_{n+1}$, i.e. for all $y \in E\setminus{\left\{{x_0,\ldots,x_{n+1}}\right\}}$, there is a path from $x^*$ to $y$ that does not pass through any of $x_0,\ldots,x_{n+1}$. Also choose $L_{n+1}$ a path from $x^*$ to $x_{n+1}$ avoiding $x_0,\ldots,x_{n}$. Necessarily the recursion terminates when $n=\# E - 1$. It is now possible to construct a graph ${\Gamma}\subseteq {\mathbb{Z}}\times S$ with ${\mathbf{P}}({\mathbf{B}}_0={\left\{{x^*}\right\}}, {\Gamma}\subseteq {\mathbf{G}}) > 0$ and when ${\Gamma}\subseteq {\mathbf{G}}$ and ${\mathbf{B}}_0 = {\left\{{x^*}\right\}}$, one has ${\mathbf{B}}_t = E$ for some $t$. Let $t_i$ be the sum of the lengths of the paths $L_0,\ldots,L_{i-1}$ for each $0 {\leqslant}i {\leqslant}\#E$, with $t_0:=0$. Let ${\Gamma}$ be the graph that for each $i$ has:
1. \[it:e1\] a path from $(t_i,x)$ to $(t_{i+1},x_i)$ with state path $L_i$ from time $t_i$ to $t_{i+1}$,
2. \[it:e2\] a path started at $(t_{i+1},x_i)$ that stays constant at $x_i$ until time $t_{\#E}$, and
3. \[it:e3\] a (possibly trivial) path started at $(t_i+1,x)$ that stays constant at $x^*$ until time $t_{i+1}$.
Note that, by construction, ${\Gamma}$ is a finite graph that is the union of edges that occur with positive probability. Moreover, the connected components of ${\Gamma}$ are formed from points \[it:e1,it:e2\] for some $i$ and \[it:e3\] from $i-1$. Whenever ${\Gamma}\subseteq {\mathbf{G}}$ and ${\mathbf{B}}_0 = {\left\{{x^*}\right\}}$, one has ${\mathbf{B}}_{t_{\#E}}= E$. This occurs with positive probability since $p_{y,y} > 0$ for all $y \in S$.
\[prop:PBx-transition-probabilities-recurrence\] Suppose ${\mathbf{G}}$ has fully independent transitions. Extend the definition of $P_{\mathbf{B}}$ to $$\begin{aligned}
P_{{\mathbf{B}}}(E,E') := {\mathbf{P}}\left({\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E}\right\}} = E'\right),
\end{aligned}$$ for all finite $E, E' \subseteq S$ containing $x^*$. Then $P_{{\mathbf{B}}}$ satisfies the following recurrence: for $E = {\left\{{x^*, x_1,\ldots, x_n}\right\}}$ and $E' = {\left\{{x^*, y_1,\ldots, y_m}\right\}}$, $$\begin{aligned}
P_{{\mathbf{B}}}(E,E')
&= \left( p_{x_n,x^*} + \sum_{i=1}^m p_{x_n,y_i}\right)P_{{\mathbf{B}}}(E\setminus {\left\{{x_n}\right\}}, E')\nonumber\\
&+\sum_{i=1}^m p_{x_n,y_i}P_{{\mathbf{B}}}(E\setminus{\left\{{x_n}\right\}}, E' \setminus
{\left\{{y_i}\right\}}),
\end{aligned}$$ with recursive depth at most $n$ and base cases $$\begin{aligned}
\begin{cases}
P_{{\mathbf{B}}}(E,E') = 0, & \#E' > \#E+1\\
P_{{\mathbf{B}}}({\left\{{x^*}\right\}}, {\left\{{x^*,y}\right\}}) = p_{x^*,y},& y \in S\\
P_{{\mathbf{B}}}(E, {\left\{{x^*}\right\}}) = \prod_{y \in E} p_{y,x^*}.
\end{cases}
\end{aligned}$$
First one justifies the extension of the definition of $P_{\mathbf{B}}$ by noting that for $E,E' \in S_{{\mathbf{B}}}$ one has $$\begin{aligned}
P_{{\mathbf{B}}}(E,E') &= {\mathbf{P}}({\mathbf{B}}_1 = E' \mid {\mathbf{B}}_0 = E)\\
&={\mathbf{P}}({\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_{1}:y\in {\mathbf{B}}_0}\right\}} = E' \mid
{\mathbf{B}}_0 = E)\\
&={\mathbf{P}}({\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_{1}:y\in E}\right\}} = E' \mid
{\mathbf{B}}_0 = E)\\
&={\mathbf{P}}({\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_{1}:y\in E}\right\}} = E' ),
\end{aligned}$$ where the last equality follows from the fact that ${\mathbf{B}}_0$ is measurable with respect to ${{({\xi_{t}})}}_{t < 0}$, whereas $F^{(0,y)}_1$ is $\xi_{0}$-measurable for each $y \in S$. The base cases for $P_{{\mathbf{B}}}$ are immediate from the definition of $P_{{\mathbf{B}}}$ and the independence structure. To see the recurrence, suppose $E = {\left\{{x^*,x_1,\ldots,x_n}\right\}}$ and $E' = {\left\{{x^*,y_1,\ldots,y_m}\right\}}$ as above. Split $P_{{\mathbf{B}}}(E,E')$ depending on the value of $F^{(0,x_n)}_1=x^*$ or $F^{(0,x_n)}_1=y_i$, and on whether ${\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E\setminus {\left\{{x_n}\right\}}}\right\}} = E'$ still or ${\left\{{x^*}\right\}} \cup{\left\{{F^{(0,y)}_1: y \in E\setminus{\left\{{x_n}\right\}}}\right\}} = E'\setminus{\left\{{y_i}\right\}}$, $$\begin{aligned}
P_{{\mathbf{B}}}(E,E')
&={\mathbf{P}}\left(F^{(0,x_n)}_1=x^*,{\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E\setminus{\left\{{x_n}\right\}}}\right\}} = E'\right)\\
&+\sum_{i=1}^m {\mathbf{P}}\left(F^{(0,x_n)}_1=y_i,{\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E\setminus{\left\{{x_n}\right\}}}\right\}} = E'\right)\\
&+\sum_{i=1}^m {\mathbf{P}}\left(F^{(0,x_n)}_1=y_i,{\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E\setminus{\left\{{x_n}\right\}}}\right\}} = E'\setminus{\left\{{y_i}\right\}}\right)
\end{aligned}$$ which, since ${\mathbf{G}}$ has fully independent transitions, equals $$\begin{aligned}
&p_{x_n,x^*}{\mathbf{P}}\left({\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E\setminus{\left\{{x_n}\right\}}}\right\}} = E'\right)\\
+&\sum_{i=1}^m p_{x_n,y_i} {\mathbf{P}}\left({\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E\setminus{\left\{{x_n}\right\}}}\right\}} = E'\right)\\
+&\sum_{i=1}^m p_{x_n,y_i}{\mathbf{P}}\left({\left\{{x^*}\right\}} \cup {\left\{{F^{(0,y)}_1: y \in E\setminus{\left\{{x_n}\right\}}}\right\}} = E'\setminus{\left\{{y_i}\right\}}\right)
\end{aligned}$$ which simplifies to $$\begin{aligned}
\left(p_{x_n,x^*}+\sum_{i=1}^m p_{x_n,y_i}\right)P_{{\mathbf{B}}}(E\setminus{\left\{{x_n}\right\}},E')
+\sum_{i=1}^m p_{x_n,y_i}P_{{\mathbf{B}}}(E\setminus{\left\{{x_n}\right\}}, E'\setminus{\left\{{y_i}\right\}}),
\end{aligned}$$ showing the recurrence holds.
Finally, the recursive depth needed to fully compute $P_{{\mathbf{B}}}(E,E')$ is at most $n$ because each application of the recurrence removes an element from $E$.
By implementing the recurrence of in, e.g. Python, one may compute $P_{{\mathbf{B}}}$ explicitly. Then, given values for the $p_{x,y}$, one may compute the staitonary distribution $\pi_{\mathbf{B}}$ of $P_{\mathbf{B}}$. For example, with $S:={\left\{{0,1,2}\right\}}$ and $x^*:=0$, and $p_{x,y} = \frac{1}{3}$ for all $x,y \in S$, one has $$\begin{aligned}
\pi_{{\mathbf{B}}}
&= \begin{bmatrix}
\pi_{{\mathbf{B}}}({\left\{{0}\right\}}) & \pi_{{\mathbf{B}}}({\left\{{0,1}\right\}}) & \pi_{{\mathbf{B}}}({\left\{{0,2}\right\}}) & \pi_{{\mathbf{B}}}({\left\{{0,1,2}\right\}})
\end{bmatrix}\\
&= \begin{bmatrix}
\frac{17}{143} & \frac{45}{143} & \frac{45}{143} & \frac{36}{143}
\end{bmatrix}.
\end{aligned}$$
It is an open question whether, in the fully independent transitions case, there is a general closed form expression for $P_{\mathbf{B}}$ in terms of $P$ or for the stationary distribution $\pi_{\mathbf{B}}$ of $P_{\mathbf{B}}$ in terms of $P$ and $\pi$.
Bibliographical Comments
========================
While this work may be the first time the Doeblin graph ${\mathbf{G}}$ has been explicitly defined and studied in its own right, it is without doubt that most, if not all, who have worked on CFTP-related research have had this picture in mind. Rather, the novelty here lies in the consideration of the bridge graph ${\mathbf{B}}$. While, to the best of the authors’ knowledge, the bridge graph ${\mathbf{B}}$ has not previously been defined or studied, it is not without ties to other objects that have been previously studied.
The first occurrence of some form of the bridge graph appears in [@borovkov1992stochastically], where Borovkov and Foss consider a family of stochastically recursive sequences started at times $0,-1,-2,\dotsc$, all with the same initial condition, and they proved the existence (under suitable conditions) of a stationary version of the SRS. They defined three notions of coupling convergence and studied when coupling convergence to the stationary SRS occurs. Their notion of strong coupling convergence to the stationary SRS is akin to the condition that ${\mathbf{B}}$ is an [[EFT]{}]{}. That is, it is the condition that all paths in ${\mathbf{B}}$ eventually merge. It is conceivable that, in the [[EFT]{}]{} case, one could derive the existence of the bi-infinite path in ${\mathbf{B}}$ from the work in [@borovkov1992stochastically], though it is not clear whether Borovkov and Foss had this in mind, and they did not make any mention of the key bi-recurrence property used in the current paper to distinguish this bi-infinite path from the potential others in ${\mathbf{G}}$.
Another occurrence of a similar object to the bridge graph may be found in [@baccelli2018renewal] in the very special case of integer-valued renewal processes. The dynamics there are slightly different, where instead of specifying a whole process started from each time, one marks each time with the time of death of an individual who is born at that time. This is akin to marking each $t \in {\mathbb{Z}}$ by the return time $\tau^{(t,x^*)}(x^*)$ of $F^{(t,x^*)}$ to $x^*$, though in [@baccelli2018renewal] these times of death are assumed to be i.i.d., whereas in the present work they have intricate dependence due to the Doeblin-type coupling. The population process defined in [@baccelli2018renewal] is then similar in nature to the sequence of cardinalities of ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ as considered in . It is proved in [@baccelli2018renewal] that, under natural conditions, the population process is a stationary regenerative process with independent cycles. In the present work, the process ${{({{\mathbf{B}}_t})}}_{t \in {\mathbb{Z}}}$ was shown in to be an irreducible, aperiodic, and positive recurrent Markov chain under suitable conditions, which therefore also admits an i.i.d. cycle decomposition. The analysis of this special case and, in particular, the identification of the I/F structure of the components has been kept in mind throughout the development of the theory of Doeblin [[EFFs]{}]{}.
Appendix
========
Postponed Proofs
----------------
Proofs that were only sketched in the main text are collected in full detail here.
For each $t \in {\mathbb{Z}}$, let $\mu_t$ be the distribution of $X_t$. It is enough to show the existence of $(\Omega', {\mathcal{F}}', {\mathbf{P}}')$ on which there is a process $X':= {{({X'_t})}}_{t \in {\mathbb{Z}}}$ and some i.i.d. $\xi':= {{({\xi'_{t}})}}_{t\in {\mathbb{Z}}}$ such that
1. \[it:c1\] $X'_t \sim \mu_t$ for all $t \in {\mathbb{Z}}$,
2. \[it:c2\] $\xi_t' \sim \xi_t$ for all $t \in {\mathbb{Z}}$,
3. \[it:c3\] $X'_t$ is independent of ${{({\xi'_s})}}_{s {\geqslant}t}$ for all $t \in {\mathbb{Z}}$, and
4. \[it:c4\]$X'_{t+1} ={h_{\mathrm{gen}}}(X_t, \xi'_{t})$ for all $t \in {\mathbb{Z}}$.
and will imply the result. Note that \[it:c1,it:c2,it:c3,it:c4\] are sufficient to characterize the joint finite dimensional distributions of ${{({X'_t})}}_{t \in {\mathbb{Z}}}$ and ${{({\xi'_{t}})}}_{t \in {\mathbb{Z}}}$. To see this fix $t_0 {\leqslant}t_1$. The joint distribution of ${{({X'_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$ and ${{({\xi'_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$ is determined because, conditional on ${{({\xi'_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$, $X'_{t_0}$ is still distributed as $\mu_{t_0}$ by \[it:c1,it:c3\], and, conditional on both $X'_{t_0}$ and ${{({\xi'_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$, one has that ${{({X'_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$ is deterministic by \[it:c4\]. Thus it suffices to show that ${{({X'_t})}}_{t \in {\mathbb{Z}}}$ and ${{({\xi'_{t}})}}_{t \in {\mathbb{Z}}}$ satisfying \[it:c1,it:c2,it:c3,it:c4\] exist. Also note that \[it:c1,it:c2,it:c3,it:c4\] with $t_0 {\leqslant}t {\leqslant}t_1$ are sufficient for determining the joint distribution of ${{({X'_s})}}_{t_0 {\leqslant}s {\leqslant}t_1}$ and ${{({\xi'_{s}})}}_{t_0 {\leqslant}s {\leqslant}t_1}$.
The proof will proceed by the Kolmogorov extension theorem. Suppose, by extending $(\Omega, {\mathcal{F}}, {\mathbf{P}})$ if necessary, that ${{({X_t})}}_{t \in {\mathbb{Z}}}$ and $\xi$ are defined on the same space and are independent of each other. Consider for each $t \in {\mathbb{Z}}$, the state path $F^{(t,X_t)}$ in ${\mathbf{G}}$ started at $(t,X_t)$. Then for all $s,t \in {\mathbb{Z}}$ with $s {\leqslant}t$ and all $x \in S$, $$\begin{aligned}
{\mathbf{P}}(F^{(s,X_s)}_t = x)
&= \sum_{y \in S} {\mathbf{P}}(X_s=y, F^{(s,X_s)}_t=x) \\
&= \sum_{y \in S} \mu_s(y) P^{t-s}(y,x)\\
&= \mu_s P^{t-s}(x),
\end{aligned}$$ where in the previous line $P$ is treated as a transition kernel with powers $P^k$ ($k=0,1,2,\ldots$). Since ${{({X_t})}}_{t \in {\mathbb{Z}}}$ exists and is a Markov chain with transition matrix $P$, one has $$\begin{aligned}
\mu_s P^{t-s} = \mu_{r}P^{s-r}P^{t-s} = \mu_{r} P^{t-r} = \mu_t
\end{aligned}$$ for all $r {\leqslant}s {\leqslant}t$. Moreover, for all $s{\leqslant}t$, $F^{(s,X_s)}_t$ is $\sigma(X_s, {{({\xi_{t'}})}}_{s {\leqslant}t' < t})$-measurable, hence it is independent of ${{({\xi_{t'}})}}_{t' {\geqslant}t}$. Now fix $s_0, t_0, t_1 \in {\mathbb{Z}}$ with $s_0 {\leqslant}t_0 {\leqslant}t_1$ and consider the joint distribution of ${{({F^{(s_0,X_{s_0})}_t})}}_{t_0{\leqslant}t {\leqslant}t_1}$ and ${{({\xi_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$. One has $F^{(s_0, X_{s_0})}$ and $\xi$ satisfy
1. \[it:c1p\] $F^{(s_0,X_{s_0})}_t \sim \mu_t$ for all $t_0 {\leqslant}t {\leqslant}t_1$,
2. \[it:c2p\] $\xi \sim \xi$,
3. \[it:c3p\] $F^{(s_0,X_{s_0})}_t$ is independent of ${{({\xi_{t'}})}}_{t' {\geqslant}t}$ for all $t_0 {\leqslant}t {\leqslant}t_1$, and
4. \[it:c4p\] $F^{(s_0,X_{s_0})}_{t+1} = {h_{\mathrm{gen}}}(F^{(s_0,X_{s_0})}_t, \xi_{t})$ for all $t_0 {\leqslant}t$.
As mentioned before, \[it:c1p,it:c2p,it:c3p,it:c4p\] are sufficient to determine the joint distribution of ${{({F^{(s_0,X_{s_0})}_t})}}_{t_0 {\leqslant}t {\leqslant}t_1}$ and ${{({\xi_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$, so the joint distribution of ${{({F^{(s_0,X_{s_0})}_t})}}_{t_0 {\leqslant}t {\leqslant}t_1}$ and ${{({\xi_{t}})}}_{t_0 {\leqslant}t {\leqslant}t_1}$ does not depend on $s_0$ as long as $s_0 {\leqslant}t_0$. Thus a consistent set of finite dimensional distributions is determined by taking $s_0, t_0 \to -\infty$ and $t_1 \to \infty$ while maintaining $s_0 {\leqslant}t_0 {\leqslant}t_1$. It follows by the Kolmogorov extension theorem that there is a space $(\Omega', {\mathcal{F}}', {\mathbf{P}}')$ and processes $X'={{({X'_{t}})}}_{t \in {\mathbb{Z}}}$ and $\xi'={{({\xi'_{t}})}}_{t \in {\mathbb{Z}}}$ satisfying \[it:c1,it:c2,it:c3,it:c4\], completing the proof.
Call $P$ [**strongly recurrent**]{} if all its recurrent classes are positive recurrent and call $P$ [**recurrent-attracting**]{} if any Markov chain with transition matrix $P$ eventually enters a recurrent state. These conditions are both automatic if $P$ is irreducible and positive recurrent.
\[prop:indep-transitions-components\] Let $S = T \cup \bigcup{{({R^i})}}_{0 {\leqslant}i< N}$ decompose $S$ into its transient states and $N\in{\mathbb{N}}\cup{\left\{{\infty}\right\}}$ recurrent communication classes for $P$. Assume that $P$ is strongly recurrent and recurrent-attracting. Let $d(i)$ be the period of $R^i$, and let $R^i = C^i_0 \cup \cdots \cup C^i_{d(i)-1}$ be a cyclic decomposition. If ${\mathbf{G}}$ has fully independent transitions, then the components ${{({\mathcal{C}^i_j})}}_{0 {\leqslant}i < N, 0 {\leqslant}j < d(i)}$ of ${\mathbf{G}}$ are in bijection with ${{({C^i_j})}}_{0 {\leqslant}i < N, 0 {\leqslant}j < d(i)}$, and for $x \in C^i_j$, $(t,x) \in V(\mathcal{C}^i_{j'})$ if and only if $j-t=j' \pmod{d(i)}$, and for $x \in T$, $(t,x) \in V(\mathcal{C}^{i}_{j})$ where $(t',y) \in V(\mathcal{C}^{i}_{j})$ is any vertex on the path of $(t,x)$ for which $y$ is recurrent. That is, $\mathcal{C}^i_{j}$ is the set of all vertices of all paths in ${\mathbf{G}}$ that pass through an element of $C^i_j$ at any time $t = 0 \pmod{d(i)}$.
Fix $i,j,t$ and let $x,y \in C^i_j$. Then $P^{d(i)}$ restricted to $C^i_j$ is irreducible, aperiodic, and positive recurrent. Thus the product chain $P^{d(i)} \otimes P^{d(i)}$ restricted to $C^i_j \times C^i_j$ is too. Strictly before the hitting time to the diagonal, ${{({F^{(t,x)}_{t+ sd(i)}, F^{(t,y)}_{t+sd(i)}})}}_{s {\geqslant}0}$ is distributed the same as the product chain $P^{d(i)} \otimes P^{d(i)}$ on $C^i_j \times C^i_j$, and thus the hitting time to the diagonal is a.s. finite because the product chain is irreducible, aperiodic, and positive recurrent. It follows that $(t,x)$ and of $(t,y)$ are in the same component of ${\mathbf{G}}$. If $x \in C^i_j$ and $y \in C^{i'}_{j'}$ with $i' \neq i$, then $F^{(t,x)}$ and $F^{(t,y)}$ cannot merge because the states of $F^{(t,x)}$ are contained in $R^i$ and the states of $F^{(t,y)}$ is contained in $R^{i'}$. If $x \in C^i_j$ and $y \in C^i_{j'}$ with $j' \neq j \pmod{d(i)}$, then $F^{(t,x)}$ and $F^{(t,y)}$ cannot merge because $F^{(t,x)}_{t+s} \in C^i_{j+s}$ but $F^{(t,y)}_{t+s} \in C^i_{j'+s}$ with indices taken modulo $d(i)$ as necessary. Thus, the set of $y \in S\setminus T$ such that $F^{(t,x)}$ eventually merges with $F^{(t,y)}$ is precisely $C^i_j$. If $x \in C^i_j, y \in C^{i'}_{j'}$ and $t {\leqslant}t'$, then $F^{(t,x)}_{t'} \in C^i_{j+(t'-t)}$, so it follows that $F^{(t,x)}$ and $F^{(t',y)}$ eventually merge if and only if $i'=i$ and $j+(t'-t) = j' \pmod{d(i)}$, or equivalently $j'-t'= j-t \pmod{d(i)}$. It follows that for any $x,y \in S\setminus T$ and any $t,t' \in {\mathbb{Z}}$, the two vertices $(t,x),(t',y) \in V({\mathbf{G}})$ are in the same component of ${\mathbf{G}}$ if and only if there are $i,j,j'$ such that $x \in C^i_j, y \in C^i_{j'}$ and $j'-t' = j-t \pmod{d(i)}$. If $x \in C^i_j$, then $F^{(t,x)}_{s} \in C^i_{j-t+s}$ for all $s {\geqslant}t$. Thus $F^{(t,x)}_{s} \in C^i_{j-t}$ for all $s = 0 \pmod{d(i)}$ with $s{\geqslant}t$. Call $C^i_{j-t}$ the [**time-zero class of $(t,x)$**]{}. Then for $x\in C^i_j, y\in C^i_{j'}$ and $t,t' \in {\mathbb{Z}}$, the condition that $j'-t' = j-t \pmod{d(i)}$ is equivalent to the fact that $(t,x)$ and $(t,y)$ have the same time-zero class. Thus the components of ${\mathbf{G}}$ are exactly the equivalence classes of vertices in the same time-zero class, except possibly ignoring $(t,x)$ for transient $x$. By the assumption that $P$ is recurrent-attracting, if $x \in T$, then $F^{(t,x)}$ eventually hits some recurrent class and so does not form a new component of ${\mathbf{G}}$, and the path $F^{(t,x)}$ is in the component of the first (and every) $(t',y)$ it hits with $y$ recurrent.
Fix $k \in {\mathbb{N}}$. For every $v \in V={\mathbb{Z}}\times S$, the event that $v \in V({\boldsymbol{{\Gamma}}})$ and $d_{{\boldsymbol{{\Gamma}}}}({\boldsymbol{o}},v) {\leqslant}k$ is measurable. Indeed, there are at most countably many paths $(v_0,v_1,v_2,\ldots,v_n)$ in $V$ with $n{\leqslant}k$, and the desired event is the union over all such paths of any length $n {\leqslant}k$ ending at $v$ of the event $$\begin{aligned}
{\left\{{{\boldsymbol{o}}=v_0}\right\}} \cap \bigcap_{i=1}^n \left({\left\{{f_V(v_i)=1}\right\}} \cap {\left\{{f_E(v_{i-1},v_{i})=1}\right\}}\right).
\end{aligned}$$ From here one sees that event that the $r$-neighborhood around ${\boldsymbol{o}}$ is exactly some fixed finite graph ${\Gamma}$ is measurable. Indeed, $${\left\{{N_{{\boldsymbol{{\Gamma}}}}({\boldsymbol{o}},r) = {\Gamma}}\right\}}
= \bigcap_{v \in V} {\left\{{(f_V(v)=1 \text{ and } d_{{\boldsymbol{{\Gamma}}}}({\boldsymbol{o}},v) {\leqslant}r) \iff v \in V({\Gamma})}\right\}}.$$ Enhancing ${\Gamma}$ with marks $\xi_u, \xi_{v,w}$ for each $u \in V({\Gamma})$ and all ${\left\{{v,w}\right\}} \in E({\Gamma})$, for any $\epsilon > 0$ and $o \in V({\Gamma})$, one sees that the event $$\begin{aligned}
D_{r,\epsilon}({\Gamma},o) := &\{
{\boldsymbol{o}}=o,\\
&N_{{\boldsymbol{{\Gamma}}}}({\boldsymbol{o}},r) = {\Gamma},\\
&\forall u \in V({\Gamma}), d_{{\Xi_{\mathrm{univ}}}}(\xi_V(u), \xi_u) < \epsilon,\\
&\forall {\left\{{v,w}\right\}} \in E({\Gamma}), d_{{\Xi_{\mathrm{univ}}}}(\xi_E(v,w), \xi_{v,w})<\epsilon\}
\end{aligned}$$ is measurable. Since $V$ is countable and ${\Gamma}$ is a finite graph, there are at most countably many rooted isomorphic copies of $({\Gamma}, o)$ that can be made with vertices in $V$. It follows that the event ${\left\{{d_{\mathcal{G}_*}([{\boldsymbol{{\Gamma}}},{\boldsymbol{o}}], [{\Gamma},o]) < \epsilon}\right\}}$ is a countable union of the events $D_{\lceil \frac{1}{\epsilon} \rceil,\epsilon}(\rho({\Gamma},o))$ with $\rho$ ranging over the countable collection of such rooted isomorphisms of $({\Gamma}, o)$. Hence $\omega \mapsto [{\boldsymbol{{\Gamma}}}(\omega),{\boldsymbol{o}}(\omega)]$ is measurable.
First suppose that $X_0$ is independent of ${\mathbf{G}}$ and uniformly distributed on a finite $S$. Let $N$ be the cardinality of $S$. Let $g:\mathcal{G}_{**} \to {\mathbb{R}}_{{\geqslant}0}$ supported on directed neighbors be given. Then $$\begin{aligned}
{\mathbf{E}}\sum_{v \in V({\mathbf{G}})} g[{\mathbf{G}},(0,X_0), v]
&= \frac{1}{N}\sum_{x \in S}{\mathbf{E}}\sum_{y\in S} g[{\mathbf{G}},(0,x),(1,y)]\\
&= \frac{1}{N}\sum_{x,y\in S}{\mathbf{E}}[g[{\mathbf{G}},(0,x),(1,y)]]\\
&= \frac{1}{N}\sum_{x,y\in S}{\mathbf{E}}[g[{\mathbf{G}},(-1,x),(0,y)]]\\
&= \frac{1}{N}\sum_{y \in S}{\mathbf{E}}\sum_{x \in S} g[{\mathbf{G}},(-1,x),(0,y)]\\
&= {\mathbf{E}}\sum_{v \in V({\mathbf{G}})} g[{\mathbf{G}},v,(0,X_0)],
\end{aligned}$$ where in the third equality time-homogeneity of ${\mathbf{G}}$ is used. It follows that in this case ${\mathbf{G}}$ is unimodular.
Next suppose $[{\mathbf{G}}, (0,X_0)]$ is unimodular. Let $\eta$ be a vertex-shift that follows the arrows in ${\mathbf{G}}$. For example, define for each network ${\Gamma}$ and $u \in V({\Gamma})$ the vertex-shift by $\eta_{\Gamma}(u):= v$ if there is a unique outgoing edge from $u$ and this edge terminates at $v$, or $\eta_{\Gamma}(u):= u$ if this condition is not met for any $v$. Since ${\mathbf{G}}$ is connected, its $\eta$-foils are ${{({{\mathbf{G}}_t})}}_{t \in {\mathbb{Z}}}$. Let the mark of a vertex $v$ be denoted $(s(v), \xi(v))$, and let $v \sim w$ denote that $v$ and $w$ are in the same $\eta$-foil. Fix $x,y \in S$ and let $g[G,v,w] := 1_{{\left\{{s(v) = x, s(w)=y, v \sim w}\right\}}}$. Then the mass transport principle implies $${\mathbf{P}}( X_0 = x) ={\mathbf{E}}\sum_{v \in V({\mathbf{G}})} g[{\mathbf{G}}, (0,X_0), v] = {\mathbf{E}}\sum_{v \in V({\mathbf{G}})} g[{\mathbf{G}}, v, (0,X_0)] = {\mathbf{P}}(X_0 = y),$$ so $X_0$ is uniformly distributed on $S$.
Next let $X_0$ be the output of the CFTP algorithm in the standard CFTP setup. Suppose $[{\mathbf{G}}, (0,X_0)]$ is unimodular. Since $(0,X_0)$ has one outgoing edge in ${\mathbf{G}}$, unimodularity implies that on average it has one incoming edge. But, being the output of the CFTP algorithm, $(0,X_0)$ a.s. has at least one incoming edge. Hence $(0,X_0)$ a.s. has exactly one incoming edge. By unimodularity, it follows that a.s. every vertex in ${\mathbf{G}}$ has exactly one incoming edge. Since ${\mathbf{G}}$ is a tree, this is only possibly if $S$ has a single element. If $S$ has only a single element unimodularity is immediate.
List of Mass Transports {#sec:list-of-mass-transports}
-----------------------
As mentioned in , the proof style of can be used to prove many equalities and inequalities in mean. A list is provided giving mass transports, followed by the results they give after applying the boilerplate proof style with these mass transports. Drawing a picture for each transport helps significantly in computing $w^+$ and $w^-$ for the given transports. In all of the following, $\beta$ is the union of all bi-recurrent paths in ${\mathbf{B}}$.
1. Send mass $1$ from each $s$ to all times $t$ strictly after $s$ and strictly before $F^{(s,x^*)}$ returns to $x^*$.
2. ${\mathbf{E}}[\#{\mathbf{B}}_0] {\leqslant}{\mathbf{E}}[\sigma^{(0,x^*)}(x^*)]$, where $\sigma^{(0,x^*)}(x^*)$ is the time until return of $F^{(0,x^*)}$ to $x^*$.
3. Fix $y \in S$. For each $s$, if $y \in {\mathbf{B}}_s$, send mass $1$ to the first time $t>s$ that $F^{(s,y)}$ hits $x^*$.
4. ${\mathbf{P}}(y \in {\mathbf{B}}_0) = {\mathbf{E}}[\#{\mathbf{R}}(0)^y]$, where ${\mathbf{R}}(0) \subseteq{\mathbf{B}}$ is the subgraph of vertices that first return to $x^*$ at time $0$, i.e., the (possibly empty) subgraph of ${\mathbf{B}}$ of all $(t,y) \in V({\mathbf{B}})$ such that $\tau^{(t,y)}(x^*) = 0$, where $\tau^{(t,y)}(x^*)$ is the return time of $F^{(t,y)}$ to $x^*$.
5. Summing over $y \in S$, one finds ${\mathbf{E}}[\#V({\mathbf{R}}(0))] = {\mathbf{E}}[\#{\mathbf{B}}_0]$.
6. \[it:f3\] Send mass $1$ from each $s$ to the first time $t>s$ that $F^{(s,x^*)}_t = F^{(s', x^*)}_t$ for some $s'>t$.
7. ${\mathbf{E}}[C(0)] = 1$, where $C(0)$ is the total number of paths $F^{(s,x^*)}$ that merge with a younger $F^{(s',x^*)}$ (i.e. with $s'>s$) for the first time at time $0$.
8. ${\mathbf{P}}(\#{\mathbf{B}}_1 {\leqslant}\#{\mathbf{B}}_0-k) {\leqslant}{\mathbf{P}}(C(1) {\geqslant}k+1) {\leqslant}\frac{1}{k+1}$ for all $k \in {\mathbb{N}}$.
9. Fix $y \in S$. For each $s$, send mass $1$ to each time $t$ that $F^{(s,x^*)}_t = y$ and $t$ is strictly before $F^{(s,x^*)}$ merges with the unique bi-recurrent path in its component of ${\mathbf{B}}$.
10. ${\mathbf{E}}[ N^{(0,x^*)}_0(y ; \beta)] = {\mathbf{E}}[1_{{\left\{{y \in {\mathbf{B}}_0\setminus\beta_0}\right\}}}\#V^{x^*}(D^{(0,y)} \cap V({\mathbf{B}}))]$, where $N^{(0,x^*)}_0(y ; \beta)$ denotes the number of visits (potentially $0$) of $F^{(0,x^*)}$ to $y$ strictly before merging with $\beta$.
11. Summing over $y \in S$, ${\mathbf{E}}[ \sigma^{(0,x^*)}_0(\beta)] = {\mathbf{E}}[\#V^{x^*}(D^{V_0({\mathbf{B}})\setminus V_0(\beta)} \cap V({\mathbf{B}}))]$, where $\sigma^{(0,x^*)}_0(\beta)$ is the number of steps (potentially $0$) before $F^{(0,x^*)}$ merges with $\beta$, and $D^{V_0({\mathbf{B}})\setminus V_0(\beta)}$ is the set of all descendants of all $v \in V_0({\mathbf{B}})\setminus V_0(\beta)$.
12. Fix $y \in S$. For each $s$, if $y \in {\mathbf{B}}_s$ send mass $1$ to the first time $t$ that $F^{(s,y)}$ is on the bi-recurrent path in its component of ${\mathbf{B}}$.
13. ${\mathbf{P}}(y \in {\mathbf{B}}_0) = {\mathbf{E}}[\# V^{y}(D^{V_0(\beta), M} \cap V({\mathbf{B}}))]$, where $D^{V_0(\beta), M}$ denotes the union of $V_0(\beta)$ with their [**mortal descendants**]{}, i.e. those descendants with only finitely many descendants and whose first ancestor in $\beta$ is at time $0$.
14. Summing over $y \in S$, one finds ${\mathbf{E}}[\#{\mathbf{B}}_0] = {\mathbf{E}}[\# (D^{V_0(\beta),M} \cap V({\mathbf{B}}))]$.
15. Fix $y \in S$ and suppose ${\mathbf{G}}$ is an [[EFT]{}]{} and ${{({\beta_t})}}_{t \in {\mathbb{Z}}}$ is the bi-recurrent path in ${\mathbf{G}}$. For each $t$, if $\beta_t = y$ send mass $1$ backwards to the most recent time $s<t$ such that $\beta_s = x^*$.
16. ${\mathbf{E}}[N^{(0,x^*)}(y;x^*) 1_{\beta_0 = x^*}] = {\mathbf{P}}(\beta_0 = y)$, where $N^{(0,x^*)}(y;x^*)$ denotes the number of visits of $F^{(0,x^*)}$ to $y$ before returning to $x^*$, including the initial visit if $y=x^*$.
17. Summing over $y \in S$, one finds ${\mathbf{E}}[\sigma^{(0,x^*)}(x^*) 1_{{\left\{{\beta_0 = x^*}\right\}}}] = 1$.
18. If ${\mathbf{G}}$ is also Markovian, then the previous points reduce to the classical cycle formulas, ${\mathbf{E}}[N^{(0,x^*)}(y;x^*)] \pi(x^*) = \pi(y)$ and ${\mathbf{E}}[\sigma^{(0,x^*)}(x^*)] \pi(x) = 1$, where $\pi$ is the stationary distribution of the Markov chain.
Instead of using the unimodularity of ${\mathbb{Z}}$ and specifying a mass transport $w=w(s,t)$ for $s,t \in {\mathbb{Z}}$, one may also use the unimodular version of ${\mathbf{B}}$ (that is, the random network with distribution ${{\mathbf{P}}^{\mathsmaller{\square}}}$) and specify a mass transport $w=w[{\Gamma}, u,v]$ for all networks ${\Gamma}$ and all $u,v \in V({\Gamma})$. Some mass transports are much easier to write in this way. For example, the mass transport in \[it:f3\] above also follows from the mass transport $w[{\Gamma},u,v] = 1$ if $v$ is the unique out-neighbor of $u$ in ${\Gamma}$. However, strictly speaking, there are no results using a mass transport on ${\mathbf{B}}$ that could not also be proved with a mass transport on ${\mathbb{Z}}$. Indeed, if $w$ is a mass transport defined for all networks ${\Gamma}$, then with $[\overline{{\mathbf{B}}},{\mathsmaller{\square}}]$ denoting the identity map under ${{\mathbf{P}}^{\mathsmaller{\square}}}$, $$\begin{aligned}
{{\mathbf{E}}^{\mathsmaller{\square}}}\left[\sum_{v \in V({\mathbf{B}})} w[\overline{{\mathbf{B}}},{\mathsmaller{\square}}, v]\right] = {{\mathbf{E}}^{\mathsmaller{\square}}}\left[\sum_{v \in V({\mathbf{B}})} w[\overline{{\mathbf{B}}}, v, {\mathsmaller{\square}}]\right]\end{aligned}$$ may be rewritten as $$\begin{aligned}
\frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]}{\mathbf{E}}\left[\sum_{t \in {\mathbb{Z}}}\hat{w}(0,t)\right]
= \frac{1}{{\mathbf{E}}[\#{\mathbf{B}}_0]}{\mathbf{E}}\left[\sum_{t \in {\mathbb{Z}}}\hat{w}(t,0)\right]\end{aligned}$$ where $$\begin{aligned}
\hat{w}(s,t) := \sum_{u \in V_s({\mathbf{B}})} \sum_{v \in V_t({\mathbf{B}})} w[\overline{{\mathbf{B}}},u,v],\qquad s,t \in {\mathbb{Z}}\end{aligned}$$ is a mass transport on ${\mathbb{Z}}$. That being said, the reader is encouraged the ponder the sequence of mass transports on ${\mathbb{Z}}$ that would be required to prove a result like the classification theorem, , for the network $[\overline{{\mathbf{B}}},{\mathsmaller{\square}}]$ directly. It seems more elegant to call upon the machinery of unimodular networks when convenient instead.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by a grant of the Simons Foundation (\#197982 to The University of Texas at Austin). The second author thanks the Research and Technology Vice-presidency of Sharif University of Technology for its support.
References
==========
[^1]: The University of Texas at Austin, [](mailto:[email protected])
[^2]: Sharif University of Technology, [](mailto:[email protected])
[^3]: The University of Texas at Austin, [](mailto:[email protected])
|
---
abstract: 'We introduce two invariants called $\mathfrak{sl}(3)$ Khovanov module and pointed $\mathfrak{sl}(3)$ Khovanov homology for spatial webs (bipartite trivalent graphs). Those invariants are related to Kronheimer-Mrowka’s instanton invariants $J^\sharp$ and $I^\sharp$ for spatial webs by two spectral sequences. As an application of the spectral sequences, we prove that $\mathfrak{sl}(3)$ Khovanov module and pointed $\mathfrak{sl}(3)$ Khovanov homology both detect the planar theta graph.'
author:
- Yi Xie
bibliography:
- 'references.bib'
title: '$\mathfrak{sl}(3)$ Khovanov module and the detection of the planar theta-graph'
---
Introduction
============
In [@Kh-Jones], Khovanov constructed a link homology which categorifies the quantum $\mathfrak{sl}(2)$ link invariant: the Jones polynomial. Later, $\mathfrak{sl}(3)$ and $\mathfrak{sl}(n)$ link homologies were introduced in [@Kh-sl3] and [@KhR].
To define the $\mathfrak{sl}(3)$ Khovanov homology of a link $L$ (in $\mathbb{R}^3$), we need to pick a diagram for $L$ and resolve the crossings by resolutions shown in Figure \[+01\] and Figure \[-01\]. A resolved diagram could be singular. In general it is a planar trivalent graph rather than a collection of circles which is the situation of the $\mathfrak{sl}(2)$ Khovanov homology. A chain complex is defined by assigning abelian groups to those planar trivalent graphs and defining a differential on the direct sum of those abelian groups. The $\mathfrak{sl}(3)$ Khovanov homology is the homology of the chain complex, which is a bi-graded abelian group.
The definition of the chain complex depends on the choice of the diagram of the link $L$. To obtain a well-defined link invariant, we need to show that the homology does not depend on the choice of the diagram. Since any two diagrams of the same link are connected by a sequence of Reidemeister moves of three types, it suffices to show that the homology of the chain complex is invariant under three types of moves.
In this paper, we consider more general objects: oriented webs embedded in $\mathbb{R}^3$ (see Definition \[web\]). Given an oriented web $\Gamma$ in $\mathbb{R}^3$, we can still pick a diagram $D$ for it. Resolving the crossings by Figure \[+01\] or Figure \[-01\], we could obtain planar webs. By assigning abelian groups to those planar webs and define a differential properly as in the link case, we obtain a chain complex $F(D)$. We define the $\mathfrak{sl}(3)$ Khovanov homology for $\Gamma$ to be the homology of this chain complex. Again the definition depends on the choice of the diagram. Any two diagrams of $\Gamma$ are connected by a sequence of *five* types of Reidemeister moves according to [@Kau-graph; @Kau-graph2]. In addition to the three types of Reidemeister moves in the link case, there are two new Reidemeister moves shown in Figures \[RIV\] and \[RV\]. To show the homology is a well-defined invariant, we need to study how it changes under the five types of moves. It turns out that the homology is still a well-defined invariant but we lose the absolute bi-grading.
Given two diagrams $D$ and $D'$ of an oriented spatial web $\Gamma$, the two chain complexes $F(D)$ and $F(D')$ are chain homotopy equivalent up to a shifting of the bi-grading. In particular, the *relatively* bi-graded abelian group $\mathcal{H}(\Gamma):=H(F(D))$ is a well-defined invariant of the oriented spatial web.
There is a $R_\Gamma$-module structure on $\mathcal{H}(\Gamma)$ where $R_\Gamma$ is a ring defined in Section \[module-str\] which does not depend on the embedding of $\Gamma$.
In [@KM-jsharp], Kronheimer and Mrowka defined two versions of instanton Floer homologies $J^\sharp$ and $I^\sharp$ for spatial webs. Their definition of webs is more general than the one used in this paper. The two instanton Floer homology theories $J^\sharp$ and $I^\sharp$ are defined only with $\mathbb{F}$-coefficients where $\mathbb{F}$ is the field of two elements. Given a (oriented) spatial web $\Gamma$, $\mathcal{H}(\Gamma;\mathbb{F})$ and $J^\sharp(\Gamma)$ are both equipped with $\mathcal{R}_\Gamma$-module structures where $\mathcal{R}_\Gamma:=R_\Gamma\otimes_\mathbb{Z}\mathbb{F}$. From [@KM-jsharp] it is known that $\mathcal{H}(\Gamma;\mathbb{F})$ and $J^\sharp(\Gamma)$ are isomorphic for any planar web $\Gamma$. More generally, we have the following.
\[s-sequence\*\] Let $\Gamma$ be an oriented spatial web. There is a spectral sequence of $\mathcal{R}_\Gamma$-modules whose $E_2$-page is the $\mathfrak{sl}(3)$ Khovanov module $\mathcal{H}(\Gamma;\mathbb{F})$ and which converges to $J^\sharp(\Gamma)$.
This spectral sequence is suggested by Kronheimer [@KM-sl3-ss]. A similar result in the $\mathfrak{sl}(2)$ Khovanov homology case is proved in [@HN-module]: they showed that Ozsváth-Szabó’s spectral sequence in [@OS-ss] respects the module structures on Khovanov homology and (hat) Heegaard Floer homology.
Given an oriented spatial web $\Gamma$ with mark points $\boldsymbol{\delta}=\{\delta_i\}$ in the interior of edges, we define a relatively bi-graded homological invariant $\mathcal{H}(\Gamma,\boldsymbol{\delta})$ by imitating the definition of pointed $\mathfrak{sl}(2)$ Khovanov homology in [@BLS]. When $\boldsymbol{\delta}=\emptyset$, it is nothing but $\mathcal{H}(\Gamma)$. We construct a spectral sequence relating $\mathcal{H}(\Gamma,\boldsymbol{\delta};\mathbb{F})$ to $I^\sharp(\Gamma)$.
\[I-ss\*\] Suppose $\Gamma$ is a *connected* oriented spatial web and ${\boldsymbol{\delta}}=\{\delta_i\}$ is a collection of points in the interior of edges of $\Gamma$ such that the homology classes of meridians around $\delta_i$ form a basis of $H_1(S^3\setminus \Gamma;\mathbb{F})$. Then there is a spectral sequence whose $E_2$-page is $\mathcal{H}(\Gamma,{\boldsymbol{\delta}};\mathbb{F})$ and which converges to $I^\sharp(\Gamma)$.
Let $\Theta$ be the planar theta graph. As an application of the above spectral sequence, we prove the following detection result.
\[theta-detection\] Suppose $\Gamma$ is a spatial theta graph and $\boldsymbol{\delta}=\{\delta_1,\delta_2\}$ are two mark points lying on two distinct edge of $\Gamma$. Then the following are equivalent:
1. $\Gamma$ is the planar theta graph;
2. $\mathcal{H}(\Gamma;\mathbb{F})$ and $\mathcal{H}(\Theta;\mathbb{F})$ are isomorphic as $\mathcal{R}_\Theta$-modules;
3. $\operatorname{rank}_\mathbb{F} \mathcal{H}(\Gamma,\boldsymbol{\delta};\mathbb{F})=4$;
4. $J^\sharp(\Gamma)$ and $\mathcal{H}(\Theta;\mathbb{F})$ are isomorphic as $\mathcal{R}_\Theta$-modules;
5. $\operatorname{rank}_\mathbb{F} I^\sharp(\Gamma)=4$.
In this article, all the Khovanov homologies are defined over $\mathbb{Z}$ unless otherwise specified. All the instanton Floer homologies are defined over $\mathbb{F}$($=\mathbb{Z}/2$) unless otherwise specified.
*Acknowledgments.* The author learned the spectral sequence in Theorem \[s-sequence\*\] from Peter Kronheimer and wishes to thank him for his generosity in sharing ideas.
$\mathfrak{sl}(3)$ Khovanov module for spatial webs {#sl3Kh}
====================================================
In this section, we first review Khovanov’s definition of $\mathfrak{sl}(3)$ link homology [@Kh-sl3], which is a bi-graded abelian group. Then we will show that the $\mathfrak{sl}(3)$ homology can be generalized for spatial bipartite trivalent graphs.
$\mathfrak{sl}(3)$ homology for links
--------------------------------------
All the contents in this subsection are from [@Kh-sl3].
A (closed) *pre-foam* consists of
- A compact 2-dimensional CW-complex $\Sigma$ such that any point on it has a neighborhood which is homeomorphic to either a 2-dimensional disk or the product of letter $Y$ and an interval. The points with neighborhood “Y”$\times I$ form a collection of circles called *seams*. The complement of the seams in $\Sigma$ is a 2-manifold whose connected components are *facets* of the foam. We require the facets are orientable.
- For each seam $C$, a cyclic order is chosen for the three facets whose closure includes $C$.
- Each facet is decorated with a number of “dots” (possibly empty).
A *pre-foam* embedded in $\mathbb{R}^3$ is called a *foam*.
It is shown in [@Kh-sl3] that
\[4axiom\] There is a unique map $F:\{\text{closed pre-foams}\}\to \mathbb{Z}$ characterized by the following four axioms
1. If $\Sigma$ is a closed orientable surface decorated with dots, then $F(\Sigma)=0$ unless
- $\Sigma$ is a 2-sphere with two dots, then $F(\Sigma)=-1$.
- $\Sigma$ is a torus without any dots, then $F(\Sigma)=3$.
2. If $\Sigma_1$ and $\Sigma_2$ are pre-foams, then $F(\Sigma_1\sqcup \Sigma_2)=F(\Sigma_1)F(\Sigma_2)$.
3. Do a surgery along a circle inside a facet of $\Sigma$ to obtain three pre-foams $\Sigma_1, \Sigma_2, \Sigma_3$ as in Figure \[Sigma123\]. Then $$-F(\Sigma)=F(\Sigma_1)+F(\Sigma_2)+F(\Sigma_3)$$
4. Let $\Theta(k_1,k_2,k_3)$ be the theta foam with $k_i$ dots on the $i$-th facet (see Figure \[theta-foam\]), then $$F(\Theta(k_1,k_2,k_3))=\left\{
\begin{array}{ll}
1, & \hbox{if $(k_1,k_2,k_3)=(0,1,2)$, up to cylic permutation;} \\
-1, & \hbox{if $(k_1,k_2,k_3)=(0,2,1)$, up to cylic permutation;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
(0,1) ellipse (1cm and 0.3cm);
(0,0) \[partial ellipse=-180:0: 1cm and 0.3cm\]; (0,0) \[partial ellipse=0:180: 1cm and 0.3cm\];
(0,-1) \[partial ellipse=-180:0: 1cm and 0.3cm\]; (0,-1) \[partial ellipse=0:180: 1cm and 0.3cm\];
(-1,-1) to (-1,1); (1,-1) to (1,1); at (0,-1.8) [$\Sigma$]{};
(3,1) ellipse (1cm and 0.3cm); (3,-1) \[partial ellipse=-180:0: 1cm and 0.3cm\]; (3,-2) \[partial ellipse=35:145: 1.22cm and 1.8cm\]; (3,-1) \[partial ellipse=0:180: 1cm and 0.3cm\]; (3,2) \[partial ellipse=-35:-145: 1.22cm and 1.8cm\]; (3.2,0.5) circle (2pt); (2.8,0.5) circle (2pt); at (3,-1.8) [$\Sigma_1$]{};
(6,1) ellipse (1cm and 0.3cm); (6,-1) \[partial ellipse=-180:0: 1cm and 0.3cm\]; (6,-2) \[partial ellipse=35:145: 1.22cm and 1.8cm\]; (6,-1) \[partial ellipse=0:180: 1cm and 0.3cm\]; (6,2) \[partial ellipse=-35:-145: 1.22cm and 1.8cm\]; (6,0.5) circle (2pt); (6,-0.5) circle (2pt); at (6,-1.8) [$\Sigma_2$]{};
(9,1) ellipse (1cm and 0.3cm); (9,-1) \[partial ellipse=-180:0: 1cm and 0.3cm\]; (9,-2) \[partial ellipse=35:145: 1.22cm and 1.8cm\]; (9,-1) \[partial ellipse=0:180: 1cm and 0.3cm\]; (9,2) \[partial ellipse=-35:-145: 1.22cm and 1.8cm\]; (9.2,-0.5) circle (2pt); (8.8,-0.5) circle (2pt); at (9,-1.8) [$\Sigma_3$]{};
(0,0) ellipse (2cm and 2.2cm); (0,0) \[partial ellipse=-180:0: 2cm and 1cm\]; (0,0) \[partial ellipse=-180:180: 2cm and 1cm\];
at (-1,0) [1]{}; at (-1,1.5) [2]{}; at (-1,-1.5) [3]{};
An oriented (closed) foam is a foam with all the facets oriented such that for any seam $C$, the induced orientations from the three nearby facets coincide and cyclic ordering of the three nearby facets are determined by the orientation of $C$ and the left-hand-rule.
We also want to define oriented foams with boundary. Before that we define the following.
\[web\] A *web* is a bipartite trivalent graph (possibly including loops without vertices). An oriented web is a directed bipartite trivalent graph such that at each vertex all the edges are either all incoming or all outgoing.
An oriented foam with boundary is the intersection of $\mathbb{R}^2\times [0,1]$ and a closed foam $\Sigma$ such that $\mathbb{R}^2\times\{0\}$ and $\mathbb{R}^2\times\{1\}$ are transversal to the facets and seams of $\Sigma$. In particular, $\Gamma_1=\mathbb{R}^2\times\{0\}\cap \Sigma$ and $\Gamma_2=\mathbb{R}^2\times\{1\}\cap \Sigma$ are two oriented planar webs. The oriented foam with boundary can be thought as a cobordism from $\Gamma_1$ to $\Gamma_2$.
By composing with the forgetful map from oriented closed foams to pre-foams, we can define $F:\{\text{oriented closed foams}\}\to \mathbb{Z}$. Here we still denote it by $F$ by abuse of notation. Next we want to extend $F$ into a (1+1)-dimensional TQFT on oriented planar webs and foams. Given two oriented planar webs $\Gamma_1$ and $\Gamma_2$, let $\operatorname{Hom}_{\text{OF}} (\Gamma_1,\Gamma_2)$ be the set of all cobordisms (oriented foams) from $\Gamma_1$ to $\Gamma_2$. Define $$F(\Gamma):=\mathbb{Z}\operatorname{Hom}_{\text{OF}} (\emptyset,\Gamma)/\{\sum_i a_i\Sigma_i| \sum_i a_i F(\Phi \circ \Sigma_i)=0~\text{for all}~\Phi \in \operatorname{Hom}_{\text{OF}} (\Gamma,\emptyset) \}$$ for an oriented planar web $\Gamma$. $F(\Gamma)$ can be equipped with a $\mathbb{Z}$-grading by defining $$\deg (\Sigma):= \chi (\partial \Sigma)-2\chi(\Sigma\setminus \text{Nbh}(\text{dots}))$$ for $\Sigma\in \operatorname{Hom}_{\text{OF}} (\emptyset,\Gamma)$ (see [@Kh-sl3]\*[Section 3.3]{} for more details). It is clear from the definition that given $\Phi\in \operatorname{Hom}_{\text{OF}} (\Gamma_1,\Gamma_2)$, there is a well-defined homomorphism $F(\Phi):F(\Gamma_1)\to F(\Gamma_2)$ of degree $\deg \Phi$.
Let $L$ be an oriented link in $\mathbb{R}^3$ and $D$ be a diagram of $L$ with $n$ crossings. A crossing is called a positive or negative crossing according to Figures \[+01\] and \[-01\]. [^1]
(1,-1) to (-1,1); (-1,-1) to (1,1); at (0,-1.2) [Positive crossing]{};
(2,-1) to \[out=45,in=270\] (2.7,0) to \[out=90,in=315\] (2,1); at (3,-1.2) [0-resolution]{}; (4,-1) to \[out=135,in=270\] (3.3,0) to \[out=90,in=225\] (4,1);
(5,-1) to (6,-0.4); (6,0.4) to (6,-0.4); (7,-1) to (6,-0.4); (6,0.4) to (5,1); (6,0.4) to (7,1); at (6,-1.2) [1-resolution]{};
\[+01\]
(1,-1) to (-1,1); (-1,-1) to (1,1); at (0,-1.2) [Negative crossing]{};
(2,-1) to (3,-0.4); (3,0.4) to (3,-0.4); (4,-1) to (3,-0.4); (3,0.4) to (2,1); (3,0.4) to (4,1); at (3,-1.2) [0-resolution]{};
(5,-1) to \[out=45,in=270\] (5.7,0) to \[out=90,in=315\] (5,1); at (6,-1.2) [1-resolution]{}; (7,-1) to \[out=135,in=270\] (6.3,0) to \[out=90,in=225\] (7,1);
\[-01\]
(1,0) to (0,0); (-1.2,-0.7) to (0,0); (-0.8,0.5) to (0,0); (-0.8,0.5) to (-0.8,2.1); (-0.8,2.1) to (-0.8, 3); (-1.2,-0.7) to (-1.2,2); (1,0) to (2,-0.7); (2,-0.7) to (2,2); (-1.2,2) to \[out=17,in=163\] (2,2); (1,0) to (2,2.5/7); (2,2.5/7) to (2.4,0.5); (2.4,0.5) to (2.4,3); (-0.8, 3) to \[out=-15, in=195\] (2.4,3);
(0.5,0) \[partial ellipse=0:180: 0.5cm and 1cm\]; (0.5,0.02) \[partial ellipse=0:180: 0.47cm and 0.97cm\];
(0.5,2.8) \[partial ellipse=197:270: 0.3cm and 1.8cm\]; (0.5,2.8) \[partial ellipse=270:358: 0.3cm and 1.8cm\];
For each crossing, there are two ways to resolve the crossing: 0-resolution and 1-resolution, see Figures \[+01\] and \[-01\]. Given any $v\in \{0,1\}^n$, we can resolve all the crossings by 0- or 1-resolution determined by $v$ and obtain an oriented planar web $D_v$. If $v,u\in \{0,1\}^n$ only differ at one crossing ($v$ assigns $0$ and $u$ assigns $1$ to this crossing), then there is a cobordism $S_{vu}$ from $D_v$ to $D_u$ which is a product cobordism away from the crossing and is the cobordism in Figure \[skein-co\] near the crossing.
Let $p_+$ and $p_-$ be the numbers of positive and negative crossings in $D$ respectively. We use $F(\Gamma)\{l\}$ to denote the graded abelian group obtained by increasing the (quantum) grading of $F(\Gamma)$ by $l$. For each vertex $v$ of the cube $\{0,1\}^n$, we assign a graded abelian group $F(D_v)\{3p_- - 2p_+ -|v|_1\}$ where $|v|_1$ is the number of 1’s in $v$. For each edge $vu$ of the cube $\{0,1\}^n$, we assign the map $F(S_{vu}):F(D_v)\{3p_- - 2p_+ -|v|_1\}\to F(D_u)\{3p_- - 2p_+ -|u|_1\}$. This map preserves the grading. After adding plus or minus sign to those maps associated with the edges appropriately, each square in the cube anti-commutes. The total complex $F(D)$ of the cube becomes a chain complex of graded abelian groups whose homological degree $i$ term is $$\label{cube}
F(D)_i:=\bigoplus_{|v|_1=i+p_-} F(D_v)\{3p_- - 2p_+ -|v|_1\}$$ Alternatively, we can define $F(D)$ inductively by Figure \[ind-cube\]. The homology of $F(D)$ is a bi-graded abelian group: one grading is the *homological grading* and the other grading (we call it the *quantum grading*) is obtained from the grading of $F(D_v)\{3p_- - 2p_+ -|v|_1\}$.
(1,-1) to (-1,1); (-1,-1) to (1,1); (1.2,0.1) to (1.6,0.1); (1.2,-0.1) to (1.6,-0.1);
at (2.1,0) [$\textbf{Tot}$]{};
(2.5,-1) to (2.5,1); (2.5,1) to (2.7,1);(2.5,-1) to (2.7,-1);
(3,-1) to \[out=45,in=270\] (3.7,0) to \[out=90,in=315\] (3,1); (5,-1) to \[out=135,in=270\] (4.3,0) to \[out=90,in=225\] (5,1); at (5.5,0) [$\{-2\}$]{}; at (4,-1.2) [0]{}; at (8.25,-1.2) [1]{};
(6,0) to (7,0);
(7.2,-1) to (8.2,-0.4); (8.2,0.4) to (8.2,-0.4); (9.2,-1) to (8.2,-0.4); (8.2,0.4) to (7.2,1); (8.2,0.4) to (9.2,1); at (9.7,0) [$\{-3\}$ ]{};
(10.4,-1) to (10.4,1); (10.4,1) to (10.2,1); (10.4,-1) to (10.2,-1); at (0,-2) ;
(1,-1) to (-1,1); (-1,-1) to (1,1); (1.2,0.1) to (1.6,0.1); (1.2,-0.1) to (1.6,-0.1);
at (2.1,0) [$\textbf{Tot}$]{};
(2.5,-1) to (2.5,1); (2.5,1) to (2.7,1);(2.5,-1) to (2.7,-1);
(3,-1) to (4,-0.4); (4,0.4) to (4,-0.4); (5,-1) to (4,-0.4); (4,0.4) to (3,1); (4,0.4) to (5,1);
at (5.5,0) [$\{3\}$]{}; at (4,-1.2) [$-1$]{}; at (8.25,-1.2) [0]{};
(6,0) to (7,0);
(7.2,-1) to \[out=45,in=270\] (7.9,0) to \[out=90,in=315\] (7.2,1); (9.2,-1) to \[out=135,in=270\] (8.5,0) to \[out=90,in=225\] (9.2,1);
at (9.7,0) [$\{2\}$ ]{};
(10.4,-1) to (10.4,1); (10.4,1) to (10.2,1); (10.4,-1) to (10.2,-1);
The diagram $D$ of the link $L$ is not unique. But any two diagrams of $L$ can be connected by a sequence of three types of Reidemeister moves.
Given two diagrams $D$ and $D'$ of the link $L$, the two chain complexes $F(D)$ and $F(D')$ are chain homotopy equivalent. In particular, the bi-graded abelian group $\mathcal{H}(L):=H(F(D))$ is a well-defined invariant of $L$.
Spatial webs and Reidemeister moves
-----------------------------------
A *spatial web* is a web embedded in $\mathbb{R}^3$. In this subsection, we want to generalized Khovanov’s link homology $\mathcal{H}(L)$ to oriented spatial webs. Similar to links, we can also choose a diagram for a spatial web and require that the crossings are disjoint from the vertices. The diagram for a spatial web is not unique. Any two diagrams for the same spatial web are connected by a sequence of *five* types of Reidemeister moves (see [@Kau-graph] or [@Kau-graph2] for more details). In addition to the three types of Reidemeister moves in the link case, there are two new Reidemeister moves shown in Figures \[RIV\] and \[RV\].
(0,1) to (0,2); (1,-1) to \[out=150, in=-90\] (-0.5,0.2); (-0.5,0.2) to \[out=90,in=210\] (0,1); (-1,-1) to \[out=30, in=-90\] (0.5,0.2); (0.5,0.2) to \[out=90,in=-30\] (0,1);
(1,0.2) to (2,0.2);
(3,1) to (3,2); (2,-1) to \[out=30, in=-90\] (3,1); (4,-1) to \[out=150, in=-90\] (3,1);
(4,0.2) to (5,0.2);
(6,1) to (6,2); (7,-1) to \[out=150, in=-90\] (5.5,0.2); (5.5,0.2) to \[out=90,in=210\] (6,1); (5,-1) to \[out=30, in=-90\] (6.5,0.2); (6.5,0.2) to \[out=90,in=-30\] (6,1);
at (-1.5,0.2) [RV:]{};
Suppose $\Gamma$ is an oriented spatial web and $D$ is a diagram for it with $n$ crossings. The we can form a chain complex $F(D)$ as in . The same argument as in the link case shows that the chain homotopy equivalence class of $F(D)$ is invariant under the first three types of Reidemeister moves. In order to obtain a well-defined invariant for $\Gamma$, we need to study the change of $F(D)$ under moves of types IV and V. We deal with RV first. Pick an orientation for a type V move as in Figure \[RVDD’\], we want to compare the chain complexes $F(D)$ and $F(D')$.
(3,2) to (3,1); (2,-1) to \[out=30, in=-90\] (3,1); (4,-1) to \[out=150, in=-90\] (3,1);
at (3,-1.5) [$D'$]{};
(4,0.2) to (5,0.2);
(6,2) to (6,1); (7,-1) to \[out=150, in=-90\] (5.5,0.2); (5.5,0.2) to \[out=90,in=210\] (6,1); (5,-1) to \[out=30, in=-90\] (6.5,0.2); (6.5,0.2) to \[out=90,in=-30\] (6,1);
at (6,-1.5) [$D$]{};
\[RVD=D’\] Let $D$ and $D'$ be the two diagrams in Figure \[RVDD’\]. Up to a shifting of the homological and quantum gradings, $F(D)$ and $F(D')$ are chain homotopy equivalent.
Before moving to the proof of this proposition, we need to state a result from [@Kh-sl3].
[@Kh-sl3]\*[Proposition 8]{}\[digon-homology\] Suppose $\Gamma_1$ and $\Gamma_2$ are two oriented planar webs such that $\Gamma_2$ is obtained by removing a digon from $\Gamma_1$ as shown in in Figure \[digon-webs\]. Then $$g=F(\Sigma_{d1})\oplus F(\Sigma_{d2}):F(\Gamma_1)\to F(\Gamma_2)\{1\}\oplus F(\Gamma_2)\{-1\}$$ is an isomorphism where $\Sigma_{d1}$ and $\Sigma_{d2}$ are the cobordisms in Figure \[digon-co\] (thought as cobordisms from the top to the bottom). Reversing the direction of the two cobordisms (i.e. regard them as cobordisms from the bottom to the top), we obtain $\overline{\Sigma}_{d1}$ and $\overline{\Sigma}_{d2}$. The compositions $g\circ F(\overline{\Sigma}_{d1})$ and $g\circ F(\overline{\Sigma}_{d2})$ are embeddings onto the $F(\Gamma_2)\{-1\}$ and $F(\Gamma_2)\{1\}$ summands respectively (be careful about the order change).
at (-1,0.5) [$\Gamma_1:$]{}; (0,2) to (0,1); (0,0) to (0,-1);
(0,0.5) \[partial ellipse=-90:90: 0.3cm and 0.5cm\]; (0,0.5) \[partial ellipse=270:90: 0.3cm and 0.5cm\];
at (2,0.5) [$\Gamma_2:$]{}; (3,2) to (3,-1);
(1,0) to (0,0); (1.5,0) \[partial ellipse=180:0: 0.5cm and 0.2cm\]; (1.5,0) \[partial ellipse=180:360: 0.5cm and 0.2cm\]; (3,0) to (2,0);
(1.5,0) \[partial ellipse=180:360: 0.5cm and 0.5cm\];
(3,0) to (3,-2); (0,0) to (0,-2);
(3,-2) to (0,-2);
at (-1,-1) [$\Sigma_{d1}:$]{};
(1,0) to (0,0); (1.5,0) \[partial ellipse=180:0: 0.5cm and 0.2cm\]; (1.5,0) \[partial ellipse=180:360: 0.5cm and 0.2cm\]; (3,0) to (2,0);
(1.5,0) \[partial ellipse=180:360: 0.5cm and 0.5cm\];
(3,0) to (3,-2); (0,0) to (0,-2);
(3,-2) to (0,-2);
(1.5,0.05) circle (2pt); at (-1,-1) [$\Sigma_{d2}:$]{};
at (-1,0) [$D_1:$]{}; (0,2) to (0,1); (0,0.5) \[partial ellipse=-90:90: 0.3cm and 0.5cm\]; (0,0.5) \[partial ellipse=270:90: 0.3cm and 0.5cm\]; (0,0) to (0,-1); (-0.5,-1.5) to (0,-1); (0.5,-1.5) to (0,-1);
Using the inductive definition in Figure \[ind-cube\], we have $$F(D)= \textbf{Tot}[F(D')\{-2\}\to F(D_1)\{-3\}]$$ where $D_1$ is shown in Figure \[RVD1\] and the map from $F(D')$ to $F(D_1)$ is induced by the cobordism in Figure \[RVD’D1-co\].
(-1.2,-0.7) to (0,0); (-0.8,0.5) to (0,0);
(1,0) to (0,0); (1.5,0) \[partial ellipse=180:0: 0.5cm and 0.2cm\]; (1.5,0) \[partial ellipse=180:360: 0.5cm and 0.2cm\]; (3,0) to (2,0);
(0.5,0) \[partial ellipse=0:180: 0.5cm and 1cm\]; (0.5,0.02) \[partial ellipse=0:180: 0.47cm and 0.97cm\];
(-1.2,-0.7) to (-1.2,1.3); (-0.8,0.5) to (-0.8,1.6); (-0.8,1.6) to (-0.8,2.5);
(2,1.3) \[partial ellipse=180:90: 3.2cm and 0.6cm\]; (2,2.5) \[partial ellipse=180:270: 2.8cm and 0.6cm\]; (3,1.9) to (2,1.9); (2,1.9) to (2,0);
(3,0) to (3,1.9);
(0.5,2.8) \[partial ellipse=213:270: 0.3cm and 1.8cm\]; (0.5,2.8) \[partial ellipse=270:332: 0.3cm and 1.8cm\];
By Lemma \[digon-homology\], we have $$F(D_1)\{-3\}=F(D')\{-2\}\oplus F(D')\{-4\}$$ The composition map $$f: F(D')\{-2\}\to F(D_1)\{-3\}\to F(D')\{-2\}$$ is induced by the composition of the digon cobordism in Figure \[digon-co\] and the cobordism in Figure \[RVD’D1-co\]. This composition cobordism can be deformed into the product cobordism (see Figure \[compo-digon-D’D1\]). Therefore $f$ is the identify map. This implies $$[F(D')\{-2\}\to F(D_1)\{-3\}]\cong [F(D')\{-2\}\to F(D')\{-2\}\oplus F(D')\{-4\}]$$ is chain homotopy equivalent to $$0\to F(D')\{-4\}$$ we have $$\textbf{Tot}[0\to F(D')\{-4\}]=F(D')\{-4\}[1]$$ where $[1]$ means increasing the homological grading by $1$. This completes the proof.
(1,0) to (0,0); (1.5,0) \[partial ellipse=180:0: 0.5cm and 0.2cm\]; (1.5,0) \[partial ellipse=180:360: 0.5cm and 0.2cm\]; (3,0) to (2,0);
(1.5,0) \[partial ellipse=180:360: 0.5cm and 0.5cm\];
(-1.2,-0.7) to (0,0); (-0.8,0.5) to (0,0);
(-0.8,0.5) to (-0.8, -0.48); (-0.8, -0.48) to (-0.8,-1.5);
(-1.2,-0.7) to (-1.2,-2.7); (3,0) to (3,-2); (0,0) to (0,-2);
(3,-2) to (0,-2); (-1.2,-2.7) to (0,-2); (-0.8,-1.5) to (0,-2);
(0.5,0) \[partial ellipse=0:180: 0.5cm and 1cm\]; (0.5,0.02) \[partial ellipse=0:180: 0.47cm and 0.97cm\];
(-1.2,-0.7) to (-1.2,1.3); (-0.8,0.5) to (-0.8,1.6); (-0.8,1.6) to (-0.8,2.5);
(2,1.3) \[partial ellipse=180:90: 3.2cm and 0.6cm\]; (2,2.5) \[partial ellipse=180:270: 2.8cm and 0.6cm\]; (3,1.9) to (2,1.9); (2,1.9) to (2,0);
(3,0) to (3,1.9);
(0.5,2.8) \[partial ellipse=213:270: 0.3cm and 1.8cm\]; (0.5,2.8) \[partial ellipse=270:332: 0.3cm and 1.8cm\];
(4.8,1.3) to (7.5,2); (5.2,2.5) to (7.5,2); (9,2) to (7.5,2);
(4.8,-2.7) to (4.8,1.3); (5.2,-1.5) to (5.2,1.6); (5.2,1.4) to (5.2,2.5); (9,2) to (9,-2);
(4.8,-2.7) to (7.5,-2); (5.2,-1.5) to (7.5,-2); (9,-2) to (7.5,-2);
(7.5,2) to (7.5,-2);
at (4,0) [=]{};
Proposition \[RVD=D’\] only deals with one choice of oriented type V move. Similar proofs work for other possible oriented type V moves, so we skip the details for other situations.
Next we deal with the type IV moves. Again we only deal with a specific oriented type IV move in detail and skip the proofs for other situations since the proofs are similar.
This time we need a “square lemma” besides the “digon lemma” used before.
(0,0) to (0,1); (0,0) to (1,0); (1,1) to (0,1); (1,1) to (1,0);
(-1,2) to (0,1); (1,1) to (2,2); (0,0) to (-1,-1); (2,-1) to (1,0);
(-5.5,2) to \[out=-60,in=90\] (-5,0.5); (-5,0.5) to \[out=-90,in=60\] (-5.5,-1); ((-2.5,-1) to \[out=120,in=-90\] (-3,0.5) to \[out=90,in=240\] (-2.5,2);
(-4,0.5) \[partial ellipse=-180:180: 0.7cm and 0.7cm\];
(-10,2) to \[out=-60,in=90\] (-9.5,0.5); (-9.5,0.5) to \[out=-90,in=60\] (-10,-1); ((-7,-1) to \[out=120,in=-90\] (-7.5,0.5) to \[out=90,in=240\] (-7,2);
(0,0.5) \[partial ellipse=-180:180: 0.4cm and 0.9cm\]; (1,0.5) \[partial ellipse=-180:180: 0.4cm and 0.9cm\];
(-4.8,0.5) \[partial ellipse=-180:180: 0.5cm and 0.9cm\]; (-3.2,0.5) \[partial ellipse=-180:180: 0.5cm and 0.9cm\];
(-2,0.5) to (-1,0.5); (-2-5+0.25,0.5) to (-1-5+0.25,0.5);
at (0.5,-1.5) [$\Gamma_{sq}$]{}; at (-4,-1.5) [$\Gamma_{1/2}$]{}; at (-8.5,-1.5) [$\Gamma_{0}$]{};
at (0.5,-1.5) [$\Gamma_{sq}$]{}; at (-4,-1.5) [$\Gamma_{1/2}$]{}; at (-8.5,-1.5) [$\Gamma_{0}$]{}; at (-10.75,0.5) [$\Sigma_{sq}:$]{};
(0,0) to (0,1); (0,0) to (1,0); (1,1) to (0,1); (1,1) to (1,0);
(-1,2) to (0,1); (1,1) to (2,2); (0,0) to (-1,-1); (2,-1) to (1,0);
(-10+4.5,2) to \[out=-30,in=180\] (-8.5+4.5,1.5) to \[out=0,in=210\] (-7+4.5,2) ; (-7+4.5,-1) to \[out=150,in=0\] (-8.5+4.5,-0.5) to \[out=180,in=30\](-10+4.5,-1);
(-4,0.5) \[partial ellipse=180:-180: 0.7cm and 0.7cm\];
(-10,2) to \[out=-30,in=180\] (-8.5,1.5) to \[out=0,in=210\] (-7,2) ; (-7,-1) to \[out=150,in=0\] (-8.5,-0.5) to \[out=180,in=30\](-10,-1);
(0.5,0) \[partial ellipse=-180:180: 0.9cm and 0.4cm\]; (0.5,1) \[partial ellipse=-180:180: 0.9cm and 0.4cm\];
(-4,-0.3) \[partial ellipse=-180:180: 0.9cm and 0.5cm\]; (-4,1.3) \[partial ellipse=-180:180: 0.9cm and 0.5cm\];
(-2,0.5) to (-1,0.5); (-2-5+0.25,0.5) to (-1-5+0.25,0.5);
at (0.5,-1.5) [$\Gamma_{sq}$]{}; at (-4,-1.5) [$\Gamma_{1/2}'$]{}; at (-8.5,-1.5) [$\Gamma_{0}'$]{}; at (-10.75,0.5) [$\Sigma_{sq}':$]{};
The square cobordism $\Sigma_{sq}$ from $\Gamma_0$ to $\Gamma_{sq}$ in Figure \[square-co\] can be described as the composition of two cobordisms. The first one is the “birth cobordism” from $\Gamma_0$ to $\Gamma_{1/2}$ which is the disjoint union of the product cobordism $\Gamma_0\times I$ and a disk (thought as a cobordism from the empty set to a circle). The second one consists of two skein cobordisms in the regions surrounded by the dashed red ellipses in Figure \[square-co\]. The square cobordism $\Sigma_{sq}'$ can be described similarly.
[@Kh-sl3]\*[Proposition 9]{}\[square-homology\] Let $\Gamma_0,\Gamma_0', \Gamma_{sq}$ be the oriented planar webs given in Figure \[square-co\]. Then $$g_{sq}:=F(\Sigma_{sq})+ F(\Sigma'_{sq}) : F(\Gamma_0)\oplus F(\Gamma_0')\to F(\Gamma_{sq})$$ is an isomorphism. Reversing the direction of the cobordisms we obtain $\overline{\Sigma}_{sq}$ and $\overline{\Sigma}_{sq}'$. The maps $F(\overline{\Sigma}_{sq})\circ g_{sq}$ and $F(\overline{\Sigma}_{sq}')\circ g_{sq}$ are projections onto the $F(\Gamma_0)$ and $F(\Gamma_0')$ summands respectively.
\[RIVD=D’\] Let $D$ and $D'$ be the two diagrams in Figure \[RIVDD’\]. Up to a shifting of the homological and quantum gradings, $F(D)$ and $F(D')$ are chain homotopy equivalent.
(-1,-1) to (0,0); (1,-1) to (0,0); (0,1)to (0,0); (-1.3,-0.6) to (1.3,-0.6);
(1.3,0) to (2.3,0);
(2.5,-1) to (3.5,0); (4.5,-1) to (3.5,0); (3.5,1)to (3.5,0); (2.2,0.6) to (4.8,0.6);
at (0,-1.5) [$D$]{}; at (3.5,-1.5) [$D'$]{};
(-1.5,-1) to (-0.5,0); (0.5,-1) to (-0.5,0); (-0.8,1.5)to (-0.8,0.8); (-1.8,0.8) to (-0.8,0.8); (-0.5,0.4) to (0.8,0.4); (-0.5,0.4) to (-0.5,0); (-0.5,0.4) to (-0.8,0.8);
(2.5,-1) to (3.5,0); (4.5,-1) to (3.5,0); (3.5,1.5)to (3.5,1.1); (2.2,0.6) to (3,0.6); (4,0.6) to (4.8,0.6);
(2.9,0.6) to \[out=0,in=90\] (3.5,0); (3.5,1.1) to \[out=-90,in=180\] (4,0.6);
(1,0) to (2,0);
at (-0.5,-1.3) [$D_0'$]{}; at (3.5,-1.3) [$D_1'$]{};
(0,1)to (0,0); (-1,-1) to (-0.7,-0.7); (-0.3,-0.3) to (0,0); (1,-1) to (0.7,-0.7); (0.3,-0.3) to (0,0);
(-1.3,-0.6) to (-0.8,-0.6); (-0.3, -0.6) to (0.3,-0.6); (0.8,-0.6) to (1.3,-0.6);
(-0.8,-0.6) to \[out=0, in=225\] (-0.3,-0.3); (-0.7,-0.7) to \[out=45, in=180\] (-0.3, -0.6); (0.7,-0.7) to \[out=135, in=180\] (0.8,-0.6); (0.3,-0.6) to \[out=0,in=-45\] (0.3,-0.3);
(4,1)to (4,0); (3,-1) to (3.3,-0.7); (3.7,-0.3) to (4,0); (5,-1) to (4.7,-0.7); (4.3,-0.3) to (4,0);
(2.7,-0.6) to (3.2,-0.6); (3.7, -0.6) to (4.3,-0.6); (4.8,-0.6) to (5.3,-0.6);
(4.7,-0.7) to \[out=135, in=180\] (4.8,-0.6); (4.3,-0.6) to \[out=0,in=-45\] (4.3,-0.3);
(4-0.8,-0.6) to \[out=0,in=45\] (4-0.7,-0.7) ; (4-0.3,-0.3) to \[out=-135,in=180\] (4-0.3, -0.6);
(3.64, -0.525) to (4-0.7,-0.65);
(0,-2)to (0,-3); (-1,-4.3) to (-0.7,-4); (-0.3,-3.3) to (0,-3); (1,-4.3) to (0.7,-4); (0.3,-3.3) to (0,-3);
(-1.3,-3.6) to (-0.8,-3.6); (-0.3, -3.9) to (0.3,-3.9); (0.8,-3.6) to (1.3,-3.6);
(-0.8,-3.6) to \[out=0, in=225\] (-0.3,-3.3); (-0.7,-4) to \[out=45, in=180\] (-0.3, -3.9);
(0.7,-4) to \[out=135, in=180\] (0.3,-3.9); (0.8,-3.6) to \[out=180,in=-45\] (0.3,-3.3);
(0.62,-3.57) to (0.57,-3.95);
(4,-2)to (4,-3); (3,-4.3) to (4-0.7,-4); (4-0.3,-3.3) to (4,-3); (5,-4.3) to (4.7,-4); (4.3,-3.3) to (4,-3);
(2.7,-3.6) to (4-0.8,-3.6); (4-0.3, -3.9) to (4.3,-3.9); (4.8,-3.6) to (5.3,-3.6);
(4-0.3, -3.9) to \[out=180, in=225\] (4-0.3,-3.3); (4-0.7,-4) to \[out=45, in=0\] (4-0.8,-3.6) ;
(4.7,-4) to \[out=135, in=180\] (4.3,-3.9); (4.8,-3.6) to \[out=180,in=-45\] (4.3,-3.3); (4.62,-3.57) to (4.57,-3.95);
(3.56,-3.75) to (3.368,-3.85);
(1.5,0) to (2.5,0); (1.5,-3) to (2.5,-3); (0,-1) to (0,-1.7); (4,-1) to (4,-1.7);
at (-1,0.7) [$D_{00}$]{}; at (3,0.7) [$D_{10}$]{}; at (-1,-2.5) [$D_{01}$]{}; at (3,-2.5) [$D_{11}$]{};
(4,1)to (4,0); (3,-1) to (3.3,-0.7); (5,-1) to (4.7,-0.7);
(2.7,-0.6) to (3.2,-0.6); (4.8,-0.6) to (5.3,-0.6);
(4.7,-0.7) to \[out=135, in=180\] (4.8,-0.6);
(4-0.8,-0.6) to \[out=0,in=45\] (4-0.7,-0.7) ;
(4,0) to \[out=-90,in=45\] (4-0.7,-0.65);
at (4,-1.3) [$D'_{10}$]{};
By Figure \[ind-cube\], we have $$\label{D'-map}
F(D')=\textbf{Tot}[F(D'_0)\{3\}\to F(D'_1)\{2\}][-1]$$ where $D_0'$ and $D_1'$ are the diagrams in Figure \[RIV-cube-D’\]. Similarly, we have $F(D)$ is the total complex of the 2-dimensional cube $$\xymatrix{
F(D_{00})\{-4\} \ar[d]_{g_1} \ar[r]^{f_1} & F(D_{10})\{-5\}\ar[d]^{g_2} \ar@{=}[r] &F(D'_{10})\{-4\}\oplus F(D'_{10})\{-6\} \\
F(D_{01})\{-5\} \ar[r]^{f_2} & F(D_{11})\{-6\} \ar@{=}[r] & F(D'_{10})\{-6\}\oplus F(D'_1)\{-6\}
}$$ where $D_{ij}$ are diagrams in Figure \[RIV-cube-D\].
By Lemma \[digon-homology\], we can identify $F(D_{10})\{-5\}$ with $$F(D'_{10})\{-4\}\oplus F(D'_{10})\{-6\}$$ where $D'_{10}$ is the diagram in Figure \[D10’\]. Notice that $D_{10}'$ can be deformed into $D_{00}$. By the proof of Proposition \[RVD=D’\], we have the composition of the projection $\pi:F(D_{10})\{-5\}\to F(D'_{10})\{-4\}$ and $f_1$ is the identify map.
By Lemma \[square-homology\] we have the identification $$F(D_{11})\{-6\}=F(D'_{10})\{-6\}\oplus F(D'_1)\{-6\}$$ The second part of Lemma \[digon-homology\] tells us that the map induced by the digon cobordism $\overline{\Sigma}_{d1}: D_{10}'\to D_{10}$ is an embedding $$F(D_{10}')\{-6\}\cong F(D_{10}')\{-6\}\subset F(D_{10})\{-5\}$$ It is not hard to see that the composition of the cobordisms $D_{10}\to D_{11}$ and $\overline{\Sigma}_{d1}: D_{10}'\to D_{10}$ is exactly the cobordism $\Sigma_{sq}$ in Figure \[square-co\]. Therefore by Lemma \[square-homology\] we have the summand $F(D'_{10})\{-6\}$ of $F(D_{10})\{-5\}$ maps isomorphically onto the summand $F(D'_{10})\{-6\}$ of $F(D_{11})\{-6\}$ under $g_2$. Now we define a subcomplex $S$ of the cube $F(D)$ in Figure \[RIV-cube-D\] as the direct sum of
- $F(D_{00})\{-4\}$,
- $F(D'_{10})\{-6\}\subset F(D_{10})\{-5\}$ and $F(D'_{10})\{-6\}\subset F(D_{11})\{-6\}$,
- $\operatorname{Im}(f_1\oplus g_1) \subset F(D_{10})\{-5\}\oplus F(D_{01})\{-5\}$.
Our knowledge on $f_1$ and $g_2$ implies that this subcomplex is null-homotopic and the quotient complex $F(D)/S$ is isomorphic to $$\label{final-map}
\textbf{Tot}[F(D_{01})\{-5\} \to F(D_1')\{-6\}] [1]$$ which is a quotient of the bottom edge in Figure \[RIV-cube-D\]. Notice that $D_{01}\cong D_0'$. Moreover, using Lemma \[square-homology\] and Figure \[compo-digon-D’D1\], one can show that the map in and the map in are induced by the same cobordism. Hence either two maps are the same or differs by a sign, which implies $$F(D')\simeq _h F(D)\{8\}[-2]$$ This completes the proof.
We can state our main result for this subsection:
Given two diagrams $D$ and $D'$ of an oriented spatial web $\Gamma$, the two chain complexes $F(D)$ and $F(D')$ are chain homotopy equivalent up to a shifting of the bi-grading. In particular, the *relatively* bi-graded abelian group $\mathcal{H}(\Gamma):=H(F(D))$ is a well-defined invariant of the oriented spatial web.
We lose the absolute bi-grading in the situation of webs. But from the proofs of Propositions \[RVD=D’\] and \[RIVD=D’\], we see that we still have an absolute $\mathbb{Z}/4$ quantum grading.
The module structure {#module-str}
--------------------
In this subsection we first review a module structure on $F(\Gamma)$ where $\Gamma$ is an oriented planar web discussed in [@Kh-sl3]\*[Section 6]{}. Then we generalize it for oriented spatial webs.
Let $U$ be a circle in $\mathbb{R}^2$, then the pair of pants cobordism equips $F(U)$ with a ring structure. From [@Kh-sl3] we have $$F(U)=\mathbb{Z}[X]/ X^3$$ where the (quantum) gradings of $1, X, X^2$ are $-2,0,2$ respectively. So the multiplication action $X: F(U) \to F(U)$ is a degree 2 operator. Let $D_i$ be a disk with $i$ dots on it. We think $D$ as a cobordism from the empty set to a circle. The generators of $F(U)$ can be described as $$X^i=F(D_i)$$ More generally, let $\Gamma$ be an oriented planar web. Merging a circle near an edge $e_i$ gives a cobordism from $U\sqcup \Gamma$ to $\Gamma$, which induces an action $X_i: F(\Gamma)\to F(\Gamma)$. Alternatively, the map $X_i$ can be defined as the map induced by the product cobordism $\Gamma_i\times I$ with a dot in the interior of $e_i\times I$. Those actions make $F(\Gamma)$ into a $R_\Gamma$ module where $R_\Gamma$ is the commutative graded ring generated by variables $X_i$ with relations $$\label{ring1}
X_i+X_i+X_k=0, X_iX_j + X_j X_k + X_k X_i=0, X_i X_j X_k=0$$ whenever $e_i, e_j, e_k$ are incident at a common vertex and $$\label{ring2}
X_i^3=0$$ when $e_i$ is a loop. The degree of $X_i$ is defined to be 2.
Now we want to generalized the above module structure for an oriented spatial web $\Gamma$. The following argument is adapted from [@Kh-module]. Let $D$ be a diagram of $\Gamma$ and $p\in \Gamma$ be a point in the interior of edge $e_i$ disjoint from all the crossings. By merging a circle at $p$, we can define a map $X_i$ on each summand of $F(D)$ as before. Put all those map together, we obtain a chain map $$X_p: F(D)\to F(D)$$ which induces a map of bi-degree $(0,2)$ on $\mathcal{H}(\Gamma)$. If another diagram $D'$ differs from $D$ by a Reidemeister move, then there is a chain homotopy equivalence (possibly shifting the bi-grading) between $F(D)$ and $F(D')$. We can also assume the move does not touch $p$: a move of an arc above or under $p$ can be made into a move of the arc across the rest of the plane or $S^2$. Under this assumption, the chain homotopy equivalence commutes with the $X_p$ action. Therefore $$X_i:=X_{p,\ast}: \mathcal{H}(\Gamma)\to \mathcal{H}(\Gamma)$$ is a well-defined operator which only depends on the edge $e_i$. The homology $\mathcal{H}(\Gamma)$ is a $R_\Gamma$-module where $R_\Gamma$ is defined in the same way as before. Notice that $R_\Gamma$ only depends on the underlying abstract web of $\Gamma$ and does not depend on the embedding.
We have the following example from [@Kh-sl3]\*[Section 6]{}:
Suppose $\Gamma$ is an oriented planar web which can be reduced into circles by removing digons in Figure \[digon-webs\]. Using induction and Lemma \[digon-homology\], one can show that $\mathcal{H}(\Gamma)$ is a free $R_\Gamma$-module of rank 1.
Pointed webs {#pointed-web}
------------
Suppose $\Gamma$ is an oriented spatial web and $$\boldsymbol{\delta}=\{\delta_i|i=1,\cdots,m\}$$ is a collection of mark points on $\Gamma\setminus V(\Gamma)$ where $V(\Gamma)$ is the set of vertices of $\Gamma$. The pair $(\Gamma,\boldsymbol{\delta})$ is called a pointed (oriented spatial) web. In [@BLS], the ($\mathfrak{sl}(2)$) Khovanov homology for a pointed link is defined. We want to define a homological invariant for $(\Gamma,\boldsymbol{\delta})$ by imitating [@BLS].
Take a diagram $D$ for $\Gamma$ as before and assume $\boldsymbol{\delta}$ is disjoint from the crossings. Let $\Lambda_{\boldsymbol{\delta}}$ be the exterior algebra generated by formal variables $\{x_i\}$. The bi-degree (homological degree and quantum degree) of $x_i$ is $(1,2)$. We define a bi-graded chain complex $$\label{PKh}
C(D,\boldsymbol{\delta}):=\Lambda_{\boldsymbol{\delta}} \otimes_{\mathbb{Z}} F(D)$$ with differential $$d_{\boldsymbol{\delta}}(x\otimes y ):=(-1)^{\operatorname{deg}_h(x)}x\otimes d(y) + \sum_{i}x_i\wedge x \otimes X_{\delta_i}(y)$$ where $d$ is the Khovanov differential on $F(D)$, $X_{\delta_i}$ is the operator defined in Section \[module-str\] and $\operatorname{deg}_h$ denotes the homological degree. From we know $C(D,\boldsymbol{\delta})$ carries a $\Lambda_{\boldsymbol{\delta}}$-module structure. It is not hard to check that the differential on $\Lambda_{\boldsymbol{\delta}}$ respects this module structure, hence the homology of $\Lambda_{\boldsymbol{\delta}}$ is a $\Lambda_{\boldsymbol{\delta}}$-module.
Alternatively, this chain complex can be described as the total complex of a cube as the definition of $F(D)$. Index the vertices of a $m$-dimensional cube by subsets $S\subset \{1,\cdots,m\}$. Assign a copy of $F(D)$ to each vertex and assign the map $X_{\delta_i}$ to the edge $S\to S\cup \{i\} $ where $i\notin S$. After adding plus or minus sign to those edge maps appropriately, the total complex becomes the chain complex $(C(D,\boldsymbol{\delta}),d_{\boldsymbol{\delta}} )$. Equivalently, $(C(D,\boldsymbol{\delta}),d_{\boldsymbol{\delta}} )$ can be defined as the iterated total complex (mapping cone) of maps $X_{\delta_i}$.
Given two diagrams $D$ and $D'$ of an pointed oriented spatial web $(\Gamma,\boldsymbol{\delta})$, the two chain complexes $C(D,\boldsymbol{\delta})$ and $C(D',\boldsymbol{\delta})$ are chain homotopy equivalent (as $\Lambda_{\boldsymbol{\delta}}$-modules) up to a shifting of the bi-grading. In particular, the *relatively* bi-graded $\Lambda_{\boldsymbol{\delta}}$-module $\mathcal{H}(\Gamma,\boldsymbol{\delta}):=H(C(D,\boldsymbol{\delta}))$ is a well-defined invariant of $(\Gamma,\boldsymbol{\delta})$.
Suppose $D$ and $D'$ differ by a Reidemeister move. By the discussion in Section \[module-str\], we can assume this move does not touch $\boldsymbol{\delta}$ and the diagram $$\xymatrix{
F(D) \ar[d] \ar[r]^{X_{\delta_i}} & F(D) \ar[d] \\
F(D') \ar[r]^{X_{\delta_i}} & F(D') }$$ commutes. The two vertical maps are the chain homotopy equivalence between $F(D)$ and $F(D')$. Therefore the mapping cone of the two horizontal maps are also chain homotopy equivalent. Since $C(D,\boldsymbol{\delta})$ and $C(D',\boldsymbol{\delta})$ are defined as the iterated mapping cone of maps $X_{\delta_i}$ ($\delta_i\in \boldsymbol{\delta} $), iterating the above argument shows $C(D,\boldsymbol{\delta})$ and $C(D',\boldsymbol{\delta})$ are chain homotopy equivalent. The $\Lambda_{\boldsymbol{\delta}}$-module structure is just the shifting of vertices of the cube, which commutes with the chain homotopy equivalence.
We can filter the cube used to define $C(D,\boldsymbol{\delta})$ by the cardinal $|S|$. Then we obtain a spectral sequence whose $E_1$-page is $2^m$ copies of $\mathcal{H}(\Gamma)$. The differential on the $E_1$-page consists of maps $X_i=X_{\delta_i,\ast}:\mathcal{H}(\Gamma) \to \mathcal{H}(\Gamma)$. The maps $X_i$ can be viewed as elements in $R_\Gamma$. Two elements $X_i$ and $X_j$ are equal if $\delta_i$ and $\delta_j$ lie on the same edge. Let $s=(X_1,\cdots, X_m)\in R_\Gamma^{\oplus m}$, then the $E_1$-page is just the Koszul complex $K(s,\mathcal{H}(\Gamma))$. To be more precise, we first define a chain complex $$\xymatrix{
K(s):=0 \to R_\Gamma \ar[r]^-{\wedge s} & \Lambda^1 R_\Gamma^{\oplus m} \ar[r]^-{\wedge s} & \Lambda^2 R_\Gamma^{\oplus m}
\ar[r]^-{\wedge s} & \cdots \ar[r]^-{\wedge s} & \Lambda^m R_\Gamma^{\oplus m}
\to 0
}$$ Then the Koszul complex is defined as $$K(s,\mathcal{H}(\Gamma)):=K(s)\otimes_{R_\Gamma}\mathcal{H}(\Gamma)$$ In summary, we have
\[koszul\] Suppose $(\Gamma,\boldsymbol{\delta}=\{\delta_i\}_{1\le i \le m})$ is a pointed oriented spatial web and $s=(X_1,\cdots, X_m)\in R_\Gamma^{\oplus m}$ as above. Then there is a spectral sequence converging to $\mathcal{H}(\Gamma,\boldsymbol{\delta})$ whose $E_1$-page is the Koszul complex $K(s,\mathcal{H}(\Gamma))$. In particular, we have $$\operatorname{rank}_{\mathbb{Z}} H(K(s,\mathcal{H}(\Gamma)))\ge \operatorname{rank}_{\mathbb{Z}} \mathcal{H}(\Gamma,\boldsymbol{\delta})$$
When $\Gamma$ is a planar web, the spectral sequence degenerates on the $E_2$-page and $$\mathcal{H}(\Gamma,\boldsymbol{\delta}) \cong H(K(s,\mathcal{H}(\Gamma)))$$
Let $\Theta$ be the planar theta graph and $\boldsymbol\delta =\{\delta_1,\delta_2\}$ where $\delta_1,\delta_2$ lie on two distinct edges. We have $$\mathcal{H}(\Theta)\cong R_\Theta \cong \mathbb{Z}[X_1,X_2]/ (X_1^2 X_2+X_1 X_2^2, X_1^2+X_2^2+X_1X_2 )$$ The homology $\mathcal{H}(\Theta,\boldsymbol{\delta})$ is defined as the homology of the cube $$\xymatrix{
R_\Theta \ar[d]_{X_2} \ar[r]^{X_1} & R_\Theta \ar[d]^{-X_2} \\
R_\Theta \ar[r]^{X_1} & R_\Theta }$$ It is straightforward to check that $\mathcal{H}(\Theta,\boldsymbol{\delta})$ is a free abelian group of rank 4.
The functor $J$ and the spectral sequence
=========================================
The instanton Floer homology for spatial trivalent graphs is introduced by Kronheimer and Mrowka in [@KM-jsharp]. In this section we first review some results we need from their work. Then we will show that there is a spectral sequence whose $E_2$-page is the $\mathfrak{sl}(3)$ Khovanov module of a spatial web and which converges to the instanton Floer homology of this spatial web. All the discussion in Section \[sl3Kh\] can be done with $\mathbb{F}$-coefficients verbatim and we use $F(\Gamma;\mathbb{F})$ to denote the functor defined in Section \[sl3Kh\] adapted with $\mathbb{F}$-coefficients.
Instanton Floer homology for webs {#functor-I}
---------------------------------
Suppose $\Gamma$ is a trivalent graph embedded in an oriented compact three-manifold $Y$. The pair $\check{Y}=(Y,\Gamma)$ can be equipped with an orbifold structure and an orbifold Riemannian metric such that the local stabilizer group $H_x\subset SO(3)$ is $\mathbb{Z}/2$ (if $x$ is not a vertex) or the Klein 4-group (if $x$ is a vertex). We call $\check{Y}$ a *bifold* and the orbifold metric on it a bifold metric. In most of the situations of this paper, $Y$ will just be $S^3$ (or $\mathbb{R}^3$).
A bifold connection over $\check{Y}$ is an orbifold $SO(3)$ connection whose asymptotic holonomy around each edge of $\Gamma$ is an order $2$ element in $SO(3)$. The underlying orbifold vector bundle of a bifold connection is called a bifold bundle. Adding a Hopf link $H$ contained in a ball $U_\mu\subset Y$ disjoint from $\Gamma$, we obtain a new bifold $(Y,\Gamma\cup H)$. Let $E_\mu\to U_\mu\setminus H$ be the $SO(3)$ bundle whose $w_2$ is dual to a small arc joining the two components of $H$. We use $\mathcal{A}$ to denote the space of bifold connections over bifold bundles which are identified with $E_\mu$ on $U_\mu\setminus \Gamma$. The gauge group $\mathcal{G}$ consists of $SO(3)$ gauge transformations $g$ such that the restriction $g|_{U_\mu\setminus \Gamma}: E_\mu\to E_\mu$ can be lifted into to a determinant-1 gauge transformation. The instanton Floer homology $J^\sharp(Y,\Gamma)$ is defined as the Morse homology (with $\mathbb{F}$-coefficients) of the Chern-Simons functional on the configuration space $\mathcal{B}^\sharp(Y,K)=\mathcal{A}/\mathcal{G}$. If $\Gamma$ is a spatial trivalent graph, then it can also be viewed as a graph in $S^3$ and we just write $J^\sharp(\Gamma)$ for $J^\sharp(S^3,\Gamma)$. See [@KM-jsharp] for more details of the definition.
There is a variant of $J^\sharp(Y,\Gamma)$ also introduced in [@KM-jsharp]: take $U_\mu$ to be the whole manifold $Y$ and repeat other parts of the definition, we obtain an invariant $I^\sharp(Y,\Gamma)$. We will not need $I^\sharp$ until the next section. The reason to introduce the Hopf link $H$ with non-trivial $w_2$ in the definition (for $J^\sharp$ or $I^\sharp$) is to avoid reducible connections.
A *generalized closed foam* $\Sigma$ in $\mathbb{R}^4$ is a 2-dimensional subcomplex decorated with dots such that each point $x$ in $\Sigma$ has a neighborhood (in $\Sigma$) which is modelled on one of the following
- A smoothly embedded disk;
- The product of an interval and the letter “Y”;
- A cone with vertex $x$ and whose base is the complete graph $K_4$ with four vertices.
The circles or arcs consist of points of the second type is called seams as before. Points of the third type are called tetrahedral points. The connected components of the complement of seams and tetrahedral points are called facets. We also require that the dots lie in the facets.
We can also define generalized foams with boundary:
A *generalized foam with boundary* is the intersection of $\mathbb{R}^3\times [0,1]$ and a generalized closed foam $\Sigma$ such that $\mathbb{R}^3\times\{0\}$ and $\mathbb{R}^3\times\{1\}$ are transversal to the facets and seams of $\Sigma$ and contain no tetrahedral points. In particular, $\Gamma_1=\mathbb{R}^3\times\{0\}\cap \Sigma$ and $\Gamma_2=\mathbb{R}^3\times\{1\}\cap \Sigma$ are two spatial trivalent graph. The generalized foam with boundary can be thought as a cobordism from $\Gamma_1$ to $\Gamma_2$.
The above definitions are more general than the ones used in Section \[sl3Kh\].
Given a cobordism $\Sigma$ from $\Gamma_1$ to $\Gamma_2$, we can also define a 4-dimensional bifold $\check{W}:=(S^3\times \mathbb{R}, \Sigma^+\cup H\times \mathbb{R})$ as in the 3-dimensional case. Here $\Sigma^+$ is obtained by adding cylindrical ends to $\Sigma$. The concepts of bifold connections, bifold bundles and configuration spaces can all be defined for the 4-dimensional case. If $\Sigma$ has no dots, there is a map $J^\sharp (\Sigma):J^\sharp (\Gamma_1)\to J^\sharp (\Gamma_2)$ defined by counting points in the 0-dimensional moduli spaces of ASD trajectories of bifold connections over $\check{W}$. If there are dots on $\Sigma$, the homology class of the meridian of each dot $\delta_i$ determines a cohomology class in $H^1(B^\sharp(S^3\times \mathbb{R},\Sigma^+);\mathbb{F})$, which can be represented by a (real) codimension 1 divisor $V(\delta_i)$ in the configuration space. The map $J^\sharp (\Sigma)$ is defined by counting points in the 0-dimensional cutting-down moduli spaces (cutting down by divisors associated with those dots). A closed generalized foam $\Sigma$ can be thought a cobordism from an empty graph to another empty graph and $J^\sharp(\Sigma)$ is a number in $\mathbb{F}$.
We summarize some properties of $J^\sharp$:
[@KM-jsharp]\[jsharp-p\] $J^\sharp$ satisfies the following properties:
1. If $\Sigma$ is a closed orientable surface (embedded standardly in $\mathbb{R}^3\subset \mathbb{R}^4$) decorated with dots, then $F(\Sigma)=0$ unless
- $\Sigma$ is a 2-sphere with two dots, then $F(\Sigma)=1$,
- $\Sigma$ is a torus without any dots, then $F(\Sigma)=1$.
2. If $\Sigma_1$ and $\Sigma_2$ are generalized foams in $\mathbb{R}^4$, then $F(\Sigma_1\sqcup \Sigma_2)=F(\Sigma_1)F(\Sigma_2)$.
3. Do a surgery along a circle inside a facet of $\Sigma$ to obtain three generalized foams $\Sigma_1, \Sigma_2, \Sigma_3$ as in Figure \[Sigma123\]. Then $$F(\Sigma)=F(\Sigma_1)+F(\Sigma_2)+F(\Sigma_3)$$
4. Let $\Theta(k_1,k_2,k_3)$ be the theta foam (embedded standardly in $\mathbb{R}^3\subset \mathbb{R}^4$) with $k_i$ dots on the $i$-th facet (see Figure \[theta-foam\]), then $$F(\Theta(k_1,k_2,k_3))=\left\{
\begin{array}{ll}
1, & \hbox{if $(k_1,k_2,k_3)=(0,1,2)$, up to permutation;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
5. Suppose $\Gamma_1$ and $\Gamma_2$ are two spatial trivalent graphs locally as shown in Figure \[digon-webs\] (no orientation is needed now), then $$J^\sharp(\Gamma_1)=J^\sharp(\Gamma_2)\oplus J^\sharp(\Gamma_2)$$
6. Suppose $\Gamma_{sq}$, $\Gamma_0$ and $\Gamma_0'$ are spatial trivalent graphs locally as shown in Figure \[square-co\] (no orientation is needed), then $$J^\sharp(\Gamma_{sq})=J^\sharp(\Gamma_0)\oplus J^\sharp(\Gamma_0')$$
7. Suppose $U$ is the unknot in $\mathbb{R}^3$, then $J^\sharp(U)\cong \mathbb{F}^3$.
With $\mathbb{F}$-coefficients, $J^\sharp$ satisfies the four axioms in Proposition \[4axiom\] for oriented foams defined in Section \[sl3Kh\]. Therefore $J^\sharp(\Sigma)=F(\Sigma;\mathbb{F})$ for any closed oriented foam $\Sigma$.
Given an oriented planar web $\Gamma$, we define $$J'(\Gamma)=\operatorname{span}\{J^\sharp(\Sigma)|\Sigma\in \operatorname{Hom}_{\text{OF}} (\emptyset,\Gamma)\}\subset J^\sharp(\Gamma)$$ We have a surjective linear map $$J'(\Gamma)\to F(\Gamma;\mathbb{F})$$ defined by $$J^\sharp(\Sigma)\mapsto [\Sigma]$$ where $\Sigma$ is a foam in $\operatorname{Hom}_{\text{OF}} (\emptyset,\Gamma)$. We need to show this map is well-defined. Suppose $$J^\sharp(\Sigma_1)=J^\sharp(\Sigma_2)$$ Then for any $\Phi \in \operatorname{Hom}_{\text{OF}} (\Gamma,\emptyset)$, we have $$J^\sharp(\Phi\circ\Sigma_1)-J^\sharp(\Phi\circ\Sigma_2)=0$$ By the functoriality of $J^\sharp$. Therefore $$[\Sigma_1]-[\Sigma_2]=0$$ by the definition of $F$ in Section \[sl3Kh\]. So the map is well-defined and $F(\Gamma;\mathbb{F})$ is a subquotient of $J^\sharp(\Gamma)$. This map turns out to be an isomorphism.
\[J=F\] Given any oriented planar web $\Gamma$, we have $$J^\sharp(\Gamma)=J'(\Gamma)$$ is isomorphic to $F(\Gamma;\mathbb{F})$. The isomorphism is natural in the sense that given two oriented planar webs $\Gamma_1$ and $\Gamma_2$, then the digram $$\xymatrix{
J^\sharp(\Gamma_1)\ar[d]_{J^\sharp(\Sigma)} \ar[r]^{\simeq} & F(\Gamma_1,\mathbb{F}) \ar[d]^{F(\Sigma,\mathbb{F})} \\
J^\sharp(\Gamma_2)\ar[r]^{\simeq} & F(\Gamma_2,\mathbb{F}) }$$ commutes for any $\Sigma\in \operatorname{Hom}_{\emph{OF}} (\Gamma_1,\Gamma_2)$.
Any oriented planar web can be reduced into a collection of circles by removing digons (Figure \[digon-webs\]) and squares (Figure \[square-co\]) (cf. [@Kh-sl3]\*[Section 2]{}). Using Lemma \[digon-homology\], Lemma \[square-homology\] and Parts (5) (6) (7) of Proposition \[jsharp-p\] to do inductions, it is easy to show that $$\dim_{\mathbb{F}}J^\sharp(\Gamma)=\dim_{\mathbb{F}}F(\Gamma;\mathbb{F})$$ for any oriented planar web $\Gamma$. Therefore we have $$J^\sharp(\Gamma)=J'(\Gamma)\cong F(\Gamma;\mathbb{F})$$ The naturality of the isomorphism follows directly from its definition.
Given any (abstract) trivalent graph $\Gamma$ with edges $\{e_i\}$, we can define a ring $\mathcal{R}_\Gamma$ generated by $X_i$ over $\mathbb{F}$ modulo relations and . Alternatively, we can define $$\mathcal{R}_\Gamma:=R_\Gamma\otimes_\mathbb{Z} \mathbb{F}$$ where $R_\Gamma$ is defined in Section \[module-str\]. Now $F(\Gamma;\mathbb{F})$ carries a $\mathcal{R}_\Gamma$-module structure.
Suppose $\Gamma$ is a spatial trivalent graph, then $J^\sharp(\Gamma)$ is also equipped with a module structure in [@KM-jsharp]. The module structure can be defined in a similar way as in Section \[module-str\]: given an edge $e_i$ of $\Gamma$, we form a generalized foam $\Sigma=\Gamma\times I$ with a dot on $e_i\times I$ and define $$X_i=J^\sharp(\Sigma):J^\sharp(\Gamma)\to J^\sharp(\Gamma)$$ This makes $J^\sharp(\Gamma)$ into a $\mathcal{R}_\Gamma$-module. The isomorphism in Proposition \[J=F\] respects the module structures on both sides.
Exact triangles
---------------
From now on we assume certain perturbations are chosen so that all the moduli spaces are regular. Denote the three links in Figure \[+01\] by $L_2, L_0, L_1$ and the three links in Figure \[-01\] by $L_2', L_0', L_1'$. The Floer chain complex used to define $J^\sharp(L_i)$ (or $J^\sharp(L_i')$) is denoted by $C_i$ ($i\in \mathbb{Z}/3)$. The following result can be read from [@KM-triangle]:
\[3-gon\] There exist maps $$f_{i,{i+k}} :C_{i}\to C_{i+k}, ~k=1,2~\text{or}~3$$ such that $$\begin{aligned}
df_{i,i+1}+f_{i,i+1}d&=& 0 \\
df_{i,i+2}+f_{i,i+2}d &=& f_{i+1,i+2}f_{i,i+1}\\
df_{i,i+3}+f_{i,i+3}d &=& f_{i+1,i+3}f_{i,i+1} + f_{i+2,i+3}f_{i,i+2}+g_i\end{aligned}$$ where $d$ is the Floer differential and $$g_i:C_{i}\to C_{i+3}=C_i$$ is a quasi-isomorphism.
By [@OS-ss]\*[Lemma 4.2]{}, the above proposition implies
\[3-gon-quasi-iso\] The map $$f_{{i},i+2}+ f_{{i+1},i+2} : \operatorname{cone}(f_{{i},i+1})=C_{i}\oplus C_{i+1}\to C_{i+2}$$ is an quasi-isomorphism.
There is a standard foam cobordism $\Sigma_{i,i+1}$ from $L_i$ to $L_{i+1}$ (or from $L_i'$ to $L_{i+1}'$). In particular, $\Sigma_{0,1}$ is the cobordism in Figure \[skein-co\]. We can make $\Sigma_{i,i+1}$ into a surface $\Sigma_{i,i+1}^+$ with cylindrical end and $\Sigma_{i,i+1}$ is a product away from a four-ball where the skein move happens. The pair $(S^3\times \mathbb{R},\Sigma_{i,i+1}^+\cup H\times \mathbb{R})$ is equipped with a family of bifold metrics parametrized by $G_{i,i+1}\cong \mathbb{R}$. The $\mathbb{R}$-translation on $S^3\times \mathbb{R}$ gives an $\mathbb{R}$-action on $G_{i,i+1}$ and the quotient space $\breve{G}_{i,i+1}$ consists of a single metric. The map $f_{i,i+1}$ is defined by counting points in the zero-dimensional moduli space over $\breve{G}_{i,i+1}$. Passing to homology, $f_{i,i+1}$ is exactly the map $J^\sharp(\Sigma_{i,i+1}):J^\sharp(L_i)\to J^\sharp(L_{i+1})$.
In general, let $\Sigma_{i,i+k}:L_i\to L_{i+k}$ be the composition of cobordisms $\Sigma_{i+k-1,i+k},\cdots, \Sigma_{i,i+1}$ and $\Sigma_{i,i+k}^+$ be the surface obtained by adding cylindrical ends to $\Sigma_{i,i+k}$. A $(k-1)$-dimensional family of bifold metrics on $(S^3\times \mathbb{R},\Sigma_{i,i+k}^+)$ parametrized by $\breve{G}_{i,i+k}$ is defined in [@KM-jsharp] for $k=1,2,3$ (see also [@KM:Kh-unknot]). Again there is a $k$-dimensional family of metrics parametrized by $G_{i,i+k}$ with an $\mathbb{R}$-action such that $\breve{G}_{i,i+k}=G_{i,i+k}/\mathbb{R}$. The map $f_{i,i+k}$ is defined by counting points in the zero-dimensional moduli spaces over $\breve{G}_{i,i+k}$ and the relations in Proposition \[3-gon\] are derived by analyzing the boundaries of 1-dimensional moduli spaces over $\breve{G}_{i,i+k}$.
The spectral sequence {#J-ss}
---------------------
In this subsection we will prove the following.
\[s-sequence\] Let $\Gamma$ be an oriented spatial web. There is a spectral sequence of $\mathcal{R}_\Gamma$-modules whose $E_2$-page is the $\mathfrak{sl}(3)$ Khovanov module $\mathcal{H}(\Gamma;\mathbb{F})$ and which converges to $J^\sharp(\Gamma)$.
The proof is very little different from the proof of a corresponding result for $\mathfrak{sl}(2)$ Khovanov homology in [@KM:Kh-unknot]. The current situation is even simpler in some aspect: since we are working in characteristic 2, there is no need to deal with the orientations and signs as in [@KM:Kh-unknot]. All the necessary ingredients of the proof are already included in [@KM:Kh-unknot; @KM-jsharp; @KM-triangle].
Suppose $\Gamma$ is an oriented spatial web and $D$ is a diagram for $\Gamma$ with $n$ crossings. Given $v\in \{0,1\}$, let $D_v$ be the planar web obtained by resolving the crossings using $v$ as in Section \[sl3Kh\]. We define $$\begin{aligned}
|v|_1 &=& \sum_i |v_i| \\
|v|_\infty &=& \sup_i |v_i|\end{aligned}$$ for any $v\in \mathbb{R}^n$. Given any $v\in \{0,1\}^n$, we use $C_v$ to denote the Floer chain complex for $J^\sharp(D_v)$ and use $d_v$ to denote the differential on $C_v$. Given $v\le u$ in $\{0,1\}^c$ (i.e. $v_i\le u_i$ for all $i$ where $v_i, u_i$ are coordinates of $v,u$), there is a cobordism $S_{vu}$ which can be made into a surface $S_{vu}^+\subset S^3\times \mathbb{R}$ with cylindrical end in the standard way. The surface $S_{vu}^+$ is a product surface away from $|v-u|_1$ four-balls where the skein moves happen. By shifting these four-balls containing the skein moves, we can define a family of metrics parameterized by $G_{vu}\cong \mathbb{R}^{|v-u|_1}$. There is an $\mathbb{R}$-action on $G_{vu}$ defined by the $\mathbb{R}$-translation on $\mathbb{R}\times S^1\times S^2$. The quotient $G_{vu}\slash \mathbb{R}$ is denoted by $\breve{G}_{vu}$. $G_{vu}$ and $\breve{G}_{vu}$ are not compact in general but can be compactified into spaces $G_{vu}^+$ and $\breve{G}_{vu}^+$ by adding broken metrics. Let $
M_{vu}(\alpha,\beta)_d
$ be the $d$-dimensional moduli space of ASD trajectories over $G_{vu}$ with limiting connection $\alpha$ on the incoming end and $\beta$ on the outgoing end. Here $\alpha$ and $\beta$ are generators for $C_v$ and $C_u$ respectively. There is an obvious map $M_{vu}\to G_{vu}$ and the $\mathbb{R}$-action on $G_{vu}$ can be lifted on $M_{vu}(\alpha, \beta)_d$. We denote the quotient by $$\breve{M}_{vu}(\alpha,\beta)_{d-1}:=M_{vu}(\alpha, \beta)_d/\mathbb{R}$$ Both $M_{vu}(\alpha,\beta)_d$ and $\breve{M}_{vu}(\alpha,\beta)_{d-1}$ can be partially compactified by adding broken trajectories lying over broken metrics in $\partial G_{vu}^+$ and $\partial\breve{G}_{vu}^+$. These are only partial compactifications because of the possible appearance of bubbles. We denote these partial compactifications by $M^+_{vu}(\alpha,\beta)$ and $\breve{M}^+_{vu}(\alpha,\beta)$ respectively. A group homomorphism $$\label{fvu}
f_{vu}: C_v\to C_u$$ can be defined by counting points in the 0-dimensional moduli space: $$\label{fvu=}
f_{vu}(\alpha):=\sum_\beta \# \breve{M}_{vu}(\alpha,\beta)_0 \cdot \beta$$ where $\beta$ runs through all the generators for $C_u$. In the case $v=u$, $m_{vv}$ is just the Floer differential.
Suppose $e_i$ is an edge of $\Gamma$ and $\delta_i\in e_i$ is a point disjoint with all the three-balls where the skein moves of $\Gamma$ happen. The homology class of the meridian around $e_i$ determines an element $\zeta_i$ in $H^1(B^\sharp(Y,\Gamma);\mathbb{F})$. Let $\nu(\delta_i)\subset S^3$ be a suitable neighborhood of $\delta_i$ that is also disjoint from all the skein moves and contains $H$. A (real) codimension 1 divisor $$V(\delta_i)\subset B^\sharp(\delta_i):= B^\sharp( \nu(\delta_i)\times (-1,1),(\Gamma\cap\nu(\delta_i))\times (-1,1))$$ is defined in [@KM-jsharp]\*[Section 3.5]{}. The pullback of $V(\delta_i)$ to $B^\sharp(Y,\Gamma)$ is the dual of $\zeta_i$. Given $v\le u$ as before, we define a map $$r_{vu}:C_v\to C_u$$ by $$\label{rvu}
r_{vu}(\alpha):= \sum_\beta \# ({M}_{vu}(\alpha,\beta)_1 \cap V(\delta_i)) \cdot \beta$$ where $({M}_{vu}(\alpha,\beta)_1 \cap V(\delta_i))$ should be understood as pulling back the divisor $V(\delta_i)$ by the restriction $r: M_{vu}\to B^\sharp(\delta_i)$. We assume the divisors $V(\delta_i)$ are generic so that all the intersections are regular. When $v=u$, $r_{vv}:C_v\to C_v$ induces the map $$X_i=J^\sharp(\dot{D}_v\times I):J^\sharp(D_v)\to J^\sharp(D_v)$$ on the homology where $\dot{D}_v\times I$ is the foam $D_v\times I$ with a dot on the facet $e_i\times I$. Now we define $$\label{CF}
\mathbf{C}=\bigoplus_{v\in \{0,1\}^n}C_v,~ \mathbf{F}:=\sum_{v\le u}f_{vu}:\mathbf{C} \to \mathbf{C}$$ and $$\label{Rrvu}
\mathbf{R}:=\sum_{v\le u}r_{vu}:\mathbf{C} \to \mathbf{C}$$
\[CF-chain\] We have $$\mathbf{F}\mathbf{F}=0, ~ \mathbf{F}\mathbf{R}+\mathbf{R}\mathbf{F}=0$$ so that $$(\mathbf{C},\mathbf{F})$$ is a chain complex and $$\mathbf{R}:\mathbf{C}\to \mathbf{C}$$ is a chain map.
The two equalities are derived by counting boundary points of 1-dimensional moduli spaces $\breve{M}_{vu}^+(\alpha,\beta)_1$ and ${M}_{vu}^+(\alpha,\beta)_2 \cap V(\delta_i)$ respectively. The boundary of $\breve{M}_{vu}^+(\alpha,\beta)_1$ consists of $$\breve{M}_{vw}(\alpha,\eta)_0 \times \breve{M}_{wu}(\eta,\beta)_0$$ where $v\le w\le u$. The moduli space $\breve{M}_{vu}^+(\alpha,\beta)_1$ may have open ends coming from bubbles. But the number of such ends is always an even number by the argument in [@KM-jsharp]\*[Section 3.3]{}. Since we are working in characteristic 2, we can ignore such ends. Therefore we have $$\sum_{w,\eta}\# \breve{M}_{vw}(\alpha,\eta)_0 \cdot \# \breve{M}_{wu}(\eta,\beta)_0=0$$ This implies the component of $\mathbf{F}\mathbf{F}$ mapping $C_v$ to $C_u$ is $0$. Since $v,u$ are arbitrary, this completes the proof of the first equality.
The boundary of ${M}_{vu}^+(\alpha,\beta)_2 \cap V(\delta_i)$ consists of $$({M}_{vw}(\alpha,\eta)_1\cap V(\delta_i)) \times \breve{M}_{wu}(\eta,\beta)_0$$ and $$\breve{M}_{vw}(\alpha,\eta)_0 \times ({M}_{wu}(\eta,\beta)_1\cap V(\delta_i))$$ where $v\le w\le u$. We still need to exclude the possible appearance of bubbles. According to [@KM-jsharp]\*[Section 3.3]{}, a codimension-2 bubble can only arise in the situation that $v=u, \alpha=\beta$ and there is a sequence of connections $A_l\in M_{vv}(\alpha,\alpha)_2$ such that $$\xymatrix{
A_l|_{S^3\times \mathbb{R}\setminus z} \ar[r] & \widetilde{\alpha}|_{S^3\times \mathbb{R}\setminus z}
}$$ where $\widetilde{\alpha}$ represents the product trajectory obtained by pulling $\alpha$ back to the product cobordism and $z$ is the bubble point on the *seam* of the orbifold points. Since $\delta_i$ is disjoint from the seam and $V(\delta_i)$ is generic, a sequence of connections in ${M}_{vu}^+(\alpha,\beta)_2 \cap V(\delta_i)$ can never converge to an ideal connection $(\widetilde{\alpha},z)$ as above. Therefore we have $$\label{M-FR+RF}
\begin{aligned}
\sum_{w,\eta}\# ({M}_{vw}(\alpha,\eta)_1\cap V(\delta_i))\cdot \# \breve{M}_{wu}(\eta,\beta)_0 + \\
\sum_{w,\eta}\#\breve{M}_{vw}(\alpha,\eta)_0 \cdot \# ({M}_{wu}(\eta,\beta)_1\cap V(\delta_i)) =0
\end{aligned}$$ This implies the component of $\mathbf{F}\mathbf{R}+\mathbf{R}\mathbf{F}$ mapping $C_v$ to $C_u$ is $0$. Since $v,u$ are arbitrary, this completes the proof of the second equality.
See [@KM:Kh-unknot] for more details on the counting argument. Also compare the proof of Proposition \[CU-chain\] where a more general situation is discussed.
\[cube-quasi-iso\] The chain complex $$(\mathbf{C},\mathbf{F})$$ is quasi-isomorphic to the Floer chain complex $C(\Gamma)$ for $J^\sharp(\Gamma)$. Moreover, the induced isomorphism $$H(\mathbf{C})\cong H(C(\Gamma))=J^\sharp(\Gamma)$$ intertwines with the induced action $\mathbf{R}_\ast$ and $X_i$.
If there is only one crossing in the diagram $D$, then the quasi-isomorphism is given in Corollary \[3-gon-quasi-iso\]. The general case can be obtained by “iterating” Corollary \[3-gon-quasi-iso\]. Let $v\in\{0,1,2\}^n$, we can resolve $\Gamma$ to obtain a spatial web $D_{v}$ where a “2-resolution” means keeping the crossing without any change. If $v,u \in\{0,1,2\}^n$ and $v\le u$, we still have a cobordism $S_{vu}:D_v\to D_u$ obtained by composing the skein cobordisms. For $v\le u$ and $|u-v|_\infty\le 1$ in $\{0,1,2\}^n$, we can define a family of metrics on $(S^3\times \mathbb{R},S_{vu}^+)$ by shifting the four-balls containing the skein move as before. Let $C_v$ be the Floer chain complex for $J^\sharp({D_v})$. Form a chain complex $$(\mathbf{C}_2,\mathbf{F}_2)=(\bigoplus_{v'\in\{0,1\}^{n-1}}C_{2v'},\mathbf{F}_2)$$ equipped with an action $\mathbf{R}_2$ by and as in the definitions of $\mathbf{F}$ and $\mathbf{R}$.
Suppose $v', u'\in\{0,1\}^{n-1}$ and $v'\le u'$, then a family of metrics parametrized by $$G_{0v',2u'}\cong \mathbb{R}^{|2u'-0v|_1}=\mathbb{R}^2\times \mathbb{R}^{|u'-v'|_1}$$ can be defined on $(S^3\times \mathbb{R}, \Sigma_{0v',2u'}^+)$. The $\mathbb{R}^{|u'-v'|_1}$ component is obtained by shifting four-ball containing the skein moves for crossings associated with $v',u'$. Take $(x,y)\in \mathbb{R}^2$. When $y-x\ge 1$, they describe the locations of the two skein moves of the first crossing. When $y-x < 1$, the metric is defined by stretching along the collar of a specific 3-dimensional sub-bifold of $(S^3\times \mathbb{R}, \Sigma_{0v',2u'}^+)$. See [@KM-triangle]\*[Section 6]{} and [@KM:Kh-unknot]\*[Section 7.2]{} for the precise definition.
Suppose $v\in\{0,1\}^{n}, u'\in\{0,1\}^{n-1}$ and $v\le 2u'$. Counting points in the 0-dimensional moduli spaces $\breve{M}_{v,2u'}$ over $\breve{G}_{v,2u'}:=G_{v,2u'}/\mathbb{R}$ gives a map $\mathbf{H}$ from $\mathbf{C}$ to $\mathbf{C}_2$. To be more precise, we define $$\mathbf{H}:=\sum h_{v,2u'}$$ where $$h_{v,2u'}:C_v\to C_{2u'}$$ is defined by $$h_{v,2u'}(\alpha)=\sum_\beta \# \breve{M}_{v,2u'}(\alpha,\beta)_0 \cdot \beta$$ The arguments in [@KM:Kh-unknot]\*[Section 7]{} and [@KM-triangle] show $\mathbf{H}$ is a quasi-isomorphism.
We define $$\mathbf{R}':\mathbf{C} \to \mathbf{C}_2, ~\mathbf{R}':=\sum r'_{v,2u'}$$ where $$r'_{v,2u'}:C_v\to C_{2u'}$$ is defined by $$r'_{v,2u'}(\alpha):=\sum_\beta \# ({M}_{v,2u'}(\alpha,\beta)_1 \cap V(\delta_i)) \cdot \beta$$ Counting boundary points of the 1-dimensional cutting-down moduli space $M^+_{v,2u'}\cap V(\delta_i)$, we obtain an equality similar to (with $u$ replaced by $2u'$), which implies $$\mathbf{R}_2\mathbf{H}+\mathbf{H}\mathbf{R}+\mathbf{F}_2\mathbf{R}'+\mathbf{R}'\mathbf{F}=0$$ This means $\mathbf{H}\mathbf{R}+\mathbf{R}_2\mathbf{H}$ is chain homotopic to zero. Hence $\mathbf{H}$ induces an isomorphism on the homologies that intertwines with $\mathbf{R}$ and $\mathbf{R}_2$.
Iterating the above argument, we finally obtain a quasi-isomorphism between $\mathbf{C}$ and $\mathbf{C}_{2,2,\cdots,2}=C(\Gamma)$. Moreover, the induced isomorphism on homologies intertwines with $\mathbf{R}_\ast$ and $\mathbf{R}_{{2,\cdots,2},\ast} =X_i:J^\sharp(\Gamma)\to J^\sharp(\Gamma)$.
Now we are ready to prove the main result of this section:
Using Proposition \[cube-quasi-iso\] and filtering the cube $\mathbf{C}$ by the sum of coordinates, we obtain a spectral sequence which converges to $J^\sharp(\Gamma)$ and whose $E_1$-page is $$\bigoplus_{v\in \{0,1\}^n} J^\sharp(D_v)$$ The differential on the $E_1$-page is $$\sum_{\substack{v<u\\|v-u|_1=1}} J^\sharp(\Sigma_{vu})$$ When $v,u\in \{0,1\}^n$ only differ at one coordinate, $\Sigma_{vu}$ is just the skein cobordism in Figure \[skein-co\]. By Proposition \[J=F\] and the definition of the Khovanov homology in Section \[sl3Kh\], it is clear that the $E_2$-page is exactly $\mathcal{H}(\Gamma;\mathbb{F})$.
On the $E_1$-page, the operator $\mathbf{R}$ becomes the operator $X_i:J^\sharp(D_v)\to J^\sharp(D_v)$. By the discussion at the end of Section \[functor-I\], we know that $\mathbf{R}$ on the $E_2$-page is the operator $X_i$ on $\mathcal{H}(\Gamma;\mathbb{F})$. This completes the proof.
Suppose $L$ is a link in $\mathbb{R}^3$ with components $L_1,\cdots, L_m$. Let $R_m$ be the ring $$\mathbb{Z}[X_1,\cdots, X_m]/(X_1^2,\cdots, X_m^2)$$ The $\mathfrak{sl}(2)$ Khovanov homology of $L$ can be equipped with an $R_m$-module structure [@Kh-module]. An $R_m$-module structure on the instanton Floer homology $I^\sharp(L;\mathbb{Z})$ is defined implicitly in [@KM:Kh-unknot]\*[Section 8.3]{}. The argument in this section can be used to show that the spectral sequence in [@KM:Kh-unknot] relating the $\mathfrak{sl}(2)$ Khovanov homology to $I^\sharp(L;\mathbb{Z})$ respects the $R_m$-module structures.
The functor $I$ and the spectral sequence
=========================================
In this section we will construct a spectral sequence relating the pointed $\mathfrak{sl}(3)$ Khovanov homology to the functor $I^\sharp$ which we briefly reviewed at the beginning of Section \[functor-I\].
Let $\Gamma$ be a *connected* spatial trivalent graph. The main difference between the definitions of $J^\sharp(\Gamma)$ and $I^\sharp(\Gamma)$ is the choice of gauge groups: we use the gauge group $\mathcal{G}$ of determinant-1 over a ball containing $H$ for $J^\sharp$ and the gauge group $\widetilde{\mathcal{G}}$ of determinant-1 over the whole manifold $S^3$ for $I^\sharp$. Since the definition of $J^\sharp$ uses a larger gauge group, we have a (normal) covering map $$\widetilde{B}^\sharp(S^3,\Gamma) \to B^\sharp(S^3,\Gamma)$$ between the configuration spaces for $I^\sharp$ and $J^\sharp$. The deck transformation group is ${\mathcal{G}}/\widetilde{\mathcal{G}}\cong H^1(S^3\setminus \Gamma;\mathbb{F})$. In particular each element in $H_1(S^3\setminus \Gamma;\mathbb{F})$ determines a cohomology class in $H^1({B}^\sharp(S^3,\Gamma);\mathbb{F})$ which is already used in Section \[J-ss\].
Given a spatial trivalent graph $\Gamma$, let $C(\Gamma)$ be the Floer chain complex for $J^\sharp(L)$. Pick edges $e_1,e_2,\cdots, e_m$ such that the meridians around those edges form a basis for $H_1(\mathbb{R}^3\setminus \Gamma, \mathbb{F})$. Use $\zeta_i$ to denote the cohomology class in $H^1({B}^\sharp(S^3,\Gamma);\mathbb{F})$ determined by $e_i$. Pick a point $\delta_i\in e_i$ for each edge $e_i$ and use $V(\delta_i)$ to denote the (real) divisor representing $\zeta_i$ used in Section \[J-ss\]. Suppose $S\subset \{1,\cdots,m\}$, we define a map $$U_S: C(\Gamma)\to C(\Gamma)$$ by $$\label{U_S}
U_S(\alpha):=\sum_\beta \sum_{[\gamma]\in \breve{M}(\alpha,\beta)_0 } (\prod_{i\in S} \sharp ([\gamma]\cap V(\delta_i))) \cdot \beta$$ We need to explain the notation in the above formula. Recall $\breve{M}(\alpha,\beta)_0={M}(\alpha,\beta)_1/\mathbb{R}$ is the 0-dimensional unparameterized moduli space with limiting connections $\alpha$ and $\beta$ on the ends, which is the quotient of the 1-dimensional moduli space of trajectories by the translation action. In the formula, $[\gamma]$ as an element in $\breve{M}(\alpha,\beta)_0$ represents a translation equivalent class in ${M}(\alpha,\beta)_1$, which is isomorphic to a real line. In the intersection $[\gamma]\cap V(\delta_i)$, $[\gamma]$ is viewed as a connected component of ${M}(\alpha,\beta)_1$. Notice that since we are working with $\mathbb{F}$-coefficients, only the parity matters in the counting. The map $U_\emptyset$ is nothing but the Floer differential. The map $U_{\{i\}}:C(\Gamma)\to C(\Gamma)$ induces $X_i:J^\sharp(\Gamma)\to J^\sharp(\Gamma)$ which is the map used to define the $\mathcal{R}_\Gamma$-module structure on $J^\sharp(\Gamma)$ in Section \[functor-I\].
[@KM-jsharp]\*[Section 8.5]{}\[J-I\] Let $(\widetilde{C}(\Gamma),\mathcal{D})$ be the chain complex defined by $$\widetilde{C}(\Gamma)=\bigoplus_{S\subset \{1,\cdots,m\}} C_S(\Gamma),~ \mathcal{D}=\sum_{A\subset B\subset \{1,\cdots,m\}} \mathcal{D}^{A,B}$$ where each $C_S(\Gamma)$ is a copy of $C(\Gamma)$ and $\mathcal{D}^{A,B}:C_A\to C_B$ is $U_{B\setminus A}$. Then $$H(\widetilde{C}(\Gamma),\mathcal{D})\cong I^\sharp(\Gamma)$$ Therefore by filtering $\widetilde{C}(\Gamma)$ by the cardinal of $S$, we obtain a spectral sequence whose $E_1$-page is $2^m$ copies of $J^\sharp(\Gamma)$ and which converges to $I^\sharp(\Gamma)$.
From this proposition, we obtain an analogue of Proposition \[koszul\].
\[koszul-J-I\] Let $\Gamma$ and $\{e_1,\cdots,e_m\}$ be given as above. There is a spectral sequence converging to $I^\sharp(\Gamma)$ whose $E_1$-page is the Koszul complex $$K(s,J^\sharp(\Gamma))$$ where $s=(X_1,\cdots,X_m)\in \mathcal{R}_\Gamma ^{\oplus m}$. In particular, we have $$\operatorname{rank}_\mathbb{F} H(K(s,J^\sharp(\Gamma)) )\ge \operatorname{rank}_\mathbb{F}I^\sharp(\Gamma))$$
Next we want to combine Proposition \[J-I\] and Proposition \[cube-quasi-iso\] to derive a new spectral sequence. Now assume $\Gamma$ is a connected oriented spatial web and a diagram $D$ with $n$ crossings for $\Gamma$ is chosen. This is exactly the same situation as in Section \[J-ss\] except the additional requirement on the connectedness of $\Gamma$. All the constructions for $\Gamma$ in Section \[J-ss\] still work and we will continue using the notation from there. Assume the the neighborhood $\nu(\delta_i)$ of point $\delta_i\in e_i$ used to define $V(\delta_i)$ is disjoint from all the skein moves as before. We define $$\widetilde{\mathbf{C}}:=\bigoplus_{S\subset \{1,\cdots,m\}}\mathbf{C}_S$$ where $\mathbf{C}_S$ is a copy of $\mathbf{C}$. Suppose $v\le u$ in $\{0,1\}^n$ and $S\subset \{1,\cdots,m\} $, we define a map $$U_{S,vu}: C_v \to C_{u}$$ by $$\label{USvu}
U_{S,vu}(\alpha):=\sum_\beta \sum_{[\gamma]\in \breve{M}_{vu}(\alpha,\beta)_0 } (\prod_{i\in S} \sharp ([\gamma]\cap V(\delta_i))) \cdot \beta$$ This definition is similar to except that we are working on moduli spaces over a family of metrics now. When $v=u$, $U_{S,vv}$ is just the map $U_S:C(D_v)\to C(D_v)$. When $S=\emptyset$, $U_{\emptyset,vu}$ is just the map $f_{vu}$ in .
Suppose $A\subset B\subset \{1,\cdots,m\}$ and $v\le u$ in $\{0,1\}^n$. Let $C_{A,v}$ and $C_{B,u}$ be the copies of $C_v$ and $C_u$ in $\mathbf{C}_A$ and $\mathbf{C}_B$ respectively. We define a map $$\mathcal{D}^{A,B}_{vu}: C_{A,v} \to C_{B,u}$$ by setting $$\mathcal{D}^{A,B}_{vu}= U_{B\setminus A, vu}$$ Now we define $$\widetilde{\mathbf{F}}:\widetilde{\mathbf{C}}\to \widetilde{\mathbf{C}}$$ by $$\widetilde{\mathbf{F}}:= \sum_{\substack{A\subset B \\ v\le u}} \mathcal{D}^{A,B}_{vu}$$ Take a component $\mathbf{C}_S$ of $\widetilde{\mathbf{C}}$, the part of $\widetilde{\mathbf{F}}$ which maps $\mathbf{C}_S$ to itself is the map $\mathbf{F}$ in .
\[CU-chain\] The pair $(\widetilde{\mathbf{C}},\widetilde{\mathbf{F}})$ is a chain complex, i.e. $$\widetilde{\mathbf{F}}\widetilde{\mathbf{F}}=0$$
The proof is very similar to the proof of Proposition \[CF-chain\]: the equality in the proposition is obtained by counting the boundary points of 1-dimensional moduli spaces. The current situation is a little bit more subtle, so we will do the counting argument more carefully here.
Take $A\subset B\subset \{1,\cdots,m\}$ and $v\le u$ in $\{0,1\}^n$. Without loss of generality, we assume $A=\emptyset$ and $B=S$. We want to show that the component of $\widetilde{\mathbf{F}}\widetilde{\mathbf{F}}$ mapping $\mathbf{C}_{\emptyset,v}$ to $\mathbf{C}_{S,u}$ is $0$. When $S=\emptyset$, this is exactly the content of Proposition \[CF-chain\]. So we assume $S\neq 0$ from now on.
The 1-dimensional boundary strata of $M_{vu}^+(\alpha,\beta)_2$ consist of
1. The strata corresponding to trajectories sliding off the incoming end of the cobordism, having the form $$\breve{M}_{vw}(\alpha,\eta)_0 \times M_{wu}^+(\eta,\beta)_1$$ where $v\le w \le u$.
2. The strata corresponding to trajectories sliding off the ongoing end of the cobordism, having the form $${M}_{vw}(\alpha,\eta)_1 \times \breve{M}_{wu}^+(\eta,\beta)_0$$ where $v\le w \le u$.
Let $N\subset M_{vu}(\alpha,\beta)_2$ be a connected component and $\breve{N}:=N/\mathbb{R}$ is non-compact. As a non-compact 1-manifold $\breve{N}$ must be an open interval. There are two possible sources for the ends of $\breve{N}$: the broken trajectories and codimension-2 bubbles. By [@KM-jsharp]\*[Section 3.3]{}, a codimension-2 bubble can only arise in the situation that $v=u, \alpha=\beta$ and there is a sequence of connections $A_l\in M_{vv}(\alpha,\alpha)$ such that $$\xymatrix{
A_l|_{S^3\times \mathbb{R}\setminus z} \ar[r] & \widetilde{\alpha}|_{S^3\times \mathbb{R}\setminus z}
}$$ where $\widetilde{\alpha}$ represents the product trajectory obtained by pulling $\alpha$ back to the product cobordism and $z$ is the bubble point on the *seam* of the orbifold points. Since $\delta_i$ is disjoint from the seam and $V(\delta_i)$ is generic, the ends from bubbling have no contribution to $\overline{N}\cap V(\delta_i)$. Let $N^+\subset M_{vu}^+(\alpha,\beta)_2$ be the partial compactification of $N$ obtained by adding broken trajectories. Then $N^+\cap V(\delta_i)$ ($i\in S$) is a compact 1-manifold possibly with boundary. If $\breve{N}$ has only one end from broken trajectory $$([\gamma_1],[\gamma_2])\in \breve{M}_{vw}(\alpha,\eta)_0 \times \breve{M}_{wu}^+(\eta,\beta)_0$$ then $N$ has two 1-dimensional boundary strata $$\{[\gamma_1]\}\times [\gamma_2] \subset \breve{M}_{vw}(\alpha,\eta)_0 \times M_{wu}^+(\eta,\beta)_1,$$ and $$[\gamma_1]\times \{[\gamma_2]\} \subset {M}_{vw}(\alpha,\eta)_1 \times \breve{M}_{wu}^+(\eta,\beta)_0.$$ Since $N^+\cap V(\delta_i)$ is a compact 1-manifold with boundaries $[\gamma_1]\cap V(\delta_i)$ and $[\gamma_2]\cap V(\delta_i)$, we have $$\label{1end}
\#[\gamma_1]\cap V(\delta_i)+\#[\gamma_2]\cap V(\delta_i)=0$$ Similarly, if both ends of $\breve{N}^+$ are broken trajectories $$([\gamma_1],[\gamma_2])~ \text{and} ~([\gamma_1'],[\gamma_2'])$$ then $$\label{2end}
\#[\gamma_1]\cap V(\delta_i)+\#[\gamma_2]\cap V(\delta_i)=\#[\gamma_1']\cap V(\delta_i)+\#[\gamma_2']\cap V(\delta_i)$$ Now implies $$\begin{aligned}
0&=\prod_{i\in S} [\#[\gamma_1]\cap V(\delta_i)+\#[\gamma_2]\cap V(\delta_i) ] \nonumber \\
0&=\sum_{S_1\sqcup S_2=S} [\prod_{i\in S_1}\#[\gamma_1]\cap V(\delta_i) \prod_{j\in S_2}\#[\gamma_2]\cap V(\delta_j)] \label{T1}\end{aligned}$$ and implies $$\prod_{i\in S} [\#[\gamma_1]\cap V(\delta_i)+\#[\gamma_2]\cap V(\delta_i)] =\prod_{i\in S} [\#[\gamma_2']\cap V(\delta_i)+\#[\gamma_2']\cap V(\delta_i) ]$$ which is the same as $$\begin{aligned}
\sum_{S_1\sqcup S_2=S} [\prod_{i\in S_1}\#[\gamma_1]\cap V(\delta_i) \prod_{j\in S_2}\#[\gamma_2]\cap V(\delta_j)] = \nonumber
\\
\sum_{S_1\sqcup S_2=S} [\prod_{i\in S_1}\#[\gamma_1']\cap V(\delta_i) \prod_{j\in S_2}\#[\gamma_2']\cap V(\delta_j)] \label{T2}\end{aligned}$$ Take the sum of or through all the connected components $N$, we obtain $$\label{FF=0}
\sum_{v\le w\le u} \sum_{S_1\sqcup S_2=S} \langle U_{S_2,wu}\circ U_{S_1,vw}(\alpha), \beta \rangle =0$$ where $ \langle a, \beta \rangle$ denotes the coefficient of the generator $\beta$ in $a$. By the definition of $\widetilde{\mathbf{F}}$, says the component of $\widetilde{\mathbf{F}}\widetilde{\mathbf{F}}$ mapping $\mathbf{C}_{\emptyset,v}$ to $\mathbf{C}_{S,u}$ is $0$. This completes the proof.
\[I-cube-iso\] The chain complex $(\widetilde{\mathbf{C}},\widetilde{\mathbf{F}})$ is quasi-isomorphic to $(\widetilde{C}(\Gamma),\mathcal{D})$ defined in Proposition \[J-I\].
The proof is similar to the proof of Proposition \[cube-quasi-iso\] and we will continue using the notation from that proof. A quasi-isomorphism $$\mathbf{H}:(\mathbf{C},\mathbf{F})\to (\mathbf{C}_2,\mathbf{F}_2)$$ is constructed in the proof of Proposition \[cube-quasi-iso\] by counting points in the 0-dimensional moduli spaces $\breve{M}_{v,2u'}$ where $v\in \{0,1\}^n, u' \in \{0,1\}^{n-1}$.
Formula can be used to define a map $$U_{S,vu}: C_v\to C_u$$ for $v\le u$ in $\{0,1,2\}^n$. Let $$\widetilde{\mathbf{C}}_2:=\bigoplus_{S\subset\{1,\cdots m\}} \mathbf{C}_{2,S}$$ where each $\mathbf{C}_{2,S}$ is a copy of $\mathbf{C}_2$. Then a differential $\widetilde{\mathbf{F}}_2$ on $\widetilde{\mathbf{C}}_2$ can be define using maps $U_{S,2v',2u'}$ where $v'\le u'$ in $\{0,1,2\}^{m-1}$ and imitating the definition of $\widetilde{\mathbf{F}}$. Moreover, we can define a map $$\widetilde{\mathbf{H}}: \widetilde{\mathbf{C}}\to \widetilde{\mathbf{C}}$$ whose component from $\mathbf{C}_{A}$ to $\mathbf{C}_{2,B}$ ($A\subset B$) is $U_{B\setminus A, v, 2u'}$ where $v\in \{0,1\}^m, u'\in \{0,1\}^{m-1}$ and $v\le 2u'$. When $A=B$, this map is just the quasi-isomorphism $\mathbf{H}$.
Now counting the boundary points in the moduli space $M_{v,2u'}^+(\alpha,\beta)_2\cap V(\delta_i)$ as in the proof of Proposition \[CU-chain\] shows that $\widetilde{\mathbf{H}}$ is a chain map. Both $\widetilde{\mathbf{C}}$ and $\widetilde{\mathbf{C}}_2$ can be filtered by the cardinal $|S|$ and $\widetilde{\mathbf{H}}$ respects the filtrations. The $E_1$-pages of the associated spectral sequences are $2^m$ copies of $H(\mathbf{C})$ and $\mathbf{C}_2$. On the $E_1$-pages, the induced map $\widetilde{\mathbf{H}}_\ast$ is $2^m$-copies of ${\mathbf{H}}_\ast$, which is an isomorphism. Therefore $\widetilde{\mathbf{H}}$ is a quasi-isomorphism.
Iterating the above argument, we finally obtain a quasi-isomorphism from $\widetilde{\mathbf{C}}$ to $\widetilde{\mathbf{C}}_{2,\cdots,2}=\widetilde{C}(L)$.
The main theorem of this section is
\[I-ss\] Suppose $\Gamma$ is a connected oriented spatial web and ${\boldsymbol{\delta}}=\{\delta_i\}$ is a collection of points in the interior of edges of $\Gamma$ such that the homology classes of meridians around $\delta_i$ form a basis of $H_1(S^3\setminus \Gamma;\mathbb{F})$. Then there is a spectral sequence whose $E_2$-page is $\mathcal{H}(\Gamma,{\boldsymbol{\delta}};\mathbb{F})$ and which converges to $I^\sharp(\Gamma)$.
By Propositions \[J-I\] and \[I-cube-iso\], we have $H(\widetilde{\mathbf{C}})\cong I^\sharp(\Gamma)$. Recall that $$\widetilde{\mathbf{C}}=\bigoplus_{v,S}C_{S,v}$$ We can filter $\widetilde{\mathbf{C}}$ by $|v|_1+|S|$. The $E_1$-page of the associated spectral spectral sequence is exactly the chain complex $$(C(D,{{\boldsymbol{\delta}}})\otimes_{\mathbb{Z}}\mathbb{F},d_{{\boldsymbol{\delta}}} )$$ used to define $\mathcal{H}(\Gamma,{\boldsymbol{\delta}};\mathbb{F})$ in Section \[pointed-web\]. Therefore the $E_2$-page is $\mathcal{H}(\Gamma,{\boldsymbol{\delta}};\mathbb{F})$.
A $\boldsymbol{\Lambda}_{\boldsymbol{\delta}}$-module structure is defined on $\mathcal{H}(\Gamma,{\boldsymbol{\delta}};\mathbb{F})$ in Section \[pointed-web\] (adapted with $\mathbb{F}$-coefficients) where $\boldsymbol{\Lambda}_{\boldsymbol{\delta}}=\Lambda^\ast[x_1,\cdots,x_m]\otimes_{\mathbb{Z}}\mathbb{F}$. We can equip $\widetilde{\mathbf{C}}$ with a $\boldsymbol{\Lambda}_{\boldsymbol{\delta}}$-module structure by requiring $$x_i:\mathbf{C}_S\to \mathbf{C}_{S\cup\{i\}}$$ is the identification map where $i\notin S$ (recall that both $\mathbf{C}_S$ and $\mathbf{C}_{S\cup\{i\}}$ are copies of $\mathbf{C}$). A similar module structure can be defined on $\tilde{C}(\Gamma)$ (hence on $I^\sharp(\Gamma)$) and the quasi-isomorphism in Proposition \[I-cube-iso\] respects the module structures. Therefore the spectral sequence in Theorem \[I-ss\] respects this module structure and the $E_2$-page is isomorphic to $\mathcal{H}(\Gamma,{\boldsymbol{\delta}};\mathbb{F})$ as $\boldsymbol{\Lambda}_{\boldsymbol{\delta}}$-modules. A priori the $\boldsymbol{\Lambda}_{\boldsymbol{\delta}}$-module structure on $I^\sharp(\Gamma)$ relies on the specific isomorphism in Proposition \[J-I\]. Indeed this module structure is intrinsic. There is a ${\mathcal{G}}/\widetilde{\mathcal{G}}\cong H^1(S^3\setminus \Gamma; \mathbb{F})$ action on $I^\sharp(\Gamma)$. Let $\{y_i\}\subset H^1(S^3\setminus \Gamma; \mathbb{F})$ be the dual basis of $\{[m_i]\}\subset H_1(S^3\setminus \Gamma; \mathbb{F})$ where $m_i$ is the meridian around $\delta_i$. Using this dual basis, we can describe the the group ring $$\mathbb{F}H^1(S^3\setminus \Gamma; \mathbb{F}) \cong \mathbb{F}[y_1,\cdots,y_m]/(y_i^2-1|i=1,\cdots,m)$$ where we view $H^1(S^3\setminus \Gamma; \mathbb{F})$ as a multiplicative group. We have an isomorphism $$\begin{aligned}
\boldsymbol{\Lambda}_{\boldsymbol{\delta}}=\Lambda^\ast[x_1,\cdots,x_m]\otimes_{\mathbb{Z}}\mathbb{F}
&\to \mathbb{F}[y_1,\cdots,y_m]/(y_i^2-1|i=1,\cdots,m) \\
x_i&\mapsto 1+y_i\end{aligned}$$
The detection of the planar theta graph
=======================================
In this section, we will prove Theorem \[theta-detection\].
For a general trivalent graph $\Gamma$ in a 3-manifold $Y$, $I^\sharp(Y,\Gamma)$ is defined with $\mathbb{F}$-coefficients. If $L$ is a link, then $I^\sharp(Y,L)$ can be defined with $\mathbb{Z}$-coefficients. This is exactly the situation in [@KM:Kh-unknot]. In this case, we use $I^\sharp(Y,L;R)$ to denote the instanton Floer homology with $R$-coefficients ($R=\mathbb{Z},\mathbb{Q},\mathbb{C}$). If we do not specify the coefficients, $I(Y,L)$ always means the Floer homology with $\mathbb{F}$-coefficients.
From now on, we take $\Gamma$ to be a spatial theta web.
\[H<4\] Suppose $\delta_1,\delta_2$ are two mark points lying on two distinct edge of $\Gamma$. If $$\mathcal{H}(\Gamma;\mathbb{F})\cong \mathcal{H}(\Theta;\mathbb{F})$$ as $\mathcal{R}_\Theta$-modules, then $$\operatorname{rank}_{\mathbb{F}} \mathcal{H}(\Gamma,\{\delta_1,\delta_2\};\mathbb{F})\le 4$$
This follows from Proposition \[koszul\] and the example at the end of Section \[pointed-web\] (adapted with $\mathbb{F}$-coefficients).
We also have a parallel proposition for $J^\sharp$ and $I^\sharp$.
\[I<4\] Suppose $\delta_1,\delta_2$ are two mark points lying on two distinct edge of $\Gamma$. If $$J^\sharp(\Gamma)\cong J^\sharp(\Theta) (\cong \mathcal{H}(\Theta;\mathbb{F}))$$ as $\mathcal{R}_\Theta$-modules, then $$\operatorname{rank}_{\mathbb{F}} I^\sharp(\Gamma)\le 4$$
This follows from Proposition \[koszul-J-I\] and the example at the end of Section \[pointed-web\] (adapted with $\mathbb{F}$-coefficients).
Recall that $$I^\sharp(\Gamma):=I(\Gamma \cup H)$$ where $H$ is a Hopf link together with an arc joining the two components representing the $w_2$. Let $B_1$ and $B_2$ be two small ball neighborhood of the two vertices of $\Gamma$ so that the boundary sphere $S_1$ (or $S_2$) meets $\Gamma$ transversally at three points. Pick a diffeomorphism $h:(S_1,\Gamma\cap S_1)\to (S_2,S_2\cap \Gamma)$ such that that $\Gamma\cap S_1$ and $\Gamma\cap S_2$ are identified by the three arcs $\Gamma\cap (S^3\setminus B_1\cup B_2)$. Move $H$ into $B_1$, cut $S^3$ along $S_1$ and $S_2$ and re-glue using $h$, we obtain $$(S^1\times S^2, L) \cup (S^3,\Theta\cup H)$$ where $L$ is a link with three components in $S^1\times S^2$. We use $S$ to denote the sphere in $S^1\times S^2$ obtained by identifying $S_1$ and $S_2$. Since $L\cap S$ consists of three points ,the non-integral condition ([@KM:Kh-unknot]\*[Definition 3.1]{}) is satisfied. Hence the instanton Floer homology $I(S^1\times S^2,L;\mathbb{Z})$ is well-defined (without adding $H$).
\[L=2gamma\] Let $\Gamma$ and $L$ be given as above, then we have $$I(S^1\times S^2,L)\otimes I^\sharp(\Theta) \cong I^\sharp(\Gamma)\oplus I^\sharp(\Gamma)$$
This is exactly the content of the proof of [@KM-jsharp]\*[Proposition 7.5]{}, which follows from an excision theorem.
\[I-theta\] We have $$I^\sharp(\Theta) \cong \mathbb{F}^4$$
We first use an argument which was used in the proof of [@KM-deform]\*[Proposition 2.1]{} to replace $\Theta$ by a Hopf link. Recall that $$I^\sharp(\Theta):=I(S^3,\Theta\cup H)$$ The Excision Theorems in [@KM-jsharp]\*[Section 4]{} can be used to show that $$I(S^3,\Theta\cup H_1)\otimes I(S^3,\Theta\cup H_2)\cong I(S^3,\Theta\cup H_1\cup H_2)
\cong I(S^3,\Theta\cup H_2)\otimes I(S^3,H_1\cup H_2)$$ where $H_1$ and $H_2$ are two disjoint copies of $H$. Since $$I(S^3,\Theta\cup H_2)=I^\sharp(\Theta)\neq 0$$ by [@KM-jsharp]\*[Section 7.1]{}, we have $$\label{theta=H}
\operatorname{rank}_\mathbb{F} I^\sharp(\Theta)=\operatorname{rank}_\mathbb{F} I(S^3,\Theta\cup H_1)=
\operatorname{rank}_\mathbb{F} I(S^3,H_1\cup H_2)$$ The Chern-Simons functional on the pair $(S^3,H_1\cup H_2)$ has a Morse-Bott critical set $SO(3)$. After a suitable generic perturbation, the restriction $F|_{SO(3)}$ of the perturbed Chern-Simons functional becomes a standard Morse function on $SO(3)$ with critical points $\alpha_i$ ($0\le i \le 3$) and the critical points of $F$ are exactly $\{\alpha_i\}$. The degree of $\alpha_i$ is just $i$ and the moduli space of trajectories $M(\alpha_i,\alpha_{i-1})_1$ ($1\le i \le 3$) approximates the Morse trajectories on $SO(3)$. Therefore the differential on $\alpha_i$ ($1\le i \le 3$) is the same as the differential in the Morse homology of $SO(3)$. Since we only have a relative $\mathbb{Z}/4$-grading on those critical points, the differential on $\alpha_0$ may be non-zero. Now we have a cyclic chain complex $$\xymatrix{
\mathbb{Z}\{\alpha_3\} \ar[r]^{0} & \mathbb{Z}\{\alpha_2\} \ar[d]^{2} \\
\mathbb{Z}\{\alpha_0\} \ar[u]_{d} & \mathbb{Z}\{\alpha_1\} \ar[l]^{0}
}$$ If $d\neq 0$, then we have $I(S^3,H_1\cup H_2;\mathbb{Q})=0$.
Let $Y_1$ and $Y_2$ be two homology 3-spheres. Fukaya’s connected sum theorem [@Fukaya-sum] relates the instanton Floer homologies $I(Y_1;\mathbb{Z}),I(Y_2;\mathbb{Z})$ to $I(Y_1\# Y_2;\mathbb{Z})$ by a spectral sequence. An elaboration of Fukaya’s theorem with $\mathbb{Q}$-coefficients can be found in [@Don:YM-Floer]\*[Section 7.4]{}. The same problem is also studied in [@Li-sum]. We want to apply Fukaya’s theorem to the connected sum $$(S^3,H_1\cup H_2)=(S^3,H_1)\#(S^3,H_2)$$ Even though this is not a connected sum of homology 3-spheres with empty links in Fukaya’s setting, the same proof still works. Our situation is even easier because we are working in an admissible case so that reducible critical points of the Chern-Simons functional do not enter into the discussion.
From [@Don:YM-Floer]\*[Section 7.4]{}, we have a spectral sequence which converges to $I(S^1\times S^2, H_1\cup H_2)$ and whose last possibly non-degenerate page is $$\begin{aligned}
\xymatrix{
I(S^3, H_1;\mathbb{Q})\otimes I(S^3, H_2;\mathbb{Q}) \ar[r]^{f} &
I(S^3, H_1;\mathbb{Q})\otimes I(S^3, H_2;\mathbb{Q}) }\end{aligned}$$ where the differential $f$ is $2\mu(x_1)\otimes 1- 1\otimes 2\mu (x_2)$ ($x_1\in S^3\setminus H_1$ and $x_2\in S^3\setminus H_2$). We use $1$ for the identify map and $\mu(x)$ for the point operator of degree 4. Our convention for $\mu$ is from [@DK].
The instanton Floer homology $I(S^3, H;\mathbb{Q})$ is 1-dimensional. Therefore the point operator $\mu(x)$ on $I(S^3, H;\mathbb{Q})$ is just a scalar product. This implies the differential $f$ is zero. We obtain $I(S^3,H_1\cup H_2;\mathbb{Q})=\mathbb{Q}^{2}$, which is a contradiction. Therefore $d$ must be $0$. We have $$I(S^3,H_1\cup H_2;\mathbb{Z})=\mathbb{Z}^2\oplus \mathbb{Z}/2,~~I(S^3,H_1\cup H_2)= \mathbb{F}^{4}$$ By , the proof is complete.
When we define $L$, we first remove two balls $B_1$ and $B_2$ from $S^3$. We can define a tangle $T:=\Gamma\cap (S^3\setminus B_1\cup B_2)$ in $(S^3\setminus B_1\cup B_2)\cong I\times S^2$. We may assume $T$ is disjoint with $I\times \{\infty\}\subset I\times S^2$ and view $T$ as a tangle in $I\times D^2$ by removing a tubular neighborhood of $I\times \{\infty\}$ in $I\times S^2$. Let $S\subset S^1\times S^2$ be a $S^2$-slice as before. In [@Street], an operator $\operatorname{\mu^{orb}}(S)$ of degree $2$ on $I(S^1\times S^2,L;\mathbb{C})$ is defined and the eigenvalues of $\operatorname{\mu^{orb}}(S)$ are shown to be $\pm 1$. As a degree $2$ operator on a $\mathbb{Z}/4$-graded vector space $I(S^1\times S^2,L;\mathbb{C})$, the generalized eigenspaces with eigenvalue $1$ or $-1$ must have the same dimension. The generalized eigenspace with eigenvalue $1$ is defined to be the odd tangle Floer homology $\operatorname{THI}^{\text{odd}}(T)$ in [@Street]\*[Definition 3.3.2]{}. From the definition we have $$\label{THI-dim}
\operatorname{rank}_{\mathbb{C}} I(S^1\times S^2,L;\mathbb{C}) = 2\operatorname{rank}_{\mathbb{C}} \operatorname{THI}^{\text{odd}}(T)$$ The odd tangle Floer homology $\operatorname{THI}^{\text{odd}}(T)$ satisfies the following.
[@AHI]\*[Theorem 3.10]{} The Floer homology $\operatorname{THI}^{\text{odd}}(T)$ is 1-dimensional if and only if $T$ is isotopic to a braid.
\[I-detection-braid\] Let $\Gamma$ and $T$ be given as above. If $$\operatorname{rank}_{\mathbb{F}}I^\sharp(\Gamma)\le 4$$ then we have $$\operatorname{THI}^{\emph{odd}}(T)\cong \mathbb{C}$$ Therefore $T$ must be a braid.
By Proposition \[L=2gamma\] and Proposition \[I-theta\], we have $$\operatorname{rank}_{\mathbb{F}} I^\sharp (S^1\times S^2, L) \le 2$$ Equality and the universal coefficients theorem imply $$\operatorname{rank}_{\mathbb{C}} \operatorname{THI}^{\text{odd}}(T) \le 1$$ Since $\operatorname{THI}^{\text{odd}}(T)$ is odd-dimensional [@AHI]\*[Proposition 4.10]{}, its dimension must be 1.
The diagram of a spatial trivalent graph can be changed by type V moves in Figure \[RV\], which changes $T$ by a generator of the braid group. After finitely many type V moves, $T$ becomes the trivial braid. So we have
\[I-detection-theta\] Let $\Gamma$ be given as above. If $$\operatorname{rank}_{\mathbb{F}} I^\sharp(\Gamma)\le 4$$ then $\Gamma$ must be the planar theta graph $\Theta$.
Now we are ready to prove the detection theorem.
By Theorem \[I-ss\], Proposition \[H<4\] and Proposition \[I<4\], any one in (b) (c) (d) (e) implies $\operatorname{rank}_{\mathbb{F}}I^\sharp(\Gamma)\le 4$. Therefore $\Gamma$ is the planar theta graph by Corollary \[I-detection-theta\]. We obtain any one in (b) (c) (d) (e) implies (a).
It is clear that (a) implies (b) (c) and (d). The last term $(e)$ is just Proposition \[I-theta\].
Yi Xie, <span style="font-variant:small-caps;">Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY 11794</span>
*E-mail address*: `[email protected]`
[^1]: The convention used here is non-standard. We use this non-standard convention in order to be consistent with [@Kh-sl3].
|
---
author:
- 'Jean-Louis Loday'
title: Dichotomy of the addition of natural numbers
---
Introduction {#introduction .unnumbered}
============
In this paper the addition of integers is split into two operations which satisfy some relations. These relations are taken so that they split the associativity relation of addition into three. Under these new operations, the unit $1$ generates elements which are in bijection with the planar binary rooted trees. More precisely, any integer $n$ splits as the disjoint union of the trees with $n$ internal vertices. The Tamari poset is a partial order structure on this set of trees. We show how the addition on trees is related to the Tamari poset structure. This first part is an elementary presentation of results contained in “Arithmetree” [@Loday02] by the author and in [@LodayRonco02] written jointly with M. Ronco.
Prompted by an unpublished page of Tamari’s thesis, we investigate various ways of realizing the Tamari poset as a polytope. In particular we show that Tamari’s way of indexing the planar binary rooted trees gives rise to a hypercube-like polytope on which the associahedron is drawn.
About the formula $1+1=2$
=========================
The equality $3+5=8$ can be seen either as $3$ acting on the left on $5$, or as $5$ acting on the right on $3$. Since adding $3$ and $5$ is both, one can imagine to “split” this sum into two pieces reflecting this dichotomy. Physically, splitting the addition symbol $+$ into two pieces gives: $$\begin{gathered}
+\\
\l \r\\
\l\quad \r\end{gathered}$$ that is, the symbols $\l$ and $\r$. Hence, since $1+1=2$, one defines two new elements $1\l 1$ and $1\r 1$ so that $$1\l 1\ \cup\ 1\r 1= 1+1=2.$$
Splitting the integers into pieces
==================================
How to go on ? A priori one can form eight elements out of three copies of $1$ and of the operations *left* $\l$ and *right* $\r$, that is $$\begin{gathered}
(1 \l 1) \l 1 \quad , \quad (1 \l 1) \r 1 \quad , \quad (1 \r 1) \l 1 \quad , \quad (1 \r 1) \r 1 \quad , \quad \\
1 \l (1 \l 1) \quad , \quad 1 \l (1 \r 1) \quad , \quad 1 \r (1 \l 1) \quad , \quad 1 \r (1 \r 1) \quad . \quad
\end{gathered}$$
But we would like to keep associativity of the operation $+$, so we want that the union of the elements on the first row is equal to the union of the elements of the second row. More generally for any component $r$ and $s$ we split the sum as $$r+s = r\l s \ \cup \ r \r s \ .$$ Taking again our metaphore of left action and right action, it is natural to choose the relations $$\begin{aligned}
&(*)\qquad\begin{cases}
(r \l s) \l t = r \l (s +t), \\
(r \r s) \l t = r \r (s \l t), \\
(r +s) \r t = r \r (s \r t), \\
\end{cases}\end{aligned}$$
since, by taking the union, we get readily $(r+s)+t = r+(s+t).$ The first relation says that “acting on the right by $s$ and then by $t$” is the same as “acting by $s+t$”. (The kowledgeable reader will remark the analogy with bimodules). Since we have three relations, our eight elements in the case $r=s=t=1$ go down to five, which are the following: $$(1\r 1)\r 1\ , \ (1\l 1)\r 1\ , \ 1\r 1 \l 1 \ , \ 1\l (1\r 1)\ , \ (1\l1)\l 1 \ .$$
Indeed, since one has $(1 \r 1) \l 1= 1 \r (1 \l 1)$, the parentheses can be discarded. On the other hand the two elements $1\r (1\r 1)$ and $(1\l 1)\l 1$ can be written respectively: $$\begin{aligned}
1\r (1\r 1) &=& (1\r 1)\r 1\ \cup \ (1\l 1)\r 1\ ,\\
\ (1\l 1)\l 1&=& 1\l (1\r 1) \ \cup \ 1\l (1\l 1).\end{aligned}$$
In conclusion, we have decomposed the integer $2$ into two components $1\l 1$ and $1\r 1$ and the integer $3$ into five components (see above), the integer $1$ has only one component, namely itself. How about the integer $n$ ? In fact, not only would we like to decompose $n$ into the union of some components, but we would also like to know how to add these components. The test will consist in checking that adding the components of $n$ with all the components of $m$, we get back the union of all the components of $m+n$.
Trees and addition on trees {#trees}
===========================
In order to understand the solution we introduce the notion of *planar binary rooted tree*, that we simply call tree in the sequel. Here are the first of them:
$$PBT_1 = \{\ \vert \ \}\ , \quad PBT_2=\big\{ \arbreA\big\}\ , \quad
PBT_3=\Big\{\arbreBA\ ,\ \arbreAB \Big\}\ ,$$
$${PBT_4=\bigg\{ \arbreABC}\ ,\ {{}\arbreBAC },\ {{}\arbreACA },\ {{}\arbreCAB },\ {{}\arbreCBA }\bigg\}\ .$$
Such a tree $t$ is completely determined by its left part $t^{l}$ and its right part $t^{r}$, which are themselves trees. The tree $t$ is obtained by joining the roots of $t^{l}$ and of $t^{r}$ to a new vertex and adding a root: $$\xymatrix@R=10pt@C=10pt{
&t^{l}\ar@{-}[dr] && t^{r}\ar@{-}[dl] \\
t=&&*{}\ar@{-}[d] &\\
&&*{}&
}$$ This construction is called the *grafting* of $t^{l}$ and $t^{r}$. One writes $t= t^{l}\vee t^{r}$.
Hence any nontrivial tree (that is different from $\vert$) is obtained from the trivial tree $\vert$ by iterated grafting. The elements of the set $PBT_{n}$ are the trees with $n$ leaves that is with $n-1$ internal vertices. The number of elements in $PBT_{n}$ is the Catalan number $c_{n-1}$. It is known that $c_{n}= \frac{(2n)!}{n!\, (n+1)!}= \frac{1}{n+1}\binom{2n}{n}$.
The solution to the splitting of natural numbers is going to be a consequence of the properties of the operations $\l$ and $\r$ on trees, which are defined as follows. For any nontrivial trees $s$ and $t$ one defines recursively the two operations $\l$ and $\r$ by the formulas $$(\ddag)\qquad s\l t := s^{l}\vee (s^{r}+t)\quad, \quad s\r t := (s+t^{l})\vee t^{r},$$ and the sum by $$s+t :=s\l t \ \ \cup \ s\r t .$$ The trivial tree is supposed to be a neutral element for the sum: $\vert = 0$, so $s\l 0= s$ and $0\r t = t$. The unique tree with one internal vertex (Y shape tree) represents $1$. Then one gets $$\arbreA \l \arbreA = \arbreBA \quad , \quad \arbreA \r \arbreA = \arbreAB \ .$$ Notice the matching between the orientation of the leaves and the involved operations: the middle leaf of the tree $\arbreBA$, resp. $\arbreAB$, is oriented to the left, resp. right, and this tree represents the element $1\l 1$, resp. $1\r 1$.
The principal properties of these two operations are given by the following statement.
[@LodayRonco02] The operations $\l$ and $\r$ satisfy the relations $(*)$. Any tree can be obtained from the initial tree $\arbreA$ by iterated application of the operations left and right.
The solution is then the following. The integer $n$ is the disjoint union of the elements of $PBT_{n+1}$, that is the trees with $n$ internal vertices. For instance: $$\begin{aligned}
0&=& \vert\\
1&=& \arbreA \\
2 &=& \arbreAB \cup \arbreBA\\
3&=& \arbreABC \cup \arbreBAC \cup \arbreACA \cup \arbreCAB \cup \arbreCBA \\\end{aligned}$$
The sum $+$ of integers can be extended to the components of these integers, that is to trees. Even better, the operations left and right can be extended to the trees. The above formulas $(\ddag)$ give the algorithm to perform the computation.
Where we show that $1+1=2$ and $2+1=3$
======================================
Here are some computation examples: $$\begin{aligned}
1\l 1 &=& \arbreA \l \arbreA = \arbreBA\quad , \quad 1\r 1 = \arbreA\r \arbreA = \arbreAB, \\
2\l 1 &=& \Big(\arbreAB \cup \arbreBA \Big)\l \arbreA = \arbreAB \l \arbreA \cup \arbreBA \l \arbreA \\
&=& \arbreACA \cup \arbreCAB \cup \arbreCBA,\\
2\r 1 &=& \Big(\arbreAB \cup \arbreBA \Big)\r \arbreA = \arbreAB \r \arbreA \cup \arbreBA \r \arbreA \\
&=& \arbreABC \cup \arbreBAC.\\\end{aligned}$$
Notice that $1+1 = 1\l 1 \ \cup \ 1\r 1 = 2$ and that $2+1 = 2\l 1 \ \cup \ 2\r 1 = 3$, since $3$ is the union of the five trees of $PBT_{4}$. They represent the five elements which can be written with three copies of $1$ (see above). Similarly one can check that $$\begin{aligned}
m+n &=& m\l n \ \cup \ m\r n\\
&=&\Big(\bigcup_{s\in PBT_{n+1}}s\ \l \ \bigcup_{t\in PBT_{m+1}}t\Big)\ \bigcup \ \Big(\bigcup_{s\in PBT_{n+1}}s\ \r \ \bigcup_{t\in PBT_{m+1}}t\Big) \\
&=&\Big(\bigcup_{s\in PBT_{n+1}, t\in PBT_{m+1}}s\ \l \ t\Big)\ \bigcup \ \Big(\bigcup_{s\in PBT_{n+1}, t\in PBT_{m+1}}s\ \r \ t\Big) \\
&= & \bigcup_{r\in PBT_{m+n+1}}r = m+n.\end{aligned}$$
Finally there are two ways to look at the $c_{n+1}$ components of the integer $n$: either through trees, or through $n$ copies of $1$ and the operations $\l$ and $\r$ duly parenthesized. Recall that this second presentation is not unique.
There are many families of sets whose number is the Catalan number (they are called Catalan sets). For each of them one can translate the algebraic structure unraveled above. In [@AvalViennot10] Aval and Viennot have performed this task for the “alternative Catalan tableaux”. It is interesting to notice that, in their description of the sum of two tableaux (see loc. cit. p. 6), there are two different kinds of tableaux. In fact the tableaux of one kind give the left operation and the tableaux of the other kind give the right operation.
The integers as molecules {#molecule}
=========================
Let us think of the integers as molecules and of its components as atoms. Then one would like to know of the ways the atoms are bonded in order to form the molecule. Since the molecule $2$ has only two atoms, we pretend that there is a bond between the two atoms:
$$\arbreAB \xymatrix{\ar@{-}[rr]&& }\arbreBA$$
For our mathematical purpose it is important to see this bond as an oriented relation (it is called a covering relation):
$$\arbreAB \xymatrix{\ar[rr]&& }\arbreBA$$
For the molecule $n$ one puts a bond between two atoms (i.e. trees) whenever one can obtain one of them from the other one by a local change as in the molecule $2$ case. Here is what we get for $n=3$:
and for $n=4$ (without mentioning the trees):
$$\KtroisFgrand$$
These drawings already appeared, under slightly different shape, in Dov Tamari’s original thesis, defended in 1951 (see the discussion below in section \[realization\]). In fact the covering relations on the set $PBT_{n}$ make it into a “partially ordered set”, usually abbreviated into “poset”. This is the *Tamari poset* on trees [@Tamari51]. The reason for introducing this poset at this point is its strong relationship with the algebraic structure that we just described on trees. It is given by the following statement proved in a joint work with María Ronco.
[@LodayRonco02] The sum of the trees $t$ and $s$ is the union of all the trees which fit in between $t/s$ and $t\backslash s$ : $$t + s = \bigcup_{t/s \leq r \leq t\backslash s} r$$ where $t/s$ is obtained by grafting the root of $t$ on the leftmost leaf of $s$ and $t\backslash s$ is obtained by grafting the root of $s$ on the rightmost leaf of $t$.
This formula makes sense because one can prove (cf. loc. cit.) that, for any trees $t$ and $s$, we have $t/s \leq t\backslash s$.
Multiplication of trees
=======================
Multiplication of natural numbers is obtained from addition since: $$n \times m := \underbrace{m+ \cdots + m}_{n \textrm{ copies}} .$$ In other words, one writes $n$ in terms of the generator $1$: $$n = \underbrace{1+ \cdots + 1}_{n \textrm{ copies}} ,$$ and then one replaces $1$ by $m$ everywhere to obtain $n�\times m$.
The very same process enables us to define the product $t\times s$ of the trees $t$ and $s$. First we write $t$ in terms of $1$ with the help of the left and right operations, and then we replace each occurence of $1$ by $s$. Here are some examples:
$$\arbreAB \times \arbreBA= \arbreBADA$$ $$\arbreAB \times \arbreAB= \arbreACAD \cup \arbreABCD.$$
The proof of the first case is as follows. Since we have $\arbreAB= 1 \r 1$, we can write: $$\arbreAB \times \arbreBA=\arbreBA \r \arbreBA = \arbreBA \vee \arbreA = \arbreBADA .$$ It is immediate to check that if $s$ has $n$ internal vertices and $t$ has $m$ internal vertices, then $s \times t$ has $nm$ internal vertices. Another relationship with the product of natural numbers is the following. Replacing $n$ by the union of trees of $PBT_{n+1}$, and $m$ by the union of trees of $PBT_{m+1}$, then $n\times m$ is actually the union of all the trees with $nm$ internal vertices.
Some of the properties of the multiplication are preserved, but not all. The associativity holds and the distributivity with respect to the left factor also holds. But right distributivity does not (and of course commutativity does not hold). This is the price to pay for such a generalization. More properties and computation can be found in [@Loday02]. The interesting paper [@BrunoYazaki08] deals with the study of prime numbers (we should say prime trees) in this framework.
Let us summarize the properties of the sum and the product of trees versus the sum and the product of integers. We let $\PP(PBT)$ be the set of non-empty subsets of $PBT_{n}$ for all $n$.
There are maps $$\NN \mono \PP(PBT) \epi \NN$$ which are compatible with the sum and the product. The composite is the identity.
Indeed, the first map sends $n$ to the union of all the trees in $PBT_{n}$. The second map sends a subset to the arity of its components.
Trees and polynomials
=====================
The algebra of polynomials (let us say with real coefficients) in one variable $x$ admits the monomials $x^{n}$ for basis. Since we know how to decompose an integer into the union of trees, we dare to write $$x^{n}= \sum_{t\in PBT_{n+1}} x^{t}.$$
More specifically, we consider the vector space spanned by the elements $x^{t}$ for any tree $t$. As usual, the sum of exponents gives rise to a product of factors: $$x^{n+m}= x^{n}x^{m}\quad , \quad x^{s+t}= x^{s}x^{t},$$ where $s$ and $t$ are trees. We use the notation $x^{\vert}= x^{0}= 1$ and $x^{\!\!\petitarbreA}=x^{1}=x$.
In fact there is no reason to consider only polynomials and one can as well consider series since the sum and the product are well-defined.
In this framework the operations $$\begin{aligned}
\begin{array}{r} \mbox{union} \\ \mbox{addition} \\ \mbox{multiplication} \end{array}
\quad \mbox{on trees} \qquad \mbox{become} \qquad
\begin{array}{r} \mbox{addition} \\ \mbox{multiplication} \\ \mbox{composition} \end{array}
\quad \mbox{on polynomials} \, .\end{aligned}$$ What about the operations $\l$ and $\r$ ? They give rise to two operations denoted $\g$ and $\d$ respectively, on polynomials. These two operations are bilinear and satisfy the relations:
$$\begin{array}{rcl}
(r \g s) \g t &=& r \g (s \g t + s\d t) \, , \\
(r \d s) \g t &=& r \d (s \g t) \, , \\
(r \g s+ r \d s) \d t &=& r \d (s \d t) \, . \\
\end{array}
$$
A vector space $A$ endowed with two bilinear operations $\g$ and $\d: A\t A \to A$, satisfying the relations just mentioned, is called a *dendriform algebra*, cf. [@Loday95; @Loday01].
The dendriform algebras show up in many topics in mathematics: higher algebra [@BurgunderRonco10; @Dotsenko09; @Ronco00], homological algebra [@Loday12], combinatorial algebra [@AvalViennot10; @AguiarSottile06; @Devadoss09; @NovelliThibon07; @PilaudSantos09; @Postnikov09], algebraic topology [@Chapoton02; @Vallette08; @Yau07], renormalization theory [@Brouder04; @BrouderFrabetti03], quantum theory [@GGL], to name a few. It is closely related to the notion of shuffles. In fact it could be called the theory of “non-commutative shuffles”.
Realizing of the associahedron {#realization}
==============================
In Dov Tamari’s seminal work “Monoides pr' eordonn' es et chaînes de Malcev” [@Tamari-these], which is his French doctoral thesis defended in 1951, the picture displayed in Fig. \[fig:Tthesis\] appears on page 12.
Unfortunately this part has not been reproduced in the published text [@Tamari51] and therefore has been forgotten for all these years. It is very interesting on three grounds. First, it is the first appearance of the *Tamari poset*. Second, the Tamari poset is portrayed in dimension 2 and 3 as a polygon and a polyhedron respectively. Third, the parenthesizings has been replaced by a code that one can consider as coordinates in the euclidean space. We now analyze these three points.
Tamari poset
------------
The Tamari poset, appearing often as the *Tamari lattice* in the literature (since it is a lattice), proved to be helpful in many places in mathematics. I mentioned earlier in this text its relevance with dendriform structures. It is playing a key role in the problem of endowing the tensor product of $\Ai$-algebras with an $\Ai$-algebra structure, cf. [@Loday12] for two reasons. First, the Tamari poset gives rise to a cell complex called the associahedron or the Stasheff polytope (see below). In 1963 Jim Stasheff showed that it encodes the notion of “associative algebra up to homotopy”, now called $\Ai$-algebras. Let us recall that such an algebra $A$ is equipped with a $k$-ary operation $m_{k}: A^{\t k} \to A, k\geq 2,$ which satisfy some universal relations describing the topological structure of the associahedron. The second reason comes as follows. For a fixed integer $n$, the associahedron $\KKK^{n}$ is a cell complex of dimension $n$. We can prove that its cochain complex $C^{\bullet}(\KKK^{n})$ can be endowed with a structure of $\Ai$-algebra. The operations $m_{k}$ can be made explicit in terms of the Tamari poset relation, cf. [@Loday12].
Associahedron and regular pentagons
-----------------------------------
The sentence following Fig. \[fig:Tthesis\] in Tamari’s thesis is the following
“G' en' eralement, on aura des hyperpolyèdres.”
But no further information is given. In fact, as we know now, we can realize the Tamari poset as a convex polytope so that each element of the poset is a vertex and each covering relation is an edge (see below). There is no harm in taking the regular pentagon in dimension 2. However, in contrast to what Fig. \[fig:Tthesis\] suggests, one cannot realize the associahedron in dimension 3 with regular pentagons. What happens is the following: the four vertices corresponding to the parenthesizings $2020, 2011, 1120$ and $1111$ do not lie in a common plane. It is a good trigonometric exercise for first year undergraduate students. If we take the convex hull of $\mathcal{M}_{4}$ of Fig. \[fig:Tthesis\] (that is keeping regular pentagons), then the faces are made up of 6 pentagons, and 6 triangles instead of the 3 quadrangles. There are 3 edges which show up and which do not correspond to any covering relation:
$$2011 - 1120, \quad 30012 - 3100, \quad 2200 -1210.$$
Realizations of the associahedron
---------------------------------
Though Tamari does not mention it, we can think of his clever way of indexing the parenthesizings as coordinates of points in the euclidean space $\RRR^{n+1}$. Let us recall briefly his method: given a parenthesizing (which is equivalent to a planar binary tree $t$) of the word $x_{0}x_{1}\ldots x_{n+1}$ we count the number of opening parentheses in front of $x_{0}$, then $x_{1}$, etc., up to $x_{n}$. For instance the word $((x_{0}x_{1}) x_{2})$ gives $2\ 0$ and the word $(x_{0}(x_{1} x_{2}))$ gives $1\ 1$. Let us denote this sequence of numbers by $$M(t)=(\aa_{0}, \ldots, \aa_{n})\in \RRR^{n+1}.$$ Since the number of parentheses depends only on the length of the word, we have $\sum\aa_{i}= n+1$ and the points $M(t)$ lie in a common hyperplane. What does the convex hull look like ? In dimension 2 we get the following pentagon:
$$\TamariDeux$$
As we see it is a quadrangle (that is a deformed square) with one point added on an edge.
In dimension 3 we get:
$$\TamariTrois$$
that is a deformed cube on which the associahedron has been drawn. In order to analyze the $n$-dimensional case, let us introduce the following notation. The convex hull of the points $M(t)$ is called the *Tamari polytope*. The *canopy* of the tree $t$ is an element of the set $\{\pm \}^{n}$ corresponding to the orientation of the interior leaves. If the leaf points to the left (resp. right), then we take $-$, resp. $+$. Of course we discard the two extremal leaves, whose orientation is fixed. We denote by $$\psi : PBT_{n+2} \to \{\pm \}^{n}$$ this map. Among the trees with a given canopy, we single out the tree which is constructed as follows. We first draw the outer part of the tree. Then for each occurence of $-$ we draw an edge which goes all the way to the right side of the tree (it is a left leaf). Then we complete the tree by drawing the right leaves. For instance: $$\sigma(-) = \arbreBA, \quad \sigma(-,+) = \arbreCAB, \quad \sigma(-,-) = \arbreCBA .$$ This construction gives a section to $\psi$ that we denote by $$\sigma : \{\pm \}^{n} \to PBT_{n+2} .$$
The Tamari polytope $KT^{n}$ is a hypercube shaped polytope, with extremal points $M(\sigma(\aa))$, for $\aa\in
\{\pm \}^{n}$. For any tree $t$ the point $M(t)$ lies on a face of this hypercube containing $M(\sigma\psi(t))$.
It is easily seen by induction that the convex hull of the points $M(\sigma(\aa))$ form a (combinatorial) hypercube.
Up to a change of orientation, this is the cubical version of the associahedron described in [@Loday02] section 2.5 (see also [@Loday11] Appendix 1). It is also described in [@SaneblidzeUmble04].
The Tamari polytope shares the following property with the standard permutohedron: all the edges have the same length.
Though Tamari himself does not consider this construction in his thesis, a close collaborator, Mrs de Fougères, worked out some variations in [@Fougeres64].
In 1963 Jim Stasheff [@Stasheff63] discovered independently the associahedron, first as a contractible cell complex, in his work on the structure of the loop spaces. It was later recognized to be realizable as a convex polytope, see for instance [@Stasheff97] Appendix B. In 2004 I gave in [@Loday04] an easy construction with integral coordinates as follows. It is usually described in terms of trees, but I will translate it in terms of parenthesized words.
Given a parenthesized word of length $n$, for instance $((x_{0} x_{1})(x_{2}x_{3}))$, we associate to it a point in the euclidean space with coordinates computed as follows. The $i$th coordinate ($i$ ranging from $0$ to $n$) is the product of two numbers $a_{i}$ and $b_{i}$. We consider the smallest subword which contains both $x_{i}$ and $x_{i+1}$. Then $a_{i}$ is the number of opening parentheses standing to the left of $x_{i}$ and $b_{i}$ is the number of closing parentheses standing to the right of $x_{i+1}$ in the subword. In the example at hand we get $1\ 4 \ 1$. It gives rise to the following polytopes in low dimension:
$$\KdeuxA\hskip3cm \KtroisA\qquad$$
$$\KKK^2\hskip5cm \KKK^3\qquad$$
Since then several interesting variations for the associahedron itself and for other families of polytopes have been given along the same lines, see for instance [@HohlwegLange07; @Devadoss09; @Forcey08a; @Forcey08b; @Postnikov09; @PilaudSantos09].
Associahedron and the trefoil knot
==================================
Let us end this paper with a surprizing relationship which is not so-well-known. If we draw a path on the 3-dimensional associahedron from the center of each quadrangle to the center of the other quadrangles via the center of the pentagons, alternating over and under as we reenter a quadrangle, then we get the trefoil knot in Fig. \[fig:trefoilknot\].
The same process applied to the 3-dimensional cube gives rise to the Borromean rings. In the cube case we know how to relate the various invariants of this link: Philip Hall identity, triple Massey product. Nothing similar is known in the associahedron case so far.
[99]{}
J.-C.Aval and X. Viennot, “The product of trees in the Loday-Ronco algebra through Catalan alternative tableaux”, *S' em. Lothar. Combin.* **63** (2010), Art. B63h, 8 pp.
M. Aguiar and F. Sottile, “Structure of the Loday-Ronco Hopf algebra of trees”, *J. Algebra* **295** (2006) 473–511.
Ch. Brouder, “Trees, renormalization and differential equations”, *BIT* **44** (2004) 425–438.
Ch. Brouder and A. Frabetti, “QED Hopf algebras on planar binary trees”, *J. Algebra* **267** (2003) 298–322.
A. Bruno and D. Yazaki, “The arithmetic of trees”, arXiv:0809.4448.
E. Burgunder and M. Ronco, “Tridendriform structure on combinatorial Hopf algebras”, *J. Algebra* **324** (2010) 2860–2883.
F. Chapoton, “Op' erades diff' erentielles gradu' ees sur les simplexes et les permutoèdres”, *Bull. Soc. Math. France* **130** (2002) 233–251.
S. Devadoss, “A realization of graph associahedra”, *Discrete Math.* **309** (2009) 27–276.
V. Dotsenko, “Compatible associative products and trees”, *Algebra Number Theory* **3** (2009) 567–586.
K. Ebrahimi-Fard, “Loday-type algebras and the Rota-Baxter relation”, *Lett. Math. Phys.* **61** (2002) 139–147.
S. Forcey, “Quotients of the multiplihedron as categorified associahedra”, *Homology, Homotopy Appl.* **10** (2008) 227–256.
S. Forcey, “Convex hull realizations of the multiplihedra”, *Topology Appl.* **156** (2008) 326–347.
D. de Fougères, “Propri' et' es g' eom' etriques li' ees aux parenth' esages d’un produit de m facteurs. Application à une loi de composition partielle”, *S' eminaire Dubreuil. Algèbre et th' eorie des nombres* **18**, no 2 (1964-65), exp. no 20, 1–21.
H. Gangl, A.B. Goncharov and A. Levin, “Multiple polylogarithms, polygons, trees and algebraic cycles”, in *Algebraic geometry, Seattle 2005*, Proc. Sympos. Pure Math. **80**, Part 2, Amer. Math. Soc., Providence, RI, 2009, 547–593.
Ch. Hohlweg and C. Lange, “Realizations of the associahedron and cyclohedron”, *Discrete Comput. Geom.* **37** (2007) 517–543.
J.-L. Loday, “Algèbres ayant deux op' erations associatives (digèbres)”, *C. R. Acad. Sci. Paris S' er. I Math.* **321** (1995) 141–146.
J.-L. Loday, “Dialgebras”, in *Dialgebras and related operads*, Springer Lecture Notes in Math. **1763** (2001) 7–66.
J.-L. Loday, “Arithmetree”, *Journal of Algebra* **258** (2002) 275–309.
J.-L. Loday, “Realization of the Stasheff polytope”, *Archiv der Mathematik* **83** (2004) 267–278.
J.-L. Loday, “The diagonal of the Stasheff polytope”, in *Higher structures in geometry and physics*, Progr. Math. **287**, Birkhäuser, Basel, 2011, 269–292.
J.-L. Loday, “Geometric diagonals for the Stasheff associahedron and products of A-infinity algebras”, preprint 2011, in preparation.
J.-L. Loday ; M.O. Ronco, “Hopf algebra of the planar binary trees”, *Adv. Math.* **139** (1998) 293–309.
J.-L. Loday and M.O. Ronco, “Order structure and the algebra of permutations and of planar binary trees”, *J. Alg. Comb.* **15** (2002) 253–270.
J.-C. Novelli and J.-Y. Thibon, “Hopf algebras and dendriform structures arising from parking functions”, *Fund. Math.* **193** (2007) 189–241.
A. Postnikov, “Permutohedra, associahedra, and beyond”, *Int. Math. Res. Not.* **2009**, no. 6, 1026–1106.
V. Pilaud and F. Santos, “Multitriangulations as complexes of star polygons”, *Discrete Comput. Geom.* **41** (2009) 284–317.
M.O. Ronco, “Primitive elements in a free dendriform algebra”, in *New trends in Hopf algebra theory (La Falda, 1999)*, Contemp. Math. **267**, Amer. Math. Soc., Providence, RI, 2000, 245–263.
S. Saneblidze and R. Umble, “Diagonals on the permutahedra, multiplihedra and associahedra”, *Homology Homotopy Appl.* **6** (2004) 363–411.
J. Stasheff, “Homotopy associativity of H-spaces. I, II”, *Trans. Amer. Math. Soc.* **108** (1963) 275–292; ibid. **108** (1963) 293–312.
J. Stasheff, “From operads to "physically” inspired theories”, in *Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995)*, Contemp. Math. **202**, Amer. Math. Soc., Providence, RI, 1997, 53–81.
D. Tamari, “Monoides pr' eordonn' es et cha�nes de Malcev”, Doctorat ès-Sciences Math' ematiques Thèse de Math' ematiques, Universit' e de Paris (1951).
D. Tamari, “Monoides pr' eordonn' es et chaînes de Malcev”, *Bull. Soc. Math. France* **82** (1954) 53–96.
B. Vallette, “Manin products, Koszul duality, Loday algebras and Deligne conjecture”, *J. Reine Angew. Math.* **620** (2008) 105–164.
D. Yau, “Gerstenhaber structure and Deligne’s conjecture for Loday algebras”, *J. Pure Appl. Algebra* **209** (2007) 739–752.
|
---
abstract: 'We show how cosmological correlation functions of massless fields can be rewritten in terms of Minkowski correlation functions, by extracting symmetry-breaking operators from the cosmological correlators. This technique simplifies some cosmological calculations. Also, known properties of Minkowski correlation functions can be translated to non-trivial properties of cosmological correlations. To illustrate this idea, inflation to Minkowski and matter bounce to Minkowski relations are presented for the interactions of general single field inflation. And a Minkowski recursion relation is translated to a novel relation for inflation.'
author:
- 'Shek Kit Chu$^{1,2}$'
- 'Mang Hei Gordon Lee$^{1}$'
- 'Shiyun Lu$^{1,2}$'
- 'Xi Tong$^{3}$'
- 'Yi Wang$^{1,2}$'
- 'Siyi Zhou$^{1,2}$'
title: Connections between Minkowski and Cosmological Correlation Functions
---
=1
Introduction
============
Symmetry plays an important role in understanding correlation functions in quantum field theory. In realistic problems, some symmetries may be broken (either hard or spontaneously), in which case we lose the ability to directly understand the correlations purely in terms of the broken symmetries. For instance, in cosmology, Lorentz invariance and time translation are spontaneously broken by the time-dependent background. As a result, some of the features of cosmological correlation functions related to spacetime symmetry become opaque. However, it is sometimes possible to rewrite an asymmetric correlation function into $$\begin{aligned}
\Big\langle \rm asymmetric\,\, correlation\,\, function \Big\rangle'
= (asymmetric\,\, operator) \times \Big\langle symmetric\,\, correlation\,\, function \Big\rangle',\end{aligned}$$ where $\langle\cdots\rangle'$ denotes correlation functions with the momentum conservation delta function stripped. Such a possibility enables us to extend the usage of symmetry and its consequences to asymmetric correlation functions. In this paper, we provide explicit examples of correlation functions in Friedmann-Robertson-Walker (FRW) cosmologies, which can be obtained by applying asymmetric operators on correlation functions with the symmetry of Minkowski space.
It was noted [@Arkani-Hamed:2015bza] that in de Sitter space, correlation functions of massless scalars with exchange of massive scalars [@Chen:2009we; @Chen:2009zp; @Baumann:2011nk; @Assassi:2012zq; @Noumi:2012vr] can be obtained by applying a differential operator on correlation functions with conformally-coupled scalars. The relation takes the form: $$\begin{aligned}
\Big \langle \mbox{massless scalars} \Big \rangle'_\mathrm{inflation}
=
{O} \Big \langle \mbox{conformal scalars} \Big \rangle'_\mathrm{inflation}~,\end{aligned}$$ where the operator $O$ is constructed by rewriting the time variables into $\partial_K$ for some combination of three-momentum $K$ and then pull this operator out of the time integration of in-in formalism. This relation connects different types of field for inflationary (approximately de Sitter) spacetime, and should be understood in a diagram-by-diagram sense.
In this paper, we extend this observation to relate the correlation functions of massless fields in FRW (inflation or matter bounce) and Minkowski backgrounds. For inflation, the relation between inflationary and flat-space correlators takes a general form: $$\begin{aligned}
\Big \langle \mbox{curvature perturbation} \Big \rangle'_\mathrm{inflation}
=
\sum_i {O}_i \Big \langle \mbox{massless scalars} \Big \rangle'_{\mathrm{flat}, i}~,\end{aligned}$$ where the index $i$ stands for different subprocesses with different contractions.
Interestingly, for some specific types of interactions such as a simple $\dot\zeta^3$ vertex (where $\zeta$ is the curvature fluctuation in comoving gauge), the operators corresponding to different contraction coincides with each other. This yields a stronger relation that we are able to pull out an overall operator acting on the Minkowski correlation to generate the correlation in cosmology: $$\begin{aligned}
\Big \langle \mbox{curvature perturbation} \Big \rangle'_\mathrm{inflation}
=
O \Big \langle \mbox{massless scalars} \Big \rangle'_{\mathrm{flat}}~.\end{aligned}$$ This type of method can be used to compute correlation functions in other kinds of cosmological background as well, such as the matter bounce cosmology.
With the relation between cosmological and Minkowski correlation functions, one can import known relations of Minkowski correlation functions to cosmology. For example, in [@Arkani-Hamed:2017fdk], a BCFW-like recursion relation [@Britto:2005fq; @ArkaniHamed:2010kv; @ArkaniHamed:2012nw; @Benincasa:2015zna] is derived for the wave function of the universe for conformally-coupled scalars. By applying the operator technique, we are able to obtain the corresponding recursion relations for massless fields in cosmology.
This paper is organized as follows. In Section \[generalmethod\], we compute the equal-time correlation functions of massless scalars in flat space, and introduce a general formalism can be used to relate these correlators to their cosmological counterpart. In Section \[applicationtoinflation\], we apply this formalism to general single-field inflation. In Section \[recursionrelation\], we apply our techniques to obtain a recursion relation between the cosmological correlation functions by using the Minkowski recursion relation. We conclude in Section \[conclusionandoutlook\].
General Method {#generalmethod}
==============
In this section, we rewrite the Minkowski correlators into the three-dimensional forms which are similar to FRW correlators, and outline how the relations between Minkowski and FRW is to be constructed. In §\[flat\], we compute equal-time three- and four-point functions in flat space. We describe the rule to translate these into inflationary correlation functions in §\[transitiontocosmology\].
Minkowski Correlator {#flat}
--------------------
We start by writing down some equal-time correlators in Minkowski spacetime, these correlators will be later used to generate equal-time correlators in inflation and bouncing cosmologies. Consider the action $$\begin{aligned}
S_0 = - \frac{1}{2}\int d\tau d^3 x\, (\partial\phi)^2 ~.\end{aligned}$$ Here the $(-,+,+,+)$ metric convention is used. The field $\phi$ is quantized as $$\begin{aligned}
\phi_{\mathbf k} (\tau) = u_k (\tau) a_{\mathbf k} + u^*_k (\tau) a^\dagger_{-\mathbf k}~,\end{aligned}$$ where $a_{\mathbf k}$ and $a^\dagger_{-\mathbf k}$ are the creation and annihilation operators satisfying the canonical commutation relations. The mode function is given by $$\begin{aligned}
\label{flatuk}
u_k(\tau) = \frac{1}{\sqrt{2k}} e^{-i k\tau}\, .\end{aligned}$$ As examples, we consider two kinds of interaction Hamiltonians of the free field, $$\begin{aligned}
H_{\phi^3} = \int d^3 x\, \lambda_3\,\phi^3\, , \quad H_{\phi^4} = \int d^3 x\, \lambda_4\,\phi^4\, ,\end{aligned}$$ where $\lambda_3$ and $\lambda_4$ are coupling constants, and use the in-in formalism to calculate the correlation in the Minkowski spacetime with $H_{I} = H_{\phi^3} + H_{\phi^4}$. The equal-time $n$-point function of $\phi$ can then be computed using the in-in formalism [@Weinberg:2005vy; @Chen:2010xka; @Wang:2013eqj] $$\begin{aligned}
\label{expression1}
\langle\phi^n(\tau)\rangle = \Big\langle \Big[ \bar T e^{i \int_{-\infty}^0d \tau'\, H_{I} (\tau')} \Big]\, \phi^n(\tau)\, \Big[ T e^{-i \int_{-\infty}^0d\tau'\, H_{I} (\tau') } \Big] \Big\rangle\, ,\end{aligned}$$ where $T$ and $\bar T$ denote the time-ordering and anti-time-ordering operators, respectively.
### Three-Point Function
The three-point function involves only the $H_{\phi^3}$ interaction at tree level. The leading-order correlation function is calculated from the first order contribution in the perturbation series $$\begin{aligned}
\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle = 2 {\rm Im} \int_{\tau_0}^{\tau} d \tau_1 \langle 0 | \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} H_{\phi^3}(\tau_1) | 0 \rangle~.\end{aligned}$$ The corresponding Feynman diagram is illustrated in Fig. \[fig:3pt\]. Taking the initial time to $-\infty$ and final time to $0$, we get $$\begin{aligned}
\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle' = -\frac{3\lambda_3}{2k_1k_2k_3k_{123}}\, ,\end{aligned}$$ where we use the notation $k_{i_1\cdots i_n}\equiv k_{i_1}+\cdots+k_{i_n}$, and the prime indicates that we strip the momentum conserving delta function $(2\pi)^3 \delta (\mathbf k_1+\mathbf k_2 + \mathbf k_3)$.
![\[fig:3pt\] The Feynman diagram associated to the three-point function.](3ptdiagram.pdf){width="3cm"}
### Four-Point Function
The four-point function involves both $H_{\phi^3}$ and $H_{\phi^4}$ interactions. The leading order correlation function is calculated from the first order and the second order in the perturbation series (as shown in Fig. \[fig:4pt\]), respectively: $$\begin{aligned}
\nonumber
\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle & = 2 {\rm Im} \int_{\tau_0}^{\tau} d \tau_1 \langle 0 | \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} H_{\phi^4}(\tau_1) | 0 \rangle \\ \nonumber
& + \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau} d\tau_2 \langle 0 | H_{\phi^3} (\tau_1) \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} H_{\phi^3} (\tau_2) | 0 \rangle \\
& - 2 {\rm Re} \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau_1} d\tau_2 \langle 0 | \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} H_{\phi^3} (\tau_1) H_{\phi^3} (\tau_2) | 0 \rangle~.\end{aligned}$$ Taking the initial time going to $-\infty$ and final time going to $0$, we get $$\begin{aligned}
\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle' = \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{(3)} + \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{(4)}~,\end{aligned}$$ where the subscripts $(3)$ and $(4)$ denote the diagrams contributed by the three-point interaction and four-point interaction, respectively. The part of contributions from three-point interaction is $$\begin{aligned}
\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{(3)} & = \frac{9\lambda_3^2}{2 k_1 k_2 k_3 k_4 k_t}\frac{k_t+k_I}{k_I(k_{12}+k_I)(k_{34}+k_I)} + \text{permutations} \, .\end{aligned}$$ where we denoted the total momentum by $k_t=k_{1234}$ and the internal momentum by $k_I=|\mathbf k_1+\mathbf k_2|$. The contact diagram gives $$\begin{aligned}
\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{(4)}=- \frac{3\lambda_4}{k_1k_2k_3k_4k_t}\, . % {\rm Im} \bigg[\frac{1}{16 k_1 k_2 k_3 k_4} \int_{-\infty}^0 d\tau_1 e^{i k_{1234} \tau_1} \bigg] + {\rm 23\,\,\, permutations}\, .\end{aligned}$$
![\[fig:4pt\]The Feynman diagrams associated to the four-point function.](4ptdiagram.pdf){width="10cm"}
From Minkowski to FRW {#transitiontocosmology}
---------------------
Here we sketch the relation between Minkowski and cosmological correlation functions for a massless field. Here we use inflation as an example (where details will be described in Section \[applicationtoinflation\] and Appendix \[appendA\]). The method can be readily generalized to matter dominated contracting universe (where details will be described in Appendix \[applicationtobounce\]), as well as other cosmologies where the solution of the massless field can be written in terms of elementary functions (such as matter dominated expansion, or radiation dominated contracting or expanding universes).
The inflation mode functions at $\tau=0$ can be related to Minkowski spacetime by $$\begin{aligned}
u_k(0)_{\rm inflation} = C u_k(0)_{\rm flat}~.\end{aligned}$$ The Hamiltonian is then related to the Minkowski Hamiltonian by $$\begin{aligned}
\int d\tau H_I(\tau)_{\rm inflation} \rightarrow O_{\mathrm{ext},i} \int d\tau H_I(\tau)_{\mathrm{ flat}} ~,\end{aligned}$$ where $i$ denotes different contractions in the in-in formalism. The subscript “ext” means that usually this operator involves integration or derivative with respect to the total momentum of the external legs associated to a certain vertex. Such a relation is possible because the inflationary mode function contains oscillatory parts and the Hamiltonian in general takes the form $\int d\tau f(\tau) e^{\pm i K \tau}$, where $K$ is a combination of external momenta to be specified in the next section. As a result, the $f(\tau)$ part of the integrand can be replaced by $f(\mp i \partial_K)$ and be extracted out of the integral. The remaining part is the Minkowski Hamiltonian up to constant normalization factors.
Putting them altogether, we have $$\begin{aligned}
\langle \cdots \rangle'_{\mathrm{inflation},i} = O_{i} \langle \cdots \rangle'_{\mathrm{flat},i} ~.\end{aligned}$$ $O_{i}$ is related to $O_{\mathrm{ext},i}$ by $$\begin{aligned}
O_{i} = O_{\mathrm{ext},i}^m ~ C^n~,\end{aligned}$$ for $n$-point function and $m$-th order in Hamiltonian. Note that this is a schematic relation. For different Hamiltonians the $O_{\mathrm{ext},i}$ operators are different, and the $C$ factors depends on the values of external momenta.
For some interactions (for example $\dot\zeta^3$ as we will show in the next section), the operators associated with different types of in-in contours degenerate, so that we can arrive at a stronger statement $$\begin{aligned}
\langle \cdots \rangle'_{\rm inflation} = O \langle \cdots \rangle'_{\rm flat} ~,\end{aligned}$$ where $O$ is related to $O_{\rm ext}$ by $$\begin{aligned}
O = O_{\rm ext}^m ~ C^n~.\end{aligned}$$ For different Hamiltonians the $O_{\rm ext}$ operators are different, but for different in-in contours these operators are the same for such specific interactions.
Application to Inflation {#applicationtoinflation}
========================
In this section, we would like to use the symmetry-breaking operator technique to calculate the three-point function in general single field inflation [@Chen:2006nt] as an illustration[^1].
The three-point function generated by single field inflation models with a canonical kinetic term is suppressed by slow-roll parameters [@Maldacena:2002vr; @Acquaviva:2002ud]. Interactions in the generalized Lagrangians [@ArmendarizPicon:1999rj; @Alishahiha:2004eh] are known to source potentially large three-point functions [@Chen:2006nt]. Here we will apply the symmetry-breaking operator method to this model. The readers not interested in the technical details can directly find the resulting inflation-Minkowski relation for three-point functions in Equations and . We will also derive the relation for four-point functions in Appendix \[appendA\].
In the following we set the Planck mass $M_{\rm pl}$ to 1. The Lagrangian for general single field inflation is $$\begin{aligned}
S = \frac{1}{2} \int d^4 x \sqrt{-g} \bigg[ R + 2 P(X,\phi) \bigg]~,\end{aligned}$$ where $\phi$ is the inflaton field and $X = -\frac{1}{2}g^{\mu\nu} \partial_\mu\phi\partial_\nu\phi$. The three slow-variation parameters are defined as $$\begin{aligned}
\epsilon = - \frac{\dot H}{H^2}, \quad \eta = \frac{\dot \epsilon}{\epsilon H}, \quad s = \frac{\dot c_s}{c_s H}~, \end{aligned}$$ where $H=\dot a/a$ is the Hubble parameter and the sound speed is $$\begin{aligned}
c_s^2\equiv \frac{P_{,X}}{P_{,X}+ 2 X P_{,XX}}~,\end{aligned}$$ where $P_{,X}$ denote the derivative of $P(X,\phi)$ with respect to $X$. There are two other useful parameters $\lambda$ and $\Sigma$ defined as [@Seery:2005wm] $$\begin{aligned}
\lambda & = X^2 P_{,XX} + \frac{2}{3} X^3 P_{,XX}~, \\
\Sigma & = X P_{,X} + 2 X^2 P_{,XX}~.\end{aligned}$$ We will focus on cases with either $c_s \ll 1$ or $|\lambda/\Sigma| \gg 1$, such that the corresponding interactions dominate over the slow roll contributions. The method can be straightforwardly generalized to these slow roll suppressed terms.
The second order action for general single field inflation is $$\begin{aligned}
S_2 = \int dt d^3 x \bigg[ a^3 \frac{\epsilon}{c_s^2} \dot{\zeta}^2 - a\epsilon (\partial \zeta)^2 \bigg]~,\end{aligned}$$ where $\zeta$ is quantized as $$\begin{aligned}
\zeta_{\mathbf k} (\tau) = u_k (\tau) a_{\mathbf k} + u^*_k (\tau) a^\dagger_{-\mathbf k}~,\end{aligned}$$ with the usual commutation relations for the creation and annihilation operators. The mode function and its time derivative are $$\begin{aligned}
\label{zetauk}
u_k (\tau) = \frac{iH}{\sqrt{4\epsilon c_s k^3}} (1+i c_s k \tau) e^{-ic_s k \tau}~,
\qquad
u'_k(\tau) = \frac{iH}{\sqrt{4\epsilon c_s k^3}} c_s^2 k^2 \tau e^{-ic_s k\tau}~.\end{aligned}$$ The two-point correlation function can be written as $$\begin{aligned}
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2} \rangle' = \frac{2\pi^2}{k^3} \Delta_\zeta^2,
\qquad
\Delta_\zeta^2 = \frac{1}{8\pi^2} \frac{H^2}{c_s \epsilon}~.\end{aligned}$$
The rule to relate external mode functions
------------------------------------------
At $\tau=0$, the difference between the mode function of $\zeta$ for inflation and the mode function of $\phi$ for Minkowski spacetime is just a constant. Using and , we have $$\begin{aligned}
u_k(0)_{\rm inflation} = \frac{iH}{\sqrt{2 \epsilon c_s} k}\, u_k(0)_{\rm flat}\, .\end{aligned}$$ For example, for a three-point function with external momenta $k_1$, $k_2$ and $k_3$, to translate the Minkowski correlation function to that of inflation, for the external lines, we have to add a factor $$\begin{aligned}
\bigg(\frac{i H}{\sqrt{2\epsilon c_s}}\bigg)^3 \frac{1}{k_1 k_2 k_3}~.\end{aligned}$$
The rule to relate Hamiltonians
-------------------------------
In order to compute the three-point function, we need to perturb the Hamiltonian to the third order. If we focus on the part with non-trivial sound speed, the third order interaction Hamiltonian is $$\begin{aligned}
H_3 = H_{\zeta'^3} + H_{\zeta' (\partial \zeta)^2} ~,\end{aligned}$$ where $$\begin{aligned}
H_{\zeta'^3} (\tau) & = \int d^3x\,2 a \frac{\lambda}{H^3} \zeta'^3~, \\
H_{\zeta' (\partial \zeta)^2} (\tau) & = - \int d^3x\,a \frac{\Sigma}{H^3} (1- c_s^2) \zeta' (\partial \zeta)^2~.\end{aligned}$$ For simplicity, in what follows we will suppress the momentum-conserving delta functions in the Hamiltonians in momentum space.
- The Minkowski Hamiltonian\
- For $H_{\phi^3}$, if all fields in it contract with the field on its left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\phi^3} \rightarrow \int_{-\infty}^0d\tau\, \frac{\lambda_3}{2 \sqrt{2} \sqrt{k_1 k_2 k_3} } e^{i k_{123} \tau}\end{aligned}$$
- The inflationary Hamiltonian\
- For $H_{\zeta'^3}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta'^3}\rightarrow \int_{-\infty}^{0} -2 \frac{\lambda}{H} \frac{i c_s^{3/2} e^{i k_{123} \tau} \sqrt{k_1 k_2 k_3} \tau^2}{8 \epsilon^{3/2} } d\tau~.
\end{aligned}$$ Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta'^3} \rightarrow \frac{i c_s k_1 k_2 k_3 \lambda}{\sqrt{2} H \epsilon^{3/2} \lambda_3 } \frac{\partial^2}{\partial k_{123}^2} H_{\phi^3}~.
\end{aligned}$$ Thus the corresponding relation between the inflationary and Minkowski three-point functions is $$\begin{aligned}
\label{eq:3pt3a}
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3} \rangle_{\rm inflation} = \bigg(\frac{i H}{\sqrt{2 \epsilon c_s}}\bigg)^3 \frac{1}{k_1 k_2 k_3} \frac{i c_s k_1 k_2 k_3 \lambda}{\sqrt{2} H \epsilon^{3/2} \lambda_3 } \frac{\partial^2}{\partial k_{123}^2} \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle_{\rm flat}~.\end{aligned}$$ If all fields in it contract with the field on the right hand side, we will get the complex conjugate instead $$\begin{aligned}
H_{\zeta'^3} \rightarrow -\frac{i c_s k_1 k_2 k_3 \lambda}{\sqrt{2} H \epsilon^{3/2} \lambda_3 } \frac{\partial^2}{\partial k_{123}^2} H_{\phi^3}~.
\end{aligned}$$ If we use it to compute the exchange diagram, since there are only two external legs attached to the vertex, we should make the substitution $k_{123}\rightarrow k_{12}$. If $k_1$ and $k_2$ contract with the field on the left hand side and $k_3$ contracts with the field on the right hand side, it gives the same operator.
- $H_{\zeta' (\partial \zeta)^2}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2} (k_1,k_2,k_3,k_{123}) \rightarrow \int_{-\infty}^{0} \frac{i(-1+c_s^2)e^{i k_{123}\tau} \sqrt{k_1} \mathbf k_2\cdot \mathbf k_3 \Sigma (1-i k_2\tau) (1-i k_3\tau) }{8 H k_2^{3/2} k_3^{3/2} \epsilon^{3/2} c_s^{1/2} } d\tau~.
\end{aligned}$$ Here $k_1$ corresponds to the momentum of $\dot\zeta$. $k_2$ and $k_3$ correspond to the momenta of the second and third $\partial \zeta$, respectively. Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2}(k_1,k_2,k_3,k_{123}) \rightarrow \frac{i (-1+c_s^2) k_1 \mathbf k_2\cdot \mathbf k_3\Sigma}{2\sqrt{2} \sqrt{c_s} H k_2 k_3 \epsilon^{3/2} \lambda_3 } \bigg( 1- (k_2+k_3) \frac{\partial}{\partial k_{123}} + k_2 k_3 \frac{\partial^2}{\partial k_{123}^2} \bigg) H_{\phi^3}~.
\end{aligned}$$ Thus the corresponding relation between the inflationary and Minkowski three-point functions is $$\begin{aligned}
\label{eq:3ptdzpzpz}
\nonumber
& \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3} \rangle_{\rm inflation} \\ \nonumber
& = \bigg(\frac{i H}{\sqrt{2 \epsilon c_s}}\bigg)^3 \frac{1}{k_1 k_2 k_3} \frac{i (-1+c_s^2) k_1 \mathbf k_2\cdot \mathbf k_3\Sigma}{2\sqrt{2} \sqrt{c_s} H k_2 k_3 \epsilon^{3/2} \lambda_3 } \bigg( 1- (k_2+k_3) \frac{\partial}{\partial k_{123}} + k_2 k_3 \frac{\partial^2}{\partial k_{123}^2} \bigg) \\
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle_{\rm flat} \times \frac{1}{6} +{5\,\,\rm permutations}~.
\end{aligned}$$ The factor of $1/6$ and the five permutations correspond to that actually this Hamiltonian is not symmetric in $k_1$, $k_2$ and $k_3$. When we want to use it in the scalar diagram, we should substitute $k_1$ with the momentum associated with the $\zeta'$ mode function whereas $k_2$ and $k_3$ with the $\partial \zeta$ mode function. $k_{123}$ should be substituted with the sum over the momenta of the external legs of this Hamiltonian.
The relation is the final result to relate the inflationary and Minkowski leading order three-point functions for the $\zeta' (\partial\zeta)^2$ interaction, because here all the fields in the interaction Hamiltonian contract with fields to the left. However, for beyond-leading order calculations, or higher-point correlation functions, we may encounter the possibility that some fields in the interaction Hamiltonian contracts to the left and some contracts to the right. For these situations, we will also need the rules below:
- $H_{\zeta' (\partial \zeta)^2}$, if all fields in it contract with the field on the right hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2} (k_1,k_2,k_3,k_{123}) \rightarrow \int_{-\infty}^{0} \frac{-i(-1+c_s^2)e^{-i k_{123}\tau} \sqrt{k_1} \mathbf k_2\cdot \mathbf k_3 \Sigma (1+i k_2\tau) (1+i k_3\tau) }{8 H k_2^{3/2} k_3^{3/2} \epsilon^{3/2} c_s^{1/2} } d\tau~.
\end{aligned}$$ Here $k_1$ corresponds to the momentum of $\dot\zeta$. $k_2$ and $k_3$ correspond to the momenta of the second and third $\partial \zeta$, respectively. Comparing with $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2}(k_1,k_2,k_3,k_{123}) \rightarrow \frac{-i (-1+c_s^2) k_1 \mathbf k_2\cdot \mathbf k_3\Sigma}{2\sqrt{2} \sqrt{c_s} H k_2 k_3 \epsilon^{3/2} \lambda_3 } \bigg( 1- (k_2+k_3) \frac{\partial}{\partial k_{123}} + k_2 k_3 \frac{\partial^2}{\partial k_{123}^2} \bigg) H_{\phi^3}~.
\end{aligned}$$ The usage of it is the same as $H_{\zeta' (\partial \zeta)^2}(k_1,k_2,k_3,k_{123})$.
- $H_{\zeta' (\partial \zeta)^2}$, two of them contracts with the field on the left and one of them contracts with the field on the right.
- $\zeta'$ contracts with right hand side $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2} (k_1,k_2,k_I,k_{12}) \rightarrow \int_{-\infty}^{0} \frac{-i(-1+c_s^2)e^{i (k_{12}-k_I)\tau} \sqrt{k_I} \mathbf k_1\cdot \mathbf k_2 \Sigma (1-i k_1\tau) (1-i k_2\tau) }{8 H k_1^{3/2} k_2^{3/2} \epsilon^{3/2} c_s^{1/2} } d\tau~.
\end{aligned}$$ Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2}(k_1,k_2,k_I,k_{12}) \rightarrow \frac{-i (-1+c_s^2) k_I \mathbf k_1\cdot \mathbf k_2\Sigma}{2\sqrt{2} \sqrt{c_s} H k_1 k_2 \epsilon^{3/2} \lambda_3 } \bigg( 1- (k_1+k_2) \frac{\partial}{\partial k_{12}} + k_1 k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) H_{\phi^3}~.
\end{aligned}$$
- $\partial\zeta$ contracts with right hand side $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2} (k_1,k_2,k_I,k_{12}) \rightarrow \int_{-\infty}^{0} \frac{-i(-1+c_s^2)e^{i (k_{12}-k_I) \tau} \sqrt{k_1} \mathbf k_2\cdot \mathbf k_I \Sigma (1-i k_2\tau) (1-i k_I\tau) }{8 H k_2^{3/2} k_I^{3/2} \epsilon^{3/2} c_s^{1/2} } d\tau~.
\end{aligned}$$ Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta' (\partial \zeta)^2}(k_1,k_2,k_I,k_{12}) \rightarrow \frac{-i (-1+c_s^2) k_1 \mathbf k_2\cdot \mathbf k_I\Sigma}{2\sqrt{2} \sqrt{c_s} H k_2 k_I \epsilon^{3/2} \lambda_3 } \bigg( 1- (k_2+k_I) \frac{\partial}{\partial k_{12}} + k_2 k_I \frac{\partial^2}{\partial k_{12}^2} \bigg) H_{\phi^3}~.
\end{aligned}$$
With these rules, one can get the relations for the four-point function. Details are left to Appendix \[appendA\]. In general, this type of method is applicable for any $n$-point correlation functions at the tree level.
For loop diagrams, we expect similar symmetry-breaking operators for the integrand. However, the symmetry-breaking operators are expected to depend on the free momentum running in the loop. Thus we cannot extract the symmetry-breaking operator out from the loop integral with the current technique.
Note that if we would like to study some parts of a Feynman diagram (for example, studying the polology of a propagator) instead of a whole diagram, we may choose to apply the symmetry-breaking operators to a selection of relevant vertices only.
Also, we present the extension of this method for bouncing cosmology in Appendix \[applicationtobounce\].
A Recursion Relation {#recursionrelation}
====================
In [@Arkani-Hamed:2017fdk], various recursion relations of conformal scalars are derived for the wave function of the de Sitter universe [@Hartle:1983ai; @Hertog:2011ky; @Bunch:1978yq; @Chernikov:1968zm]. Here we first use their method to obtain a recursion relation for massless fields in flat space thanks to the similarity between these two cases. Then, we apply symmetry breaking operators to obtain the corresponding recursion relation for massless fields for inflation.
For Minkowski space with an artificial “future boundary” at $\tau\rightarrow 0$, the wave function of the universe takes the form $$\begin{aligned}
\nonumber
\Psi = {\rm exp} \bigg[ & \frac{1}{2!} \int d^3 z_1 \int d^3 z_2 \phi (z_1) \phi(z_2) \hat \psi_2 (z) \\ \nonumber
+ & \frac{1}{3!} \int d^3 z_1 \int d^3 z_2 \int d^3 z_3 \phi (z_1) \phi(z_2) \phi (z_3) \hat \psi_3 (z) \\
+ & \frac{1}{4!} \int d^3 z_1 \int d^3 z_2 \int d^3 z_3 \int d^3 z_4 \phi (z_1) \phi(z_2) \phi(z_3) \phi(z_4) \hat \psi_4 (z) \bigg].\end{aligned}$$ In the following, we work in the momentum space and $\hat\psi_n$ denotes the corresponding quantity in the momentum space with momentum conservation delta function striped. The $n$-point correlation functions at a fixed time slice $\tau_c$ can be obtained once we know the form of the wave function in the following way $$\begin{aligned}
\langle \phi_{\mathbf k_1} \cdots \phi_{\mathbf k_n} \rangle = \frac{\int\prod_{\mathbf k} d\phi_{\mathbf k} |\Psi[\phi_{\mathbf k},\tau_c]|^2 \phi_{\mathbf k_1}\cdots\phi_{\mathbf k_n}}{\int\prod_{\mathbf k} d\phi_{\mathbf k} |\Psi[\phi_{\mathbf k},\tau_c]|^2}\end{aligned}$$ Using the Gaussian integral, the following dictionary can be established between the inflationary correlation function and the wave function of the universe, $$\begin{aligned}
\langle \phi_{\mathbf k} \phi_{-\mathbf k} \rangle' = \frac{1}{-2{\rm Re} \hat \psi_2 }~.\end{aligned}$$ The three-point function can be calculated as $$\begin{aligned}
\langle \phi_{\mathbf k_1} \phi_{\mathbf k_2} \phi_{\mathbf k_3} \rangle' = \frac{2{\rm Re} \hat \psi_3}{ (-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2) }~.\end{aligned}$$ The four-point function with a scalar exchange of $\mathbf k_I$ can be calculated as $$\begin{aligned}
\nonumber
& \langle \phi_{\mathbf k_1} \phi_{\mathbf k_2} \phi_{\mathbf k_3} \phi_{\mathbf k_4} \rangle' \\ \label{4ptwavefunctionvscorrelation}
& = \frac{2{\rm Re} \hat \psi_4 }{(-2{\rm Re} \hat \psi_2) (-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2) } + \frac{2{\rm Re} \hat \psi_3 2{\rm Re} \hat \psi_3}{ (-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2)(-2{\rm Re} \hat \psi_2) }~.\end{aligned}$$ $\hat \psi_4$ is contributed by the scalar exchange diagram and be calculated as $$\begin{aligned}
\hat \psi_4 = \int_{-\infty}^0 d\tau_1 \int_{-\infty}^0 d\tau_2 H_{k_1}(\tau_1)H_{k_2}(\tau_1)H_{k_3}(\tau_2)H_{k_4}(\tau_2)G_{k_I}(\tau_1,\tau_2)~,\end{aligned}$$ where $H_k(\tau)$ and $G_k(\tau_1,\tau_2)$ are bulk-boundary and bulk-bulk propagators defined as $$\begin{aligned}
H_k (\tau) & = e^{i k \tau} \\
G_k (\tau_1,\tau_2) & = \frac{1}{2 k} [e^{-i k (\tau_1-\tau_2)}\Theta(\tau_1-\tau_2) + e^{i k(\tau_1-\tau_2)}\Theta(\tau_2-\tau_1) - e^{i k (\tau_1+\tau_2)} ]~,\end{aligned}$$ where the purpose of the last term in the bulk-bulk propagator is to force its value vanish on the boundary. The $\hat \psi_4$ can be calculated in the following way $$\begin{aligned}
\hat \psi_4 = \int_{-\infty}^0 d\tau_1 \int_{-\infty}^0 d\tau_2 e^{i k_{34}\tau_2} e^{i k_{12}\tau_1} \frac{1}{2k_I} \bigg[ e^{-i k_I (\tau_1-\tau_2)} \Theta(\tau_1-\tau_2) + e^{i k_I(\tau_1-\tau_2)} \Theta(\tau_2-\tau_1) -e^{i k_I (\tau_1+\tau_2)} \bigg] ~.\end{aligned}$$ If we integrate $\tau_1$ first, we have $$\begin{aligned}
\label{masteridentity}
\hat \psi_4 = \frac{1}{k_{12}^2- k_I^2} \bigg(\int_{-\infty}^{0} d\tau_2 \frac{1}{i} e^{i(k_{12}+k_{34})\tau_2} - \int_{-\infty}^{0} d\tau_2 \frac{1}{i} e^{i(k_I+k_{34})\tau_2} \bigg)~.\end{aligned}$$ In [@Arkani-Hamed:2017fdk], it is noted that the right hand side is just the difference of two three-point vertices of the wave function $\hat\psi_3$, with shifted momenta. And this observation can be applied recursively to obtain a relation between the $n$-point vertex of the wave function and the difference of many three-point vertexes with shifted momenta. Thus, one can regard $\hat\psi_3$ as a fundamental building block of the high-point vertices of the wave function. This is in analogous to the fact that in CFT, the three-point function is the fundamental building block; in AdS/CFT, the cubic Witten diagram (triple-K integral [@Bzowski:2013sza; @Bzowski:2015pba; @Bzowski:2015yxv]) is the fundamental building block; and in the scattering amplitude literature, the MHV amplitude is the building block. Thus the significance of finding an inflationary correlation counter part of this kind of recursion relation not only lies in the interest of simplifying calculations, but also in theoretically understanding the mathematical structure of the correlation functions.
Making use of , Equation can be written as $$\begin{aligned}
\nonumber
& \frac{ \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'}{2 \langle \phi_{\mathbf k_1}\phi_{-\mathbf k_1} \rangle'\langle \phi_{\mathbf k_2}\phi_{-\mathbf k_2} \rangle'\langle \phi_{\mathbf k_3}\phi_{-\mathbf k_3} \rangle'\langle \phi_{\mathbf k_4}\phi_{-\mathbf k_4} \rangle'} = \frac{ \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_I}\rangle'\langle\phi_{-\mathbf k_I}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle' }{2\langle \phi_{\mathbf k_1}\phi_{-\mathbf k_1} \rangle'\langle \phi_{\mathbf k_2}\phi_{-\mathbf k_2} \rangle'\langle \phi_{\mathbf k_3}\phi_{-\mathbf k_3} \rangle'\langle \phi_{\mathbf k_4}\phi_{-\mathbf k_4} \rangle'\langle \phi_{\mathbf k_I}\phi_{-\mathbf k_I} \rangle'} \\ \nonumber
& + \frac{1}{k_I^2 - k_{12}^2} \bigg[ \frac{-\langle \phi_{\mathbf k_1}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'}{2\langle \phi_{\mathbf k_1}\phi_{-\mathbf k_1}\rangle' \langle \phi_{\mathbf k_3}\phi_{-\mathbf k_3} \rangle'\langle \phi_{\mathbf k_4}\phi_{-\mathbf k_4} \rangle'} \bigg|_{\lambda_3\rightarrow \lambda_3^2, k_1\rightarrow k_{12}} - \frac{-\langle \phi_{\mathbf k_3} \phi_{\mathbf k_4} \phi_{\mathbf k_I} \rangle'}{2\langle \phi_{\mathbf k_3} \phi_{-\mathbf k_3} \rangle'\langle \phi_{\mathbf k_4} \phi_{-\mathbf k_4} \rangle'\langle \phi_{\mathbf k_I} \phi_{-\mathbf k_I} \rangle'}\bigg|_{\lambda_3\rightarrow \lambda_3^2 } \bigg]\\
& + {\rm permutations}~.\end{aligned}$$ Let’s take $H_{\zeta'^3}$ interaction as an example, acting $\frac{H^2 k_I^2 \lambda^2}{8 \epsilon^5 \lambda_3^2}\frac{\partial^2}{\partial k_{12}^2}\frac{\partial^2}{\partial k_{34}^2}$ on both sides of the correlation, and after some simplification, we have[^2] $$\begin{aligned}
\nonumber
& \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle'_{\rm inflation} = \frac{ \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_I}\rangle'_{\rm inflation}\langle\zeta_{-\mathbf k_I}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle'_{\rm inflation} }{ \langle \zeta_{\mathbf k_I}\zeta_{-\mathbf k_I} \rangle'_{\rm inflation} } \\ \nonumber
& + \bigg( \frac{k_I^2\lambda c_s^{3/2}}{\epsilon\lambda_3^2 H^2} \frac{\partial^2}{\partial k_{12}^2} \bigg\{ \frac{1}{k_I^2-k_{12}^2} \frac{-\langle\zeta_{\mathbf k_{12}} \zeta_{\mathbf k_3} \zeta_{\mathbf k_4} \rangle'_{\rm inflation}}{\langle \zeta_{\mathbf k_{12}}\zeta_{-\mathbf k_{12}} \rangle'_{\rm inflation} } \frac{1}{k_{12}^2} \bigg\} - H \frac{\lambda c_s^{3/2}}{\epsilon \lambda_3^2 H^2} \frac{\partial^2}{\partial_{k_{12}}^2} \bigg\{ \frac{1}{k_I^2-k_{12}^2} \frac{-\langle \zeta_{\mathbf k_3} \zeta_{\mathbf k_4} \zeta_{\mathbf k_I}\rangle'_{\rm inflation}}{ \langle \zeta_{\mathbf k_I} \zeta_{-\mathbf k_I} \rangle'_{\rm inflation}} \bigg\} \bigg) \\ \label{recursionrelation2}
& \langle \zeta_{\mathbf k_1}\zeta_{-\mathbf k_1} \rangle'_{\rm inflation} \langle \zeta_{\mathbf k_2}\zeta_{-\mathbf k_2} \rangle'_{\rm inflation} {k_1^2 k_2^2} + {\rm permutations} ~.\end{aligned}$$ Here $\langle \zeta_{\mathbf k_{12}}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle'$ is not a physical correlation function. This is because of the lack of momentum conservation: $\mathbf{k}_{12}+\mathbf{k}_3+\mathbf{k}_4\neq 0$. Nevertheless, for the particular $\dot\zeta^3$ interaction (and in general interactions without dot products of spatial derivatives), the three-point function does not depend on the direction of its momenta. Thus $\langle \zeta_{\mathbf k_{12}}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle' = \langle \zeta_{ k_{12}}\zeta_{ k_3}\zeta_{ k_4} \rangle'$. As a result, to test this recursion relation, one can search for correlation with momenta of size $k_{12}$, $k_3$ and $k_4$ in the sky as long as they satisfy the triangular inequalities.
On the other hand, one can also restrict the momenta configuration to the collinear limit[^3] where two external momenta (say, $\mathbf{k}_1$ and $\mathbf{k}_2$) are in the same direction. In this limit, $\mathbf{k}_1+\mathbf{k}_2 \rightarrow \mathbf{k}_{12}$ and thus the vectors $\mathbf k_{12}$, $\mathbf k_3$, $\mathbf k_4$ satisfy momentum conservation. In this limit, the factor $k_I^2-k_{12}^2$ in the denominator blows up. Thus we have to Taylor-expand the denominator and the numerator (similar to the L’Hospital’s rule, but the existence of the operator $\partial_{k_{12}}^2$ forces us to expand to third order in the Taylor series to get a meaningful result). The recursion relation then becomes $$\begin{aligned}
\nonumber
& \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle'_{\rm inflation} = \frac{ \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_I}\rangle'_{\rm inflation}\langle\zeta_{-\mathbf k_I}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle'_{\rm inflation} }{ \langle \zeta_{\mathbf k_I}\zeta_{-\mathbf k_I} \rangle'_{\rm inflation} } \\ \nonumber
&+ \frac{\lambda c_s^{3/2}}{\epsilon\lambda_3^2} \bigg\{ \frac{1}{-4 k_I} \bigg[ (- k_I) f^{(2)}_{\rm inflation} + f^{(1)}_{\rm inflation} + \frac{2}{3} k_I^2 f^{(3)}_{\rm inflation} \bigg] \bigg\} + {\rm permutations} ~,\end{aligned}$$ where $$\begin{aligned}
f^{(n)}_{\rm inflation} = \frac{\partial}{\partial\varepsilon^n} \bigg[ \frac{-\langle \zeta_{\mathbf k_{12}}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle'_{\rm inflation}}{ \langle \zeta_{\mathbf k_{12}}\zeta_{-\mathbf k_{12}}\rangle'_{\rm inflation} \frac{k_{12}^2}{H^2} } \bigg|_{k_{12}\rightarrow k_I+\varepsilon} \bigg] \langle \zeta_{\mathbf k_1}\zeta_{-\mathbf k_1} \rangle'_{\rm inflation} \langle \zeta_{\mathbf k_2}\zeta_{-\mathbf k_2} \rangle'_{\rm inflation} \frac{k_{1}^2}{H^2}\frac{k_{2}^2}{H^2}~.\end{aligned}$$ Note that the curvature perturbation is massless. Thus in flat space, the collinear limit corresponds to the situation where an internal propagator is on-shell if the external lines are on-shell. In the context of cosmology, the on-shell condition is not obvious because we are forced to consider space and time differently. Also we are considering correlation functions without making them into scattering amplitudes. Nevertheless, our method of extracting the symmetry-breaking operator allows us to find out such a collinear relation. It remains interesting to see if the result can be further connected to recent studies of scattering amplitudes (see for example [@Nandan:2016ohb] and references therein for the related flat space amplitudes).
Although we have been working in a particular model, the recursion relation has a range of generality:
- The relation is only sensitive to the vertex connecting $k_1$ and $k_2$ to internal propagators. No information on the interaction structure of the rest part of the Feynman diagram is needed. Thus we can have similar relations from a complicated $n$-point correlation function with general interactions, as long as the vertex attached to $k_1$ and $k_2$ has the same interaction structure as above.
- Here we have considered the interaction $H_{\zeta'^3}$ for the corresponding vertex. If a different Hamiltonian is used, a similar relation may apply, where different kinds of operators are applied onto the correlation functions.
We plan to leave a detailed study of this recursion relation, its generalizations and cosmological implications to a future work.
Conclusion and Outlook {#conclusionandoutlook}
======================
We have shown that massless cosmological correlation functions can be obtained by acting symmetry breaking operators to Minkowski correlation functions. We have derived the operators corresponding to general single field inflation as an example. Once written in terms of the Minkowski correlation function, properties in Minkowski spacetime can be translated to relations in cosmology.
There are many interesting possibilities to explore. Here we list a few examples. We hope to address some of these possibilities in the future.
- We have not taken advantages of Minkowski symmetries. It would be helpful to write the targeting Minkowski correlation functions in obviously 4-dimensional covariant formalism. More structures on the cosmological correlation functions may be uncovered due to the obvious symmetries.
- We have focused on massless scalar fields. It is interesting to generalize the method to include massless fields with higher spins, especially primordial gravitational waves. Also, acting an operator on a scalar exchange diagram can give us the result of a high spin massless field exchange diagram, which simplifies the previous result of [@Seery:2008ax]. Similar technique can also be applied to AdS/CFT to simplify the calculation about Witten diagram [@Hiroshi].
- We have focused on correlation functions for Minkowski and cosmological types. However, in flat space, much more relations are known for scattering amplitudes compared to general correlation functions. Conceptually, we expect that similar techniques should apply for cosmology. This is because on super-Hubble scales, the fluctuations get frozen and become classical. The classical fluctuations should be considered on-shell as their quantum components become irrelevant. It is thus valuable to construct a LSZ-type formalism to pick up the on-shell component of the correlation function and take advantage of the on-shell condition for further analysize the structure of the cosmological correlation functions, for example, unitarity, analyticity, causality [@Adams:2006sv; @Baumann:2015nta] and so on.
- Our current method only applies for massless fields (in fact also conformal fields but that is almost trivial). It is important to seek for alternative ways to relate cosmology with the Minkowski correlation function, with the generality to include fields with arbitrary mass. One hope is by taking the $k_t\rightarrow 0$ limit [@Arkani-Hamed:2015bza; @Arkani-Hamed:2017fdk] in the analytically continued momentum space. The reason is that in the very early universe deep inside the Hubble radius, the spacetime is approximately Minkowski. In AdS/CFT, this is famously known as a bulk point singularity [@Raju:2012zr]. It is interesting to see how far one can proceed in this direction and how much cosmological information can get recovered from this approach.
- We have relied on perturbation theory in searching for the symmetry breaking operators. It is interesting to seek for a non-perturbative description which is not organized in diagram-by-diagram basis. Such non-perturbative operators, if identified, may help in understanding the nonlinear physics of the large scale structure based on enhanced symmetries.
- It remains interesting to find a geometric implication of the known relations in cosmology. The polytope structure [@Arkani-Hamed:2017fdk] offers us this possibility. Especially the de Sitter dilatation symmetry and special conformal symmetry can be visualized as some transformation on the cosmological polytope. A more ambitious goal is to find an object that automatically taking into account all the diagrams in the perturbation series like what is discovered about the amplituhedron [@Arkani-Hamed:2013jha; @Arkani-Hamed:2013kca; @Arkani-Hamed:2017vfh] in Minkowski spacetime.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Dionysios Anninos, Andrew Cohen, Hayden Lee, Shing Yan Li, Yubin Li and Toshifumi Noumi for useful discussions. This work is supported in part by ECS Grant 26300316 and GRF Grant 16301917 from the Research Grants Council of Hong Kong. XT is supported by the Qian San-Qiang Class in the University of Science and Technology of China. SZ is supported by the Hong Kong PhD Fellowship Scheme (HKPFS) issued by the Research Grants Council (RGC) of Hong Kong.
Four-Point Function {#appendA}
===================
In this appendix, we calculate the four-point function of general single field inflation which was originally obtained in [@Chen:2009bc; @Arroja:2009pd]. We used the symmetry-breaking operator formalism and show the simplifications therein.
For the four-point function, we have a factor contributed by the external legs, $$\begin{aligned}
\bigg(\frac{i H}{\sqrt{2\epsilon c_s}}\bigg)^4 \frac{1}{k_1 k_2 k_3 k_4}~.\end{aligned}$$ Also we need the following form of the Hamiltonian for $H_{\phi^4}$. If all field in it contract with the field on its left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\phi^4} \rightarrow \int_{-\infty}^0 \frac{\lambda_4}{4 \sqrt{k_1 k_2 k_3 k_4} } e^{i k_{1234} \tau} d\tau~.\end{aligned}$$
Contact-interaction Diagram
---------------------------
We compute the four-point correlation function contributed by the contact-interaction diagram. This type of diagram was originally calculated in [@Huang:2006eha; @Arroja:2008ga]. We define the following parameter $$\begin{aligned}
\mu \equiv \frac{1}{2} X^2 P_{,XX} + 2 X^3 P_{,XXX} + \frac{2}{3} X^4 P_{,XXXX}~.\end{aligned}$$ The forth order Hamiltonian takes the form $$\begin{aligned}
H_{4} = H_{\zeta'^4} + H_{(\partial \zeta)^2 \zeta'^2} + H_{(\partial\zeta)^4 }~,\end{aligned}$$ where $$\begin{aligned}
H_{\zeta'^4} (\tau) & = \frac{1}{H^4} \bigg(-\mu + 9 \frac{\lambda^2}{\Sigma}\bigg) \zeta'^4~, \\
H_{(\partial \zeta)^2 \zeta'^2} (\tau) & = \frac{1}{H^4} (3\lambda c_s^2 - \Sigma(1-c_s^2)) (\partial \zeta)^2 \zeta'^2~, \\
H_{(\partial\zeta)^4 } (\tau) & = \frac{1}{4H^4} \Sigma (- c_s^2 + c_s^4) (\partial\zeta)^4 ~.\end{aligned}$$ Now we derive the rules relating the inflationary Hamiltonian and the Minkowski Hamiltonian. For the inflationary correlation functions contributed by different interactions, we all use $\langle\cdots \rangle'_{\rm inflation}$ to avoid complexity of notation. The different contractions are classified in the following cases:
- $H_{\zeta'^4}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta'^4} \rightarrow \int_{-\infty}^{0} - \frac{c_s e^{i k_{1234} \tau } \sqrt{k_1 k_2 k_3 k_4} (-9\lambda^2+\mu\Sigma) \tau^4 }{16 \epsilon^2 \Sigma} d\tau ~.
\end{aligned}$$ Comparing with the integral expression of $H_{\phi^4}$, we have $$\begin{aligned}
H_{\zeta'^4} \rightarrow - \frac{c_s k_1 k_2 k_3 k_4(-9\lambda^2+\mu\Sigma)}{4 \epsilon^2 \lambda \Sigma} \frac{\partial^4}{\partial k_{1234}^4} H_{\phi^4}~.
\end{aligned}$$ Thus the corresponding relation between the inflationary and Minkowski four-point functions is $$\begin{aligned}
\label{eq:4ptdz4}
\nonumber
& \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} \\
& = \bigg(\frac{i H}{\sqrt{2\epsilon c_s}}\bigg)^4 \frac{1}{k_1 k_2 k_3 k_4} \bigg(- \frac{c_s k_1 k_2 k_3 k_4(-9\lambda^2+\mu\Sigma)}{4 \epsilon^2 \lambda \Sigma} \frac{\partial^4}{\partial k_{1234}^4}\bigg) \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle_{\rm flat}~.
\end{aligned}$$
- $H_{(\partial \zeta)^2 \zeta'^2}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{(\partial \zeta)^2 \zeta'^2} (k_1,k_2,k_3,k_4) \rightarrow \int_{-\infty}^{0} \frac{e^{i k_{1234}\tau} \sqrt{k_1} \sqrt{k_2} \mathbf k_3\cdot \mathbf k_4 (-\Sigma+c_s^2(3\lambda+\Sigma)) }{16 c_s k_3^{3/2} k_4^{3/2} \epsilon^2 } (\tau_1^2(-1) (1-i k_3\tau)(1-i k_4\tau) ) d\tau~.
\end{aligned}$$ Comparing with the integral expression of $H_{\phi^4}$, we have $$\begin{aligned}
H_{(\partial \zeta)^2 \zeta'^2} (k_1,k_2,k_3,k_4) \rightarrow \frac{k_1 k_2 \mathbf k_3\cdot \mathbf k_4 (-\Sigma+c_s^2(3\lambda+\Sigma)) }{4 c_s k_3 k_4 \epsilon^2 \lambda_4} \frac{\partial^2}{\partial k_{1234}^2} \bigg(1-(k_3+k_4)\frac{\partial}{\partial k_{1234}} + k_3 k_4 \frac{\partial^2}{\partial k_{1234}^2} \bigg) H_{\phi^4} ~.
\end{aligned}$$ Thus the corresponding relation between the inflationary and Minkowski four-point functions is $$\begin{aligned}
\label{eq:4ptpz2dz2}
\nonumber
& \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} \\ \nonumber
= &\bigg(\frac{i H}{\sqrt{2\epsilon c_s}}\bigg)^4 \frac{1}{k_1 k_2 k_3 k_4} \frac{k_1 k_2 \mathbf k_3\cdot \mathbf k_4 (-\Sigma+c_s^2(3\lambda+\Sigma)) }{4 c_s k_3 k_4 \epsilon^2 \lambda_4} \\
& \times\frac{\partial^2}{\partial k_{1234}^2} \bigg(1-(k_3+k_4)\frac{\partial}{\partial k_{1234}} + k_3 k_4 \frac{\partial^2}{\partial k_{1234}^2} \bigg) \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle_{\rm flat} \times\frac{1}{24} +{\rm 23\,\, permutations}~.
\end{aligned}$$
- $H_{(\partial\zeta)^4}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{(\partial\zeta)^4} \rightarrow \int_{-\infty}^{0} (-1) \frac{(-1+c_s^2) e^{i k_{1234} \tau} \mathbf k_1\cdot\mathbf k_2 \mathbf k_3\cdot \mathbf k_4 (1-i k_1\tau)(1-i k_2\tau)(1-i k_3\tau)(1-i k_4\tau) }{64 k_1^{3/2}k_2^{3/2}k_3^{3/2}k_4^{3/2} \epsilon^2 } d\tau ~.
\end{aligned}$$ Comparing with the integral expression of $H_{\phi^4}$, we have $$\begin{aligned}
\nonumber
H_{(\partial\zeta)^4} \rightarrow &-\frac{(-1+c_s^2)\mathbf k_1\cdot \mathbf k_2 \mathbf k_3\cdot \mathbf k_4 \Sigma}{16 k_1 k_2 k_3 k_4 \epsilon \lambda_4}\\
&\times \bigg[ 1 - (k_1+k_2+k_3+k_4) \frac{\partial}{\partial k_{1234}} + \sum_{i\neq j} k_i k_j \frac{\partial^2}{\partial k_{1234}^2} - \sum_{i\neq j\neq k} k_i k_j k_k \frac{\partial^3}{\partial k_{123}^3} + k_1 k_2 k_3 k_4 \frac{\partial^4}{\partial k_{1234}^4} \bigg] H_{\phi^4}~.
\end{aligned}$$ Thus the corresponding relation between the inflationary and Minkowski four-point functions is $$\begin{aligned}
\label{eq:4ptpz4}
\nonumber
& \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} \\ \nonumber
= &\bigg(\frac{i H}{\sqrt{2\epsilon c_s}}\bigg)^4 \frac{1}{k_1 k_2 k_3 k_4} \bigg(-\frac{(-1+c_s^2)(\mathbf k_1\cdot \mathbf k_2) (\mathbf k_3\cdot \mathbf k_4) \Sigma}{16 k_1 k_2 k_3 k_4 \epsilon \lambda_4}\bigg)\\\nonumber
& \times \bigg[ 1 - k_{1234}\frac{\partial}{\partial k_{1234}} + \sum_{i\neq j} k_i k_j \frac{\partial^2}{\partial k_{1234}^2} - \sum_{i\neq j\neq k} k_i k_j k_k \frac{\partial^3}{\partial k_{123}^3} + k_1 k_2 k_3 k_4 \frac{\partial^4}{\partial k_{1234}^4} \bigg]
\\ &
\times \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle_{\rm flat}
\times\frac{1}{24} +{\rm 23\,\, permutations}~.
\end{aligned}$$
Scalar Exchange Diagram
-----------------------
We evaluate the relation between inflationary four-point function contributed by the scalar exchange diagram and that of the Minkowski spacetime using the tools we already obtained in Section \[applicationtoinflation\].
- Contribution from $H_{\zeta'^3}$ and $H_{\zeta'^3}$ $$\begin{aligned}
\nonumber
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} & = \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau} d\tau_2 \langle 0 | H_{\zeta'^3} (\tau_1) \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} H_{\zeta'^3} (\tau_2) | 0 \rangle \\
& - 2 {\rm Re} \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau_1} d\tau_2 \langle 0 | \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} H_{\zeta'^3} (\tau_1) H_{\zeta'^3} (\tau_2) | 0 \rangle~,
\end{aligned}$$ we take the initial time going to $-\infty$ and final time going to $0$. The relation between the four-point function contributed by the $H_{\zeta'^3}$ and $H_{\zeta'^3}$ interaction and that of the Minkowski correlation function contributed by $H_{\phi^3}$ interaction is $$\begin{aligned}
\nonumber
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} & = \frac{c_s H^2 k_I^2 \lambda^2}{8\epsilon^5}
\frac{\partial^2}{\partial k_{12}^2} \frac{\partial^2}{\partial k_{34}^2} \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{ {\rm flat noP}} \times 4 + {\rm permutations}~,
\end{aligned}$$ where the subscript “noP" denotes the $\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat}$ without 23 permutations.
- Contribution from $H_{\zeta' (\partial \zeta)^2}$ and $H_{\zeta'^3}$ $$\begin{aligned}
\nonumber
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} & = \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau} d\tau_2 \langle 0 | H_{\zeta' (\partial \zeta)^2} (\tau_1) \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} H_{\zeta'^3} (\tau_2) | 0 \rangle \\
& - 2 {\rm Re} \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau_1} d\tau_2 \langle 0 | \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} H_{\zeta' (\partial \zeta)^2} (\tau_1) H_{\zeta'^3} (\tau_2) | 0 \rangle~,
\end{aligned}$$ we take the initial time going to $-\infty$ and final time going to $0$. Then we have the relation between the four-point function contributed by the $H_{\zeta' (\partial \zeta)^2}$ and $H_{\zeta'^3}$ interaction and that of the Minkowski correlation function contributed by $H_{\phi^3}$ interaction
- $\zeta'$ contracts with $\zeta'$ $$\begin{aligned}
\nonumber
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} & = \frac{(-1+c_s^2)H^2 \mathbf k_1\cdot \mathbf k_2 k_I^2 \lambda \Sigma}{3\times 16 k_1^2 k_2^2 c_s \epsilon^5} \frac{\partial^2}{\partial k_{34}^2} \bigg(1-k_{12}\frac{\partial}{\partial k_{12} } + k_1 k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~.
\end{aligned}$$
- $\partial\zeta$ contracts with $\zeta'$
- Non-time-ordered part $$\begin{aligned}
\nonumber
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} & = \frac{2\times (-1+c_s^2)H^2 \mathbf k_I \cdot \mathbf k_2 \lambda \Sigma}{3\times 16 k_2^2 c_s \epsilon^5} \frac{\partial^2}{\partial k_{34}^2} \bigg(1-(k_{2}+k_I)\frac{\partial}{\partial k_{12} } + k_I k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~.
\end{aligned}$$
- Time-ordered part $$\begin{aligned}
\nonumber
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} & = \frac{2\times (-1+c_s^2)H^2 \mathbf k_I \cdot \mathbf k_2 \lambda \Sigma}{3\times 16 k_2^2 c_s \epsilon^5} \frac{\partial^2}{\partial k_{34}^2} \bigg(1-(k_{2}-k_I)\frac{\partial}{\partial k_{12} } - k_I k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~.
\end{aligned}$$
- Contribution from $H_{\zeta' (\partial \zeta)^2}$ and $H_{\zeta' (\partial \zeta)^2}$ $$\begin{aligned}
\nonumber
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} & = \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau} d\tau_2 \langle 0 | H_{\zeta' (\partial \zeta} (\tau_1) \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} H_{\zeta' (\partial \zeta} (\tau_2) | 0 \rangle \\
& - 2 {\rm Re} \int_{\tau_0}^{\tau} d\tau_1 \int_{\tau_0}^{\tau_1} d\tau_2 \langle 0 | \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} H_{\zeta' (\partial \zeta} (\tau_1) H_{\zeta' (\partial \zeta} (\tau_2) | 0 \rangle~,
\end{aligned}$$ we take the initial time going to $-\infty$ and final time going to $0$. Then we have the relation between the four-point function contributed by the $H_{\zeta' (\partial \zeta)^2}$ and $H_{\zeta' (\partial \zeta)^2}$ interaction and that of the Minkowski correlation function contributed by $H_{\phi^3}$ interaction
- $\zeta'$ contracts with $\zeta'$\
$$\begin{aligned}
\nonumber
&\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} \\\nonumber
& = \frac{ (-1+c_s^2)^2 H^2 \mathbf k_1 \cdot \mathbf k_2 \mathbf k_3 \cdot \mathbf k_4 k_I^2 \Sigma^2 }{9\times 32 k_1^2 k_2^2 k_3^2 k_4^2 c_s^3 \epsilon^5} \bigg(1-(k_1+k_{2})\frac{\partial}{\partial k_{12} } + k_1 k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \bigg(1-(k_3+k_{4})\frac{\partial}{\partial k_{34} } + k_3 k_4 \frac{\partial^2}{\partial k_{34}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~,
\end{aligned}$$
- $\zeta'$ contracts with $\partial\zeta$\
$$\begin{aligned}
\nonumber
&\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} \\ \nonumber
& = \frac{ 2\times 2 (-1+c_s^2)^2 H^2 \mathbf k_1 \cdot \mathbf k_2 \mathbf k_I \cdot \mathbf k_4 \Sigma^2 }{9\times 32 k_1^2 k_2^2 k_4^2 c_s^3 \epsilon^5} \bigg(1-(k_1+k_{2})\frac{\partial}{\partial k_{12} } + k_1 k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \bigg(1-(k_{4}+k_I)\frac{\partial}{\partial k_{34} } + k_I k_4 \frac{\partial^2}{\partial k_{34}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~,
\end{aligned}$$
- $\partial\zeta$ contracts with $\zeta'$\
- Non-time-ordered part $$\begin{aligned}
\nonumber
&\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} \\ \nonumber
& = \frac{ 2\times 2 (-1+c_s^2)^2 H^2 \mathbf k_3 \cdot \mathbf k_4 \mathbf k_I \cdot \mathbf k_2 \Sigma^2 }{9\times 32 k_3^2 k_2^2 k_4^2 c_s^3 \epsilon^5} \bigg(1-(k_{2}+k_I)\frac{\partial}{\partial k_{12} } + k_I k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \bigg(1-(k_3+k_{4})\frac{\partial}{\partial k_{34} } + k_3 k_4 \frac{\partial^2}{\partial k_{34}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~,
\end{aligned}$$
- Time-ordered part $$\begin{aligned}
\nonumber
&\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} \\ \nonumber
& = \frac{ 2\times 2 (-1+c_s^2)^2 H^2 \mathbf k_3 \cdot \mathbf k_4 \mathbf k_I \cdot \mathbf k_2 \Sigma^2 }{9\times 32 k_3^2 k_2^2 k_4^2 c_s^3 \epsilon^5} \bigg(1-(k_{2}-k_I)\frac{\partial}{\partial k_{12} } - k_I k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \bigg(1-(k_3+k_{4})\frac{\partial}{\partial k_{34} } + k_3 k_4 \frac{\partial^2}{\partial k_{34}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~,
\end{aligned}$$
- $\partial\zeta$ contracts with $\partial\zeta$
- Non-time-ordered part $$\begin{aligned}
\nonumber
&\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} = \frac{2\times 2 (-1+c_s^2)^2 \mathbf k_I \cdot \mathbf k_2 \mathbf k_I \cdot \mathbf k_4 \Sigma^2 }{9\times 32 k_2^2 k_4^2 k_I^2 \epsilon^5} \\\nonumber
& \bigg(1-(k_{2}+k_I)\frac{\partial}{\partial k_{12} } + k_I k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \bigg(1-(k_{4}+k_I)\frac{\partial}{\partial k_{34} } + k_I k_4 \frac{\partial^2}{\partial k_{34}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~,
\end{aligned}$$
- Time-ordered part $$\begin{aligned}
\nonumber
&\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4} \rangle_{\rm inflation} = \frac{2\times 2 (-1+c_s^2)^2 \mathbf k_I \cdot \mathbf k_2 \mathbf k_I \cdot \mathbf k_4 \Sigma^2 }{9\times 32 k_2^2 k_4^2 k_I^2 \epsilon^5} \\\nonumber
& \bigg(1-(k_{2}-k_I)\frac{\partial}{\partial k_{12} } - k_I k_2 \frac{\partial^2}{\partial k_{12}^2} \bigg) \bigg(1-(k_{4}+k_I)\frac{\partial}{\partial k_{34} } + k_I k_4 \frac{\partial^2}{\partial k_{34}^2} \bigg) \\\nonumber
& \langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3}\phi_{\mathbf k_4} \rangle'_{\rm flat{\rm noP}} \times 4 + {\rm permutations}~.
\end{aligned}$$
Application to Bouncing Cosmology {#applicationtobounce}
=================================
In this appendix, we apply the symmetry breaking operator technique to matter bounce cosmology with a general single field content [@Li:2016xjb]. The idea is that the universe underwent a matter contraction phase before the Big Bang [@Wands:1998yp; @Finelli:2001sr] (for a review, see [@Cai:2009zp; @Brandenberger:2012zb]). We still consider the action of the form $\displaystyle \int(R+2P(X,\phi))$ as in the inflationary scenario. But for the matter bounce, the scale factor is given by $$\begin{aligned}
a(\tau) = \bigg(\frac{\tau-\tilde \tau_B}{\tau_B - \tilde \tau_B}\bigg)^2 ~.\end{aligned}$$ where $\tau_B$ corresponds to the conformal time at the beginning of the bounce phase and $\tilde \tau_B$ corresponds to the time of the bouncing singularity (if there is no new physics to resolve the singularity; otherwise a non-singular bounce is obtained and $\tilde \tau_B$ is indeed close to the time of the bounce). The Hubble parameter is $$\begin{aligned}
H = \frac{a'}{a^2} = \frac{2(\tau_B-\tilde \tau_B)^2}{(\tau-\tau_B)^3}~.\end{aligned}$$ The mode function and its derivative is $$\begin{aligned}
u_k(\tau)=-\frac{iA[1+ic_sk(\tau-\tilde{\tau}_B)]}{2\sqrt{\epsilon c_sk^3}(\tau-\tilde{\tau}_B)^3}e^{-ic_sk(\tau-\tilde{\tau}_B)}
\end{aligned}$$ and its derivative with respect to $\tau$ is $$\begin{aligned}
u_k'(\tau)=\frac{iA}{2\sqrt{\epsilon c_sk^3}}\left(\frac{3\left[1+ic_sk(\tau-\tilde{\tau}_B)\right]}{(\tau-\tilde{\tau}_B)^4}-\frac{c_s^2k^2}{(\tau-\tilde{\tau}_B)^2}\right)e^{-ic_sk(\tau-\tilde{\tau}_B)} ~,
\end{aligned}$$ where $A=(\tau_B-\tilde{\tau}_B)^2$ and $M_{\rm{Pl}}$ is set to be $1$. Thus the dimensionless power spectrum is $$\begin{aligned}
\Delta_\zeta^2=\frac{1}{12\pi^2c_s(\tau_B-\tilde{\tau}_B)^2}~.
\end{aligned}$$
To evaluate the three-point correlation function, the third order action is [@Li:2016xjb] $$\label{action3}
\begin{aligned}
S^{(3)}=\int d\tau\,d^3x\bigg\{&-\frac{a}{H^3}\left[\Sigma\left(1-\frac{1}{c_s^2}\right)+2\lambda\right]\zeta'^3
+a^2\left[\frac{\epsilon}{c_s^4}(\epsilon-3+3c_s^2)-\frac{\epsilon^3}{2}\right]\zeta\zeta'^2
+\frac{a^2\epsilon}{c_s^2}(\epsilon-2s+1-c_s^2)\zeta\partial_i\zeta\partial^i\zeta\\
&-\frac{2a\epsilon}{c_s^2}\zeta'\partial_i\zeta\partial^i\chi
+\frac{a^2\epsilon}{2c_s^2}\frac{d}{d\tau}\left(\frac{\eta}{c_s^2}\right)\zeta^2\zeta'\
+\frac{\epsilon}{2}\zeta(\partial_i\partial_j\chi)(\partial^i\partial^j\chi)
+2f(\zeta)\frac{\delta L}{\delta\zeta}\bigg|_1\bigg\}~,
\end{aligned}$$ where $\chi$ is defined via $\partial^2\chi=a\epsilon\zeta'$, and $$\begin{aligned}
f(\zeta)&=\frac{\eta}{4c_s^2}\zeta^2+\frac{1}{c_s^2aH}\zeta\zeta'+\frac{1}{4a^2H^2}\left(-\partial_i\zeta\partial^i\zeta+\partial^{-2}\left[\partial_i\partial_j\left(\partial^i\partial^j\zeta\right)\right]\right)+\frac{1}{2a^2H}\{\partial_i\zeta\partial^i\chi-\partial^{-2}[\partial_i\partial_j(\partial^i\zeta\partial^j\chi)]\}~,\\
\frac{\delta L}{\delta\zeta}\bigg|_1&=a\left(\frac{d}{dt}\partial^2\chi+H\partial^2\chi-\epsilon\partial^2\zeta\right)~.
\end{aligned}$$ $\partial^2$ and $\partial^{-2}$ are the Laplacian and the inverse of Laplacian, respectively. If we only consider the case that $c_s$ is nearly constant, that is, $s\approx0$, we have $\lambda/\Sigma\approx$ constant, and the first term can be rewritten as $$\begin{aligned}
-\frac{\epsilon}{c_s^2}\left(1-\frac{1}{c_s^2}+\frac{2\lambda}{\Sigma}\right)\frac{a}{H}\zeta'^3~.
\end{aligned}$$
The $c_s=1$ case is considered in [@Cai:2009fn], whereas the general case $c_s\neq 1$ is considered in [@Li:2016xjb]. The last term in this action is removed by performing the field redefinition $$\begin{aligned}
\zeta\rightarrow\tilde\zeta+f(\tilde\zeta)~,\end{aligned}$$ where $\tilde\zeta$ denotes the field after redefinition.
Rule to relate external mode function
-------------------------------------
For matter bounce, we have the following relation between the mode functions in cosmology and in Minkowski spacetime $$\begin{aligned}
u_k(\tau_B)_{\rm MB} = u_k(c_s(\tau_B-\tilde{\tau}_B))_{\rm flat}\left(-\frac{i\left[1+ic_sk(\tau_B-\tilde{\tau}_B)\right]}{\sqrt{2 \epsilon c_s} k(\tau_B-\tilde{\tau}_B)}\right)~.\end{aligned}$$ For three-point function, we have an extra factor of $$\begin{aligned}
\frac{i}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3(\tau_B-\tilde{\tau}_B)^3}\prod_{i=1}^3\left[1+ic_sk_i(\tau_B-\tilde{\tau}_B)\right]~.\end{aligned}$$
Rule to relate Hamiltonian
--------------------------
- The Minkowski Hamiltonian\
- $H_{\phi^3}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\phi^3} \rightarrow \int_{-\infty}^{c_s(\tau_B-\tilde{\tau}_B)} \frac{\lambda_3}{2 \sqrt{2} \sqrt{k_1k_2k_3} } e^{i k_{123} \tau} d\tau
\end{aligned}$$
- Matter bounce contraction phase Hamiltonian\
- $H_{\zeta'^3}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta'^3}\rightarrow-\int_{-\infty}^{\tau_B-\tilde{\tau}_B}d\tau\;\left(1-\frac{1}{c_s^2}+\frac{2\lambda}{\Sigma}\right)\frac{i(\tau_B-\tilde{\tau}_B)^2}{16\sqrt{\epsilon c_s^7k_1^3k_2^3k_3^3}}\tau^5\prod_{i=1}^3\left[\frac{3\left(1-ic_sk_i\tau\right)}{\tau^4}-\frac{c_s^2k_i^2}{\tau^2}\right]e^{ic_sk_{123}\tau}
\end{aligned}$$ Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta'^3
} \rightarrow \left(1-\frac{1}{c_s^2}+\frac{2\lambda}{\Sigma}\right)\frac{i(\tau_B-\tilde{\tau}_B)^2}{4\sqrt{2\epsilon c_s^7}k_1k_2k_3\lambda_3}O_{\zeta'^3}H_{\phi^3}~,
\end{aligned}$$ where $O_{\zeta'^3}$ is defined as $$\begin{aligned}
O_{\zeta'^3}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)\equiv(ic_s)^7\bigg[&27I^7+27k_{123}I^6+\frac{9}{2}\bigg(2\sum_{i=1}^3k_i^2+3\sum_{i\neq j}k_ik_j\bigg)I^5+9\bigg(\sum_{i\neq j}k_i^2k_j+3k_1k_2k_3\bigg)I^4\\
&+\frac{3}{2}\bigg(\sum_{i\neq j}k_1^2k_j^2+3\sum_{i\neq j\neq k}k_i^2k_jk_k\bigg)I^3+\frac{3}{2}\bigg(\sum_{i\neq j\neq k}k_i^2k_j^2k_k\bigg)I^2+k_1^2k_2^2k_3^2I\bigg]~.
\end{aligned}$$ and $I^n$ means integrating $n$ times with respect to $k_{123}$. Thus the corresponding part of the three-point function is $$\begin{aligned}
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3} \rangle_{\rm MB} &= \frac{i\prod_{i=1}^3\left[1+ic_sk_i(\tau_B-\tilde{\tau}_B)\right]}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3(\tau_B-\tilde{\tau}_B)^3}\left(1-\frac{1}{c_s^2}+\frac{2\lambda}{\Sigma}\right)\frac{i(\tau_B-\tilde{\tau}_B)^2}{4\sqrt{2\epsilon c_s^7}k_1k_2k_3\lambda_3}\\
&\quad\times O_{\zeta'^3}\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle_{\rm flat}\\
&+\text{5 permutations}
\end{aligned}$$
- $H_{\zeta \zeta'^2}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta\zeta'^2}(k_1,k_2,k_3)\rightarrow&\int_{-\infty}^{\tau_B-\tilde{\tau}_B}d\tau\;\left[\frac{\epsilon}{c_s^4}(\epsilon-3+3c_s^2)-\frac{\epsilon^3}{2}\right]\frac{i(\tau_B-\tilde{\tau}_B)^2}{8\sqrt{\epsilon c_sk_1k_2k_3}^3}\\
&\times\tau^4\left(\frac{1-ic_sk_1\tau}{\tau^3}\right)\prod_{i=2}^3\left[\frac{3\left(1-ic_sk_i\tau\right)}{\tau^4}-\frac{c_s^2k_i^2}{\tau^2}\right]e^{ic_sk_{123}\tau}
\end{aligned}$$ Here $k_1$ corresponds to the momentum of $\zeta$, $k_2$ and $k_3$ corresponds to the momentum of the second or third $\zeta'$, respectively. Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta\zeta'^2}(k_1,k_2,k_3)\rightarrow&\left[\frac{\epsilon}{c_s^4}(\epsilon-3+3c_s^2)-\frac{\epsilon^3}{2}\right]\frac{i(\tau_B-\tilde{\tau}_B)^2}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3\lambda_3}O_{\zeta\zeta'^2}H_{I3}
\end{aligned}$$ where $O_{\zeta\zeta'^2}$ is defined as $$\begin{aligned}
O_{\zeta\zeta'^2}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)&\equiv9I^7+9k_{123}I^6+3(3k_1k_2+3k_2k_3+3k_3k_1+k_2^2+k_3^2)I^5\\
&\quad\;+3\left(k_1k_2^2+k_1k_3^2+k_2k_3^2+k_2^2k_3+3k_1k_2k_3\right)I^4+\left(3 k_1 k_2^2 k_3 + 3 k_1 k_2 k_3^2 + k_2^2 k_3^2\right)I^3\\
&\quad\;+\left(k_1 k_2^2 k_3^2\right)I^2~.
\end{aligned}$$ Thus the corresponding part of the three-point function is $$\begin{aligned}
&\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3} \rangle_{\rm MB} \\
=&\frac{i\prod_{i=1}^3\left[1+ic_sk_i(\tau_B-\tilde{\tau}_B)\right]}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3(\tau_B-\tilde{\tau}_B)^3}\left[\frac{\epsilon}{c_s^4}(\epsilon-3+3c_s^2)-\frac{\epsilon^3}{2}\right]\frac{i(\tau_B-\tilde{\tau}_B)^2}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3\lambda_3}\\
&\times O_{\zeta\zeta'^2}\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle_{\rm flat}\\
&+\text{5 permutations}
\end{aligned}$$
- $H_{\zeta(\partial\zeta)^2}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta(\partial\zeta)^2}(k_1,k_2,k_3)\rightarrow&\int_{-\infty}^{\tau_B-\tilde{\tau}_B}d\tau\;\frac{i(\epsilon+1-c_s^2)(\tau_B-\tilde{\tau}_B)^2}{8\sqrt{\epsilon c_s^7k_1^3k_2^3k_3^3}}\mathbf{k_2}\cdot\mathbf{k_3}\tau^4\prod_{i=1}^3\left[\frac{3\left(1-ic_sk_i\tau\right)}{\tau^4}-\frac{c_s^2k_i^2}{\tau^2}\right]e^{ic_sk_{123}\tau}
\end{aligned}$$ Here $k_1$ corresponds to the momentum of the first $\zeta$, $k_2$ and $k_3$ corresponds to the momentum of the second or third $\zeta$, respectively. Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta(\partial\zeta)^2}(k_1,k_2,k_3)\rightarrow&\frac{i(\epsilon+1-c_s^2)(\tau_B-\tilde{\tau}_B)^2}{2\sqrt{2\epsilon c_s^7}k_1k_2k_3\lambda_3}O_{\zeta(\partial\zeta)^2}H_{\phi^3}
\end{aligned}$$ where ${O}_{\zeta(\partial\zeta)^2}$ is defined as $$\begin{aligned}
{O}_{\zeta(\partial\zeta)^2}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)&\equiv(ic_s)^5\bigg(I^5+k_{123}I^4+\frac{1}{2}\sum_{i\neq j}k_ik_jI^3+k_1k_2k_3I^2\bigg)~.
\end{aligned}$$
Thus the corresponding part of the three-point function is $$\begin{aligned}
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3} \rangle_{\rm MB} =&\frac{i\prod_{i=1}^3\left[1+ic_sk_i(\tau_B-\tilde{\tau}_B)\right]}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3(\tau_B-\tilde{\tau}_B)^3}\frac{i(\epsilon+1-c_s^2)(\tau_B-\tilde{\tau}_B)^2}{2\sqrt{2\epsilon c_s^7}k_1k_2k_3\lambda_3}\\
&\times O_{\zeta(\partial\zeta)^2}\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle_{\rm flat}\\
&+\text{5 permutations}
\end{aligned}$$
- $H_{\zeta' \partial\zeta \partial\chi}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta' \partial\zeta \partial\chi}(k_1,k_2,k_3)\rightarrow&-\int_{-\infty}^{\tau_B-\tilde{\tau}_B}d\tau\;\frac{i\sqrt{\epsilon}(\tau_B-\tilde{\tau}_B)^2}{4\sqrt{c_s^7k_1^3k_2^3k_3^3}}\frac{\mathbf{k_2}\cdot\mathbf{k_3}}{k_3^2}\tau^4\left(\frac{1-ic_sk_2\tau}{\tau^3}\right)\prod_{i=1,3}\left[\frac{3\left(1-ic_sk_i\tau\right)}{\tau^4}-\frac{c_s^2k_i^2}{\tau^2}\right]e^{ic_sk_{123}\tau}
\end{aligned}$$ Here $k_1$ and $k_2$ corresponds to the momentum of the first or second $\zeta$, $k_3$ corresponds to the momentum of $\chi$, respectively. Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta' \partial\zeta \partial\chi}(k_1,k_2,k_3)\rightarrow&-\frac{i\sqrt{\epsilon}(\tau_B-\tilde{\tau}_B)^2}{\sqrt{2c_s^7}k_1k_2k_3\lambda_3}O_{\zeta' \partial\zeta \partial\chi}H_{\phi^3}
\end{aligned}$$ where $O_{\zeta' \partial\zeta \partial\chi}$ is defined as $$\begin{aligned}
O_{\zeta'\partial\zeta\partial\chi}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)&\equiv\frac{\mathbf{k}_2\cdot\mathbf{k}_3}{k_3^2}{O}_{\zeta\zeta'^2}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)~.
\end{aligned}$$ Thus the corresponding part of the three-point function is $$\begin{aligned}
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3} \rangle_{\rm MB} =&-\frac{i\prod_{i=1}^3\left[1+ic_sk_i(\tau_B-\tilde{\tau}_B)\right]}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3(\tau_B-\tilde{\tau}_B)^3}\frac{i\sqrt{\epsilon}(\tau_B-\tilde{\tau}_B)^2}{\sqrt{2c_s^7}k_1k_2k_3\lambda_3}\\
&\times O_{\zeta' \partial\zeta \partial\chi}\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle_{\rm flat}\\
&+\text{5 permutations}
\end{aligned}$$
- $H_{\zeta(\partial_i\partial_j\chi)^2}$, if all fields in it contract with the field on the left hand side, it corresponds to the following integral, $$\begin{aligned}
H_{\zeta(\partial_i\partial_j\chi)^2}(k_1,k_2,k_3)\rightarrow&\int_{-\infty}^{\tau_B-\tilde{\tau}_B}d\tau\;\frac{i\sqrt{\epsilon^3}(\tau_B-\tilde{\tau}_B)^2}{8\sqrt{c_s^3k_1^3k_2^3k_3^3}}\left(\frac{\mathbf{k_2}\cdot\mathbf{k_3}}{k_2k_3}\right)^2\tau^4\left(\frac{1-ic_sk_1\tau}{\tau^3}\right)\\
&\quad\;\prod_{i=2}^3\left[\frac{3\left(1-ic_sk_i\tau\right)}{\tau^4}-\frac{c_s^2k_i^2}{\tau^2}\right]e^{ic_sk_{123}\tau}
\end{aligned}$$ Here $k_1$ corresponds to the momentum of $\zeta$, $k_2$ and $k_3$ corresponds to the momentum of the second or third $\chi$, respectively. Comparing with the integral expression of $H_{\phi^3}$, we have $$\begin{aligned}
H_{\zeta(\partial_i\partial_j\chi)^2}(k_1,k_2,k_3)\rightarrow&\frac{i\sqrt{\epsilon^3}(\tau_B-\tilde{\tau}_B)^2}{2\sqrt{2c_s^3}k_1k_2k_3\lambda_3}O_{\zeta(\partial_i\partial_j\chi)^2}H_{\phi^3}
\end{aligned}$$ where $O_{\zeta(\partial_i\partial_j\chi)^2}$ is defined as $$\begin{aligned}
O_{\zeta(\partial_i\partial_j\chi)^2}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)&\equiv\left(\frac{\mathbf{k}_2\cdot\mathbf{k}_3}{k_2k_3}\right)^2\mathcal{O}_{\zeta\zeta'^2}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)~.
\end{aligned}$$ Thus the corresponding part of the three-point function is $$\begin{aligned}
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3} \rangle_{\rm MB} =&-\frac{i\prod_{i=1}^3\left[1+ic_sk_i(\tau_B-\tilde{\tau}_B)\right]}{2\sqrt{2\epsilon^3c_s^3}k_1k_2k_3(\tau_B-\tilde{\tau}_B)^3}\frac{i\sqrt{\epsilon^3}(\tau_B-\tilde{\tau}_B)^2}{2\sqrt{2c_s^3}k_1k_2k_3\lambda_3}\\
&\times O_{\zeta(\partial_i\partial_j\chi)^2}\langle \phi_{\mathbf k_1}\phi_{\mathbf k_2}\phi_{\mathbf k_3} \rangle_{\rm flat}\\
&+\text{5 permutations}~.
\end{aligned}$$
- Secondary contributions: The contribution from the term $$\frac{a^2\epsilon}{2c_s^2}\frac{d}{d\tau}\left(\frac{\eta}{c_s^2}\right)\zeta^2\zeta'\nonumber$$ in equation is exactly zero since $\eta=0$ during the matter contraction. We can also neglect the contribution from the term $$\frac{a\epsilon}{c_\mathrm{s}^2}(\epsilon-2s+1-c_\mathrm{s}^2)\zeta(\partial\zeta)^2 \nonumber$$ since the leading order term of the resulting three-point function is proportional to $c_\mathrm{s}^2k_i^2(\tau_B-\tilde{\tau}_B)^2$, which means that this term is suppressed outside the sound horizon.
[999]{}
N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 \[hep-th\]. X. Chen and Y. Wang, “Large non-Gaussianities with Intermediate Shapes from Quasi-Single Field Inflation,” Phys. Rev. D [**81**]{}, 063511 (2010) \[arXiv:0909.0496 \[astro-ph.CO\]\]. X. Chen and Y. Wang, “Quasi-Single Field Inflation and Non-Gaussianities,” JCAP [**1004**]{}, 027 (2010) \[arXiv:0911.3380 \[hep-th\]\]. D. Baumann and D. Green, “Signatures of Supersymmetry from the Early Universe,” Phys. Rev. D [**85**]{}, 103520 (2012) \[arXiv:1109.0292 \[hep-th\]\].
V. Assassi, D. Baumann and D. Green, “On Soft Limits of Inflationary Correlation Functions,” JCAP [**1211**]{}, 047 (2012) \[arXiv:1204.4207 \[hep-th\]\]. T. Noumi, M. Yamaguchi and D. Yokoyama, “Effective field theory approach to quasi-single field inflation and effects of heavy fields,” JHEP [**1306**]{}, 051 (2013) \[arXiv:1211.1624 \[hep-th\]\]. N. Arkani-Hamed, P. Benincasa and A. Postnikov, “Cosmological Polytopes and the Wavefunction of the Universe,” arXiv:1709.02813 \[hep-th\]. R. Britto, F. Cachazo, B. Feng and E. Witten, “Direct proof of tree-level recursion relation in Yang-Mills theory,” Phys. Rev. Lett. [**94**]{}, 181602 (2005) \[hep-th/0501052\]. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, “The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM,” JHEP [**1101**]{}, 041 (2011) \[arXiv:1008.2958 \[hep-th\]\]. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, “Grassmannian Geometry of Scattering Amplitudes,” arXiv:1212.5605 \[hep-th\]. P. Benincasa, “On-shell diagrammatics and the perturbative structure of planar gauge theories,” arXiv:1510.03642 \[hep-th\]. S. Weinberg, “Quantum contributions to cosmological correlations,” Phys. Rev. D [**72**]{}, 043514 (2005) \[hep-th/0506236\]. X. Chen, “Primordial Non-Gaussianities from Inflation Models,” Adv. Astron. [**2010**]{}, 638979 (2010) \[arXiv:1002.1416 \[astro-ph.CO\]\].
Y. Wang, “Inflation, Cosmic Perturbations and Non-Gaussianities,” Commun. Theor. Phys. [**62**]{}, 109 (2014) \[arXiv:1303.1523 \[hep-th\]\].
X. Chen, M. x. Huang, S. Kachru and G. Shiu, “Observational signatures and non-Gaussianities of general single field inflation,” JCAP [**0701**]{}, 002 (2007) \[hep-th/0605045\]. A. Strominger, “The dS / CFT correspondence,” JHEP [**0110**]{}, 034 (2001) \[hep-th/0106113\]. A. Strominger, “Inflation and the dS / CFT correspondence,” JHEP [**0111**]{}, 049 (2001) \[hep-th/0110087\]. P. McFadden and K. Skenderis, “The Holographic Universe,” J. Phys. Conf. Ser. [**222**]{}, 012007 (2010) \[arXiv:1001.2007 \[hep-th\]\]. P. McFadden and K. Skenderis, “Holographic Non-Gaussianity,” JCAP [**1105**]{}, 013 (2011) \[arXiv:1011.0452 \[hep-th\]\]. H. Isono, T. Noumi, G. Shiu, S. S. C. Wong and S. Zhou, “Holographic non-Gaussianities in general single-field inflation,” JHEP [**1612**]{} (2016) 028 \[arXiv:1610.01258 \[hep-th\]\]. Y. S. Piao, arXiv:1112.3737 \[hep-th\]. J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP [**0305**]{}, 013 (2003) \[astro-ph/0210603\]. V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto, “Second order cosmological perturbations from inflation,” Nucl. Phys. B [**667**]{}, 119 (2003) \[astro-ph/0209156\]. C. Armendariz-Picon, T. Damour and V. F. Mukhanov, “k - inflation,” Phys. Lett. B [**458**]{}, 209 (1999) \[hep-th/9904075\]. M. Alishahiha, E. Silverstein and D. Tong, “DBI in the sky,” Phys. Rev. D [**70**]{}, 123505 (2004) \[hep-th/0404084\]. D. Seery and J. E. Lidsey, “Primordial non-Gaussianities in single field inflation,” JCAP [**0506**]{}, 003 (2005) \[astro-ph/0503692\]. J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,” Phys. Rev. D [**28**]{}, 2960 (1983). T. Hertog and J. Hartle, “Holographic No-Boundary Measure,” JHEP [**1205**]{}, 095 (2012) \[arXiv:1111.6090 \[hep-th\]\]. T. S. Bunch and P. C. W. Davies, “Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting,” Proc. Roy. Soc. Lond. A [**360**]{}, 117 (1978). A. Bzowski, P. McFadden and K. Skenderis, “Implications of conformal invariance in momentum space,” JHEP [**1403**]{}, 111 (2014) \[arXiv:1304.7760 \[hep-th\]\]. A. Bzowski, P. McFadden and K. Skenderis, “Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies,” JHEP [**1603**]{}, 066 (2016) \[arXiv:1510.08442 \[hep-th\]\]. A. Bzowski, P. McFadden and K. Skenderis, “Evaluation of conformal integrals,” JHEP [**1602**]{}, 068 (2016) \[arXiv:1511.02357 \[hep-th\]\]. N. A. Chernikov and E. A. Tagirov, “Quantum theory of scalar fields in de Sitter space-time,” Ann. Inst. H. Poincare Phys. Theor. A [**9**]{}, 109 (1968). D. Seery, M. S. Sloth and F. Vernizzi, “Inflationary trispectrum from graviton exchange,” JCAP [**0903**]{}, 018 (2009) \[arXiv:0811.3934 \[astro-ph\]\].
H. Isono, T. Noumi and G. Shiu, to appear
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,” JHEP [**0610**]{}, 014 (2006) \[hep-th/0602178\]. D. Baumann, D. Green, H. Lee and R. A. Porto, “Signs of Analyticity in Single-Field Inflation,” Phys. Rev. D [**93**]{}, no. 2, 023523 (2016) \[arXiv:1502.07304 \[hep-th\]\]. S. Raju, “New Recursion Relations and a Minkowski Limit for AdS/CFT Correlators,” Phys. Rev. D [**85**]{}, 126009 (2012) \[arXiv:1201.6449 \[hep-th\]\]. N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP [**1410**]{}, 030 (2014) \[arXiv:1312.2007 \[hep-th\]\]. N. Arkani-Hamed and J. Trnka, “Into the Amplituhedron,” JHEP [**1412**]{}, 182 (2014) \[arXiv:1312.7878 \[hep-th\]\]. N. Arkani-Hamed, H. Thomas and J. Trnka, “Unwinding the Amplituhedron in Binary,” JHEP [**1801**]{}, 016 (2018) \[arXiv:1704.05069 \[hep-th\]\]. X. Chen, B. Hu, M. x. Huang, G. Shiu and Y. Wang, “Large Primordial Trispectra in General Single Field Inflation,” JCAP [**0908**]{} (2009) 008 \[arXiv:0905.3494 \[astro-ph.CO\]\]. F. Arroja, S. Mizuno, K. Koyama and T. Tanaka, “On the full trispectrum in single field DBI-inflation,” Phys. Rev. D [**80**]{}, 043527 (2009) \[arXiv:0905.3641 \[hep-th\]\]. X. Chen, M. x. Huang and G. Shiu, “The Inflationary Trispectrum for Models with Large Non-Gaussianities,” Phys. Rev. D [**74**]{}, 121301 (2006) \[hep-th/0610235\]. F. Arroja and K. Koyama, “Non-gaussianity from the trispectrum in general single field inflation,” Phys. Rev. D [**77**]{}, 083517 (2008) \[arXiv:0802.1167 \[hep-th\]\]. D. Nandan, J. Plefka and W. Wormsbecher, “Collinear limits beyond the leading order from the scattering equations,” JHEP [**1702**]{}, 038 (2017) \[arXiv:1608.04730 \[hep-th\]\]. H. Jiang and Y. Wang, “Towards the physical vacuum of cosmic inflation,” Phys. Lett. B [**760**]{}, 202 (2016) \[arXiv:1507.05193 \[hep-th\]\]. H. Jiang, Y. Wang and S. Zhou, “On the initial condition of inflationary fluctuations,” JCAP [**1604**]{}, no. 04, 041 (2016) \[arXiv:1601.01179 \[hep-th\]\]. Y. F. Cai, W. Xue, R. Brandenberger and X. Zhang, “Non-Gaussianity in a Matter Bounce,” JCAP [**0905**]{}, 011 (2009) \[arXiv:0903.0631 \[astro-ph.CO\]\]. Y. B. Li, J. Quintin, D. G. Wang and Y. F. Cai, “Matter bounce cosmology with a generalized single field: non-Gaussianity and an extended no-go theorem,” JCAP [**1703**]{}, no. 03, 031 (2017) \[arXiv:1612.02036 \[hep-th\]\]. D. Wands, “Duality invariance of cosmological perturbation spectra,” Phys. Rev. D [**60**]{}, 023507 (1999) \[gr-qc/9809062\]. F. Finelli and R. Brandenberger, “On the generation of a scale invariant spectrum of adiabatic fluctuations in cosmological models with a contracting phase,” Phys. Rev. D [**65**]{}, 103522 (2002) \[hep-th/0112249\]. Y. F. Cai, E. N. Saridakis, M. R. Setare and J. Q. Xia, “Quintom Cosmology: Theoretical implications and observations,” Phys. Rept. [**493**]{}, 1 (2010) \[arXiv:0909.2776 \[hep-th\]\]. R. H. Brandenberger, “The Matter Bounce Alternative to Inflationary Cosmology,” arXiv:1206.4196 \[astro-ph.CO\].
[^1]: From the perspective of symmetry, there are other efforts along the line of the dS/CFT correspondence [@Strominger:2001pn; @Strominger:2001gp] which makes the symmetry structure of inflationary correlation functions manifest [@McFadden:2010na; @McFadden:2010vh; @Isono:2016yyj]. A nearly Minkowski spacetime with slow expansion can also be related to inflation by a cosmic duality [@Piao:2011bz].
[^2]: Here the direction of $\mathbf{k}_{12}$ is not defined or actually used.
[^3]: In cosmology, the collinear relations (which reduce to folded relations when considering the three-point functions) is known to appear in the context of non-Bunch-Davies vacua [@Chen:2006nt; @Chen:2009bc] and its decay [@Jiang:2015hfa; @Jiang:2016nok]. This is because the decay of non-Bunch-Davies modes and the resulting cosmological correlations are dominated by early times deep inside the Hubble radius, where the metric can be approximated by the Minkowski metric implicitly. Here, we are considering Bunch-Davies initial condition thus we are getting a different observation in the collinear limit.
|
---
abstract: 'A new explicitly correlated functional form for expanding the wave function of an $N$-particle system with arbitrary angular momentum and parity is presented. We develop the projection-based approach, numerically exploited in our previous work \[J. Chem. Phys. [**149**]{}, 184105 (2018)\], to explicitly correlated Gausssians with one-axis shifted centers and derive the matrix elements for the Hamiltonian and the angular momentum operators by analytically solving the integral projection operator. Variational few-body calculations without assuming the Born-Oppenheimer approximation are presented for several rotationally excited states of three- and four-particle systems. We show how the new formalism can be used as a unified framework for high-accuracy calculations of properties of small atoms and molecules.'
author:
- Andrea Muolo
- Markus Reiher
date: 'June 8, 2020'
title: 'Analytically projected rotationally symmetric explicitly correlated Gaussian functions with one-axis shifted centers'
---
Introduction {#SEC:intro}
============
Highly accurate bound states of the Schrödinger equation for small atoms and molecules can be constructed by expanding the wave function in terms of basis functions depending explicitly on inter-particle distances [@ECG-history1960; @ECG-history1960_1; @svm-history1977; @randTempe-1; @FECG-integrals-advances1978; @Szalewicz_1979; @Szalewicz_II_1983; @randTempe-2; @randTempe-3; @Szalewicz_1990; @ECG-history1993; @Drake1997; @Korobov2000; @Adamowicz2003a; @Matyus2012; @Pachucki2012; @Adamowicz2013; @Pachucki2015]. Non-separable functions with respect to the particle coordinates are tailored to describe particle-particle correlations, especially to accurately reproduce the exact wave function for infinitesimally short distances and in the long range limit. Furthermore, they allow for a unified treatment of different kinds of particles, [*[e.g.]{}*]{} of electrons and nuclei. Within this framework, two- and three-electron atoms can be very accurately calculated employing Hylleraas-type functions [@Hylleraas1928; @Hylleraas1929a; @Hylleraas1929b; @Hylleraas1930a; @Hylleraas1930b; @Drake1997] that explicitly include powers of the inter-electronic distances $r_{ij}=|\bm{r}_i-\bm{r}_j|$. However, the difficulties of the analytical calculation of their matrix elements prevent application of this approach to larger systems [@Perkins1973; @Hyl3ele_1987; @Langer2011]. Generality with respect to the particle number and accessible analytical Hamiltonian matrix elements are achievable through powers of the quadratic form of the inter-particle distances that define explicitly correlated Gaussian-type (ECG) functions [@ECG-history1960; @ECG-history1960_1]. Plain explicitly correlated Gaussian (pECG) functions for $N_p$ interacting particles $$\begin{aligned}
\phi_I^{\text{pECG}} = \exp\left[-\sum_{i<j=1}^{N_p}{A_I}_{ij}\bm{r}_i\cdot\bm{r}_j\right] ~,\end{aligned}$$ are the simplest functions of this type and have been successfully employed to describe a number of diverse physical systems, from small atoms and molecules to light nuclei, hadrons, quantum dots, and Efimov systems [@Matyus2012; @Adamowicz2013; @Adamowicz2013_rev]. pECG functions are also manifestly spherically symmetric, i.e. invariant under rotation, as they are eigenfunctions of the total angular momentum squared operator with eigenvalue zero. Additional and important higher angular momentum contributions originate from the cross terms of the exponential part, i.e. $\exp(-{A_I}_{ij}\bm{r}_i\cdot\bm{r}_j)$ which, when expanded into a power series, contain terms of the form $$\begin{aligned}
(\bm{r}_i\cdot\bm{r}_j)^n = \sum_{2k+l=n} \frac{4\pi(2k+l)!}{2^kk!(2k+2l+1)!!}
~ {|r_i|}^{2k} {|r_j|}^{2k}
\sum_{m=-l}^{l} \mathcal{Y}_{lm}(\bm{r}_i) \mathcal{Y}_{lm}(\bm{r}_j) ~,\end{aligned}$$ which are associated with different solid spherical harmonics $\mathcal{Y}_{lm}$ for the coordinates $\bm{r}_i$ and $\bm{r}_j$.
Although these advantages made ECG-type functions very popular in high accuracy calculations [@svm-history1977; @FECG-integrals-advances1978; @ECG-history1993; @Varga1995; @Korobov2000; @Adamowicz2003a], the spherical symmetry limits the applicability of plain ECGs to ground rotational states only. Different approaches [@Varga1996; @Adamowicz2013_rev] have been developed to extend ECGs to nonspherical problems, i.e. for calculating states with non-zero total spatial angular momentum quantum numbers $N$.
In general, the ECGs are being multiplied with a nonspherical function $\theta_{NM_N}(\bm{r})$ of the collective position vectors $\bm{r}$ that for one particle in a central potential would just reduce to a solid spherical harmonic $\mathcal{Y}(\bm{r}_1)$. The generalization to the $N_p$-particle case is a vector-coupled product of the solid spherical harmonics of the relative coordinates, $$\begin{aligned}
\theta_{NM_N}(\bm{r}) = \sum_{\kappa=\{m_1,m_2,\ldots,m_{N_p}\}} \mathcal{C}_{\kappa} \prod_{i=1}^{N_p} \mathcal{Y}_{l_im_i}(\bm{r}_i) ~,
\label{eq:partWaveDecomp}\end{aligned}$$ where $\mathcal{C}_{\kappa}$ is a product of Clebsch–Gordan coefficients, $$\begin{aligned}
\mathcal{C}_{\kappa} =& \langle l_1m_1l_2m_2|L_{12}m_1+m_2\rangle \langle L_{12}m_1+m_2l_3m_3|L_{123}m_1+m_2+m_3\rangle \nonumber \\
& \cdots \langle L_{12\ldots N_p-1}m_1+m_2+\ldots+m_{N_p-1}l_{N_p}m_{N_p}|NM_N\rangle ~,\end{aligned}$$ that couples the orbital angular momenta sequentially to the specified total quantum numbers ($N,M_N$). Since the angular momentum of the relative motion is not a conserved quantity, it is important for an accurate description to include several sets of orbital angular momenta ($l_1,l_2,\ldots,l_{N_p};L_{12},L_{123},\ldots$) weighted by $\mathcal{C}_{\kappa}$. Eq. (\[eq:partWaveDecomp\]) is a partial-wave expansion whose direct implementation is cumbersome since the matrix elements for this choice of $\theta_{NM_N}(\bm{r})$ will become very complicated. Moreover the algebraic complexity of the integral matrix elements is not invariant with respect to the number of particles, and hence, analytical expressions must be derived for each different system.
One viable alternative to the full partial wave decomposition is to consider only limited coupling schemes “specializing” the basis functions for a given $N$ while the relative matrix elements are explicitly derived. For example, Refs. [@Komasa2001; @Adamowicz2008_zCECG; @Adamowicz2009_N1; @Adamowicz2013_N1; @Adamowicz2015_zECG] focused on ECG functions specifically tailored for $N=1$ states considering the sets of orbital angular momenta ($l_1=0,\ldots,l_i=1,\ldots,l_{N_p}=0$). Ref. [@Adamowicz2010_N2; @Adamowicz2011_N2; @Adamowicz2011_N2_Ryd1; @Adamowicz2011_N2_Ryd2]) tackled $N=2$ states analogously with lowest-order angular momentum couplings.
Alternatively, representations of $\theta_{NM_N}(\bm{r})$ including the orientation of a global vector $\bm{v}$ formed as a linear combination of all particle coordinates $\{\bm{r}_i\}$, have been successfully employed in high-accuracy calculations of properties of small atoms and molecules [@suzukivarga1998; @Matyus2012]. This approach is based on an equivalence condition between the global vector representation of $\theta_{NM_N}(\bm{r})$ and the partial-wave expansion for a given orientation of the global vector. Under the assumption of a smooth energy landscape in parameter space, the global vector orientation can be recovered variationally through the minimization of the energy with respect to its real-valued parameters. Although this approach is appealing because it yields analytical matrix elements for quantum mechanical operators that are form invariant with respect to the angular momentum quantum numbers $N$ and $M_N$, and the number of particles $N_p$, the variational optimization of the global vector parameters is difficult and not every $\theta_{NM_N}(\bm{r})$ can be represented. These alternative formulations are strictly derived from the partial wave expansion as a result of having truncated or variationally approximated Eq. (\[eq:partWaveDecomp\]).
In this work, we extend our numerical projection scheme onto irreducible representations of the rotational-inversion O(3) group presented in our previous work [@Muolo2018b], focusing on a special case where the integral projector can now be solved analytically. In Ref. [@Muolo2018b], we considered explicitly correlated Gaussian functions with centers shifted by a vector in the three dimensional Euclidean space, $\bm{s}\in\mathbb{R}^3$. Numerically exact eigenfunctions of the squared total spatial angular momentum operator $\hat{\bm{N}}^2$ and the parity operator $\hat{p}$ were then constructed with explicit projection onto the corresponding eigenspace. We relied on numerical quadrature schemes for the calculation of integral matrix elements which introduced noticeable computational cost in the variational iterative steps. In practice, numerical projection precludes large basis sets from being optimized variationally and limits the applicability of the developed formalism. Here, we consider solving exactly the projection operator for a subset of floating ECG functions having shifted centers along only one axis. We devise analytical integral matrix elements for projected functions for the overlap, kinetic, Coulomb, and angular momentum operators. We illustrate the validity of this novel functional form by studying the first three rotational states of the dihydrogen molecular ion, H$_2^+=\{$p$^+,$p$^+,$e$^-\}$ treated explicitly as a three-particle system and the dyhydrogen molecule, H$_2=\{$p$^+,$p$^+,$e$^-,$e$^-\}$ treated explicitly as a four-particle system.
Theory {#SEC:theory}
======
We consider a non-relativistic Coulombic Hamiltonian for $N_p$ particles $$\label{nonrel-H}
\hat{H}_{\text{lab}} = -\bm{\nabla_{r}}^T M \bm{\nabla_r} +
\sum_{i=1}^{N_p} \sum_{j>i}^{N_p} \frac{q_iq_j}{\left|\bm{r}_i-\bm{r}_j\right|} ~,$$ with the position vector $\bm{r}_i$ of the $i$th particle in the laboratory fixed Cartesian coordinates (LFCC), its mass $m_i$ and its charge $q_i$. $\bm{\nabla_r}$ is the gradient with respect to $\bm{r}_i$ and $M$ is a $N_p\times N_p$ matrix with elements $M_{ij}=\delta_{ij}/2m_i$.
As we are interested in bound states, the motion of the center of mass (CM) can be discarded. This is usually realized by a linear transformation of the coordinates $$\label{prima-transf}
U_x\bm{r}=\left(\bm{x}_1,\bm{x}_2,\ldots,\bm{x}_{N_p-1},\bm{x}_{\text{CM}}\right)^T$$ in which the $\bm{x}_{\text{CM}}=\sum_{i=1}^{N_p}m_i\boldsymbol{r}_i/(\sum_{i=1}^{N_p}m_i)$ are the center-of-mass Cartesian coordinates and $\bm{x}\equiv(\bm{x}_1,\ldots,\bm{x}_{N_p-1})$ denotes the translationally invariant Cartesian coordinates (TICC) corresponding to the internal coordinates of the system generated through the relative tranformation matrix $U_x$. A transformation of the Hamiltonian in Eq. (\[nonrel-H\]) separates the kinetic energy term for the center of mass from the internal Hamiltonian [@suzukivarga; @Adamowicz2013_rev]: $$\begin{aligned}
\hat{H}_{\text{int}} = - \nabla_{\bm{x}}^\text{T}\,\mu\,\nabla_{\bm{x}} +
\sum_{i=1}^{N_p-1} \sum_{j>i}^{N_p-1} \frac{q_iq_j}{|(\bm{f}_{ij}\otimes\mathbb{1}_3)\bm{x}|} ~,
\label{eq:TI-Hamiltonian}\end{aligned}$$ where $$\begin{aligned}
\mu =& U_x^{-T}MU_x ~,\end{aligned}$$ and $$\begin{aligned}
(\bm{f}_{ij})_k =& (U^{-1}_x)_{ik} - (U^{-1}_x)_{jk} ~.\end{aligned}$$ This separation of the center-of-mass coordinate requires transforming both the Hamiltonian and the state function and has been exploited in practice [@Adamowicz2003a; @Matyus2012].
By contrast, here we solely transform the basis functions in a given TICC set without transforming quantum mechanical operators following the method described in our previous work [@Benjamin2013; @Muolo2018a]. In this approach, the matrix-element calculations are carried out naturally in the LFCC set and the center-of-mass contamination is rigorously subtracted from the expectation values. While handling state functions in a TICC set is very appealing because of the restriction of the parameter space to only $N_p-1$ internal coordinates, we avoid the difficulties arising from matrix elements for transformed operators and instead retain the algebraic simpler and intuitive LFCC set for the integral evaluation. We employ the heavy-particle centered, the center-of-mass centered, and Jacobian Cartesian coordinate sets, allowing the basis functions to cycle through these TICC representations in order to describe efficiently different ”groupings“ of particles (e.g., pairs and triples of particles).
Basis functions {#SEC:basisfunction}
===============
Given the total spin quantum number and its projection on the $z$-axis $S$ and $M_s$, respectively, the wave function representing is expanded as a linear combination of (anti-)symmetrized floating explicitly correlated Gaussian (FECG) functions $$\label{eq:wavefun}
\Psi(\mathbf{r})=\sum_{I=1}^{N_b}c_{I} \, \bm{\chi}_{I}^{S,M_{S}} \, \hat{Y}\phi_{I}^{{\text{FECG}}}
(\bm{r};A_I^{(r)},\bm{s}_I^{(r)}) ~,$$ where $c_I$ are the expansion coefficients, $\bm{\chi}_{I}^{S,M_{S}}$ are spin functions, and $\hat{Y}$ is the Young operator that accounts for the appropriate permutation symmetry of sets of identical particles as described by Kinghorn [@Kinghorn1996]. FECGs have the following general form $$\begin{aligned}
\phi_I^{\text{FECG}} (\bm{r};A_I^{(r)},\bm{s}_I^{(r)})
&= \exp\left[-(\bm{r}-\bm{s}_I^{(r)})^T (A_I^{(r)}\otimes\mathbb{1}_3) (\bm{r}-\bm{s}_I^{(r)}) \right] ~.
\label{eq:fecg}\end{aligned}$$ Here, $A_I^{(r)}$ is an $N_p\times N_p$ symmetric matrix of the $\frac{1}{2}N_p(N_p+1)$ variational parameter, with the subscript $I$ indicating that the matrix is unique for each basis function and the superscript indicating that the variational parameters refer to the LFCC set. It is $\bm{r}(A_I^{(r)}\otimes\mathbb{1}_3)\bm{r}>0 \,\,\forall\,\, \bm{r}\in\mathbb{R}^{3N_p}$, that is $A_I^{(r)}$ must be positive definite, to ensure square integrability of the $\phi_I^{[{\text{FECG}}]}$ basis function. A necessary and sufficient condition for a symmetric real matrix to be positive definite is that all eigenvalues must be positive. Here $\bm{r}-\bm{s}_I^{(r)}$ stands for a set of vectors $\{\bm{r}_1-\bm{s}_{I\,1}^{(r)},\ldots,\bm{r}_{N_p}-\bm{s}_{I\,N_p}^{(r)}\}$ that correspond to shifted particle coordinates with the $3N_p$-dimensional vector $\bm{s}_I^{(r)}$ composed of parameters to be optimized in a variational procedure.
Note that the floating spherical Gaussian orbitals (FSGO) approach introduced by Frost in 1967 [@Frost1967] is based on one-particle functions (orbitals) and is therefore a limiting case of our approach for diagonal (and not dense) Gaussian parameter matrices $A_I$. In fact, this special case reduces our FECG basis functions to a product of exponential functions, each of which being spherically symmetric about its origin. By contrast, FECG basis functions with dense $A_I$ Gaussian parameter matrices, include partial waves contributions from many higher angular momentum states (see the Introduction).
In the following sections we explicitly work out the integral matrix elements in the simple LFCC frame.
Projection technique {#SEC:projection}
====================
The FECGs in Eq. (\[eq:fecg\]) define Gaussian functions with shifted centers to allow for suitable deformations of the ansatz for the all-particle wave function that are predominantly needed for polyatomic systems [@Adamowicz2013_rev; @Muolo2018a]. A general FECG function is, however, neither an eigenfunction of the squared total angular momentum operator $\bm{N}^2$, nor an eigenfunction of the space inversion operator $\hat{p}$. As the rotation-inversion symmetry must be restored variationally in the limit of a complete basis set, these basis functions gives rise to poor energy convergence.
To alleviate this problem, we recently proposed an integral projection operator, $\hat{P}_{M_N}^{[N,p]}$ [@Muolo2018b], to ensure the correct spatial rotation-inversion symmetry corresponding to $N$ and $M_N$, the total spatial angular momentum quantum numbers, and the parity quantum number $p$: $$\label{eq:projOp}
\hat{P}_{M_N}^{[N,p]} = \hat{P}_{M_NM_N}^{[N]} \, \hat{P}_{C_I}^{[p]} ~,$$ with $$\begin{aligned}
\hat{P}_{M_1M_2}^{[N]} =& \int \frac{d\Omega}{4\pi^3}
\,\, D^{[N]}_{M_1M_2}\left(\Omega\right)^* \hat{R}\left(\Omega\right) ~,\end{aligned}$$ and $$\begin{aligned}
\hat{P}_{C_I}^{[p]} =& \hat{\mathcal{E}} + p\cdot \hat{\mathcal{I}} ~,\end{aligned}$$ where $\hat{\mathcal{E}}$ is the identity operator, $\hat{\mathcal{I}}$ is the spatial inversion operator, and $D_{M_1M_2}^{[N]}$ is the element of the $N$-th Wigner $D$-matrix $$\begin{aligned}
D^{[N]}_{M_1M_2} = \exp(-iM_1\alpha) \, d^{[N]}_{M_1M_2}(\beta) \exp(-iM_2\gamma) ~,\end{aligned}$$ with the Wigner (small) $d$-matrix being $$\begin{aligned}
d^{[N]}_{M_1M_2}(\beta) =& \big[(N+M_1)!(N-M_1)!(N+M_2)!(N-M_2)!\big]^{\frac{1}{2}}
\nonumber \\[0.1cm]
& \times \sum_s \Bigg[ \frac{ (-1)^{M_1-M_2+s} \left(\cos\frac{\beta}{2}\right)^{2N+M_2-M_1-2s}
\left(\sin\frac{\beta}{2}\right)^{M_1-M_2+2s} } { (N+M_2-s)!s!(M_1-M_2+s)!(N-M_1-s)! } \Bigg] ~.\end{aligned}$$ $\hat{R}(\Omega)$ is the quantum mechanical rotation operator over the Euler angles $\Omega\equiv\{\alpha,\beta,\gamma\}$ [@Rose:AngularMomentum], $$\begin{aligned}
\hat{R}(\alpha,\beta,\gamma) = \exp(-i\alpha N_z) \exp(-i\beta N_y) \exp(-i\gamma N_z) ~.
\label{eq:RotOp1}\end{aligned}$$ The effect of the projector operator in Eq. (\[eq:projOp\]) on a state $|N\,M_N\rangle$ is $$\begin{aligned}
\hat{P}_{M_1M_2}^{[N_1]} |N_2M_2\rangle = |N_1M_1\rangle \,\delta_{N_1N_2} \,\delta_{M_1M_2} ~,\end{aligned}$$ with $|NM_N\rangle$ being angular momentum eigenstates. Note that our original implementation [@Muolo2018b] of the projection scheme was purely numerical, which we overcome in this work for the special case of projection on one spatial axis, for which an analytical expression can be derived.
The form of the rotation operators in Eq. (\[eq:RotOp1\]) is not a convenient operational definition because they require an explicit expression of the angular momentum components $N_i$ that is not entirely straightforward in our all-particle explicitly-correlated formulation. Nonetheless, exactly the same symmetry operation will be realized if we rotate the physical system itself or if we rotate the coordinate axis in the opposite direction, $$\begin{aligned}
\hat{R}(\Omega)\phi^{{\text{FECG}}}_I\big(\bm{r};A_I^{(r)},\bm{s}_I^{(r)}\big) & =
\phi^{{\text{FECG}}}_I\big(U(\Omega)^{-1}\,\bm{r};A_I^{(r)},\bm{s}_I^{(r)}\big)
\nonumber \\
& = \exp\left[-\big(U(\Omega)^{-1}\,\bm{r}-\bm{s}_I^{(r)}\big)^T
\big(\bar{A}_I^{(r)}\otimes\mathbb{1}_3\big)
\big(U(\Omega)^{-1}\,\bm{r}-\bm{s}_I^{(r)}\big)\right] \nonumber \\
& = \exp\left[-\big(\bm{r}-U(\Omega)\bm{s}_I^{(r)}\big)^T
\big(\bar{A}_I^{(r)}\otimes \tilde{U}(\Omega)^{-T}\tilde{U}(\Omega)^{-1}\big)
\big(\bm{r}-U(\Omega)\bm{s}_I^{(r)}\big)\right] \nonumber \\
& = \phi_I^{{\text{FECG}}}\big(\bm{r};A_I^{(r)},U(\Omega)\bm{s}_I^{(r)}\big) ~,
\label{eq:FormInvariance}\end{aligned}$$ where $U(\Omega)=\mathbb{1}_{N_p}\otimes\tilde{U}(\Omega)$ represents the coordinate transformation generalized to a system of $N_p$ particles with $$\begin{aligned}
\tilde{U}(\Omega) =
\left(\begin{array}{ccc}
\cos\alpha\cos\beta\cos\gamma-\sin\alpha\sin\gamma & -\cos\gamma\sin\alpha-\cos\alpha\cos\beta\sin\gamma & -\cos\alpha\sin\beta \\
\cos\beta\cos\gamma\sin\alpha+\cos\alpha\sin\gamma & \cos\alpha\cos\gamma-\cos\beta\sin\alpha\sin\gamma & -\sin\alpha\sin\beta \\
\cos\gamma\sin\beta & \sin\beta\sin\gamma & \cos\beta
\end{array}\right) ~.
\label{eq:RotOp}\end{aligned}$$
The properties of the rotation operator are summarized in four commutation relations: $$\begin{aligned}
&\left[\hat{R}(\Omega),\hat{H}\,\right] \, = \,\, 0 ~, \label{eq:projOpProp1} \\
&\left[\hat{R}(\Omega),\hat{N}^2\right] = \,\, 0 ~, \label{eq:projOpProp2} \\
&\left[\hat{R}(\Omega),\hat{N}_z\right] \ne \,\, 0 ~, \label{eq:projOpProp3} \\
&\left[\hat{R}(\Omega),\hat{p}\,\right] \, = \,\, 0 ~. \label{eq:projOpProp4}\end{aligned}$$ Furthermore, the $\hat{P}_{M_NM_N}^{[N]}$ projection operator is idempotent and Hermitian: $$\begin{aligned}
(\hat{P}_{M_NM_N}^{[N]})^2 =& \hat{P}_{M_NM_N}^{[N]} \label{eq:projOpProp5} \\
(\hat{P}_{M_NM_N}^{[N]})^{\dagger} =& \hat{P}_{M_NM_N}^{[N]} \label{eq:projOpProp6} ~.\end{aligned}$$
Properties in Eqs. (\[eq:projOpProp1\])-(\[eq:projOpProp6\]) are employed in the remainder of this work for the calculation of quantum mechanical expectation values.
Matrix elements {#SEC:matrixelements}
===============
In this section, we present analytically projected [FECG]{}s matrix elements for important operators in the special case of unidimensional shift vectors, that is, employing $\bm{s}_I$ shift vectors of the form $$\label{zFECG_0}
\bm{s}_I^{(r)} = \bm{u}_I^{(r)} \otimes \bm{e}_z ~,$$ where $\bm{u}_I^{(r)}$ is a vector of length $N_p$ and $\bm{e}_z=\left(0,0,1\right)^T$. From this choice of the $\bm{s}_I^{(r)}$ vectors we obtain the fundamental relation $$\label{zFECG_0_1_fun2}
\bm{e}_z^T\tilde{U}(\Omega)\bm{e}_z = \cos\beta ~.$$
Eq. (\[zFECG\_0\_1\_fun2\]) is employed throughout this work to derive analytical matrix elements for the overlap, kinetic, Coulomb, and angular momentum operators. For the matrix element of these operators we start from the analytical expressions derived for plain FECG by Cafiero and Adamowicz [@Adamowicz2002_FECG]. Conversely, angular momentum matrix elements are derived from the analytical expressions for plain FECG presented in our previous work [@Muolo2018b]. The unprojected and analytically projected $z$-shifted floating explicitly correlated Gaussian functions are abbreviated with zFECGs and apzFECGs, respectively.
Given a quantum mechanical operator $\hat{O}$ commuting with the projector operator, the matrix element $IJ$ for apzFECGs reads as follows: $$\begin{aligned}
\mathcal{O}_{IJ[N,M_N,p]}^{\text{apzFECG}} =
& \left\langle \phi^{\text{apzFECG}}_{I[N,M_N,p]} \big| \hat{O} \big| \phi^{\text{apzFECG}}_{J[N,M_N,p]} \right\rangle
= \left\langle \phi_I^{\text{zFECG}} \big| \hat{O} \big| \hat{P}^{[N,p]}_{M_N} \phi_J^{\text{zFECG}} \right\rangle ~,
\label{eq:MatEl}\end{aligned}$$ where the Hermiticity and idempotency of the projection operator, Eqs. (\[eq:projOpProp5\]) and (\[eq:projOpProp6\]), were exploited to simplify the integral expression. In the following, analytical matrix elements for a variety of quantum mechanical operators are derived. For the sake of brevity, the projection onto the parity states $\hat{P}_{C_I}^{[p]}$ is omitted.
Overlap integral
----------------
The matrix elements of the identity operator for plain FECGs are given by [@Adamowicz2002_FECG] $$\begin{aligned}
\left\langle \phi_I^{\text{FECG}} | \phi_J^{\text{FECG}} \right\rangle
=\, \tilde{S}_{IJ} \, \exp\left[2 \bm{s}_I^{(r)^T}
A_I^{(r)} A_{IJ}^{{(r)}^{-1}} A_{J}^{(r)} \bm{s}_J^{(r)}\right] ~,
\label{eq:overlap_plainFECG}\end{aligned}$$ where $A_{IJ}^{(r)}=A_I^{(r)}+A_J^{(r)}$ and $$\begin{aligned}
\tilde{S}_{IJ} =& \left(\frac{\pi^{N_p}}
{\left|\bar{A}_I^{(r)}+\bar{A}_J^{(r)}\right|}\right)^{\frac{3}{2}}
\exp\left[- \bm{s}_I^{(r)^T} A_I^{(r)} \bm{s}_I^{(r)}
-\bm{s}_J^{(r)^T} A_J^{(r)} \bm{s}_J^{(r)}\right] \nonumber \\
& \times \exp\left[+ \bm{s}_I^{(r)^T} A_I^{(r)} A_{IJ}^{{(r)}^{-1}}
A_{I}^{(r)} \bm{s}_I^{(r)}
+\bm{s}_J^{(r)^T} A_J^{(r)} A_{IJ}^{(r)^T}
A_{J}^{(r)} \bm{s}_J^{(r)}\right] ~.\end{aligned}$$
In Eq. (\[eq:overlap\_plainFECG\]) we have separated $\tilde{S}_{IJ}$, the term unaffected by the action of the rotation operator on the shift vector $\bm{s}_J^{(r)}$. The remaining term must be investigated since it involves the angular integration over the Euler angles. For apzFECGs the overlap matrix element reads $$\begin{aligned}
S^{\text{apzFECG}}_{IJ[N,M_N,p]}
=& \left\langle \phi_I^{\text{zFECG}} (\bm{r};A_I^{(r)},\bm{s}_I^{(r)}) \big|
\hat{P}^{[N,p]}_{M_N} \phi_J^{\text{zFECG}} (\bm{r};A_J^{(r)},\bm{s}_J^{(r)}) \right\rangle ~,\end{aligned}$$ and writing explicitly the projection operator leads to $$\begin{aligned}
\label{eq:SapzFECG}
S_{IJ[N,M_N,p]}^{\text{apzFECG}}
=& \int \frac{d\Omega}{4\pi^3} \,\, D^{[N]}_{M_NM_N}(\Omega)^*
\left\langle \phi_I (\bm{r};A_I^{(r)},\bm{s}_I^{(r)} \big|
\phi_J (\bm{r};A_J^{(r)},U(\Omega)\bm{s}_J^{(r)} ) \right\rangle ~,\end{aligned}$$ where we again drop the projector onto the parity state for the sake of brevity. Because $\tilde{S}_{IJ}$ is invariant under the action of $\hat{P}^{[N,p]}_{M_N}$, Eq. (\[eq:SapzFECG\]) can be written as $$\begin{aligned}
S_{IJ[N,M_N,p]}^{\text{apzFECG}} = \, \tilde{S}_{IJ} \, \Upsilon^{N}_{M_N} ~,\end{aligned}$$ with $$\begin{aligned}
\Upsilon^{N}_{M_N} = \int \frac{d\Omega}{4\pi^3} \,\,
D^{[N]}_{M_NM_N}(\Omega)^* \, \exp\left[2\, \bm{s}_I^{(r)^T} A_I^{(r)} A_{IJ}^{{(r)}^{-1}}
A_{J}^{(r)} U(\Omega) \bm{s}_J^{(r)} \right] ~.
\label{zFECG_0_1}\end{aligned}$$ Since $U(\Omega)=\mathbb{1}_{N_p}\otimes\tilde{U}(\Omega)$, we have $$\label{zFECG_0_1_fun1}
U(\Omega) \bm{s}_J^{(r)} = \bm{u}_J^{(r)} \otimes\tilde{U}(\Omega)\bm{e}_z ~,$$ where Eq. (\[zFECG\_0\]) and the definition of $U(\Omega)$ in Eq. (\[eq:RotOp\]) have been exploited.
Considering Eqs. (\[zFECG\_0\]), (\[zFECG\_0\_1\_fun1\]), and (\[zFECG\_0\_1\_fun2\]) and that $A_K^{(r)}=\bar{A}_K^{(r)}\otimes\mathbb{1}_3$ with $K\in\{I,J,IJ\}$, we have $$\exp\left[2\, \bm{s}_I^{(r)^T} A_I^{(r)} A_{IJ}^{{(r)}^{-1}}
A_{J}^{(r)} U(\Omega) \bm{s}_J^{(r)} \right]
= \exp\left[C\,\bm{e}_z^T\tilde{U}(\Omega)\bm{e}_z\right]
= \exp\left[C\,\cos{\beta}\right] ~,
\label{zFECG_0_2}$$ with $C$ given as $$\begin{aligned}
C = 2\, \bm{u}_I^{(r)^T} \bar{A}_I^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_{J}^{(r)} \bm{u}_J^{(r)} ~.
\label{eq:zFECG_Cval}\end{aligned}$$
Finally, the angular integration reduces to $$\begin{aligned}
\Upsilon^{N}_{M_N} = \frac{1}{4\pi^3} \int_0^{2\pi}d\alpha & \int_0^{\pi}d\beta \int_0^{2\pi}d\gamma
\,\,\, \sin(\beta) \, D^{[N]^*}_{M_NM_N}(\Omega) \exp\left[C\cos(\beta)\right] ~,
\label{eq:UpsilonNMN}\end{aligned}$$ To analytically solve the triple integration over Euler angles, we first note that the elements $D^{[N]}_{00}(\beta)$ of the Wigner $D$-matrices corresponding to $M_N=0$ are polynomial of $\cos\beta$ of degree $N$ with coefficients $a_\mu^{[N]}$ ([*[e.g.]{}*]{}, $a_0^{[0]}=1$, $a_0^{[1]}=0$, $a_1^{[1]}=1$), $$\label{eq:WignerN00}
D^{[N]}_{00}(\Omega) = D^{[N]}_{00}(\beta) =
\sum_{\mu=0}^{N} a_\mu^{[N]} \left(\cos{\beta}\right)^{\mu} ~.$$ Therefore, for apzFECGs with $M_N=0$, the integration over $\alpha$ and $\gamma$ Euler angles is trivial and Eq. (\[eq:UpsilonNMN\]) becomes $$\Upsilon^{N}_{0} = \frac{1}{\pi} \sum_{\mu=0}^{N} \int_0^{\pi}d\beta
\,\,\, \sin(\beta) [\cos(\beta)]^{\mu} \exp\left[C\cos(\beta)\right] ~.
\label{eq:UpsilonNMN_1}$$ Furthermore, since apzFECG functions do not depend on Euler angles $\alpha$ and $\gamma$, the integration of the $D^{[N]^*}_{M_NM_N}(\Omega)$ yields zero for every $N\in\mathbb{N}_0$ and $M_N\ne0$. The results of the integration over the Euler angle $\beta$ in Eq. (\[eq:UpsilonNMN\_1\]) for the spherically symmetric ground state as well as the two lowest rotationally excited states are then written as $$\begin{aligned}
\Upsilon^{N}_{M_N} =
& \left\{
\begin{array}{lc}
\displaystyle{\frac{2}{\pi C}} \sinh (C)
& N=0 \,,\, M_N=0 \\[0.3cm]
\displaystyle{\frac{2}{\pi C}} \cosh (C) - \displaystyle{\frac{2}{\pi C^2}} \sinh (C)
& N=1 \,,\, M_N=0 \\[0.3cm]
\displaystyle{\frac{2}{\pi C^3}} \Big[\left(C^2+3\right) \sinh (C)-3 C \cosh (C)\Big]
& N=2 \,,\, M_N=0 \\[0.3cm]
0 & \forall N\in\mathbb{N}_0 \,,\, M_N\ne0
\end{array}
\right. ~,
\label{zFECG_0_3}\end{aligned}$$ For a list of $\Upsilon^{N}_{M_N}$ up to $N=10$ see the Appendix.
Kinetic integral
----------------
The kinetic integral for plain FECGs reads [@Adamowicz2002_FECG] $$\begin{aligned}
\left\langle \phi_I^{\text{FECG}} | -\bm{\nabla}_{\bm{r}}^TM\bm{\nabla}_{\bm{r}} | \phi_J^{\text{FECG}} \right\rangle
= \, \tilde{S}_{IJ} \, \Big[ 4\,\big( \bm{s}_I^{(r)}-\bm{s}_J^{(r)} \big)^T B
\big(\bm{s}_I^{(r)}-\bm{s}_J^{(r)} \big) +6\operatorname{Tr}\left(M \bar{A}_J^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_I^{(r)} \right) \Big] ~,
\label{eq:kinetic_plainFECG}\end{aligned}$$ where $$\begin{aligned}
B = & \,\, 4 \, A_J^{(r)} A_{IJ}^{{(r)}^{-1}} A_I^{(r)} M
A_J^{(r)} A_{IJ}^{{(r)}^{-1}} A_I^{(r)} ~. \end{aligned}$$ For apzFECGs we have $$T_{IJ[N,M_N,p]}^{\text{apzFECG}} = \left\langle \phi_I^{\text{zFECG}} \big|
\hat{P}^{[N,p]}_{M_N} \phi_J^{\text{zFECG}} \right\rangle
= \,\, \tilde{S}_{IJ} \, \Sigma^{N}_{M_N} ~,$$ where the angular integral is written as $$\begin{aligned}
\Sigma^{N}_{M_N} =& \,\, \int \frac{d\Omega}{4\pi^3} \,\, D^{[N]}_{M_NM_N}(\Omega)^* \exp\left[ C\cos\beta \right] \nonumber \\[0.1cm]
& \times \left[ -\bm{s}_I^{(r)^T} B \bm{s}_I^{(r)}
-\bm{s}_J^{(r)^T} B \bm{s}_J^{(r)}
+2\bm{s}_I^{(r)^T} B\, U(\Omega) \bm{s}_J^{(r)}
+6\operatorname{Tr}\left(M \bar{A}_J^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}} \bar{A}_I^{(r)} \right) \right] ~.
\label{zFECG_1}\end{aligned}$$
We define $$\omega = -\bm{s}_I^{(r)^T} B \bm{s}_I^{(r)}
-\bm{s}_J^{(r)^T} B \bm{s}_J^{(r)}
+6\operatorname{Tr}\left(M \bar{A}_J^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}} \bar{A}_I^{(r)} \right) ~,$$ and $$\sigma= 2 \, \bm{u}_I^{(r)\,T} \bar{B} \bm{u}_J^{(r)} ~,$$ so that Eq. (\[zFECG\_1\]) can be cast in the compact form $$\begin{aligned}
\Sigma^{N}_{M_N} =& \,\,
\int \frac{d\Omega}{4\pi^3} \,\, D^{[N]}_{M_NM_N}(\Omega)^* \, \left( \omega+\sigma\cos\beta \right) \, \exp\left[ C\cos\beta \right] ~, \end{aligned}$$ With Eq. (\[eq:WignerN00\]), the integration over Euler angles can be reduced to the single integration over $\beta$ for which these analytical results follow $$\Sigma^{N}_{M_N} =
\left\{ \begin{array}{lc}
\displaystyle{\frac{2}{\pi C^2}} \Big[\sinh(C) (C\omega -\sigma)+C\sigma\cosh(C)\Big]
& N=0 \,,\, M_N=0 \\[0.3cm]
\displaystyle{\frac{2}{\pi C^3}} \Big[\sinh(C) \left(\left(C^2+2\right) \sigma -C\omega \right)+C\cosh(C) (C\omega -2\sigma)\Big]
& N=1 \,,\, M_N=0 \\[0.3cm]
\displaystyle{\frac{2}{\pi C^4}} \Big[\sinh(C) \left(C \left(C^2+3\right) \omega -\left(4 C^2+9\right) \sigma\right)
& \\[0.3cm]
\hspace{1cm} +C \cosh(C)\left(\left(C^2+9\right) \sigma -3C\omega \right)\Big]
& N=2 \,,\, M_N=0 \\[0.3cm]
0 & \forall N\in\mathbb{N}_0 \,,\, M_N\ne0 \\
\end{array}\right. ~.$$ For a list of $\Sigma^{N}_{M_N}$ up to $N=10$ see the Appendix.
Coulomb integral
----------------
From Ref. [@Adamowicz2002_FECG] we retrieve the Coulomb matrix element for plain FECGs as follows: $$\begin{aligned}
\left\langle \phi_I^{\text{FECG}} \left| \frac{1}{\left|\bm{r}_i-\bm{r}_j\right|} \right| \phi_J^{\text{FECG}} \right\rangle
= \tilde{S}_{IJ} \, \left(\frac{1}{{\bm{S}}^TJ_{ij}{\bm{S}}}\right)^{\frac{1}{2}}
\operatorname{erf}\left[\left(\frac{{\bm{S}}^TJ_{ij}{\bm{S}}}{\operatorname{Tr}\left(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \right)}\right)^{\frac{1}{2}}\right] ~,\end{aligned}$$ where the vector $\bm{S}$ is defined as $$\begin{aligned}
{\bm{S}} = A_{IJ}^{{(r)}^{-1}} \left( A_I^{(r)}
\bm{s}_I^{(r)} +A_J^{(r)} \bm{s}_J^{(r)} \right) ~,\end{aligned}$$ and $$\begin{aligned}
J_{ij} = \left\{ \begin{array}{ll} E_{ii} & {\text{if}} \,\,\, i=j \\ E_{ii}+E_{jj}-E_{ij}-E_{ji} & {\text{if}} \,\,\, i\ne j \end{array} \right. ~,\end{aligned}$$ with ${\left(E_{ij}\right)}_{\alpha\beta}=\delta_{\alpha\beta}$ being an $N_p\times N_p$ matrix.
We now define the matrix elements for apzFECG functions as $$\begin{aligned}
V_{IJ[N,M_N,p]}^{\text{apzFECG}} = \left\langle \phi_I^{\text{zFECG}} \left| \frac{1}{\left|\bm{r}_i-\bm{r}_j\right|}
\right| \hat{P}^{[N,p]}_{M_N} \phi_J^{\text{zFECG}} \right\rangle
= \,\, \tilde{S}_{IJ} \, \Lambda^{N}_{M_N} ~,\end{aligned}$$ where $$\begin{aligned}
\Lambda^{N}_{M_N} =& \,\, \int \frac{d\Omega}{4\pi^3} \,\, D^{[N]}_{M_NM_N}(\Omega)^* \, e^{C\cos\beta}
\left(\frac{1}{\tilde{\bm{S}}^TJ_{ij}\tilde{\bm{S}}}\right)^{\frac{1}{2}}
\operatorname{erf}\left[\left(\frac{\tilde{\bm{S}}^TJ_{ij}\tilde{\bm{S}}}{\operatorname{Tr}\left(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \right)}\right)^{\frac{1}{2}}\right] ~.
\label{zFECG_2}\end{aligned}$$
Here, we adopt the notation of Cafiero and Adamowicz [@Adamowicz2002_FECG] which is corrected in order to account for the rotated $\bm{s}_J$ vector $$\begin{aligned}
\tilde{\bm{S}} = A_{IJ}^{{(r)}^{-1}} \left( A_I^{(r)} \bm{s}_I^{(r)}
+A_J^{(r)} U(\Omega)\bm{s}_J^{(r)} \right) ~.\end{aligned}$$
In order to make $\beta$ explicit and solve the angular integration, we consider the following substitution $$\begin{aligned}
\tilde{\bm{S}}^TJ_{ij}\tilde{\bm{S}}
=& \tau_{ij} + 2\, \bm{s}_I^{(r)^T} A_I^{(r)} A_{IJ}^{{(r)}^{-1}}
J_{ij} A_{IJ}^{{(r)}^{-1}} A_J^{(r)} U(\Omega) \bm{s}_J^{(r)}
\nonumber \\[0.1cm]
=& \tau_{ij} + F_{ij} \left(\bm{e}_z^T\tilde{U}(\Omega)\bm{e}_z\right)
\nonumber \\
=& \tau_{ij} + F_{ij} \, \cos\beta ~,\end{aligned}$$ with $$\begin{aligned}
\tau_{ij} = & \bm{s}_I^{(r)^T} A_I^{(r)} A_{IJ}^{{(r)}^{-1}}
J_{ij} A_{IJ}^{{(r)}^{-1}} A_I^{(r)} \bm{s}_I^{(r)}
+\bm{s}_J^{(r)^T} A_J^{(r)} A_{IJ}^{{(r)}^{-1}}
J_{ij} A_{IJ}^{{(r)}^{-1}} A_J^{(r)} \bm{s}_J^{(r)} ~, \\[0.1cm]
F_{ij} = & 2\cdot \bm{u}_I^{(r)^T} \bar{A}_I^{(r)}
\bar{A}_{IJ}^{{(r)}^{-1}} \bar{J}_{ij} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_J^{(r)} \bm{u}_J^{(r)} ~.\end{aligned}$$ The angular integration in Eq. (\[zFECG\_2\]) is now written as $$\begin{aligned}
\Lambda^{N}_{M_N} =&
\int \frac{d\Omega}{4\pi^3} \,\, D^{[N]}_{M_NM_N}(\Omega)^* \, e^{C\cos\beta}
\left(\frac{1}{\tau_{ij} + F_{ij} \cdot \cos\beta}\right)^{\frac{1}{2}}
\operatorname{erf}\left[\left(\frac{\tau_{ij} + F_{ij} \cdot \cos\beta}{\operatorname{Tr}\left(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \right)}\right)^{\frac{1}{2}}\right] ~.
\label{zFECG_2.1}\end{aligned}$$ While the integration with respect to $\alpha$ and $\gamma$ is trivial to integrate over $\beta\in[0,\pi)$, we change the variable, $y\equiv\tau_{ij}+F_{ij}\cos\beta$ so that Eq. (\[zFECG\_2.1\]) becomes $$\begin{aligned}
\Lambda^{N}_{M_N} =& \frac{e^{-\frac{\tau_{ij}\cdot C}{F_{ij}}}}{\pi F_{ij}} \int_{\tau_{ij}-F_{ij}}^{\tau_{ij}+F_{ij}} dy \,\,
D^{[N]}_{M_NM_N}(y) \,\, y^{-\frac{1}{2}} \,\, e^{\frac{C}{F_{ij}}\,y} \,\,
\operatorname{erf}\left[\left(\frac{y}{\operatorname{Tr}\left(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \right)}\right)^{\frac{1}{2}}\right] ~.
\label{zFECG_2.2} \end{aligned}$$ To change the variable of the Wigner $D$-matrix we recall Eq. (\[eq:WignerN00\]), namely that the elements $D^{[N]}_{00}(\beta)$ for any $N$ are polynomial of $\cos\beta$ of degree $N$. Therefore, after changing the variable, the zeroth diagonal element of the Wigner $D$-matrix can be written as $$\begin{aligned}
D^{[N]}_{00}(y)
=& \sum_{\mu=0}^{N} a_{\mu}^{[N]} \left( \frac{y-\tau_{ij}}{F_{ij}} \right)^{\mu} \nonumber \\[0.1cm]
=& \sum_{\mu=0}^{N} \sum_{k=0}^{\mu} \frac{\mu!\,a_\mu^{[N]}}{(\mu-k)!k!}
\label{zFECG_Dpoly}\end{aligned}$$ where in the second line the power of the binomial is written explicitly. By inserting Eq. (\[zFECG\_Dpoly\]), the polynomial form of the Wigner $D$-matrix, Eq. (\[zFECG\_2.2\]) reads $$\begin{aligned}
\Lambda^{N}_{0} =& \,\, \frac{e^{-\frac{\tau_{ij}\cdot C}{F_{ij}}}}{\pi F_{ij}}
\sum_{\mu=0}^{N} \sum_{k=0}^{\mu} \frac{\mu!\,a_\mu^{[N]}}{(\mu-k)!k!}
\left(-\frac{\tau_{ij}}{F_{ij}}\right)^{\mu-k} \left(\frac{1}{F_{ij}}\right)^k \nonumber \\[0.1cm]
& \times \int_{\tau_{ij}-F_{ij}}^{\tau_{ij}+F_{ij}} dy \,\, y^{-\frac{1}{2}+k} \,\,
e^{\frac{C}{F_{ij}}\,y} \,\, \operatorname{erf}\left[\left(\frac{y}{\operatorname{Tr}\left(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \right)}\right)^{\frac{1}{2}}\right] ,
\label{zFECG_2.21} \end{aligned}$$ whereas expanding the exponential in a Taylor series yields $$\begin{aligned}
\Lambda^{N}_{0} =& \,\, \frac{e^{-\frac{\tau_{ij}\cdot C}{F_{ij}}}}{\pi F_{ij}}
\sum_{\mu=0}^{N} \sum_{k=0}^{\mu} \frac{\mu!\,a_\mu^{[N]}}{(\mu-k)!k!}
\left(-\frac{\tau_{ij}}{F_{ij}}\right)^{\mu-k} \left(\frac{1}{F_{ij}}\right)^k \nonumber \\[0.1cm]
& \times \sum_{n=0}^{\infty} \frac{1}{n!} \left(\frac{C}{F_{ij}}\right)^{n}
\int_{\tau_{ij}-F_{ij}}^{\tau_{ij}+F_{ij}} dy \,\,
\,\, y^{-\frac{1}{2}+k+n} \,\, \operatorname{erf}\left[\left(\frac{y}{\operatorname{Tr}\left(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \right)}\right)^{\frac{1}{2}}\right] ~.
\label{zFECG_2.22}\end{aligned}$$ The integral over $y$ possesses an analytical solution, $$\begin{aligned}
\Lambda^{N}_{0} =& \frac{e^{-\frac{\tau_{ij}\cdot C}{F_{ij}}}}{\pi F_{ij}}
\sum_{\mu=0}^{N} \sum_{k=0}^{\mu} \frac{\mu!\,a_\mu^{[N]}}{(\mu-k)!k!}
\left(-\frac{\tau_{ij}}{F_{ij}}\right)^{\mu-k} \left(\frac{1}{F_{ij}}\right)^k
\sum_{n=0}^{\infty} \frac{2}{(2 k+2 n+1) n!} \left(\frac{C}{F_{ij}}\right)^{n} \nonumber \\[0.1cm]
& \, \times \Bigg[ -\operatorname{erf}(\sqrt{t_2}) \, (\tau_{ij}-F_{ij})^{k+n+\frac{1}{2}}
+\operatorname{erf}(\sqrt{t_1}) \, (F_{ij}+\tau_{ij})^{k+n+\frac{1}{2}} \nonumber \\[0.1cm]
& \hspace{0.6cm} + \frac{\operatorname{Tr}\big(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \big)^{k+n+\frac{1}{2}}}{\sqrt{\pi}} \,\,
\bigg( \Gamma (k+n+1,t_1) \,-\, \Gamma (k+n+1,t_2) \bigg) \Bigg] ~,
\label{zFECG_2.3}\end{aligned}$$ with $$\begin{aligned}
t_1 = & \frac{\tau_{ij}+F_{ij}}{\operatorname{Tr}\big(\bar{J}_{ij}\bar{A}_{IJ}^{{(r)}^{-1}}\big)} ~, \\[0.1cm]
t_2 = & \frac{\tau_{ij}-F_{ij}}{\operatorname{Tr}\big(\bar{J}_{ij}\bar{A}_{IJ}^{{(r)}^{-1}}\big)} ~.\end{aligned}$$ If the resulting series in Eq. (\[zFECG\_2.3\]) is considered separately for each term, the first two can be evaluated exactly in terms of the lower incomplete Gamma function $\gamma(n,b)$, while the latter is simplified according to the properties of the incomplete Gamma functions $$\begin{aligned}
\Lambda^{N}_{0} =& \frac{e^{-\frac{\tau_{ij}\cdot C}{F_{ij}}}}{\pi F_{ij}}
\sum_{\mu=0}^{N} \sum_{k=0}^{\mu} \,\, \frac{\mu!\,a_\mu^{[N]}}{(\mu-k)!k!} \,
\left(-\frac{\tau_{ij}}{F_{ij}}\right)^{\mu-k} \, \left(\frac{1}{F_{ij}}\right)^k \nonumber \\[0.1cm]
& \Bigg[ \left(-\frac{C}{F_{ij}}\right)^{-k-\frac{1}{2}} \operatorname{erf}(\sqrt{t_1})
\,\,\, \gamma\left(k+\frac{1}{2},-\frac{C(F_{ij}+\tau_{ij})}{F_{ij}}\right) \nonumber \\[0.1cm]
& -\left(-\frac{C}{F_{ij}}\right)^{-k-\frac{1}{2}} \operatorname{erf}(\sqrt{t_2})
\,\,\, \gamma\left(k+\frac{1}{2},\frac{C(F_{ij}-\tau_{ij})}{F_{ij}}\right) \nonumber \\[0.1cm]
& + \frac{2}{\sqrt{\pi}} \sum_{n=0}^{\infty} \frac{\Gamma (k+n+1,t_1,t_2)}{n! \,\, (2 k+2 n+1)}
\left(\frac{C}{F_{ij}}\right)^n \operatorname{Tr}\big(\bar{J}_{ij}
\bar{A}_{IJ}^{{(r)}^{-1}} \big)^{k+n+\frac{1}{2}} \Bigg] ~,
\label{zFECG_2.4}\end{aligned}$$ where the last remaining series converges factorially and only requires the generalized incomplete Gamma functions $\Gamma(n,a,b)$, with $n\in\mathbb{N}^+$, that can be efficiently calculated in closed form as $$\begin{aligned}
\Gamma(n,t_1,t_2) = \Gamma(n) \left( e^{-t_1} \sum_{k=0}^{n-1} \frac{t_1^k}{k!}
-e^{t_2} \sum_{k=0}^{n-1} \frac{t_2^k}{k!} \right) ~.\end{aligned}$$ While Eq. (\[zFECG\_2.4\]) provides a general $N$-formula toward the calculation of Coulomb matrix elements, a closed formula can be obtained with the ’differentiation under the integral’ technique from Eq. (\[zFECG\_2.21\]) $$\begin{aligned}
\Lambda^{N}_{0} =& \frac{e^{-\frac{\tau_{ij}\cdot C}{F_{ij}}}}{\pi F_{ij}}
\sum_{\mu=0}^{N} \sum_{k=0}^{\mu} \,\, \frac{\mu!\,a_\mu^{[N]}}{(\mu-k)!k!} \,
\left(-\frac{\tau_{ij}}{F_{ij}}\right)^{\mu-k} \, \left(\frac{1}{F_{ij}}\right)^k \nonumber \\[0.2cm]
& \times 2 \, T^{\frac{1}{2}}\,F_{ij}^k \,\, \frac{\partial^k}{\partial C^k}
\int_{\sqrt{(\tau_{ij}-F_{ij})/T}}^{\sqrt{(\tau_{ij}+F_{ij})/T}}
dx \,\, e^{\frac{C\,T}{F_{ij}}\,x^2} \,\, \operatorname{erf}\left[x\right] ~,
\label{zFECG_2.5} \end{aligned}$$ where $T=\operatorname{Tr}(\bar{J}_{ij} \bar{A}_{IJ}^{{(r)}^{-1}})$, the integration variable is changed according to $(y/T)^{\frac{1}{2}}=x$, and the $k$-th derivative with respect to $C$ is considered. The integral in Eq. (\[zFECG\_2.5\]) possesses an analytical solution, $$\begin{aligned}
& \int_a^b dx \,\, e^{-q\,x^2} \,\, \operatorname{erf}\left[x\right] = \, 2\sqrt{\frac{\pi}{q}}
\Bigg[ T\Big(a\sqrt{2q},\frac{1}{\sqrt{q}}\Big) - T\Big(b\sqrt{2q},\frac{1}{\sqrt{q}}\Big) \Bigg] ~,
\label{zFECG_2.6}\end{aligned}$$ where $T(h,x)$ is the Owen’s T function.
Squared total angular momentum expectation value
------------------------------------------------
To solve $\langle\phi_I^{\text{zFECG}}|\hat{N}^2|\phi_J^{\text{apzFECG}}\rangle$, the squared total angular momentum expectation value for projected zFECG functions, we start from the matrix elements for FECGs derived in our previous work [@Muolo2018b] $$\begin{aligned}
\langle\phi_I^{\text{FECG}}|\hat{N}^2|\phi_J^{\text{FECG}}\rangle =& \epsilon_{ijk}'
\Bigg[ 2 \bigg( \bm{s}_I^{(r)^T} \omega_I^{(j,k)^T}
A_{IJ}^{{(r)}^{-1}} \omega_{J}^{(j,k)} \bm{s}_J^{(r)} \bigg)
\nonumber \\[0.2cm]
& + 4 \bigg( {\bm{w}}^T A_{IJ}^{{(r)}^{-1}} \omega_J^{(j,k)} \bm{s}_J^{(r)} \bigg)
\bigg( \bm{w}^T A_{IJ}^{{(r)}^{-1}} \omega_{I}^{(j,k)} \bm{s}_I^{(r)} \bigg) \Bigg]
S_{IJ}^{\text{FECG}} ~,\end{aligned}$$ where $\bm{w}=A_I^{(r)}\bm{s}_I^{(r)}+A_J^{(r)}\bm{s}_J^{(r)}$, $\epsilon_{ijk}'$ is the Levi-Civita symbol for which only the negative entries are set to zero, and $$\begin{aligned}
\label{zFECG_4}
\omega^{(x,y)}_K = \bar{A}_K^{(r)} \otimes
\left( E_{xy}-E_{yx} \right) \quad {\text{with}} \,\,\, K\in\{I,J\} ~,\end{aligned}$$ with $(E_{ij})_{xy}=\delta_{ix}\delta_{jy}$. Note that the $i,j,$ and $k$ indices are summed with Einstein’s summation convention. We recall that for [apzFECG]{} functions, the vector $\bm{s}_K^{(r)}$ ($K\in\{I,J\}$) must obey the constraint introduced in Eq. (\[zFECG\_0\]) and $\bm{s}_J^{(r)}$ is subject to the rotation operator $\hat{R}(\Omega)$ involving the transformation matrix $U(\Omega)$. Considering Eqs. (\[zFECG\_0\]), (\[zFECG\_0\_1\_fun1\]), (\[zFECG\_0\_1\_fun2\]) and (\[zFECG\_4\]) we have $$\begin{aligned}
\langle\phi_I^{\text{zFECG}}|\hat{N}^2|\phi_{J[N,M_N,p]}^{\text{apzFECG}}\rangle = & \, \tilde{S}_{IJ} \,\, \Xi^{N}_{M_N} ~,\end{aligned}$$ where $$\begin{aligned}
\Xi^{N}_{M_N} = & \, \epsilon_{ijk}' \int \frac{d\Omega}{4\pi^3} \,\, D_{M_NM_N}^{[N]}(\Omega)^* \, e^{C\,\cos\beta} \nonumber \\[0.1cm]
& \Bigg[ 2 \big( \bm{u}_I^{(r)^T} \bar{A}_I^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_J^{(r)} \bm{u}_J^{(r)} \big)
\Big(\bm{e}_z^T(E_{jk}-E_{kj})^T(E_{jk}-E_{kj})\tilde{U}(\Omega)\bm{e}_z\Big) \nonumber \\[0.1cm]
& +4 \bigg( \big( \bm{u}_I^{(r)^T} \bar{A}_I^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_J^{(r)} \bm{u}_J^{(r)} \big)
\big(\bm{e}_z^T(E_{jk}-E_{kj})\tilde{U}(\Omega)\bm{e}_z\big) \nonumber \\[0.1cm]
& + \big(\bm{u}_J^{(r)^T} \bar{A}_J^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_J^{(r)} \bm{u}_J^{(r)} \big)
\big(\bm{e}_z^T\tilde{U}(\Omega)^T(E_{jk}-E_{kj})\tilde{U}(\Omega)\bm{e}_z\big) \bigg) \nonumber \\[0.1cm]
& \times \bigg( \big( \bm{u}_I^{(r)^T} \bar{A}_I^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_I^{(r)} \bm{u}_I^{(r)} \big)
\big(\bm{e}_z^T(E_{jk}-E_{kj})\bm{e}_z\big) \nonumber \\[0.1cm]
& + \big(\bm{u}_J^{(r)^T} \bar{A}_J^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_I^{(r)} \bm{u}_I^{(r)} \big)
\big(\bm{e}_z^T\tilde{U}(\Omega)^T(E_{jk}-E_{kj})\bm{e}_z\big) \bigg) \Bigg] ,
\label{zFECG_4_1}\end{aligned}$$ where $C$ has been defined in Eq. (\[eq:zFECG\_Cval\]).
Furthermore, provided that $(j,k)\in\{(2,3),(3,1),(1,2)\}$ (see Ref. [@Muolo2018b] for a detailed demonstration), we have $$\begin{aligned}
& \bm{e}_z^T(E_{23}-E_{32})^T(E_{23}-E_{32})\tilde{U}(\Omega)\bm{e}_z = \cos\beta \label{eq:zFECG_transfProp1} ~, \\[0.1cm]
& \bm{e}_z^T(E_{31}-E_{13})^T(E_{31}-E_{13})\tilde{U}(\Omega)\bm{e}_z = \cos\beta \label{eq:zFECG_transfProp2} ~, \\[0.1cm]
& \bm{e}_z^T(E_{23}-E_{32})\tilde{U}(\Omega)\bm{e}_z = +\sin\alpha \, \sin\beta \label{eq:zFECG_transfProp3} ~, \\[0.1cm]
& \bm{e}_z^T(E_{31}-E_{13})\tilde{U}(\Omega)\bm{e}_z = -\cos\alpha \, \sin\beta \label{eq:zFECG_transfProp4} ~, \\[0.1cm]
& \bm{e}_z^T\tilde{U}(\Omega)^T(E_{23}-E_{32})\tilde{U}(\Omega)\bm{e}_z = 0 \label{eq:zFECG_transfProp5} ~, \\[0.1cm]
& \bm{e}_z^T\tilde{U}(\Omega)^T(E_{31}-E_{13})\tilde{U}(\Omega)\bm{e}_z = 0 \label{eq:zFECG_transfProp6} ~, \\[0.1cm]
& \bm{e}_z^T(E_{23}-E_{32})\bm{e}_z = 0 \label{eq:zFECG_transfProp7} ~, \\[0.1cm]
& \bm{e}_z^T(E_{31}-E_{13})\bm{e}_z = 0 \label{eq:zFECG_transfProp8} ~, \\[0.1cm]
& \bm{e}_z^T\tilde{U}(\Omega)^T(E_{23}-E_{32})\bm{e}_z = -\sin\alpha \, \sin\beta \label{eq:zFECG_transfProp9} ~, \\[0.1cm]
& \bm{e}_z^T\tilde{U}(\Omega)^T(E_{31}-E_{13})\bm{e}_z = +\cos\alpha \, \sin\beta \label{eq:zFECG_transfProp10} ~,\end{aligned}$$ while it can be shown that for $(j,k)=(1,2)$ all these expressions evaluate to zero.
Eq. (\[zFECG\_4\_1\]) can now be written as $$\begin{aligned}
\Xi^{N}_{M_N} =& \int \frac{d\Omega}{4\pi^3} \,\, D_{M_NM_N}^{[N]}(\Omega)^* \,
\exp\big[{C\cos\beta}\big]
\nonumber \\[0.2cm]
& \times \Bigg[ 2 \big( C\,\cos\beta \big) + \big(C\sin\alpha\,\sin\beta\big) \big(-C\sin\alpha\,\sin\beta\big) \nonumber \\
& + \big(-C\cos\alpha\,\sin\beta\big) \big(C\cos\alpha\,\sin\beta\big) \Bigg] ~,\end{aligned}$$ and its analytical solution to the angular integration for $N=0,1,$ and $2$ yields $$\begin{aligned}
\Xi^{N}_{M_N} =&
\left\{ \begin{array}{lc}
0 & \quad {\text{if}} \quad N=0 \,,\, M_N=0 \\
2 \,\, \Upsilon^1_0 & \quad {\text{if}} \quad N=1 \,,\, M_N=0 \\
6 \,\, \Upsilon^2_0 & \quad {\text{if}} \quad N=2 \,,\, M_N=0 \\
0 & \quad \forall \, N\in\mathbb{N}_0 \,,\, M_N\ne0
\end{array} \right. ~,\end{aligned}$$ where $\Upsilon^{N}_{M_N}$ are the solution of the overlap angular integration given in Eq. (\[zFECG\_0\_3\]). This is in accordance with the expected eigenvalue for the squared total spatial angular momentum $N(N+1)$ in Hartree atomic units. For a list of $\Xi^{N}_{M_N}$ up to $N=5$ see the Appendix.
Projection of the angular momentum onto the $z$ axis
----------------------------------------------------
We recall the $\langle\hat{N}_z\rangle_{IJ}$ matrix elements for FECG functions [@Muolo2018b] $$\begin{aligned}
\langle\phi_I^{\text{FECG}}|\hat{N}_z|\phi_J^{\text{FECG}}\rangle =
\frac{2}{i} \left( \bm{w}^T A_{IJ}^{{(r)}^{-1}} \omega_J^{(1,2)} \bm{s}_J^{(r)} \right)
\left\langle\phi_I|\phi_j\right\rangle ~.\end{aligned}$$ Here, we cannot simplify the expectation value for apzFECGs since $[\hat{R}(\Omega),\hat{N}_z]\ne0$. The term in parenthesis then becomes $$\begin{aligned}
\bm{w}^T A_{IJ}^{{(r)}^{-1}} \omega_J^{(1,2)} \bm{s}_J^{(r)}
=& \bm{s}_I^{(r)^T} A_I^{(r)} A_{IJ}^{{(r)}^{-1}}
\omega_J^{(1,2)} \bm{s}_J^{(r)}
+ \bm{s}_J^{(r)^T} A_J^{(r)}
A_{IJ}^{{(r)}^{-1}} \omega_J^{(1,2)} \bm{s}_J^{(r)} \nonumber \\[0.2cm]
=& \left( \bm{u}_I^{(r)^T} \bar{A}_I^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_J^{(r)} \bm{u}_J^{(r)} \right)
\left(\bm{e}_z^T\tilde{U}(\Omega')(E_{21}-E_{12})\tilde{U}(\Omega)\bm{e}_z\right) \nonumber \\[0.2cm]
& + \left( \bm{u}_J^{(r)^T} \bar{A}_J^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_J^{(r)} \bm{u}_J^{(r)} \right)
\left(\bm{e}_z^T\tilde{U}(\Omega)^T(E_{21}-E_{12})\tilde{U}(\Omega)\bm{e}_z\right) =0 ~.\end{aligned}$$ It follows from Eqs. (\[eq:zFECG\_transfProp5\]) and (\[eq:zFECG\_transfProp6\]) that the latter term is zero, i.e., $\bm{e}_z^T(E_{21}-E_{12})\tilde{U}(\Omega)\bm{e}_z=0$, while the former one is $$\begin{aligned}
\bm{e}_z^T\tilde{U}(\Omega')(E_{21}-E_{12})\tilde{U}(\Omega)\bm{e}_z =
\cos\alpha'\sin\alpha\sin\beta\sin\beta' - \cos\alpha\sin\alpha'\sin\beta\sin\beta' ~.\end{aligned}$$ The resulting expectation value for apzFECG functions reads $$\begin{aligned}
\langle\phi_{I[N,M_N,p]}^{\text{apzFECG}}|\hat{N}_z|\phi_{J[N,M_N,p]}^{\text{apzFECG}}\rangle =&
\, \tilde{S}_{IJ} \int\frac{d\Omega}{4\pi^3} \int\frac{d\Omega'}{4\pi^3} \,\, D_{M_NM_N}^{[N]}(\Omega)^* D_{M_NM_N}^{[N]}(\Omega')^*
\exp\left[C\,\cos\beta\right] \nonumber \\[0.2cm]
& \times \left( \bm{u}_I^{(r)^T} \bar{A}_I^{(r)} \bar{A}_{IJ}^{{(r)}^{-1}}
\bar{A}_J^{(r)} \bm{u}_J^{(r)} \right)
\left[\sin(\alpha-\alpha')(\sin\beta)^2(\sin\beta')^2\right] ~, \end{aligned}$$ which evaluates to zero for every $N$, $M_N$ pairs: $$\begin{aligned}
\langle\phi_{I[N,M_N,p]}^{\text{apzFECG}}|\hat{N}_z|\phi_{J[N,M_N,p]}^{\text{apzFECG}}\rangle
= 0 \quad \forall \,\, N \,\, | \,\, N=(0,1,2,\ldots) , M_N=(-N,\ldots,+N) ~.\end{aligned}$$
This shows that apzFECG functions have zero projection of the total angular momentum on the $z$ axis. The results in this section can be expanded by noting that not only the expectation value of $\hat{N}_z$ is zero, but also the corresponding eigenvalue of the apzFECG functions, $$\begin{aligned}
\hat{N}_z \phi_{I\,[N,M_N,p]}^{\text{apzFECG}} = 0 ~.
\label{eq:apzFECG_eigenfun_Nz}\end{aligned}$$ The derivation of Eq. (\[eq:apzFECG\_eigenfun\_Nz\]) follows from the definition of $\phi_{I\,[N,M_N,p]}^{\text{apzFECG}}$, $\hat{N}_z$, and $P^{[N,p]}_{M_N}$ $$\begin{aligned}
\hat{N}_z \phi_{I\,[N,M_N,p]}^{\text{apzFECG}} =& \hat{N}_z P^{[N,p]}_{M_N}
\phi_I^{\text{zFECG}} \big(\bm{r};A_I^{(r)},\bm{s}_I^{(r)}\big)
\nonumber \\[0.1cm]
=& \hat{N}_z \int \frac{d\Omega}{4\pi^3} ~ D_{M_NM_N}^{[N]}(\Omega)^*
\phi_I^{\text{zFECG}} \Big(\bm{r};A_I^{(r)},U(\Omega)\bm{s}_I^{(r)}\Big)
\nonumber \\[0.1cm]
=& \frac{2}{i} \int \frac{d\Omega}{4\pi^3} ~ D_{M_NM_N}^{[N]}(\Omega)^*
\Big[\bm{r}^T\omega_I^{(x,y)}U(\Omega)\bm{s}_I^{(r)}\Big]
\phi_I^{\text{zFECG}} \Big(\bm{r};A_I^{(r)},U(\Omega)\bm{s}_I^{(r)}\Big) ~,
\label{eq:apzFECG_eigenfun_Nz_1}\end{aligned}$$ and by noting that $$\begin{aligned}
U(\Omega)\bm{s}_I^{(r)} = \bm{u}_I^{(r)} \otimes \tilde{U}(\Omega)\bm{e}_z
= \bm{u}_I^{(r)} \otimes \left(
\begin{array}{c} -\cos{\alpha}\sin{\beta} \\ -\sin{\alpha}\sin{\beta}\cos{\beta} \\ \cos{\beta} \end{array}
\right) ~.\end{aligned}$$ Since $D_{M_NM_N}^{[N]}\propto\exp(-\text{i}M_N\gamma)$ and the right-hand side of Eq. (\[eq:apzFECG\_eigenfun\_Nz\_1\]) do not depend on $\gamma$, the integration over the Euler angles yields zero for all $M_N\ne0$. This shows that Eq. (\[eq:apzFECG\_eigenfun\_Nz\]) is correct, and additionally, we have $$\begin{aligned}
\phi_{I\,[N,M_N,p]}^{\text{apzFECG}}
= \hat{P}_{M_N}^{[N,p]} \phi_I^{\text{zFECG}}
= 0 ~~~ \forall ~ M_N\ne0 ~,\end{aligned}$$ i.e., there is no component of $\phi_I^{\text{zFECG}}$ on the $M_N\ne0$ eigenspaces.
Elimination of center-of-mass contamination
-------------------------------------------
Contributions from the center of mass are eliminated from the expectation values according to the protocol devised in Refs. [@Benjamin2013; @Muolo2018a]. First, the variational matrices $A^{(r)}$ and the variational vectors $\bm{s}^{(r)}$ are manipulated in a given TICC, $A^{(x)}$ and $\bm{s}^{(x)}$, respectively, and defined in block diagonal form $$\begin{aligned}
\bar{A}_I^{(r)} =& \, U_x^{T} \left(
\begin{array}{cc} \mathcal{A}_I^{(x)} & 0 \\ 0 & {c_A} \end{array} \right) U_x ~, \\[0.2cm]
\bm{s}_I^{(r)} =& U_x \left(\begin{array}{cc} \bm{s}_I^{(x)} \\ {\bm{c}_S} \end{array} \right)
= U_x \left(\begin{array}{cc} \bm{u}_I^{(x)} \\ {c_S}_z \end{array}\right)
\otimes \bm{e}_z ~,\end{aligned}$$ where the $N_p-1\times N_p-1$ matrix $\mathcal{A}_I^{(x)}$ and the $N_p-1$ vector $\bm{u}_I^{(x)}$ are related to the internal coordinates, while $c_A$ and ${c_S}_z$ are scalar parameters associated with the center of mass. Note the superscript distinguishing the LFCC set $\{r\}$ from a generic TICC set $\{x\}$. Although the choice of zero for both $c_A$ and ${c_S}_z$ for all $I\in\{1,\ldots,N_b\}$ would systematically cancel center-of-mass contributions from every expectation value, $c_A=0$ leads to a singular matrix $A_I$, which violates the square-integrable and positive-definiteness requirements for the basis functions.
We note that the choice of $c_{A}=1$ and ${c_S}_z=0$, implies that every FECG, zFECG, or apzFECG function is exactly factorizable into a spherical Gaussian centered at the origin for the center-of-mass coordinate, and an FECG function for the $N_p-1$ internal coordinates. In fact, the FECG in (transformed) TICC coordinates $\{x\}$ can be written as $$\begin{aligned}
\phi_I^{\text{FECG}} =& \exp\Bigg[ -
\left(\begin{array}{cc} \bm{x}-\bm{s}_I^{(x)} \\ \bm{x}_{\text{CM}}-{\bm{c}_S} \end{array} \right)^T
\left( \begin{array}{cc} \mathcal{A}_I^{(x)} & 0 \\ 0 & {c_A} \end{array} \right)
\left(\begin{array}{cc} \bm{x}-\bm{s}_I^{(x)} \\ \bm{x}_{\text{CM}}-{\bm{c}_S} \end{array} \right)
\Bigg] \nonumber \\[0.2cm]
=& \exp\left[ -(\bm{x}-\bm{s}_I^{(x)})^T \mathcal{A}_I^{(x)} (\bm{x}-\bm{s}_I^{(x)}) \right]
\exp\Big[-\bm{x}_{\text{CM}}^2\Big] ~.\end{aligned}$$
We chose not to evaluate the integral matrix elements with basis functions and operators in a (transformed) TICC set. Instead, we carry out the integrations straightforwardly in the simple LFCC set and correct *a posteriori* the resulting expression by subtracting center-of-mass dependent terms as described in our previous work. Hence, elimination of center-of-mass contaminations is equivalent to subtraction of the residual $c_A$-terms [@Benjamin2013; @Muolo2018a].
We start detecting $c_{A}$-dependent terms from the $C$ factor. To this aim, we transform it to the TICC sets $\{x\}$ and $\{y\}$, for the $I$-th and $J$-th basis functions, respectively, $$\begin{aligned}
C =& \,2\,\bm{u}_I^{(r)\,T}\bar{A}_I^{(r)}\bar{A}_{IJ}^{{(r)}^{-1}}\bar{A}_{J}^{(r)}\bm{u}_J^{(r)}
\nonumber \\[0.2cm]
=& \,2\,\bm{u}_I^{(r)\,T} \Big[ U_x^{T}\bar{A}_I^{(x)}U_x\bar{A}_{IJ}^{{(r)}^{-1}}
U_y^{T}\bar{A}_I^{(y)}U_y \Big] \bm{u}_J^{(r)}
\nonumber \\[0.2cm]
=& \,2\,\bm{u}_I^{(r)\,T} \left[ U_x^T
\left(\begin{array}{cc} \mathcal{A}_{I}^{(x)} & 0 \\ 0 & {c_A} \end{array}\right)
\left(\begin{array}{cc} \mathcal{A}_{IJ}^{-1} & 0 \\ 0 & \frac{1}{2c_A} \end{array}\right)
\left(\begin{array}{cc} \mathcal{A}_{J}^{(y)} & 0 \\ 0 & {c_A} \end{array}\right)
U_x \right] \bm{u}_J^{(r)}
\nonumber \\[0.2cm]
=& \,2\, \big( \bm{u}_I^{(x)} ~~ {c_S}_z \big)
\left(\begin{array}{cc} \mathcal{A}_{I}^{(x)}\mathcal{A}_{IJ}^{-1}\mathcal{A}_{J}^{(y)}
& 0 \\ 0 & \frac{c_A}{2} \end{array}\right)
\left(\begin{array}{c} \bm{u}_J^{(y)} \\ {c_S}_z \end{array}\right) ~,
\label{eq:C_TICC}\end{aligned}$$ where $\mathcal{A}_{IJ}=\mathcal{A}_I+\mathcal{A}_J$. In the third step, the following mathematical relation is employed [@Muolo2018a] $$\begin{aligned}
U_x\bar{A}_{IJ}^{-1}U_y^T =
\left(\begin{array}{cc} \mathcal{A}_{IJ}^{-1} & 0 \\ 0 & \frac{1}{2c_A} \end{array}\right) ~.\end{aligned}$$ From Eq. (\[eq:C\_TICC\]) it follows that the center-of-mass contributions to $C$ are zero for ${c_S}_z=0$. For this reason, since the expectation value of the total angular momentum squared operator depends solely on $C$ terms, we conclude that it is free of center-of-mass contaminations.
The only center-of-mass dependent term arising in the analytical kinetic energy integral with the favorable choice ${c_S}_z=0$, is the $R$ term defined as $$\begin{aligned}
R = \operatorname{Tr}\left(M A_J^{(r)} A_{IJ}^{{(r)}^{-1}} A_I^{(r)}\right) ~.\end{aligned}$$ The translational contamination can now be eliminated by replacing $$\begin{aligned}
R_{\text{corr.}} = R - \frac{1}{4} {c_A}{c_M} ~,\end{aligned}$$ with $c_M=\sum_{i=0}^{N_p}m_i$ being the total mass of the system. We emphasize that minimization of the energy with respect to translationally invariant parameters only excludes the center-of-mass coordinate, and hence, reduces the original problem for $N_p$ particles to a simpler optimization problem for $N_p-1$ pseudo-particles with lower complexity.
Numerical stability {#SEC:NumStab}
-------------------
We investigate the numerical stability of the analytical matrix elements in finite-precision arithmetic. A naive implementation of the integral expressions results in ill-conditioned overlap and Hamiltonian matrices because of the hyperbolic functions. To restore numerical stability, we introduce normalization for the basis functions, defined as $$\Phi_{I\,[N,M_N,p]}^{\text{apzFECG}} = \frac{\hat{P}^{[N,p]}_{M_N}\phi^{\text{zFECG}}_I}{|\phi_{I\,[N,M_N,p]}^{\text{apzFECG}}|} ~,$$ where the normalization factor is $$\begin{aligned}
|\phi^{[N,M_N]}_I|=\langle
\hat{P}^{[N,p]}_{M_N}\phi^{\text{apzFECG}}_{I[N,M_N,p]}|
\hat{P}^{[N,p]}_{M_N}\phi^{\text{apzFECG}}_{I[N,M_N,p]}\rangle^{\frac{1}{2}} ~.\end{aligned}$$ Matrix elements $\mathcal{O}_{IJ}^{\text{apzFECG}}$ for a generic operator $\hat{O}$ are then evaluated as $$\langle\Phi^{\text{apzFECG}}_{I[N,M_N,p]}|\hat{O}|\Phi^{\text{apzFECG}}_{J[N,M_N,p]}\rangle =
\frac{\langle\hat{P}^{[N,p]}_{M_N}\phi^{\text{zFECG}}_{I[N,M_N,p]} |\hat{O}| \hat{P}^{[N,p]}_{M_N}\phi^{\text{zFECG}}_{J[N,M_N,p]}\rangle}
{|\phi^{\text{apzFECG}}_{I[N,M_N,p]}| |\phi^{\text{apzFECG}}_{J[N,M_N,p]}|} ~.$$ Although the normalization of apzFECGs assures well-conditioned representation matrices for the quantum mechanical operators, extreme $C$ values cause overflow of the hyperbolic sine and cosine functions as well as cancellation errors in the kinetic energy terms because of the high powers of $C$. To remedy these two sources of errors, we differentiate the integral evaluation scheme for different orders of magnitude of $C$ by allowing higher-precision arithmetic to be employed when needed. In particular, we detected possible sources of numerical instabilities for $|C|>700$ when working in double precision floating point arithmetic. However, quadruple precision suffices for achieving the desired accuracy for every test calculations with unconstrained optimization of the variational parameters. While basis functions yielding $|C|>700$ can also be discarded, we prefer the latter strategy to keep the energy function continuous with respect to the variational parameters.
The accuracy and convergence of special functions, i.e., the hyperbolic sine and cosine functions and the generalized incomplete Gamma functions, converge to $0.9~\varepsilon$ for every point without the need to resort to higher precision arithmetics. The latter we implemented for the handling of particularly difficult cases following Ref. [@GenIncompGamma_1994; @GenIncompGamma_1996].
Comparing apzFECGs for $N=0$ and the spherically symmetric (simple) ECG functions, we note that the former require systematically less function evaluations to reach a given accuracy. Simple ECG functions are plagued by problems of linear dependence in the basis during energy optimization of a polyatomic system. In diatomics, there exists a large nuclear density at a distance to the origin in relative coordinates. Simple ECG functions account for this by requiring nearly overlapping terms in the linear combinations with large matching linear coefficients of opposite sign. This near-linear dependency in the basis complicates optimization and yields numerically unstable eigensystems with ill-conditioned Hamiltonian matrices. Conversely, we did not encounter such severe near-linear dependencies with apzFECG functions because these functions can effectively separate the proton densities along an axis.
Numerical results {#SEC:numericalresults}
=================
The formulae derived we implemented in a `C++` computer program. These analytical expressions allow us to calculate matrix elements reliably. Other sources of error such as numerical integration or truncation of infinite series are eliminated by our approach.
As test examples for the novel basis function presented in this work we chose the dihydrogen molecular ion, H$_2^+=\{$p$^+,$p$^+,$e$^-\}$, and dihydrogen, H$_2=\{$p$^+,$p$^+,$e$^-,$e$^-\}$ treated explicitly as three and four-particle systems, respectively. The Born-Oppenheimer approximation is not invoked, i.e., nuclei and electrons are described on equal footing. The energies obtained for the first three rotational states are shown in Tables \[TAB:proj\_zFECG\_1\] and \[TAB:proj\_zFECG\_3\], respectively. For each state, we optimized a different basis sets consisting of $400$ and $600$ zFECG functions, respectively. Matrix elements were calculated as discussed in Sec. \[SEC:matrixelements\] where the projection operator was applied to the ket function. The virial coefficient, $\eta=|1+\langle\Psi|\hat{V}|\Psi\rangle/(2\langle\Psi|\hat{T}|\Psi\rangle)|$ vanishes for the exact solution [@suzukivarga], so that it represents a diagnostic for the overall quality of the variationally optimized wave function. The basis set size was gradually increased following the competitive selection method [@suzukivarga] for which the newer basis functions entering the basis set are selected from a large pool of randomly generated trial functions. A simultaneous refinement of the non-linear variational parameters was crucial to achieve efficient energy convergence. This optimization problem of minimizing the energy with respect to the set of non-linear parameters is a difficult problem as the objective function is non-convex, non-separable, and often (Sec. \[SEC:NumStab\]) ill-conditioned. We relied on two derivative-free algorithms: the Subplex algorithm by Rowan [@Subplex] and the Principal Axis method discussed by Brent [@Praxis]. In our computer implementation of both methods, we used the `NLopt` package [@nlopt]. We employed our multi-channel optimization approach presented in our previous work [@Muolo2018a] and we have included every possible set of Jacobi coordinates, the heavy-particle-centered coordinates, and the center-of-mass-centered coordinates. The construction of the Gaussian parameters through different $U_a^{\text{TICC}}$ maps allows us to explore the parameter space faster and to describe different groupings of the particles with the most appropriate TICC set. These calculations were carried out using message passing interface (MPI) parallelization on six multiprocessor computer platforms (AMD Opteron$^{\texttt{TM}}$ Processor 6376).
We compare the results for H$_2^+$ and H$_2$ with Ref. [@H2+_1] and [@Pachucki2009], respectively. Earlier results obtained with unprojected FECG and numerically projected FECG functions (with three-dimensional shifted centers) for H$_2$ with a basis set size of $N_b=1560$ are $1.162739$ E$_{\text{h}}$ and $1.163998$ E$_{\text{h}}$, respectively [@Muolo2018b]. The wall time of these earlier calculations was about three months. Our best result with only $600$ linearly combined apzFECGs for the rotational ground state of H$_2$ is $-1.16402502482$ E$_{\text{h}}$. Accordingly, the wall time of the calculation was reduced to about two months yielding a result of higher accuracy. Investigating the results in Tables \[TAB:proj\_zFECG\_1\] and \[TAB:proj\_zFECG\_3\], we observe that the energies are well converged with the number of basis functions. The optimized basis-function parameters are deposited in the supplementary material.
Conclusions
===========
Projection techniques increase the effectiveness of variational basis function optimization carried out in the desired eigenspace. The formalism developed in this paper analytically solves the projection based approach for the subset of explicitly correlated floating Gaussian functions having shift vectors aligned on one axis. We have derived analytical expressions of important matrix elements for projected zFECGs with arbitrary angular momentum and parity configurations. The resulting analytically projected zFECGs can potentially target any rotational state. This can be done efficiently because they are eigenfunctions of the total (nuclei plus electrons) squared spatial angular momentum operator $\hat{N}^2$ with eigenvalue $N$ and of $\hat{N}_z$ with eigenvalue $M_N=0$. Since only states with zero total spatial angular momentum projection onto the $z$ axis can be accessed, among the $2N+1$ degenerate states with $M_N=-N,\ldots,+N$, these functions are not suited in applications for which these degeneracies are lifted, e.g., in the presence of external magnetic fields. Despite this limitation, projected zFECGs address the problem of targeting rotationally excited states exactly, whereas other explicitly correlated basis functions either specialize on one specific $N$ considering only lowest-order angular momentum couplings for the ease of the Hamiltonian matrix elements, or resemble the correct partial wave decomposition only for very high linear combinations and in the variational limit with the so-called global vector representation. The numerical examples presented demonstrate the correctness of the derived formulae and the applicability of the approach to excited rotational states of small molecules.
Particularly interesting will be the application of our new analytical projection method to shift vectors lying on a plane and the extension to floating Gaussian functions with pre-exponential factors which can well represent the radial nodes of, for example, pure vibrational states. Such calculations are beyond the scope of the present paper and are therefore deferred to future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been financially supported by ETH Zurich.
List of integrals
=================
This appendix reviews the solutions to the principal integrals of the overlap, kinetic, and total angular momentum squared integral matrix elements for apzFECG functions. All formulas have been checked for consistency against multiple implementations and known special cases ($\bm{s}_I=0$, $C=0$). The list of analytical solutions to the principal integrals for $N\in [0,10]$ is as follows:
\^[0]{}\_[0]{} =& (C) &&
\^[1]{}\_[0]{} =& (C) - (C) &&
\^[2]{}\_[0]{} =& &&
\^[3]{}\_[0]{} =& &&
\^[4]{}\_[0]{} =& &&
\^[5]{}\_[0]{} =& &&
\^[6]{}\_[0]{} =& &&
\^[7]{}\_[0]{} =& &&
\^[8]{}\_[0]{} =& &&
\^[9]{}\_[0]{} =& &&
\^[10]{}\_[0]{} =& &&
\^[0]{}\_[0]{} =& &&
\^[1]{}\_[0]{} =& &&
\^[2]{}\_[0]{} =& &&
\^[7]{}\_[0]{} =& &&
\^[8]{}\_[0]{} =& ( (C) (C (C\^8+630 C\^6+51975 C\^4+945945 C\^2+2027025)\
& -(37 C\^8+8820 C\^6+530145 C\^4+8648640 C\^2+18243225) )\
& +C (C) ((C\^8+702 C\^6+79695 C\^4+2567565 C\^2+18243225)\
& -9 C (4 C\^6+770 C\^4+30030 C\^2+225225) ) \] &&
\^[9]{}\_[0]{} =& &&
\^[10]{}\_[0]{} =& ((C) ( C (C\^[10]{}+1485 C\^8+315315 C\^6+18918900 C\^4\
& +310134825 C\^2+654729075) -(56 C\^[10]{}+30195 C\^8+4414410 C\^6+224324100 C\^4\
& +3445942500 C\^2+7202019825) ) +C (C) ( (C\^[10]{}+1595 C\^8+418275 C\^6\
& +35945910 C\^4+1045269225 C\^2+7202019825)\
& -55 C (C\^8+468 C\^6+51597 C\^4+1670760 C\^2+11904165) ) ) &&
\^[0]{}\_[0]{} =& 0 &&
\^[1]{}\_[0]{} =& 2 \^1\_0 &&
\^[2]{}\_[0]{} =& 6 \^2\_0 &&
\^[3]{}\_[0]{} =& 12 \^3\_0 &&
\^[4]{}\_[0]{} =& 20 \^4\_0 &&
\^[5]{}\_[0]{} =& 30 \^5\_0 &&
[54]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase 10.1098/rspa.1960.0195) [****, ()](\doibase 10.1098/rspa.1960.0196) @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevA.15.1) @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevA.19.2360) [****, ()](\doibase 10.1063/1.445672) [****, ()](\doibase 10.1063/1.451543) [****, ()](\doibase 10.1063/1.452951) [****, ()](\doibase 10.1063/1.458755) [****, ()](\doibase http://dx.doi.org/10.1063/1.464293) @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevA.61.064503) @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevA.86.052514) [****, ()](\doibase
10.1021/cr200419d) [****, ()](\doibase 10.1103/PhysRevA.92.062501) [****, ()](\doibase 10.1007/BF01340013) [****, ()](\doibase 10.1007/BF01506430) [****, ()](\doibase 10.1007/BF01375457) [****, ()](\doibase 10.1007/BF01421744) [****, ()](\doibase 10.1007/BF01397263) [****, ()](\doibase 10.1103/PhysRevA.8.700) [****, ()](\doibase 10.1103/PhysRevA.36.1013) [****, ()](\doibase 10.1063/1.3569565) [****, ()](\doibase 10.1103/RevModPhys.85.693) [****, ()](\doibase 10.1103/PhysRevC.52.2885) [****, ()](http://www.ncbi.nlm.nih.gov/pubmed/9913088) [****, ()](\doibase
https://doi.org/10.1016/S0009-2614(01)00563-2) [****, ()](\doibase 10.1063/1.2894866) [****, ()](\doibase
10.1103/PhysRevA.80.062510) [****, ()](\doibase 10.1063/1.4826450) @noop [****, ()]{} [****, ()](\doibase 10.1063/1.3419931) [****, ()](\doibase 10.1063/1.3523348) [****, ()](\doibase 10.1063/1.3591836) [****, ()](\doibase 10.1103/PhysRevA.83.012506) @noop [****, ()]{} [****, ()](\doibase 10.1063/1.5050462) @noop [**]{} (, ) @noop [****, ()]{} [****, ()](\doibase 10.1063/1.5009465) [****, ()](\doibase
10.1002/(SICI)1097-461X(1996)57:2<141::AID-QUA1>3.0.CO;2-Y) [****, ()](\doibase 10.1063/1.1701524) @noop [**]{} (, ) [****, ()](\doibase 10.1063/1.1457435) [****, ()](\doibase https://doi.org/10.1016/0377-0427(94)90187-2) [****, ()](\doibase https://doi.org/10.1016/0377-0427(95)00018-6) @noop [**]{} (, ) @noop [**]{} (, ) @noop [ ]{} [****, ()](\doibase 10.1103/PhysRevA.74.052506) @noop [****, ()]{}
|
---
abstract: 'We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional if there does not exist any $m$-infinity element in $\mathfrak B(V)$; (ii) Nichols Lie algebra $\mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.'
address:
- |
Weicai Wu\
Department of Mathematics\
Hunan University\
Changsha 410082\
P.R. China\
[email protected]
- |
Shouchuan Zhang\
Department of Mathematics\
Hunan University\
Changsha 410082\
P.R. China\
[email protected]
- |
Yao-Zhong Zhang\
School of Mathematics and Physics\
The University of Queensland\
Brisbane 4072\
Australia\
[email protected]
author:
- 'Weicai Wu, Shouchuan Zhang and Yao-Zhong Zhang'
title: Relationship between Nichols braided Lie algebras and Nichols algebras
---
Introduction {#s0}
============
The theory of Lie superalgebras has been developed systematically, which includes the representation theory and classifications of simple Lie superalgebras and their varieties [@Ka77] . In many physical applications or in pure mathematical interest, one has to consider not only $\mathbb Z_2$– or $\mathbb Z$– grading but also G-grading of Lie algebras, where G is an abelian group equipped with a skew symmetric bilinear form given by a 2-cocycle. Lie algebras in symmetric and more general categories were discussed in [@GRR95; @Gu86; @ZZ04]. A sophisticated multilinear version of the Lie bracket was considered in [@Kh99a; @Pa98]. Various generalized Lie algebras have already appeared under different names, e.g. Lie color algebras, $\epsilon $ Lie algebras [@Sc79], quantum and braided Lie algebras [@Ma94; @KS97], generalized Lie algebras [@BFM96] and H-Lie algebras [@BFM01]. In [@Ar11], a Milnor–Moore type theorem for primitively generated braided bialgebras was obtained by means of braided Lie algebras. The question of finite-dimensionality of Nichols algebras dominates an important part of the recent developments in the theory of (pointed) Hopf algebras. The interest in this problem comes from the Lifting Method by Andruskiewitsch and Schneider to classify finite (Gelfand-Kirillov) dimensional pointed Hopf algebras, which are generalizations of quantized enveloping algebras of semi-simple Lie algebras. The classification of finite dimensional pointed Hopf algebras was studied in [@AS02; @AHS08; @AS10; @He05; @He06a; @He06b; @WZZ].
This paper provides a new method to determine whether a Nichols algebra is finite dimensional or not.
Let $\mathfrak B(V)$ be the Nichols algebra of vector space $V$. Let $\mathfrak L(V)$ , $\mathfrak L^-(V)$ and $\mathfrak L_c(V)$ denote the braided Lie algebras generated by $V$ in $\mathfrak B(V)$ under Lie operations $[x, y] = yx - p_{yx}xy$, $[x, y]^- = xy - yx$ and $[x, y]_c = xy - p_{xy}yx$, respectively, for any homogeneous elements $x, y \in \mathfrak B(V)$. $(\mathfrak L(V), [\ ])$, $(\mathfrak L^-(V), [\ ]^-)$ and $(\mathfrak L_c(V), [\ ]_c)$ are called Nichols braided Lie algebra, Nichols Lie algebra and Nichols braided m-Lie algebra of $V$, respectively. It is clear that $(\mathfrak L(V), [ \ ])$ and $(\mathfrak L_c(V), [ \ ]_c)$ are equivalent as vector spaces. If $\mathfrak B(V)$ is finite dimensional then $\mathfrak B(V)$ is nilpotent, so $(\mathfrak L(V), [\ ])$ and $(\mathfrak L^-(V), [\ ]^-)$ also are nilpotent.
In this paper we prove the following two results: (i) $\mathfrak B(V)$ is finite-dimensional if and only if $\mathfrak L(V)$ is finite-dimensional when there does not exist any $m$-infinity element; (ii) $\mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.
This paper is organized as follows. In section \[s1\] we recall some results on Nichols algebras and fix the notation. In section \[s2\] we show that $\mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. In section \[s3\] we prove that $\mathfrak B(V)$ is finite-dimensional if and only if $\mathfrak L(V)$ is finite-dimensional when there does not exist any $m$-infinity element in $\mathfrak B(V)$. In section \[s4\] we present the condition for $\mathfrak B(V) = F\oplus \mathfrak L(V).$ In section \[s5\] we give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.
Throughout, $\mathbb Z =: \{x | x \hbox { is an integer }\}.$ $\mathbb N_0 =: \{x | x \in \mathbb Z, x\ge 0\}.$ $\mathbb N =: \{x | x \in \mathbb Z, x>0\}.$ $F$ denotes the base field of characteristic zero.
Preliminaries {#s1}
==============
In this section we recall some results on Nichols algebras ( see [@AHS08]).
\[1\] (see [@AHS08])If $(V, \alpha, \delta )$ is a $FG$-[YD]{} module, then tensor algebra $T(V)$ over $V$ is a $FG$-${\rm YD}$ module.
.1in If $\{x_1, \cdots, x_n\}$ is a basis of vector space $V$ and $C(x_i\otimes x_j) = q_{ij} x_j\otimes x_i$ with $q_{ij} \in F$, then $V$ is called a braided vector space of diagonal type, $\{x_1, \cdots, x_n\}$ is called canonical basis and $(q_{ij})_{n\times n}$ is called braided matrix. Throughout this paper all of braided vector spaces are connected and of diagonal type without special announcement. Let $G= \mathbb Z^n$ and $E=\{e_{1}, e_{2}, \cdots, e_{n}\}, e_i =: ( \stackrel{i} { \overbrace{0, \cdots 0, 1}}, \cdots 0)\in G$ , $1\le i \le n.$ Let $\chi$ be a bicharacter of $G$ such that $\chi (e_i, e_j) = p_{ij}$ and $C(x_i\otimes x_j) = \chi (e_i, e_j) x_j\otimes x_j$. Let $S_{m}\in End_{k}(T(V)^{m})$ and $S_{1, j}\in End_{k}(T(V)^{j+1})$ denote the maps $S_{m}=\prod \limits_{j=1}^{m-1}(id^{\bigotimes m-j-1}\bigotimes S_{1, j})$ , $S_{1, j}=id+C_{12}^{-1}+C_{12}^{-1}C_{23}^{-1}+\cdots+C_{12}^{-1}C_{23}^{-1}\cdots C_{j, j+1}^{-1}$ (in leg notation) for $m\geq 2$ and $j\in \mathbb N$. Then the subspace $S=\bigoplus \limits _{m=2}^{\infty} ker S_{m}$ of the tensor $T(V)=\bigoplus \limits _{m=0}^{\infty} T(V)^{\bigotimes m}$ is a two-sided ideal, and algebra $\mathfrak B(V)=T(V)/S$ is termed the Nichols algebra associated to $(V, C)$. Define linear map $p$ from $\mathfrak B(V) \otimes \mathfrak B(V) $ to $F$ such that $p(u\otimes v) = \chi (deg (u), deg (v)), $ for any homogeneous element $u, v \in \mathfrak B(V).$ For convenience, $p(u\otimes v)$ is denoted by $p_{uv}$. Let $A =: \{x_1, x_2, \cdots, x_n\}$ be an alphabet, $A^*$ the set of all of words in $A$ and $A^+ =: A^*\setminus 1$. Define $x_1 < x_2< \cdots < x_n$ and the order on $A^*$ is the lexicographic ordering. For the concept of words refer [@Lo83]. Let $| u| $ denote the length of word $u$.
\[2\] () A word $u$ is called a Lyndon word if $| u| =1$ or $| u| \geq2$, and for each representation $u=u_1u_2$, where $u_1$and $u_2$ are nonempty words, the inequality $u<u_2u_1$ holds.
Any word $u\in A^*$ has a unique decomposition into the product of non-increasing sequence of Lyndon words by . If $u$ is a Lyndon word with $| u | >1$, then there uniquely exist two Lyndon words $v$ and $w$ such that $u =vw$ and $v$ is shortest (see [@Lo83 Prop. 5.1.3] and [@He07]) (the composition is called the Shirshov decomposition of $u$).
\[3\] We inductively define a linear map $[\ \ ]$ from $A^+$ to $\mathfrak B(V)$ as follows: (1) $[u] =: u$ when $u$ is a letter; (2) $[u] =: [w][v]-p_{wv}[v][w]$ when $u$ is a Lyndon word with $| u | >1$ and $u =vw$ is a Shirshov decomposition; (3) $[u] =: [[[l_1, l_2], l_3] \cdots l_m] $, when $u = l_1l_2 \cdots l_m$ is a non-increasing product of Lyndon words, i.e. $l_1 \ge l_2 \ge l_3 \ge \cdots \ge l_m $, and $l_i$ is a Lyndon word for any $1\le i \le m$.
Similarly, we inductively define a linear map $[\ \ ]^-$ from $A^+$ to $\mathfrak B(V)$ as follows: (1) $[u]^- = : u$ when $u$ is a letter; (2) $[u]^- = : [w]^-[v]^- -[v]^-[w]^-$ when $u$ is a Lyndon word with $| u | >1$ and $u = vw$ is a Shirshov decomposition; (3) $[u]^- = : [[[l_1, l_2]^-, l_3]^- \cdots l_m]^- $, when $u = l_1l_2 \cdots l_m$ is a non-increasing product of Lyndon words.
$[u]$ is called a nonassociative word for any $u \in A^+$, $[u]$ is called a standard nonassociative word if $u$ is a Lyndon word. Every standard nonassociative word is also called a super-letter.
\[5\] () A super-letter $[u]$ is said to be hard if it is not a linear combination of products $[u_1][u_2]\cdot\cdot\cdot[u_i], i\in
\mathbb N$, where $[u_j]$ are super-letter with $[u]<[u_j]$, $1\le j \le i$.
\[6\] ([@Kh99b Def. 7] or ) We say that the height of a super-letter $[u]$ with degree $d$ equals a natural number $h$ if $h$ is least with the following properties:
\(1) $p_{uu}$ is a primitive root of unity of degree $t\ge 1, $ and $h=t$ ;
\(2) super-word $[u]^h$ is a linear combination of super-words of degree $hd$ in greater super-letters than $[u].$
If the number $h$ with above properties does not exist then we say that the height of $[u]$ is infinite.
Let $h_u$ denote the height of $u$. Let ${\rm ord } (p_{uu})$ denote the order of $p_{uu}$ with respect to multiplication. $D =: \{[u] \mid [u] \hbox { is a hard super-letter }\}$. If $[u] \in D$ and ${\rm ord } (p_{u, u}) =m>1$ with $h_u = \infty$, then $[u]$ is called an $m$-infinity element. $P = :\{[u_{1}]^{k_1}[u_{2}]^{k_2}\cdots [u_{s}]^{k_s}\ \mid \ [u_{i}]\in D,
k_i, s \in \mathbb N_0; 0 \le k_i < h_{u_i}; 1\le i \le s; u_s<u_{s-1}<\cdots< u_1\}$. $\Delta ^+(\mathfrak B(V)): = \{ \deg (u) \mid [u]\in D\}$. $ \Delta (\mathfrak B(V)) := \Delta ^+(\mathfrak B(V)) \cup \Delta ^-(\mathfrak B(V))$, which is called the root system of $V.$ If $ \Delta (\mathfrak B(V))$ is finite, then it is called an arithmetic root system. Let $E_e' =: \frac {1}{2} \mid \{ ([u], [v])\in D\times D \mid [u], [v] \in D, p_{u, v} p_{v, u} \not=1\}\mid $. Let $D^- : = \{ [u]^- \mid [u] \in D\}$ and ${\deg } (D^-) : = \{ \deg ([u]^- ) \mid [u] \in D\}$. Let $E_e$ denote the number of edges of generalized Dynkin diagram. If $u=vw$ is the shirshov decomposition of $[u] \in D$, then $[v]$, $[w] \in D$, which are called sons of $u$. If $[u_1], [u_2], \cdots, [u_m]\in D$ and $u_{i+1}$ is a son of $u_i$ for $1\le i \le m-1$, then $u_2, u_3, \cdots, u_m$ are called descendants of $u_1.$
There does not exist any $m$-infinity element in $\mathfrak B(V)$ if and only if Property (P) in holds.
\[8\] ( or ) $P$ is a basis of $\mathfrak B (V)$.
Relationship between Nichols algebras and Nichols Lie algebras {#s2}
==============================================================
In this section it is proved that Nichols Lie algebra $\mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite.
\[63\] Assume that $u = vw$ is a Shirshov decomposition of $u$. If $[u]\in D$, then $[[v], [w]]^-\neq0$. Furthermore, if $[v], [w] \in \mathfrak L ^-(V)$ (e.g. $\mid u \mid = 2$), then $[u]^- \not = 0.$
If $[[v], [w]]^- = 0$, then $[v][w] = [w][v]$, we know $[[v], [w]] = [w][v]-p_{wv}[v][w] = (1-p_{wv})[w][v]$, it contradicts to $[u] \in P$ and $ [w][v]\in P$.
\[63’\] (i) $\dim \mathfrak L^-(V) \ge \mid \deg (D^-)\mid -1 \ge n + E_e -1$, where $E_e$ is the number of edges in generalized Dynkin diagram of $V$. (ii) If $D^-$ is infinite, then $ \dim \mathfrak L^-(V) = \infty$.
If $u_1, u_2, \cdots, u_m$ in $D^- \setminus 0$ with different degrees, then $u_1, u_2, \cdots, u_m$ is linearly independent. It is clear that $D^- \subseteq L^-(V).$ Consequently, $\dim L^-(V) \ge \mid \deg ( D^-)\mid -1 $. Obviously, there exists a line between $x_i$ and $x_j$ if and only if $[x_ix_j]\in D$ with $i<j$, which implies $\mid \deg (D^-)\mid -1 \ge n + E_e -1$.
Relationship between Nichols algebras and Nichols braided Lie algebras {#s3}
======================================================================
In this section it is proved that $\mathfrak B(V)$ is finite-dimensional if and only if $\mathfrak L(V)$ is finite-dimensional when there does not exist any $m$-infinity element in $\mathfrak B(V)$. Let $l_u(v) := [u, v]$ and $r_u(v) := [v, u]$ for any $u, v\in \mathfrak B(V).$
\[7\] If $[u]$ is a nonassociative word, then $[u] \in \mathfrak L(V)$.
By the definition of nonassociative words, we have $[u] \in \mathfrak L(V)$.
If $| D | =\infty$, then $\dim \mathfrak L(V)=\infty$.
\[11\] If $[u]$ is a nonassociative word with $ t \in \mathbb N$, $t \le ord (p_{uu})$, then $[u]^t \in \mathfrak L(V)$.
Let $l_{[u]}^{0}[u]=:[u], l_{[u]}^{i}[u]=:[ [u], l_{[u]}^{i-1}[u]], i\geq1$. Obviously $l_{[u]}^{i}[u] \in \mathfrak L(V)$. It is clear $l_{[u]}^{1}[u]=[[u],
[u]]=[u]^2-p_{uu}[u]^2=(1-p_{uu})[u]^2$. By means of induction, we obtain $l_{[u]}^k[u]=(1-p_{uu})(1-p_{uu}^2)\cdot\cdot\cdot(1-p_{uu}^{k})[u]^{k+1}, \forall \ 1<k\in
\mathbb N $. We have $(1-p_{uu})(1-p_{uu}^2)\cdot\cdot\cdot(1-p_{uu}^{t-1}) \not=0$ since $t \le ord (p_{uu})$, which implies $[u]^t \in \mathfrak L(V).$
\[9\] If there does not exist any $m$-infinity element in $\mathfrak B(V)$ and $1<ord (p_{uu}) <\infty$ for any $u \in D$, then the following conditions are equivalent: (i) $\mathfrak B(V)$ is finite-dimensional; (ii) $\mathfrak L(V)$ is finite-dimensional; (iii) $\Delta (\mathfrak B(V))$ is an arithmetic root system.
It follows from that (i) and (iii) are equivalent. $(i)\Longrightarrow (ii)$. Assume that $\mathfrak B(V)$ is finite-dimensional. Since $\mathfrak L(V)\subseteq \mathfrak B(V)$, we have that $\mathfrak L(V)$ are finite-dimensional. $(ii)\Longrightarrow (i)$. Assume that $\mathfrak L(V)$ is finite-dimensional. By Lemma \[7\], $D \subseteq \mathfrak L(V).$ Obviously, $D \subseteq P$. Therefore $D$ is linearly independent and $| D | \le \dim \mathfrak L(V) < \infty,$ $h_{u} <\infty$ since $1<ord (p_{uu}) <\infty$ for $[u] \in D$. It follows from Theorem \[8\] that $\dim \mathfrak B(V) < \infty$.
\[63”’\] Assume that $V$ is a Cartan type with generalized Cartan matrix $(a_{ij})_{n\times n}$ and $1< ord (p_{uu}) < \infty $ for any $[u] \in D$. If there does not exist any $m$-infinity element in $\mathfrak B(V)$, then the following conditions are equivalent. (i) $\mathfrak L(V)$ is finite dimensional; (ii) $(a_{ij})_{n\times n}$ is a Cartan matrix; (iii) $\dim \mathfrak B(V)< \infty.$
It follows from [@He05 Th. 2.10.2], Theorem \[9\] and Lemma \[7\].
\[16\] If there exists $ [u]\in D$ such that ${\rm ord } (p_{uu})=\infty$, then $\dim \mathfrak B(V)=\infty$ and $\dim \mathfrak L(V)=\infty$.
By Theorem \[8\], $\dim \mathfrak B(V)=\infty$. By Lemma \[11\], $\dim \mathfrak L(V)=\infty$.
\[21\] If there exist $[u],[v]$ such that $p_{uu}^{i}p_{uv}p_{vu}\neq 1$ for $\ \forall \ 0\leq i\leq 2k-2$, $\ \forall \ k\in \mathbb N $, then $[v][u]^k,[u][v][u]^{k-1},\ldots,[u]^{k}[v]\in \mathfrak L(V)$.
We first show $$\begin{aligned}
\label {e5} &&p_{uu}^{k-t+i}p_{uv}r_{[u]}^{t-i-1}([u]^{i}l_{[u]}^{k-t+1}[v])+r_{[u]}^{t-i}([u]^{i}l_{[u]}^{k-t}[v])
\nonumber \\
&&~~~~~~~~~~~~~~~~~~~~= (1-p_{uu}^{2(k-t)+i}p_{uv}p_{vu})r_{[u]}^{t-i-1}([u]^{i+1}l_{[u]}^{k-t}[v])\end{aligned}$$ for $\ \forall \ 1\leq t \leq k$ , $\ \forall \ 0\leq i \leq t-1$. In fact, $$\begin{aligned}
{\rm left~ hand~ side ~ of~ } (\ref {e5}) &=& p_{uu}^{k-t+i}p_{uv}r_{[u]}^{t-i-1}([u]^{i}[[u],
l_{[u]}^{k-t}[v]])+r_{[u]}^{t-i-1}([[u]^{i}l_{[u]}^{k-t}[v], [u]])\\
&=&r_{[u]}^{t-i-1}\left( p_{uu}^{k-t+i}p_{uv}[u]^{i}l_{[u]}^{k-t}[v][u]
-p_{uu}^{k-t+i}p_{uv}[u]^{i}l_{[u]}^{k-t}[v][u]\right. \\
& &\left.-p_{uu}^{k-t+i}p_{uv}p_{uu}^{k-t}p_{vu}[u]^{i+1}l_{[u]}^{k-t}[v]+[u]^{i+1}l_{[u]}^{k-t}[v]
\right)\\
&=& (1-p_{uu}^{2(k-t)+i}p_{uv}p_{vu})r_{[u]}^{t-i-1}([u]^{i+1}l_{[u]}^{k-t}[v])\\
&=& {\rm right~ hand~ side~ of ~} (\ref {e5}).\end{aligned}$$
Let $B^{(i)}=(b^{(i)}_{rs})_{(k-i+1)\times (k-i+1)}$ be real matrices such that
$ \left(\begin{array}{ccccccc}
[u]^{i}l_{[u]}^{k-i}[v] \\
r_{[u]}^{1}([u]^{i}l_{[u]}^{k-i-1}[v])\\
r_{[u]}^{2}([u]^{i}l_{[u]}^{k-i-2}[v]) \\
\cdots\\
r_{[u]}^{k-1}([u]^{i}[v]) \\
\end{array}\right) $ $ $= $ B^{(i)} $ $\left( \begin{array}{c}
[u]^{i}[v][u]^{k-i}\\
[u]^{i+1}[v][u]^{k-i-1}\\
[u]^{i+2}[v][u]^{k-i-2}\\
\cdots \\
[u]^{k}[v] \\
\end{array}\right)$ ${}
$ and $b_{11}^{(i)}=1$ for $0\le i \le t-1$.
We know $| B^{(i)}| =\prod \limits _{t=i+1}^{k}(1-p_{uu}^{2(k-t)+i}p_{uv}p_{vu})| B^{(i+1)}| $ by (\[e5\]). Consequently, $| B^{(0)}| =\prod \limits _{i=0}^{k-1}\prod \limits _{t=i+1}^{k}(1-p_{uu}^{2(k-t)+i}p_{uv}p_{vu})$ and $| B^{(0)}| =0$ is equivalent to $\prod \limits _{i=0}^{2k-2}(1-p_{uu}^{i}p_{uv}p_{vu})=0$. This completes the proof.
.1in According to the above Proposition, we obtain immediately,
\[22\] If there exist $[u], [v] \in D$ such that $p_{u, u}=1$ and $p_{uv}p_{vu}\neq 1$, then $\dim \mathfrak B(V)=\infty, \ \dim \mathfrak L(V)= \infty$ .
\[23\] If $[u], [v]\in D,$ such that $p_{u, u}=1$ and $p_{uv}p_{vu}=1$, then $d_{1}d_{2}\cdots d_{k}[v]\in Fl_{[u]}^{k}[v]$ for $ \forall \ d_{i}=l_{[u]}$ or $r_{[u]}, 1\leq i \leq k$.
We know $$\begin{aligned}
\label {e7}
p_{uv}r_{[u]}^{s}l_{[u]}^{k-s}[v]+r_{[u]}^{s+1}l_{[u]}^{k-s-1}[v]=(1-p_{uv}p_{vu})[u]r_{[u]}^{s}l_{[u]}^{k-s-1}[v]\end{aligned}$$ for $\ \forall \ 0\leq s < k$ by Definition \[5\]. Then $l_{[u]}^{k}[v]=-p_{vu}r_{[u]}^{1}l_{[u]}^{k-1}[v]=\ldots=(-p_{vu})^{k}r_{[u]}^{k}[v]. $ On the other hand, $r_{[u]}l_{[u]}[v]=l_{[u]}r_{[u]}[v]$. This proves the corollary.
\[41\] If there exists $[u]\in D$ such that $p_{uv}p_{vu}=1$ for $ \ \forall v \in D $ with $ [u]\neq [v]$, then $\Delta(\mathfrak B(V)) $ is not an arithmetic root system while $\mathfrak B(V) $ is connected Nichols algebra of diagonal type with $\dim V >1$. Moreover, $\dim \mathfrak B(V)=\infty, \ \dim \mathfrak L(V)= \infty$.
There exists a basis $\pi$ of $\Delta ( \mathfrak B (V))$ such that $\deg u \in \pi.$ Since $\mathfrak B(V) $ is connected Nichols algebra and $n = \mid \pi \mid >1,$ there exists $\beta \in \pi\setminus \{\deg u\}$ such that $\chi (\deg u, \beta) \chi (\beta, \deg u) \not=1.$ By the definition of $\Delta ( \mathfrak B (V))$ there exists $[v] \in D\setminus \{u\}$ with $\deg v \in \{ \beta, - \beta\}.$ This yields a contradiction to $p_{uv}p_{vu} =1.$
\[51\] If $\mathfrak B(V) $ is connected Nichols algebra of diagonal type with $\dim V>1$ and there does not exist any $m$-infinity elements, then $\mathfrak B(V)$ is finite-dimensional if and only if $\mathfrak L(V)$ is finite-dimensional.
It follows from Proposition \[9\], Proposition \[16\] , Corollary \[22\] and Proposition \[41\].
.1in By [@ZZ04], $(\mathfrak B(V), [\ ]_c) $ is a braided m-Lie algebra and we have the braided Jacobi identity as follows: $$\begin{aligned}
\label {e2}
[ [u, v], w]= [u, [v, w]] +p_{vw}^{ -1} [ [u, w], v]
+(p_{wv} -p_{vw}^{ - 1}) v\cdot [u, w].\end{aligned}$$
\[12\] If $u$ and $ v$ are homogeneous elements in $\mathfrak L(V)$ with $p_{uv}p_{vu}\not = 1$, then $uv, vu \in \mathfrak L(V)$. Furthermore, if $u, v \in \mathfrak L (V)$, then $ [u, v]^- \in \mathfrak L(V).$
$[u, v] = vu -p_{v, u} uv$ and $[v, u] = uv -p_{u, v} vu$, which implies that $uv$ and $vu$ are a linear combination of $[u, v]$ and $[v, u]$.
\[3.33\] $\dim \mathfrak L(V) \ge \sum _{[u] \in D} (h_u -1) + E_e'. $
It follows from Lemma \[11\] and Lemma \[7\].
.1in Recall the dual $\mathfrak B(V^*) $ of Nichols algebra $\mathfrak B(V) $ of rank $n$ in [@He05 Section 1.3] and [@He06b]. Let $y_{i}$ be a dual basis of $x_{i}$. $\delta (y_i) = g_i ^{-1} \otimes y_i$, $g_i \cdot y_j = p_{ij}^{-1} y_j $ and $\Delta (y_i) = g_i ^{-1} \otimes y_i +y_i \otimes 1.$ There exists a bilinear map $<, >$ from $(\mathfrak B(V^*) \# kG) \times \mathfrak B(V) $ to $\mathfrak B(V) $ such that $<y_i, uv> = <y_i, u>v +g_i^{-1}.u<y_i, v>$ and $<y_i, <y_j, u>> = <y_iy_j, u>$ for any $u, v\in \mathfrak B(V) $. Furthermore, for any $u\in \oplus _{i=1}^\infty \mathfrak B(V)_{(i)}$, one has that $u=0$ if and only if $<y_i, u> = 0$ for any $1\leq i \leq n.$ Let $i$ denote $x_{i}$ in short, sometimes.
\[14\] Let $l_{i}^{0}[j]=[j]$ , $l_{i}^{k}[j]=[i, l_{i}^{k-1}[j]]$ , $r_{i}^{0}[j]=[j]$ , $r_{i}^{k}[j]=[r_{i}^{k-1}[j], i]$ , $k\geq 1$ . Then we have
\(i) $<y_j, l_{i}^{k}[ j]>=0, <y_i, r_{i}^{k}[ j]>=0, \forall\ k\geq 1$;
\(ii) the following conditions are equivalent: (1) $l_{i}^{k}[j]=0$ ; (2) $r_{i}^{k}[j]=0$ ; (3) $(k)_{p_{ii}}^{!}\prod \limits _{t=0}^{k-1}(p_{ii}^{t}p_{ji}p_{ij}-1)=0$ .
\(i) It is clear $<y_j, l_{i}^{k}[ j]>=0$, $ <y_i, r_{i}^{k}[ j]>=<y_i,
ir_{i}^{k-1}[ j]-p_{ii}^{k-1}p_{ij}r_{i}^{k-1}[ j]i>
=r_{i}^{k-1}[ j]-p_{ii}^{k-1}p_{ij}p_{ii}^{-(k-1)}p_{ij}^{-1}r_{i}^{k-1}[ j]=0$.
\(ii) By means of induction, we obtain $<y_{i}, l_{i}^{k}[j]>=p_{ii}^{-(k-1)}p_{ij}^{-1}(1-p_{ii}^{k-1}p_{ij}p_{ji})(1+p_{ii}
+\cdots p_{ii}^{k-1})l_{i}^{k-1}[j]$, then $<y_{j}y_{i}^k, l_{i}^{k}[j]>=p_{ii}^{-\sum \limits_{i=1}^{k-1}i}p_{ij}^{-k}\prod
\limits _{t=0}^{k-1}(1-p_{ii}^{t}p_{ij}p_{ji})(1+p_{ii}+\cdots p_{ii}^{t})$, (1) is equivalent to (3) by (i). On the other hand, $<y_{j}, r_{i}^{k}[j]>=p_{ji}^{-k}\prod \limits _{t=0}^{k-1}(1-p_{ii}^{t}p_{ij}p_{ji})[i]^k$, $<y_{i}^{k}y_{j}, r_{i}^{k}[j]>=p_{ii}^{-\sum \limits_{i=1}^{k-1}i}p_{ji}^{-k}\prod
\limits _{t=0}^{k-1}(1-p_{ii}^{t}p_{ij}p_{ji})(1+p_{ii}+\cdots p_{ii}^{t})$. One knows that (2) is equivalent to (3) by (i). This proves the lemma.
Conditions for $\mathfrak B(V) = F\oplus \mathfrak L(V).$ {#s4}
=========================================================
In this section we give the sufficient conditions for $\mathfrak B(V) = F\oplus \mathfrak L(V).$
\[13\] () [(i)]{} If $| u | = | v |$, then $u<v$ if and only if $uw < vw.$ [(ii)]{} If $u=vw$ is the Shirshov decomposition of Lyndon word $u$ and $[u]$ is hard, then both $[v]$ and $[w]$ are hard too.
\[10\] If there exist $x_{i}, x_{j}, i\neq j$ such that $p_{ij}p_{ji}=1$, then $\mathfrak B(V) \neq F\oplus \mathfrak L(V).$
It is clear $[x_{i}, x_{j}]=[x_{j}, x_{i}]=0$ and $x_{i}x_{j}=p_{ij}x_{j}x_{i}\neq 0$. Then $x_{i}x_{j}$ or $x_{j}x_{i}\in P$ and $x_{i}x_{j}, x_{j}x_{i}\notin \mathfrak L(V).$
\[60\] If $\mathfrak B(V) $ is connected Nichols algebra of rank $>3$ of diagonal type and $\Delta(\mathfrak B(V)) $ is arithmetic root systems, then $\mathfrak B(V) \neq F\oplus \mathfrak L(V).$
It is clear from , , and .
\[50\] If $\begin{picture}(100, 20)
\put(10, 1) {\makebox(0, 0) [t]{$\bullet$}}
\put(80, 1){\makebox(0, 0) [t]{$\bullet$}}
\put(10, -1) {\line(1, 0) {70}}
\put(5, 9) {$\zeta$}
\put(40, 4) {$-\zeta$}
\put(75, 9) {$-1$}
\put(90, 1) {$, $}\ \ \ \put(95, 1) {$ \zeta \in R_3$,}
\end{picture}$\
then $D=\{ [x_1], [x_2], [x_1, x_2], [x_1, [x_1, x_2]]\}, \dim \mathfrak B(V) =2^{2}3^{2}$ and $\mathfrak B(V) = F\oplus \mathfrak L(V)$.
Assume that $ [u]$ is a hard super-letter or zero and $u=vw$ is the Shirshov decomposition of $u$ when $[u] \not=0$. We show $ [u] \in D$ step by step for the length $| u | $ of $u$.
[(a)]{} $| u| =2$, then $u= [1, 2]$ by Lemma \[14\].
[(b)]{} $| u| =3$, then $ [u]=
[1, [1, 2]] $ and $[[1 , 2], 2]=0$ by Lemma \[14\].
[(c)]{} $| u| =4$. then $
[1, [1, [1, 2]] =0$ by Lemma \[14\].
[(d)]{} $| u| =5$, then $$\begin{aligned}
<y_1, [[1, [1, 2]], [1, 2]]>&=&<y_1, [1, 2][1, [1, 2]]-p_{11}^{2}p_{12}p_{21}^{2}p_{22}[1, [1, 2]][1, 2]>\\
&=&p_{12}^{ -1}(1-p_{12}p_{21})2[1, [1, 2]]\\
& & +p_{11}^{-1}p_{12}^{-1}[1, 2]p_{11}^{-1}p_{12}^{-1}(1-p_{11}p_{12}p_{21})(1+p_{11})[1, 2]\\
& & - p_{11}^{2}p_{12}p_{21}^{2}p_{22}p_{11}^{-1}p_{12}^{-1}(1-p_{11}p_{12}p_{21})
(1+p_{11})[1, 2][1, 2] \\
& & -p_{11}^{2}p_{12}p_{21}^{2}p_{22} p_{11}^{-2}p_{12}^{-1}[1,[1, 2]]p_{12}^{ -1}(1-p_{12}p_{21})2\\
&=& p_{12}^{ -1}(1-p_{12}p_{21})[[1, [1, 2]], 2]\\
& & +(p_{11}^{-1}p_{12}^{-1}-p_{11}^{2}p_{12}p_{21}^{2}p_{22})p_{11}^{-1}p_{12}^{-1}\\
& &~~~\times (1-p_{11}p_{12}p_{21})(1+p_{11})[1, 2]^2\\
&=&p_{12}^{ -1}(1-p_{12}p_{21})\left\{p_{12}^{-1}p_{22}^{-1}[[1, 2],[1, 2]]\right.\\
& &~~ \left.+(p_{21}p_{22}-p_{12}^{-1}p_{22}^{-1})[1, 2]^2\right\}\\
& &+(1-p_{11}^{3}p_{12}^{2}p_{21}^{2}p_{22})p_{11}^{-2}p_{12}^{-2}(1-p_{11}p_{12}p_{21})
(1+p_{11})[1, 2]^2\\
&=&p_{12}^{ -1}(1-p_{12}p_{21})(p_{21}p_{22}-p_{11}p_{21})[1, 2]^2\\
& &+(1-p_{11}^{3}p_{12}^{2}p_{21}^{2}p_{22})p_{12}^{-2}\\
& &~~~\times (p_{11}^{-2}+p_{11}^{-1}-p_{11}^{-1}p_{12}p_{21}-p_{12}p_{21})[1, 2]^2\\
&=&p_{12}^{ -2}\{p_{12}p_{21}(1-p_{12}p_{21})(p_{22}-p_{11})\\
& &+(1-p_{11}^{3}p_{12}^{2}p_{21}^{2}p_{22})\\
& &~~~\times (p_{11}^{-2}+p_{11}^{-1}-p_{11}^{-1}p_{12}p_{21}-p_{12}p_{21})\}[1, 2]^2\\
&=&p_{12}^{ -2}\left(p_{11}^{ -2}(1-p_{11}p_{12}p_{21})(1-p_{11}^{2}
p_{12}p_{21})(1+p_{11}^{ 2}p_{12}p_{21}p_{22})\right.\\
& &\left.+p_{11}^{-1}(1-p_{11}^{ 2}p_{12}p_{21})(1-p_{11}p_{12}^{ 2}p_{21}^{ 2}p_{22})\right)[1, 2]^2\\
&=&p_{12}^{ -2}(1-p_{11}^{2}p_{12}p_{21})\{p_{11}^{ -2}(1-p_{11}p_{12}p_{21})
(1+p_{11}^{ 2}p_{12}p_{21}p_{22})\\
& & +p_{11}^{-1}(1-p_{11}p_{12}^{ 2}p_{21}^{ 2}p_{22})\}[1, 2]^2\\
&=&p_{12}^{ -2}(1-p_{11}^{2}p_{12}p_{21})\{p_{11}^{ -2}(1-p_{11}p_{12}p_{21}+p_{11})\\
& &+p_{12}p_{21}p_{22}(1-p_{11}p_{12}p_{21}-p_{12}p_{21})\}[1, 2]^2\\
&=&p_{12}^{ -2}(1-p_{11}^{2}p_{12}p_{21})(1+p_{11})(p_{11}-p_{11}^{2}p_{12}p_{21}
+p_{12}^{2}p_{21}^{2})[1, 2]^2\\
&=&p_{12}^{ -2}(1-p_{11}^{2}p_{12}p_{21})(1+p_{11})(\zeta+\zeta^2\zeta+\zeta^2)[1, 2]^2=0.\end{aligned}$$
[(e)]{} $| u| =6$. $ [u] $ does not exist. Then we show that $D=\{ [u_{4}]=[1], [u_{1}]=[2], [u_{2}]=[1, 2],
[u_{3}]=[1, [1, 2]]\}$ , $p_{u_{1}
u_{2}}p_{u_{2}u_{1}}=-\zeta, p_{u_{1}u_{3}}p_{u_{3}u_{1}}=\zeta^2, p_{u_{1}u_{4}}p_{u_{4}u_{1}}=-\zeta,
p_{u_{2}u_{3}}p_{u_{3}u_{2}}=-\zeta, p_{u_{2}u_{4}}p_{u_{4}u_{2}}=-1,
p_{u_{3}u_{4}}p_{u_{4}u_{3}}=-\zeta^2, $ and $p_{u_{i}u_{i}}= -1, \zeta^2, -1, \zeta, \ ord (p_{u_{i}u_{i}})= 2, 3, 2, 3 , \ i=1, 2, 3, 4$. Considering Lemma \[12\], we have $$\begin{aligned}
P\setminus \{1 \}
&=&\left\{u_{1},u_{2},u_{2}^2=u_2 u_2,u_{3},u_{4},u_{4}^2=u_4 u_4,u_{1}u_{2},u_{1}u_{2}^2,u_{1}u_{3},u_{1}u_{4},
u_{1}u_{4}^2,u_{2}u_{3},\right.\\
& & u_{2}^{2}u_{3},u_{2}u_{4},u_{2}^{2}u_{4}=u_2(u_2u_4),u_{2}u_{4}^2=(u_2u_4)u_4,u_{2}^{2}u_{4}^2=(u_2^2u_4)u_4,u_{3}u_{4},\\
& & u_{3}u_{4}^2,(u_{1}u_{2})u_{3},u_{1}u_{2}^{2}u_{3}=(u_{1}u_{2}^2)u_{3},(u_{1}u_{2})u_{4},(u_{1}u_{2}^{2})u_{4},
(u_{1}u_{2})u_{4}^2,\\
& &u_{1}(u_{2}^{2}u_{4}^2),u_{1}(u_{3}u_{4}),u_{1}(u_{3}u_{4}^2),(u_{2}u_{3})u_{4},(u_{2}^2u_{3})u_{4},(u_{2}u_{3})u_{4}^2,
(u_{2}^2u_{3})u_{4}^2,\\
& & \left. (u_{1}u_{2}u_{3})u_{4},u_{1}(u_{2}u_{3}u_{4}^2),u_{1}(u_{2}^2u_{3}u_{4}),u_{1}u_{2}^2u_{3}u_{4}^2
= (u_{1}u_{2})u_{2}u_{3}u_{4}^2\right\}.\end{aligned}$$ Thus $\mathfrak B(V) = F\oplus \mathfrak L(V)$.
\[62\] Assume that $\mathfrak B(V) $ is connected Nichols algebra of diagonal type and $\Delta(\mathfrak B(V)) $ is arithmetic root systems.If $u, v, w \in D$ with $\deg u = \deg v + \deg w$( specially, if $u = vw$ is the Shirshov decomposition of $u \in D$), then $p_{wv}p_{vw}\neq1$ except the following cases: (i) $p_{ww} = p_{vv}, p_{vv} \not = \pm 1$; (ii) $p_{ww} = -p_{vv}^{-1}, p_{vv} \not = \pm 1$; (iii) $p_{vv} = -p_{ww}^{2}, p_{ww}\in {R_{18}}$; (iv) $p_{vv} = -p_{ww}^{-4}, p_{ww}\in {R_{18}}$; (v) $p_{vv} = -p_{ww}^{-4}, p_{ww}\in {R_{10}}$; (vi) $p_{ww} = -p_{vv}^{2}, p_{vv}\in {R_{18}}$; (vii) $p_{ww} = -p_{vv}^{-4}, p_{vv}\in {R_{18}}$; (viii) $p_{ww} = -p_{vv}^{-4}, p_{vv}\in {R_{10}}$.
\(i) $\deg (u)\in \Delta(\chi;\deg(v), \deg(w))$ $\begin{picture}(100, 20)
\put(10, 1) {\makebox(0, 0) [t]{$\bullet$}}
\put(80, 1){\makebox(0, 0) [t]{$\bullet$}}
\put(10, -1) {\line(1, 0) {70}}
\put(5, 9) {$p_{vv}$}
\put(35, 4) {$p_{vw}p_{wv}$}
\put(75, 9) {$p_{ww}$}
\put(90, 1) {$. $}\ \ \ \put(95, 1) { }
\end{picture}$ By and , it is clear $p_{wv}p_{vw}\neq1$.
\(ii) If exist some $k\in\mathbb N$ such that $\deg(v)-k\deg(w)\in \mathbb N\cdot\Delta^+(\mathfrak B(V))$, let $k_{1}\in\mathbb N$ be the maximum integer such that $\deg(v)-k_{1}\deg(w): = k_{2}\deg(v_{1})\in \mathbb N\cdot\Delta^+(\mathfrak B(V))$. We know $\deg(v_{1})-k\deg(w)\notin \mathbb N\cdot\Delta(\mathfrak B(V))$ for $\ \forall k\in\mathbb N$ by the maximality of $k_{1}$. Then we obtain $\deg (u)\in \Delta(\chi;\deg(v_{1}), \deg(w))$ with $\begin{picture}(100, 20)
\put(10, 1) {\makebox(0, 0) [t]{$\bullet$}}
\put(80, 1){\makebox(0, 0) [t]{$\bullet$}}
\put(10, -1) {\line(1, 0) {70}}
\put(5, 9) {$p_{v_{1}v_{1}}$}
\put(35, 4) {$p_{v_{1}w}p_{wv_{1}}$}
\put(75, 9) {$p_{ww}$}
\put(90, 1) {$ {} $}\ \ \ \put(95, 1) { }
\end{picture}$ by and . In this case, let $\alpha = \deg (v_1)$. If $\deg(v)-k\deg(w)\notin \mathbb N\cdot\Delta^+(\mathfrak B(V))$ for $\ \forall k\in\mathbb N$ and there exists some $k\in\mathbb N$ such that $\deg(v)-k\deg(w)\in \mathbb N\cdot\Delta^-(\mathfrak B(V))$. Let $k_{1}\in\mathbb N$ be the maximum integer such that $\deg(v)-k_{1}\deg(w): = -k_{2}\deg(v_{1})\in \mathbb N\cdot\Delta^-(\mathfrak B(V))$. We know $-\deg(v_{1})-k\deg(w)\notin \mathbb N\cdot\Delta(\mathfrak B(V))$ for $\ \forall k\in\mathbb N$ by the maximality of $k_{1}$. Then we obtain $\deg (u)\in \Delta(\chi;-\deg(v_{1}), \deg(w))$ $\begin{picture}(100, 20)
\put(10, 1) {\makebox(0, 0) [t]{$\bullet$}}
\put(80, 1){\makebox(0, 0) [t]{$\bullet$}}
\put(10, -1) {\line(1, 0) {70}}
\put(5, 9) {$p_{v_{1}v_{1}}$}
\put(35, 4) {$p_{v_{1}w}p_{wv_{1}}$}
\put(75, 9) {$p_{ww}$}
\put(90, 1) {${} $}\ \ \ \put(95, 1) { }
\end{picture}$ by and . In these cases, $\deg(v) = -k_{2}\deg(v_{1})+k_{1}\deg(w)$, $\deg(u) = -k_{2}\deg(v_{1})+(k_{1}+1)\deg(w)$, and $2 \nmid k_{2}$ by . In this case, let $\alpha = -\deg (v_1)$.
.1in (iii) Set $\deg(w) = e_{2}, \alpha = e_{1}$. Then $p_{vw}p_{wv} = p_{ww}^{2k_{1}}(p_{vw}p_{wv})^{k_{2}} = p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}}$.
T4(1). $p_{0} = p_{12}p_{21}p_{11}\in R_{12}$, $p_{11} = p_{0}^4$, $p_{22} = -p_{0}^2$, $p_{12}p_{21} = p_{0}p_{11}^{-1} = p_{0}^{-3} = -p_{0}^{3}$. $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (-p_{0}^{2})^{2k_{1}}(-p_{0}^{3})^{k_{2}} = (-1)^{k_{2}}(p_{0})^{4k_{1}+3k_{2}}\neq1$ since $2 \nmid k_{2}$.
T4(2). $p_{12}p_{21}\in R_{12}$, $p_{11} = p_{22} = -(p_{12}p_{21})^2$, $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}$ $(-(p_{12}p_{21})^2)^{2k_{1}} = (p_{12}p_{21})^{4k_{1}+k_{2}}\neq1$ since $2 \nmid k_{2}$.
T5(1). $p_{12}p_{21}\in R_{12}$, $p_{11} = -(p_{12}p_{21})^2$, $p_{22} = -1$, $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$.
T5(2). $p_{0} = p_{12}p_{21}p_{11}\in R_{12}$, $p_{11} = p_{0}^4$, $p_{22} = -1$, $p_{12}p_{21} = p_{0}p_{11}^{-1} = p_{0}^{-3} = -p_{0}^{3}$. $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (-p_{0}^{3})^{k_{2}} = (-1)^{k_{2}}(p_{0})^{3k_{2}}\neq1$ since $2 \nmid k_{2}$.
T7(1). $p_{11}\in R_{12}$, $ p_{12}p_{21} = p_{11}^{-3}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11}^{-3})^{k_{2}} = p_{11}^{-3k_{2}}\neq1$ since $2 \nmid k_{2}$.
T7(2). $p_{12}p_{21}\in R_{12}$, $p_{11} = (p_{12}p_{21})^{-3}$, $p_{22} = -1$. $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$;
T8(2)$_{1}$. $ (p_{12}p_{21})^4 = -1$, $p_{22} = -1$, $p_{12}p_{21} = -p_{11}$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$.
T8(2)$_{2}$. $ (p_{12}p_{21})^4 = -1$, $p_{22} = -1$, $p_{11} = (p_{12}p_{21})^{-2}$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$.
T8(3)$.~~ (p_{12}p_{21})^4 = -1$, $p_{11} = (p_{12}p_{21})^2$, $p_{22} = (p_{12}p_{21})^{-1}$;\
$p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}(p_{12}p_{21})^{-2k_{1}} = (p_{12}p_{21})^{k_{2}-2k_{1}}\neq1$ since $2 \nmid k_{2}$.
T10. $p_{12}p_{21}\in R_{24}$, $p_{11} = (p_{12}p_{21})^{-6}$, $p_{22}= (p_{12}p_{21})^{-8}$;\
$p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}((p_{12}p_{21})^{-8})^{2k_{1}} = (p_{12}p_{21})^{-16k_{1}+k_{2}}\neq1$ since $2 \nmid k_{2}$.
T11(2). $p_{11}\in R_{20}$, $p_{12}p_{21} = p_{11}^{-3}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11}^{-3})^{k_{2}} = p_{11}^{-3k_{2}}\neq1$ since $2 \nmid k_{2}$.
T12. $p_{11}\in{R_{30}}$, $ p_{12}p_{21} = p_{11}^{-3}$, $p_{22} = -p_{11}^5$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (-p_{11}^5)^{2k_{1}}(p_{11})^{-3k_{2}}$ $= (p_{11})^{10k_{1}-3k_{2}}\neq1$ since $2 \nmid k_{2}$.
T13. $p_{12}p_{21}\in R_{24}$, $p_{11} = (p_{12}p_{21})^6$, $p_{22} = (p_{12}p_{21})^{-1}$;\
$p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}(p_{12}p_{21})^{-2k_{1}} = (p_{12}p_{21})^{-2k_{1}+k_{2}}\neq1$ since $2 \nmid k_{2}$.
T15. $p_{12}p_{21}\in R_{30}$, $p_{11} = -(p_{12}p_{21})^{-3}$, $p_{22}= (p_{12}p_{21})^{-1}$;\
$p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}(p_{12}p_{21})^{-2k_{1}} = (p_{12}p_{21})^{-2k_{1}+k_{2}}\neq1$ since $2 \nmid k_{2}$.
T16(2). $p_{12}p_{21}\in R_{20}$, $p_{11} = (p_{12}p_{21})^{-4}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$.
T17. $p_{12}p_{21}\in R_{24}$, $p_{11} = -(p_{12}p_{21})^4$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$.
T18. $p_{12}p_{21}\in R_{30}$, $p_{11} = -(p_{12}p_{21})^5$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$.
T19. $p_{11}\in R_{14}$, $ p_{12}p_{21} = p_{11}^{-3}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11}^{-3})^{k_{2}} = p_{11}^{-3k_{2}}\neq1$ since $2 \nmid k_{2}$.
T20. $p_{12}p_{21}\in R_{30}$, $p_{11} = (p_{12}p_{21})^{-6}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{12}p_{21})^{k_{2}}\neq1$ since $2 \nmid k_{2}$.
T21 and T22. $p_{11}\in R_{24}$ or $p_{11}\in R_{14}$, $ p_{12}p_{21} = p_{11}^{-5}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11}^{-5})^{k_{2}} = p_{11}^{-5k_{2}}\neq1$ since $2 \nmid k_{2}$.
T2. $\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, e_{1}+e_{2}\}$.
T3. $\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, e_{1}+e_{2}, 2e_{1}+e_{2}\}$.
T6. $p_{11}\in{R_{18}}$, $ p_{12}p_{21} = p_{11}^{-2}$, $p_{22} = -p_{11}^3$, $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11}^{-2})^{k_{2}}(-p_{11}^{3})^{2k_{1}} = p_{11}^{6k_{1}-2k_{2}}$.\
$\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, e_{1}+e_{2}, 2e_{1}+e_{2}, e_{1}+2e_{2}, 3e_{1}+2e_{2}\}$ by . If $\deg(v) = e_{1}+e_{2}$, it is clear $\deg(u) = e_{1}+2e_{2}, (p_{11})^{6k_{1}-2k_{2}} = (p_{11})^{4}\neq1$,
T8(1). $ p_{12}p_{21} = p_{11}^{-3}$, $p_{22} = p_{11}^3$, $p_{11} \in \cup _{m = 4}^\infty R_{m}$. $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11}^{-3})^{k_{2}}(p_{11}^{3})^{2k_{1}} = p_{11}^{6k_{1}-3k_{2}}$. $\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, e_{1}+e_{2}, 2e_{1}+e_{2}, 3e_{1}+e_{2}, 3e_{1}+2e_{2}\}$. If $\deg(v) = 3e_{1}+e_{2}$, then it is clear $\deg(u) = 3e_{1}+2e_{2}$ and $p_{11}^{6k_{1}-3k_{2}} = (p_{11})^{-3}\neq1$.
T9. $p_{12}p_{21}\in R_{9}$, $p_{11} = (p_{12}p_{21})^{-3}$, $p_{22} = -1$; $\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, e_{1}+e_{2}, 2e_{1}+e_{2}, 4e_{1}+3e_{2}, 3e_{1}+2e_{2}\}$ by ,
T11(1). $p_{11}\in R_{5}$, $ p_{12}p_{21} = p_{11}^{-3}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11}^{-3})^{k_{2}} = p_{11}^{-3k_{2}}$; $\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, 3e_{1}+e_{2}, 2e_{1}+e_{2}, 5e_{1}+3e_{2}, 4e_{1}+3e_{2}, 3e_{1}+2e_{2}, e_{1}+e_{2}\}$ by . If $\deg(v) = 3e_{1}+e_{2}$, it is clear $\deg(u) = 3e_{1}+2e_{2}$, then $(p_{11})^{-3k_{2}} = (p_{11})^{-9}\neq1$.
T14. $p_{11}\in R_{18}$, $ p_{12}p_{21} = p_{11}^{-4}$, $p_{22} = -1$; $\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, e_{1}+e_{2}, 2e_{1}+e_{2}, 4e_{1}+e_{2}, 3e_{1}+e_{2}\}$ by .
T16(1). $p_{11}\in R_{10}$, $ p_{12}p_{21} = p_{11}^{-4}$, $p_{22} = -1$; $p_{22}^{2k_{1}}(p_{12}p_{21})^{k_{2}} = (p_{11})^{-4k_{2}} = (p_{11})^{-4k_{2}}$. $\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{2}, 3e_{1}+e_{2}, 2e_{1}+e_{2}, 5e_{1}+2e_{2}, 4e_{1}+e_{2}, 3e_{1}+2e_{2}, e_{1}+e_{2}\}$ by . If $\deg(v) = 3e_{1}+e_{2}$, it is clear $\deg(u) = 3e_{1}+2e_{2}, (p_{11})^{-4k_{2}} = (p_{11})^{-12}\neq1$.
.1in (iv) Set $\deg(w) = e_{1}, \alpha = e_{2}$. In these cases, $\deg(v) = k_{1}e_1+k_{2}e_2$ and $2 \nmid k_{2}$ by . It is clear that $|u|\geq 3$ and $|v|\geq 2$. Then $p_{vw}p_{wv} = p_{ww}^{2k_{1}}(p_{vw}p_{wv})^{k_{2}} = p_{11}^{2k_{1}}(p_{12}p_{21})^{k_{2}}$.
Arguments for T4 - T16 are similar to those above except for the following additional cases:
T3(1)$_1$. $p_{12}p_{21} =p_{11}^{-2}$, $p_{22} = p_{11}^2$, $p_{11} \not = \pm 1$, if $\deg(v)=e_{1}+e_{2}$, it is clear $\deg(u)=2e_{1}+e_{2}$, then $p_{11}^{2}(p_{12}p_{21})^{1}=1$.
T3(1)$_2$. $p_{12}p_{21} =p_{11}^{-2}$, $p_{22} =-1$, $p_{11} \not = \pm 1$, if $\deg(v)=e_{1}+e_{2}$, it is clear $\deg(u)=2e_{1}+e_{2}$, then $p_{11}^{2}(p_{12}p_{21})^{1}=1$.
T6. $p_{11}\in{R_{18}}$, $ p_{12}p_{21} = p_{11}^{-2}$, $p_{22} = -p_{11}^3$, If $\deg(v)=e_{1}+e_{2}$, it is clear $\deg(u)=2e_{1}+e_{2}$, then $(p_{11})^{2k_{1}-2k_{2}}=(p_{11})^{0}=1$.
T14. $p_{11}\in R_{18}$, $ p_{12}p_{21} = p_{11}^{-4}$, $p_{22} = -1$; If $\deg(v)=2e_{1}+e_{2}$, it is clear $\deg(u)=3e_{1}+e_{2}$, then $(p_{11})^{2k_{1}-4k_{2}}=(p_{11})^{0}=1$.
T16(1). $p_{11}\in R_{10}$, $ p_{12}p_{21} = p_{11}^{-4}$, $p_{22} = -1$; If $\deg(v)=2e_{1}+e_{2}$, it is clear $\deg(u)=3e_{1}+e_{2}$, then $(p_{11})^{2k_{1}-4k_{2}}=(p_{11})^{0}=1$.
.1in Similarly, we have (v)–(viii).
\(i) If $[u][v]\neq 0$, then the following conditions are equivalent: (1) $[[u], [v]]=0$ , $[[v], [u]]=0$; (2) $1-p_{uv}p_{vu}=0$ , $[[v], [u]]=0$; (3) $1-p_{uv}p_{vu}=0$ , $[[u], [v]]=0$.
\(ii) If $[[u], [v]]=0$, then $[[v], [u]]=(1-p_{uv}p_{vu})[u][v]$.
There is a braided m-Lie algebra which is not a Nichols braided Lie algebra or Nichols braided m-Lie algebra. In fact, let $L = L_{\bar 0} + L_{\bar 1}$ be a super-Lie algebra with $L_{\bar 0} = sl (2)$, $ L_{\bar 1} =0$. It is clear that $L$ is a finite dimensional m-braided Lie algebra of diagonal type in $^{k \mathbb Z_2}_{k \mathbb Z_2} { \mathcal YD}$. Because $L$ is not nilpotent and every finite dimensional $\mathfrak L(V)$ is nilpotent, $L$ is not a Nichols braided Lie algebra or Nichols braided m-Lie algebra.
Cartan type {#s5}
===========
In this section we give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations. Basic field $F$ is the complex field $\mathbb C.$
Let $\Phi^+$ denote the positive root system of simple Lie algebras. $E_e = n-1, n-1, n-1, n-1, 5, 6, 7, 3, 1$ in $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$, respectively, i.e. $E_e = {\rm rank } \Phi -1$. In fact, $\Phi^+ = \Delta ^+(\mathfrak B(V))$ by [@He06b]. Let $\epsilon _1, \epsilon _2, \cdots, \epsilon_n$ be a normal orthogonal basis of $\mathbb R^n$; $I = : \sum _{i = 1}^n\mathbb Z
\epsilon _i$ and $I' = I + \mathbb Z (\epsilon _1+ \epsilon _2 +\cdots + \epsilon_n)/2. $ Let $\Delta = \{e_1, \cdots, e_n\}$ be a prime root system.
\[4.1\] Let $\Omega$ be the class of connected components of $V$. then $\mathfrak{B} (V) = \oplus _{J\in \Omega }\mathfrak{B}(V_J)$ and $\mathfrak{L} (V) = \oplus _{J\in \Omega }\mathfrak{L}(V_J).$
\[4.1”\] Assume that $V$ is a braided vector space of diagonal type with braided matrix $(q_{ij})_{n\times n}$ and basis $x_1, x_2, \cdots, x_n$. Then ${\rm ord } ( q_{ij})$ is finite for any $1\le i, j \le n$ if and only if there exists a finite abelian group $G$ such that $V$ becomes a $kG$- [YD]{} module.
The necessity is clear. The sufficiency. Let $N$ be the least common multiple of $\{ ord q_{ij} \mid 1\le i, j \le n \}$ and $G = (g_1) \times (g_2) \times \cdots \times (g_n)$ with ${\rm ord } ( g_i) =N$ for $1\le i\le n$. Let $\chi (g_1 ^{k_1} \cdots g_n ^{k_n},
g_1 ^{k_1'} \cdots g_n ^{k_n'} ):= \prod _{1\le i, j \le n}q_{ij} ^{k_ik_j'}$. It is clear that $\chi $ is a bicharacter on $G \times G$. Therefore $V$ becomes a $kG$- [YD]{} module.
\[4.1’\] Assume that $V$ is a braided vector space of diagonal type with braided matrix $(q_{ij})_{n\times n}$. If $\dim \mathfrak B(V)<\infty$, then there exists a braided matrix $(q_{ij}')_{n \times n}$, which is twisted equivalent to $(q_{ij})_{n\times n}$, such that ${\rm ord } ( q_{ij}' ) <\infty$ for any $1\le i, j \le n.$
We show this by two steps.
\(i) ${\rm ord } ( q_{ij}q_{ji}) < \infty$ for any $1\le i, j \le n$. In fact, it is clear ${\rm ord } (q_{ii} ^2 )< \infty$ for any $1\le i \le n.$ If there exist $i$ and $j$ with $i < j$ such that ${\rm ord } ( q_{ij}q_{ji}) = \infty$. Obviously, $[u]:= [x_i, x_j] \in D.$ Consequently, ${\rm ord } ( p_{u, u}) =\infty$ and $\dim \mathfrak B(V)=\infty$, which is a contradiction. (ii) Set $q_{ij}' := \sqrt{q_{ij} q_{ji}}$ for any $1\le i, j \le n$.
\[4.222’\] Assume that $(V, (q_{ij})_{n\times n})$ is of connected Cartan type with Cartan matrix $(a_{ij})_{n\times n}$. Then
\(i) For $A_n, D_n, E_8, E_7, E_6$: $p_{\alpha, \alpha} = q $, $p_{\alpha, \gamma}p_{ \gamma, \alpha} = q^{(\alpha, \gamma)}$, $p_{\alpha, \beta} p_{ \beta, \alpha} = q^{\pm 1}$ or $1$ for any $\alpha, \beta, \gamma\in \Delta^+(\mathfrak B(V))$ with $\alpha \not= \beta$.
\(ii) For $B_n$:
$\begin{picture}(100, 15)
\put(27, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(60, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(93, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(159, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(192, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(225, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(28, -1){\line(1, 0){33}}
\put(61, -1){\line(1, 0){30}}
\put(130, -1){\makebox(0, 0)[t]{$\cdots\cdots\cdots\cdots$}}
\put(160, -1){\line(1, 0){30}}
\put(193, -1){\line(1, 0){30}}
\put(22, -15){1}
\put(58, -15){2}
\put(91, -15){3}
\put(157, -15){n-2}
\put(191, -15){n-1}
\put(224, -15){n}
\put(22, 10){$q^{2}$}
\put(58, 10){$q^{2}$}
\put(91, 10){$q^{2}$}
\put(157, 10){$q^{2}$}
\put(191, 10){$q^{2}$}
\put(224, 10){$q$}
\put(40, 5){$q^{-2}$}
\put(73, 5){$q^{-2}$}
\put(172, 5){$q^{-2}$}
\put(205, 5){$q^{-2}$}
\end{picture}$ .1in $$\begin{aligned}
\Delta^+(\mathfrak B(V)) &=& \left \{\sum \limits_{i = n}^{n}e_{i}, \sum \limits_{i = n-1}^{n}e_{i},
\sum \limits_{i = n-2}^{n}e_{i}, \cdots, \sum \limits_{i = 1}^{n}e_{i}\right \}\\
& & \cup ~\left \{ \sum \limits_{i = 1}^{1}e_{i}+\sum \limits_{i = 2}^{n}2e_{i},
\sum \limits_{i = 1}^{2}e_{i}+\sum \limits_{i = 3}^{n}2e_{i}, \sum \limits_{i = 1}^{3}e_{i}+\sum \limits_{i = 4}^{n}2e_{i}, \right. \\
& & \cdots,\sum \limits_{i = 1}^{n-1}e_{i}+\sum \limits_{i = n}^{n}2e_{i},
\sum \limits_{i = 2}^{2}e_{i}+\sum \limits_{i = 3}^{n}2e_{i}, \sum \limits_{i = 2}^{3}e_{i}+\sum \limits_{i = 4}^{n}2e_{i}, \\
& & \left.\cdots,\sum \limits_{i = 2}^{n-1}e_{i}+\sum \limits_{i = n}^{n}2e_{i}, \cdots,
\sum \limits_{i = n-1}^{n-1}e_{i}+\sum \limits_{i = n}^{n}2e_{i} \right\}\\
& &\cup~ \left\{ \sum \limits_{i = 1}^{1}e_{i}, \sum \limits_{i = 1}^{2}e_{i},
\sum \limits_{i = 1}^{3}e_{i}, \cdots, \sum \limits_{i = 1}^{n-1}e_{i}, \sum \limits_{i = 2}^{2}e_{i}, \sum \limits_{i = 2}^{3}e_{i}, \right.\\
& & \left. \cdots,\sum \limits_{i = 2}^{n-1}e_{i}, \cdots, \sum \limits_{i = n-2}^{n-2}e_{i},
\sum \limits_{i = n-2}^{n-1}e_{i}, \sum \limits_{i = n-1}^{n-1}e_{i}\right\}: = Q \cup S\cup T,\end{aligned}$$ $p_{\alpha, \alpha } = q$ for $\ \forall \alpha \in Q$ and $p_{\alpha, \alpha} = q^2$ for $\ \forall \alpha \in S$ or $T$.
\(iii) For $C_n$:
$\begin{picture}(100, 15)
\put(27, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(60, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(93, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(159, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(192, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(225, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(28, -1){\line(1, 0){33}}
\put(61, -1){\line(1, 0){30}}
\put(130, -1){\makebox(0, 0)[t]{$\cdots\cdots\cdots\cdots$}}
\put(160, -1){\line(1, 0){30}}
\put(193, -1){\line(1, 0){30}}
\put(22, -15){1}
\put(58, -15){2}
\put(91, -15){3}
\put(157, -15){n-2}
\put(191, -15){n-1}
\put(224, -15){n}
\put(22, 10){$q$}
\put(58, 10){$q$}
\put(91, 10){$q$}
\put(157, 10){$q$}
\put(191, 10){$q$}
\put(224, 10){$q^2$}
\put(40, 5){$q^{-1}$}
\put(73, 5){$q^{-1}$}
\put(172, 5){$q^{-1}$}
\put(205, 5){$q^{-2}$}
\end{picture}$ .1in $$\begin{aligned}
\Delta^+(\mathfrak B(V)) &=& \left \{\sum \limits_{i = 1}^{n-1}2e_{i}+e_{n},
\sum \limits_{i = 2}^{n-1}2e_{i}+e_{n}, \cdots, \sum \limits_{i = n-1}^{n-1}2e_{i}+e_{n}, e_{n}\right \}\\
& & \cup ~ \left\{\sum \limits_{i = 1}^{1}e_{i}+\sum \limits_{i = 2}^{n-1}2e_{i}+e_{n},
\sum \limits_{i = 1}^{2}e_{i}+\sum \limits_{i = 3}^{n-1}2e_{i}+e_{n}, \right.\\
& & \sum \limits_{i = 1}^{3}e_{i}+\sum \limits_{i = 4}^{n-1}2e_{i}+e_{n}, \cdots,
\sum \limits_{i = 1}^{n-1}e_{i}+e_{n}, \sum \limits_{i = 2}^{2}e_{i}+\sum \limits_{i = 3}^{n-1}2e_{i}+e_{n}, \\
& & \left. \sum \limits_{i = 2}^{3}e_{i}+
\sum \limits_{i = 4}^{n-1}2e_{i}+e_{n}, \cdots, \sum \limits_{i = 2}^{n-1}e_{i}+e_{n}, \cdots,
\sum \limits_{i = n-1}^{n-1}e_{i}+e_{n}\right \} \\
& & \cup ~ \left\{\sum \limits_{i = 1}^{1}e_{i}, \sum \limits_{i = 1}^{2}e_{i},
\sum \limits_{i = 1}^{3}e_{i}, \cdots, \sum \limits_{i = 1}^{n-1}e_{i}, \sum \limits_{i = 2}^{2}e_{i}, \sum \limits_{i = 2}^{3}e_{i}, \right. \\
& &\left. \cdots,\sum \limits_{i = 2}^{n-1}e_{i}, \cdots, \sum \limits_{i = n-2}^{n-2}e_{i},
\sum \limits_{i = n-2}^{n-1}e_{i}, \sum \limits_{i = n-1}^{n-1}e_{i}\right \} : = Q\cup S\cup T,\end{aligned}$$ $p_{\alpha, \alpha } = q^2$ for $\ \forall \alpha \in Q$ and $p_{\alpha, \alpha} = q$ for $\ \forall \alpha \in S$ or $T$.
\(iv) For $F_4$:
$\begin{picture}(100, 15)
\put(27, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(60, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(93, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(126, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(28, -1){\line(1, 0){33}}
\put(61, -1){\line(1, 0){30}}
\put(94, -1){\line(1, 0){30}}
\put(22, -15){1}
\put(58, -15){2}
\put(91, -15){3}
\put(124, -15){4}
\put(22, 10){$q^2$}
\put(58, 10){$q^2$}
\put(91, 10){$q$}
\put(124, 10){$q$}
\put(40, 5){$q^{-2}$}
\put(73, 5){$q^{-2}$}
\put(106, 5){$q^{-1}$}
\end{picture}$ .1in $$\begin{aligned}
\Delta^+(\mathfrak B(V)) & = & \{ e_2+2 e_3+ 2 e_4, e_1+ e_2+2 e_3+ 2 e_4, e_1+ 2e_2+2 e_3+ 2 e_4,\\
& & e_1,e_1+e_2,e_2, 2e_1+ 3e_2+4 e_3+ 2 e_4,e_1+ 3e_2+4 e_3+ 2 e_4,\\
& & e_1+ 2e_2+4 e_3+ 2 e_4,e_1+ 2e_2+2e_3,e_1 + e_2+2e_3,e_2+2e_3\}\\
& & \cup ~\{e_1+ 2e_2+3 e_3+ e_4, e_2+2 e_3+ e_4,e_1+ e_2+2 e_3+ e_4,\\
& &e_1+ 2e_2+2e_3+ e_4,e_3+ e_4,e_2+ e_3+ e_4,e_1+ e_2+ e_3+ e_4,e_4,\\
& & e_1+ 2e_2+3 e_3+ 2 e_4,e_1+ e_2+ e_3,e_2+ e_3,e_3 \} : = Q\cup S,\end{aligned}$$ $p_{\alpha, \alpha } = q^2$ for $\ \forall\alpha \in Q $ and $p_{\alpha, \alpha }
= q$ for $\ \forall\alpha \in S $.
\(v) For $G_2$: .1in $\begin{picture}(100, 15)
\put(27, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(60, 1){\makebox(0, 0)[t]{$\bullet$}}
\put(28, -1){\line(1, 0){33}}
\put(22, -15){1}
\put(58, -15){2}
\put(22, 10){$q$}
\put(58, 10){$q^3$}
\put(40, 5){$q^{-3}$}
\end{picture}$\
\
$\Delta^+(\mathfrak B(V)) = \{e_{1}, e_{1}+e_{2}, 2e_{1}+e_{2}\} \cup \{ 3e_{1}+e_{2}, 3e_{1}+2e_{2}, e_{2}\}: = Q \cup S.$ $p_{\alpha\alpha} = q$ for $\ \forall \alpha\in Q$ and $p_{\alpha\alpha} = q^3$ for $\ \forall \alpha \in S$.
\(i) By [@Hu72 Section 12.1], the root system $\Delta^+(\mathfrak B(V)) = \{ \beta \in I \mid (\beta, \beta ) = 2 \}.$ Let $\alpha = k_1e_1 + \cdots + k _n e_n, \gamma = k_1'e_1 + \cdots + k _n' e_n \in \Delta^+(\mathfrak B(V))$. $$\begin{aligned}
\label {e6.1.2} (\alpha, \gamma ) = \sum _{i = 1}^n 2k_ik'_i + \sum _{i \not= j} a_{ij}k_ik_j'\end{aligned}$$ and $$\begin{aligned}
\label {e6.1.3}
p_{\alpha, \gamma }p _{\gamma, \alpha} = q ^ { \sum _{i = 1}^n 2k_ik'_i + \sum _{i \not= j} a_{ij}k_ik_j'} = q ^{(\alpha, \gamma)}.\end{aligned}$$ Consequently, $p_{\alpha, \alpha} = q $ by (\[e6.1.3\]). By [@Hu72 Section 9.4, Table 1], $(\alpha, \beta ) = 1 $ or $-1$ or $0.$
\(ii) - (v) are clear.
.1in Recall that $(T(V), [\ ])$ is a braided Lie algebra. The braided Lie algebra $(FL(V), [\ ])$ generated by $V$ in $(T(V), [\ ])$ is called the free braided Lie algebra of $V.$ If $f_1, f_2, \cdots, f_r \in FL(V)$ and $I$ is an ideal $I$ generated by $f_1, f_2, \cdots, f_r$ in $(FL(V), [\ ])$, then $(FL(V)/ I, [\ ])$ is called the braided Lie algebra generated by $x_1, x_2, \cdots, x_n$ with the defining relations $f_1, f_2, \cdots, f_r$.
\[4.2’\] Assume that $(V, (q_{ij})_{n\times n})$ is a connected braided vector space of finite Cartan type with Cartan matrix $(a_{ij})_{n\times n}$. If ${\rm ord } (q_{ii})$ is prime to $3$ when $q_{ij} q_{ji} \in \{ q_{ii}^ {3}, q_{jj} ^{3}\}$, then ${\rm ord } ( p_{\alpha, \alpha } )=N$ for root vector $x_\alpha$ with $\alpha \in \Delta^+(\mathfrak B(V))$ and $N = ord (q_{11})$, where root vectors were defined in [@Lu90].
By [@AS00 Th.1.1(i)], ${\rm ord } ( q_{ii}) =N$ for $1\le i \le n.$ $ord (p_{\alpha, \alpha })= N$ for any root $\alpha$ by Lemma \[4.222’\] and $x_\alpha^i \in \mathfrak L(V) $ for $1\le i \le N$ by Lemma \[11\].
.1in Let $ad _c xy := [x, y]_c = xy - p_{x, y} yx.$
\[4.3\] If $V$ is a finite Cartan type with Cartan matrix $(a_{ij})_{n\times n}$ and the following conditions satisfied for any $1\le i, j \le n:$ (i) ${\rm ord } (q_{ii})$ is odd; (ii) ${\rm ord } (q_{ii})$ is prime to $3$ when $q_{ij} q_{ji} \in \{ q_{ii}^ {3}, q_{jj} ^{3}\};$ (iii) ${\rm ord } ( q_{ij})< \infty$. then Nichols braided Lie algebra $\mathfrak{L} (V)$ is a homomorphic image of the braided Lie algebra generated by $x_1, x_2, \cdots, x_n$ with defining relations: (iv) $ad _c x_i ^{1- a_{ij}}x_j$, $i\not = j.$ (v) $x_\alpha ^N$ for any $\alpha \in \Delta ^+(\mathfrak B(V)), $ where $N$ is order of $q_{11}$.
By , $x_\alpha \in FL(V)$. It follows from Lemma \[4.2’\] and Lemma \[11\] that $x_\alpha ^N \in FL(V)$. Let $I$ and $J$ denote ideals generated by elements of (iv) and (v) in $T(V)$ as algebras and in $FL(V)$ as braided Lie algebras with bracket $[\ \ ]$. Consequently, using , we have that the map from $FL(V)/ J$ to $\mathfrak L(V)$ by sending $x +J$ to $x+I$ is a epimorphism.
\[4.2\] If $\dim \mathfrak{B} (V)< \infty$ and the following conditions satisfied for any $1\le i, j \le n:$ (i) ${\rm ord } (q_{ii})$ is odd; (ii) ${\rm ord } (q_{ii})$ is prime to $3$ when $q_{ij} q_{ji} \in \{ q_{ii}^ {3}, q_{jj} ^{3}\};$ (iii) ${\rm ord } (q_{ii})>3$; (iv) ${\rm ord } ( q_{ij})< \infty$. Then $V$ is a finite Cartan type with Cartan matrix $(a_{ij})_{n\times n}$ and Nichols braided Lie algebra $\mathfrak{L} (V)$ is a homomorphic image of the braided Lie algebra generated by $x_1, x_2, \cdots, x_n$ with the defining relation, (v) $ad _c x_i ^{1- a_{ij}}x_j$, $i\not= j$; (vi) $x_\alpha ^N$ for any $\alpha \in \Delta ^+(\mathfrak B(V)), $ where $N$ is order of $q_{11}$.
By the proof of , $V$ is a finite Cartan type. Using Theorem \[4.3\] we complete the proof.
\[8”\] Under the conditions of Theorem \[4.2\] or Theorem \[4.3\], for any $\alpha \in\Delta ^+(\mathfrak B(V))$, there exists a unique hard super-letter $[u]$ such that $\deg (u) = \alpha$.
Considering the dimensional formulas of $\mathfrak B(V)$ in Theorem \[8\] and [@AS00 Th. 1.1(i)], we complete the proof.
\[8”’\] Assume that $(V, (q_{ij})_{n\times n})$ is of a connected Cartan type.If $\alpha, \beta, \gamma \in\Delta ^+(\mathfrak B(V))$ with $\alpha + \beta = \gamma$, then $p_{\alpha, \beta} p_{ \beta, \alpha} =1 $ if and only if $(\alpha, \beta)$ or $( \beta, \alpha) \in X$ and $X$ is defined in the following cases:
\(i) For $F_4$, $X := \{ (\epsilon _i, \epsilon _j) \mid $ $i\not= j \}$ $ \cup ~ \{ (\frac {1}{2} (\epsilon_1 + (-1)^{k_2} \epsilon_2+ (-1)^{k_3} \epsilon_3+ (-1)^{k_4}\epsilon_4 ),
\frac {1}{2} (\epsilon_1 + (-1)^{k_2'} \epsilon_2+ (-1)^{k_3'} \epsilon_3+ (-1)^{k_4'}\epsilon_4 ) )
\mid (-1)^{k_2}+ (-1)^{k_2'} ~{\rm or} ~ (-1)^{k_3}+(-1)^{k_3'}~{\rm or}~
(-1)^{k_4}+(-1)^{k_4'}~ {\rm is~ } 1 \}$;
\(ii) For $B_n$, $X= \{ (\epsilon _i, \epsilon _j \mid 1\le i \not= j \le n\}$;
\(iii) For $C_n$, $X:= \{ (\epsilon _i -\epsilon _j, \epsilon _i+\epsilon _j)\mid 1\le i< j \le n \}$.
By Proposition \[62\], it is clear that $p_{\alpha, \beta} p_{ \beta, \alpha} =1$ if and only if $$\begin{aligned}
\label {e5.1}
p_{\gamma, \gamma} = q^2, p_{\alpha, \alpha} = p_{\beta, \beta}=q;\end{aligned}$$ $$\begin{aligned}
\label {e5.2}
\hbox { or } \ \ \ p_{\gamma, \gamma} = q, p_{\alpha, \alpha} = p_{\beta, \beta}=q^2, q^3 =1.\end{aligned}$$ By Lemma \[4.2’\], $ p_{\alpha, \alpha} =q$ and $p_{\alpha, \beta} p_{ \beta, \alpha} \not=1 $ for $A_n, D_n, E_8, E_7, E_6$. We can complete the proof by simple computation.
\[5.6\] Assume $[u]\in D$. (i) If $p_{vw}p_{wv} \not= 1$ for any two descendants $v$ and $w$ of $u$, then $[u]^- \in {\mathfrak L}(V)$; (ii) If connected $(V, (q_{ij})_{n\times n})$ is of $A_n, D_n, E_6, E_7, E_8, G_2$, then $[u]^- \in {\mathfrak L}(V)$.
\(i) By induction and Lemma \[12\], we obtain (i). (ii) follows from (i) and Lemma \[8”’\].
\[5.6’\] Under the conditions of Theorem \[4.2\] or Theorem \[4.3\], assume that connected $(V, (q_{ij})_{n\times n})$ is of $A_n, D_n, E_6, E_7, E_8, G_2$, and $q= q_{ii}$ with $N := {\rm ord } (q_{11})$. (i) If $u, v\in D,$ then $p_{uu}^{i} p_{u, uv}p_{uv, u}\not= 1$ for $1\le i \le 2([\frac {N}{2}]-1)-2$. (ii) $ \dim {\mathfrak L}(V) \ge (N-1)^{ \mid \Phi^+ \mid } + ( [\frac {N} {2}]-1)\frac { \mid \Phi^+ \mid ( \mid \Phi^+ \mid-1 )}{2}$.
\(i) By Lemma \[4.2’\] (i), $p_{uu}^{i} p_{u, uv}p_{uv, u} = p_{uu} ^ip_{uu}^2 p_{uv}p_{vu}= q ^{i +2+ ( \deg (u), \deg (v))} \not= 1 $ for $1\le i \le 2([\frac {N}{2}]-1)-2$. (ii) Let $D:= \{ u_1, u_2, \cdots, u_r\}$ with $u_r < u_{r-1} < \cdots < u_1 $. By Lemma \[8”\], $r = \mid \Phi ^+ \mid.$ It follows from Lemma \[21\] that $u^i uv \in \mathfrak L(V)$ for any $u, v\in D$ and $1\le i \le [\frac{N}{2}] -1. $ Consequently, $B:= \{ u_1^j, u_2^j, $ $\cdots, u_r^j; u_1^iu_1u_2, u_1^iu_1u_3, $ $ \cdots,
u_1^iu_1u_r; u_2^iu_2u_3, u_2^iu_2u_4, \cdots, u_2^iu_2u_r; $ $ \cdots ; $ $ u_{r-1}^iu_{r-1}u_r \} $ $\subset P \cap {\mathfrak L}(V) $, which implies $\dim {\mathfrak L}(V) \ge (N-1)^{ \mid \Phi^+ \mid } + ( [\frac {N} {2}]-1)\frac { \mid \Phi^+ \mid ( \mid \Phi^+ \mid-1 )}{2}$.
.3in [**Acknowledgments**]{} .1in YZZ acknowledges the partial financial support from the Australian Research Council through Discovery Projects DP110103434 and DP140101492. We would like to thank the referee for many suggestions which lead to the substantial improvement of the paper.
[200]{} N. Andruskiewitsch and H. J. Schneider, Finite quantum groups over abelian groups of prime exponent, Ann. Sci. Ec. Norm. Super. [**35**]{} (2002), 1-26.
N. Andruskiewitsch and H. J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math. [**154**]{} (2000), 1-45.
N. Andruskiewitsch, I. Heckenberger and H. J. Schneider, The Nichols algebra of a semisimple Yetter-Drinfeld module, Amer. J. Math. [**132**]{} (2010), 1493-1547.
N. Andruskiewitsch and H. J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. [**171**]{} (2010), 375-417.
I. Angiono, Nichols algebras of unidentified diagonal type, arXiv:1108.5157.
A. Ardizzoni, A Milnor-Moore type theorem for primitively generated braided bialgebras, J. Alg. [**327**]{} (2011), 337-365.
Y. Bahturin, D. Fishman and S. Montgomery, On the generalized Lie structure of associative algebras, J. Alg. [**96**]{} (1996), 27-48.
Y. Bahturin, D. Fischman and S. Montgomery, Bicharacter, twistings and Scheunert’s theorem for Hopf algebra, J. Alg. [**236**]{} (2001), 246-276.
D. Gurevich, A. Radul and V. Rubtsov, Noncommutative differential geometry related to the Yang-Baxter equation, J. Math. Sci. [**77**]{} (1995), 3051-3062.
D. I. Gurevich, The Yang-Baxter equation and the generalization of formal Lie theory, Dokl. Akad. Nauk SSSR [**288**]{} (1986), 797-801.
I. Heckenberger, Nichols algebras of diagonal type and arithmetic root systems, Habilitation thesis, Leipzig, 2005.
I. Heckenberger, Classification of arithmetic root systems, Adv. Math. [**220**]{} (2009), 59-124.
I. Heckenberger, The Weyl-Brandt groupoid of a Nichols algebra of diagonal type, Invent. Math. [**164**]{} (2006), 175-188.
I. Heckenberger, Examples of finite-dimensional rank 2 Nichols algebras of diagonal type, Compos. Math. [**143**]{} (2007), 165-190.
J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9, Springer-Verlag, 1972.
V. G. Kac, Lie Superalgebras, Adv. Math. [**26**]{} (1977), 8-96.
V. K. Kharchenko, An existence condition for multilinear quantum operations, J. Alg. [**217**]{} (1999), 188-228.
V. K. Kharchenko, A Quantum analog of the poincar$\acute{e}$-Birkhoff-Witt theorem, Algebra and Logic [**38**]{} (1999), 259-276.
A. Klimyk and K. Schmüdgen, Quantum groups and their representations, Springer-Verlag, Heidelberg, 1997.
M. Lothaire, Combinatorics on words, Cambridge University Press, London, 1983.
G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata [**35**]{} (1990), 89-114.
S. Majid, Quantum and braided Lie algebras, J. Geom. Phys. [**13**]{} (1994), 307-356.
B. Pareigis, On Lie algebras in the category of Yetter-Drinfeld modules, Appl. Categ. Structures [**6**]{} (1998), 151-175.
M. Scheunert, Generalized Lie algebras, J. Math. Phys. [**20**]{} (1979), 712-720.
S. Zhang and Y.-Z. Zhang, Braided m-Lie algebras. Lett. Math. Phys. [**70**]{} (2004), 155-167.
W. Wu, S. Zhang and Y.-Z. Zhang, Finite dimensional Nichols algebras over finite cyclic groups. J. Lie Theory [**24**]{} (2014), 351-372.
|
---
abstract: 'We examine Bousso’s covariant entropy bound conjecture in the context of radiation filled, spatially flat, Friedmann-Robertson-Walker models. The bound is violated near the big bang. However, the hope has been that quantum gravity effects would intervene and protect it. Loop quantum cosmology provides a near ideal setting for investigating this issue. For, on the one hand, quantum geometry effects resolve the singularity and, on the other hand, the wave function is sharply peaked at a quantum corrected but smooth geometry which can supply the structure needed to test the bound. We find that the bound is respected. We suggest that the bound need not be an essential ingredient for a quantum gravity theory but may emerge from it under suitable circumstances.'
author:
- Abhay Ashtekar
- 'Edward Wilson-Ewing'
title: The covariant entropy bound and loop quantum cosmology
---
Introduction {#s1}
============
Over 30 years ago, Bekenstein made the seminal suggestion that black holes carry an entropy equal to a quarter of their surface area in Planck units [@bek1; @bek2]. The discovery of the first law of black hole thermodynamics by Bardeen, Carter and Hawking [@bch; @akrev], coupled with Hawking’s discovery of black hole radiance [@swh] made this suggestion compelling. The ensuing generalized second law of thermodynamics [@bek3] in turn motivated the more recent holographic principle by ’t Hooft and Susskind [@'tH; @suss] as well as several entropy bound conjectures. It has been suggested that the holographic principle is a powerful hint and should be used as an essential building block for any quantum gravity theory, much as the principle of equivalence is fundamental to general relativity [@'tH; @suss; @bousso2].
Perhaps the most promising of the entropy bound conjectures is due to Bousso [@bousso1; @bousso2]. It is formulated in precise geometric terms and limits the entropy flux through a surface’s “light-sheet” by the area of that surface. More precisely, let $A$ be the area of an arbitrary spatial 2-surface $\B$. A three-dimensional null hypersurface $\L$ is a light-sheet of $\B$ if it is generated by nonexpanding light rays that begin at $\B$ and extend orthogonally away from $\B$. Let $S$ denote the total entropy flux of the matter fields across any light-sheet $\L$ of $\B$. Then (in units where c=k=1) \[conj\] S . In general, $S$ is difficult to compute unless an entropy current $s^a$ can be introduced, in which case S=\_[Ł]{} s\^a \_[abcd]{}, where $\epsilon_{abcd}$ is the space-time volume 4-form. Although the existence of an entropy current is not necessary for the bound to be meaningful, it often is not clear how to compute $S$ if $s^a$ does not exist.
The conjecture has several curious features. First, for it to be easily tested, matter should admit an entropy current. This is a strong requirement since a well-defined expression of entropy current is generally not available unless the matter can be represented as a fluid. Therefore, standard space-times with “fundamental” matter fields —such as scalar, Maxwell, and gauge fields— that one often considers in general relativity cannot readily be tested. Thus, circumstances where the conjecture can be tested are rather limited. However, it is still a highly nontrivial question as to whether the bound holds for matter sources that do admit a well-defined entropy current. Remarkably, the answer is in the affirmative provided the entropy current and stress-energy tensor satisfy certain inequalities along $\L$ [@fmw; @bfm]. Moreover, these inequalities can be motivated from statistical physics of standard bosons and fermions provided one remains away from the Planck scale. Finally, there is also a necessary and sufficient condition [@ap] for the validity of the bound, based on a plausible assumption on the time scale on which thermodynamic equilibrium can be reached.
A second feature of the conjecture is that its current proofs [@fmw; @bfm] require that matter fields satisfy the dominant (or at least null) energy condition; otherwise it would be possible to add entropy across $\L$ without adding energy and appreciably changing the area of $\B$. This assumption is completely reasonable within classical general relativity and indeed constitutes a building block of many of the hard results in this theory. However, since the right side of (\[conj\]) becomes infinite in the limit $\hbar$ goes to zero, the conjecture is nontrivial only in the realm of quantum physics. But in quantum field theory, expectation values of the matter stress-energy tensor fail to satisfy the dominant (or the null) energy condition. Thus there is a clear tension between the classical formulation of the covariant entropy bound and the quantum world it is trying to represent. What would happen in full quantum gravity? Would a suitable generalization of the conjecture survive? As remarked above, it has been suggested that an appropriate generalization should in fact be a building block of any quantum gravity theory.
Finally, as we will see in Sec. \[s2\], even in the k=0, radiation dominated Friedmann-Robertson-Walker (FRW) space-time —a classical solution of Einstein’s equation, which admits a perfect fluid as source— the conjecture is violated near the big bang.[^1] (In this regime the inequalities assumed in [@fmw; @bfm; @ap] fail.) It is natural to suppose that this violation is spurious, because quantum gravity effects would be dominant in this regime and modify the dynamics of general relativity. But one’s first expectation would be that in this Planck regime the space-time metric would fluctuate violently,[^2] making it impossible to speak of a spacelike surface $\B$ and especially its light-sheet $\L$. If so, the basic ingredients in the very statement of the conjecture would be unavailable. However, loop quantum cosmology (LQC) provides a rather unexpected situation [@aps-lett; @aps2; @apsv]: Einstein’s equations do get modified, the classical singularity *is resolved* and replaced by a quantum bounce, but all this physics is well captured by a quantum corrected metric, which is again a smooth tensor field (although its expression contains $\hbar$ corrections, which dominate the dynamics near the bounce). Therefore, the structure required for Bousso’s formulation is available. It is then natural to ask: Does the bound hold? We will show that the answer is in the affirmative.
This result by itself is not a strong statement in favor of either the covariant bound or LQC. Had it been violated, one could have argued that the fluid approximation is inappropriate near the bounce where the matter density is close to the Planck density, or that LQC, with its focus on homogeneous and isotropic situations, is too restrictive. Nonetheless, the fact that there is a coherence between ideas that were independently developed from completely different motivations and perspectives is quite striking. There may well be a deeper underlying reason behind this apparent unity. Our viewpoint, motivated by results of loop quantum gravity (LQG), is the following. In the Planck regime, space-time geometry is quantum mechanical and has a fundamental, in-built discreteness (see, e.g., [@alrev; @crbook; @ttbook]). This discreteness is directly responsible for many of the novel features of the theory, including the resolution of the big bang singularity through a quantum bounce [@aps-lett; @aps2; @apsv]. Because of this discreteness, the true degrees of freedom of nonperturbative quantum gravity are very different from those of standard quantum field theories on background space-times. This fact has important consequences which have a “holographic character” —e.g. the derivation of the entropy of black holes and cosmological horizons [@abck-lett; @abk; @aev]. Bousso’s covariant entropy bound could thus emerge from LQG in suitable circumstances, including some in which the assumptions made in [@fmw; @bfm; @ap] are violated. Thus, rather than being an essential building block for quantum gravity, the bound could *emerge* from a background independent, nonperturbative theory which appropriately incorporates the quantum nature of geometry in the Planck regime.
The paper is organized as follows. In Sec. \[s2\], we show that in the k=0, radiation filled FRW cosmology, the covariant entropy bound is violated near the big bang singularity. In Sec. \[s3\], we first recall the basic results from LQC and then show that the bound is in fact respected in the quantum corrected “effective” space-time geometry that descends from LQC. In the classical as well the LQC analysis of the bound, we only consider “round” 2-spheres $\B$. As in other discussions —e.g., of marginally trapped surfaces— in general relativity, treatment of general surfaces would increase the technical complexity very significantly. Section \[s4\] summarizes the results and discusses related issues.
FRW Universe {#s2}
============
The line element of a flat FRW universe is given by \[metric\] ds\^2 = -d\^2 + a()\^2 (dr\^2 + r\^2 d\^2), and the Friedmann equations describing the evolution of this universe are \[fried\] ()\^2 = , \[cont\] + 3(+P) =0.We will restrict ourselves to a radiation-dominated universe where the required entropy current can be constructed using standard statistical mechanical considerations. The equation of state $P=\f{1}{3}\rho$ of the fluid gives the solution \[aFRW\] a() = (\^2)\^[1/4]{} () = , where $K_0$ is an integration constant without direct physical significance, which drops out of the expressions of measurable quantities.
We will take the radiation fluid to be a photon gas and assume that the universe is always in instantaneous equilibrium. Then, from standard statistical mechanics we have \[therm1\] =T\^4, where $T$ is the temperature. From (\[aFRW\]) we then conclude as usual that the temperature has the following time dependence: T=()\^[1/4]{}.
The entropy density $s$ of a photon gas is given by \[therm2\] s=. Therefore, the entropy flux through a light-sheet $\L$ is given by S = \_[Ł]{} s\^a\_[abcd]{}, where the entropy density 4-vector is $s^a= su^a$ with $u^a$, the unit vector normal to constant $\tau$ slices and where, as before, $\epsilon_{abcd}$ is the space-time volume 4-form. For our calculations, we will choose our spatial 2-surface $\B$ to be a metric 2-sphere at time $\tau_f$. Since the FRW space-time does not admit a trapped surface, $\B$ must admit a past light-sheet. Furthermore, since the FRW space-times are conformally flat and $\B$ is a round 2-sphere, $\L$ would either terminate on the big bang singularity or be the future null cone of a point. Without loss of generality we can assume that $\L$ is generated by the null vector $k^a=(-1,-\f{1}{a},0,0)$. Then, if $\L$ is the null cone of a point at $\tau=\tau_i > 0$, we find that \[S\] S = ()\^[1/4]{}R\_f\^3, where \[Rf\] R\_f = \_[\_i]{}\^[\_f]{} is the radius of $\B$. $R_f$, and hence $S$, is well defined so long as $\tau_i>0$. Since the area of $\B$ is simply \[A\] A=4 a(\_f)\^2R\_f\^2 , we obtain =()\^[1/4]{} (1-) as the ratio of the entropy flux through $\L$ to the area of $\B$. Clearly this ratio diverges as $\tau_f$ approaches zero (keeping $\tau_i < \tau_f$). Therefore, one can violate the covariant entropy bound by an arbitrary amount by choosing $\B$ sufficiently close to the big bang.
However, by plugging in numbers it is easy to show that the bound does hold if $\tau_f \gtrsim 0.06 t_{\pl}$ or, equivalently, the matter density satisfies $\rho \lesssim 8.3\rho_{\pl}$. Thus, the bound is violated only in the deep Planck regime where quantum gravity effects are expected to be dominant. Indeed, since the phenomenological fluid approximation is likely to fail for much larger values of $\tau_f$, it is surprising that the bound has such a large domain of validity! Nonetheless, since it does fail, it is natural to ask whether the quantum gravity effects that resolve the big bang singularity can also restore the validity of the bound. In the next section, we will analyze this issue in the context of LQC and show that the answer is in the affirmative.
Quantum bounce and the entropy bound {#s3}
====================================
This section is divided into two parts. In order to make the paper self-contained, in the first part we provide a concise summary of ideas that underlie LQC and results that have emerged from it, focusing on those features that are most relevant to our main result. Readers who are already familiar with LQC will find the streamlined summary of “effective” equations useful. Readers who are interested only in the entropy bound can skip this material and go directly to the next subsection where we analyze the covariant entropy bound in the quantum corrected, “effective” space-time that emerges from LQC.
Loop Quantum Cosmology {#s3.1}
----------------------
Loop quantum cosmology is an application of the techniques of LQG [@alrev; @crbook; @ttbook] to space-times that are homogenous and isotropic. Currently, symmetry reduction is carried out at the classical level; LQC is yet to be derived systematically from LQG. However, one quantizes the reduced system by mimicking full LQG as closely as possible. Therefore, the mathematical structure of LQC closely resembles that of full LQG, thus differing from the older Wheeler-DeWitt quantum cosmology *already at the kinematical level*.[^3] In LQG the gravitational phase space is the same as that in the $\SU(2)$ Yang-Mills theory. Thus, the configuration variable is an $\SU(2)$ connection $A_a^i$ and its conjugate momentum is a vector density $E^a_i$ of weight 1 which also takes values in the Lie algebra of $\SU(2)$.[^4] However, $A_a^i$ is now the *gravitational* connection used to parallel propagate chiral spinors. Similarly, the “electric field” $E^a_i$ now has a direct geometrical interpretation: it represents a (density weighted) orthonormal triad that determines the spatial Riemannian geometry. The fundamental quantum algebra $\mathfrak{a}$ is generated by holonomies $h_e$ defined by the gravitational spin-connection $A_a^i$ along (1-dimensional curves or) edges $e$, and fluxes $E_S$ of the “electric fields” $E^a_i$ across 2-surfaces $S$. The key task in quantum kinematics is to find an appropriate representation of $\mathfrak{a}$ on a Hilbert space $\Hkin$. Somewhat surprisingly, it turns out that the requirement of background independence selects the representation uniquely [@lost]! This representation provides the arena to formulate quantum dynamics. Now, the classical dynamics of the gravitational and matter fields is generated by a set of first class constraints. These classical constraints have to be promoted to well-defined self-adjoint operators on $\Hkin$. Finally, the physical Hilbert space is built from suitable solutions to these operator constraints. As in any background independent system, these physical states already encode quantum dynamics which can be made explicit by choosing one of the degrees of freedom —such as a matter field or, in cosmology, the scale factor— as an internal clock with respect to which other degrees of freedom evolve.
When one symmetry reduces general relativity by imposing homogeneity and isotropy, it turns out simplest to gauge fix and solve the so-called Gauss and vector constraints, which respectively generate the internal $\SU(2)$ rotations of triads and spatial diffeomorphisms. One is then left just with the Hamiltonian constraint, which generates evolution (in proper time). In the k=0, FRW model, it is given by + = 0,&[where]{}&\
= - \^[-2]{} \^[ij]{}\_[k]{} F\_[ab]{}\^k,&[and]{}& = , where $\gamma$ is the so-called Barbero-Immirzi parameter, $F_{ab}^k$ the field strength of the connection $A_a^i$, $\rho$, the matter density and $E$, the determinant of $E^a_i$ (or equivalently, of the spatial metric $q_{ab}$ determined by $E^a_i$). Let us focus on the gravitational part $\Hgrav$ of this constraint. The term containing triad can be taken over to a quantum operator in a natural fashion, first spelled out by Thiemann [@tt; @ttbook]. Therefore, the key problem in passage to the quantum theory lies in the construction of an operator $\hat{F}_{ab}^k$ representing field strength. For, while the unique representation of $\mathfrak{a}$ selected by the requirement of background independence admits an operator $\hat{h}_e$ representing holonomies, these fail to be continuous in the edge $e$, whence there is no operator corresponding to the connection $A_a^i$ itself. The strategy then is to use holonomy operators to obtain $\hat{F}_{ab}^k$.
Now, it is well known that in classical geometry, the field strength can be recovered as the limit of the ratio of the holonomy around an appropriate closed loop divided by the area of the loop, as one shrinks the loop thereby sending its area to zero. We can use the same idea for quantization of $F_{ab}^k$. However, now because of quantum geometry, eigenvalues of the area operator are discrete. Therefore, to define the action of $\hat{F}_{ab}^k$ on a given LQC state, we are led to shrink the loop only until the physical area it encloses reaches the minimum nonzero eigenvalue, say $\a_o\lp^2$ (in the class of LQG states compatible with the LQC state under consideration). The resulting $\hat{F}_{ab}^k$ is a self-adjoint operator on the kinematical Hilbert space $\Hkin^{\rm LQC}$ with a built-in nonlocality at a scale $\a_o$. The viewpoint in LQC is that this nonlocality is fundamental and the more familiar, local expression of the field strength arises only in the classical limit. (For details, see [@abl; @aps2; @aa-rev].) This nonlocality is important for quantum dynamics: it is the principal reason why the LQC Hamiltonian constraint is qualitatively different from the Wheeler-DeWitt differential equation.
Solutions to the LQC Hamiltonian constraints were first obtained numerically [@aps2] and an analytical understanding was developed more recently [@acs]. If one begins with a suitable quantum state that is sharply peaked on a classical trajectory at late times and evolves it both forward and backward in time, the trajectory remains sharply peaked for all times; the theory admits “dynamical coherent states”. However the trajectory on which they remain peaked are classical solutions only till the matter density $\rho$ reaches about 1% of the Planck density $\rho_{\pl}$. Then, the quantum evolution departs from the classical trajectory. Rather than evolving into the singularity in the backwards evolution, the trajectory undergoes a quantum bounce. The density then starts decreasing again. Once it falls below 1% of $\rho_{\pl}$, classical general relativity again becomes an excellent approximation. Thus, the effect of quantum geometry is to produce an effective *repulsive force* which is negligible till $\rho \sim 0.01
\rho_{\rm Pl}$ but rises *extremely quickly* once the density further increases, so much so that it is able to overwhelm the classical gravitational attraction, thereby triggering the bounce and avoiding the classical singularity. It is this short but critical interval that will be important for our entropy considerations in the next subsection.
The LQC quantum evolution is extremely well approximated by quantum corrected “effective” equations. These are obtained [@jw] using a geometrical formulation of quantum mechanics.[^5] We will conclude this subsection by sketching the structure of these equations. Recall first that in the FRW models one generally introduces a fiducial frame $\e^a_i$ and the corresponding co-frame $\w_a^i$ and expresses the dynamical variables in terms of them. Homogeneity and isotropy imply that the connection $A_a^i$ is proportional to $\w_a^i$ —$A_a^i\, \sim\,\, c\,
\omega_a^i$— and the triad $E^a_i$ to $\e^a_i$ —$E^a_i\, \sim\,\, p\, \e^a_i$— where, for simplicity we have omitted some kinematical factors that are irrelevant for dynamics. $c,p$ is then the basic canonically conjugate pair. The kinematical factors make $c$ dimensionless, endow $p$ with dimensions of ${\rm (length)}^2$, and provide the following Poisson bracket: $\{c,p\} = 8\pi \gamma G/3$. The geometrical meaning of these variables is the following: $c \sim \gamma
\dot{a}$ and $|p| \sim a^2$ where, as in (\[metric\]), $a$ is the scale factor (where, again, we have omitted some kinematical proportionality factors that are irrelevant for dynamics). The Hubble parameter is given by $H :={\dot{a}}/{a} = {c}/{\gamma
\sqrt{|p|}}$.
In terms of these variables the effective Hamiltonian constraint turns out to be: \[heff\] = - \^2 + |p|\^[3/2]{} = 0 .As in the classical theory, the canonical transformation generated by this constraint yields evolution equations (in proper time). However, $\Heff$ has a dependence on Planck’s constant through $\lp=\sqrt{G\hbar}$. The classical Hamiltonian constraint results when we take the limit $\hbar \rightarrow 0$. Both $\a_o$ and the Barbero-Immirzi parameter $\gamma$ disappear in this limit, as they must. Note that only the gravitational part of $\Heff$ has acquired quantum corrections; as one would expect, they arise from the fundamental nonlocality of $\hat{F}_{ab}^k$ in LQC. However, to compare with the standard form of the classical constraint —the Friedmann equation— it is convenient to rewrite this equation in a slightly different form. Using the equation of motion := {p, } = for $p$, the fact that $H^2 = {(\dot{p})^2}/{4p^2}$, and (\[heff\]) we obtain \[mfried\] H\^2 = (1 - ), = . Thus, using the equation of motion $\Heff$ generates, we have moved the quantum correction from the “geometric” to the “matter” side. (This procedure is quite general and holds so long as $\Hmatt$ does not depend on the gravitational connection $c$.) As $\hbar$ goes to zero, $\rcr$ diverges and the second term in the parenthesis on the right-hand side disappears; we recover the classical Friedmann equation, Eq. (\[fried\]). The quantum correction is negligible when $\rho \ll \rcr \sim \rho_{\rm Pl}$. But it dominates dynamics when $\rho \sim \rcr$. In particular, when $\rho =\rcr$ the right side vanishes, whence $\dot{a} =0$ and we have a quantum bounce.
The expression of the critical density contains two dimensionless parameters, $\gamma$ and $\a_o$. In LQG, the value of the Barbero-Immirzi parameter $\gamma$ is fixed through a black hole entropy calculation [@abck-lett; @abk] and turns out to be $\gamma \approx 0.2375$. The second parameter is $\alpha_o$. In the LQC literature to date it has been taken to be the minimum nonzero eigenvalue of the area operator, $\alpha_o =
2\sqrt{3}\pi\gamma$, on gauge invariant states (see, e.g., [@aps2; @apsv; @acs]). The corresponding area eigenstate has two edges: an edge $e_1$, which terminates *transversally* at a vertex $v$ on the surface whose area is being computed, and meets there an edge $e_2$ that is *tangential* to the surface (each edge carrying a spin-label $j=1/2$). However, we recently realized that if one sets up a semiheuristic correspondence between LQG and LQC quantum states, such states would not occur in homogeneous models: at $v$, $e_1$ would meet an edge $e_2$, *which is also transversal* to the surface. Therefore in LQC one should take the minimum area eigenvalue *within this class*. That minimum is $\alpha_o =
4\sqrt{3}\pi\gamma \approx 5.166$.[^6] Thus, in this paper we will set $\gamma= 0.2375$ and $\alpha_o =
5.166$.
Entropy bound in the LQC-corrected FRW model {#s3.2}
--------------------------------------------
Since we have restricted ourselves to the k=0, isotropic homogeneous situation, the effective space-time metric in LQC again has the form \[lqc-metric\] ds\^2=-d\^2+a()\^2(dr\^2+r\^2d\^2). Any symmetric, second rank tensor field $T_{ab}$ that is invariant under these symmetries can be written as $T_{ab} = \rho \nabla_a\tau
\nabla_b \tau + P (g_{ab} \,+\, \nabla_a \tau \nabla_b\tau)$. Hence the stress energy tensor is necessarily that of a perfect fluid. Since we are working with a radiation filled universe, $P=\f{1}{3}\rho$. As usual we can model it using kinetic theory on a curved space-time. The resulting conservation equation $\nabla^aT_{ab}=0$ again yields the continuity equation: \[cont2\] +3(+P)=0.Thus the form of the metric and the continuity equations are the same as in the classical theory. However as we saw in Sec. \[s3.1\], the Friedmann equation is now replaced by: \[mfried2\] H\^2 = (1 - ), = . We can repeat the procedure followed in Sec. \[s2\] and use these three equations to solve for $a(\tau)$. The result is: \[aLQC\] a()=(\^2+)\^[1/4]{}, () = where, as before, $K_o$ is an integration constant which does not have a direct physical significance. This functional form of $a(\tau)$ is very similar to Eq. (\[aFRW\]) in the classical case, and reduces to it in the limit $\hbar \rightarrow 0$ as well as in the limit $\tau \rightarrow \pm \infty$. The critical density $\rcr$ at which the universe bounces is a measure of the strength of the effects of quantum gravity. The smaller $\rcr$ is (or, the larger the area gap $\a_o$ is), the earlier the onset of the quantum regime will be. Now, in the classical theory, the ratio $S/A$ could be made arbitrarily large by choosing the surface $\B$ sufficiently close to the singularity. However this is precisely the regime in which LQC effects become dominant. Therefore we are now led to ask:
- Do these effects naturally bound the ratio $S/A$ under consideration?
- If so, for the values of parameters $\gamma$ and $\alpha_o$ that are currently used in LQG and LQC, is the bound less than $0.25/\lp^2 $?
These questions can now be answered by a straightforward calculation. Let us again choose a round 2-sphere $\B$ at an instant of time $\tau_f$ and consider its past lightsheet. Since the space-time metric (\[lqc-metric\]) is conformally flat *and nonsingular*, these null rays necessarily converge to a point $p$, say at time $\tau_i$, so that $\B$ is a cross section of the (future) light cone of $p$.[^7] However as one moves along the light cone from $\B$ to $p$, the expansion may become positive somewhere, in which case the light sheet of $\B$ would only be a *portion* of the (future) light cone of $p$. If not, the light-sheet is the entire light cone from $p$ to $\B$. Both these possibilities occur. But in either case, the area of $\B$ is just A = 4\^2R\_f\^2 R\_f = \_[\_i]{}\^[\_f]{} . Using the space-time metric it is easy to calculate the entropy current $s^a = [(\rho+P)/T]u^a$ through *the full light cone*, where $u^a$ is the unit normal to the $\tau={\rm const}$ slices. We obtain: = ()\^[1/4]{} , where $R_f$ is determined by a hypergeometric function: R\_f(\_i,\_f) =\_[\_i]{}\^[\_f]{} . If the light-sheet of $\B$ is only a portion of the light cone, the $S/A$ relevant for the covariant entropy bound will be smaller.
![ The $(\tau_i, \tau_f)$ region (in Planck units) where LQC corrections are significant. Since $\tau_i <\tau_f$, the striped portion is excluded. For $(\tau_i,\tau_f)$ in the black region, surfaces $\B$ do not admit past light-sheets. In the grey region, the expansion is negative near $\B$ but then becomes positive so that the light-sheet $\L$ of $\B$ is incomplete. Points in the white region are the most interesting ones for testing the entropy bound. For $(\tau_i,\tau_f)$ in this region, light-sheets of surfaces $\B$ are complete.[]{data-label="fig1"}](expansion.eps){width="4in"}
As noted in Sec. \[s3.1\], the LQC solution is extremely close to the classical FRW solution once the matter density $\rho$ falls to about $1\%$ of the Planck density $\rho_{\pl}$. Since the covariant entropy bound is respected in the classical solution in this regime, it suffices to examine a neighborhood of the bounce in which $\rho > 10^{-2}\, \rho_{\pl}$. In the conventions used above, the bounce occurs at $\tau=0$, and the matter density is approximately $1.86\times 10^{-3}\rho_{pl}$ at $\tau= \pm
4t_{\pl}$. Therefore, it is sufficient to examine the region $\tau_i,\:\tau_f\in \left[-4t_{\pl},4 t_{\pl} \right]$.
Because $\tau_i$ is necessarily less than $\tau_f$, the allowed range of these parameters excludes the striped region in Fig. \[fig1\]. For $(\tau_i, \tau_f)$ in the allowed region, consider the (future) null cone of any point $p$ at $\tau=\tau_i$ till it intersects the surface $\tau=\tau_f$ in a round 2-sphere $\B$. If the point $(\tau_i, \tau_f)$ lies in the black region, the expansion of the light rays is positive at $\B$, whence no portion of the null cone is a light-sheet of $\B$. (Thus, all $\B$ in the black region are future trapped surfaces.) Therefore the black portion is irrelevant for the conjecture. If a point $(\tau_i, \tau_f)$ lies in the grey area, then the expansion is negative near $\B$ but becomes positive as we move further back in time. So, in this case the light-sheet is only a portion of the light cone. An explicit calculation shows that, because of this truncation, the light-sheet $\L$ is too short for the entropy flux through it to violate the bound. Thus, to test the entropy bound, the nontrivial portion of the $(\tau_i, \tau_f)$ plane is just the white region. The corresponding surfaces $\B$ have complete past light-sheets.
Figure \[fig2\]a shows the ratio $S/A$ in Planck units for $(\tau_i,\tau_f)$ in this region, while Fig. \[fig2\]b focuses on the region where the ratio reaches its maximum. The maximum attained is ,for $\tau_i\cong -2.035\,t_{\pl}$ and $\tau_f\cong0.0619\,
t_{\pl}$. Thus the ratio does approach the 1/4$\lp^2$ specified in the covariant bound but does not quite reach it.
To conclude, let us return to the two questions raised earlier in this subsection. In the above calculations we used the specific values of $\gamma= 0.2375$ and $\a_o = 5.166$ for the parameters $\gamma$ and $\a_o$ that appear in the expression of the quantum corrected space-time metric. The value of $\gamma$ is well motivated by black hole entropy considerations [@abk] but an independent check is still lacking. The basis for the value of $\a_o$ is more tentative, because we do not have a systematic procedure to arrive at the Hamiltonian constraint of LQC from a specific, well-defined proposal in LQG. However, since the quantum corrected geometry is well defined for any value of these parameters, it follows that $S/A$ would remain bounded even if these values were to shift. Thus, the affirmative answer to the first question is robust. The second question, on the other hand, refers to specific values of the parameters. The fact that the bound is satisfied but almost saturated near the bounce brings out an unforeseen coherence, thereby providing independent circumstantial evidence that the current values of $\gamma$ and $\alpha_o$ may be correct to a good approximation.
Discussion {#s4}
==========
In Sec. \[s2\] we found that in the classical, radiation filled FRW model, the covariant entropy bound is violated near the big bang singularity. However, the violation occurs in the region where space-time curvature and matter density are of Planck scale. Therefore one would expect quantum gravity effects to be dominant. The question is whether they can protect the bound. A priori this appears to be a very difficult question to address because quantum gravity effects would typically introduce large fluctuations of geometry in the Planck regime, making the notion of a light-sheet $\L$ of a spacelike surface $\B$ —and hence the very statement of the bound— ill defined.
LQC provides a near ideal setting to investigate this issue because of two features. First, the singularity is resolved: quantum states can be evolved “through” the putative singularity. The universe simply undergoes a quantum bounce and LQC equations provide a deterministic evolution from a pre-big-bang branch to our current post-big-bang branch. In FRW models, these results are robust: they hold irrespective of whether there is a cosmological constant [@bp], whether we have a k=0 or k=1 universe [@aps2; @apsv], and the singularity resolution is valid for all states [@acs]. Second, the LQC wave function remains sharply peaked on a smooth geometry even near the bounce point. Therefore, there is a quantum corrected, “effective” metric —obtained by taking expectation values— which reproduces the full quantum dynamics to an excellent degree of approximation. Although its coefficients involve $\hbar$, the metric is smooth and can be used to introduce spacelike surfaces $\B$ and their light-sheets $\L$. So, we can now ask: Is the covariant entropy bound respected in this quantum corrected geometry? We cannot simply use one of conditions [@fmw; @bfm; @ap] that ensures the satisfaction of the bound because Einstein’s equations do not hold on the quantum corrected space-time. However, a direct calculation in Sec. \[s3.2\] showed that the answer is in the affirmative.
While this calculation provides an interesting convergence of very different ideas related to quantum gravity, it has some important limitations that stem from the current formulation of the bound. First, the statement of the entropy bound requires a well-defined notion of entropy current $s^a$. In practice this means that the matter be described in hydrodynamical terms. Matter described by “fundamental” classical and quantum fields does not readily admit such a description. Furthermore, even in the hydrodynamic context, entropy current is typically well-defined only in equilibrium. These restrictions led us to use radiation fluid as matter and assume that it is in instantaneous equilibrium throughout the history of the universe. These approximations are suspect especially in the Planck regime. The fact that the entropy bound persists in spite of these seemingly crude approximations is intriguing. Indeed even in the classical Friedmann universe, the bound is respected after $\tau \approx 0.06 \tau_{\pl}$, a time at which the matter density is about $8.3 \rho_{\pl}$. Why does the bound continue to hold in such circumstances which are clearly beyond the scope of approximations that are made? Such “unreasonable” successes have led to the suggestion that the bound may have truly fundamental significance and should be a cornerstone of any quantum gravity theory.
The most serious limitation of the current formulation of the bound is that it requires a smooth classical geometry and, even on such a geometry, one does not readily allow quantum matter fields. As discussed in Sec. \[s1\], since quantum matter violates the dominant (or null) energy condition, the available proofs break down. Can one somehow incorporate quantum violations of energy conditions? In the context of 2-dimensional, semiclassical space-times which include the back reaction of the Hawking radiation, there is an interesting proposal: modify the right-hand side of the bound by addition of a term corresponding to “entanglement entropy” which could protect the bound in spite of quantum violations [@st]. Perhaps the arguments available in the classical theory [@fmw; @bfm; @ap] can be generalized to prove a version of an appropriately modified bound also in four dimensions. However the next step, dropping reference to a smooth classical geometry, would be *significantly* more difficult. What is to become of the light-sheet $\L$ or indeed of a spacelike surface $\B$ in situations [@aa-large; @atv] in which, unlike quantum cosmology, there are large fluctuations of geometry ?
Our overall viewpoint on the bound can be summarized as follows. The bound is strongly motivated by the generalized second law (which also does not have a well-defined, definitive formulation). Now, already the standard second law of thermodynamics is a deep fact of Nature but it has a “fuzziness” which is not shared by other deep laws such as the conservation of energy-momentum and angular momentum. In particular, the second law requires a coarse graining in an essential way. It is not a statement about the evolution of micro-states; in a fundamental theory their dynamics is always time reversible (leaving aside, for simplicity, quantum measurements). Rather, it is a statement about how the number of micro-states compatible with a prespecified coarse graining changes in time. For instance if we have a gas that is confined to the left half of a box and we open the partition and wait till equilibrium is again reached, the entropy increases. Any one of the final micro-states (occupying the entire box) of these experiments will, under time-reversal, evolve back to a micro-state that is confined to the left half. Nonetheless, entropy increases because the *total* number of states compatible with the final coarse graining —most of which would not have resulted as end points of an *actual* evolution in our experiment— is enormously larger than that corresponding to the initial coarse graining. Thus, while an increase of entropy can be calculated using statistical mechanics, it has little relevance to the fundamental dynamics of micro-states. It is also not an input in the construction of statistical mechanics. In the same vein, we believe that the covariant entropy bound —and its appropriate generalizations that could encompass quantum field theory processes even on “quantum corrected” but smooth space-times— should *emerge* from a fundamental quantum gravity theory. These bounds would be valuable but are not essential ingredients in the *construction* of such a theory. The distinguishing feature about LQC, for example, is the underlying quantum geometry. The covariant entropy bound was never an input or even a motivation. Nonetheless, it emerged on the quantum corrected space-time as a consequence of LQC.
Acknowledgements: {#acknowledgements .unnumbered}
=================
We would like to thank Martin Bojowald, Éanna Flanagan, Ian Hinder, Jainendra Jain, Don Marolf, Tomasz Pawlowski and Parampreet Singh for helpful discussions. This work was supported in part by the NSF Grant No. PHY04-56913, The Natural Sciences and Engineering Research Council of Canada and Le Fonds québécois de la recherche sur la nature et les technologies, the Alexander von Humboldt Foundation, and the Eberly research funds of Penn State.
[99]{}
J. D. Bekenstein, [Black holes and the second law]{}, Nuovo Cim. Lett. **4**, 737 (1972).
J. D. Bekenstein, [Black holes and entropy]{}, Phys. Rev **D7**, 2333 (1973).
J. M. Bardeen, B. Carter and S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. **31**, 161-170 (1973).
Ashtekar A and Krishnan B (2004) Isolated and dynamical horizons and their applications *Living Rev. Rel.* **7**, 10, `arXiv:gr-qc/0407042`
S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. **43**, 199-220 (1975)
J. D. Bekenstein, [Generalized second law of thermodynamics in black hole physics]{}, Phys. Rev. **D9**, 3292 (1974).
G. ’t Hooft, [Dimensional reduction in quantum gravity]{}, (1993), `arXiv:gr-qc/9310026v1`.
L. Susskind, [The world as a hologram]{}, J. Math. Phys. **36**, 6377 (1995), `arXiv:9409089`.
R. Bousso, [The holographic principle]{}, Rev. Mod. Phys. **74**, 825-874 (2002), `arXiv:hep-th/0203101`.
R. Bousso, [A covariant entropy conjecture]{}, JHEP **07**, 004 (1999), `arXiv:hep-th/9905177`.
E. E. Flanagan, D. Marolf and R. M. Wald, [Proof of the classical versions of the Bousso entropy bound and of the Generalized Second Law]{}, Phys. Rev. **D62**, 084035 (2000), `arXiv:hep-th/9908070`.
R. Bousso, E. E. Flanagan and D. Marolf, [Simple sufficient conditions for the generalized covariant entropy bound]{}, Phys. Rev. **D68**, 064001 (2003), `arXiv:hep-th/0305149`.
A. Pesci, From Unruh temperature to generalized Bousso bound, Class. Quantum Grav. **24**, 6219-6225 (2007), `arXiv:0708.3729`.
W. Fischler and L. Susskind, Holography and Cosmology, (1998), `arXiv:hep-th/9806039`.
A. Ashtekar, Large quantum gravity effects: Unexpected limitations of the classical theory, Phys. Rev. Lett. **77**, 4864-4867 (1996), `arXiv:gr-qc/9610008`.
A. Ashtekar, V. Taveras, M. Varadarajan, [Information is not lost in the evaporation of 2-dimensional black holes]{}, Phys. Rev. Lett. **100**, 211302 (2008) `arXiv:0801.1811`
A. Ashtekar, An introduction to loop quantum gravity through cosmology, Nuovo Cimento **B122**, 135 (2007), `arXiv:gr-qc/0702030`.
A. Ashtekar, T. Pawlowski and P. Singh, [Quantum nature of the big bang]{}, Phys. Rev. Lett. **96**, 141301 (2006), `arXiv:gr-qc/0602086`.
A. Ashtekar, T. Pawlowski and P. Singh, [Quantum nature of the big bang: Improved dynamics]{}, Phys. Rev. **D74**, 084003 (2006), `arXiv:gr-qc/0607039`.
A. Ashtekar, T. Pawlowski, P. Singh and K. Vandersloot, [Loop quantum cosmology of $k$=1 FRW models]{}, Phys. Rev. **D75**, 024035 (2007), `arXiv:gr-qc/0612104`.
A. Ashtekar and J. Lewandowski, [Background independent quantum gravity: A status report]{}, Class. Quant. Grav. [**21**]{}, R53-R152 (2004), `arXiv:gr-qc/0404018`.
C. Rovelli [*Quantum Gravity*]{}, (CUP, Cambridge, 2004).
T. Thiemann [*Modern Canonical Quantum General Relativity*]{}, (CUP, Cambridge, 2007).
A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Quantum Geometry and Black Hole Entropy, Phys. Rev. Lett. **80**, 904-907 (1998), `arXiv:gr-qc/9710007`.
A. Ashtekar, J. Baez, and K. Krasnov, Quantum Geometry of Isolated Horizons and Black Hole Entropy, Adv. Theor. Math. Phys. **4**, 1-94 (2000).\
Domagala M and Lewandowski J 2004 Black hole entropy from quantum geometry, Class. Quant. Grav. **21**, 5233-5244 (2004) `arXiv:gr-qc/0407051`.\
Meissner K A 2004 Black hole entropy in loop quantum gravity, Class. Quant. Grav. **21**, 5245-5252 (2004) `arXiv:gr-qc/0407052`.
A. Ashtekar, J. Engle and C. Van Den Broeck, Quantum horizons and black hole entropy: Inclusion of distortion and rotation, Class. Quant. Grav. 22, L27-L34 (2005)
J. Lewandowski, A. Okolow, H. Sahlmann and T. Thiemann, [Uniqueness of diffeomorphism invariant states on holonomy flux algebras]{}, Comm. Math. Phys. **267**, 703-733 (2006) `arXiv:gr-qc/0504147`;\
C. Fleishchack, [Representations of the Weyl algebra in quantum geometry]{}, `arXiv:math-ph/0407006`.
T. Thiemann, Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity, *Phys. Lett.* **B380**, 257–264 (1996);\
Quantum spin dynamics (QSD), [Class. Quant. Grav.]{} **15** 839–873 (1998);\
QSD V : Quantum gravity as the natural regulator of matter quantum field theories, [Class. Quant. Grav.]{} **15**, 1281–1314 (1998).
A. Ashtekar, M. Bojowald, J. Lewandowski, [Mathematical structure of loop quantum cosmology]{}, Adv. Theo. Math. Phys. **7**, 233-268 (2003), `arXiv:gr-qc/0304074`.
M. Bojowald, Large scale effective theory for cosmological bounces, Phys. Rev. **D75**, 081301 (2007), `arXiv:gr-qc/0608100`;\
A. Ashtekar, A. Corichi and P. Singh, Robustness of key features of quantum cosmology, Phys. Rev. **D77**,024046 (2008), `arXiv:0710.3565`
E. Bentevigna and T. Pawlowski, Anti-deSitter universe dynamics in LQC, `arXiv:0803.4446`
A. Ashtekar and T. Schilling, Geometrical formulation of quantum mechanics, In: *On Einstein’s path*, A. Harvery, ed (Springer-Verlag, New York, 1998); `arXiv:gr-qc/9706069`. T. Schilling, Geometry of quantum mechanics, Ph.D. Dissertation, Penn State (1996), http://www.gravity.psu.edu/research/archives/thesis/index.shtml
J. Willis, [On the low energy ramifications and a mathematical extension of loop quantum gravity]{}, Ph.D. Dissertation, The Pennsylvania State University (2004), http://www.gravity.psu.edu/research/archives/thesis/index.shtml\
A. Ashtekar, M. Bojowald and J. Willis, [Corrections to Friedmann equations induced by quantum geometry]{}, IGPG preprint (2004)\
V. Taveras, IGPG preprint (2008).
A. Strominger and D. Thompson, A quantum Bousso bound, (2003), `arXiv:hep-th/0303067`.
[^1]: Such problems in the deep Planck regime very near the big bang were encountered also in the earlier versions of entropy bounds [@fs].
[^2]: Explicit examples of such phenomena are provided by some symmetry reduced models. See, e.g., [@aa-large; @atv].
[^3]: For a relatively short review of LQC and its relation to LQG, see, e.g., [@aa-rev].
[^4]: Thus, indices $a,b,c,\ldots$ refer to the tangent space of the “spatial” 3-manifold and $i,j,k,\ldots$ to the Lie algebra of $\SU(2)$.
[^5]: In this formulation, the space of quantum states is represented by an infinite dimensional phase space $\Gamma_{\rm
Quan}$ on which the exact quantum dynamics defines a Hamiltonian flow (see, e.g., [@as]). The quantum corrected equations are obtained by projecting this flow on a suitably chosen subspace which is isomorphic to the classical phase space $\Gamma_{\rm Cl}$. To distinguish them from effective equations obtained from other methods, these equations should really be referred to as “geometric quantum mechanics projected equations”. However, for brevity we will refer to them just as “effective” equations.
[^6]: These considerations become quite important in a systematic treatment of the Hamiltonian constraint in non-isotropic models.
[^7]: It is not necessary to consider, in addition, past light cones because the quantum corrected “effective” solution is symmetric about the bounce point at $\tau=0$.
|
---
abstract: 'Holomorphic chains on a Riemann surface arise naturally as fixed points of the natural ${\mathbb{C}}^*$-action on the moduli space of Higgs bundles. In this paper we associate a new quiver bundle to the $\operatorname{Hom}$-complex of two chains, and prove that stability of the chains implies stability of this new quiver bundle. Our approach uses the Hitchin-Kobayashi correspondence for quiver bundles. Moreover, we use our result to give a new proof of a key lemma on chains (due to Álvarez-Cónsul–Garc[í]{}a-Prada–Schmitt), which has been important in the study of Higgs bundle moduli; this proof relies on stability and thus avoids the direct use of the chain vortex equations.'
address:
- |
Centro de Matemática da Universidade do Porto\
Faculdade de Ciências da Universidade do Porto\
Rua do Campo Alegre, s/n\
4169-007 Porto\
Portugal
- |
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional\
Faculdade de Ciências da Universidade de Lisboa\
Edf. C6, Campo Grande\
1749-016 Lisboa\
Portugal
author:
- 'P. B. Gothen'
- 'A. Nozad'
title: Quiver bundles and wall crossing for chains
---
Introduction
============
A *holomorphic $(m+1)$-chain* on a compact Riemann surface $X$ of genus $g{\geqslant}2$ is a diagram $$\xymatrix{
C:\mbox{ }E_m\ar[r]^{\phi_m}&E_{m-1}\ar[r]^{\phi_{m-1}}&\cdots\ar[r]^{\phi_2}&E_1\ar[r]^{\phi_1}&E_0,\\
}$$ where each $E_i$ is a holomorphic vector bundle and $\phi_i: E_i\longrightarrow E_{i-1}$ is a holomorphic map. Moduli spaces for holomorphic chains have been constructed by Schmitt [@Schmitt:2005] using GIT and, as is usual for decorated bundles, depend on a stability parameter $\boldsymbol{\alpha}=(\alpha_0,\dots,\alpha_m)$, where $\alpha_i\in{\mathbb{R}}$.
One important application of holomorphic chains stems from the fact that, for a specific value of the stability parameter, their moduli can be identified with fixed loci for the natural ${\mathbb{C}}^*$-action on the moduli space of Higgs bundles. Thus, knowledge of moduli spaces of chains can be used to study the moduli space of Higgs bundles. The basic idea (in the case of rank 2 Higgs bundles) goes back to the seminal paper of Hitchin [@hitchin:1987a].
For higher rank Higgs bundles, knowledge of the moduli of chains becomes in itself difficult to come by, and a successful strategy for this has been to study the variation of the moduli of chains under changes in the parameter, using wall crossing arguments. This approach goes back to the work of Thaddeus [@thaddeus:1994] (used for rank 3 Higgs bundles in [@gothen:1994]). Recent important examples of the study of wall crossing of chains and applications to moduli of Higgs bundles include the work of García-Prada–Heinloth–Schmitt [@garcia-heinloth-schmitt:2014], García-Prada–Heinloth [@garcia-heinloth:2013], and Heinloth (see also Bradlow–García-Prada–Gothen–Heinloth [@bradlow-etal:2017] for an application to ${\mathrm{U}}(p,q)$-Higgs bundles). We should mention here that recently alternative approaches to the study of the cohomology of Higgs bundle moduli have been highly succesful: see Schiffman [@schiffmann:2016], Mozgovoy–Schiffman [@mozgovoy-schiffman:2017] and Mellit [@mellit:2017]; also, Maulik–Pixton have announced a proof of a conjecture of Chuang–Diaconescu–Pan [@CDP] which leads to a calculation of the motivic class of the moduli space of twisted Higgs bundles.
All the aforementioned results on chains rely on a key result of Álvarez-Cónsul–Garc[í]{}a-Prada–Schmitt [@Consul-Prada-Schmitt:2006 Proposition 4.14] which, in particular, is used in estimating codimensions of flip loci under wall crossing. The proof of this result is analytic in nature and relies on the solutions to the chain vortex equations, whose existence is guaranteed by the Hitchin–Kobayashi correspondence for holomorphic chains (see Álvarez-Cónsul–Garc[í]{}a-Prada [@Prada:2001; @Prada:2003]).
In this paper, given a pair of chains, we associate to them a new quiver bundle which extends and refines the $\operatorname{Hom}$-complex of the chains; we call it the *extended $\operatorname{Hom}$-quiver*. Moreover, we show that polystability of the chains implies polystability of this extended $\operatorname{Hom}$-quiver (see Theorem \[relating polystability\]). We then use our result to give a new and simpler proof of the key result [@Consul-Prada-Schmitt:2006 Proposition 4.14] mentioned above (see Theorem \[thm:AGS\]). The main merit of our argument is that it is algebraic, in the sense that it only uses stability of the extended $\operatorname{Hom}$-quiver and avoids direct use of the chain vortex equations. Thus, though our proof of Theorem \[relating polystability\] does ultimately rely on the Hitchin–Kobayashi correspondence (through Lemma \[lem:hom-complex-vortex\]), the roles of the correspondence and of stability are clarified. Our result can be viewed as a generalization of a result of [@bradlow-garcia-prada-gothen:2004] for length two chains (also known as triples), though in this case the extended $\operatorname{Hom}$-quiver is itself a chain.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Steve Bradlow for useful discussions and we thank the referee for insightful comments which helped improve the exposition.
Definitions and basic results {#sec:basic-results}
=============================
In this section we recall definitions and relevant facts on quiver bundles, from [@Gothen:2005] and [@Prada:2003].
Quiver bundles
--------------
A *quiver* $Q$ is a directed graph specified by a set of vertices $Q_0$, a set of arrows $Q_1$ and head and tail maps $h,t : Q_1\to Q_0$. We shall assume that $Q$ is finite.
A holomorphic *quiver bundle*, or simply a *$Q$-bundle*, is a pair $\mathcal{E} = (V, \varphi)$, where $V$ is a collection of holomorphic vector bundles $V_i$ on $X$, for each $i \in Q_0$, and $\varphi$ is a collection of morphisms $\varphi_a: V_{ta}\to V_{ha}$, for each $a\in Q_1$.
The notions of $Q$-subbundles and quotient $Q$-bundles, as well as simple $Q$-bundles are defined in the obvious way. The subobjects $(0,0)$ and $\mathcal{E}$ itself are called the *trivial subobjects*. The *type* of a $Q$-bundle $\mathcal{E}= (V, \varphi)$ is given by $$t(\mathcal{E})=(\operatorname{rk}(V_i); \deg(V_i))_{ i\in Q_0},$$ where $\operatorname{rk}(V_i)$ and $\deg(V_i)$ are the rank and degree of $V_i$, respectively. We sometimes write $\operatorname{rk}(\mathcal{E})= \operatorname{rk}(\bigoplus V_i)$ and call it the *rank of $\mathcal{E}$*. Note that the type is independent of $\varphi$.
Stability
---------
Fix a tuple $\boldsymbol{\alpha}=(\alpha_i)\in\mathbb{R}^{|Q_0|}$ of real numbers. For a non-zero $Q$-bundle $\mathcal{E}=(V,\varphi)$, the associated *$\boldsymbol{\alpha}$-slope* is defined as $$\mu_{\boldsymbol{\alpha}}(\mathcal{E})=\frac{\underset{i\in Q_0}{\sum}\big(\alpha_i \operatorname{rk}(V_i)+\deg(V_i)\big)}{\underset{i\in Q_0}{\sum}\operatorname{rk}(V_i)}.$$
A $Q$-bundle $\mathcal{E}=(V,\varphi)$ is said to be *$\boldsymbol{\alpha}$-(semi)stable* if, for all non-trivial subobjects $\mathcal{F}$ of ${\mathcal{E}}$, $\mu_{\boldsymbol{\alpha}}(\mathcal{F})
<({\leqslant})\mu_{\boldsymbol{\alpha}}(\mathcal{E})$. An *$\boldsymbol{\alpha}$-polystable* $Q$-bundle is a finite direct sum of $\boldsymbol{\alpha}$-stable $Q$-bundles, all of them with the same $\boldsymbol{\alpha}$-slope.
A $Q$-bundle $\mathcal{E}$ is *strictly $\boldsymbol{\alpha}$-semistable* if there is a non-trivial subobject $\mathcal{F}\subset \mathcal{E}$ such that $\mu_{\boldsymbol{\alpha}}(\mathcal{F})=\mu_{\boldsymbol{\alpha}}(\mathcal{E})$.
In fact, the most general stability condition for quiver bundles involves additional parameters, see [@Consul-Prada-Schmitt:2006]. Since $\boldsymbol{\alpha}$ is the parameter which has been used in the literature for the study of moduli of chains via wall crossing, we confine ourselves to considering this parameter.
The gauge theory equations
--------------------------
Let $\mathcal{E}=(V,\varphi)$ be a $Q$-bundle on $X$. A Hermitian metric on $\mathcal{E}$ is a collection $H$ of Hermitian metrics $H_i$ on $V_i$, for each $i\in Q_0$. To define the gauge equations on $\mathcal{E}$, we note that $\varphi_a:V_{ta}\to V_{ha}$ has a smooth adjoint morphism $\varphi_a^\ast:V_{ha}\to V_{ta}$ with respect to the Hermitian metrics $H_{ta}$ on $V_{ta}$ and $H_{ha}$ on $V_{ha}$, for each $a\in Q_1$, so it makes sense to consider the compositions $\varphi_a\circ\varphi_a^\ast$ and $\varphi_a^\ast\circ\varphi_a$.
Let $\boldsymbol{\alpha}$ be the stability parameter. Define $\boldsymbol{\tau}$ to be the vector of real numbers $\tau_i$ given by $$\tau_i=\mu_{\boldsymbol{\alpha}}(\mathcal{E})-\alpha_i, \mbox{ }i\in Q_0.\label{relation}$$ Since the stability condition does not change under a global translation $\boldsymbol{\alpha}$ can be recovered from $\boldsymbol{\tau}$ as follows $$\begin{aligned}
\alpha_i=\tau_{0}-\tau_i, \mbox{ } i\in Q_0.\end{aligned}$$
A Hermitian metric $H$ satisfies the *quiver $\boldsymbol{\tau}$-vortex equations* if $$\sqrt{-1}\Lambda F(V_i)+\underset{i=ha}{\sum}\varphi_a\varphi_a^{\ast}-\underset{i=ta}{\sum}\varphi_a^{\ast}\varphi_a=\tau_i \mathrm{Id}_{V_i}\label{vortex equations1}$$ for each $i\in Q_0$, where $F(V_i)$ is the curvature of the Chern connection associated to the metric $H_i$ on the holomorphic vector bundle $V_i$, and $ \Lambda: \Omega^{i,j}(M)\to \Omega^{i-1,j-1}(M)$ is the contraction operator with respect to a fixed Kähler form $\omega$ on $X$.
The following is the *Hitchin–Kobayashi correspondence* between the twisted quiver vortex equations and the stability condition for holomorphic twisted quiver bundles, given by Álvarez-Cónsul and García-Prada [@Prada:2003 Theorem 3.1]:
\[Hitchin-Kobayashi\] A holomorphic $Q$-bundle $\mathcal{E}$ is $\boldsymbol{\alpha}$-polystable if and only if it admits a Hermitian metric $H$ satisfying the quiver $\boldsymbol{\tau}$-vortex equations $(\ref{vortex equations1})$, where $\boldsymbol{\alpha}$ and $\boldsymbol{\tau}$ are related by $(\ref{relation})$.
Note that the definitions and facts can be specialized for holomorphic chains.
The $\operatorname{Hom}$-complex for chains
-------------------------------------------
Fix two holomorphic chains $C''$ and $C'$, given by $$\xymatrix{
C':\mbox{ }E'_m\ar[r]^{\phi'_m}&E'_{m-1}\ar[r]^{\phi'_{m-1}}&\cdots\ar[r]^{\phi'_2}&E'_1\ar[r]^{\phi'_1}&E'_0\\
}$$ $$\xymatrix{
C'':\mbox{}E''_m\ar[r]^{\phi''_m}&E''_{m-1}\ar[r]^{\phi''_{m-1}}&\cdots\ar[r]^{\phi''_2}&E''_1\ar[r]^{\phi''_1}&E''_0\\
}$$ Consider the following two terms complex of sheaves $$\label{deformation complex of chains}
\mathcal{H}^{\bullet}(C'',C'): \mathcal{H}^0\overset{d}{\longrightarrow} \mathcal{H}^{1}$$ with terms $$\begin{aligned}
& &\mathcal{H}^0=\bigoplus_{i-j=0} \operatorname{Hom}(E''_i,E'_j),\mbox{ }\mathcal{H}^{1}=\bigoplus_{i-j=1}\operatorname{Hom}( E''_i, E'_j),\end{aligned}$$ and the map $d$ is defined by $$d(g_0,\ldots,g_m)=(g_{i-1}\circ\phi''_i-\phi'_i\circ g_i),\mbox{
for }g_i\in \operatorname{Hom}(E''_i,E'_i).$$ The complex $\mathcal{H}^{\bullet}(C'',C')$ is called the *$\operatorname{Hom}$-complex*. It governs the homological algebra of chains; in particular $\mathcal{H}^{\bullet}(C,C)$ is the deformation complex of a chain $C$.
The extended $\operatorname{Hom}$-quiver
========================================
Here we introduce a $Q$-bundle, associated to two chains, and show that solutions to the vortex equations on the holomorphic chains produce a solution on the corresponding quiver bundle. The basic idea is the following: to the chains $C''$ and $C'$ we associate the vector bundles $E''$ and $E'$, obtained as the direct sum of the individual bundles in the chains. The quiver bundle structure on the chains then induces a natural quiver bundle structure on the bundle $\operatorname{Hom}(E'',E')$. Thus our construction can be seen as a kind of extension of structure group and it becomes natural to expect that a solution to the vortex equations on the chains should give a solution on the induced quiver bundle. Indeed, this is exactly the content of our Lemma \[lem:hom-complex-vortex\] below. This in turn implies the main result of this section, Theorem \[relating polystability\], which says that $\tilde{\mathcal{H}}(C'',C')$ is (poly)stable for suitable values of the parameter.
We note that, since there are algebraic proofs of results saying that stability is preserved under extension of structure group (in the setting of principal bundles by Ramanan–Ramanathan [@RR] and for Hitchin pairs by Balaji–Parameswaran [@BP2012]) this might indicate the possibility of an algebraic proof of our result as well, though we do not pursue this possibility here.
We also point out that one might attempt to generalize our construction to more general quiver bundles than chains; see [@GN Section 4.2] for the case of ${\mathrm{U}}(p,q)$-Higgs bundles.
Let $C'$ and $C''$ be chains of length $m$. The *extended $\operatorname{Hom}$-quiver* $\tilde{\mathcal{H}}(C'',C')$ is a quiver bundle defined as follows:
- For each $(i,j)$ with $0{\leqslant}i,j{\leqslant}m$, there is a vertex to which we associate the bundle $\operatorname{Hom}(E_i'',E_j')$, of *weight $k=i-j$*.
- For each $\operatorname{Hom}(E_i'',E_j')$, of weight $k=i-j$, there are maps $$\begin{aligned}
\delta_{ij}^-\colon\operatorname{Hom}(E_i'',E_j') &\to \operatorname{Hom}(E_i'',E_{j-1}'),\\
f &\mapsto -\phi_j'\circ f,
\end{aligned}$$ and $$\begin{aligned}
\delta_{ij}^+\colon\operatorname{Hom}(E_i'',E_j') &\to \operatorname{Hom}(E_{i+1}'',E_{j}'),\\
f &\mapsto f\circ\phi_{i+1}''.
\end{aligned}$$
In other words, $\tilde{\mathcal{H}}(C'',C')$ is defined by associating to $(E''=\bigoplus E_i'',\phi''=\sum_i\phi_i'')$ and $(E'=\bigoplus E_i',\phi'=\sum_i\phi_i')$ the bundle $\operatorname{Hom}(E'',E')$ and the map $$\begin{aligned}
\operatorname{Hom}(E'',E') &\to \operatorname{Hom}(E'',E'),\\
f &\mapsto f\circ \phi''-\phi'\circ f,\end{aligned}$$ and then taking the quiver bundle induced from the splitting $\operatorname{Hom}(E'',E') = \bigoplus_{i,j} \operatorname{Hom}(E_i'',E_j')$. We can picture this construction as follows:
$${\refstepcounter{equation}\tag{\theequation}}\label{quiver}
\resizebox{\linewidth}{!}{
\xymatrix{
& & & && \operatorname{Hom}(E''_0,E_0')\ar[dr]& & & &\\
& & & &\operatorname{Hom}(E_{0}'', E_1')\ar[ur]\ar[dr]& &\operatorname{Hom}(E_1'',E_0')\ar[dr]& & &\\
& & & \iddots\ar[ur]\ar[dr]& \vdots& \operatorname{Hom}(E''_1,E_1')\ar[ur]& \vdots& \ddots\ar[dr]&& \\
& &\operatorname{Hom}(E_0'',E'_{m-1})\ar[ur]\ar[dr]& &\operatorname{Hom}(E_{i-1}'', E_i')\ar[dr]&\vdots &\operatorname{Hom}(E_{i}'',E_{i-1}')\ar[ur] \ar[dr]&&\operatorname{Hom}(E_{m-1}'',E'_0)\ar[dr] \\
&\operatorname{Hom}(E_0'',E'_{m})\ar[ur]\ar[dr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots& &\operatorname{Hom}(E_i'', E_i')\ar[ur]\ar[dr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots & &\operatorname{Hom}(E_m'', E'_0)\\
& & \operatorname{Hom}(E_1'',E'_{m})\ar[ur]\ar[dr]&\vdots&\operatorname{Hom}(E_{i}'',E_{i+1}')\ar[ur]&\vdots & \operatorname{Hom}(E_{i+1}'',E_i') \ar[ur]\ar[dr]&& \operatorname{Hom}(E_m'',E'_{1})\ar[ur]& \\
& & &\ddots\ar[dr]\ar[ur] &&\operatorname{Hom}(E_{m-1}'',E'_{m-1})\ar[dr]&& \mbox{ }\mbox{ } \iddots \mbox{ }\mbox{ }\ar[ur] && \\
& & & &\operatorname{Hom}(E_{m-1}'', E'_m)\ar[ur]\ar[dr]&& \operatorname{Hom}( E_{m}'',E'_{m-1})\ar[ur]&&&\\
& & &&&\operatorname{Hom}(E_m'',E'_m)\ar[ur]& & &&}}$$
Note that if we take the direct sums of the middle two columns $${\textstyle\underset{i-j=0}{\bigoplus}\operatorname{Hom}(E_i'',E'_j)
\xrightarrow{\delta^++\delta^-}
\underset{i-j=1}{\bigoplus}\operatorname{Hom}(E_i'', E'_j)}$$ we obtain the $\operatorname{Hom}$-complex of the chains $C''$ and $C'$, defined in $(\ref{deformation complex of chains})$.
\[lem:hom-complex-vortex\] Let $C'$ and $C''$ be holomorphic chains and suppose we have solutions to the $(\tau'_0,\ldots,\tau'_m)$-vortex equations on $C'$ and the $(\tau''_0,\ldots,\tau_m'')$-vortex equations on $C''$. Then the induced Hermitian metric on the extended $\operatorname{Hom}$-quiver $\tilde{\mathcal{H}}(C'',C')$ pictured in $(\ref{quiver})$ satisfies the quiver $\widetilde{\boldsymbol{\tau}}$-vortex equations, for $\widetilde{\boldsymbol{\tau}}
=(\tilde{\tau}_{ij})
=(\tau'_j-\tau^{''}_{i})$.
To show that the induced Hermitian metric satisfies the equation at $\operatorname{Hom}(E_{i}'',E'_j)$ of weight $k$, for $-m{\leqslant}k{\leqslant}m$, first recall that we have the following identity of curvature operators: $$F\big(\operatorname{Hom}(E_{i}'', E'_j)\big)(f)=F(E'_j)\circ f -f\circ
F(E''_{i}).$$ Also, the vortex equations for $C'$ and $C''$ are $$\sqrt{-1}\Lambda F(E_i')+\phi_{i+1}'\phi_{i+1}^{'\ast}-\phi_i^{'\ast}\phi'_i=\tau_i' \operatorname{Id}_{E'_i},\mbox{ }i=0,\ldots,m$$ $$\sqrt{-1}\Lambda F(E^{''}_i)+\phi^{''}_{i+1}\phi^{''\ast}_{i+1}-\phi_i^{\ast''}\phi_i^{''}=\tau_i'' \operatorname{Id}_{E^{''}_i}\mbox{, }i=0,\ldots,m.$$ Now, considering the quiver $\tilde{\mathcal{H}}(C'',C')$ at $\operatorname{Hom}(E_{i}'',E'_j)$ we have $$\xymatrix@=0.5em@1@R=1.5em{
&\operatorname{Hom}(E_{i-1}'',E'_j)\ar[dr]^{\delta_c}& &\operatorname{Hom}(E_{i}'',E'_{j-1})\\
&&\operatorname{Hom}(E_{i}'',E'_{j})\ar[ur]^{\delta_b}\ar[dr]_{\delta_a}&\\
&\operatorname{Hom}(E_{i}'',E'_{j+1})\ar[ur]_{\delta_d}& &\operatorname{Hom}(E_{i+1}'',E'_j)}$$ where for ease of notation we have written $$\begin{aligned}
\delta_a(f) &=& \delta_{ij}^+(f)= f\circ\phi_{i+1}''\\
\delta_b(f) &=& -\delta_{ij}^-(f)= \phi'_j\circ f\\
\delta_c(g) &=& \delta_{i-1,j}^+(g)=g\circ\phi_{i}''\\
\delta_d(h) &=& -\delta_{i,j+1}^-(h)= \phi_{j+1}'\circ h
\end{aligned}$$ A straightforward calculation gives the following $$\begin{aligned}
\delta_a^\ast(g)&=& g\circ\phi_{i+1}^{''\ast}\\
\delta_b^\ast(h)&=& \phi^{'\ast}_j\circ h\\
\delta_c^\ast(f)&=&f\circ\phi_{i}^{''\ast}\\
\delta_d^\ast(f)&=& \phi_{j+1}^{'\ast}\circ f
\end{aligned}$$ therefore $$\begin{aligned}
\left(\delta_c\delta_c^\ast+\delta_d\delta_d^\ast-\delta_a^\ast\delta_a-\delta_b^\ast\delta_b\right)(f)
&=&
\delta_c(f\circ\phi_{i}^{''\ast})+\delta_d(\phi_{j+1}^{'\ast}\circ f)-\delta_a^\ast(f\circ\phi_{i+1}'')-\delta_b^\ast(\phi'_j\circ f)\\
&=&
f\circ\phi_{i}^{''\ast}\circ\phi_{i}''+\phi_{j+1}'\circ\phi_{j+1}^{'\ast}\circ f-f\circ\phi_{i+1}''\phi_{i+1}^{''\ast}-\phi^{'\ast}_j\phi'_j\circ f\end{aligned}$$ Hence, using the vortex equations for $C'$ and $C''$ and the above identity of curvature operators, we have for $f\in\operatorname{Hom}(E_{i+k}'', E'_i)$: $$\begin{aligned}
\lefteqn{(\sqrt{-1}\Lambda F(\operatorname{Hom}(E_{i}'', E'_j))+\delta_c\delta_c^\ast+\delta_d\delta_d^\ast-\delta_a^\ast\delta_a-\delta_b^\ast\delta_b)(f)}\\
&=&
\Big(\big(\sqrt{-1}\Lambda F(E'_j)+\phi^{''}_{j+1}\phi_{j+1}^{''\ast}-\phi_j^{''\ast}\phi^{''}_j\big)\circ f-f\circ\big( \sqrt{-1}\Lambda F(E''_{i})-\phi_{i}'\phi_{i}^{'\ast}+\phi_{i+1}^{'\ast}\phi_{i+1}'\big)\Big)\\
&=&
(\tau_j'-\tau''_{i})f.\end{aligned}$$ This finishes the proof.
\[relating polystability\] Let $C'$ and $C''$ be $\boldsymbol{\alpha}'=(\alpha'_1,\ldots,\alpha'_m)$ and $\boldsymbol{\alpha}''=(\alpha''_1,\ldots,\alpha''_m)$-polystable holomorphic chains, respectively. Then the extended $\operatorname{Hom}$-quiver $\tilde{\mathcal{H}}(C'',C')$, as in $(\ref{quiver})$, is $\boldsymbol{\widetilde{\alpha}}=(\widetilde{\alpha}_{ij})$-polystable for $
\widetilde{\alpha}_{ij}=\alpha''_m+\alpha'_j-\alpha''_{i}$.
Since the holomorphic chains $C'$ and $C''$ are $\boldsymbol{\alpha}'$- and $\boldsymbol{\alpha}''$-polystable, it follows from Proposition \[Hitchin-Kobayashi\] that both the $(\tau'_0,\ldots,\tau'_m)$- and the $(\tau''_0,\ldots,\tau''_m)$-vortex equations have a solution. Then, by Lemma \[lem:hom-complex-vortex\] the extended $\operatorname{Hom}$-quiver $\tilde{\mathcal{H}}(C'',C')$ satisfies the quiver $(\tau'_j-\tau^{''}_{i})$-vortex equations and therefore the Hitchin–Kobayashi correspondence implies that $\tilde{\mathcal{H}}(C'',C')$ is $\widetilde{\boldsymbol{\alpha}}$-polystable for $$\begin{aligned}
\widetilde{\alpha}_{ij}&=&\tau'_0-\tau''_m-(\tau'_{j}-\tau^{''}_i)=\tau'_0-\tau'_{j}+\tau''_0-\tau''_m+\tau''_i-\tau''_0=\alpha_m''+\alpha_{j}'-\alpha_i''.\end{aligned}$$
Application to wall crossing for chains
=======================================
As an application of Theorem \[relating polystability\] we give a simplified and more conceptual proof of a result of Álvarez-Cónsul, Garc[í]{}a-Prada and Schmitt in [@Consul-Prada-Schmitt:2006], showing how it follows from stability of the quiver bundle (\[quiver1\]). This result is a key ingredient in wall crossing arguments for holomorphic chains, which have had a number of important applications lately as explained in the introduction. First we state a particular case of our main theorem which will be used in the proof.
If we take $\boldsymbol{\alpha}=\boldsymbol{\alpha}'=\boldsymbol{\alpha}''$ in Theorem \[relating polystability\], then the stability parameter at every vertex in the middle column of is $\tilde{\alpha}_{ii} = \alpha_m+\alpha_i-\alpha_i=\alpha_m$. Hence we can collapse the central column in the quiver into a single vertex, to which we associate the direct sum of the corresponding bundles and obtain the following quiver bundle: $$\label{quiver1}
\resizebox{\linewidth}{!}{
\xymatrix{
& & & &\operatorname{Hom}(E_{0}'', E_1')\ar[dddr] & &\operatorname{Hom}(E_1'',E_0')\ar[dr]& & &\\
& & & \iddots\ar[ur]\ar[dr]& \vdots & & \vdots & \ddots\ar[dr] & &\\
& & \operatorname{Hom}(E_0'',E'_{m-1})\ar[ur]\ar[dr]& &\operatorname{Hom}(E_{i-1}'', E_i')\ar[dr]& &\operatorname{Hom}(E_{i}'',E_{i-1}')\ar[ur] \ar[dr]&&\operatorname{Hom}(E_{m-1}'',E'_0)\ar[dr] \\
& \operatorname{Hom}(E_0'',E'_{m})\ar[ur]\ar[dr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots& & \underset{i-j=0}{\bigoplus}\operatorname{Hom}(E_i'', E_j')\ar[ur]\ar[dr]\ar[uuur]\ar[dddr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots & &\operatorname{Hom}(E_m'', E'_0)\\
& & \operatorname{Hom}(E_1'',E'_{m})\ar[ur]\ar[dr]&& \operatorname{Hom}(E_{i}'',E_{i+1}')\ar[ur]& & \operatorname{Hom}(E_{i+1}'',E_i')\ar[ur]\ar[dr]&& \operatorname{Hom}(E_m'',E'_{1})\ar[ur]& \\
& & &\ddots\ar[ur] \ar[dr]&& && \iddots\ar[ur] && \\
& & & &\operatorname{Hom}(E_{m-1}'', E'_m)\ar[uuur]&& \operatorname{Hom}( E_{m}'',E'_{m-1})\ar[ur]&& }}$$ The next theorem says that this will be an $\bar{\boldsymbol\alpha}$-polystable quiver bundle for the corresponding collapsed stability parameter $\bar{\boldsymbol\alpha}$.
\[cor:relating-polystability\] Let $C'$ and $C''$ be $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_m)$-polystable holomorphic chains. Then the quiver bundle pictured in is $\bar{\boldsymbol\alpha}$-semistable, where the stability parameter $\bar{\boldsymbol\alpha}$ is defined by $\bar{\alpha}_{ij}=\alpha_m+\alpha_j-\alpha_{i}$ (at the central vertex we mean by this that the parameter is $\alpha_m$).
Any quiver subbundle $F$ of induces a quiver subbundle of by collapsing the middle column and, by our assumption on the stability parameters, the $\bar{\boldsymbol\alpha}$-slope of the collapsed quiver bundle $F$ equals the $\boldsymbol\alpha$-slope of the original quiver bundle. Thus quiver subbundles $F$ of obtained from quiver subbundles of by collapsing the middle column satisfy the $\bar{\boldsymbol\alpha}$-semistability condition. This in fact suffices to prove the result by using a standard argument (see, e.g., [@Prada:2001 Proposition 3.11] or [@gothen:1994 Lemma 2.2]): the idea is to use that any quiver subbundle of can be obtained by successive extensions of quiver subbundles of .
An alternative proof of Theorem \[cor:relating-polystability\] (allowing to conclude polystability rather than semistability) can be given using the Hitchin–Kobayashi correspondence. Simply note that by Lemma \[lem:hom-complex-vortex\] a solution to the vortex equations on $C'$ and $C''$ gives a solution on the quiver bundle . Under the assumption on the parameters this, in turn, gives a solution on the collapsed quiver bundle .
\[thm:AGS\] Let $C'$ and $C^{''}$ be $\boldsymbol{\alpha}$-polystable holomorphic chains and let $\alpha_{i}-\alpha_{i-1}{\geqslant}2g-2$ for all $i=1,
\cdots, m$. Then the following inequalities hold $$\begin{aligned}
\label{kernel(d)}
\mu(\ker(d))&{\leqslant}\mu_{\boldsymbol\alpha}(C^{'})-\mu_{\boldsymbol\alpha}(C^{''}),\end{aligned}$$ $$\begin{aligned}
\label{cokernel(d)}
\mu(\operatorname{coker}(d))&{\geqslant}\mu_{\boldsymbol\alpha}(C^{'})-\mu_{\boldsymbol\alpha}(C^{''})+2g-2\end{aligned}$$ where $d: \mathcal{H}^0\longrightarrow \mathcal{H}^{1}$ is the morphism in the $\operatorname{Hom}$-complex $\mathcal{H}^\bullet(C^{''},C')$, defined in .
Denote the quiver bundle by $\bar{\mathcal{E}}$. Using $\ker(d)$, define a subobject of $\bar{\mathcal{E}}$ as follows: $$\xymatrix@=2em@1@R=2em{
& & & &0\ar[dddr]& &0\ar[dr]& & &\\
& & & \iddots\ar[ur]\ar[dr]& \vdots& & \vdots& \ddots\ar[dr]&& \\
& & 0\ar[ur]\ar[dr]& &0\ar[dr]& &0\ar[ur] \ar[dr]&&0\ar[dr] \\
& 0\ar[ur]\ar[dr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots& & \ker(d)\ar[ur]\ar[dr]\ar[uuur]\ar[dddr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots & &0\\
& & 0\ar[ur]\ar[dr]& & 0\ar[ur]& & 0\ar[ur]\ar[dr]&& 0\ar[ur]& \\
& & &\ddots\ar[ur] \ar[dr]&& && \iddots\ar[ur] && \\
& & & &0\ar[uuur]&& 0\ar[ur]&& }$$ By Theorem \[cor:relating-polystability\] $\bar{\mathcal{E}}$ is $\bar{\boldsymbol\alpha}$-semistable, and a simple calculation shows that $\mu_{\bar{\boldsymbol\alpha}}(\bar{\mathcal{E}})
=\mu_{\boldsymbol\alpha}(C^{'})-\mu_{\boldsymbol\alpha}(C^{''})+\alpha_m$. Hence $$\begin{aligned}
\mu(\ker(d))+\alpha_m{\leqslant}\mu_{\boldsymbol\alpha}(C^{'})-\mu_{\boldsymbol\alpha}(C^{''})+\alpha_m,
\end{aligned}$$ which is equivalent to $(\ref{kernel(d)})$.
To prove $(\ref{cokernel(d)})$ consider the following quotient quiver bundle $$\label{eq:quotient}
\xymatrix@=1.8em@1@R=1.8em{
& & & &0\ar[dddr]& &\operatorname{coker}(d_1)\ar[dr]& & &\\
& & & \iddots\ar[ur]\ar[dr]& \vdots& & \vdots& \ddots\ar[dr]&& \\
& & 0\ar[ur]\ar[dr]& &0\ar[dr]& &\operatorname{coker}(d_{i})\ar[ur] \ar[dr]&&0\ar[dr] \\
& 0\ar[ur]\ar[dr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots& & 0\ar[ur]\ar[dr]\ar[uuur]\ar[dddr]& & \mbox{ }\mbox{ }\mbox{ }\mbox{ }\cdots & &0\\
& & 0\ar[ur]\ar[dr]& & 0\ar[ur]& & \operatorname{coker}(d_{i+1})\ar[ur]\ar[dr]&& 0\ar[ur]& \\
& & &\ddots\ar[ur] \ar[dr]&& && \iddots\ar[ur] && \\
& & & &0\ar[uuur]&& \operatorname{coker}(d_{m})\ar[ur]&& }$$ where $d_i$ is defined as the composition: $$d_i: {\textstyle\underset{i-j=0}{\bigoplus}\operatorname{Hom}(E_i'',E'_j)
\xrightarrow{\delta^++\delta^-}
\underset{i-j=1}{\bigoplus}\operatorname{Hom}(E_i'', E'_j)}\to \operatorname{Hom}(E_i'', E'_{i-1}).$$ By $\bar{\boldsymbol\alpha}$-semistability of $\bar{\mathcal{E}}$ the $\bar{\boldsymbol\alpha}$-slope of is greater than or equal to $\mu_{\bar{\boldsymbol\alpha}}(\bar{\mathcal{E}})$, which means that $$\mu(\operatorname{coker}(d))+\alpha_m+\frac{\sum_{i=1}^{m}(\alpha_{i}-\alpha_{i-1})\operatorname{rk}(\operatorname{coker}(d_i))}{\sum_{i=1}^{m}\operatorname{rk}(\operatorname{coker}(d_i)}{\geqslant}\mu_{\boldsymbol\alpha}(C^{'})-\mu_{\boldsymbol\alpha}(C^{''})+\alpha_m$$ and therefore $$\begin{aligned}
\mu(\operatorname{coker}(d))&{\geqslant}\mu_{\boldsymbol\alpha}(C^{'})-\mu_{\boldsymbol\alpha}(C^{''})+\frac{\sum_{i=1}^{m}(\alpha_{i}-\alpha_{i-1})\operatorname{rk}(\operatorname{coker}(d_i))}{\sum_{i=1}^{m}\operatorname{rk}(\operatorname{coker}(d_i)}\\
&{\geqslant}\mu_{\boldsymbol\alpha}(C^{'})-\mu_{\boldsymbol\alpha}(C^{''})+2g-2,
\end{aligned}$$ which gives .
[99]{} L. Álvarez Cónsul and O. Garc[í]{}a-Prada, *Dimensional reduction, $\mathrm{SL}(2,{\mathbb{C}})$-equivariant bundles and stable holomorphic chains*, Internat. J. Math. **12** (2001), 159–201.
, *Hitchin-Kobayashi correspondence, quivers, and vortices*, Commun. Math. Phys. **238** (2003), 1–33.
V. Balaji and A. Parameswaran, *Tensor product theorem for Hitchin pairs—an algebraic approach*, Ann. Inst. Fourier (Grenoble) **61** (2011), 2361–-2403.
W. Chuang, D.-E. Diaconescu, and G. Pan, *Wallcrossing and cohomology of the moduli space of Hitchin pairs*, Commun. Number Theory Phys. **5** (2011), 1–56.
L. Álvarez Cónsul, O. Garc[í]{}a-Prada and A. H. W. Schmitt, *On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces*, Internat. Math. Res. Papers, Art ID 73597 (2006), 1–82.
S. B. Bradlow, O. Garc[í]{}a-Prada, and P. B. Gothen, *Moduli spaces of holomorphic triples over compact [R]{}iemann surfaces*, Math. Ann. **328** (2004), 299–351.
S. Bradlow, O. Garcia-Prada, P. Gothen, and J. Heinloth, *Irreducibility of moduli of semistable chains and applications to $\mathrm{U}(p,q)$-Higgs bundles*, 2017, <http://arxiv.org/abs/1703.06168>.
O. Garc[í]{}a-Prada and J. Heinloth, *The [$y$]{}-genus of the moduli space of [${\rm PGL}_n$]{}-[H]{}iggs bundles on a curve (for degree coprime to [$n$]{})*, Duke Math. J. **162** (2013), no. 14, 2731–2749.
O. Garc[í]{}a-Prada, J. Heinloth, and A. Schmitt, *On the motives of moduli of chains and [H]{}iggs bundles*, J. Eur. Math. Soc. **16** (2014), no. 12, 2617–2668.
P. B. Gothen, *The [B]{}etti numbers of the moduli space of rank 3 [H]{}iggs bundles*, Internat. J. Math. **5** (1994), 861–875.
P.B. Gothen and A.D. King, *Homological algebra of twisted quiver bundles*, J. London Math. Soc. **71** (2005), 85–99.
P. B. Gothen and A. Nozad, *Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles*, Revista Matemática Complutense **30** (2017), 91–128.
N. J. Hitchin, *The self-duality equations on a [R]{}iemann surface*, Proc. London Math. Soc. (3) **55** (1987), 59–126.
A. Mellit, *Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures)*, 2017, <http://arxiv.org/abs/1707.04214>.
S. Mozgovoy and O. Schiffmann, *Counting Higgs bundles and type A quiver bundles*, 2017, <http://arxiv.org/abs/1705.04849>.
S. Ramanan and A. Ramanathan, *Some remarks on the instability flag*, Tohoku Math. J. (2) **36** (1984), 269-–291. O. Schiffmann, *Indecomposable vector bundles and stable higgs bundles over smooth projective curves*, Ann. Math. **183** (2016), 297––362.
A. H. W. Schmitt, *Moduli for decorated tuples for sheaves and representation spaces for quivers*, Proc. Indian Acad. Sci. Math. Sci. **115** (2005), 15–49.
M. Thaddeus, *Stable pairs, linear systems and the [V]{}erlinde formula*, Invent. Math. **117** (1994), 317–353.
|
---
author:
- Takashi Hotta and Akira Shudo
title: 'Chaos in Jahn-Teller Rattling'
---
Recently, a peculiar magnetically robust heavy-electron phenomenon observed in Sm-based filled skutterudite compound [@Sanada] has triggered active investigations on cage-structure compounds, in which a guest ion contained in a cage composed of relatively light atoms oscillates with large amplitude in an anharmonic potential. Such a local vibration with large amplitude is called [*rattling*]{} and exotic magnetism and superconductivity induced by rattling have attracted much attention in the research field of condensed matter physics. As easily understood from the above explanation, rattling is considered to be one of typical nonlinear phenomena, but unfortunately, such a viewpoint has not been recognized at all in the research field of nonlinear physics.
However, we emphasize an important point of contact between rattling phenomena and nonlinear physics through a concept of [*chaos*]{}. For nonlinear physicists, it will be quite natural to expect the appearance of chaos in the two-dimensional oscillator in a potential with plural numbers of minima. In fact, such a situation is expected to occur in cage-structure compounds, if we consider a Jahn-Teller ion vibrating in a cubic anharmonic potential, [@Hotta] in the course of the research of the Kondo effect with phonon origin. [@Kondo1; @Kondo2; @Vladar1; @Vladar2; @Vladar3; @Yu-Anderson; @Matsuura1; @Matsuura2; @Yotsuhashi; @Hattori1; @Hattori2; @Mitsumoto1; @Mitsumoto2; @Hotta1; @Hotta2; @Hotta3; @Hotta4; @Hotta5; @Hotta6; @Yashiki1; @Yashiki2; @Yashiki3; @Hattori3; @Hotta7; @Fuse1; @Fuse2; @Fuse3; @Fuse4; @Fuse5; @Fuse6] It is worth to point out that such a cubic anharmonic term of Jahn-Teller vibration just indicates the Hénon-Heiles potential which has been discussed for the appearance of chaos in the early stage.[@HH]
Actually, apart from the rattling problem, the appearance of chaos in the vibronic state, i.e., the complicated electron-vibration coupled state, has been already pointed out by several groups. [@Cederbaum; @JTC0; @JTC1a; @JTC1b; @JTC2; @JTC3; @JTC4] However, we hit upon an idea that [*chaos originates from anharmonic Jahn-Teller vibration*]{}, not from the vibronic state composed of electron and anharmonic Jahn-Teller vibration. It is important to confirm that the origin of chaos exists in the anharmonic Jahn-Teller oscillator, since it will provide us a realistic model in condensed-matter physics for the research of chaos in nonlinear physics.
In this Letter, we clarify the chaotic property in anharmonic Jahn-Teller vibration. In order to confirm the appearance of chaos, we evaluate the nearest-neighbor level-spacing distribution $P(s)$, indicating that $P(s)$ changes from the Poisson to the Wigner distribution with the increase of cubic anharmonicity. We also discuss the energy region in which chaotic behavior occurs. In order to consider a possible way to detect the chaotic behavior, we propose the measurement of specific heat in cage-structure materials. It is pointed out that the peak structure in the temperature dependence of specific heat could be a signal of chaotic behavior.
Let us consider a Jahn-Teller oscillator in an anharmonic potential. The Hamiltonian is given by [@unit] $$H=(P_1^2+P_2^2)/(2M)+V(Q_1,Q_2),$$ where $M$ is the reduced mass of Jahn-Teller oscillator, $Q_1$ and $Q_2$ denote normal coordinates of $(3z^2-r^2)$- and $(x^2-y^2)$-type Jahn-Teller oscillation, respectively, $P_1$ and $P_2$ indicate corresponding canonical momenta, and $V(Q_1,Q_2)$ is the potential for the Jahn-Teller oscillator. The potential is given by $V(Q_1, Q_2)$=$A(Q_1^2+Q_2^2)$+$B(Q_1^3-3 Q_1 Q_2^2)$+$C(Q_1^2+Q_2^2)^2$, where $A$ is the quadratic term of the potential, while $B$ and $C$ denote the coefficients for third- and fourth-order anharmonic terms, respectively. As mentioned above, the third-order term is just the Hénon-Heiles potential.[@HH] Note also that we include only the anharmonicity which maintains the cubic symmetry. Among the coefficients, $A$ and $C$ are taken as positive, while $B$ is set as negative in this research.
In order to understand the properties of the potential, it is convenient to introduce the non-dimensional distortion as $q_1$=$\sqrt{2M\omega}Q_1$ and $q_2$=$\sqrt{2M\omega} Q_2$, where $\omega$ is the phonon energy given by $\omega$=$\sqrt{2A/M}$. By introducing $q$ and $\theta$ through the relations of $q_1$=$q \cos \theta$ and $q_2$=$q \sin \theta$, we obtain $V$ as $$\label{eq:pot}
V(q, \theta)=\omega (q^2/4+\beta q^3 \cos 3\theta/3
+\gamma q^4/8),$$ where non-dimensional anharmonicity parameters are defined by $\beta$=$3B/[(2M)^{3/2}\omega^{5/2}]$ and $\gamma$=$2C/(M^2\omega^3)$. Note that the energy scale of the potential is given by $\omega$. Thus, in the following, we set the energy unit as $\omega$=$1$.
In this potential, for $|\beta|$$\le$$\sqrt{\gamma}$, there is a single minimum at $q$=$0$, while for $|\beta|$$>$$\sqrt{\gamma}$, there appear three minima for $q$$\ne$$0$ in addition to the shallow minimum at $q$=$0$. In Fig. 1(a), we plot $V(q,\theta)$ vs. $q$ for several values of $\theta$ for the case of $\beta$=$-2$ and $\gamma$=$1$. Along the direction of $\theta$=$0$, we find a deep minimum, while we find a saddle point along the direction of $\theta$=$\pi/3$. The potential structure is gradually changed with the increase of $\theta$. In Fig. 1(b), we show the contour plot of the potential $V$ for $\beta$=$-2$ and $\gamma$=$1$. Here we find three minima along the directions of $\theta$=$0$, $2\pi/3$, and $4\pi/3$, corresponding to $(3z^2-r^2)$-, $(3x^2-r^2)$-, and $(3y^2-r^2)$-type Jahn-Teller distortions, respectively. Note that there still remains trigonal symmetry, as easily understood from the term of $\cos 3\theta$ in eq. (\[eq:pot\]), since we consider the cubic anharmonicity.
{width="8.0truecm"}
For the later discussion, here we define the potential depth $V_0$ as $V_0$=$V(q_-,\theta=0)$$-$$V(q_+,\theta=0)$, where $q_{\pm}$ denotes the position of extrema, given by $q_{\pm}$=$(-\beta \pm \sqrt{\beta^2-\gamma})/\gamma$. Then, we obtain $V_0$ as $$\label{eq:v0}
V_0=2|\beta|(\beta^2-\gamma)^{3/2}/(3\gamma^3).$$ Note that $V_0$ is defined for $|\beta|$$\ge$$\sqrt{\gamma}$.
In order to discuss the local phonon state, it is necessary to perform the quantization procedure through the relations of $q_1$=$a_1$+$a_1^{\dag}$ and $q_2$=$a_2$+$a_2^{\dag}$, where $a_1$ and $a_2$ are annihilation operators of phonons for Jahn-Teller oscillations. In order to unveil the conserved quantities in the Hamiltonian $H$, it is useful to introduce the transformation of phonon operators as $a_{\pm}$=$(a_1 \pm {\rm i}a_2)/\sqrt{2}$,[@Takada] where the sign in this equation intuitively indicates the rotational direction in the potential. With the use of these operators, the Hamiltonian is rewritten as $$\begin{split}
H & = a_{+}^{\dag}a_{+}+a_{-}^{\dag}a_{-}+1 \\
& +(\beta/3) [(a_{+} + a_{-}^{\dag})^3
+(a_{-} + a_{+}^{\dag})^3] \\
& +(\gamma/2) (a_{+}^{\dag}a_{+}+a_{-}^{\dag}a_{-}+1
+a_{+}^{\dag} a_{-}^{\dag}+a_{+}a_{-})^2.
\end{split}$$ Note again that the energy unit is set as $\omega$=$1$.
In order to diagonalize the Hamiltonian, we prepare the phonon basis $|L; n\rangle$, given by $$|L; n \rangle = \left \{
\begin{array}{ll}
|L+n, n \rangle & L \ge 0 \\
|n, n+|L| \rangle & L <0,
\end{array}
\right.$$ where the phonon basis $|n_+,n_- \rangle$ is given by $|n_+,n_- \rangle$= $(1/{\sqrt{n_+! n_-!}})
(a_+^{\dag})^{n_+}(a_-^{\dag})^{n_-}|0\rangle$ with the vacuum $|0\rangle$. In actual numerical calculations to solve the eigenvalue problem, the phonon basis $|L; n \rangle$ is truncated at a finite number $N_{\rm ph}$ and a maximum angular momentum $L_{\rm max}$. In order to check the convergence, we have performed the numerical calculations for $N_{\rm ph}$ and $L_{\rm max}$ up to 250 and 125, respectively.
For $\beta$=$0$, we find that the quantum phonon state is labelled by the angular momentum $L$, given by $L$=$a_{+}^{\dag}a_{+}$$-$$a_{-}^{\dag}a_{-}$. Note that $L$ commutes with $H$ for $\beta$=$0$. Thus, when we diagonalize the Hamiltonian for $\beta$=$0$, we prepare the phonon basis for a fixed value of $L$, since the states with different $L$ are not mixed. When the potential has continuous rotational symmetry for $\beta$=$0$, the angular momentum should be the conserved quantity in the quantum mechanics.
When we include the effect of $\beta$, namely, cubic anharmonicity, the situation is changed. As easily understood from eq. (\[eq:pot\]), there occurs the trigonal term in the potential. In such a case, $L$ is no longer the good quantum number, but there still exists conserved quantity concerning $L$. In order to clarify such a point, we express $L$ as $$L=3\ell+L_0,$$ where $L_0$ takes the values of $0$ and $\pm 1$. It is found that $L_0$ is the good quantum number.
The states with $L_0$=$\pm 1$ is expressed by $$|\Phi^{(\pm 1)}_{k} \rangle
=\sum_{\ell,n} \varphi_{\ell,n}^{(k,\pm 1)} |3\ell \pm 1; n\rangle,$$ where $|\Phi^{(L_0)}_{k} \rangle$ denotes the $k$-th eigenstate characterized by quantum number $L_0$ and $\varphi$ is the coefficient of the eigenstate. The corresponding eigenenergy is expressed by $E^{(L_0)}_{k}$.
Note that for the case of $L_0$=$0$, there exists extra conserved quantity of parity, concerning the change of $\ell \rightarrow -\ell$. It is simply understood from the fact that the bonding and anti-bonding states of $|3\ell;n\rangle$ and $|-3\ell;n\rangle$ are not mixed with each other. Then, the parity for the bonding (even) or anti-bonding (odd) state is another good quantum number. Note that the bonding state of $|3\ell;n\rangle+|-3\ell;n\rangle$ is mixed with $|0;n\rangle$, but the anti-bonding state $|3\ell;n\rangle-|-3\ell;n\rangle$ is not.
The state for $L_0$=$0$ with even parity is given by $$\begin{split}
|\Phi^{(0{\rm e})}_{k} \rangle
&= \sum_n \varphi_{0,n}^{(k,0{\rm e})} |0; n\rangle \\
&+\sum_{\ell>0,n} \varphi_{\ell,n}^{(k,0{\rm e})}
(|3\ell; n\rangle +|-3\ell; n\rangle)/\sqrt{2},
\end{split}$$ while the state for $L_0$=$0$ with odd parity is given by $$|\Phi^{(0{\rm o})}_{k} \rangle
=\sum_{\ell>0,n} \varphi_{\ell,n}^{(k,0{\rm o})}
(|3\ell; n\rangle -|-3\ell; n\rangle)/\sqrt{2}.$$ In short, we classify the eigenstates for the case of $\beta$$\ne$$0$ into four groups with the labels of “$+1$”, “$-1$”, “$0$e”, and “$0$o”.
In Fig. 1(c), we plot $E^{(0{\rm e})}_{0}$, $E^{(\pm 1)}_{0}$, and $E^{(0{\rm o})}_{0}$, which denote the lowest eigenenergies of the groups of “$0$e”, “$\pm 1$”, and “$0$o”, respectively. We observe that the ground state is always given by $|\Phi^{(0{\rm e})}_{0} \rangle$ and the first excited state is doubly degenerate, given by $|\Phi^{(\pm 1)}_{0} \rangle$. Note that these three states seem to be almost degenerate in the region of $\beta$$<$$-1.7$. The state of $|\Phi^{(0{\rm o})}_{0} \rangle$ appears in the relatively high-energy region.
The lowest-energy state of $L_0$=$0$ with even parity includes the significant contribution of $L$=$0$ and it corresponds to the zero-point oscillation. On the other hand, the state with $L_0$=$\pm 1$ has excitation with finite angular momentum. Around at $\beta$$\approx$$-1.7$, the zero-point energy is found to be less than the potential depth $V_0$, suggesting that the oscillation states begin to be localized in the potential minima. In such a situation, the potential minima become deep and the quantum tunneling among potential minima is suppressed. Thus, the energy difference due to rotational motion becomes very small and the excitation energy is extremely reduced.
{width="8.0truecm"}
Now we explain a way to extract information on chaos from the energy spectrum.[@Bohigas] First we prepare the eigenenergies $\{ E_k \}$ for each quantum number. Note that we cannot obtain correct distribution, if the eigenstates with different symmetry are mixed. Next we introduce the average counting function $\langle N(E) \rangle$, where $N(E)$ denotes the number of energy levels less than $E$, and perform the procedure of “unfolding” by the mapping $x_k$=$\langle N(E_k) \rangle$ with the unfolded level $x_k$. Then, we evaluate the distribution of nearest-neighbor level-spacing $P(s)\Delta s$ by counting the number of spacings satisfying $s$$<$$x_k$$-$$x_{k-1}$$<$$s$$+$$\Delta s$ with an appropriate mesh $\Delta s$.
In Fig. 2, we show $P(s)$ obtained from $E_k^{(+1)}$ with $\Delta s$=$0.1$ for $\beta$=$0^{+}$, $-2$, and $-5$. Here $0^{+}$ indicates the infinitesimal small positive number. Note that there is no significant difference in the distribution $P(s)$, if we use the numerical data of $E_k^{(-1)}$ and $E_k^{(0{\rm e})}$. For $\beta$=$0^+$, we observe the Poisson distribution $P_{\rm P}(s)$=$e^{-s}$, while for $\beta$=$-5$, we find the Wigner distribution $P_{\rm W}(s)$=$(\pi s/2)e^{-\pi s^2/4}$. For $\beta$=$-2$, the mixture of $P_{\rm P}(s)$ and $P_{\rm W}(s)$ is observed. It is well known that when the classical system exhibits chaos, eigenenergies of the corresponding quantum system shows the Wigner distribution. Thus, we conclude that the chaotic behavior appears when we increase the cubic anharmonicity $\beta$. We also emphasize that the chaotic behavior observed in the vibronic state in the previous research should be considered to originate from chaos in the anharmonic Jahn-Teller vibration.
{width="8.0truecm"}
Here we have a naive question: In which energy region chaos predominates ? In order to reply to this question, we divide the sequence of eigenenergies into several sectors and evaluate $P(s)$ of each sector. We consider the eigenenergies with $L_0$=$+1$ for $\beta$=$-5$ and $\gamma$=$1$, where $V_0$=$160\sqrt{6}$=$392$ from eq. (\[eq:v0\]). We prepare the following sequence of integer: $k_0$=$0$, $k_1$=$550$, and $k_j$=$3000$$\times$$(j-1)$ for $j \ge 2$. Note that $k_1$ is determined so as to satisfy the relation of $E^{(+1)}_{k_1}$=$V_0$. We define the sector $j$ including the eigenenergies from $k_{j-1}$ to $k_j$. Then, for each sector $j$, we evaluate $P(s)$.
In Fig. 3, we plot $P(\Delta s/2)$ vs. $E_{\rm mid}$ for $\Delta s$=$0.1$, where $E_{\rm mid}$ indicates the energy just at the center of the sector $j$. The horizontal line denotes the value of $P_{\rm W}(\Delta s/2)$. We find that for small $E_{\rm mid}$ comparable with $V_0$, $P(\Delta s/2)$ is apparently larger than $P_{\rm W}(\Delta s/2)$, suggesting that the chaotic nature is weak. This is understood from the fact that the oscillation is harmonic near the bottom of the potential well, since the potential is quadratic near the potential minimum. For large $E_{\rm mid}$, we also observe that $P(\Delta s/2)$ significantly deviates from $P_{\rm W}(\Delta s/2)$. For the energy region much larger than $V_0$, the potential is dominated by the fourth-order term with $q^4$ and the system asymptotically approaches the integrable system, suggesting that the chaotic nature should disappear.
In the energy region for moderately larger than $V_0$, we find the relation of $P(\Delta s/2)$$\approx$$P_{\rm W}(\Delta s/2)$, although some deviations occur due to the statistical property. In one word, the result clearly defines the energy region with chaotic behavior. Namely, the chaotic nature comes from the eigenenergies between $a_{\rm L}V_0$ and $a_{\rm H}V_0$, where $a_{\rm L}$ is a number of the order of unity, while $a_{\rm H}$ is in the order of hundred. It seems natural that the energy of the chaotic region is related with the potential depth $V_0$, but such energy region spreads over a few hundred times larger than $V_0$. The width of the chaotic region is much larger than we have naively expected.
{width="8.0truecm"}
Now we discuss a possible way to detect the chaotic nature in observables. For the purpose, we evaluate the specific heat $C$, given by $C$=$(\langle H^2 \rangle-\langle H \rangle^2)/T^2$, where $T$ is a temperature and $\langle H^m \rangle$=$\sum_k e^{-E_k/T} E_k^m/Z$ with the partition function $Z$=$\sum_k e^{-E_k/T}$. Since the evaluation of $C$ is done by the numerical calculation with the use of finite numbers of phonon bases, we should note that $C$ may exhibit unphysical behavior at high temperatures.
In Fig. 4, we show the specific heat $C$ vs. temperature $T$ for several values of $\beta$. Note that the unit of $C$ is $k_{\rm B}$, which is set as unity. [@unit] For $\beta$=$0$ and $-1$, $C$ is increased rapidly around at $T$$\sim$$1$ and it becomes almost a constant value, corresponding to the Dulong-Petit law. Note that the result agrees well with the broken curve of $2C_1$ in the high-temperature region, where $C_1$ denotes the specific heat of one-dimensional anharmonic oscillator in the potential of $V_1(q)$=$\omega(q^2/4$$+$$q^4/8)$ with non-dimensional length $q$. At high enough temperatures, all the results should approach the broken curve, since the potential is dominated by the fourth-order term in the high-energy region. However, in the actual calculations with finite numbers of phonon bases, it is inevitable that $C$ is deviated from the constant value at some temperature, as denoted by a shaded region in Fig. 4.
For $\beta$=$-2$, we find a hump at $T$$\sim$$3.3$. At low temperatures, we find a Schottky peak determined by the first excitation energy $\Delta E$=$E_0^{(\pm 1)}$$-$$E_0^{(0{\rm e})}$, which is the difference between lowest two curves in Fig. 1(c). Note that the Schottky peak cannot be observed for $\beta$=$0$ and $-1$, since $\Delta E$ is larger than unity for both cases. On the other hand, for $\beta$$<$$-2$, the Schottky peak exists, but the peak position is smaller than $10^{-4}$.
With the increase of $|\beta|$, the hump found in $\beta$=$-2$ grows and it eventually becomes the robust peak structure. The temperature at the peak $T_{\rm p}$ is found to be given by $T_{\rm p}$=$11$, $33$, and $77$ for $\beta$=$-3$, $-4$, and $-5$, respectively. These values are well scaled by $V_0$ in eq. (\[eq:v0\]). For large values of $|\beta|$ such as $\beta$=$-4$ and $-5$, we have carefully checked that the value of $T_{\rm p}$ converges even in the present numerical calculations, although it is difficult to reproduce the Dulong-Petit law consistent with $2C_1$ in the high-temperature region. Note, however, that we observe a shoulder in the position of $2C_1$ for $\beta$=$-3$.
From Figs. 3 and 4, the appearance of the peak structure with large width in the specific heat over the broken curve in the high-temperature region seems to characterize the chaotic nature, since we intuitively consider that an entropy is expected to be enhanced due to uniform spreading of the eigenfunctions in the phase space for the energy region with chaotic behavior.[@Berry; @Voros] Here we note that $C$ is related with the entropy $S$ as $C$=$T (\partial S/\partial T)$, suggesting that $C$ forms a peak when the entropy is rapidly increased with the increase of $T$. Thus, we deduce that the robust peak structure in the specific heat becomes a signal of the emergence of chaos.
In the experiments of cage-structure materials, the specific heat has been usually measured. In many cases, experimentalists have plotted the value of $C/T^3$ as a function of $T$, since the Debye specific heat is in proportion to $T^3$ at low temperatures. In the plot of $C/T^3$ vs. $T$, we obtain the peak structure corresponding to the characteristic frequency of the Einstein phonon for rattling. However, our proposal is to seek for the peak in $C$, not in $C/T^3$, as the enhancement of $C$ due to the chaotic nature of anharmonic Jahn-Teller vibration. A candidate material is cage-structure compound with off-center rattling such as clathrate. Note that we consider the specific heat in the temperature region higher than a room temperature.
In summary, we have clarified the chaotic nature of Jahn-Teller rattling. From the evaluation of $P(s)$, we have confirmed the occurrence of chaos in the anharmonic Jahn-Teller oscillation and the energy scale of the chaotic region. It has been emphasized that the chaotic behavior in the vibronic state of dynamical Jahn-Teller system originates from the anharmonic Jahn-Teller oscillation. We have proposed to observe the peak structure in the specific heat of cage-structure materials as a signal of the chaotic nature. It is a novel possibility to detect chaos in nonlinear physics by the standard experiment in condensed matter physics.
This work has been supported by JSPS KAKENHI Grant Numbers 25400405 and 24540379. The computation has been done using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo.
[99]{}
S. Sanada, Y. Aoki, H. Aoki, A. Tsuchiya, D. Kikuchi, H. Sugawara, and H. Sato: J. Phys. Soc. Jpn. [**74**]{} (2005) 246.
T. Hotta: arXiv:1405.7462.
J. Kondo: Physica B+C [**84**]{} (1976) 40.
J. Kondo: Physica B+C [**84**]{} (1976) 207.
K. Vladár and A. Zawadowski: Phys. Rev. B [**28**]{} (1983) 1564.
K. Vladár and A. Zawadowski: Phys. Rev. B [**28**]{} (1983) 1582.
K. Vladár and A. Zawadowski: Phys. Rev. B [**28**]{} (1983) 1596.
C. C. Yu and P. W. Anderson: Phys. Rev. B [**29**]{} (1984) 6165.
T. Matsuura and K. Miyake: J. Phys. Soc. Jpn. [**55**]{} (1986) 29.
T. Matsuura and K. Miyake: J. Phys. Soc. Jpn. [**55**]{} (1986) 610.
S. Yotsuhashi, M. Kojima, H. Kusunose, and K. Miyake: J. Phys. Soc. Jpn. [**74**]{} (2005) 49.
K. Hattori, Y. Hirayama, and K. Miyake: J. Phys. Soc. Jpn. [**74**]{} (2005) 3306.
K. Hattori, Y. Hirayama, and K. Miyake: J. Phys. Soc. Jpn. [**75**]{} (2006) Suppl. 238.
K. Mitsumoto and Y. [Ō]{}no: Physica B [**403**]{} (2008) 859.
K. Mitsumoto and Y. [Ō]{}no: J. Phys. Soc. Jpn [**79**]{} (2010) 054707.
T. Hotta: Phys. Rev. Lett. [**96**]{} (2006) 197201.
T. Hotta: J. Phys. Soc. Jpn. [**76**]{} (2007) 023705.
T. Hotta: J. Phys. Soc. Jpn. [**76**]{} (2007) 084702.
T. Hotta: Physica B [**403**]{} (2008) 1371.
T. Hotta: J. Phys. Soc. Jpn. [**77**]{} (2008) 103711.
T. Hotta: J. Phys. Soc. Jpn. [**78**]{} (2009) 073707.
S. Yashiki, S. Kirino, and K. Ueda: J. Phys. Soc. Jpn. [**79**]{} (2010) 093707.
S. Yashiki, S. Kirino, K. Hattori, and K. Ueda: J. Phys. Soc. Jpn. [**80**]{} (2011) 064701.
S. Yashiki and K. Ueda: J. Phys. Soc. Jpn. [**80**]{} (2011) 084717.
K. Hattori: Phys. Rev. B [**85**]{} (2012) 214411.
T. Hotta and K. Ueda: Phys. Rev. Lett. [**108**]{} (2012) 247214.
T. Fuse and Y.Ōno: J. Phys. Soc. Jpn. [**79**]{} (2010) 093702.
T. Fuse and Y.Ōno: J. Phys. Soc. Jpn. [**80**]{} (2011) SA136.
T. Fuse, Y. Ōno, and T. Hotta: J. Phys. Soc. Jpn. [**81**]{} (2012) 044701.
T. Fuse and T. Hotta: J. Phys.: Conf. Ser. [**428**]{} (2013) 012013.
T. Fuse and T. Hotta: J. Korean Phys. Soc. [**62**]{} (2013) 1874.
T. Fuse and T. Hotta: To appear in the Proceedings of SCES2013. M. Hénon and C. Heiles: Astrophysical Journal [**69**]{} (1964) 73.
H. Köppel, W. Domcke, and L. S. Cederbaum: Adv. Chem. Phys. [**57**]{} (1984) 59.
R. S. Markiewicz: Phys. Rev. E [**64**]{} (2001) 026216.
H. Yamasaki, Y. Natsume, A. Terai, and K. Nakamura: Phys. Rev. E [**68**]{} (2003) 046201.
H. Yamasaki, Y. Natsume, A. Terai, and K. Nakamura: J. Phys. Soc. Jpn. [**73**]{} (2004) 1415.
E. Majern[' i]{}kov[' a]{} and S. Shpyrko: Phys. Rev. E [**73**]{} (2006) 057202.
E. Majern[' i]{}kov[' a]{} and S. Shpyrko: J. Molecular Structure [**838**]{} (2007) 22.
E. Majern[' i]{}kov[' a]{} and S. Shpyrko: J. Phys. A: Math. Theor. [**44**]{} (2011) 065101.
In this paper, we use such units as $\hbar$=$k_{\rm B}$=$1$.
Y. Takada: Phys. Rev. B [**61**]{} (2000) 8631.
O. Bohigas: “Random matrix theories and chaotic dynamics”, in [*Chaos and Quantum Physics*]{}, Les Houches, Session LII, 1989 (Eds. M. -J. Giannoni, A. Voros, and J. Zinn-Justin), pp. 87-199, Elsevier Sci. Publ., Amsterdam (1991).
M. V. Berry: J. Phys. A [**10**]{} (1977) 2083.
A. Voros: “Semi-classical ergodicity of quantum eigenstates in the Wigner representation”, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lectures Notes in Physics Vol. 93 (Eds. G. Casati and J. Ford), pp. 326-333, Springer, Berlin (1979).
|
---
abstract: 'The unique Steiner triple system of order $7$ has a point-block incidence graph known as the Heawood graph. Motivated by questions in combinatorial matrix theory, we consider the problem of constructing a faithful orthogonal representation of this graph, i.e., an assignment of a vector in $\mathbb C^d$ to each vertex such that two vertices are adjacent precisely when assigned nonorthogonal vectors. We show that $d=10$ is the smallest number of dimensions in which such a representation exists, a value known as the *minimum semidefinite rank* of the graph, and give such a representation in $10$ real dimensions. We then show how the same approach gives a lower bound on this parameter for the incidence graph of any Steiner triple system, and highlight some questions concerning the general upper bound.'
address:
- 'Department of Mathematics, Quinnipiac University, Hamden, CT 06518, USA.'
- 'Department of Mathematics, Brigham Young University, Provo, UT 84602, USA'
author:
- Louis Deaett
- 'H. Tracy Hall'
bibliography:
- 'Orthrep-Steiner-arXiv.bib'
title: |
Orthogonal representations of Steiner triple system\
incidence graphs
---
Introduction {#sec:intro}
============
Fundamental to what follows is the idea of assigning a vector to each vertex of a graph so that the inner products among the vectors in some way reflect the adjacency relation on the vertices. A geometric representation of this sort may then be useful in studying properties of the graph. This approach dates back at least to the celebrated work of Lovász [@lovasz] in determining the Shannon capacity of the $5$-cycle; see also [@parsons] for a unifying discussion. The following definition provides one realization of this idea.
\[def:orth\_rep\] Let $G$ be a graph and $X$ be an inner product space. An *orthogonal representation of $G$ in $X$* is a function $r:V(G) \rightarrow X$ such that two vertices of $G$ are adjacent if and only if they are mapped by $r$ to nonorthogonal vectors, i.e., for any distinct $u,v \in V(G)$, $$\label{eqn:orth_rep_condition}
\langle r(u), r(v) \rangle \neq 0
\, \Longleftrightarrow \,
\{u,v\} \in E(G).$$
We note that the notion requiring only the forward direction of has received a good deal of attention; many authors would refer to the notion set out in Definition \[def:orth\_rep\] as that of a *faithful* orthogonal representation. Also, what some authors would consider an orthogonal representation of $G$ would be considered by others to be an orthogonal representation of the complement of $G$. Variations of this sort must be kept in mind when considering the related literature; some results derived in the context of different such choices are surveyed in each of [@haynes] and [@parsons].
The particular notion of an orthogonal representation given by Definition \[def:orth\_rep\] has relevance to combinatorial matrix theory in the context of certain variants of the *minimum rank problem*, which, broadly construed, calls for finding the smallest possible rank among all matrices meeting a given combinatorial description. Instances of this problem arise naturally in applications such as computational complexity theory [@deaett_srinivasan] and quantum information theory [@scarpa_severini]. Often, a matrix is first required to be symmetric (or Hermitian) and then further conditions are imposed in terms of the graph whose edges correspond to the locations of the off-diagonal nonzero entries of the matrix. This is made precise by the following definition. Note that, in all that follows, we denote by $A_{ij}$ the entry in row $i$ and column $j$ of matrix $A$.
Let $A$ be an $n\times n$ Hermitian matrix. The *graph of $A$* is the unique simple graph on vertices $v_1,\ldots,v_n$ such that, for every $i\not= j$, vertices $v_i$ and $v_j$ are adjacent if and only if $A_{ij} \neq 0$.
The associated minimum rank problem is to determine the smallest rank among the Hermitian (or real symmetric) matrices with a fixed graph, a value known as the *minimum rank* of the graph. This problem has received considerable attention in combinatorial matrix theory; see [@handbook_min_rank_chapter] for a survey. The present work bears on a variant of the problem in which only positive semidefinite Hermitian matrices are considered. In particular, we study the graph invariant defined as follows.
\[def:msr\] Let $G$ be a simple graph on $n$ vertices. The *minimum semidefinite rank* of $G$ is the smallest rank among all positive semidefinite Hermitian matrices with graph $G$. This value is denoted by ${{\rm msr}(G)}$.
The following observation connects this variant of the minimum rank problem with the notion of an orthogonal representation. It is a simple consequence of the characterization of positive semidefinite matrices as Gram matrices.
The smallest $d$ such that $G$ has an orthogonal representation in $\mathbb C^d$ is $d={{\rm msr}(G)}$.
The smallest $d$ allowing an orthogonal representation of $G$ in $\mathbb R^d$ may also be of interest, in which case the minimum of Definition \[def:msr\] can be taken over the real symmetric matrices; we refer to this value as the *minimum semidefinite rank of $G$ over $\mathbb R$*.
The ordinary Laplacian matrix of a graph $G$ on $n$ vertices shows ${{\rm msr}(G)}$ to be well-defined and at most $n-1$. In addition, the minimum semidefinite rank is additive on the connected components of a graph, so that it is sufficient to consider connected graphs only. The question of how combinatorial properties of a graph relate to its minimum semidefinite rank has received a good deal of interest; see, e.g., [@booth_et_al_2011; @booth_et_al_2008] and [@handbook_min_rank_chapter Section 46.3]. One simple result is the following.
\[thm:random\_msr\_facts\] If $G$ is a cycle on $n$ vertices, then ${{\rm msr}(G)} = n-2$.
The motivation for the present work begins with a result of [@rosenfeld] that can be recast as follows.
\[thm:rosenfeld\] If $G$ is a connected triangle-free graph on $n \ge 2$ vertices, then ${{\rm msr}(G)} \ge \frac 12 n$.
The question of how Theorem \[thm:rosenfeld\] might generalize has received some attention. For instance, the implications of replacing the triangle-free condition with a larger upper bound on the clique number are explored in [@furedi]. In [@deaett_msr], it was observed that (except in trivial cases) a graph meeting the lower bound of Theorem \[thm:rosenfeld\] must have a girth of $4$, suggesting that, in seeking a generalization, the condition that the graph be triangle-free be viewed as a lower bound on its girth. In particular, the following conjecture was put forward.
\[conj:girth\_conjecture\] Suppose $G$ is a connected graph on $n \ge 2$ vertices and $k$ is an integer with $k \ge 4$. If $G$ has girth at least $k$, then ${{\rm msr}(G)} \ge \left(\frac{k-2}k\right)\!n$.
In light of Theorem \[thm:random\_msr\_facts\], this conjecture may be viewed as asserting that, among the connected graphs of girth at least $k$, the minimum semidefinite rank as a fraction of the number of vertices is minimized by the $k$-cycle. While Conjecture \[conj:girth\_conjecture\] was found to hold for all graphs on at most $7$ vertices, it was suggested that a revealing test case might be provided by the *cage graphs*, defined as follows.
A graph that has girth $g$ in which each vertex has degree $d$ is called a *$(d,g)$-cage* when no graph on fewer vertices has both of those properties.
The well-known Petersen graph, with $10$ vertices, is the unique $(3,5)$-cage. Its minimum semidefinite rank is $6$, meeting the lower bound of Conjecture \[conj:girth\_conjecture\]. There is also a unique $(3,6)$-cage, known as the Heawood graph, with $14$ vertices, shown in Figure \[fig:heawood\_graph\]. Conjecture \[conj:girth\_conjecture\] would require its minimum semidefinite rank to be at least $10$.
\_mark\_radius [1.75]{}
in [0,...,6]{} [ (A) at ([360/ + 80]{} : ); (B) at ([360/ + 100]{} : ); (A) circle\[radius=\_mark\_radius pt\]; (B) circle\[radius=\_mark\_radius pt\]; ]{}
in [0,...,6]{} [ (A) – (B); let 1=[int(mod(+2,7))]{} in (A) – (B1); let 1=[int(mod(+6,7))]{} in (A) – (B1); ]{}
Unfortunately, the problem of determining for a given graph the value of its minimum rank or minimum semidefinite rank may be very difficult. One of the few general techniques (introduced in [@original_ZF_paper]) that is available for lower-bounding the minimum rank involves computing a *zero forcing parameter* for the graph. Such a parameter gives an upper bound on the dimension of the null space, and hence a lower bound on the rank, of a matrix by exploiting how the graph of the matrix constrains the zero-nonzero patterns (i.e., supports) that may occur among its null vectors.
A variant of the zero forcing technique specific to the positive semidefinite case was introduced in [@barioli_et_al]. Namely, the *positive semidefinite zero forcing number* of $G$, denoted ${Z_+(G)}$, is defined for any graph $G$. While a precise definition of ${Z_+(G)}$ is beyond the scope of this paper, we note that the definition is purely combinatorial, so that ${Z_+(G)}$ may be considered from a strictly graph-theoretic perspective. (See, e.g., [@psd_zf_paper].) Nevertheless, ${{\rm msr}(G)} \ge n - {Z_+(G)}$ for every graph $G$, though the gap may be arbitrarily large [@mitchell_et_al].
With $H$ denoting the Heawood graph, computer calculation using [@sage_min_rank_library] gives $Z_+(H) = 5$, implying that ${{\rm msr}(H)} \ge 9$. The results developed in this paper show that in fact ${{\rm msr}(H)}=10$. This is accomplished via a geometric approach that may be applied to the graph describing the incidence structure of any Steiner triple system. The Heawood graph is one such graph; we treat the general case in Section \[sec:generalization\].
The remainder of this paper is organized as follows. Section \[sec:heawood\] presents the necessary background regarding the Heawood graph and its relevant connections with other mathematical objects. Section \[sec:heawood\_msr\] develops the main results of the paper, establishing upper and lower bounds on the minimum semidefinite rank of the Heawood graph. Section \[sec:generalization\] explores how the same approach may be applied to the incidence graph of any Steiner triple system, and exhibits further bounds derived by this method. Finally, Section \[sec:conclusion\] highlights some questions suggested by this work and possible directions for future research.
The Heawood graph {#sec:heawood}
=================
The Heawood graph, shown in Figure \[fig:heawood\_graph\], has served as an important example in the study of minimum rank problems. For example, the complement of this graph was used in [@barioli_et_al_JGT] to give separation between various minimum rank parameters and corresponding combinatorial bounds. Also, [@canto] presented the first example of a zero-nonzero pattern for which the minimum rank (over the reals) was unequal to a combinatorial lower bound known as the *triangle number*, and the pattern given was exactly that of the biadjacency matrix (defined in Section \[sec:heawood\_msr\]) of the Heawood graph.
The minimum rank of the Heawood graph may be obtained as follows. First, the ordinary zero forcing number (not the positive semidefinite variant) gives a lower bound of $8$. Meanwhile, the adjacency matrix $A$ of the graph has the eigenvalue $\sqrt 2$ with multiplicity $6$, so that $\operatorname{rank}(A - \sqrt 2 I)=14-6=8$ gives a corresponding upper bound. Hence, the minimum rank of the Heawood graph is $8$.
In the context of minimum semidefinite rank, interest in the Heawood graph emerged due to properties making it an attractive test case for Conjecture \[conj:girth\_conjecture\], as outlined in Section \[sec:intro\]. For what follows, the most important way to view the Heawood graph is through its connection with the *Fano plane*, the finite projective plane of order $2$, illustrated in Figure \[fig:fano\_plane\]. This is a finite geometry comprising seven points and seven lines in which each line contains exactly three points and each point lies on exactly three lines. Hence, its points and lines give a Steiner triple system (in fact, the unique one) of order $7$. We return to this connection in Section \[sec:generalization\]; for now, we need to note only the following.
\_mark\_radius [1.75]{}
\(O) at (0,0); (O) circle\[radius=\_mark\_radius pt\];
\(O) circle ();
in [1,2,3]{}
\(A) at ([120 \* - 90]{} : ); (B) at ($(O)!{-2*\r cm}!(A\s)$); (A) – (B);
\(A) circle\[radius=\_mark\_radius pt\]; (B) circle\[radius=\_mark\_radius pt\];
(B1) – (B2) – (B3) – (B1);
(B3) node\[above=0.5ex\] [$\sf 2$]{}; (B2) node\[below=1.5ex,right=0.0ex\] [$\sf 4$]{}; (B1) node\[below=1.5ex,left=0.0ex\] [$\sf 7$]{}; (A1) node\[above=0.5ex,right=0.25ex\] [$\sf 5$]{}; (A3) node\[below=0.25ex\] [$\sf 6$]{}; (A2) node\[above=0.5ex,left=0.25ex\] [$\sf 3$]{}; (O) node\[above=1.9ex,right=-0.6ex\] [$\sf 1$]{};
\[obs:lines\_of\_fano\_plane\] The points of the Fano plane may be identified with the integers $1,2,\ldots,7$ so that the set of its lines becomes $$\label{eqn:fano_plane_lines}
\{\{1,2,6\}, \{2,3,7\}, \{1,3,4\},\{2,4,5\},\{3,5,6\},\{4,6,7\}, \{1,5,7\}\}.$$ Figure \[fig:fano\_plane\] shows the points of the Fano plane labeled to reflect such an identification.
The Heawood graph is the *point-edge incidence graph* of the Fano plane. That is, its vertices can be partitioned into two independent sets, one in correspondence with the points of the Fano plane, and the other in correspondence with its lines, such that a point and a line are incident precisely when the corresponding vertices are adjacent. Through this connection, many of the properties of the Heawood graph that we will need follow from properties of the Fano plane.
One such property concerns the smallest size of a set from which one may color the points of the Fano plane without inducing a *monochromatic line*, a line with all of its points colored the same. In particular, the following simple fact (a special case of a result of [@rosa]; see Section \[sec:generalization\]) is straightforward to verify.
\[lem:fano\_plane\_not\_2\_colorable\] Every $2$-coloring of the points of the Fano plane induces a monochromatic line.
The minimum semidefinite rank of the Heawood graph {#sec:heawood_msr}
==================================================
The goal of this section is to establish that the minimum semidefinite rank of the Heawood graph is 10, and that this in fact holds over $\mathbb R$ as well. We begin by noting that if a graph is bipartite, then a special attack is possible on the problem of determining its minimum semidefinite rank. A brief argument in the case of the Heawood graph follows; for a general discussion, see [@deaett_msr Theorem 5.3] or [@jiang_et_al Proposition 3.1].
\[lm:condition\_on\_terminal\_Us\] Let $F$ be $\mathbb R$ or $\mathbb C$. The Heawood graph has an orthogonal representation in $F^{7+n}$ if and only if some matrix $A \in M_{7+n,7}(F)$ with mutually orthogonal columns has the form $$\label{eq:heawood_matrix_with_terminal_u_vectors}
{\left[ \begin{array}{lllllllll}
* & * & 0 & 0 & 0 & * & 0 \\
0 & * & * & 0 & 0 & 0 & * \\
* & 0 & * & * & 0 & 0 & 0 \\
0 & * & 0 & * & * & 0 & 0 \\
0 & 0 & * & 0& * & * & 0 \\
0 & 0 & 0 & * & 0 & * & * \\
* & 0 & 0 & 0 & * & 0 & * \\ \hline
u_1 & u_2 & u_3 & u_4 & u_5 & u_6 & u_7
\end{array} \right]},$$ where each $\ast$ denotes a nonzero entry, and each $u_i$ is a (column) vector in $F^n$.
Let $B$ be a *biadjacency matrix* for the Heawood graph; that is, $B$ is a $(0,1)$-matrix with rows in correspondence with one of its partite sets and columns in correspondence with the other such that $B_{ij}=1$precisely when the vertices corresponding to row $i$ and column $j$ are adjacent. Then, subject to an appropriate ordering of its rows and columns, the zero-nonzero pattern of $B$ is given by the upper $7\times 7$ submatrix of . That is, $A \in M_{7+n,7}(F)$ is of the form if and only if the upper $7\times 7$ submatrix of $A$ has the zero-nonzero pattern of $B$. Thus, if such a matrix $A$ exists with mutually orthogonal columns, then the columns of $A$ together with the initial $7$ unit coordinate vectors in $F^{7+n}$ form an orthogonal representation of the Heawood graph.
Conversely, given an orthogonal representation of the Heawood graph in $F^{7+n}$, it may be assumed (subject to an appropriate unitary transformation) that one of the partite sets is assigned the first $7$ standard coordinate vectors. Taking the vectors assigned to the other partite set as the columns of a matrix $A$ then gives $A \in M_{7+n,7}(F)$ of the form with mutually orthogonal columns.
Hence, the minimum semidefinite rank of the Heawood graph is seen to be the smallest value of $7+n$ such that some $A \in M_{7+n,7}(\mathbb C)$ of the form has mutually orthogonal columns. Lemma \[lm:condition\_on\_vectors\] gives a useful reformulation of this condition; its proof relies on the following observation.
\[obs:combinatorics\_of\_fano\_plane\_upper\_7\_by\_7\] Let $A$ be a matrix of the form . In particular, each of the seven lines of the Fano plane, as given in , gives the locations of the nonzero entries within one of the first seven rows of $A$. Since each pair of points of the Fano plane lies on exactly one line, it follows that, for every pair of distinct columns $i$ and $j$ of $A$, there exists a unique $k \in \{1,2,\ldots,7\}$ such that both columns have a nonzero entry in row $k$. Hence, when $A$ has entries from $\mathbb C$, the two columns are orthogonal if and only if $A_{ki}\overline{A_{kj}} = -\langle u_i, u_j \rangle$.
Note that, combinatorially, Observation \[obs:combinatorics\_of\_fano\_plane\_upper\_7\_by\_7\] derives from the fact that the Heawood graph is the incidence graph of the Fano plane, precisely because the lines of the Fano plane form a Steiner triple system.
\[lm:condition\_on\_vectors\] Suppose $u_1,u_2,\ldots, u_7 \in \mathbb C^n$. Then the following are equivalent.
1. \[cond:terminal\_vector\_condition\] The vectors $u_1,u_2,\ldots,u_7$ occur as the $u_i$ of in some matrix $A \in M_{7+n,7}(\mathbb C)$ of that form having mutually orthogonal columns.
2. \[cond:inner\_product\_condition\] The product $\langle u_i, u_j \rangle\langle u_j, u_k \rangle\langle u_k, u_i\rangle$ is real and negative whenever $\{i,j,k\}$ is a line in the Fano plane, i.e., whenever $\{i,j,k\}$ is contained in the set .
Moreover, if condition is satisfied with each $u_i \in \mathbb R^n$, then a matrix $A$ witnessing condition exists with $A \in M_{7+n,7}(\mathbb R)$.
Suppose first that condition is satisfied. Then there exists some $A \in M_{7+n,7}(\mathbb C)$ with mutually orthogonal columns such that $$A = {\left[ \begin{array}{lllllllll}
a & b & 0 & 0 & 0 & c & 0 \\
0 & * & * & 0 & 0 & 0 & * \\
* & 0 & * & * & 0 & 0 & 0 \\
0 & * & 0 & * & * & 0 & 0 \\
0 & 0 & * & 0& * & * & 0 \\
0 & 0 & 0 & * & 0 & * & * \\
* & 0 & 0 & 0 & * & 0 & * \\ \hline
u_1 & u_2 & u_3 & u_4 & u_5 & u_6 & u_7
\end{array} \right]},$$ where $a$, $b$, $c$ and each $\ast$ entry are nonzero. Since the columns indexed by the set $\{1,2,6\}$ are mutually orthogonal, it follows from Observation \[obs:combinatorics\_of\_fano\_plane\_upper\_7\_by\_7\] that $$\label{eqn:inner_product_equations}
a = \frac {-\langle u_1,u_2\rangle}{\overline b},
\quad
b = \frac {-\langle u_2, u_6 \rangle} {\overline c}
\quad
\text{ and }
\quad
c = \frac {-\langle u_6, u_1 \rangle} {\overline a}.$$ Combining the first and third of these equations yields $$\label{eqn:expression_for_c}
c = \frac {-\langle u_6, u_1 \rangle} {\overline a} = -\langle u_6, u_1 \rangle \frac b {-\langle u_2,u_1\rangle} = \frac {\langle u_6, u_1 \rangle} {\langle u_2,u_1\rangle} b,$$ and combining this with the second equation of gives $$b = \frac {-\langle u_2, u_6 \rangle} {\overline c} = \frac {-\langle u_2, u_6 \rangle}{\overline b} \frac {\langle u_1, u_2 \rangle} {\langle u_1, u_6\rangle},$$ which implies that $$\label{eqn:equation_in_terms_of_b}
0 > -|b|^2 = \frac {\langle u_1, u_2 \rangle\langle u_2, u_6 \rangle} {\langle u_1, u_6\rangle}
\frac{\langle u_6, u_1\rangle}{\langle u_6, u_1\rangle} = \frac {\langle u_1, u_2 \rangle\langle u_2, u_6 \rangle\langle u_6, u_1\rangle } {|\langle u_1, u_6\rangle|^2}.$$ In particular, then, $\langle u_1, u_2 \rangle\langle u_2, u_6 \rangle\langle u_6, u_1\rangle$ is real and negative. This same argument may be applied to every row of $A$, and hence condition holds.
Conversely, suppose condition holds. Equations analogous to , and then yield values for the nonzero entries in each of the initial $7$ rows of a matrix $A$ of the form . Explicitly, for each $m \in \{1,2,\ldots,7\}$, if the three nonzero entries in row $m$ fall in columns $i$, $j$ and $k$ with $i < j < k$, then values for those entries may be taken as $$\label{eqn:entries_from_inner_products}
A_{mj} =
\sqrt{-\frac {\langle u_i, u_j \rangle\langle u_j, u_k \rangle\langle u_k, u_i\rangle } {|\langle u_i, u_k\rangle|^2}},
\quad
A_{mi} = -\frac{\langle u_i,u_j\rangle}{A_{mj}},
\quad \text{and}\quad
A_{mk} = A_{mj}\frac{\langle u_k, u_i \rangle}{\langle u_j,u_i\rangle}.$$ It then follows from Observation \[obs:combinatorics\_of\_fano\_plane\_upper\_7\_by\_7\] that $A$ has mutually orthogonal columns. Hence, condition holds.
Lemmas \[lm:condition\_on\_terminal\_Us\] and \[lm:condition\_on\_vectors\] together show that the minimum semidefinite rank of the Heawood graph is the smallest value of $7+n$ such that vectors $u_1,u_2,\ldots,u_7 \in \mathbb{C}^n$ exist satisfying condition of Lemma \[lm:condition\_on\_vectors\]. In particular, to establish an upper bound of $10$ on this value, it suffices to construct vectors in $\mathbb C^3$ satisfying this condition; we next show that in fact such vectors can be constructed in $\mathbb R^3$.
\[lm:heptagonal\_construction\] There exist vectors $u_1,u_2,\ldots,u_7 \in \mathbb R^3$ such that $\langle u_i, u_j \rangle\langle u_j, u_k \rangle\langle u_k, u_i\rangle$ is real and negative whenever $\{i,j,k\}$ is a line in the Fano plane, i.e., whenever $\{i,j,k\}$ is contained in the set .
Given a positive real number $\alpha$, let $$u_j = \left(\cos(2\pi j/7),\sin(2\pi j/7), \sqrt\alpha\, \right)$$ for each $j\in\{1,2,\ldots,7\}$. Then, for any $j$ and $k$, $$\begin{aligned}
\langle u_j,u_k \rangle &=& \cos(2\pi j/7)\cos(2\pi k/7) + \sin(2\pi j/7)\sin(2\pi k/7) + \alpha \\
&=& \cos( 2\pi(j-k)/7) + \alpha,\end{aligned}$$ so that $\langle u_j,u_k\rangle$ is completely determined by the difference $j-k$ modulo $7$. But the set of pairwise differences modulo $7$ is the same for every set contained in ; explicitly, it is $\{1,4,5\}$. Hence, it suffices to ensure that the conclusion holds for any one such set, e.g., to guarantee that $\langle u_1, u_2 \rangle\langle u_2, u_6 \rangle\langle u_6, u_1\rangle$ is real and negative. This can be achieved by choosing $\alpha$ such that $\cos(3\pi/7) < \alpha < \cos(\pi/7)$, as then $$\begin{aligned}
{7}
\langle u_1,u_2 \rangle &=& \cos(2\pi/7) + \alpha &>& 0,\phantom{\cos(2\pi/7) + \alpha} \\
\langle u_2,u_6 \rangle &=& \cos(8\pi/7) + \alpha &=& -\cos(\pi/7) + \alpha & < 0, & \text{ and}\\
\langle u_6,u_1 \rangle &=& ~\cos(10\pi/7)+ \alpha &=& ~-\cos(3\pi/7)+\alpha & > 0. &
\qedhere\end{aligned}$$
This yields an upper bound on the minimum semidefinite rank of the Heawood graph.
\[prop:real\_upper\_bound\] The minimum semidefinite rank of the Heawood graph is at most $10$.
By Lemma \[lm:heptagonal\_construction\], there exist $u_1,u_2,\ldots,u_7 \in \mathbb R^3$ satisfying condition of Lemma \[lm:condition\_on\_vectors\], and hence appearing as the $u_i$ of for some matrix $A \in M_{10,7}(\mathbb R)$ with orthogonal columns. Hence, by Lemma \[lm:condition\_on\_terminal\_Us\], the Heawood graph has an orthogonal representation in $\mathbb R^{10}$, and hence in $\mathbb{C}^{10}$.
To establish our main result, we turn now to the requisite lower bound, namely that the minimum semidefinite rank of the Heawood graph is at least $10$. By the discussion above, it suffices to show that no vectors from $\mathbb C^2$ exist satisfying the conditions of Lemma \[lm:condition\_on\_vectors\]. Our approach can be summarized as follows. Assuming to the contrary that such vectors do exist, we identify them with points on the Riemann sphere. We then argue that the two conditions on these vectors shown to be equivalent by Lemma \[lm:condition\_on\_vectors\] are themselves equivalent to a condition on the corresponding points on the sphere that, when satisfied, implies that these points must be arranged in such a way as to induce a $2$-coloring of the Fano plane with no monochromatic line, in contradiction to Lemma \[lem:fano\_plane\_not\_2\_colorable\].
Our first task is to establish the appropriate correspondence between vectors in $\mathbb C^2$ and points on the appropriate sphere in $\mathbb R^3$. We begin with a crucial observation.
\[obs:scaling\_invariance\] Each of the two conditions of Lemma \[lm:condition\_on\_vectors\] is unaffected by multiplying any individual vector by an arbitrary nonzero complex scalar.
In light of Observation \[obs:scaling\_invariance\], we may regard the conditions of Lemma \[lm:condition\_on\_vectors\] as applying to points on the projective line $\mathbb CP^1$, which can be thought of as the extended complex plane, $\mathbb C \cup \{\infty\}$. Through the usual stereographic projection, the extended complex plane can be transformed bijectively to a sphere in $\mathbb R^3$. The image of such an identification is typically referred to as the *Riemann sphere*. (See [@needham Section 3.IV] for details.) Any sphere in $\mathbb R^3$ can be made the image of such an identification; for the sake of making our computations explicit in what follows, we will choose the sphere of radius $1/2$ centered at $(0,0,1/2)$, namely $$\label{eq:def_of_sphere}
S = \{ (x,y,z) : x^2+y^2+(z-1/2)^2 = (1/2)^2 \} = \{ (x,y,z) : x^2+y^2+z^2 = z \}.$$ Again for the sake of explicit computation, we now define a function $\varphi$ that effects the identification outlined above, mapping points in $\mathbb C^2$ to points on $S$.
\[def:def\_of\_phi\_map\] Let $\varphi:\mathbb C^2 \rightarrow \mathbb R^3$ be defined as follows. First, let $\varphi_1$ map $\mathbb C^2$ to $\mathbb CP^1$ in the usual way, i.e., $$\varphi_1(z_1,z_2) = \begin{cases} z_1/z_2 & \text{if } z_2\not= 0, \\ \infty & \text{otherwise}. \end{cases}$$ Next, apply the familiar stereographic projection to map the image of $\varphi_1$ to the sphere $S$ defined in . Specifically, identify $\infty$ with the “pole” of the sphere at $(0,0,1)$, and identify $a+bi \in \mathbb C$ with the unique point at which $S\setminus\{(0,0,1)\}$ intersects the line parameterized by $$t(a,b,0) + (1-t)(0,0,1),\quad t \in \mathbb R.$$ It follows from that this point of intersection is $\frac 1{1+a^2+b^2}(a,b,a^2+b^2)$. Thus, we let $$\varphi_2(z) = \begin{cases}
\frac 1{1+a^2+b^2}(a,b,a^2+b^2) & \text{if } z=a+bi, \text{ and} \\
(0,0,1) & \text{if } z = \infty.
\end{cases}$$ Finally, let $\varphi = \varphi_2 \circ \varphi_1$.
Having identified each vector in $\mathbb C^2$ with a point on the sphere $S$, the conditions of Lemma \[lm:condition\_on\_vectors\] applied to triples of vectors in $\mathbb C^2$ can be reinterpreted as applying to triples of points on $S$. The next lemma provides two equivalent such interpretations.
\[lm:condition\_on\_points\_of\_S2\] Let $C$ denote the center of the sphere $S$ defined in , i.e., $C=(0,0,1/2)$. For any $u_i$, $u_j, u_k \in \mathbb C^2$, the following are equivalent.
1. \[cond:ip\_cond\] The product $\langle u_i,u_j \rangle \langle u_j,u_k \rangle \langle u_k,u_i \rangle$ is real and negative.
2. \[cond:center\_in\_convex\_hull\] No two of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ are antipodal on $S$, but the convex hull of all three contains $C$.
3. \[cond:separating\_plane\_condition\] Every plane passing through $C$ that contains none of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ separates one of those latter three points from the other two. Moreover, each of those three points is separated from the other two by some such plane.
We start with some simplifying assumptions. First, subject to the appropriate scaling, we may assume that each of $u_i$, $u_j$ and $u_k$ is equal either to $(1,0)$ or to $(z,1)$ for some $z\in\mathbb C$. (That is, we may work projectively.) Since it is clear from Definition \[def:def\_of\_phi\_map\] that the image of a point under $\varphi$ is determined only by the line through the origin in $\mathbb C^2$ on which the point lies, this cannot affect conditions or , while by Observation \[obs:scaling\_invariance\] it does not affect condition .
Next, observe that some pair of rotations of the sphere can be applied sequentially to move $\varphi(u_i)$ to the origin and $\varphi(u_j)$ to a point on the $xz$-plane with a nonnegative $x$-coordinate. This clearly leaves conditions and unaffected. Moreover, such a rigid motion of the sphere corresponds to a unitary transformation of $\mathbb C^2$ [@needham Section 6.II] and hence preserves condition . Hence, we may assume that $\varphi(u_i)=(0,0,0)$, so that, equivalently, $u_i = (0,1)$, and also that for some nonnegative real number $s$, $$\varphi(u_j)={\textstyle \frac 1{1+s^2}}(s,0,s^2), \text{ so that, equivalently, } u_j = (s,1).$$ These assumptions are illustrated in Figure \[fig:sphere\_with\_points\].
cos() [ [ ]{} [ ]{} ]{} [ [ ]{} [ ]{} ]{}
(0,0) circle (); (0,0) circle ();
\(O) at (0,0); (C) at (0,0); (S) at (0,-);
(0.65\*,1.5) coordinate\[mark coordinate\] (XE);
[ [ [ ]{} [ ]{} ]{} (:1) arc (:+180:1); (-180:1) arc (-180::1); ]{}
(-1\*,0) – (01\*,0) node\[below\] [$x$]{}; (0,-1\*) – (0,1\*) node\[below\] [$y$]{}; (0,-) – (0,1.15\*) node\[above\] [$z$]{};
(0,1.5) – (0.65\*,1.5); (0.65\*,1.5) – (0.65\*,-1\*);
\(C) +(1.4ex,0.1ex) node\[below\] [$C$]{}; (S) +(4.1ex,2.1ex) node\[below\] [$\varphi(u_i)$]{}; (XE) +(1.25ex,0.5ex) node\[above left\] [$\varphi(u_j)$]{};
Finally, for any $z \in \mathbb C^2$, let $\overline \varphi(z)$ denote the point on $S$ antipodal to $\varphi(z)$. In particular, $$\label{eqn:phi_of_uj_expression}
\overline\varphi(u_j)= {\textstyle\frac 1{1+s^2}} \left( -s,0, 1 \right).$$ We now begin the proof by showing conditions and to be equivalent. Suppose first that holds. This is incompatible with $u_k=(1,0)$, since $u_i=(0,1)$. Therefore $u_k=(t,1)$ for some $t\in\mathbb C$. Hence, $\langle u_i,u_j \rangle \langle u_j,u_k \rangle \langle u_k,u_i \rangle = 1+st$ is real and negative by , and so $t \in \mathbb R$ with $t < 0$ and $s > 0$. Thus, $$\label{eqn:phi_of_uk_expression}
\varphi(u_k)={\textstyle\frac 1{1+t^2}}(t,0,t^2),$$ and so $\varphi(u_k)$ lies on the $xz$-plane with a negative $x$-coordinate. Moreover, $$1+st < 0 \implies |st| > 1 \implies s^2t^2 > 1 \implies 1+t^2 < t^2(1+s^2) \implies {\textstyle\frac 1{1+s^2}} < {\textstyle\frac {t^2}{1+t^2}}.$$ By and , this shows that the $z$-coordinate of $\varphi(u_k)$ exceeds that of $\overline\varphi(u_j)$, so that $C$ is in the convex hull of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$, and also shows that $\varphi(u_j) \not= \overline\varphi(u_k)$. Moreover, neither $\varphi(u_j)$ nor $\varphi(u_k)$ may equal $\overline\varphi(u_i)=(0,0,1)$. Hence, the three points $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ do not contain an antipodal pair. Thus, condition holds.
Now suppose that holds. Then $s>0$, as $s=0$ would give $\varphi(u_j)=\varphi(u_i)$, requiring $\varphi(u_k) = \overline\varphi(u_i)$ in order that $C$ lie in the convex hull of the three points. Further, since $\varphi(u_k) \not= \overline\varphi(u_i)$, we cannot have $u_k=(1,0)$. Therefore, $u_k=(t,1)$ for some $t \in \mathbb C$. The fact that $C$ is in the convex hull of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ implies that $\varphi(u_k)$ lies on the plane containing $\varphi(u_i)$, $\varphi(u_j)$ and $C$, namely the $xz$-plane. Thus, we have $t\in \mathbb R$, so that $$\varphi(u_k) = {\textstyle\frac 1{1+t^2}}(t,0,t^2).$$ Moreover, since $\varphi(u_k)$ and $\varphi(u_j)$ must lie on opposite sides of the $yz$-plane, we have $t<0$. Finally, the $z$-coordinate of $\varphi(u_k)$ must exceed that of $\overline\varphi(u_j)$, so that $\frac{t^2}{1+t^2} > \frac 1{1+s^2}$. This gives $t^2+t^2s^2 = t^2(1+s^2) > 1+t^2$. Hence, $s^2t^2 > 1$, so that $|st| > 1$. Combining this with the fact that $t<0$ and $s > 0$ so that $st < 0$, we have $\langle u_i,u_j \rangle \langle u_j,u_k \rangle \langle u_k,u_i \rangle=1+st < 0$, so that condition holds.
Having shown that conditions and are equivalent, we now complete the proof by proving the equivalence of conditions and . Assume first that holds. Then $C$ is in the convex hull of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$, so that all three points lie on the $xz$-plane. Now consider a plane $P$ passing through $C$ that contains none of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$. As $P$ may not coincide with the $xz$-plane, its intersection with the $xz$-plane is a line $L$ passing through $C$. Since $C$ is in the convex hull of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$, these three points cannot lie all on the same side of $L$. This implies that $L$, and hence $P$, separates one of the three points from the other two.
It remains to show that each of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ is separated from the other two by some plane containing $C$. By symmetry, it suffices to prove that $\varphi(u_i)$ is separated from $\varphi(u_j)$ and $\varphi(u_k)$ by some such plane. Toward that end, consider the plane $P$ perpendicular to the $xz$-plane and passing through $\varphi(u_j)$ and $\overline\varphi (u_j)$. Since $C$ is in the convex hull of the three points, $\varphi(u_k)$ cannot lie on the same side of $P$ as does $\varphi(u_i)$. Moreover, $\varphi(u_k)$ cannot lie on the plane $P$, as this would imply $\varphi(u_k)=\overline\varphi(u_j)$, which forbids. Hence, $\varphi(u_i)$ and $\varphi(u_k)$ must lie on opposite sides of $P$. Since $P$ contains the line through $C$ that is perpendicular to the $xz$-plane, it follows that rotating $P$ about this line by a sufficiently small angle produces a plane through $C$ separating $\varphi(u_i)$ from $\varphi(u_j)$ and $\varphi(u_k)$, as desired.
Now suppose holds. If any two points among $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ were antipodal, then those two points could not be separated from the third by any plane containing $C$, which would contradict . Hence, $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ do not contain an antipodal pair, and it remains to show that $C$ is in their convex hull.
First, since $\varphi(u_i)$ and $\varphi(u_j)$ are not antipodal, they lie on the same side of some line in the $xz$-plane passing through $C$. If $\varphi(u_k)$ were not in the $xz$-plane, then rotating the $xz$-plane about this line by some small angle would produce a plane relative to which all three of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ would lie on the same side, contradicting . Hence, $\varphi(u_k)$ lies on the $xz$-plane along with $\varphi(u_i)$, $\varphi(u_j)$ and $C$.
We have by that $\varphi(u_i)$ is separated from $\varphi(u_j)$ and $\varphi(u_k)$ by some plane $P$ that contains none of those points but does contain $C$. As $P$ may not coincide with the $xz$-plane, it intersects the $xz$-plane in some line $L$ passing through $C$. Let $A$ be the point at which the line through $\varphi(u_i)$ and $\varphi(u_j)$ intersects $L$ and let $B$ be the point at which the line through $\varphi(u_i)$ and $\varphi(u_k)$ intersects $L$. (See Figure \[fig:circle\_diagram\].)
\(C) at (1.0\*,0) ; (O) at (20:0\*); (O) node\[left\] [$\varphi(u_i)$]{};
\(C) at (1\*,0); (C) node\[below=1ex\] [$C$]{};
(0,1\*) node\[above\] [$x$]{} – (0,-1\*); (O) – (2.5\*,0) node\[right\] [$z$]{};
(phi\_uj) at (1.8\*,0.6\*); (phi\_uj) node\[above=1ex,right=1ex\] [$\varphi(u_j)$]{};
(phi\_uk) at (1.1\*,-1\*); (phi\_uk) node\[below=1ex,right=0.25ex\] [$\varphi(u_k)$]{};
\(C) ++(:1.55\*) coordinate (L\_end1); (C) ++(+180:1.65\*) coordinate (L\_end2); (C) – (L\_end1) node\[left=2ex\] [$L\vphantom{^|}$]{}; (C) – (L\_end2);
\(O) – (phi\_uj); (O) – (phi\_uk);
\(A) at (intersection of O–phi\_uj and C–L\_end1); (A) node\[below=1ex\] [$A$]{};
\(B) at (intersection of O–phi\_uk and C–L\_end1); (B) node\[below=1ex\] [$B$]{};
Note that $\varphi(u_j)$ and $\varphi(u_k)$ must lie on opposite sides of the $yz$-plane, as otherwise rotating that plane by some small angle about the line through $C$ that is perpendicular to the $zx$-plane would produce a plane containing $C$ on one side of which would lie all three of the points $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$, contradicting . It follows that $A$ and $B$ lie on opposite sides of the $yz$-plane as well. This implies that $C$ is on the line segment with endpoints $A$ and $B$. Since $A$ and $B$ were chosen within the convex hull of $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$, it follows that $C$ lies in the convex hull of those points as well. Hence, condition holds.
We now have that any collection of vectors in $\mathbb C^2$ satisfying the algebraic condition of Lemma \[lm:condition\_on\_vectors\] corresponds to a collection of points on the sphere $S$ arranged such that every triple of points corresponding to a line in the Fano plane satisfies the geometric conditions and of Lemma \[lm:condition\_on\_points\_of\_S2\]. We next show that such an arrangement gives rise to an impossible coloring of the Fano plane, a contradiction that yields our desired lower bound.
\[prop:complex\_lower\_bound\] The minimum semidefinite rank of the Heawood graph is at least $10$.
Suppose to the contrary that the Heawood graph has an orthogonal representation in $\mathbb C^9$. Then, by Lemma \[lm:condition\_on\_terminal\_Us\], there exist vectors $u_1,u_2,\ldots,u_7 \in \mathbb C^2$ satisfying the equivalent conditions of Lemma \[lm:condition\_on\_vectors\]. By Lemma \[lm:condition\_on\_points\_of\_S2\], these vectors induce points $\varphi(u_1),\varphi(u_2),\ldots,\varphi(u_7)$ on the sphere $S$ defined in such that whenever $\{i,j,k\}$ is a line in the Fano plane, i.e., whenever $\{i,j,k\}$ is contained in the set , the triple of points $\varphi(u_i)$, $\varphi(u_j)$ and $\varphi(u_k)$ satisfies condition of Lemma \[lm:condition\_on\_points\_of\_S2\].
Let $P$ be any plane through the center of $S$ that contains none of the points $\varphi(u_1),\varphi(u_2),\ldots,\varphi(u_7)$. Then $P$ divides $S$ into hemispheres. Choose one of the hemispheres, and color red every point $i$ of the Fano plane such that $\varphi(u_i)$ lies on that hemisphere. Color the other points of the Fano plane green. By Corollary \[lem:fano\_plane\_not\_2\_colorable\], this coloring must result in some line of the Fano plane, say $\{r,s,t\}$, all of whose points are colored the same. But this means that $\varphi(u_r)$, $\varphi(u_s)$ and $\varphi(u_t)$ lie all on the same hemisphere of $S$, contradicting condition of Lemma \[lm:condition\_on\_points\_of\_S2\].
Our main result now follows from the combination of Propositions \[prop:real\_upper\_bound\] and \[prop:complex\_lower\_bound\].
The minimum semidefinite rank of the Heawood graph is $10$.
Incidence graphs of Steiner triple systems {#sec:generalization}
==========================================
We now identify the Heawood graph as one of a general family of graphs to which the approach of Section \[sec:heawood\_msr\] may be applied. Recall the following definition from combinatorial design theory; see, e.g., [@comb_design_handbook Chapter 2].
\[def:steiner\_system\] A *Steiner triple system* of order $v$ consists of a set $X$, whose elements are called the *points* of the system, such that $|X|=v$, together with a collection of $3$-subsets of $X$, called the *triples* of the system, such that every $2$-subset of $X$ is contained in exactly one triple.
It follows from Observation \[obs:lines\_of\_fano\_plane\] that the lines of the Fano plane form a Steiner triple system of order $7$. (Actually it is the unique Steiner triple system of that order.) A fact crucial to the proof of Proposition \[prop:complex\_lower\_bound\] was previously noted as Lemma \[lem:fano\_plane\_not\_2\_colorable\], namely that every $2$-coloring of the points of the Fano plane induces a monochromatic line. More generally, the *weak chromatic number* of a Steiner triple system is the smallest number of colors from which the points of the system may be colored such that no triple is left with all of its points colored the same; Lemma \[lem:fano\_plane\_not\_2\_colorable\] is a special case of the following result of [@rosa].
\[thm:steiner\_3\_colorable\] Every Steiner triple system of order $7$ or greater has a weak chromatic number of at least $3$.
Every Steiner triple system is represented by a bipartite graph in the same sense in which the Fano plane is represented by the Heawood graph.
The *incidence graph* of a Steiner triple system is the graph $G$ whose vertices can be partitioned into two sets, one in correspondence with the points of the system and the other in correspondence with its triples, such that two vertices are adjacent precisely when they correspond to a point and a triple containing that point.
Hence, the Heawood graph is the incidence graph of the unique Steiner triple system of order $7$. The approach developed in Section \[sec:heawood\_msr\] to establish a lower bound on the minimum semidefinite rank of the Heawood graph can be adapted to do the same for the incidence graph of any Steiner triple system of order at least $7$.
\[thm:general\_steiner\_triple\_system\_analog\] Let $G$ be the incidence graph of a Steiner triple system of order $v \ge 7$, let $b$ be the number of triples of the system, and let $n$ be the number of vertices of $G$. Then $b = { \textstyle \frac 13{v \choose 2} }$, $n= b + v = \textstyle \frac 16 (v^2+5v)$, and $$\label{eq:steiner_system_general_bound}
{{\rm msr}(G)} \geq\, b + 3 = \textstyle\frac 16 (v^2-v+18).$$
That $b = { \textstyle \frac 13{v \choose 2} }$ follows immediately from Definition \[def:steiner\_system\]. The claim that $n=b+v$ is trivial. By definition, $G$ has a biadjacency matrix $M$ of size $b \times v$. With the zero-nonzero pattern of $M$ playing the role of the upper portion of , a result analogous to Lemma \[lm:condition\_on\_terminal\_Us\] is obtained by the same argument. It follows from Definition \[def:steiner\_system\] that, for any matrix whose initial $b$ rows have a zero-nonzero pattern matching that of $M$, a statement analogous to Observation \[obs:combinatorics\_of\_fano\_plane\_upper\_7\_by\_7\] holds. Hence, the statement and proof of Lemma \[lm:condition\_on\_vectors\] can be adapted in a straightforward way, and Lemma \[lm:condition\_on\_points\_of\_S2\] can then be applied without modification.
The conclusion of the argument then proceeds as in the proof of Proposition \[prop:complex\_lower\_bound\]. That is, any supposed orthogonal representation of $G$ in fewer than $b+3$ dimensions gives rise to $v$ points on the Riemann sphere arranged so as to induce a $2$-coloring of the points of the Steiner triple system in which at least two different colors occur within every triple, contradicting Theorem \[thm:steiner\_3\_colorable\].
We wish to point out the limitations of Theorem \[thm:general\_steiner\_triple\_system\_analog\], so as to make clear why we did not attempt to develop our main results in such general terms. To this end, note that the incidence graph $G$ of a Steiner triple system of order $v$ has an independent set (corresponding to the triples of the system) of size $b=\frac 13{v \choose 2}$. This trivially implies that ${{\rm msr}(G)} \ge b = \frac 13{v \choose 2} = \frac 16(v^2 - v)$, and Theorem \[thm:general\_steiner\_triple\_system\_analog\] provides only a slight improvement on this bound. In the case of the Heawood graph, what is interesting is that this improved bound is sharp. It is unclear whether this remains the case for the incidence graphs of larger Steiner triple systems, however. Nevertheless, there are many (see [@comb_design_handbook p. 15]) Steiner triple systems of small order, and the application of Theorem \[thm:general\_steiner\_triple\_system\_analog\] to their incidence graphs may be illuminating.
Conclusion and open questions {#sec:conclusion}
=============================
Theorem \[thm:general\_steiner\_triple\_system\_analog\] gives a lower bound on ${{\rm msr}(G)}$ whenever $G$ is the incidence graph of a Steiner triple system. It is natural to compare this bound with that obtained from the positive semidefinite zero forcing number ${Z_+(G)}$ referenced in Section \[sec:intro\]. Table \[tab:msr\_vs\_zf\_for\_sts\] details the result of this comparison for each Steiner triple system of order $v$, with $v > 3$ to avoid the trivial case, up to $v = 15$. (The next order for which any Steiner triple systems exist is $v=19$, but in this case it would be computationally expensive to determine ${Z_+(G)}$ for even just one of these, and there are altogether $11{,}084{,}874{,}829$ of them [@kaski].)
[C[0.475in]{}@C[0.475in]{}C[0.8in]{}C[0.62in]{}C[0.8in]{}C[0.8in]{}C[1.1in]{}]{} & Number of Steiner triple systems & Number of vertices in $G$ & Positive semidefinite zero forcing number & Bound from zero forcing & Bound from Theorem \[thm:general\_steiner\_triple\_system\_analog\]\
(lr)[1-2]{}(lr)[4-4]{}(lr)[5-5]{}(lr)[6-6]{}(lr)[7-7]{}
$v$ & $b$ & & $n=v+b$ & ${Z_+(G)}$ & $n-{Z_+(G)}$ & $b+3 = \frac 13{v \choose 2} + 3$\
7 & 7 & 1 & 14 & 5 & 9 & 10\
9 & 12 & 1 & 21 & 7 & 14 & 15\
13 & 26 & 2 & 39 & 11 & 28 & 29\
15 & 35 & 80 & 50 & 13 & 37 & 38\
Table \[tab:msr\_vs\_zf\_for\_sts\] invites some observations. The first is that in each case the lower bound obtained from Theorem \[thm:general\_steiner\_triple\_system\_analog\] exceeds the lower bound provided by the zero forcing number by exactly one. This happens to be the case because, for each graph $G$ of the $84$ detailed in the table, ${Z_+(G)}$ turns out to be $2$ less than the order of the corresponding Steiner triple system. The question as to whether this holds in general is outside the scope of the present work, but seems interesting.
\[q:pattern\_in\_table\_continues\] Does ${Z_+(G)} = v - 2$ whenever $G$ is the incidence graph of a Steiner triple system of order $v$?
An affirmative answer to Question \[q:pattern\_in\_table\_continues\] would imply that the positive semidefinite zero forcing number of the incidence graph of a Steiner triple system of order $v$ is determined by $v$ alone. It is open as well whether this may be the case for the positive semidefinite minimum rank itself.
\[q:noniso\_STS\_with\_differing\_msr\] Do there exist two nonisomorphic Steiner triple systems of the same order whose incidence graphs differ in their minimum semidefinite rank?
In particular, although the lower bound provided by Theorem \[thm:general\_steiner\_triple\_system\_analog\] is met by the Heawood graph, we do not expect that this is uniformly the case for the incidence graphs of Steiner triple systems of larger order. Nevertheless, the question remains open even for the unique Steiner triple system of order $9$.
Is the Heawood graph the only incidence graph of a Steiner triple system for which the lower bound of Theorem \[thm:general\_steiner\_triple\_system\_analog\] is met? In particular, with $G$ the incidence graph of the unique Steiner triple system of order $9$, does an orthogonal representation of $G$ in $\mathbb C^{15}$ exist?
Given a lower bound on the minimum semidefinite rank, the problem of establishing a corresponding upper bound is often handled via some appropriate geometric construction. Here this is done for the Heawood graph via Lemma \[lm:heptagonal\_construction\]. For the incidence graphs of larger Steiner triple systems, however, the problem remains open.
Can properties of Steiner triple systems in general be exploited to construct low-dimensional orthogonal representations for their incidence graphs?
Of course, the questions explored here for Steiner triple systems may be considered for the incidence graphs of other families of combinatorial designs. By definition, such graphs are bipartite, and so a natural analog of Lemma \[lm:condition\_on\_terminal\_Us\] is always available. No appropriate generalization of Lemma \[lm:condition\_on\_vectors\], however, seems forthcoming in any case beyond that of a Steiner triple system.
For the case of Steiner triple systems, Lemma \[lm:condition\_on\_points\_of\_S2\] gives a useful geometric interpretation of the conditions on the vectors $u_i$ of Lemma \[lm:condition\_on\_vectors\], and this was crucial to the approach used to obtain the lower bound of Theorem \[thm:general\_steiner\_triple\_system\_analog\]. This raises the question as to whether there can be found some analogous geometric interpretation for these conditions as they apply to vectors in $\mathbb C^k$ for $k \ge 3$. Such an interpretation might provide an avenue toward generalizing the lower bound established here for the Heawood graph to the minimum semidefinite ranks of the incidence graphs of other Steiner triple systems.
Acknowledgments
===============
The present work developed through a collaboration of the authors that began at the 2010 NSF-CBMS Regional Research Conference entitled *The Mutually Beneficial Relationship of Matrices and Graphs*, supported by the IMA and by the NSF through grant number DMS-0938261. The authors wish to thank those organizations as well as Iowa State University, which hosted the meeting.
|
---
author:
- |
First Author\
Affiliation / Address line 1\
Affiliation / Address line 2\
Affiliation / Address line 3\
[email@domain]{}\
Second Author\
Affiliation / Address line 1\
Affiliation / Address line 2\
Affiliation / Address line 3\
[email@domain]{}\
title: 'Diversity driven attention model for query-based abstractive summarization'
---
|
---
abstract: 'The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s= {\mbox{$\textstyle{\frac{1}{2}}$}}$. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region ${\rm Re}\,s<1$ by allowing [*operator-valued*]{} zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by $\zeta(s)$.'
author:
- 'Carl M. Bender$^1$ and Dorje C. Brody$^{2}$'
title: 'Operator-valued zeta functions and Fourier analysis'
---
[**1**]{}. In the so-called Hilbert-Pólya programme, one attempts to establish the Riemann hypothesis by (a) finding an operator (possibly a differential operator) along with a boundary condition such that the eigenvalues of the operator correspond to the nontrivial zeros of the zeta function, and (b) showing that the operator is self-adjoint. A ${\cal PT}$-symmetric operator fulfilling the requirement (a) has recently been identified [@r1] and its eigenvalues were subsequently shown to be real [@r2]. However, these findings have not yet brought us closer to a proof of the Riemann hypothesis because the span of the eigenfunctions of the operator satisfying the boundary condition may not be fully contained within the rigged Hilbert space upon which the self-adjointness of the operator is established. To be specific, although a Hilbert space has been identified, one is automatically restricting the analysis to the critical line ${\rm Re}\,s={\mbox{$\textstyle{\frac{1}{2}}$}}$ of the zeta function $\zeta(s)$ so that little can be inferred about the zeros off the critical line, if there are any. This may explain why the Hilbert-Pólya programme has not yet led to a proof of the Riemann hypothesis.
With this in mind, we propose here an alternative way to investigate the properties of the Riemann zeta function by analysing [*operator-valued*]{} zeta functions; that is, we examine the behaviour of $\zeta(\hat s)$, where $\hat s$ is an operator, such as the dilation operator. It is unclear whether this approach will lead to a deep understanding of the zeta function. However, it is shown here that one can use this approach to establish some properties of Riemann zeta function very easily. Specifically, we can calculate $\zeta(s)$ for some integer values of $s$ without performing an explicit analytic continuation.
In the Hilbert-Pólya programme one investigates operators whose eigenvalues correspond to the locations of the zeros of the zeta function but we propose here to investigate operators whose eigenvalues are the values of the zeta function itself. For example, in the Hilbert-Pólya programme one might consider the properties of the operator ${\mbox{$\textstyle{\frac{1}{2}}$}}(1-{{\rm i}}{\hat h}_{\rm BK})$, where ${\hat h}_{\rm BK}={\hat x}{\hat p}+{\hat p}{\hat x}$ denotes the Berry-Keating Hamiltonian [@r3], but here we consider the operator $\zeta({\mbox{$\textstyle{\frac{1}{2}}$}}(1-{{\rm i}}{\hat
h}_{\rm BK}))$. We investigate such an operator by letting it act on trigonometric functions. We will show that $$\zeta\left({\mbox{$\textstyle{\frac{1}{2}}$}}(1-{{\rm i}}{\hat h}_{\rm BK})\right)\sin x=\frac{\sin x}{2(1-\cos x)}
\label{eq:1}$$ for $x\in(0,\pi]$, from which we can deduce that $\zeta(-n)=(-1)^nB_{n+1}/(n+1)$ for $n$ a positive-odd integer, where $\{B_n\}$ are the Bernoulli numbers. There are numerous similar relations, from which further properties of the zeta function can be inferred. Another example is $$\zeta\left({\mbox{$\textstyle{\frac{1}{2}}$}}(3-{{\rm i}}{\hat h}_{\rm BK})\right)\sin x=\frac{\pi-x}{2}
\label{eq:2}$$ for $x\in[0,\pi]$, from which we can deduce that $\zeta(s)$ vanishes for negative-even integers $s$ (these are the trivial zeros) without analytically continuing the functional equation $\zeta(s)=2^s\pi^{s-1}\sin(\pi s/2)\Gamma(1-s
)\zeta(1-s)$. From (\[eq:2\]) we can also deduce that $\zeta(0)=-\frac{1}{2}$, and that $\zeta(s)$ has a pole at $s=1$. We remark that operators of the form $\zeta({\mbox{$\textstyle{\frac{1}{2}}$}}\pm{\mbox{$\textstyle{\frac{1}{2}}$}}{{\rm i}}{\hat h}_{\rm BK})$ were mentioned briefly in Ref. [@r3].
[**2**]{}. This paper is based in part on the following relations in Fourier analysis [@r4]: $$\sum_{n=1}^\infty\frac{\cos(nx)}{n^{2m}}=\frac{(-1)^{m-1}(2\pi)^{2m}}{2(2m)!}\,
B_{2m}\left(\frac{x}{2\pi}\right)
\label{eq:3}$$ and $$\sum_{n=1}^\infty\frac{\sin(nx)}{n^{2m-1}}=\frac{(-1)^{m}(2\pi)^{2m-1}}
{2(2m-1)!}\,B_{2m-1}\left(\frac{x}{2\pi}\right)
\label{eq:4}$$ for $m=1,2,\ldots$, where $B_m(x)$ denotes the Bernoulli polynomial of order $m$. Note that these equations are only valid for [*real*]{} $x\in[0,2\pi]$; the Fourier series diverge for complex $x$. If we set $m=1$ in (\[eq:4\]), we obtain $(\pi-x)/2$, which is the right side of (\[eq:2\]). Similar series were investigated by Clausen [@r5].
In this paper we reinterpret these results by using quantum-mechanical operator techniques. We show that it is possible to infer properties of $\zeta(s)$ by studying actions of the operator $\zeta({\hat s})$ on functions when ${\hat s}$ is an operator. (Bernoulli polynomials play a major role in the theory of the Riemann zeta function, so it is not surprising that some properties of the zeta function on the real line can be inferred in the context of Fourier series.)
Series of the form (\[eq:3\]) or (\[eq:4\]) can be extended to cases for which $m$ is $0$ or a negative integer. Such series can be summed by using Euler summation. For example, we have $$\sum_{n=1}^\infty{{\rm e}}^{{\rm i}nx}=\lim_{r\to1^-}\sum_{n=1}^\infty\left(r{{\rm e}}^{{\rm
i}x}\right)^n=\lim_{r\to1^-}\frac{r{{\rm e}}^{{\rm i}x}}{1-r{{\rm e}}^{{\rm i}x}}=
\frac{1}{{{\rm e}}^{-{\rm i}x}-1}.
\label{eq:5}$$ Taking the imaginary part, we deduce that $$\sum_{n=1}^\infty\sin(nx)=\frac{\sin x}{2(1-\cos x)},
\label{eq:6}$$ which is the right side of (\[eq:1\]). This result can also be obtained by complex analysis; from the analytic continuation of the Lerch zeta function $$L(s,x)=\sum_{n=1}^\infty\frac{{{\rm e}}^{{\rm i}nx}}{n^s}=\frac{\Gamma(1-s)}{2\pi{{\rm i}}}
\int_C\frac{{{\rm e}}^t\,t^{s-1}}{1-{{\rm e}}^{t+{\rm i}x}}\,{{\rm d}}t,
\label{eq:7}$$ Apostol deduced sums such as (\[eq:6\]) [@r6]. Here, the integration path $C$ is a Hankel contour that encircles the negative-$t$ axis in the positive direction. Thus, we obtain (\[eq:5\]) by setting $s=0$ in (\[eq:7\]) and using the residue at $t=0$ to evaluate the integral.
[**3**]{}. Let us proceed to establish relations such as (\[eq:2\]) above. For this purpose we require the notion of the dilation operator. The generator of the dilation is ${\hat x}{\hat p}$, where ${\hat p}=-{{\rm i}}\,{{\rm d}}/{{\rm d}}x$, so that for a smooth function $f(x)$ we have $${{\rm e}}^{{\rm i}\lambda{\hat x}{\hat p}}f(x)=f({{\rm e}}^\lambda x).$$ It follows that $$\sin(nx)=n^{{\rm i}{\hat x}{\hat p}}\,\sin x.$$ Therefore, ignoring for now the question of the convergence of the sum, we deduce that $$\sum_{n=1}^\infty\frac{\sin(nx)}{n}=\sum_{n=1}^\infty\frac{n^{{\rm i}{\hat x}{
\hat p}}}{n}\,\sin x=\zeta\left(1-{{\rm i}}{\hat x}{\hat p}\right)\,\sin x.
\label{eq:10}$$ Thus, the action of the Riemann dilation operator $\zeta(1-{{\rm i}}{\hat x}{\hat p})$ on a trigonometric function generates a Fourier series. \[Note that we do not define an operator of the form $\zeta(z+{{\rm i}}{\hat x}{\hat p})$ as a Taylor expansion of $\zeta(s)$ about $s=z$ in powers of ${{\rm i}}{\hat x}{\hat p}$. Such an expansion may diverge.\] In this example, the left side is the Fourier representation for the linear function $(\pi-x)/2$, and from the relation ${\hat h}_{\rm BK}=2{\hat x}{\hat p}-{{\rm i}}$ we observe that $1-{{\rm i}}{\hat x}{\hat p}={\mbox{$\textstyle{\frac{1}{2}}$}}(3-{{\rm i}}{\hat h}_{\rm BK})$. We therefore deduce the identity (\[eq:2\]). Hence if the operator $\zeta\left(1-{{\rm i}}{\hat x}{\hat p}
\right)$ were invertible, we would expect the relation $$\frac{1}{\zeta\left(1-{{\rm i}}{\hat x}{\hat p}\right)}\,\frac{\pi-x}{2}=\sin x$$ to hold. However, since ${\hat x}{\hat p}$ is the dilation generator, it cannot change the power of $x$ on the left side, so we arrive at a contradiction. This suggests that the operator $\zeta\left(1-{{\rm i}}{\hat x}{\hat p}\right)$ [*cannot be inverted*]{} because its spectrum contains at least one zero eigenvalue.
Before we proceed to inspect the locations of the zeros, let us check the consistency of (\[eq:2\]) without relying on the summation representation of the zeta function. For this purpose we use the integral representation $$\zeta(s)=\frac{\Gamma(1-s)}{2\pi{{\rm i}}}\int_C\frac{t^{s-1}}{{{\rm e}}^{-t}-1}\,{{\rm d}}t
\label{eq:12}$$ for the zeta function and $$\frac{1}{\Gamma(1-s)}=\frac{1}{2\pi{{\rm i}}}\int_C{{\rm e}}^{t}t^{s-1}\,{{\rm d}}t$$ for the reciprocal of the Gamma function. Because $s$ appears in two different ways in (\[eq:12\]) our strategy is to check the validity of $$\frac{1}{\Gamma({{\rm i}}{\hat x}{\hat p})}\,\frac{\pi-x}{2}=\frac{\zeta\left(1-{{\rm i}}{
\hat x}{\hat p}\right)}{\Gamma({{\rm i}}{\hat x}{\hat p})}\,\sin x$$ to infer (\[eq:2\]). For the left side we deduce that $$\frac{1}{\Gamma({{\rm i}}{\hat x}{\hat p})}\,\frac{\pi-x}{2}=\frac{1}{2\pi{{\rm i}}}
\int_C{{\rm e}}^{t}t^{-{\rm i}{\hat x}{\hat p}}\left(\frac{\pi-x}{2}\right){{\rm d}}t=\frac
{1}{2\pi{{\rm i}}}\int_C{{\rm e}}^t\left(\frac{\pi-t^{-1}x}{2}\right){{\rm d}}t=-\frac{1}{2}x.$$ The constant term $\pi/2$ has been annihilated here because of the pole of $\zeta(s)$ at $s=1$. On the other hand, expanding $\sin x$ in a power series, we deduce from $$\frac{\zeta\left(1-{{\rm i}}{\hat x}{\hat p}\right)}{\Gamma({{\rm i}}{\hat x}{\hat p})}\,
x^n=\frac{1}{2\pi{{\rm i}}}\int_C\frac{t^{-{\rm i}{\hat x}{\hat p}}}{{{\rm e}}^{-t}-1}x^n\,
{{\rm d}}t=\frac{1}{2\pi{{\rm i}}}\int_C \frac{t^{-n}}{{{\rm e}}^{-t}-1}x^n\,{{\rm d}}t
=\frac{\zeta(1-n)}{\Gamma(n)}\,x^n$$ that $$\frac{\zeta\left(1-{{\rm i}}{\hat x}{\hat p}\right)}{\Gamma({{\rm i}}{\hat x}{\hat p})}\,
\sin x=\sum_{n=1}^\infty\frac{\zeta(2(1-n))}{(2n-1)!\Gamma(2n-1)}\, x^{2n-1}.
\label{eq:17}$$ Since the right side of (\[eq:17\]) must equal $-\frac{1}{2}x$, we infer that $\zeta(0)=-\frac{1}{2}$, and that $\zeta(-2)=\zeta(-4)=\cdots=0$. Conversely, from these elementary facts about the zeta function we infer the consistency of (\[eq:10\]).
An essentially identical line of argument leads to the observation that $$\zeta\left(2-{{\rm i}}{\hat x}{\hat p}\right)\,\cos x=
\frac{\pi^2}{6}-\frac{\pi x}{2}+\frac{x^2}{4},
\label{eq:18}$$ $$\zeta\left(3-{{\rm i}}{\hat x}{\hat p}\right)\,\sin x=
\frac{\pi^2x}{6}-\frac{\pi x^2}{4}+\frac{x^3}{12},
\label{eq:19}$$ and so on. Thus, for each of the Clausen functions in (\[eq:3\]) and (\[eq:4\]) we obtain a corresponding representation in the form of an operator $\zeta\left( N-{{\rm i}}{\hat x}{\hat p}\right)$ acting on a trigonometric function, for $N$ a positive integer. Each of these relations reveals some information about the values of $\zeta(s)$ for real integral values of $s$.
[**4**]{}. As a slightly shorter way to do the analysis above, we observe that since ${{\rm i}}{\hat x}{\hat p}\,x^\alpha=\alpha x^\alpha$, and since $\zeta(s)$ is analytic except for a simple pole at $s=1$, we have $\zeta(N-{{\rm i}}{\hat x}{\hat
p})\,x^n=\zeta(N-n)\,x^n$. However, one must be careful about the existence of the pole. To illustrate this, we consider the example $\zeta(1-{{\rm i}}{\hat x}{\hat
p})\,\sin x$. Expanding the sine series, and assuming the interchangeability of the two limits, we obtain $$\zeta(1-{{\rm i}}{\hat x}{\hat p})\sum_{n=1}^\infty\frac{x^{2n-1}}{(2n-1)!}=\sum_{
n=1}^\infty\zeta(1-{{\rm i}}{\hat x}{\hat p})\frac{x^{2n-1}}{(2n-1)!}=\sum_{n=1
}^\infty\zeta(2-2n)\frac{x^{2n-1}}{(2n-1)!}=-\frac{1}{2}x,$$ which shows that term-by-term application of the differential operator $\zeta(
1-{{\rm i}}{\hat x}{\hat p})$ is not permissible because we have missed the constant term $\pi/2$ associated with the pole of $\zeta(s)$. In fact, for each of the examples discussed above, interchanging the limits leaves out just one term corresponding to the pole of $\zeta(s)$; that is, one term on the left side that is annihilated by $\Gamma({{\rm i}}{\hat x}{\hat p}+1-N)^{-1}$. This term is the only parity-violating term; while each term on the left side of (\[eq:4\]) has odd parity, one term on the right side has even parity. Similarly, while each term on the left side of (\[eq:3\]) has even parity, one term on the right side has odd parity. Thus, in (\[eq:18\]) the term $\pi x/2$ on the right side violates parity, and similarly in (\[eq:19\]) the term $\pi x^2/4$ on the right side violates parity. Hence, the commutator of the two limits gives the parity-breaking term resulting from summing the series.
We remark that the term-by-term application of the operator respects both parity and analyticity. To illustrate this, we consider the series in (\[eq:6\]). Observe that each term in the series on the left side has odd parity and the right side is also odd, so that there is no violation of parity. One might expect that term-by-term application of $\zeta(-{{\rm i}}{\hat x}{\hat p})$ on the power-series expansion of $\sin x$ is permissible. However, while each term in the series on the left side is analytic and vanishes at $x=0$, the right side diverges like $1/x$ as $x\to0$. Indeed, $$\begin{aligned}
\sum_{n=0}^\infty\zeta(-{{\rm i}}{\hat x}{\hat p})\frac{(-1)^n}{(2n+1)!}x^{2n+1} &=&
\sum_{n=0}^\infty\zeta(-2n-1)\frac{(-1)^n}{(2n+1)!}\,x^{2n+1}\nonumber\\
&=& \sum_{n=0}^\infty(-1)^{2n+1}\frac{B_{2n+2}}{2n+2}\frac{(-1)^n}{(2n+1)!}\,
x^{2n+1}\nonumber\\
&=& \frac{1}{x}\sum_{n=0}^\infty\frac{{{\rm i}}^{2n+2}}{(2n+2)!}\,B_{2n+2}\,x^{2n+2}
\nonumber\\
&=&\frac{1}{x}\sum_{k=2}^\infty\frac{1}{k!}\,B_k\,({{\rm i}}x)^{k}\nonumber\\
&=& \frac{1}{x}\left[\sum_{k=0}^\infty\frac{1}{k!}\,B_k\,({{\rm i}}x)^k-1-{{\rm i}}B_1x
\right].
\label{eq:21}\end{aligned}$$ Therefore, from the generating function $\sum_{k=0}^\infty B_k\,x^k/k!=x/({{\rm e}}^x-
1)$ with $B_1=-\frac{1}{2}$ we get $$\sum_{n=0}^\infty\zeta(-{{\rm i}}{\hat x}{\hat p})\frac{(-1)^n}{(2n+1)!}x^{2n+1}=
\frac{\sin x}{2(1-\cos x)}-\frac{1}{x}.$$ Remarkably, we recover the right side of (\[eq:1\]), but with its singularity removed. Moreover, the singular term in the right side of (\[eq:1\]) corresponds to the pole of $\zeta(s)$ at $s=1$. This is the only term that is annihilated by the action of $\Gamma(1+{{\rm i}}{\hat x}{\hat p})^{-1}$.
Analogous results can be seen in other examples, for instance, in $$\zeta\left(-1-{{\rm i}}{\hat x}{\hat p}\right)\,\cos x=\sum_{n=1}^\infty n\cos(nx)=-
\frac{1}{2(1-\cos x)}.$$ Once again, there is no parity violation but the right side is singular at $x=0$ and behaves like $-1/x^2$, while each of the summands in the middle term is well behaved. On the other hand, by interchanging the order of differentiation and summation associated with the Taylor expansion of $\cos x$ we obtain $$\sum_{n=0}^\infty\zeta\left(-1-{{\rm i}}{\hat x}{\hat p}\right)\frac{(-1)^n}{(2n)!}
x^{2n}=-\sum_{n=0}^\infty\frac{(2n+1)({{\rm i}}x)^{2n}}{(2n+2)!}\,B_{2n+2}.$$ Then a calculation like that in (\[eq:21\]) leads to the same conclusion that $$\sum_{n=0}^\infty\zeta\left(-1-{{\rm i}}{\hat x}{\hat p}\right)\frac{(-1)^n}{(2n)!}
x^{2n}=-\frac{1}{2(1-\cos x)}+\frac{1}{x^2},$$ and the singularity at the origin has been removed.
[**5**]{}. The analysis presented here can be extended to more general Dirichlet $L$-functions. These are functions expressible in the form $$L_\chi(s) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ for ${\rm Re}(s)>1$, and otherwise can be defined by their analytic continuations. Here $\chi(n)$ denotes a Dirichlet character, which is a function from integers to complex numbers satisfying the multiplicative property that $\chi(mn)=\chi(m)
\chi(n)$, the periodicity that $\chi(n)=\chi(n+k)$ for some positive $k$, and the condition that if $n$ and $k$ are relative primes then $\chi(n)\neq0$ but otherwise $\chi(n)=0$. Thus, for $k=1$ we have $\chi(n)=1$ for all $n$ and $L_\chi(s)$ reduces to the Riemann zeta function.
As a simple example other than the Riemann zeta function, let us consider the Dirichlet beta function arising from considering the period $k=4$. Specifically, for ${\rm Re}(s)>1$ the Dirichlet beta function is defined by the series $$\beta(s) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^s} ,$$ from which we deduce that $$\beta(-{{\rm i}}{\hat x}{\hat p})\, \sin x = \sum_{n=0}^\infty (-1)^n
(2n+1)^{{\rm i}{\hat x}{\hat p}} \sin x
= \sum_{n=0}^\infty (-1)^n \sin\big( (2n+1)x\big)
=0,$$ where the vanishing of the alternating sine series here can be deduced by using Euler summation. On the other hand, interchanging the order of summation and differentiation in the series expansion of $\sin x$ gives $$\beta(-{{\rm i}}{\hat x}{\hat p})\, \sin x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} \,
\beta\big(-(2n+1)\big) \, x^{2n+1},$$ from which we deduce that $\beta(-n)=0$ for all positive odd $n$, without explicitly relying on analytic continuation. Note that the interchange of the limits is permissible in this example because there is no pole contribution.
An analogous calculation shows that $$\beta(-{{\rm i}}{\hat x}{\hat p})\, \cos x = \sum_{n=0}^\infty (-1)^n
(2n+1)^{{\rm i}{\hat x}{\hat p}} \cos x
= \sum_{n=0}^\infty (-1)^n \cos\big( (2n+1)x\big)
=\frac{1}{2\cos x},$$ whereas by interchanging the limits we find that $$\beta(-{{\rm i}}{\hat x}{\hat p})\, \cos x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} \,
\beta(-2n) \, x^{2n}.$$ Comparing these two we deduce that $\beta(-n)=E_n/2$ for all positive even $n$. This result can also be obtained by considering $$\beta(1-{{\rm i}}{\hat x}{\hat p})\, \sin x = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}
\sin\big( (2n+1)x\big)={\mbox{$\textstyle{\frac{1}{2}}$}}{{\rm i}}\big[\tan^{-1}({{\rm e}}^{-{\rm i}x})-\tan^{-1}({{\rm e}}^{{\rm i}x})
\big] ,$$ and comparing this with $$\beta(1-{{\rm i}}{\hat x}{\hat p})\, \sin x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} \,
\beta(-2n) \, x^{2n+1} .$$
[**6**]{}. In the foregoing analysis we have only considered one class of operator-valued zeta functions, namely, zeta functions evaluated at a linear function of the dilation operator. This class of operators is suitable in the context of Fourier analysis [@r7]. It appears that the action of this class of operators on trigonometric functions only yields information about $\zeta(s)$ for real $s$ although further study is required to clarify this point. In this connection, we note that the matrix elements of, for example, $\zeta(1-{{\rm i}}{\hat x}{\hat p})$, viewed as an operator acting on the Hilbert space of square-integrable functions on $[0,\pi]$, in the standard sine basis $\{\sqrt{2/\pi}\sin(nx)\}$, is given by $$\zeta_{mn}=\left\{\begin{array}{ll} n/m & \quad{\rm if}~n~{\rm divides}~m,\\
0 & \quad{\rm otherwise}.\end{array}\right.$$ Thus, the matrix $\{\zeta_{mn}\}$ encodes the information about factorisation of integers. This suggests that it might be possible to extract more information by studying further properties of the class of operator-valued zeta functions considered here.
Evidently, there are many other operator-valued zeta functions that one might consider. For instance, the action of $\zeta({\hat p}^2+{\hat x}^2)$ on Hermite polynomials might yield further results on the zeta function. As another example, if we let ${\hat a}=({\hat x}+{{\rm i}}{\hat p})/\sqrt{2}$ denote the standard annihilation operator and $|s\rangle$, $s\in{\mathds C}$, a coherent state, we then have $\zeta({\hat a})|s\rangle=\zeta(s)|s\rangle$. Thus, if the action of the operator $\zeta({\hat a})$ were implementable in a laboratory, then one would see the coherent light being absorbed whenever $s$ is a zero of the zeta function.
To conclude, we have shown that by studying the action of Riemann dilation operators on trigonometric functions, we are able to infer some properties of the Riemann zeta function. Of course, the properties of $\zeta(s)$ inferred here are already known. Nevertheless, we were able to determine, for example, the locations of the trivial zeros from elementary Fourier analysis without relying explicitly on the analytic continuation of the zeta function. This suggests that further research into actions of operator-valued zeta functions may yield interesting new results.
[**Acknowledgement**]{}
DCB thanks the Russian Science Foundation for support (project 16-11-10218). The authors thank J. Keating for suggesting the idea of examining other Dirichlet $L$-functions.
[999]{}
Bender, C. M., Brody, D. C. & Müller, M. P. (2017) Hamiltonian for the zeros of the Riemann zeta function. [*Physical Review Letters*]{} [**118**]{}, 130201.
Bender, C. M. & Brody, D. C. (2018) Asymptotic analysis of a pseudo-Hermitian Riemann-zeta Hamiltonian. [*Journal of Physics*]{} A [**51**]{}, 135203.
Berry, M. V. & Keating, J. P. (1999) $H=xp$ and the Riemann zeros. In [*Supersymmetry and Trace Formulae: Chaos and Disorder*]{}. Edited by I. V. Lerner [*et al*]{}. (New York: Kluwer Academic / Plenum).
Abramowitz, M. & Stegun, I. A. (1983) [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*]{}. 23.1.17 and 23.1.18 (New York: Dover). p. 1005.
Clausen, T. (1832) Über die Function $\sin\phi+(1/2^2)\sin 2\phi+(1/3^2)\sin 3\phi+$ etc. [*Journal für die reine und angewandte Mathematik*]{} [**8**]{}, 298-300.
Apostol, T. M. (1951) On the Lerch zeta function. [*Pacific Journal of Mathematics*]{} [**1**]{}, 161-167.
Brody, D. C. (2018) Biorthogonal systems on unit interval and zeta dilation operators. [*Journal of Physics*]{} A [**51**]{}, 285202.
|
---
abstract: 'We study the stability of hexaquark systems containing two heavy quarks and four light quarks within a simple quark model. No bound or metastable state is found. The reason stems on a delicate interplay between chromoelectric and chromomagnetic effects. Our calculation provides also information about anticharmed pentaquarks that are seemingly unbound in simple quark models.'
author:
- 'J. Vijande'
- 'A. Valcarce'
- 'J.-M. Richard'
- 'P. Sorba'
title: 'Search for doubly-heavy dibaryons in a quark model'
---
Introduction {#se:intro}
============
Many interesting hadrons have been discovered recently with hidden heavy flavor, the $XYZ$ mesons and the LHCb pentaquarks. Some reviews are available, the latest ones including the hidden-charm pentaquarks and a few of them discussing also doubly-charm hadrons [@Swanson:2006st; @Chen:2016qju; @Lebed:2016cna; @Ali:2016gli; @Hambrock:2013tpa; @Nielsen:2009uh].
The sector of doubly heavy-flavor will certainly call up a major experimental activity, in particular for a confirmation of the long-awaited doubly-charm baryons [@Broadsky:2012rw] and for the search of doubly-charm mesons [@Hyodo:2012pm; @Che11] and other flavor-exotic states [@Cho11].
In the case of $(\bar q \bar q QQ)$ with spin-parity $J^P=1^+$ and isospin $I=0$, where $Q$ stands for $c$ or $b$ and $q$ for a light quark, there is the fortunate cooperation of two effects. First, the chromoelectric interaction (CE), even if alone, gives stability below the $(\bar q Q)+(\bar q Q)$ threshold if the quark-to-antiquark mass ratio is large enough, as it takes advantage of the deeper binding of the $QQ$ pair [@Ader:1981db; @Zouzou:186qh; @Heller:1986bt; @Carlson:1988hh]. Second, the chromomagnetic interaction (CM) between the light quarks is also favorable [@Semay:1994ht; @Lee:2009rt]. Janc and Rosina predicted the stability of $(\bar u\bar d cc)$ using a quark model fitting ordinary hadrons [@Jan04]. Their calculation was confirmed and improved by Vijande [*et al.*]{} [@Vijande:2007rf]. This configuration is also pointed out as a good candidate for a stable exotic in other approaches such as a $D D{}^*$ molecule [@Manohar:1992nd; @Ericson:1993wy; @Tornqvist:1993ng; @Ohkoda:2011vj; @CarXX], lattice QCD [@Green:1998nt; @Bali:2010xa; @Bicudo:2015kna] or QCD sum rules [@Dias:2011mi].
One hardly finds another configuration with both spin-independent and spin-dependent effects cooperating to privilege a collective multiquark state rather than a splitting into two hadrons. In this paper, we consider a multiquark system with two thresholds that might be nearly degenerate, and study to which extent the mixing of color and spin configurations can stabilize the multiquark. The system is $(qqqqQQ')$, where $Q$ or $Q'$ denotes a heavy quark and $q$ stands for a light quark. The dissociation thresholds are either of the $(qqq)+(qQQ')$ type, which benefits from the $QQ'$ CE interaction, or $(qqQ)+(qqQ')$ which can be shifted down by CM effects if $qq$ is in a spin singlet state. The question is whether $(qqqqQQ')$ can combine CE and CM dynamics coherently to build a bound state.
In writing down the formalism and discussing the results, some other states will be evoked, such as the $H$ particle $(uuddss)$ in the limit of light $Q$ and $Q'$, or the pentaquark $(qqqq\bar Q)$ which is very similar in the limit where $Q$ and $Q'$ are clustered in a compact diquark.
The paper is organized as follows. Section \[se:model\] contains the model and the method to calculate the relevant spin-color states and the matrix elements within this basis. In Sec. \[se:var-cal\] we present the variational calculation and its application in the case of states suspected to be either bound or weakly bound. The results are presented and discussed in Sec. \[se:results\], while Sec. \[se:outlook\] is devoted to some conclusions and perspectives.
The model {#se:model}
=========
We consider $(qqqqQQ')$, where $Q$ and $Q'$ are heavy quarks which are different, hence no Pauli constraints apply, but carry the same mass $M$ for simplicity. Giving $Q$ and $Q'$ different masses would not change our conclusions. We also take the SU(3)$_{\text F}$ limit in the light sector, with the same mass $m$ for $q=u,\,d$ or $s$. For each baryon involved in the threshold and for the dibaryon, we search the ground state of the Hamiltonian $H=T+V$, where $T$ is the kinetic-energy operator and $V$ the interaction. With $m= 0.4\,$GeV, $M=1.3\,$GeV, and a potential $$\begin{gathered}
\label{eq:pot}
V=-\frac{3}{16}\sum_{i<j} \ll{i}{j} \left( - \frac{a}{r_{ij}} + b\,r_{ij}\right.\\
{}+ \left.\frac{c}{m_i\,m_j}\left(\frac{\mu}{\pi}\right)^{3/2}\exp(-\mu\,r_{ij}^2)\,\ss{i}{j}\right)\, ,\end{gathered}$$ where $a=0.4$, $b=0.2$, $c=2.0$, and $\mu=1.0$, in appropriate powers of GeV, one obtains a satisfactory account for the baryon masses entering the thresholds, both in the SU(3)$_\text{F}$ limit and with SU(3)$_\text{F}$ broken. A large negative constant is omitted in the above potential, but it can be disregarded, as it affects equally the thresholds and the multiquark energies. We do not elaborate here on the validity of a model with pairwise forces and color factors, which has been discussed already by several authors (see, e.g., [@Stanley:1980fe; @Badalian:1987gg]). As it is, this is the simplest tool for such exploratory study.
There is already an abundant literature on the wave functions of six-quark systems and the algebra of the spin, color, and spin-color operators entering the quark model [@Hogaasen:1978xs; @Mulders:1980vx; @Park:2016cmg; @Wang:1995kp; @Leandri:1995zm; @Leandri:1997ge; @Pepin:1998ih], with a careful account for the antisymmetrization constraints. We thus restrict ourselves here to a brief summary of our notation. To construct the basis of color and spin states, we formally consider the system as a set of three two-quark subsystems, $(qq)(qq)(QQ')$, with color $\bar 3$ or $6$ and spin 0 or 1. We built the most general basis compatible with an overall color singlet and spin 0 state. The requirements of antisymmetrization are strictly enforced for all states which are shown.
For an overall scalar, one can combine either three spins 0, two spins 1 and one spin 0, or three spins 1, say $$\label{eq:spin-states}
\begin{gathered}
S_1=(000)~,\quad
S_2=(011)~,\quad
S_3=(101)~, \\
S_4=(110)~,\quad
S_5=(111)~.
\end{gathered}$$
The simplest color singlet is $(\bar3\bar3\bar3)$ similar to any antibaryon made of three antiquarks. Another possibility consists of coupling two $\bar3$ diquarks into a color antisextet, and then to get an overall singlet with the third diquark being in a color sextet. The last possibility is to couple three $6$ diquarks into a singlet. In short, $$\label{eq:col-states}
\begin{gathered}
C_1=(666)~,\quad
C_2=(6\bar3\bar3)~,\quad
C_3=(\bar36\bar3)~, \\
C_4=(\bar3\bar36)~,\quad
C_5=(\bar3\bar3\bar3)~.
\end{gathered}$$ For the sake of cross-checking, the color states have been listed explicitly, and rearranged using the SU(3) Clebsch-Gordan coefficients given in [@Alex:2010wi].
The matrix elements of the spin, color and color-spin operators are obvious in this basis for the pairs $(1,2)$, $(3,4)$ or $(5,6)$. For the others, the crossing matrices corresponding to suited transpositions have been used. For instance, the spin crossing matrix corresponding to $$\label{eq:13-overlap}
(12)(23)(34)\leftrightarrow (13)(24)(56)~,$$ is $$\label{eq:spin-X}
\frac{1}{2}\,
\begin{pmatrix}
1 & 0 & 0 & \sqrt{3} & 0 \\
0 & 1 & -1 & 0 & -\sqrt{2} \\
0 & -1 & 1 & 0 & -\sqrt{2} \\
\sqrt{3} & 0 & 0 & -1 & 0 \\
0 & -\sqrt{2} & -\sqrt{2} & 0 & 0 \\
\end{pmatrix}$$ and gives access to $\ss{1}{3}$ and $\ss{2}{4}$. Its color analogue, $$\label{eq:colour-X}
\frac{1}{2}\,
\begin{pmatrix}
-1 & 0 & 0 & \sqrt{3} & 0 \\
0 & -1 & 1 & 0 & -\sqrt{2} \\
0 & 1 & -1 & 0 & -\sqrt{2} \\
\sqrt{3} & 0 & 0 & 1 & 0 \\
0 & -\sqrt{2} & -\sqrt{2} & 0 & 0 \\
\end{pmatrix}$$ allows for the calculation of $\ll{1}{3}$ and $\ll{2}{4}$.
Several checks can be made on the matrix elements $\ss{i}{j}$, $\ll{i}{j}$ and $\llss{i}{j}$. For instance, the Casimir operators such as $\sum \ss{i}{j}$ or $\sum \ll{i}{j}$ depend only on the overall spin or color value. In the case of the $H$, the maximal value $\sum \llss{i}{j}=24$ is recovered, which exceeds the value $16$ corresponding to the $\Lambda\Lambda$ threshold, as first shown by Jaffe [@Jaffe:1976yi]. Similarly, if the sum is restricted to the light sector, the maximal value $\sum \llss{i}{j}=16$ is obtained as in [@Gignoux:1987cn; @Lipkin:1987sk], corresponding to more attraction than the value $8$ of the single baryon entering the lowest threshold.
We note in Eq. the smearing of the spin-spin interaction, instead of a mere delta function when this term is treated at first order. Here the smearing parameter $\mu$ is the same for all pairs, unlike some more elaborate models, where it depends on the masses [@Ono:1982ft]. For consistency, the stability is discussed with respect to the threshold computed within the same model.
Variational calculation {#se:var-cal}
=======================
We solved the 6-body problem using a Gaussian expansion $$\label{eq:Gauss1}
\phi(\tilde x) =\sum_i \gamma_i \exp[-(\tilde x^\dagger. A^{(i)}. \tilde x)/2]~,$$ where $\tilde x=\{{\boldsymbol{x}}_1, \dots {\boldsymbol{x}}_5\}$ is a set of five Jacobi variables describing the relative motion, ${\boldsymbol{x}}_1={\boldsymbol{r}}_2-{\boldsymbol{r}}_1$, etc., and $A^{(i)}$ a $5\times 5$ definite positive matrix. For a given choice of the matrices $A^{(i)}$, the weight factors $\gamma_i$ are given by a generalized eigenvalue problem. The overall variational energy is obtained by a numerical minimization over $A^{(i)}$.
The calculation was started using a single spin-color channel, say $$\label{eq:Gauss2}
\psi_{a,b} =\phi_{a,b} \,|C_a\rangle|S_b\rangle~,$$ with $\phi_{a,b}$ given by Eq. . It was later on extended to account for the coupling among the spin-color states, mainly due to the CM term, i.e., $$\label{eq:Gauss3}
\Psi =\sum_{a,b}\psi_{a,b}~,$$ where the summation is extended to the states sharing the same symmetry properties.
For estimating the energy of a deeply bound state, a straightforward strategy consists of choosing first a few diagonal matrices $A^{(i)}_{a,b}$ containing the range parameters, and then to add some non-diagonal terms and to increase the number of matrices.
If binding does not show up, an alternative strategy consists of choosing Gaussians that describe two baryons times a relative function, for instance, $$\label{eq:Gauss4}
\Psi=\phi_{123}\,\phi_{456} \sum_j \delta_j \,\exp(-\eta_j \,{\boldsymbol{r}}_{123-456}^2/2)~,$$ where $\phi_{ijk}$ is a Gaussian approximation to the baryon containing the $\{i,j,k\}$ quarks and ${\boldsymbol{r}}_{123-456}$ links the centers of mass of the two baryons. Then the partitioning can be extended to other baryon-baryon configurations, say, in an obvious notation, $$\label{eq:Gauss5}
\Psi=\sum_{\substack{a=\{i,j,k\}\\ b=\{i',j',k'\}}}\hskip -9pt\phi_a\,\phi_b \sum_j \delta_j^{(a,b)} \,\exp(-\eta_j^{(a,b)} \,{\boldsymbol{r}}_{a-b}^2/2)~.$$ This is similar to the method of Kamimura [*et al.*]{} [@Hiyama:2003cu], which has been applied successfully to a variety of few-body systems.
Results and discussion {#se:results}
======================
We first show the energies of the baryons constituting the thresholds in Table \[tab1\].
$qqQ (\Sigma)$ $qqQ' (\Lambda)$ $qqq (\Sigma)$ $QQ'q(\Sigma)$
---------------- ------------------ ---------------- ----------------
1.372 1.258 1.461 1.109
: \[tab1\] Energy (in GeV) of the baryons involved in the thresholds within the model . $\Sigma$ stands for a baryon where the first two quarks are in a spin 1 state, and $\Lambda$ in a spin 0 state.
We show in Table \[tab2\] the results for the scalar state $J^P=0^+$ with isospin $I=1/2,3/2$, that would be degenerate because the potential in Eq. (\[eq:pot\]) does not depend on the total isospin. This would stand, for example, for a flavor content $(uudscc)$[^1]. In this case thirteen different color-spin vectors are allowed by antisymmetry requirements, with the notation of Eqs. and they will be: $C_1S_1$, $C_2S_1$, $C_3S_4$, $C_2S_2$, $C_3S_3$, $C_3S_5$, $C_1S_2$, $C_4S_3$, $C_4S_4$, $C_4S_5$, $C_5S_3$, $C_5S_4$, and $C_5S_5$. The two thresholds allowed for the dissociation of the $J^P=0^+$ six-quark state would have energies: $qqQ(\Sigma) + qqQ'(\Lambda)=$ 2.630 GeV and $QQ'q(\Sigma) + qqq(\Sigma)=$ 2.570 GeV.
Color-spin vector E (GeV)
------------------- --------------
$C_1S_1$ 3.079
$C_2S_1$ 2.829
$C_3S_4$ 2.831
$C_2S_2$ 3.030
$C_3S_3$ 3.030
$C_3S_5$ 2.908
$C_1S_2$ 2.995
$C_4S_3$ 2.835
$C_4S_4$ 3.080
$C_4S_5$ 3.016
$C_5S_3$ 2.891
$C_5S_4$ 2.997
$C_5S_5$ 3.034
Coupled 2.767
Thresholds 2.570, 2.630
: \[tab2\]Six-quark energies of the different color-spin vectors contributing to the $J^P=0^+$ state, together with the coupled channel result and the energies of the allowed thresholds.
We also give in Table \[tab3\] the probabilities of the different channels contributing to the coupled channel calculation (those that are not listed have probabilities smaller than $10^{-6}$).
Channel $C_1S_2$ $C_2S_1$ $C_3S_4$ $C_4S_3$
------------- ---------- ---------- ---------- ----------
Probability 0.004 0.539 0.456 0.001
: \[tab3\]Probabilities of the different six-body channels contributing to the $J^P=0^+$ six-quark state.
The calculations using the variational wave function for a single channel, and for the case of coupled channels always give values above the threshold, which go down very slowly when the Gaussian basis is augmented. As already said, this is the sign of either the absence of a bound state, or, at most, of a very tiny binding. This is confirmed by the use of the alternative bases, Eq. or Eq. , where one always finds a 6-body energy equal to the sum of the two baryon energies, obtained in the approximation of the Gaussian expansion $\phi_{ijk}$ and $\phi_{i'j'k'}$. This means that neither the residual color-singlet exchange between the two clusters, nor the coupling of the different baryon-baryon thresholds is sufficient to bind the system.
We have checked that in the infinite mass limit for the mass of the heavy quarks the system gets binding with respect to the upper threshold, $(qqQ)+(qqQ')$, but it is always above the lowest one $(QQ'q)+(qqq)$. For example, for $M=10$ GeV and $m=0.4$ GeV we get 2.326 GeV for the energy of the six-quark state in the coupled channel calculation, while the thresholds come given by $E(QQ'q)+E(qqq)=$ 2.162 GeV and $E(qqQ)+E(qqQ')=$ 2.477 GeV. The six-quark state, that it is now in between the two thresholds, is described by the same color-spin vectors shown in Table \[tab3\], $C_2S_1$ and $C_3S_4$, where the two-heavy quarks are in a $\bar 3$ color state, see Eq. , that would split into the lowest threshold. In other words, the two-heavy quarks control the mass of the six-body state in the infinite mass limit. As mentioned above, by making use of the variational wave function of Eq. or Eq. one obtains exactly the two-baryon mass in the six-body calculation.
We now try to explain why these results are plausible. For the CM part, the subject is already well documented, with the discussions around the $H$ dibaryon or the 1987-vintage pentaquark. See, for instance, [@Oka:1983ku; @Rosner:1985yh; @Karl:1987cg]. The effects of SU(3)$_{\rm F}$ breaking, a different mass for the strange quark, tends to spoil the promises of binding based on the sole spin-color algebra, and, more important, the short-range correlation factors (the expectation values of $\exp(-\mu\,r_{ij}^2)$ in our model) are significantly smaller in a multiquark than in baryons.
As for the CE part, a superficial analysis would argue that, as soon as $-\sum \ll{i}{j}$ is locked to $16$ in any spin-color channel $|C_a\rangle|S_b\rangle$, the CE part of the binding will remain basically untouched, independent of the combination of the $|C_a\rangle|S_b\rangle$ dictated by the CM part. However, this is not the case. For equal masses, the deepest CE binding is obtained when the distribution of CE strength factors $\{-\ll{i}{j}\}$ is the most asymmetric [@Ric11], which favors the threshold against a compact multiquark. For a mass distribution such as $(qqqqQQ')$, CE dynamics favors the $QQ'$ two-quark state being in a color $\bar 3$ state. Once this is enforced, the best CE energy is obtained when the $Qq$ and $Q'q$ pairs receive the largest strength, and they come with a larger reduced mass than $qq$. This can be checked explicitly in a simple solvable model with an interaction proportional to $ -\ll{i}{j}\,r_{ij}^2$. However, CM effects are optimized when the light sector receives the largest color strengths. Hence, there is somewhat a conflict between CE and CM effects, and this explains the lack of bound states in our model.
Summary and outlook {#se:outlook}
===================
In this paper, we have used a simple quark model with CE and CM components to search for possible bound states of $(qqqqQQ')$ configurations below their lowest threshold. The answer is negative: no bound state is found, nor any metastable state in the continuum with a mass below the highest threshold and a suppressed decay to the lowest threshold.
Our calculation provides also some information about the anticharmed pentaquark, or beauty analog $P=(qqqq \bar Q)$. In the limit of large $Q$ and $Q'$ masses, the $QQ'$ pair in $(qqqqQQ')$ behaves as a single antiquark. >From our results, $(qqqq \bar Q)$ is seemingly unbound in simple quark models, while a simple CM counting suggests that this configuration is bound [@Gignoux:1987cn; @Lipkin:1987sk]. While the $H$ has been much studied, in particular within lattice QCD [@Bea10; @Ino10; @Bea11; @Ino12; @Yamazaki:2015nka], the $P$ has received less attention.
It is worth to emphasize how our study illustrates the difficulty to get multiquark bound states in constituent quark models. Other approaches have suggested some ways out, such as:
- Another spin-dependent part for our potential . For instance, a chiral quark model was considered in [@Pepin:1998ih] for the $H$ and for $(uuddsQ)$, but no bound state was found.
- Multibody potentials, generalizing the $Y$-shape potential for baryons, provide more attraction than the color-additive model [@Vijande:2011im], but in the minimization of the flux tube configurations, several color states are mixed, and this is delicate in the case of identical quarks, where color is constrained by the requirement of antisymmetry [@Vijande:2013qr].
- Diquarks, whose clustering is motivated by CM effects but not really demonstrated, might lead to dibaryon states [@Fredriksson:1981mh; @Maiani:2015iaa].
- String dynamics [à]{} la Rossi-Veneziano suggests the existence of states containing more junctions that the lowest thresholds, and thus metastable, as the internal annihilation of junctions is suppressed by an extension of the Zweig rule [@Rossi:2016szw].
- Molecular dynamics, doubly-heavy dibaryons have been also recently approached in a molecular picture [@Lee11; @Meg11; @Liz12; @Oka13; @Hua14; @Car15]. The main motivation of these studies originates from the reduction of the kinetic energy due to large reduced mass as compared to systems made of light baryons. However, such molecular states that have been intriguing objects of investigations and speculations for many years, are usually the concatenation of several effects and not just a fairly attractive interaction. The coupling between close channels or the contribution of non-central forces used to play a key role for their existence. When comparing to similar problems in the strange sector the mass difference between the two competing channels $(qqQ)+(qqQ')$ and $(qqq)+(qQQ')$ increases, making the coupled channel effect less important. Thus, without the strong transition potentials reported in the QDCSM model of Ref. [@Hua14] or the strong tensor couplings occurring in the hadronic one-pion exchange models of Refs. [@Meg11; @Liz12], it seems difficult to get a molecular bound state of two heavy baryons, as has been recently reported in Ref. [@Car15].
On the experimental side, the search for doubly-charm dibaryons can be made together with the search for doubly-charm baryons [@Chn14; @Che14; @Zhe16] and doubly-charm exotic mesons [@Hyodo:2012pm; @Cho11; @Che11] as they share some triggers.
We benefited from fruitful discussions with Emiko Hiyama and Makoto Oka. This work has been partially funded by Ministerio de Educación y Ciencia and EU FEDER under Contracts No. FPA2013-47443 and FPA2015-69714-REDT, by Junta de Castilla y León under Contract No. SA041U16, and by Generalitat Valenciana PrometeoII/2014/066. A.V. is thankful for financial support from the Programa Propio XIII of the University of Salamanca.
[99]{}
E. S. Swanson, Phys. Rept. [**429**]{}, 243 (2006).
H. -X. Chen, W. Chen, X. Liu, and S. -L. Zhu, Phys. Rept. in print, (2016), arXiv:1601.02092.
R. F. Lebed, arXiv:1605.07975.
A. Ali, arXiv:1605.05954.
C. Hambrock, PoS Beauty2013, 044 (2013).
M. Nielsen, F. S. Navarra, and S. H. Lee. Phys. Rept. [**497**]{}, 41 (2010).
S. Brodsky, G. de Teramond, and M. Karliner, Ann. Rev. Nucl. Part. Sci. [**62**]{}, 1 (2012); Ann. Rev. Nucl. Part. Sci. [**62**]{}, 2082 (2011).
T. Hyodo, Y. -R. Liu, M. Oka, K. Sudoh, and S. Yasui, Phys. Lett. B [**721**]{}, 56 (2013).
Y. -Q. Chen and S.- Z. Wu, Phys. Lett. B [**705**]{}, 93 (2011).
S. Cho, T. Furumoto, T. Hyodo, D. Jido, C. M. Ko, S. H. Lee, M. Nielsen, A. Ohnishi, T. Sekihara, S. Yasui, and K. Yazaki (ExHIC Collaboration), Phys. Rev. Lett. [**106**]{}, 212001 (2011).
J.-P. Ader, J.-M. Richard, and P. Taxil, Phys. Rev. D [**25**]{}, 2370 (1982).
S. Zouzou, B. Silvestre-Brac, C. Gignoux, and J.-M. Richard, Z. Phys. C [**30**]{}, 457 (1986).
L. Heller and J. A. Tjon, Phys. Rev. D [**35**]{}, 969 (1987).
J. Carlson, L. Heller, and J. A. Tjon, Phys. Rev. D [**37**]{}, 744 (1988).
C. Semay and B. Silvestre-Brac, Z. Phys. C [**61**]{}, 271 (1994).
S. H. Lee and S. Yasui, Eur. Phys. J. C [**64**]{}, 283 (2009).
D. Janc and M. Rosina, Few Body Syst. [**35**]{}, 175 (2004).
J. Vijande, E. Weissman, A. Valcarce, and N. Barnea, Phys. Rev. D [**76**]{}, 094027 (2007).
A. V. Manohar and M. B. Wise, Nucl. Phys. B [**399**]{}, 17 (1993).
T. E. O. Ericson and G. Karl, Phys. Lett. B [**309**]{}, 426 (1993).
N. A. T[ö]{}rnqvist, Z. Phys. C [**61**]{}, 525 (1994).
S. Ohkoda, Y. Yamaguchi, S. Yasui, K. Sudoh, and A. Hosaka, Phys. Rev. D [**86**]{}, 014004 (2012).
T. F. Caramés, A. Valcarce and J. Vijande, Phys. Lett. B [**699**]{}, 291 (2011).
A. M. Green and P. Pennanen, Phys. Rev. C [**57**]{}, 3384 (1998).
G. Bali and M. Hetzenegger, PoS LATTICE2010, 142 (2010).
P. Bicudo, K. Cichy, A. Peters, and M. Wagner, Phys. Rev. D [**93**]{}, 034501 (2016).
J. M. Dias, S. Narison, F. S. Navarra, M. Nielsen, and J.-M. Richard, Phys. Lett. B [**703**]{}, 274 (2011).
D. P. Stanley and D. Robson, Phys. Rev. Lett. [**45**]{}, 235 (1980).
A. M. Badalian, Phys. Lett. B [**199**]{}, 267 (1987).
H. Hogaasen and P. Sorba, Nucl. Phys. B [**150**]{}, 427 (1979).
P. J. Mulders, A. T. Aerts and J. J. de Swart, Phys. Rev. D [**21**]{}, 2653 (1980).
W. Park, A. Park and S. H. Lee, Phys. Rev. D [**93**]{}, 074007 (2016).
F. Wang, J. -l. Ping and T. Goldman, Phys. Rev. C [**51**]{}, 1648 (1995).
J. Leandri and B. Silvestre-Brac, Phys. Rev. D [**51**]{}, 3628 (1995).
J. Leandri and B. Silvestre-Brac, Few Body Syst. [**23**]{}, 39 (1997).
S. Pepin and Fl. Stancu, Phys. Rev. D [**57**]{}, 4475 (1998).
A. Alex, M. Kalus, A. Huckleberry, and J. von Delft, J. Math. Phys. [**52**]{}, 023507 (2011).
R. L. Jaffe, Phys. Rev. Lett. [**38**]{}, 195 (1977) \[Erratum Phys. Rev. Lett. [**38**]{}, 617(E) (1977)\].
C. Gignoux, B. Silvestre-Brac, and J.-M. Richard, Phys. Lett. B [**193**]{}, 323 (1987).
H. J. Lipkin, Phys. Lett. B [**195**]{}, 484 (1987).
S. Ono and F. Schöberl, Phys. Lett. B [**118**]{}, 419 (1982).
E. Hiyama, Y. Kino, and M. Kamimura, Prog. Part. Nucl. Phys. [**51**]{}, 223 (2003).
M. Oka, K. Shimizu, and K. Yazaki, Phys. Lett. B [**130**]{}, 365 (1983).
J. L. Rosner, Phys. Rev. D [**33**]{}, 2043 (1986).
G. Karl and P. Zenczykowski, Phys. Rev. D [**36**]{}, 2079 (1987).
J.-M. Richard, Few Body Syst. [**50**]{}, 137 (2011).
S. R. Beane E. Chang, W. Detmold, B. Joo, H. W. Lin, T. C. Luu, K. Orginos, A. Parreño, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), Phys. Rev. Lett. [**106**]{}, 162001 (2011).
T. Inoue, N. Ishii, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), Phys. Rev. Lett. [**106**]{}, 162002 (2011).
S. R. Beane E. Chang, W. Detmold, B. Joo, H. W. Lin, T. C. Luu, K. Orginos, A. Parreño, M. J. Savage, A. Torok, and A. Walker-Loud (NPLQCD Collaboration), Mod. Phys. Lett. A [**26**]{}, 2587 (2011).
T. Inoue, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, N. Ishii, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), Nucl. Phys. A [**881**]{},28 (2012).
T. Yamazaki, PoS LATTICE [**2014**]{}, 009 (2015).
J. Vijande, A. Valcarce and J.-M. Richard, Phys. Rev. D [**85**]{}, 014019 (2012).
J. Vijande, A. Valcarce and J.-M. Richard, Phys. Rev. D [**87**]{}, 034040 (2013).
S. Fredriksson and M. Jandel, Phys. Rev. Lett. [**48**]{}, 14 (1982).
L. Maiani, A. D. Polosa, and V. Riquer, Phys. Lett. B [**750**]{}, 37 (2015).
G. Rossi and G. Veneziano, JHEP [**1606**]{}, 041 (2016).
N. Lee, Z. -G. Luo, X. -L. Chen, and S. -L. Zhu, Phys. Rev. D [**84**]{}, 014031 (2011).
W. Meguro, Y. -R. Liu, and M. Oka, Phys. Lett. B [**704**]{}, 547 (2011).
N. Li and S. -L. Zhu, Phys. Rev. D [**86**]{}, 014020 (2012).
M. Oka, Nucl. Phys. A [**914**]{}, 447 (2013).
H. Huang, J. Ping, and F. Wang, Phys. Rev. C [**89**]{}, 035201 (2014).
T. F. Caramés and A. Valcarce, Phys. Rev. D [**92**]{}, 034015 (2015).
G. Chen, X. -G. Wu, J. -W. Zhang, H. -Y. Han, and H. -B. Fu, Phys. Rev. D [**89**]{}, 074020 (2014).
G. Chen, X. -G. Wu, Z. Sun, Y. Ma, and H. -B. Fu, JHEP [**1412**]{}, 018 (2014).
X. -C. Zheng, C. -H. Chang, and Z. Pan, Phys. Rev. D [**93**]{}, 034019 (2016).
[^1]: Other channels and flavor contents have been studied with similar results.
|
---
abstract: 'We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold) and the second order asymptotics (at small deviations from the threshold). We apply our results to find operational interpretations of various Rényi information measures. In case the alternative hypothesis is comprised of bipartite product distributions, we find that the optimal error and strong converse exponents are determined by variations of Rényi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by variations of Rényi conditional mutual information. In either case the relevant notion of Rényi mutual information depends on the precise choice of the alternative hypothesis. As such, our work also strengthens the view that different definitions of Rényi mutual information, conditional entropy and conditional mutual information are adequate depending on the context in which the measures are used.'
author:
- 'Marco Tomamichel$^{\ddag}$, [*Member, IEEE*]{} and Masahito Hayashi$^{\dagger*}$, [*Senior Member, IEEE*]{} [^1] [^2] [^3] [^4]'
bibliography:
- 'library.bib'
title: Operational Interpretation of Rényi Information Measures via Composite Hypothesis Testing Against Product and Markov Distributions
---
composite hypothesis testing, error exponent, strong converse exponent, second order, Rényi divergence, Rényi entropy, mutual information, conditional entropy, conditional mutual information
Introduction
============
Let us first consider simple hypothesis testing. Here the null hypothesis states that a random variable $X^n$ follows the independent and identical (i.i.d.) law $P^{\times n}$ and the alternative hypothesis states that $X^n$ follows the i.i.d. law $Q^{\times n}$, where $P$ and $Q$ are probability mass functions on a discrete set ${\mathcal{X}}$. We write this as follows: [ $$\begin{aligned}
\textnormal{{null hypothesis}:} \quad & \textnormal{$X^n \sim P^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \sim Q^{\times n}$} . \end{aligned}$$ ]{}
Assume now that our test $T^n$ is given as a randomized function from the observed event in ${\mathcal{X}}^n$ to $\{0,1\}$. We are particularly interested in the asymmetric case where two kind of errors are treated differently. The *type-I error*, given as $\alpha_n = P^{\times n}[T^n(X^n) = 0]$, is the probability that the test rejects the null hypothesis even if it is correct. The *type-II error*, given as $\beta_n = Q^{\times n}[T^n(X^n) = 1]$, is the probability that the test confirms the null hypothesis when the alternative hypothesis is correct. (See Section \[sec:binaryhypo\] for formal definitions of these quantities.)
On the one hand, if we impose a constant constraint on the type-I error, namely if we require that $\alpha_n \leq \varepsilon$ for some $\varepsilon \in (0,1)$, then the there exists a sequence of tests such that $\beta_n$ goes zero exponentially fast in $n$. The exponent is known to be the relative entropy, $D(P\|Q)$. This is Stein’s lemma [@chernoff56] (see also [@dembo98; @bucklew90]) and we also call this exponent the *threshold rate* of the problem. (See Section \[sec:opquant\] for definitions of the relevant operational quantities.) Further, Strassen [@strassen62] derived the second order expansion of the optimal exponent as $$\begin{aligned}
- \log \beta_n = n D(P\|Q) - \sqrt{n V(P\|Q)} \Phi^{-1}(\varepsilon) + O(\log n)\,, \label{eq:intro1}\end{aligned}$$ where $\Phi$ is the cumulative standard normal distribution function and $V$ is the variance of the logarithmic likelihood ratio. (See Section \[sec:renyi\] for definitions of the relevant information quantities.)
On the other hand, if we impose an exponential constraint on the type-II error, namely if we require that $\beta_n \leq \exp(-n R)$ for some rate $R \in (0, D(P\|Q))$, we find that the optimal type-I error decreases exponentially fast to zero with $$\begin{aligned}
- \log \alpha_n = n
\sup_{0 < s < 1} \frac{1-s}{s} \big(D_{s}(P\|Q)-R\big) + o(n) \,, \label{eq:intro2}\end{aligned}$$ where $D_{s}(P\|Q)$ is the Rényi relative entropy. This is known as Hoeffding’s bound [@hoeffding65], and the exponent is called *error exponenent* in the following. Moreover, if the rate $R$ exceeds the threshold rate $D(Q\|P)$, the minimum probability of the second error goes to $1$ exponentially fast with $$\begin{aligned}
- \log (1-\alpha_n) = n
\sup_{s > 1} \frac{s-1}{s} \big(R - D_{s}(P\|Q) \big) + o(n) \,. \label{eq:intro3}\end{aligned}$$ This is known as Han-Kobayashi bound [@han89], and the exponent is called *strong converse exponent* in the following. (The form of is due to Ogawa-Nagaoka [@ogawa00]. Moreover, Nakagawa-Kanaya [@nakagawa93] first treat the case of large $R$.)
These results can partially be extended to the case when the null hypothesis is composite as a consequence of Sanov’s theorem [@sanov61]. In contrast, our goal is to extend the above results, in particular Eq. –, to the setting where the alternative hypothesis is composite. More precisely, we want to consider a set ${\mathcal{Q}}$ of distributions on ${\mathcal{X}}$ and the maximal type-II error $\beta_n = \max_{Q \in {\mathcal{Q}}} Q^{\times n}[T^n(X^n) = 1]$. We write the corresponding hypothesis testing problem as follows: [ $$\begin{aligned}
\textnormal{{null hypothesis}:} \quad & \textnormal{$X^n \sim P^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \sim Q^{\times n}$ for some $Q \in {\mathcal{Q}}$} . \end{aligned}$$ ]{}
Sanov’s theorem allows the alternative hypothesis to be the set ${\mathcal{Q}}= \{Q :\, D(P\|Q) > R\}$ for a given real number $R>0$. More precisely, when the first kind of error probability is restricted to $\alpha_n \leq \varepsilon$, the optimal exponent for $\beta_n$ is given as $- \log \beta_n = n R + o(n)$. Moreover, the Hoeffding bound was extended to certain classes of composite hypotheses which are composed of i.i.d. distributions [@unnikrishnan11; @shayevitz11].
### Our Contributions {#our-contributions .unnumbered}
Our first main result establishes that, if the alternative hypothesis satisfies certain axioms discussed in Section \[sec:axioms\], the above results, Eq. -, hold as stated after we substitute $$\begin{aligned}
D(P\|Q) \to \min_{Q \in {\mathcal{Q}}} D(P\|Q) = D(P\|\hat{Q}) , \quad V(P\|Q) \to V(P\|\hat{Q}), \quad D_s(P\|Q) \to \min_{Q \in {\mathcal{Q}}} D_s(P\|Q),\end{aligned}$$ where $\hat{Q} \in {\mathcal{Q}}$ is the distribution that minimizes the relative entropy. Hence, we generalize Stein’s lemma, Strassen’s second order expansion, the Hoeffding bound and the Han-Kobayashi bound to the case of a composite alternative hypothesis. Moreover, we do not need to restrict the alternative hypothesis to i.i.d. distributions but can allow permutation invariant or even more general distributions. We formally state all of our results in Section \[sec:main\].
Our second main result, which is an application of the first, is to give an operational interpretation to various measures of Rényi mutual information, Rényi conditional entropy, and Rényi conditional mutual information. A complete discussion of this can be found in Section \[sec:operational\], and here we exhibit a few representative examples:
1. Let $(X, Y)$ be two random variables governed by a joint probability distribution $P_{XY}$ with marginal $P_X$. We find that the hypothesis testing problem [ $$\begin{aligned}
\textnormal{{null hypothesis}:} \quad & \textnormal{ $(X^n,Y^n) \sim P_{XY}^{\times n}$ } \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{ $X^n \sim P_{X}^{\times n}$ independent of $Y^n$\label{eq:hpoly}} , \end{aligned}$$ ]{} originally proposed by Polyanskiy [@polyanskiy13 Sec. II] in the context of channel coding, leads to an operational interpretation of Sibson’s [@sibson69] definition (see also [@csiszar95 p. 27]) of Rényi mutual information, $$\begin{aligned}
I_s^{{\textnormal{\tiny$\uparrow\!\downarrow$}}}(X\!:\!Y) = \min_{Q_Y} D_s(P_{XY} \| P_X \times Q_Y) \,.
\end{aligned}$$ A similar hypothesis testing problem where the alternative hypothesis further requires that $X^n$ is uniform leads to an operational interpretation of Arimoto’s definition of Rényi conditional entropy [@arimoto75].
2. We further treat the problem of detecting correlations in a collection of random variables. Specifically, consider a null hypothesis that the random random variables $(X_1, X_2, \ldots, X_k)$ are governed by a specific distribution $P_{X_1 X_2 \ldots X_k}$ and compare this to the alternative hypothesis that these random variables are independent, which is a natural formulation from the viewpoint of statistics. This can be phrased as the hypothesis testing problem [ $$\begin{aligned}
\textnormal{{null hypothesis}:} \quad & \textnormal{ $(X_1^n, X_2^n, \ldots, X_k^n) \sim P_{X_1 X_2 \ldots X_k}^{\times n}$ } \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{ $(X_1^n, X_2^n, \ldots, X_k^n) \sim \big( Q_{X_1} \times Q_{X_2} \times \ldots \times Q_{X_k} \big)^{\times n}$ for some $Q_{X_i}$\label{eq:hcomp}} . \end{aligned}$$ ]{}
The quantity $D(P_{X_1 X_2 \ldots X_k} \| P_{X_1} \times P_{X_2} \times \ldots \times P_{X_k})$ is a measure of correlations for $k$-partite systems. We show that it attains operational significance as a threshold rate for the above problem. We also derive error exponents and strong converse exponents for this problem as long as $R$ is close enough to the threshold rate. These are determined by the quantities $$\begin{aligned}
\min_{ Q_{X_1}, Q_{X_2}, \ldots Q_{X_k} } D_s(P_{X_1 X_2 \ldots X_k} \| Q_{X_1} \times Q_{X_2} \times \ldots \times Q_{X_k}) \,.
\end{aligned}$$
Our test depends on the specific distribution $P_{X_1 X_2 \ldots X_k}$, so it is not able to detect arbitrary correlations in these random variables.
3. Let $P_{XYZ}$ be a joint probability distribution. We find that the hypothesis testing problem [ $$\begin{aligned}
\textnormal{{null hypothesis}:} \quad & \textnormal{ $(X^n,Y^n,Z^n) \sim P_{XYZ}^{\times n}$ } \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{ $(X^n,Y^n,Z^n) \sim Q_{XYZ}^{\times n}$ for some $Q_{XZY}$ that is Markov $X \leftrightarrow Y \leftrightarrow Z$ } , \end{aligned}$$ ]{} yields an operational interpretation for the conditional mutual information, $I(X\!:\!Z|Y)$, as the threshold rate. Moreover, the error exponents are determined by a certain Rényi conditional mutual information, $$\begin{aligned}
I_s^{{\textnormal{\tiny$\downarrow\!\downarrow\!\downarrow$}}}(X\!:\!Z|Y) = \min_{Q_{XYZ} } D_s(P_{XYZ} \| Q_Y \times Q_{X|Y} \times Q_{Z|Y}) \,.
\end{aligned}$$ However, this definition of Rényi conditional mutual information is by no means the only definition that attains operational significance. If we vary the problem slightly and only consider alternative hypothesis with a fixed marginal $(X^n, Y^n) \sim P_{XY}^{\times n}$ we recover the same threshold rate but different exponents determined by $$\begin{aligned}
I_s^{{\textnormal{\tiny$\uparrow\!\uparrow\!\downarrow$}}}(X\!:\!Z|Y) &= \min_{Q_{Z|Y}} D_s(P_{XYZ} \| P_{XY} \times Q_{Z|Y}) \, .
\end{aligned}$$ Our result thus yield an operational interpretation for this definition of Rényi conditional mutual information for all positive $s$, and we provide a closed form of this quantity in . To the best of our knowledge this definition has not appeared in literature before. From the operational perspective we have chosen here, it is a natural extension of Sibson’s definition of Rényi mutual information to the conditional setting, and we expect it to have other applications in information theory.
On a technical level, our work introduces the concept of an universal distribution and an universal channel. The purpose of the former is to dominate any i.i.d. product (or permutation invariant) distribution in terms of the relative entropy and the Rényi relative entropy (cf. Lemmas \[lm:universal-limit\] and \[lm:uni-dist\]). More formally, we show that there exists a sequence of distributions $U^n$ on ${\mathcal{X}}^n$ such that $D_s(P^n \| Q^{\times n}) \geq D_s(P^n \| U^n) + O(\log n)$ for any $P^n$ and $Q$. Similarly, the output distribution of the universal channel dominates the output distribution of any memoryless product (or permutation covariant) channel, whenever both channels are given the same input (cf. Lemma \[lm:uni-channel\]).
The universal distribution is the classical analogue of the universal state originally introduced in the quantum setting by one of the authors in [@hayashi09b] and [@hayashi09c]. The latter paper also introduced universal classical-quantum channels.[^5] Clarke and Barron [@clarke90] showed that Bayesian mixture well approximates any independent and identical distribution in terms of relative entropy. (This was extended in a recent paper [@hayashi15] to the approximation in terms of the Rényi relative entropy, even in the continuous case.) Although the universal distribution is similar to the Bayesian mixture of i.i.d. distributions, our universal distribution is easier to analyze in the discrete case.
Another contribution is the axiomatic approach we have taken to the problem. We derive a sufficient condition for the hypotheses testing problems to derive analogues of Strassen’s bound, Hoeffding’s bound and the Han-Kobayashi bound. Since this approach accepts hypotheses containing non-i.i.d. distributions, it has a wide applicability.
### Outline {#outline .unnumbered}
The remainder of the paper is structured as follows. In Section \[sec:framework\] we introduce the axiomatic framework for composite hypothesis testing that we build on, and also define our information quantities, the Rényi divergences. In Section \[sec:main\] we present our main results and treat some generic examples, with the proofs deferred to the later sections. The examples discussed in the introduction, which yield an operational interpretation of various notions of Rényi conditional entropy, mutual information, and conditional mutual information, are then treated in detail in Section \[sec:operational\]. In fact, Section \[sec:operational\] can be understood without referring to Section \[sec:main\]. The proofs for the Hoeffding bound are discussed in Section \[sec:hoeffding\], the Han-Kobayashi bound follows in Section \[sec:sc\] and the second order analysis of Stein’s lemma is found in Section \[sec:second\]. We conclude our work with a discussion and outlook in Section \[sec:conc\].
A Framework for Composite Hypothesis Testing {#sec:framework}
============================================
In this section we introduce a general framework for composite hypothesis that encompasses, but is not restricted to, the examples mentioned in the introduction.
Binary Hypothesis Testing with Composite Alternative Hypothesis {#sec:binaryhypo}
---------------------------------------------------------------
We restrict our attention to discrete (and finite) alphabets. Let ${\mathcal{X}}$ be such an alphabet. The set of probability mass functions (in the following often just called distributions) on ${\mathcal{X}}$ is denoted by ${\mathcal{P}}({\mathcal{X}})$ and comprised of positive valued functions on ${\mathcal{X}}$ with $\sum_{x \in {\mathcal{X}}} P(x) = 1$. For $P \in {\mathcal{P}}({\mathcal{X}})$ and a random variable $X$ on ${\mathcal{X}}$, we write $X \sim P$ to denote that $X$ is distributed according to the law $P$. we use ${\mathcal{X}}^n$ to denote the $n$-fold Cartesian product of ${\mathcal{X}}$ and its elements by vectors $x^n = (x_1, x_2, \ldots, x_n)$. For any $P \in {\mathcal{P}}({\mathcal{X}})$, we use $P^{\times n}$ to denote the identical and independent distribution (i.i.d.) given by $P^{\times n}(x^n) = \prod_{i=1}^n P(x_i)$.
We consider hypothesis testing problems with a composite alternative hypothesis of the following form.
\[def:hypo\] A sequence of *hypothesis testing problems with composite alternative hypothesis* is determined by a triple $\mathbb{H} = \big({\mathcal{X}}, P, \{{\mathcal{Q}}_n\}_{n \in \mathbb{N}} \big)$, comprised of a finite set ${\mathcal{X}}$, a distribution $P \in {\mathcal{P}}({\mathcal{X}})$, and a sequence of sets ${\mathcal{Q}}_n \subseteq {\mathcal{P}}({\mathcal{X}}^n)$ for all $n \in \mathbb{N}$. This determines a hypothesis testing problem for each $n \in \mathbb{N}$. Namely, we observe $n$ instances of a random variable $X$ on ${\mathcal{X}}$ and consider the following two hypotheses. [ $$\begin{aligned}
\textnormal{$\mathbb{H}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$X^n \sim P^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \sim Q^n$ for some $Q^n \in {{\mathcal{Q}}}_n$} . \end{aligned}$$ ]{}
For convenience we employ the shorthand notation ${\mathcal{Q}}\equiv {\mathcal{Q}}_1$ and we use $\overline{{\mathcal{Q}}}_n$ to denote the convex hull of ${\mathcal{Q}}_n$. We will analyze this problem for sequences of sets ${\mathcal{Q}}_n$ that satisfy certain axioms (cf. Section \[sec:axioms\]).
Consider probabilistic hypothesis tests, given by a function $T: {\mathcal{X}}\to [0,1]$, and define the [type-I error]{} probability and [type-II error]{} probability, respectively, as follows: $$\begin{aligned}
\alpha(T;P) :=
\sum_{x \in {\mathcal{X}}} P(x)(1-T(x)),
\qquad \textrm{and} \qquad
\beta(T;{\mathcal{Q}}) :=
\sup_{Q \in {\mathcal{Q}}} \
\sum_{x \in {\mathcal{X}}} Q(x) T(x) \,. \label{eq:def-beta}\end{aligned}$$
In this work we focus on asymmetric hypothesis testing. In this context it is convenient to define the quantity $\hat{\alpha}(\mu; P\| {\mathcal{Q}})$ as the minimum type-I error probability when the type-II error probability is below a threshold $\mu \geq 0$, i.e. we consider the following optimization problem: $$\begin{aligned}
\hat{\alpha}(\mu;P\| {\mathcal{Q}}) :=
\min_{T} \big\{ \alpha(T;P) \,\big|\,
\beta(T;{\mathcal{Q}}) \leq \mu \big\} , \qquad \textrm{for } \mu \in \mathbb{R} \,.\end{aligned}$$ Note that $\mu \mapsto \hat{\alpha}(\mu;P\| {\mathcal{Q}})$ is monotonically decreasing and evaluates to $0$ for $\mu \geq 1$. Note that we always have $$\begin{aligned}
\hat{\alpha}(\mu;P\| {\mathcal{Q}}) = \hat{\alpha}(\mu;P\| \overline{{\mathcal{Q}}}) \label{eq:convex-hull}\end{aligned}$$ since the expression we optimize in is linear in $Q$ and the maximum is thus achieved on the boundary.
Operational Quantities for Asymmetric Hypothesis Testing {#sec:opquant}
--------------------------------------------------------
Let us now discuss the main operational quantities that we want to investigate.
### Threshold Rate
We will study the following properties of asymmetric composite hypothesis tests. The first concept concerns the threshold rate of a sequence of such tests.
Let $\mathbb{H} = \big({\mathcal{X}}, P, \{{\mathcal{Q}}_n\}_{n \in \mathbb{N}} \big)$ be a sequence of hypothesis testing problems. We define the *threshold rate* of the sequence $\mathbb{H}$ as $$\begin{aligned}
{R_{\rm th}}(\mathbb{H}) := \sup \bigg\{ R \in \mathbb{R} : \limsup_{n \to \infty} \hat{\alpha}\big( \exp(-n R); P^{\times n} \| {\mathcal{Q}}_n\big) = 0 \bigg\} \,.
\end{aligned}$$ Similarly, we define the *strong converse threshold rate* of the sequence $\mathbb{H}$ as (cf. [@han02 Def. 4.3.1] and [@nagaoka07]) $$\begin{aligned}
{R_{\rm th}}^*(\mathbb{H}) = \inf \bigg\{ R \in \mathbb{R} : \liminf_{n \to \infty} \hat{\alpha}\big( \exp(-n R); P^{\times n} \| {\mathcal{Q}}_n\big) = 1 \bigg\} \,.
\end{aligned}$$
Clearly ${R_{\rm th}}(\mathbb{H}) \leq {R_{\rm th}}^*(\mathbb{H})$ always holds. Moreover, we note that the threshold rate is always nonnegative because $\hat{\alpha}\big( \exp(-n R); P^{\times n} \| {\mathcal{Q}}_n\big)$ vanishes for $R \leq 0$. The problems we study in this paper are particularly well-behaved and we will find that the threshold rate and the strong converse threshold rate agree.
### Error and Strong Converse Exponents
Moreover, if we choose a rate $R$ below ${R_{\rm th}}(\mathbb{H})$ then we will observe that $\hat{\alpha}\big( \exp(-n R); P^{\times n} \| {\mathcal{Q}}_n\big)$ converges exponentially fast to $0$ as $n$ increases. The exponent characterizing this decrease is called the *error exponent* of the sequence with regards to $R$.
Let $\mathbb{H} = \big({\mathcal{X}}, P, \{{\mathcal{Q}}_n\}_{n \in \mathbb{N}} \big)$ be a sequence of hypothesis testing problems. For every $R > 0$, the *error exponent* of $\mathbb{H}$ with regards to $R$ is defined as $$\begin{aligned}
\mathrm{e}_R(\mathbb{H}) := \liminf_{n \to \infty} - \frac{1}{n} \log \hat{\alpha}\big( \exp(-n R); P^{\times n} \| {\mathcal{Q}}_n\big) \,,\end{aligned}$$ if this limit exists, or $+ \infty$ otherwise.
If we choose a rate $R$ exceeding ${R_{\rm th}}(\mathbb{H})$ then we will instead observe that $\hat{\alpha}\big( \exp(-n R); P^{\times n} \| {\mathcal{Q}}_n\big)$ converges exponentially fast to $1$ as $n$ increases. The exponent characterizing this convergence is called the *strong converse exponent* of the sequence with regards to $R$.
Let $\mathbb{H} = \big({\mathcal{X}}, P, \{{\mathcal{Q}}_n\}_{n \in \mathbb{N}} \big)$ be a sequence of hypothesis testing problems. For every $R > 0$, the *strong converse exponent* of $\mathbb{H}$ with regards to $R$ is defined as $$\begin{aligned}
\mathrm{sc}_R(\mathbb{H}) := \limsup_{n \to \infty} - \frac{1}{n} \log \Big( 1- \hat{\alpha}\big( \exp(-n R); P^{\times n} \| {\mathcal{Q}}_n\big) \Big) \,,\end{aligned}$$ if this limit exists, or $+ \infty$ otherwise.
Information Measures for Asymmetric Hypothesis Testing {#sec:renyi}
------------------------------------------------------
All our results will be stated as an equivalence between one of the above-mentioned operational quantities and an information measures derived from the Rényi divergence. We formally define the Rényi divergence [@renyi61] here.
Let $P \in {\mathcal{P}}({\mathcal{X}})$ and $Q : {\mathcal{X}}\to \mathbb{R}_0^+$. For $s \in (0,1) \cup (1,\infty)$, define $$\begin{aligned}
\label{eq:renyi-g}
g_s(P\|Q) := \lim_{\varepsilon \to 0}\ \sum_{x \in {\mathcal{X}}} P(x)^{s} \big( Q(x) + \varepsilon \big)^{1-s} = \sum_{\substack{x \in {\mathcal{X}}\\ P(x) > 0}} P(x)^{s} Q(x)^{1-s} ,\end{aligned}$$ where the latter expression is finite if $s < 1$ or if $Q(x) = 0 \implies P(x) = 0$ for all $x \in {\mathcal{X}}$, and otherwise we set $g_s(P\|Q) = +\infty$. The Rényi divergence of $P$ with regards to $Q$ of order $s$ is then defined as $$\begin{aligned}
\label{eq:renyi}
D_{s}(P \| Q) &:= \frac{\log g_{s}(P \| Q)}{s-1}\end{aligned}$$ For $s \in \{0,1,\infty\}$ the Rényi divergence is defined as the corresponding limit.
See [@vanerven14] for a comprehensive discussion of its properties. The Rényi divergence is lower semi-continuous and diverges to $+\infty$ if the support condition is violated and $s > 1$ or if $s < 1$ and $P$ and $Q$ do not share any support, in which case $g_s(P\|Q) = 0$. Our sets ${\mathcal{Q}}_n$ may contain elements that violate these conditions. Nonetheless, sets consistent with our axioms will always contain at least one element that satisfies the support conditions.
One of the most important properties of the latter functional is that $g_{s}(\cdot\|\cdot)$ is jointly concave for $s \in (0,1)$ and jointly convex for $s \in (1,\infty)$. Moreover, the function $s \mapsto \log g_{s}(P\|Q)$ is convex and as a consequence $s \mapsto D_{s}(P\|Q)$ is monotonically increasing. In particular, the Kullback-Leibler divergence is given by $$\begin{aligned}
D(P\|Q) := D_1(P\|Q) = \sum_x P(x) \big( \log P(x) - \log Q(x) \big) \,.\label{eq:kullback}\end{aligned}$$ For two positive valued functions $Q$ and $Q'$ on ${\mathcal{X}}$, we observe that $Q(x) \leq Q'(x)$ for all $x \in {\mathcal{X}}$ implies $D_{s}(P\|Q) \geq D_{s}(P\|Q')$. Furthermore, for any scalar $v$, we have $D_{s}(P\|vQ) = D_{s}(P\|Q) + \log v$.
To express our second order results we need to introduce some additional quantities. The *variance of the logarithmic likelihood ratio* is given by $$\begin{aligned}
V(P\|Q) := \sum_{x \in {\mathcal{X}}} P(x) \big( \log P(x) - \log Q(x) - D(P\|Q) \big)^2 \,. \label{eq:optimal-crit}\end{aligned}$$ This variance is proportional to the derivative of the Rényi divergence at $s = 1$, a consequence of the fact that the Rényi divergence is proportional to the cumulant generating function of the logarithmic likelihood ratio. More precisely, the first order Taylor expansion of $D_s(P\|Q)$ around $s = 1$ is given by $$\begin{aligned}
D_s(P\|Q) = D(P\|Q) + \frac{s}{2} V(P\|Q) + O(s^2) \label{eq:taylor} \,.
\end{aligned}$$
Finally, we define the Rényi divergence of $P$ with regards to a set ${\mathcal{Q}}$ of positive valued functions as $$\begin{aligned}
D_{s}(P \| {\mathcal{Q}}) := \inf_{Q \in {\mathcal{Q}}} D_{s}(P \| Q) \,. \label{eq:minimizers}\end{aligned}$$ The minimizer, if it is unique, is defined as $$\begin{aligned}
\hat{Q}^{s} := \operatorname*{\arg\min}_{Q \in {\mathcal{Q}}} D_{s}(P\|Q) \,. \label{eq:mini2}\end{aligned}$$ We define $V(P\|{\mathcal{Q}}) := V(P\|\hat{Q}^1)$. Similarly, taking note of the sign of $(s-1$), we define $$\begin{aligned}
g_{s}(P\|{\mathcal{Q}}) := \exp\big( (s-1) D_{s}(P\|{\mathcal{Q}}) \big) = \begin{cases} \displaystyle \sup_{Q \in {\mathcal{Q}}} g_{s}(P \| Q) & \textrm{if } s \in (0,1) \\
\displaystyle \inf_{Q \in {\mathcal{Q}}} g_{s}(P \| Q) & \textrm{if } s \in (1,\infty)
\end{cases}.\end{aligned}$$
Axioms for Alternative Hypotheses {#sec:axioms}
---------------------------------
Let us fix a probability distribution $P \in {\mathcal{P}}(X)$. We present a collection of axioms that the sets $\{ {\mathcal{Q}}_n \}_{n \in \mathbb{N}}$ must satisfy in order for our main results to hold. The first axiom ensures that the base set, ${\mathcal{Q}}$, is convex.
\[ax:convex\] The set ${\mathcal{Q}}\subseteq {\mathcal{P}}({\mathcal{X}})$ is compact convex. Moreover, for all $s > 0$ the minimizer $\hat{Q}^s$ in is unique and lies in the interior of ${\mathcal{Q}}$.
The second axiom ensures that i.i.d. products of distributions in ${\mathcal{Q}}$ are elements of ${\mathcal{Q}}_n$.
\[ax:prod\] The set ${\mathcal{Q}}_n$ contains the element $Q^{\times n}$ for every $Q \in {\mathcal{Q}}$.
As a direct consequence of Axiom \[ax:prod\] and the additivity of the Rényi divergence for product distributions, we find the following lemma.
\[lm:subadd\] Assuming Axiom \[ax:prod\], we have $D_{s}(P^{\times n} \| {\mathcal{Q}}_n) \leq n D_{s}(P\|{\mathcal{Q}})$ for all $s > 0$ and $n \in \mathbb{N}$.
The purpose of the next axiom is to ensure that this is in fact an equality.
\[ax:add\] For all $s > 0$ and $n \in \mathbb{N}$, we have $D_{s}(P^{\times n} \| {\mathcal{Q}}_n) \geq n D_{s}(P\|{\mathcal{Q}})$.
The next axiom concerns the existence of a sequence of universal distributions. Before we state it, let us introduce some additional notation. We denote the [symmetric group]{} of permutations of $n$ elements by $S_n$. This group has a natural representation as bistochastic matrices. For every $\pi \in S_n$, the matrix $W^n[\pi]$ is defined by the relation $P^n W^n[\pi] (x_1, x_2, \ldots, x_n) = P^n (x_{\pi(1)}, x_{\pi(2)}, \ldots, x_{\pi(n)})$. We say that a probability distribution $P^n \in {\mathcal{P}}({\mathcal{X}}^n)$ is [permutation invariant]{} if $P^n W^n[\pi] = P^n$ holds for all $\pi \in S_n$. The set of all permutation invariant distributions on ${\mathcal{X}}^n$ is denoted ${\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}}^n)$ and permutation invariant distributions in ${\mathcal{Q}}_n$ comprise the subset ${\mathcal{Q}}_n^{{\textnormal{sym}}}$.
\[ax:universal\] There exists a sequence of probability mass functions $\{ U^n \}_{n \in \mathbb{N}}$ with $U^n \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}}^n)$ and a polynomial $v(n)$ such that the following relation holds. For all $n \in \mathbb{N}$ and $Q \in {\mathcal{Q}}_n^{{\textnormal{sym}}}$, $$\begin{aligned}
Q(x^n) \leq v(n)\, U^n(x^n), \quad
\forall \, x^n \in {\mathcal{X}}^n, \qquad \textrm{and} \qquad D_{s}(P^{\times n}\| U^n) \geq D_{s}(P^{\times n}\| {\mathcal{Q}}_n) \,. \label{eq:ax-uni}
\end{aligned}$$ Moreover, the set ${\mathcal{Q}}_n$ is closed under symmetrization, i.e. if $P^n \in {\mathcal{Q}}_n$ then $\frac{1}{n!} \sum_{\pi \in S_n} P^n W^n[\pi] \in {\mathcal{Q}}_n^{{\textnormal{sym}}}$.
The latter condition in is automatically satisfied if $U^n \in {\mathcal{Q}}_n^{{\textnormal{sym}}}$, but this is not necessary. The above axioms have the following immediate consequence.
\[lm:universal-limit\] Assume Axioms \[ax:add\] and \[ax:universal\] hold. Then, for all $s > 0$ $$\begin{aligned}
\lim_{n \to \infty} \frac{1}{n} D_{s}(P^{\times n} \| U^n) = D_{s}(P \| {\mathcal{Q}}) \,, \label{eq:uni1}
\end{aligned}$$ and the map $s \mapsto \phi(s) := \log g_s(P\|{\mathcal{Q}})$ is convex.
We first show . Additivity implies that $\frac{1}{n} D_{s}(P^{\times n}\| U^n) \geq \frac{1}{n} D_{s}(P^{\times n}\| {\mathcal{Q}}_n) = D_{s}(P\|{\mathcal{Q}})$. To establish the other direction, we use Axiom \[ax:universal\]. For any $Q \in {\mathcal{Q}}$, we have $$\begin{aligned}
\frac{1}{n} D_{s}(P^{\times n}\| U^n) \leq
\frac{1}{n} \Big( D_{s}(P^{\times n} \| Q^{\times n} ) + \log v(n) \Big)
= D_{s}(P\|Q) + \frac{1}{n} \log v(n) \label{eq:step1}
\end{aligned}$$ Hence, minimizing over all such $Q$ we can replace $D_{s}(P\|Q)$ with $D_{s}(P\|{\mathcal{Q}})$ on the right hand side. Moreover, using the property that $v(n) = \textrm{poly}(n)$, we find $\limsup_{n\to\infty} \frac{1}{n} D_{s}(P^{\times n}\| U^n) \leq D_{s}(P\|{\mathcal{Q}})$. Finally, since $s \mapsto \log g_s(P\|{\mathcal{Q}})$ is the point-wise limit of convex functions, it is also convex.
Finally, we note that convexity in Axiom \[ax:convex\] is quite a strong requirement and not satisfied by some of the examples we consider. Instead of requiring that the set ${\mathcal{Q}}$ is convex, it suffices to assume that there exists a convenient convex parametrization of the set such that concavity and convexity of $g_{s}(P\|\cdot)$ are preserved.
\[ax:para\] There exists a compact convex set $\Theta$ in a finite-dimensional vector space, a twice continuously differentiable ($C^2$) function $\Theta \ni \theta \mapsto Q_{\theta} \in {\mathcal{Q}}$, and an open interval $(a,b) \ni \{1\}$ such that the following holds:
- We have ${\mathcal{Q}}= \{ Q_{\theta} : \theta \in \Theta \}$.
- The map $\theta \mapsto g_{s}(P\|Q_{\theta})$ is concave for $s \in (a,1)$.
- The map $\theta \mapsto g_{s}(P\|Q_{\theta})$ is convex for $s \in (1,b)$.
- For all $s \in (a,b)$, the minimizer $\hat{\theta}^{s} := \operatorname*{\arg\min}_{\theta \in \Theta} D_{s}(P\|Q_{\theta})$ is unique and lies in the interior of $\Theta$.
Clearly, since the map in Axiom \[ax:para\] is assumed to be $C^2$ and $(s, Q) \mapsto D_{s}(P\|Q)$ is smooth, Axiom \[ax:para\] implies that $(s,\theta) \mapsto D_{s}(P\|Q_{\theta})$ is $C^2$ as well. Further note that Axiom \[ax:convex\] implies Axiom \[ax:para\] using the trivial parametrization $\Theta = {\mathcal{Q}}$ and $(a,b) = (0,\infty)$.[^6] If we assume Axiom \[ax:para\] instead of Axiom \[ax:convex\] we must also relax the additivity property. Namely, additivity in Axiom \[ax:add\] is only required in the interval $s \in (a,b)$ and Lemma \[lm:universal-limit\] only holds for $s \in (a,b)$.
Main Results and Examples {#sec:main}
=========================
Statement of Main Results
-------------------------
Our first result considers the asymptotic situation where the type-II error probability goes to zero exponentially with a rate below $D(P\|{\mathcal{Q}})$. In this case, we find that type-I error probability converges to zero exponentially fast, and the exponent is determined by the Rényi divergence, $D_{s}(P\|{\mathcal{Q}})$ with $s < 1$.
To state our result we need the following concept.
\[def:critical\] Fix $P \in {\mathcal{P}}({\mathcal{X}})$ and ${\mathcal{Q}}\subseteq {\mathcal{P}}({\mathcal{X}})$. For any $c \geq 0$, the $c$-critical rate is defined as $$\begin{aligned}
R_c := \lim_{s \to c} \big( s\phi'(s) - \phi(s) \big)
\end{aligned}$$ with $\phi(s) = (s-1) D_s(P\|{\mathcal{Q}})$ as defined in Lemma \[lm:universal-limit\].
The map $c \mapsto R_c$ on $(a, b)$ is monotonically increasing (cf. Lemma \[lm:hmono\]), and furthermore we find $R_0 = D_0(P\|{\mathcal{Q}})$ and $R_1 = D(P\|{\mathcal{Q}})$, as well as $R_{\infty} \geq D_{\infty}(P\|{\mathcal{Q}})$.
\[th:hoeffding\] Let $\mathbb{H} = \big({\mathcal{X}}, P, \{{\mathcal{Q}}_n\}_{n \in \mathbb{N}} \big)$ be such that Axioms \[ax:prod\]–\[ax:para\] are satisfied on $(a,1]$. Then, for any $R \geq 0$, $$\label{eq:hoeffding-thm}
\mathrm{e}_R(\mathbb{H}) \geq \sup_{s \in (a, 1)} \left\{ \frac{1-s}{s} \big( D_{s}(P\|{\mathcal{Q}}) - R \big) \right\}.$$ Moreover, if $R \geq R_a$ equality holds.
The proof is given in Section \[sec:hoeffding\]. The case where ${\mathcal{Q}}_n = \{ Q^{\times n} \}$ are singletons is attributed to Hoeffding [@hoeffding65]. Note that if $R \geq D(P\|{\mathcal{Q}})$ the right hand side of evaluates to zero, revealing that in this case the error of the first kind will decay slower than exponential in $n$. Otherwise the right hand side is always positive.
Our second result considers the case where type-II error probability goes to zero exponentially with a rate exceeding the mutual information $D(P\|{\mathcal{Q}})$. In this case, we find that type-I error probability converges to 1 exponentially fast, and the exponent is determined by the Rényi divergence $D_s(P\|{\mathcal{Q}})$, with $s > 1$.
\[th:sc\] Let $\mathbb{H} = \big({\mathcal{X}}, P, \{{\mathcal{Q}}_n\}_{n \in \mathbb{N}} \big)$ be such that Axioms \[ax:prod\]–\[ax:para\] are satisfied on $[1, b)$. Then, for any $R \geq 0$, $$\label{eq:sc-thm}
\mathrm{sc}_R(\mathbb{H}) \geq \sup_{s \in (1, b)} \left\{ \frac{s-1}{s} \big( R - D_{s}(P\|{\mathcal{Q}}) \big) \right\}.$$ Moreover, if $R \leq R_b$ equality holds.
The proof is given in Section \[sec:sc\]. The case where ${\mathcal{Q}}_n$ are singletons was shown by Han-Kobayashi [@han89]. Note that even in the singleton case, the original results do not apply for $R > R_{\infty}$. In fact Nakagawa-Kanaya [@nakagawa93] showed that in this setting the above optimal exponent can be attained only by a randomized test. We will not further discuss this setting in this paper.
Again we note that if $R \leq D(P\|{\mathcal{Q}})$ the right hand side of evaluates to zero, otherwise it is strictly positive. The threshold rates are thus determined by the above results, and we find the following corollary of Theorems \[th:hoeffding\] and \[th:sc\].
Let $\mathbb{H} = \big({\mathcal{X}}, P, \{{\mathcal{Q}}_n\}_{n \in \mathbb{N}} \big)$ be such that Axioms \[ax:prod\]–\[ax:para\] are satisfied with any $(a,b) \ni \{1\}$. Then, $$\begin{aligned}
{R_{\rm th}}(\mathbb{H}) = {R_{\rm th}}^*(\mathbb{H}) = D(P\|{\mathcal{Q}}) \,.
\end{aligned}$$
For completeness, we also investigate the second order behavior, namely we investigate the error of the first kind when the error of the second kind vanishes as $\exp(- n D(P \|{\mathcal{Q}}) - \sqrt{n}r)$. This analysis takes a step beyond Stein’s lemma and extends Strassen’s work for simple alternative hypotheses [@strassen62].
\[th:second\] Assume Axioms \[ax:prod\]–\[ax:para\] hold for any $(a,b) \ni \{1\}$. Then, for any $r \in \mathbb{R}$, we have $$\begin{aligned}
\lim_{n \to \infty} \hat{\alpha}\Big( \exp\big(- n D(P \|{\mathcal{Q}}) - \sqrt{n}r\big) ; P^{\times n} \Big\| {\mathcal{Q}}_n \Big) = \Phi \left( \frac{r}{\sqrt{V(P\|{\mathcal{Q}})}} \right) ,
\end{aligned}$$ where $\Phi(x) := (2\pi)^{-\frac12} \int_{-\infty}^{x} e^{-\frac{y^2}{2}} \mathrm{d}y$ and $V(P\|{\mathcal{Q}})$ is defined after .
The proof is given in Section \[sec:second\]. The achievability proof is of a different flavor than previous proofs of the singleton case and relies on Lévy’s continuity theorem.
Examples {#sec:examples}
--------
In the following we will discuss various examples of hypothesis testing problems that can be tackled with the above framework. The cases we treat here in particular cover the examples in Section \[sec:operational\].
### Product distributions with a fixed marginal {#sec:ex1}
Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be two discrete sets. Consider a pair of random variables $(X, Y) \sim P_{XY}$ that are governed by a joint probability distribution $P_{XY} \in {\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}})$. We denote by $P_{X|Y=y}$ the distribution of $X$ conditioned on the event $Y=y$. The conditional distribution $P_{X|Y}$ is interpreted as a stochastic matrix mapping or channel from ${\mathcal{Y}}$ to ${\mathcal{X}}$. In particular, we write $P_{XY} = P_Y \times P_{X|Y}$ and $P_X = P_Y P_{X|Y}$. We assume without loss of generality that $P_X$ and $P_Y$ have full support, i.e. we restrict the sets ${\mathcal{X}}$ and ${\mathcal{Y}}$ to the support of the marginals $P_X$ and $P_Y$, respectively. Moreover, let $T_X \in {\mathcal{P}}({\mathcal{X}})$ be such that $T_X$ has full support as well. Now consider the sets $$\begin{aligned}
{\mathcal{Q}}_n = \big\{ T_X^{\times n} \times Q_{Y^n} :\ Q_{Y^n} \in {\mathcal{P}}({\mathcal{Y}}^n) \big\} \,. \label{eq:set-one}\end{aligned}$$ We emphasize that the distributions $Q_{Y^n}$ are unstructured, in particular they do not have to be $n$-fold i.i.d. products.
\[pr:examples-one\] The sequence of tests $\big( {\mathcal{X}}\times {\mathcal{Y}}, P_{XY}, \{{\mathcal{Q}}_n \}_{n \in \mathbb{N}} \big)$ with ${\mathcal{Q}}_n$ in satisfies Axioms \[ax:convex\]–\[ax:universal\]. Moreover, this still holds if we restrict ${\mathcal{Q}}_n$ to permutation invariant or i.i.d. product distributions.
The proof is given in Appendix \[sec:proof-examples-1\] and relies on the following Lemma, which is of independent interest.
\[lm:uni-dist\] The exists a sequence of distributions $\{ U_{X^n}^n \}_{n \in \mathbb{N}}$ with $U_{X^n}^n \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}}^n)$ such that, for every $n \in \mathbb{N}$ and $Q_{X^n} \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}}^n)$, we have $Q_{X^n}(x^n) \leq |{\mathcal{T}}_n({\mathcal{X}})|\, U_{X^n}^n(x^n)$ for all $x^n \in {\mathcal{X}}^n$.
This is the classical analogue of the universal state originally introduced in the quantum setting [@hayashi09b; @hayashi09c]. The proof uses the method of types [@csiszar98], and the result is only sensible for (finite) discrete sets.
The universal distributions are given by $$\begin{aligned}
U_{X^n}^n(x^n) = \sum_{\lambda \in {\mathcal{T}}_n({\mathcal{X}})} \frac{1}{| {\mathcal{T}}_n({\mathcal{X}}) |} \frac{ 1 }{ |\lambda| } \, 1 \{ x^n \textrm{ is of type } \lambda \} \, ,
\end{aligned}$$ where we use ${\mathcal{T}}_n({\mathcal{X}})$ to denote the set of ${\mathcal{X}}$-types of length $n$ and $\frac{1}{|\lambda|} 1 \{ x^n \textrm{ is of type } \lambda \}$ is the uniform distribution over all sequences of type $\lambda$. Every permutation invariant distribution in $Q_{X^n} \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}})$ has to be flat over sequences of the same type. Namely, it has to be of the form $$\begin{aligned}
Q_{X^n}(x^n) = \sum_{\lambda \in {\mathcal{T}}_n({\mathcal{Y}})} \frac{ q(\lambda) }{ |\lambda| } \, 1 \{ x^n \textrm{ is of type } \lambda \} \label{eq:universal1} \,
\end{aligned}$$ for some distribution $q \in {\mathcal{P}}({\mathcal{T}}_n({\mathcal{Y}}))$. The desired bound can now be verified easily.
### General (permutation invariant) product distributions {#sec:ex2}
Consider discrete sets ${\mathcal{X}}_1$, ${\mathcal{X}}_2$, …, ${\mathcal{X}}_k$ and a distribution $P_{X_1X_2\ldots X_k Y} \in {\mathcal{P}}({\mathcal{X}}_1 \times {\mathcal{X}}_2 \times \ldots \times {\mathcal{X}}_k \times {\mathcal{Y}})$. Without loss of generality we assume that all the marginals of $P_{X_1X_2\ldots X_k Y}$ have full support. Then, consider $$\begin{aligned}
{\mathcal{Q}}_n = \big\{ Q_{X_1^n} \times Q_{X_2^n} \times \ldots \times Q_{X_k^n} \times Q_{Y^n} :\ Q_{X_i^n} \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}}_i^n), i \in [k] \textrm{ and } Q_{Y^n} \in {\mathcal{P}}({\mathcal{Y}}^n) \big\} , \label{eq:set-two} \,.\end{aligned}$$ Note that these sets are not convex, so Axiom \[ax:convex\] is certainly violated. Moreover, note that the restriction that the $Q_{X_i^n}$ be permutation invariant is necessary. Without such a restriction, even a correlated null hypothesis lies in the convex hull of the set of alternative hypothesis since ${\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}})$ equals the convex hull of ${\mathcal{P}}({\mathcal{X}}) \times {\mathcal{P}}({\mathcal{Y}})$. Clearly it is then no longer possible to distinguish these two hypotheses.
\[pr:examples-two\] The sequence of tests $\big( {\mathcal{X}}_1 \times {\mathcal{X}}_2 \times \ldots \times {\mathcal{X}}_k \times {\mathcal{Y}}, P_{X_1X_2\ldots X_k Y}, \{{\mathcal{Q}}_n \}_{n \in \mathbb{N}} \big)$ with ${\mathcal{Q}}_n$ in satisfy Axioms \[ax:prod\]–\[ax:para\] with $(a,b) = \big(\frac{k}{k+1}, \infty\big)$. Moreover, this still holds if we restrict ${\mathcal{Q}}_n$ to i.i.d. product distributions.
The proof is given in Appendix \[sec:proof-examples-2\].
### Recovered and other Markov distributions {#sec:ex3}
Let $P_{XYZ} \in {\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}}\times {\mathcal{Z}})$ be a joint probability distribution with marginals $P_X$ and $P_Y$ and $P_Z$. An natural test considers alternative hypothesis comprised of Markov distributions where only two marginals are fixed. We can see this as the problem of distinguishing a fixed tripartite distribution $P_{XYZ}$ from the set of distributions that can be “recovered” from its marginal $P_{XY}$ via a probabilistic operation.
We assume without loss of generality that $P_X(x)$ and $P_Y(y)$ and $P_Z(z)$ have full support, i.e. we restrict the sets ${\mathcal{X}}$, ${\mathcal{Y}}$, ${\mathcal{Z}}$ to the support of the marginals $P_X$, $P_Y$ and $P_Z$, respectively. The set of conditional probability distributions of $X$ given $Y$, or channels from ${\mathcal{Y}}$ to ${\mathcal{Z}}$, is denoted by ${\mathcal{P}}({\mathcal{Z}}|{\mathcal{Y}})$. Consider the sets $$\begin{aligned}
{\mathcal{Q}}_n = \big\{ P_{XY}^{\times n} \times Q_{Z^n|Y^n} : Q_{Z^n|Y^n} \in {\mathcal{P}}({\mathcal{Z}}^n|{\mathcal{Y}}^n) \big\} \,. \label{eq:set-three}\end{aligned}$$
\[pr:examples-three\] The sequence of tests $\big( {\mathcal{X}}\times {\mathcal{Y}}\times {\mathcal{Z}}, P_{XYZ}, \{{\mathcal{Q}}_n \}_{n \in \mathbb{N}} \big)$ with ${\mathcal{Q}}_n$ in satisfy Axioms \[ax:convex\]–\[ax:universal\]. Moreover, this still holds if we restrict ${\mathcal{Q}}_n$ to permutation invariant or i.i.d. product distributions.
This proposition relies on the following lemma, which is of independent interest.
\[lm:uni-channel\] There exists a sequence of channels, $\{ U^n_{Y^n|X^n} \}_{n \in \mathbb{N}}$, where $U^n_{Y^n|X^n} \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{Y}}^n|{\mathcal{X}}^n)$ such that the following holds. For every $n \in \mathbb{N}$, $Q_{Y^n|X^n} \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{Y}}^n|{\mathcal{X}}^n)$ and $P_{X^n} \in {\mathcal{P}}({\mathcal{X}}^n)$, we have $$\begin{aligned}
P_{X^n} \times Q_{Y^n|X^n} (x^n, y^n) \leq | {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}}) | \, P_{X^n} \times U^n_{Y^n|X^n} (x^n, y^n), \qquad \forall x^n \in {\mathcal{X}}^n, y^n \in {\mathcal{Y}}^n \,.
\end{aligned}$$
Both of the above statements are proven in Appendix \[sec:proof-examples-3\].
Another natural question is to distinguish between a null hypothesis $P_{XYZ}$ and all Markov distributions $X \leftrightarrow Y \leftrightarrow Z$, i.e. distributions $Q_{XYZ} = Q_Y \times Q_{X|Y} \times Q_{Z|Y}$. Consider the set $$\begin{aligned}
{\mathcal{Q}}_n = \big\{ Q_{X^nY^nZ^n} \in {\mathcal{P}}^{{\textnormal{sym}}}\big(({\mathcal{X}}\!\times\!{\mathcal{Y}}\!\times\!{\mathcal{Z}})^n\big) : Q_{X^nY^nZ^n} = Q_{Y^n} \times Q_{X^n|Y^n} \times Q_{Z^n|Y^n} \big\} \,. \label{eq:set-four}\end{aligned}$$
\[pr:examples-four\] The sequence of tests $\big( {\mathcal{X}}\times {\mathcal{Y}}\times {\mathcal{Z}}, P_{XYZ}, \{{\mathcal{Q}}_n \}_{n \in \mathbb{N}} \big)$ with ${\mathcal{Q}}_n$ in satisfy Axioms \[ax:prod\]–\[ax:para\] with $(a,b) = (\frac23,\infty)$. Moreover, this still holds if we restrict ${\mathcal{Q}}_n$ to i.i.d. product distributions.
Again note that every distribution is contained in the convex hull of all Markov distributions, and hence some restrictions on the set are necessary. It is possible to slightly weaken the condition that $(X^n,Y^n,Z^n)$ is permutation invariant, but we will not discuss this here. This is shown in Appendix \[sec:proof-examples-4\].
Operational Interpretation of Rényi Information Measures {#sec:operational}
========================================================
In this section we present the main application of our results, finding operational interpretations of various measures of Rényi mutual information, conditional entropy and conditional mutual information.
Rényi Mutual Information: Testing Against Independent Distributions {#sec:examples-mi}
-------------------------------------------------------------------
It is well known that the mutual information can be expressed in terms of the Kullback-Leibler divergence in several ways. Consider two random variables $X$ and $Y$ and a joint distribution $P_{XY} \in {\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}})$ with marginals $P_X$ and $P_Y$. We are interested in the identities $$\begin{aligned}
I(X \!:\! Y) &= D(P_{XY} \| P_X \times P_Y) \label{eq:mi1} \\
&= \min_{Q_Y \in {\mathcal{P}}(Y)} D(P_{XY} \| P_X \times Q_Y) \label{eq:mi2} \\
&= \min_{Q_X \in {\mathcal{P}}(X),\, Q_Y \in {\mathcal{P}}(Y)} D(P_{XY} \| Q_X \times Q_Y) \label{eq:mi3}\end{aligned}$$
Each of these identities gives rise to a different hypothesis testing problem and a different notion of Rényi mutual information. In the following we treat these three problems in the above order.
### All marginals fixed {#sec:examples-mi-zero}
As a warmup consider the following (simple) hypothesis testing problem. [ $$\begin{aligned}
\textnormal{$\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\uparrow\!\uparrow$}}}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n) \sim P_{XY}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \sim P_X^{\times n}$ and $Y^n \sim P_Y^{\times n}$ are independent} . \end{aligned}$$ ]{} That is, we set ${\mathcal{Q}}_n = \{ P_X^{\times n} \times P_Y^{\times n} \}$ for all $n$ in Defintion \[def:hypo\]. Stein’s Lemma and its strong converse ensure that $$\begin{aligned}
{R_{\rm th}}(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\uparrow\!\uparrow$}}}) = {R_{\rm th}}^*(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\uparrow\!\uparrow$}}})
= D(P_{XY} \| P_X \times P_Y) = I(X\!:\!Y) \,.\end{aligned}$$ Moreover, the Hoeffding [@hoeffding65] and Han-Kobyashi [@han89] bounds give an operational interpretation for the following Rényi mutual information: $$\begin{aligned}
I_s^{{\textnormal{\tiny$\uparrow\!\uparrow$}}}(X\!:\!Y) := D_s(P_{XY} \| P_X \times P_Y)
= \frac{1}{s - 1} \log \left( \sum_{y \in {\mathcal{Y}}} P_Y(y) \sum_{x \in {\mathcal{X}}} P_{X|Y=y}(x)^s P_X(x)^{1-s} \right) \,.\end{aligned}$$ As an example, in wire-tap channel coding, this kind of Rényi mutual information is used to express the exponents of leaked mutual information [@hayashi11b; @han14].
### One marginal fixed {#sec:examples-mi-one}
Here we consider a hypothesis test where the alternative hypothesis is comprised of product distributions where one marginal is fixed. This is the example discussed in Section \[sec:ex1\] with $T_X = P_X$.
This hypothesis test figures prominently when analyzing the converse to various channel coding questions in the fixed error regime, for example for second-order analysis of the discrete memoryless channels [@polyanskiy10; @hayashi09] and beyond [@tomamicheltan12]. In this context, Polyanskiy [@polyanskiy13 Sec. II] raised the following hypothesis testing problem: [ $$\begin{aligned}
\textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n) \sim P_{XY}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \sim P_X^{\times n}$ independent of $Y^n$\label{eq:hchannel}} . \end{aligned}$$ ]{}
This problem has the same threshold, $I(X\!:\!Y)$, but gives an operational interpretation for Sibson’s [@sibson69] definition of Rényi mutual information, which is given by $$\begin{aligned}
I_s^{{\textnormal{\tiny$\uparrow\!\downarrow$}}}(X\!:\!Y) &:= \min_{Q_Y \in {\mathcal{P}}({\mathcal{Y}})} D_s(P_{XY} \| P_X \times Q_Y) \\
&\,= \frac{s}{s-1} \log \left( \sum_{y \in {\mathcal{Y}}} P_Y(y) \Bigg( \sum_{x \in {\mathcal{X}}} P_{X|Y=y}(x)^s P_{X}(x)^{1-s} \Bigg)^{\frac{1}{s}} \right) \\
&\,= \frac{s}{s-1} E_0\left( \frac{s-1}{s} , P_X\! \right) ,\end{aligned}$$ where $E_0$ is Gallager’s error exponent function [@gallager68].[^7] The explicit form of the distribution $Q_{Y}$ that achieves the minimum is given by Sibson’s identity (cf. Appendix \[sec:proof-examples\]).
Our results for the optimal error and strong converse exponents then read $$\begin{aligned}
\mathrm{e}_R(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\uparrow\!\downarrow$}}}) = \sup_{s \in (0,1)} \frac{1-s}{s} \big( I_s^{{\textnormal{\tiny$\uparrow\!\downarrow$}}}(X\!:\!Y) - R \big)
\quad \textrm{and} \quad
\mathrm{sc}_R(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\uparrow\!\downarrow$}}}) = \sup_{s > 1} \frac{s-1}{s} \big( R - I_s^{{\textnormal{\tiny$\uparrow\!\downarrow$}}}(X\!:\!Y) \big) \,,\end{aligned}$$ In the setting of channel coding with constant composition codes of type $P_X$, the exponents $\mathrm{e}_R(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\uparrow\!\downarrow$}}})$ and $ \mathrm{sc}_R(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\uparrow\!\downarrow$}}})$ are equal to the error exponent [@gallager68] and the strong converse exponents [@arimoto73], respectively. In wire-tap channel coding, it is used for expressing the exponents of leaked information when the leaked information is measured in terms of the variational distance [@hayashi13 Thm. 5].
### Arbitrary product distributions, permutation invariant {#sec:examples-mi-two}
Let us now consider the most general problem, [ $$\begin{aligned}
\textnormal{$\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\downarrow\!\downarrow$}}}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n) \sim P_{XY}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n$ and $Y^n$ independent, $(X^n, Y^n)$ permutation invariant} . \end{aligned}$$ ]{} This is covered by the example in Section \[sec:ex2\]. In fact, it is sufficient to require that either $X^n$ or $Y^n$ are permutation invariant (cf. Proposition \[pr:examples-two\]).
We find that for $R$ sufficiently close to the threshold $I(X\!:\!Y)$, we have $$\begin{aligned}
\mathrm{e}_R(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\downarrow\!\downarrow$}}}) = \sup_{s \in (\frac12,1)} \frac{1-s}{s} \big( I_s^{{\textnormal{\tiny$\downarrow\!\downarrow$}}}(X\!:\!Y) - R \big)
\quad \textrm{and} \quad
\mathrm{sc}_R(\mathbb{H}^{\rm mi}_{{\textnormal{\tiny$\downarrow\!\downarrow$}}}) = \sup_{s > 1} \frac{s-1}{s} \big( R - I_s^{{\textnormal{\tiny$\downarrow\!\downarrow$}}}(X\!:\!Y) \big) \,,\end{aligned}$$ with a different definition of Rényi mutual information, $$\begin{aligned}
I_s^{{\textnormal{\tiny$\downarrow\!\downarrow$}}}(X\!:\!Y) := \min_{Q_X \in {\mathcal{P}}({\mathcal{X}}),\, Q_Y \in {\mathcal{P}}({\mathcal{Y}})} D_s(P_{XY} \| Q_X \times Q_Y) \,.\end{aligned}$$ Our result gives an operational interpretation for this definition of Rényi mutual information, which we have not found in the existing literature. However, this operational interpretation only applies for $s \geq \frac12$. In fact, it is unclear if our results can be extended to smaller $s$. Furthermore, we do know of a closed form expression for this quantity.
Rényi Conditional Entropy: Testing Against Uniform and Independent Distribution {#sec:examples-ce}
-------------------------------------------------------------------------------
The conditional entropy can be expressed in terms of the Kullback-Leiber divergence very similarly to the mutual information, with the difference that we require one marginal to be fixed to a uniform distribution. This leads to the following two expressions: $$\begin{aligned}
H(X|Y) &= \log |{\mathcal{X}}| - D( P_{XY} \| R_X \times P_Y) \label{eq:c1} \\
&= \log |{\mathcal{X}}| - \min_{Q_Y \in {\mathcal{P}}({\mathcal{Y}})} D( P_{XY} \| R_X \times Q_Y), \label{eq:c2}\end{aligned}$$ where $R_X$ is the uniform distribution over $X$ and, as in the previous section, $X$ and $Y$ are two random variables governed by a joint distribution $P_{XY} \in {\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}})$ with marginals $P_X$ and $P_Y$. Note that the term $\log |{\mathcal{X}}|$ can easily be incorporated in the relative entropy term if we do not require the second argument to be a normalized probability distribution but instead allow arbitrary positive distributions. Our result extend to this more general setup but we will restrict to normalized distributions as otherwise the corresponding hypothesis testing problems are unnatural.
### Fixed marginal distribution
Consider the following hypothesis testing problem: [ $$\begin{aligned}
\textnormal{$\mathbb{H}^{\rm c}_{{\textnormal{\tiny$\uparrow$}}}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n) \sim P_{XY}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \sim R_X^{\times n}$ and $Y^n \sim P_Y^{\times n}$ are independent} . \end{aligned}$$ ]{} We find that the threshold rate is $D(P_{XY} \| R_X \times P_Y) = \log |{\mathcal{X}}| - H(X|Y)$ and the Hoeffding [@hoeffding65] and Han-Kobyashi [@han89] bounds establish an operational meaning for the Rényi conditional entropy $$\begin{aligned}
H_s^{{\textnormal{\tiny$\downarrow$}}}(X|Y) := \log |{\mathcal{X}}| - D_s(P_{XY} \| R_X \times P_Y)
= \frac{1}{1 - s} \log \left( \sum_{y \in {\mathcal{Y}}} P_Y(y) \sum_{x \in {\mathcal{X}}} P_{X|Y=y}(x)^{s} \right) \,.\end{aligned}$$ This definition of conditional Rényi entropy is quite natural. It is for example used for expressing the leaked modified mutual information in the secure random number generation [@hayashi11b Thm. 2] and [@hayashitan15 Thm. 2].
### Arbitrary marginal distribution {#sec:examples-ce-one}
We consider the following problem, in analogy with Section \[sec:examples-mi-one\]: [ $$\begin{aligned}
\textnormal{$\mathbb{H}^{\rm c}_{{\textnormal{\tiny$\downarrow$}}}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n) \sim P_{XY}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \sim R_X^{\times n}$ independent of $Y^n$\label{eq:hcond}} . \end{aligned}$$ ]{} This is covered by the example in Section \[sec:ex1\] with $T_X$ the uniform distribution. Again we determine the threshold $\log |{\mathcal{X}}| - H(X|Y)$ and the error and strong converse exponents given operational significance to Arimoto’s [@arimoto75] definition of Rényi conditional entropy, $$\begin{aligned}
H_s^{{\textnormal{\tiny$\uparrow$}}}(X|Y) := \log |{\mathcal{X}}| - \min_{Q_Y \in {\mathcal{Q}}({\mathcal{Y}})} D_{s}(P_{XY} \| R_X \times Q_Y)
= \frac{s}{1-s} \log \left( \sum_{y \in {\mathcal{Y}}} P_Y(y) \left( \sum_{x \in {\mathcal{X}}} P_{X|Y=y}^{s} \right)^{\frac{1}{s}} \right) .\end{aligned}$$ The minimum was evaluated using Sibson’s identity (cf. Lemma \[lm:sibson\] in Appendix \[sec:proof-examples\]).
This definition has recently be reviewed in [@fehr14] and compares favorably to other definitions of Rényi conditional entropy that have recently been put forward [@teixeira12; @iwamoto13]. For example, it has an operational interpretation determining the moments of the number of rounds required to guess $X$ from $Y$ [@arikan96], and relatedly as an exponent for task encoding with side information [@bunte14]. More precsiely, we have $$\begin{aligned}
\mathrm{e}_R(\mathbb{H}^{\rm c}_{{\textnormal{\tiny$\uparrow\!\downarrow$}}}) = \sup_{s \in (0,1)} \frac{1-s}{s} \big(\log |{\mathcal{X}}|- H_s^{{\textnormal{\tiny$\uparrow\!\downarrow$}}}(X\!:\!Y) - R \big)
\quad \textrm{and} \quad
\mathrm{sc}_R(\mathbb{H}^{\rm c}_{{\textnormal{\tiny$\uparrow\!\downarrow$}}})
= \sup_{s > 1} \frac{s-1}{s} \big( R -\log |{\mathcal{X}}|+ H_s^{{\textnormal{\tiny$\uparrow\!\downarrow$}}}(X\!:\!Y) \big) ,\end{aligned}$$ As a further example, in secure random number extraction, $ \mathrm{sc}_{\log |{\mathcal{X}}|-R}(\mathbb{H}^{\rm c}_{{\textnormal{\tiny$\uparrow\!\downarrow$}}})$ expresses the error exponent under the universal composability criterion [@hayashi13 Thm. 4] and [@hayashiwatanabe15 Thm. 30].
Rényi Conditional Mutual Information: Testing Against Markov Distributions {#sec:examples-cmi}
--------------------------------------------------------------------------
Let $X, Y$ and $Z$ be three random variables with a joint distribution $P_{XYZ} \in {\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}}\times {\mathcal{Z}})$. The conditional mutual information, $I(X\!:\!Z|Y)$, can be seen as a measure of how close the distribution $P_{XYZ}$ is to a Markov chain $X \leftrightarrow Y \leftrightarrow Z$. For example, we can write $$\begin{aligned}
I(X\!:\!Z|Y) &= D(P_{XYZ} \| P_Y \times P_{X|Y} \times P_{Z|Y}) \label{eq:cmi1} \\
&= \min_{Q_{Z|Y} \in {\mathcal{P}}(Z|Y) } D(P_{XYZ} \| P_Y \times P_{X|Y} \times Q_{Z|Y}) \label{eq:cmi2} \\
&= \min_{\substack{Q_{XYZ} \in {\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}}\times {\mathcal{Z}}) \\ Q_{XYZ} = Q_{X|Y} \times Q_{Z|Y} \times Q_Z}} D(P_{XYZ} \| Q_Y \times Q_{X|Y} \times Q_{Z|Y}) \label{eq:cmi3} \,,\end{aligned}$$ where the latter optimization is over all distributions satisfying the Markov condition $Q_{XYZ} = Q_{X|Y} \times Q_{Z|Y} \times Q_Z$. These are only a few of all the possible expressions for the conditional mutual information, but we will focus our attention on these examples and follow a similar discussion as with the mutual information.
### All marginals fixed {#sec:examples-cmi-1}
Again we first consider a simple alternative hypothesis. [ $$\begin{aligned}
\textnormal{$\mathbb{H}^{\mathrm{cmi}}_{{\textnormal{\tiny$\uparrow\!\uparrow\!\uparrow$}}}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n, Z^n) \sim P_{XYZ}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \leftrightarrow Y^n \leftrightarrow Z^n$ is Markov, $(X^n, Y^n) \sim P_{XY}^{\times n}$, and $(Y^n, Z^n) \sim P_{YZ}^{\times n}$} . \end{aligned}$$ ]{}
This corresponds to the sets ${\mathcal{Q}}_n := \big\{ (P_{Y} \times P_{X|Y} \times P_{Z|Y} )^{\times n} \big\}$. The threshold rate for this problem is the conditional mutual information, $I(X\!:\!Z|Y)$. Furthermore, the Hoeffding [@hoeffding65] and Han-Kobyashi [@han89] bounds yield an operational interpretation of the Rényi conditional mutual information given as $$\begin{aligned}
I_s^{{\textnormal{\tiny$\uparrow\!\uparrow\!\uparrow$}}}(X\!:\!Z|Y) &:= D_s(P_{XYZ} \| P_Y \times P_{X|Y} \times P_{Z|Y}) \\
&\,= \frac{1}{s-1} \log \left( \sum_{y \in {\mathcal{Y}}} P_Y(y) \left( \sum_{z \in {\mathcal{Z}}} P_{Z|Y=y}(z) \left( \sum_{x \in {\mathcal{X}}} P_{X|Y=y,Z=z}(x)^s P_{X|Y=y}(x)^{1-s} \right) \right) \right) \,.\end{aligned}$$
In the special case where $P_{XYZ} = P_{YZ} \times P_{X|Z}$, this kind of Rényi mutual information describes the exponent for leaked mutual information when we employ a superposition code in the wire-tap channel [@watanabe15 Lem. 16] and [@hayashi12b Thm. 20].
### Two marginals fixed, recovery channels {#sec:examples-cmi-rec}
The expression in corresponds to the following problem: [ $$\begin{aligned}
\textnormal{$\mathbb{H}^{\mathrm{cmi}}_{{\textnormal{\tiny$\uparrow\!\uparrow\!\downarrow$}}}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n, Z^n) \sim P_{XYZ}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \leftrightarrow Y^n \leftrightarrow Z^n$ is Markov, $(X^n, Y^n) \sim P_{XY}^{\times n}$\label{eq:hcmi}} . \end{aligned}$$ ]{}
This problem is discussed in Section \[sec:ex3\]. We show that the threshold for this test is again given by $I(X\!:\!Z|Y)$. Moreover, the optimal error and strong converse exponents for $R$ close to the threshold are given by $$\begin{aligned}
\mathrm{e}_R(\mathbb{H}^{\rm cmi}_{{\textnormal{\tiny$\uparrow\!\uparrow\!\downarrow$}}}) = \sup_{s \in (0,1)} \frac{1-s}{s} \big( I_s^{{\textnormal{\tiny$\uparrow\!\uparrow\!\downarrow$}}}(X\!:\!Z|Y) - R \big)
\quad \textrm{and} \quad
\mathrm{sc}_R(\mathbb{H}^{\rm cmi}_{{\textnormal{\tiny$\uparrow\!\uparrow\!\downarrow$}}}) = \sup_{s > 1} \frac{s-1}{s} \big( R - I_s^{{\textnormal{\tiny$\uparrow\!\uparrow\!\downarrow$}}}(X\!:\!Z|Y) \big) \,,\end{aligned}$$ where we have introduced a new definition of the Rényi conditional mutual information. This is given by $$\begin{aligned}
I_s^{{\textnormal{\tiny$\uparrow\!\uparrow\!\downarrow$}}}(X\!:\!Z|Y) &:= \min_{Q_{Z|Y} \in {\mathcal{P}}({\mathcal{Z}}|{\mathcal{Y}})} D_s(P_{XYZ} \| P_Y \times P_{X|Y} \times Q_{Z|Y}) \\
&\,= \frac{1}{s-1} \log \left( \sum_{y \in {\mathcal{Y}}} P_Y(y) \left( \sum_{z \in {\mathcal{Z}}} P_{Z|Y=y}(z) \left( \sum_{x \in {\mathcal{X}}} P_{X|Y=y,Z=z}(x)^s P_{X|Y=y}(x)^{1-s} \right)^{\!\frac{1}{s}} \right)^{\!\!s\,} \right) . \label{eq:cmi-closed-form}\end{aligned}$$ The minimum was evaluated using Sibson’s identity in Lemma \[lm:sibson\] for the distribution $Q_{Z|Y=y}$ seperately for each $y \in {\mathcal{Y}}$. (See Appendix \[sec:proof-examples-3\] for details.) The resulting expression can be regarded as a conditional version of the Gallager function by replacing $s$ with $\frac{\rho}{\rho-1}$. In the special case where $P_{XYZ} = P_{YZ} \times P_{X|Z}$, this quantity is used in superposition coding to describe the error exponent [@kaspi11 Sec. II] and the exponent of leaked information [@hayashi12b Thm. 22].
### Arbitrary Markov distribution, permutation invariant {#sec:examples-cmi-full}
The most general alternative hypothesis that we consider is comprised of all distributions that have a Markov structure $X \leftrightarrow Y \leftrightarrow Z$. More precisely, the following problem: [ $$\begin{aligned}
\textnormal{$\mathbb{H}^{\mathrm{cmi}}_{{\textnormal{\tiny$\downarrow\!\downarrow\!\downarrow$}}}$} : \quad \qquad \textnormal{{null hypothesis}:} \quad & \textnormal{$(X^n, Y^n, Z^n) \sim P_{XYZ}^{\times n}$} \,, \nonumber \\
\textnormal{{alternative hypothesis}:} \quad & \textnormal{$X^n \leftrightarrow Y^n \leftrightarrow Z^n$ is Markov, $(X^n,Y^n,Z^n)$ permutation invariant} . \end{aligned}$$ ]{} This is covered in Section \[sec:ex3\]. The threshold is again $I(X\!:\!Z|Y)$ and the optimal error and strong converse exponents for $R$ close to the threshold are given by $$\begin{aligned}
\mathrm{e}_R(\mathbb{H}^{\rm cmi}_{{\textnormal{\tiny$\downarrow\!\downarrow\!\downarrow$}}}) = \sup_{s \in (\frac23,1)} \frac{1-s}{s} \big( I_s^{{\textnormal{\tiny$\downarrow\!\downarrow\!\downarrow$}}}(X\!:\!Z|Y) - R \big)
\quad \textrm{and} \quad
\mathrm{sc}_R(\mathbb{H}^{\rm cmi}_{{\textnormal{\tiny$\downarrow\!\downarrow\!\downarrow$}}}) = \sup_{s > 1} \frac{s-1}{s} \big( R - I_s^{{\textnormal{\tiny$\downarrow\!\downarrow\!\downarrow$}}}(X\!:\!Z|Y) \big) \,,\end{aligned}$$ with yet another Rényi conditional mutual information, $$\begin{aligned}
I_s^{{\textnormal{\tiny$\downarrow\!\downarrow\!\downarrow$}}}(X\!:\!Z|Y) &:= \min_{Q_{XYZ} \in {\mathcal{P}}({\mathcal{X}}\times{\mathcal{Y}}\times{\mathcal{Z}}) } D_s(P_{XYZ} \| Q_Y \times Q_{X|Y} \times Q_{Z|Y}) \,.\end{aligned}$$ Note that we only have an operational interpretation of this quantity for $s > \frac23$, and, moreover, we do not know of a closed form expression.
Proofs: Hoeffding Bound {#sec:hoeffding}
=======================
The proof of Theorem \[th:hoeffding\] is split into two parts, achievability and optimality, which both rely on different Axioms.
Some Properties of $\phi(s)$ {#sec:convex1}
----------------------------
Before we state our result, let us introduce some helpful notation. Recall that $\phi(s) := (s-1) D_{s}(P\|{\mathcal{Q}}) = \log g_s(P\|{\mathcal{Q}})$ and define $$\begin{aligned}
\bar{\phi}_{s_0}(s) := (s-1) D_{s}(P\|\hat{Q}^{s_0}) = \log g_s(P\|\hat{Q}^{s_0}) \,,\end{aligned}$$ where $\hat{Q}^{s_0} := Q_{\hat{\theta}^{s_0}}$. Clearly, $\phi(s_0) = \bar{\phi}_{s_0}(s_0)$ by definition of $\hat{\theta}^s$ in Axiom \[ax:para\]. An important consequence of this Axiom is the following lemma, which shows that the first derivative of $\phi$ and $\bar{\phi}$ agree as well at $s = s_0$.
\[lm:derivative\] Assume Axiom \[ax:para\] holds on $(a,b)$. Then, for $s_0 \in (a,b)$, we have $$\begin{aligned}
\frac{{\mathrm{d}}}{{\mathrm{d}}s} D_{s}(P\|{\mathcal{Q}}) \Big|_{s = s_0} = \frac{{\mathrm{d}}}{{\mathrm{d}}s} D_{s}(P\|Q_{\hat{\theta}^{s_0}}) \Big|_{s=s_0} \,.
\end{aligned}$$ Moreover, the function $\phi(s)$ on $(a,b)$ is continuously differentiable and satisfies $\phi'(s_0) = \bar{\phi}_{s_0}'(s_0)$.
Let us define $f(s,\theta) = g_{s}(P\|Q_{\theta})$. Write $\theta = (\theta_1, \ldots, \theta_d)$ as a $d$-dimensional real vector. The point $\hat{\theta}^s$ is determined by the implicit functions $$\begin{aligned}
F_i(s,\theta) := \frac{\partial}{\partial \theta_i} f(s,\theta) = 0 , \qquad \forall i \in \{1, 2, \ldots, d\} \,.
\end{aligned}$$ Moreover, the Hesse matrix $H_s$ of $\theta \mapsto f(s,\theta)$ at $\hat{\theta}^s$ is given by $$\begin{aligned}
(H_s)_{i,j} = \frac{\partial^2}{\partial \theta_i \partial \theta_j} f(s,\theta) \big|_{\theta = \hat{\theta}^{s}} \,.
\end{aligned}$$ The map $s \mapsto H_s$ is continuous since $(s,\theta) \mapsto f(s,\theta)$ is $C^2$ by Axiom \[ax:para\].
Let us first treat the case $s_0 > 1$. From the fact that $f$ is convex and the minimizer $\hat{\theta}^{s_0}$ is uniquely achieved in the interior of $\Theta$, we conclude that $H$ is positive definite and invertible. Then, the implicit function theorem yields $$\begin{aligned}
\frac{ \partial \hat{\theta}_i^{s} }{\partial s} \bigg|_{s=s_0} = - \sum_{j=1}^d \big( H_{s_0}^{-1} \big)_{i,j} \cdot\frac{ \partial F_j(s,\hat{\theta}^{s_0}) }{\partial s} \bigg|_{s = s_0 } \label{eq:thetas}
\end{aligned}$$ and in particular $s \mapsto \hat{\theta}^s$ is continuously differentiable. We further find that $$\begin{aligned}
\frac{{\mathrm{d}}}{{\mathrm{d}}s} f(s, \hat{\theta}^s) \Big|_{s = s_0}
= \frac{\partial}{\partial s} f(s, \hat{\theta}^{s_0}) \Big|_{s = s_0} + \sum_{i=1}^d F^i(s_0,\hat{\theta}^{s_0}) \cdot \frac{ \partial \hat{\theta}_i^{s} }{\partial s} \bigg|_{s=s_0}
= \frac{\partial}{\partial s} f(s, \hat{\theta}^{s_0}) \Big|_{s = s_0} \,,
\end{aligned}$$ where we used that $F^i(s_0,\hat{\theta}^{s_0}) = 0$ by definition of $\hat{\theta}^{s_0}$. This establishes the result for $s_0 \in (1,b)$. An analogous argument, using concavity instead of convexity, yields the same result for $s_0 \in (a,1)$. For $s_0 = 1$ we instead choose $f(s,\theta) = D_s(P\|Q_{\theta})$. Again the Hesse matrix for the derivative with regards to $\theta$ is strictly positive, and the remainder of the argument proceeds as before.
Finally, since $s \mapsto D_s(P\|Q)$ is smooth for fixed $P, Q \in {\mathcal{P}}(X)$ and $\theta \mapsto Q_{\theta}$ as well as $s \mapsto \hat{\theta}^s$ are continuous, we deduce that the functions $s \mapsto D_s(P\|{\mathcal{Q}})$ and $\phi(s) = (s-1) D_{s}(P\|{\mathcal{Q}})$ are continuously differentiable.
Some Properties of Convex $C^1$ Functions {#sec:convex2}
-----------------------------------------
In this section let ${\tilde{\phi}}(s)$ be a general convex $C^1$ function on $(a, b) \subseteq \mathbb{R}_0^+$ and define ${\tilde{\psi}}(s) := s {\tilde{\phi}}'(s) - {\tilde{\phi}}(s)$. Moreover, define $\tilde{R}_{c} := \lim_{s \to c} {\tilde{\psi}}(s)$ for every $c \in [a, b]$.
\[lm:hmono\] The function ${\tilde{\psi}}(s)$ is continuous and monotonically increasing on $(a,b)$.
Let $a < s_0 < s_1 < b$. By the mean value theorem there exists an $s \in [s_0, s_1]$ such that $(s_1 - s_0) {\tilde{\phi}}'(s) = {\tilde{\phi}}(s_1) - {\tilde{\phi}}(s_0)$. Thus, $$\begin{aligned}
{\tilde{\psi}}(s_1) - {\tilde{\psi}}(s_0) &= s_1 {\tilde{\phi}}'(s_1) - s_0 {\tilde{\phi}}'(s_0) - \big ( {\tilde{\phi}}(s_1) - {\tilde{\phi}}(s_0) \big) \\
&= s_1 {\tilde{\phi}}'(s_1) - s_0 {\tilde{\phi}}'(s_0) - (s_1 - s_0) {\tilde{\phi}}'(s) \\
&= s_1 \big( {\tilde{\phi}}'(s_1) - {\tilde{\phi}}'(s) \big) + s_0 \big( {\tilde{\phi}}'(s) - {\tilde{\phi}}'(s_0) \big) \geq 0 \,.
\end{aligned}$$ The inequality follows from the assumption that ${\tilde{\phi}}(s)$ is convex, and ${\tilde{\phi}}'(s)$ thus monotonically increasing.
\[lm:supmax\] Let $R \in (\tilde{R}_a, \tilde{R}_b)$. Then, there exists an $\hat{s} \in (a, b)$ such that ${\tilde{\psi}}(\hat{s}) = R$ and $$\begin{aligned}
\sup_{s \in (a,b)} \frac{(s-1)R - {\tilde{\phi}}(s)}{s} = \frac{(\hat{s} - 1)R - {\tilde{\phi}}(\hat{s})}{\hat{s}}
\end{aligned}$$
By continuity, for every $R \in (\tilde{R}_a, \tilde{R}_b)$, there exists (at least one) value $\hat{s} \in (a, b)$ such that ${\tilde{\psi}}(\hat{s}) = R$. Let us first calculate the derivative of $g(s) := \frac{(s-1) R - {\tilde{\phi}}(s)}{s}$. This yields $$\begin{aligned}
g'(s)\, =
\frac{s R - s {\tilde{\phi}}'(s) - (s-1) R + {\tilde{\phi}}(s) }{s^2}
= \frac{ R - {\tilde{\psi}}(s) }{s^2} \,.
\end{aligned}$$ Note that the numerator is monotonically decreasing in $s$ due to Lemma \[lm:hmono\] and vanishes at $s = \hat{s}$. In particular, we find that $g'(s) \geq 0$ for $a < s < \hat{s}$ and $g'(s) \leq 0$ for $\hat{s} < s < b$. We conclude that $\hat{s}$ maximizes $g(s)$ on $(a, b)$.
Proof of Achievability
----------------------
Achievability for Theorem \[th:hoeffding\] follows from the following statement.
\[pr:hoeffding-achieve\] Assume Axioms \[ax:add\] and \[ax:universal\] hold with parameter $a$. Then, we have $$\label{eq:hoeffding-achieve}
\liminf_{n \to \infty} - \frac{1}{n} \log
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\| \,\overline{{\mathcal{Q}}}_n \Big)
\geq \sup_{s \in (a, 1)} \left\{ \frac{1-s}{s} \big( D_{s}(P\|{\mathcal{Q}}) - R \big) \right\}.$$
Note that the expression on the right hand side of is zero if $R \geq D(P\|{\mathcal{Q}})$ and the inequality thus holds trivially for that case. We assume that $R < D(P\|{\mathcal{Q}})$ for the remainder of this proof.
Let us fix any $s \in (a,1)$ for the moment. Moreover, let us define the sequence of tests $$\begin{aligned}
T_n(x^n) :=
\begin{cases}
1 & \textrm{ if } P^{\times n}(x^n) \geq \exp(\lambda_n) U^n(x^n) \\
0 & \textrm{ otherwise}
\end{cases}
, \label{eq:thetest}\end{aligned}$$ where $U^n$ is the universal distribution of Axiom \[ax:universal\]. We also choose the sequence $\{\lambda_{n} \}_{n \in \mathbb{N}}$ of real numbers as $$\begin{aligned}
\lambda_n = \frac{1}{s} \Big( \log v(n) + n R + (s-1) D_{s} (P^{\times n} \| U^n) \Big) .
\label{eq:sc-mu22}\end{aligned}$$
Axiom \[ax:universal\] implies that ${\mathcal{Q}}_n$ is closed under symmetrization. Moreover, the test $T_n$ is permutation invariant. Hence, for all $\pi \in S_n$, we have $Q^n[T_n] = Q^n[T_n W_{\pi}^n] = Q^n W_{\pi^{-1}}^n[T^n]$ and we can in particular replace $Q^n$ with its symmetrization. This yields $$\begin{aligned}
\beta(T_n; {\mathcal{Q}}_n)
&= \max_{Q^n \in {\mathcal{Q}}_n} Q^n \left[ P^{\times n}(X^n)
\geq \exp(\lambda_n) U^n(X^n) \right] \label{eq:optimize-linear} \\
&= \max_{Q^n \in {\mathcal{Q}}_n^{{\textnormal{sym}}}} Q^n \left[ P^{\times n}(X^n)
\geq \exp(\lambda_n) U^n(X^n) \right] \,. \label{eq:optimize-sym}\end{aligned}$$ Next we use the universal distribution in Axiom \[ax:universal\] to further bound $$\begin{aligned}
\beta(T_n; {\mathcal{Q}}_n)
&= v(n) \sum_{x^n \in {\mathcal{X}}^n} \!\! U^n(x^n) \, 1 \left\{ P^{\times n}(x^n) \geq \exp(\lambda_n) U^n(x^{n}) \right\} \\
&\leq v(n) \exp(-s \lambda_n) \sum_{x^n \in {\mathcal{X}}^n} \!\! \big(U^n(x) \big)^{1-s} \big( P^{\times n}(x) \big)^{s} \, 1 \left\{ P^{\times n}(x^n) \geq \exp(\lambda_n) U^n(x^{n}) \right\} \\
&\leq v(n) \exp(-s \lambda_n) \sum_{x^n \in {\mathcal{X}}^n} \!\! \big(U^n(x) \big)^{1-s} \big( P^{\times n}(x) \big)^{s} \\
&= v(n) \exp\left(-s \lambda_n\right) \exp\left( (s-1) D_{s}( P^{\times n} \| U^n ) \right) . \end{aligned}$$ Hence, the requirement that $\beta(T_n; {\mathcal{Q}}_n) \leq \exp(-n R)$ is satisfied by the choice of $\lambda_n$ in . Note that this statement can directly be extended to the convex hull due to .
Let us now take a closer look at the error of the first kind. Using a similar development as above, we find $$\begin{aligned}
\hat{\alpha}\big(\exp(-n R); P^{\times n} \big\|\overline{{\mathcal{Q}}}_n \big)
\leq \alpha(T_n; P^{\times n})
&= P^{\times n} \left[ P^{\times n}(X^n) < \exp(\lambda_n) U^n(X^{n}) \right] \\
&\leq \exp\big( (1-s) \lambda_n \big) \exp \big( (s-1) D_{s} (P^{\times n}\| U^n) \big) \\
&= \exp \Big( \frac{1-s}{s} \big( \log v(n) + nR - D_{s} (P^{\times n}\| U^n) \big) \Big)\end{aligned}$$ where we substituted $\lambda_n$ from in the last step. Further using the additivity property of Axiom \[ax:add\], we find that $$\begin{aligned}
D_{s} (P^{\times n}\| U^n) \geq D_{s} (P^{\times n} \| {\mathcal{Q}}_n) = n D_s(P \| {\mathcal{Q}}),\end{aligned}$$ and thus we arrive at the bound $$\begin{aligned}
\log \hat{\alpha}\big(\exp(-n R); P^{\times n} \big\|\overline{{\mathcal{Q}}}_n \big) \leq \frac{1-s}{s} \big( nR - nD_s(P \| {\mathcal{Q}}) + \log v(n) \big) \label{6-22-1} \,.
\end{aligned}$$ Since $\log v(n) = O(\log n)$, taking the limit $n \to \infty$ yields $$\begin{aligned}
\liminf_{n \to \infty} - \frac{1}{n} \log
\hat{\alpha}\big(\exp(-n R); P^{\times n} \big\|\overline{{\mathcal{Q}}}_n\big)
\geq
\frac{1-s}{s} \big(D_{s}(P\|{\mathcal{Q}}) - R\big) \,.
\end{aligned}$$ Finally, since this derivation holds for all $s \in (a,0)$, we established the direct part.
Proof of Optimality
-------------------
To show optimality, we will directly employ the converse of the Hoeffding bound.
Assume Axioms \[ax:prod\]–\[ax:para\] hold with parameter $a$. Then, for any $R > R_a$, we have $$\label{eq:hoeffding-conv}
\limsup_{n \to \infty} - \frac{1}{n} \log
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\| {\mathcal{Q}}_n \Big)
\leq \sup_{s \in (a, 1)} \left\{ \frac{1-s}{s} \big( D_{s}(P\|{\mathcal{Q}}) - R \big) \right\}.$$
Let us first consider the case $R \in (R_a, D(P\|{\mathcal{Q}}))$. We will use the results of Sections \[sec:convex1\] and \[sec:convex2\]. Take $\hat{s} \in (a, 1)$ to be the optimizer in Lemma \[lm:supmax\] for the function ${\tilde{\phi}}= \phi$ and ${\tilde{\psi}}= \psi$ on $(a,1)$. Then we have that $\psi(\hat{s}) = \hat{s} \phi'(\hat{s}) - \phi(\hat{s}) = R$ and $$\begin{aligned}
\sup_{s \in (0, 1)} \frac{1-s}{s} \left( D_{s}(P \| {\mathcal{Q}}) - R \right) = \frac{1-\hat{s}}{\hat{s}} \left( D_{\hat{s}}(P \| {\mathcal{Q}}) - R \right) \label{eq:supmax1}\end{aligned}$$ The following consequence of Lemma \[lm:derivative\] is crucial. Recall that $$\begin{aligned}
\bar{\phi}(s) = \log g_s(P\|Q_{\hat{\theta}^{\hat{s}}}) \qquad \textrm{and} \qquad \bar{\psi}(s) = s \bar{\phi}'(s) - \bar{\phi}(s) \,,\end{aligned}$$ where $\hat{\theta}^{\hat{s}}$ is the optimal $\theta$ at $s = \hat{s}$. Then, Lemma \[lm:derivative\] implies that $\phi'(\hat{s}) = \bar{\phi}'(\hat{s})$, and hence, $\bar{\psi}(\hat{s}) = \psi(\hat{s}) = R$.
Next we note that $$\begin{aligned}
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\| {\mathcal{Q}}_n \Big)
\geq \hat{\alpha}\Big(\exp(-n R); P^{\times n} \Big\| Q^{\times n}\Big) \end{aligned}$$ for any $Q \in {\mathcal{Q}}$. At this point we can apply the converse of the Hoeffding bound (we take the formulation in [@nagaoka06 Thm. 1]) to the expression on the right-hand side, which yields $$\begin{aligned}
&\limsup_{n \to \infty} \left\{ - \frac{1}{n} \log \hat{\alpha}\Big(\exp(-n R); P^{\times n} \Big\| {\mathcal{Q}}_n \Big) \right\} \label{eq:lhs1} \\
&\qquad \qquad \leq
\limsup_{n \to \infty} \left\{ - \frac{1}{n} \log \hat{\alpha}\Big(\exp(-n R); P^{\times n} \Big\| Q_{\hat{\theta}^{\hat{s}}}^{\times n}\Big) \right\}\\
&\qquad \qquad =
\sup_{s \in (0, 1)} \frac{1-s}{s} \left( D_{s}(P \| Q_{\hat{\theta}^{\hat{s}}}) - R \right) .\end{aligned}$$ Next we apply Lemma \[lm:supmax\] for the functions ${\tilde{\phi}}= \bar{\phi}$ and ${\tilde{\psi}}= \bar{\psi}$ on $(0,1)$. We find that $$\begin{aligned}
\sup_{s \in (0, 1)} \frac{1-s}{s} \left( D_{s}(P \| Q_{\hat{\theta}^{\hat{s}}}) - R \right) = \frac{1-\hat{s}}{\hat{s}} \left( D_{\hat{s}}(P \| {\mathcal{Q}}) - R \right) ,\end{aligned}$$ which proves the result together with .
If $R \geq D(P\|{\mathcal{Q}})$ the right hand side evaluates to zero. Moreover, there exists at least one $Q \in {\mathcal{Q}}$ such that $R \geq D(P\|Q)$. The result then follows from the converse of Hoeffding’s bound since $$\begin{aligned}
\limsup_{n \to \infty} - \frac{1}{n} \log
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\| {\mathcal{Q}}_n \Big)
\leq \limsup_{n \to \infty} - \frac{1}{n} \log
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\| Q^{\times n} \Big) = 0 \,.
\end{aligned}$$
Proofs: Strong Converse Exponents {#sec:sc}
=================================
Again we treat achievability and optimality with separate proofs that rely on different axioms.
Proof of Achievability
----------------------
Our proof relies on a variant of the Gärtner-Ellis theorem of large deviation theory (see, e.g., [@dembo98 Sec. 2 and Sec. 3.4] for an overview), which we recall here. Given a sequence of random variables $\{ Z_n \}_{n \in \mathbb{N}}$ we introduce its asymptotic *cumulant generating function* as $$\begin{aligned}
\Lambda_Z(t) &:= \lim_{n \to \infty} \left\{ \frac{1}{n} \log \big( \mathbb{E} \left[ \exp ( n t Z_n ) \right] \big) \right\} ,
\label{eq:cum}\end{aligned}$$ if it exists. For our purposes it is sufficient to use the following variant of the Gärtner-Ellis theorem due to Chen [@chen00 Thm. 3.6] (see also [@mosonyi14 Lem. A.2]).
\[pr:mosonyi\] Let us assume that $t \mapsto \Lambda_Z(t)$ as defined in exists and is differentiable in some interval $(a, b)$. Then, for any $z \in \big( \lim_{t \searrow a} \Lambda_Z'(t), \lim_{t \nearrow b} \Lambda_Z'(t) \big)$, we have $$\begin{aligned}
\limsup_{n \to \infty} \left\{ - \frac{1}{n} \log \Pr [ Z_n \geq z] \right\} \leq \sup_{t \in (a, b)} \left\{ t z - \Lambda_Z(t) \right\} \,.
\end{aligned}$$
Achievability follows from the following statement.
Assume Axioms \[ax:add\]–\[ax:para\] hold with parameter $b$. For any $R \in (0, R_b)$, we have $$\label{eq:sc-achieve}
\lim_{n \to \infty} - \frac{1}{n} \log \bigg( 1 -
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\|\, \overline{{\mathcal{Q}}}_n \Big) \bigg)
\leq \sup_{s \in (1, b)} \left\{ \frac{s-1}{s} \big( R - D_{s}(P\|{\mathcal{Q}}) \big) \right\}.$$
Let us first assume that $R \in (D(P\|{\mathcal{Q}}), R_b)$. Using Lemma \[lm:supmax\] with ${\tilde{\phi}}= \phi$ on $(1,b)$, we find the value $\hat{s} \in (1,b)$ that satisfies $$\begin{aligned}
\sup_{s \in (1,b)} \frac{s-1}{s} \big( R - D_{s}(P\|{\mathcal{Q}}) \big) = \frac{\hat{s}-1}{\hat{s}} \big( R - D_{\hat{s}}(P\|{\mathcal{Q}}) \big)
\end{aligned}$$ and $\psi(\hat{s}) = \hat{s} \phi'(\hat{s}) - \phi(\hat{s}) = R$. We use the same sequence of tests $T_n$ as in and the sequence $\lambda_n$ of , substituting $\hat{s}$ for $s$. This ensures that $\beta(T_n; \overline{{\mathcal{Q}}}_n) \leq \exp(-nR)$, as shown in the proof of Proposition \[pr:hoeffding-achieve\]. Moreover, $$\begin{aligned}
1 - \hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\|\, \overline{{\mathcal{Q}}}_n \Big)
&\geq 1 - \alpha(T_n; P^{\times n}) \label{eq:inserthere} \\
&= P^{\times n} \left[ P^{\times n}(X^n) \geq \exp(\lambda_n) U^n \right] = \Pr[Z_n \geq 0] \,,
\end{aligned}$$ where we defined the sequence of random variables $Z_n(X^n)$ following the law $X^n \leftarrow P^{\times n}$ and $$\begin{aligned}
Z_n(x^n)
&= \frac{1}{n} \left( \log \frac{P^{\times n}(x^n)}{U^n(x^n)} - \lambda_n \right) \\
&= \frac{1}{n} \left( \log \frac{P^{\times n}(x^n)}{U^n(x^n)}
- \frac{\log v(n)}{\hat{s}} - \frac{\hat{s}-1}{\hat{s}} D_{\hat{s}}(P^{\times n}\|U^n) - \frac{nR}{\hat{s}} \right) \,.
\end{aligned}$$ Its asymptotic cumulant generating function evaluates to $$\begin{aligned}
\Lambda_Z(t-1) =&\ \lim_{n \to \infty} \left\{ \frac{1}{n} \log \big( \mathbb{E} \left[ \exp ( n (t-1) Z_n ) \right] \big) \right\} \\
=&\ \lim_{n \to \infty} \left\{ \frac{1}{n} \log \mathbb{E} \left[ \frac{P^{\times n}(X^n)^{t-1}}{U^n(X^n)^{t-1}} \right] - (t-1) \left( \frac{\log v(n)}{n \hat{s}} + \frac{\hat{s}-1}{n\hat{s}} D_{\hat{s}}(P^{\times n}\|U^n) + \frac{R}{n\hat{s}} \right) \right\} \\
=&\ (t-1) \lim_{n \to \infty} \left\{ \frac{1}{n} D_{t}(P^{\times n}\| U^n) - \frac{\hat{s}-1}{n\hat{s}} D_{\hat{s}}(P^{\times n}\|U^n) - \frac{R}{\hat{s}} \right\} \\
=&\ \phi(t) - \frac{t-1}{\hat{s}} ( \phi(\hat{s}) + R ) = \phi(t) - (t-1) \phi'(\hat{s}) \,.
\end{aligned}$$ Note that $t \mapsto \Lambda_Z(t-1)$ is differentiable on $(1, b)$ due to Lemma \[lm:derivative\].
Moreover, in order to apply Lemma \[pr:mosonyi\] with $z=0$, we need to verify the following two inequalities: $$\begin{aligned}
\lim_{t \to 1} \left\{ \frac{{\mathrm{d}}}{{\mathrm{d}}t} \Lambda_Z(t-1) \right\}
= \phi'(1) - \phi'(\hat{s}) < 0 \quad \textrm{and} \qquad
\lim_{t \to b} \left\{ \frac{{\mathrm{d}}}{{\mathrm{d}}t} \Lambda_Z(t-1) \right\}
= \phi'(b) - \phi'(\hat{s}) > 0 \,. \label{eq:ineq}
\end{aligned}$$ We cannot invoke strict convexity of $\phi(s)$ to verify the above bounds; instead, note that $D(P\|{\mathcal{Q}}) < R$, and thus $$\begin{aligned}
\phi'(1) - \phi'(\hat{s}) &= D(P\|{\mathcal{Q}}) - \frac{\hat{s}-1}{\hat{s}} D_{\hat{s}}(P\|{\mathcal{Q}}) - \frac{1}{\hat{s}} R \\
&< \frac{\hat{s}-1}{\hat{s}} \big( D(P\|{\mathcal{Q}}) - D_{\hat{s}}(P\|{\mathcal{Q}}) \big) \leq 0 \,.
\end{aligned}$$ To prove the second inequality in , we use the fact that $R_b > R$ to show $$\begin{aligned}
\phi'(b) - \phi'(\hat{s}) &= \frac{1}{b} R_b + \frac{b-1}{b} D_b(P\|{\mathcal{Q}}) - \frac{1}{\hat{s}} R - \frac{\hat{s}-1}{\hat{s}} D_{\hat{s}}(P\|{\mathcal{Q}}) \\
&> \frac{1}{b} R + \frac{b-1}{b} D_b(P\|{\mathcal{Q}}) - \frac{1}{\hat{s}} R - \frac{\hat{s}-1}{\hat{s}} D_{\hat{s}}(P\|{\mathcal{Q}}) \\
&= \frac{\hat{s}-1}{\hat{s}} \big( R - D_{\hat{s}}(P\|{\mathcal{Q}}) \big) - \frac{b-1}{b} \big( R - D_b(P\|{\mathcal{Q}}) \big) \geq 0 \,,
\end{aligned}$$ where the last inequality follows by the definition of $\hat{s}$.
We have now verified the conditions of Lemma \[pr:mosonyi\] with $z=0$, which yields $$\begin{aligned}
\lim_{n \to \infty} -\frac{1}{n} \log \Pr[Z_n \geq 0] &= \sup_{t \in (1,b)} (t-1)\phi'(\hat{s}) - \phi(t) \label{eq:supneeded}\\
&= (\hat{s}-1)\phi'(\hat{s}) - \phi(\hat{s}) \label{eq:supdone}\\
&= \frac{\hat{s}-1}{\hat{s}} \bigg( \hat{s} \phi'(\hat{s}) - \phi(\hat{s}) + \frac{1}{\hat{s}-1} \phi(\hat{s}) \bigg) = \frac{\hat{s}-1}{\hat{s}} \big( R - D_{\hat{s}}(P\|{\mathcal{Q}}) \big) \,.
\end{aligned}$$ To evaluate the supremum in , we note that the objective function $t \mapsto (t-1)\phi'(\hat{s}) - \phi(t)$ is concave in $t$ and its derivative vanishes at $t = \hat{s}$. This establishes . Combining this with concludes the proof.
For $R \leq D(P\|{\mathcal{Q}})$ the right hand side of evaluates to zero. Since the expression on the left hand side is clearly monotonically increasing in $R$ we deduce that, for all such $R$, $$\begin{aligned}
\lim_{n \to \infty} - \frac{1}{n} \log \bigg( 1 -
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\|\, \overline{{\mathcal{Q}}}_n \Big) \bigg)
\leq \inf_{R > D(P\|{\mathcal{Q}})} \sup_{s \in (1, b)} \left\{ \frac{s-1}{s} \big( R - D_{s}(P\|{\mathcal{Q}}) \big) \right\} = 0 \,.
\end{aligned}$$
Proof of Optimality
-------------------
Optimality follows as a corollary of Han and Kobayashi’s [@han89] derivation of the strong converse exponent.
Assume Axiom \[ax:prod\] holds. For any $R \geq 0$, we have $$\liminf_{n \to \infty} - \frac{1}{n} \log \bigg( 1 -
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\| {\mathcal{Q}}_n \Big) \bigg)
\geq \sup_{s > 1} \left\{ \frac{s-1}{s} \big( R - D_{s}(P\|{\mathcal{Q}}) \big) \right\}.$$
If $R < D(P\|{\mathcal{Q}})$ the bound holds trivially. Otherwise, analogous to the optimality proof for Theorem \[th:hoeffding\], we first fix $Q \in {\mathcal{Q}}$ and apply the Han-Kobayashi converse bound [@han89] (in the form of [@ogawa00 Ch. VI] and [@mosonyiogawa13 Thm. IV.9]). This yields $$\begin{aligned}
&\liminf_{n \to \infty} \left\{ - \frac{1}{n} \log \bigg( 1 -
\hat{\alpha}\Big(\exp(-n R);P^{\times n} \Big\| {\mathcal{Q}}_n \Big) \bigg) \right\} \\
&\qquad \quad \geq \liminf_{n \to \infty} \left\{ - \frac{1}{n} \log \bigg( 1- \hat{\alpha}\Big(\exp(-n R); P^{\times n} \Big\| Q^{\times n} \Big) \bigg) \right\}\\
&\qquad \quad =
\sup_{s > 1} \left\{ \frac{s-1}{s} \big( R - D_{s}(P \Big\| Q) \big) \right\} \label{eq:rhs2}.
\end{aligned}$$ As this holds for all $Q \in {\mathcal{Q}}$, we maximize the expression in over $Q$ to arrive at the desired result.
Proofs: Second Order Asymptotics {#sec:second}
================================
The following result refines Lemma \[lm:universal-limit\], and is the key ingredient of our proof.
\[lm:universal2\] Assume Axioms \[ax:prod\]–\[ax:para\] holds on $(a,b) \supset \{ 1 \}$. For all $t \in \mathbb{R}$, $$\begin{aligned}
\lim_{n \to \infty} \left\{ \frac{t}{\sqrt{n}} \Big( D_{1 + \frac{t}{\sqrt{n}}}(P^{\times n} \| U^n) - n D(P\|{\mathcal{Q}}) \Big) \right\} = \frac{t^2}{2} V(P\|{\mathcal{Q}}) \,. \label{eq:uni2}
\end{aligned}$$
By combining Lemma \[lm:derivative\] at $s_0 = 1$ with the Taylor expansion in , we find $D_{1+s}(P\|{\mathcal{Q}}) = D(P\|{\mathcal{Q}}) + \frac{s}{2} V(P\|{\mathcal{Q}}) + O(s^2)$. From we learn that $$\begin{aligned}
D_{1+\frac{t}{\sqrt{n}}}(P^{\times n}\| U^n)
\leq n D_{1+\frac{t}{\sqrt{n}}}(P\|{\mathcal{Q}}) + \log v(n)
= n D(P\|{\mathcal{Q}}) + \frac{t\sqrt{n}}{2} V(P\|{\mathcal{Q}}) + O(\log n) \,. \label{eq:second1}
\end{aligned}$$ Furthermore, employing additivity from Axiom \[ax:add\] yields $$\begin{aligned}
D_{1+\frac{t}{\sqrt{n}}}(P^{\times n}\| U^n)
\geq n D_{1+\frac{t}{\sqrt{n}}}(P\|{\mathcal{Q}})
= n D(P\|{\mathcal{Q}}) + \frac{t\sqrt{n}}{2} V(P\|{\mathcal{Q}}) + O(\log n) \,. \label{eq:second2}
\end{aligned}$$ Combining and yields the desired statement.
We prove the direct and converse part of Theorem \[th:second\] together.
We first show the converse statement. Choosing the optimal distribution $\hat{Q} = \hat{Q}^1 \in {\mathcal{Q}}$ as defined in , we find $$\begin{aligned}
\hat{\alpha}\big( \exp(- n D(P \|{\mathcal{Q}}) - \sqrt{n}r) ; P^{\times n} \big\| {\mathcal{Q}}_n \big) \geq \hat{\alpha}\big( \exp(- n D(P \|\hat{Q} ) - \sqrt{n}r) ; P^{\times n} \big\| \hat{Q}^{\times n} \big) ,
\end{aligned}$$ and the limiting statement then follows using Strassen’s result [@strassen62].
To show achievability we again rely on the test given in and set $\lambda_n = n D(P \| {\mathcal{Q}}) + \sqrt{n} r + \log v(n)$. Then by Axiom \[ax:universal\] and using the argument leading to to establish the first identity, we have $$\begin{aligned}
\beta(T_n; {\mathcal{Q}}_n) &= \max_{Q^n \in {\mathcal{Q}}_n^{{\textnormal{sym}}}} Q^n\big[ P^{\times n}(X^n) \geq \exp(\lambda_n) U^n(X^n) \big] \\
&\leq v(n) U^n \big[ P^{\times n}(X^n) \geq \exp(\lambda_n) U^n(X^n) \big] \\
&\leq v(n) \exp(-\lambda_n) P^{\times n} \big[ P^{\times n}(X^n) \geq \exp(\lambda_n) U^n(X^n)\big] \\
&\leq \exp \big(- n D(P\|{\mathcal{Q}}) - \sqrt{n} r \big) \,.
\end{aligned}$$
Furthermore, we find $$\begin{aligned}
\alpha(T_n; P^{\times n}) &= P^{\times n} \big[ \log P^{\times n}(X^n) - \log U^n(X^n) < n D(P\|{\mathcal{Q}}) + \sqrt{n} r + \log v(n) \big] \\
&= \Pr [ Y_n(X^n) < r] \,,
\end{aligned}$$ where $X^n \sim P^{\times n}$ and we defined the following sequence of random variables as $$\begin{aligned}
Y_n(X^n) := \frac{1}{\sqrt{n}} \big( \log P^{\times n}(X^n) - \log U^n(X^n) - n D(P\|{\mathcal{Q}}) - \log v(n) \big) \,.
\end{aligned}$$
Lemma \[lm:universal2\] then implies that the asymptotic cumulant generating function of the sequence $\{ Y_n \}_n$ converges to $$\begin{aligned}
\Lambda_Y(t) &= \lim_{n \to \infty} \left\{ \frac{1}{n} \log \left( \mathbb{E}\big[ \exp(n t Y^n) \big] \right) \right\} \\
&= \lim_{n \to \infty} \left\{ \frac{t}{\sqrt{n}} \big( D_{1+\frac{t}{\sqrt{n}}}(P^{\times n}\|U^n) - n D(P \|{\mathcal{Q}}) - \log v(n) \big) \right\} \\
&= \frac{t^2}{2} V(P\|{\mathcal{Q}}) \,.
\end{aligned}$$ Hence, by Lévi’s continuity theorem (see, e.g., [@fristedt37 Ch. 14, Thm. 21]), the sequence of random variable $\{ Y_n \}_n$ converges in distribution to a random variable $Y$ with cumulant generating function $\Lambda_Y(t)$, i.e., a Gaussian random variable with zero mean and variance $V(P\|{\mathcal{Q}})$. In particular, this yields $$\begin{aligned}
\lim_{n \to \infty} P^{\times n} \left[ Y_n < r \right]
= \Pr \left[ Y < r \right]
= \Phi \left( \frac{r}{\sqrt{V(P\|{\mathcal{Q}})}} \right) .
\label{eq:sec2}\end{aligned}$$ Since $\hat{\alpha}\big( \exp(- n D(P \|{\mathcal{Q}}) - \sqrt{n}r) ; P^{\times n} \big\| {\mathcal{Q}}_n \big) \leq \alpha(T_n; P^{\times n})$, this concludes the proof.
Conclusion {#sec:conc}
==========
We have introduced a general framework to treat binary hypothesis testing with a composite alternative hypothesis. In this general framework we show analogues of Stein’s Lemma, Hoeffding’s optimal error exponents and Han-Kobyashi’s optimal strong converse exponents. We have discussed several concrete examples that lead to operational interpretations of various Rényi information measures.
The coincidence between our obtained exponents for the hypothesis testing problem in and the corresponding exponents for channel coding is quite interesting. A similar coincidence has been observed for the case of source coding with side information and . These facts seem to indicate a deep relation between coding and the composite alternative hypotheses given in and . Its further clarification is an interesting future direction of study, for example one could try to find coding problems that are closely related to .
In statistics, the $\chi^2$ test is used in an asymptotic setting similar to . The test assumes i.i.d. distributions and is used for the case when both hypotheses are composite (see, e.g., [@lehmann05]). For small samples and $k=2$, Fisher’s exact test [@fisher22] can be used to replace the $\chi^2$ test. Recently, the setting of small samples and general $k$ has been studied using a Gröbner basis approach [@diaconis98; @sakata05]. In contrast to their formulation, we have not assumed i.i.d. structure for the independent case; instead, we only require permutation invariance for each random variable. Our result suggests that we can replace the i.i.d. condition by a permutation invariant condition when testing independence, which can be expected to have wider applications.
The key ingredient of our derivation is the axiomatic approach based on the universal distribution. Due to its generality, we can treat many composite hypothesis testing problems without i.i.d. assumption (for the composite hypothesis), and it will be interesting to explore further examples that fit into our framework. Moreover, because the universal distribution plays an important role in universal channel coding [@hayashi09], we can expect that it will play an important role when analyzing universal protocols for other problems in information theory.
As explained in Section \[sec:ex2\], we cannot remove the permutation invariance condition in that example, an essential difference to the example given in Section \[sec:ex1\]. This kind of difference sheds light on the difference between channel coding and secure random number generation. Originally, for the channel coding, the meta converse was introduced using simple hypothesis testing [@nagaoka01 Sec. 3] and [@polyanskiy10]. Polyanskiy [@polyanskiy13 Sec. II] then extended it to the composite hypothesis testing of the form . Although this improvement does not effect the exponents and the second coding rate, it can improve the bound in the finite blocklength regime. Recently, Tyagi-Watanabe [@tyagi14; @tyagi14b] introduced a converse bound for secure random number generation by using simple hypothesis testing between a true joint distribution and an arbitrary product distribution. Although in their converse bound, we can choose an arbitrary product distribution as the alternative hypothesis, we cannot replace the alternative hypothesis by a composite hypothesis composed of all of product distributions. Hence, for secure random number generation, we cannot extend their bound to a bound based on the composite hypothesis as in [@polyanskiy13].
In prior work [@hayashitomamichel14] the present authors have analyzed composite hypothesis testing in the non-commutative (quantum) regime and found an operational interpretation for various definitions of quantum Rényi mutual information and quantum Rényi conditional entropy [@tomamichel13]. Finding appropriate definitions for Rényi conditional mutual information in the non-commutative setting is an ongoing topic of research [@bertawilde14]. It is possible that an adaption of our analysis to quantum hypothesis testing will lead to further progress in this direction. However, some caution is advised since the definitions of the regular conditional mutual information in – are not equivalent in the quantum case.
#### Acknowledgements {#acknowledgements .unnumbered}
MH is partially supported by a MEXT Grant-in-Aid for Scientific Research (A) No. 23246071, and by the National Institute of Information and Communication Technology (NICT), Japan. MT is funded by an University of Sydney Postdoctoral Fellowship and acknowledges support from the ARC Centre of Excellence for Engineered Quantum Systems (EQUS).
Verification of the Axioms for Examples in Section \[sec:examples\] {#sec:proof-examples}
===================================================================
Sibson’s identity {#sec:sibson}
-----------------
This appendix proves that the examples satisfy our axioms. One of the main ingredients is Sibson’s identity [@sibson69], as presented in [@csiszar95 Eq. (11)–(13)].
\[lm:sibson\] For any distributions $P_{XY} \in {\mathcal{P}}({\mathcal{X}}\times {\mathcal{Y}})$, $T_X \in {\mathcal{P}}({\mathcal{X}})$ and $Q_Y \in {\mathcal{P}}({\mathcal{Y}})$, and any $s \in (0,1) \cup (1, \infty)$, $$\begin{aligned}
D_{s}(P_{XY} \| T_X \times Q_Y) = D_{s}(P_{XY} \| T_X \times \hat{Q}_Y^s) + D_{s}(\hat{Q}_Y^s \| Q_Y) \, \label{eq:sibson}\end{aligned}$$ where the optimal distribution $\hat{Q}_Y^s \in {\mathcal{P}}({\mathcal{Y}})$ is given by $$\begin{aligned}
\hat{Q}_Y^s(y) = \frac{P_Y(y) g_s(P_{X|Y=y}\| T_X)^{\frac1s}}{\sum_y P_Y(y) g_s(P_{X|Y=y}\| T_X)^{\frac1s}} \,.\end{aligned}$$ Thus, in particular, $\operatorname*{\arg\min}_{Q_Y \in {\mathcal{P}}({\mathcal{Y}})} D_{\alpha}(P_{XY} \| T_X \times Q_Y) = \{ Q_Y^* \}$.
We rewrite as $g_{s}(P_{XY} \| T_X \times Q_Y) = g_s(P_{XY} \| T_X \times \hat{Q}_Y^s) \cdot g_{s}(\hat{Q}_Y^s \| Q_Y)$, at which point the equality can be verified by close inspection. The fact that $\hat{Q}_Y^s$ is the unique minimizer is then a consequence of the positive definiteness of the Rényi divergence.
Proof of Proposition \[pr:examples-one\] {#sec:proof-examples-1}
----------------------------------------
Clearly ${\mathcal{Q}}$ is compact convex and we explicitly find the optimizer using Sibson’s identity in Lemma \[lm:sibson\]. Up to normalization it is given by $$\begin{aligned}
\hat{Q}_Y^s(y) &\sim P_Y(y) g_s ( P_{X|Y=y} \| T_X )^{\frac1s}, \label{eq:Qopt}
\end{aligned}$$ and thus Axiom \[ax:convex\] is verified. Axiom \[ax:prod\] holds by definition and Axiom \[ax:add\] can be verified by noting that $\hat{Q}^s$ in takes on an i.i.d. product form when both $P_{XY}^{\times n}$ and $T_X^{\times n}$ are i.i.d. products.
Next, note that ${\mathcal{Q}}_n$ is closed under permutations and convex. The universal distributions are $$\begin{aligned}
U_{X^nY^n}^n = T_X^{\times n} \times U_{Y^n}^n, \qquad \textrm{with} \qquad U_{Y^n}^n(y^n) = \sum_{\lambda \in {\mathcal{T}}_n({\mathcal{Y}})} \frac{1}{| {\mathcal{T}}_n({\mathcal{Y}}) |} \frac{ 1 }{ |\lambda| } \, 1 \{ y^n \textrm{ is of type } \lambda \} \, ,
\end{aligned}$$ as in Lemma \[lm:uni-dist\]. Clearly $U_{X^nY^n}^n \in {\mathcal{Q}}_n^{{\textnormal{sym}}}$ and thus. we find that Axiom \[ax:universal\] is satisfied with $v(n) = |{\mathcal{T}}_n(Y)| = {\mathrm{poly}}(n)$.
Finally note that all the above remains true if we restrict ${\mathcal{Q}}_n$ to permutation invariant or i.i.d. product distributions, denoted ${\mathcal{Q}}_n'$, except that now $U_{X^nY^n}^n \notin {\mathcal{Q}}_n'$. However, we still have $$\begin{aligned}
D_s(P_{XY}^{\times n} \| U_{X^nY^n}^n) \geq D_s(P_{XY}^{\times n} \| {\mathcal{Q}}_n) = D_s(P_{XY}^{\times n} \| {\mathcal{Q}}_n') ,
\end{aligned}$$ since additivity property guarantees that the minimum in $D_s(P_{XY}^{\times n} \| {\mathcal{Q}}_n)$ is taken by a product distribution.
Proof of Proposition \[pr:examples-two\] {#sec:proof-examples-2}
----------------------------------------
We give the proof for the case $k=1$ and set ${\mathcal{X}}_1 = {\mathcal{X}}$. The generalization to larger $k$ does not require further conceptual insights, and we will remark in a footnote where nontrivial changes are necessary.
Axiom \[ax:prod\] holds by definition. To verify Axiom \[ax:universal\] we first note that the joint permutations of $X^n$ and $Y^n$ separate as $W_{X^nY^n}^n[\pi] = W_{X^n}^n[\pi] \times W_{Y^n}^n[\pi]$. Thus, we can write $$\begin{aligned}
\sum_{\pi \in S_n} \frac{1}{n!} \big(Q_{X^n} \times Q_{Y^n} \big) W_{X^nY^n}^n[\pi] = \sum_{\pi \in S_n} \frac{1}{n!} Q_{X^n} W_{X^n}^n[\pi] \times Q_{Y^n} W_{Y^n}^n[\pi] = Q_{X^n} \times \sum_{\pi \in S_n} \frac{1}{n!} Q_{Y^n} W_{Y^n}^n[\pi] \,,
\end{aligned}$$ where we used that $Q_{X^n} \in {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}}^n)$ to establish the last equality. Clearly the resulting distribution lies in ${\mathcal{Q}}_n$. Next, consider the universal distribution $U_{X^nY^n}^n = U_{X^n}^n \times U_{Y^n}^n$ with $U_{X^n}^n$ and $U_{Y^n}^n$ given as in . Clearly, since ${\mathcal{Q}}_n^{{\textnormal{sym}}} = {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{X}}^n) \times {\mathcal{P}}^{{\textnormal{sym}}}({\mathcal{Y}}^n)$ we then find that every symmetric distribution satisfies $$\begin{aligned}
Q_{X^n} \times Q_{Y^n} \leq |{\mathcal{T}}_n({\mathcal{X}})| U_{X^n}^n \times |{\mathcal{T}}_n({\mathcal{Y}})| U_{Y^n}^n\end{aligned}$$ and Axiom \[ax:universal\] holds with $v(n) = |{\mathcal{T}}_n({\mathcal{X}})| |{\mathcal{T}}_n({\mathcal{Y}})| = {\mathrm{poly}}(n)$.
For Axiom \[ax:para\] we chose the following parametrization. Since ${\mathcal{P}}({\mathcal{X}})$ is convex subset of $\mathbb{R}^{|{\mathcal{X}}|-1}$, there exists a natural smooth parametrization $\Theta_1 \ni \theta_1 \mapsto Q_{X,\theta_1} \in {\mathcal{P}}({\mathcal{X}})$ where $\Theta_1$ is a convex subset of $\mathbb{R}^{|{\mathcal{X}}|-1}$, and similarly for ${\mathcal{P}}({\mathcal{Y}})$. Combining these two parameterizations, we introduce a $\Theta \subset \mathbb{R}^{|{\mathcal{X}}| + |{\mathcal{Y}}| - 2}$ such that $$\begin{aligned}
\Theta \ni \theta = (\theta_1, \theta_2) \mapsto Q_{X,\theta_1} \times Q_{Y,\theta_2} = Q_{XY,\theta} \in {\mathcal{Q}}\,.
\end{aligned}$$ The set $\Theta$ is evidently convex. Let us next verify the required convexity and concavity properties. First note that the map $f(x,y) = x^{1-s} y^{1-s}$ for $x, y \geq 0$ is strictly jointly concave when $s \in (\frac12, 1)$ and strictly jointly convex when $s > 1$. [^8] From this we conclude that the map $(Q_X, Q_Y) \mapsto g_s(P_{XY}\|Q_X \times Q_Y)$ is strictly jointly concave for $s \in (\frac12, 1)$ and strictly jointly convex for $s > 1$. Hence the desired concavity and convexity properties for the map $\theta \mapsto g_s(P_{XY}\|Q_{XY,\theta})$ hold. In particular, the minimizer, if it exists in the interior of ${\mathcal{Q}}$, is unique.
Let us assume that $D_s(P_{XY} \| {\mathcal{Q}}) = D_s(P_{XY} \| \hat{Q}_X^s \times \hat{Q}_Y^s)$ for some optimal distributions $\hat{Q}_X^s$ and $\hat{Q}_Y^s$. To verify Axiom \[ax:add\] we note that distributions are optimal if and only if they satisfy the self-consistency relation (cf. Eq. ) $$\begin{aligned}
\hat{Q}_X^s(x) \sim P_X(x) g_s ( P_{Y|X=x} \| \hat{Q}_Y^s ) \quad \textrm{and} \quad
\hat{Q}_Y^s(y) \sim P_Y(y) g_s ( P_{X|Y=y} \| \hat{Q}_X^s ) \,. \label{eq:self-consistency}
\end{aligned}$$ Therefore this solution is in the interior of ${\mathcal{Q}}$.
Furthermore, we find that the product distributions $\hat{Q}_X^s \times \hat{Q}_X^s$ and $\hat{Q}_Y^s \times \hat{Q}_Y^s$ satisfy the self-consistency relations for the optimal solution of $D_s(P_{XY}^2 \| {\mathcal{Q}}_2)$. More precisely, we find $$\begin{aligned}
\hat{Q}_X^s(x_1)\hat{Q}_X^s(x_2) \sim P_X(x_1)P_X(x_2) g_s ( P_{Y|X=x_1} \times P_{Y|X=x_2} \| \hat{Q}_Y^s \times \hat{Q}_Y^s) ,
\end{aligned}$$ and vice versa. Hence, we can conclude that these product distributions are optimal as well, namely $$\begin{aligned}
D_{s}(P_{XY}^{\times 2} \| {\mathcal{Q}}_2 ) = D_{s}\big(P_{XY}^{\times 2} \big\| (\hat{Q}_{X} \times \hat{Q}_{Y})^{\times 2}\big) = 2 D_s(P_{XY} \| {\mathcal{Q}}) \,.
\end{aligned}$$ Applying this argument inductively yields the condition of Axiom \[ax:prod\].
Proof of Proposition \[pr:examples-three\] {#sec:proof-examples-3}
------------------------------------------
Before we commence with the proof we need to introduce some additional concepts and auxiliary results. Let us introduce the following representation of channels, which is reminiscent of the Choi-Jamio[ł]{}kowski isomorphism in quantum information theory. Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be discrete sets. For any channel $Q_{Y|X} \in {\mathcal{P}}({\mathcal{Y}}|{\mathcal{X}})$ we define a vector representation $\tilde{Q}_{XY} = 1_{X} \times Q_{Y|X}$, where $1_X$ is the identity vector for the Schur (element-wise) product of vectors, i.e. $1_{X}(x) = 1$ for all $x \in {\mathcal{X}}$. More concretely, the vector is given by $\tilde{Q}_{XY}(x,y) = Q_{Y|X}(y|x)$. Note that $\tilde{Q}_{XY}$ is not a probability distribution but clearly we must have $$\begin{aligned}
\label{eq:normalize-choi}
\sum_{y \in {\mathcal{Y}}} \tilde{Q}_{XY}(x,y) = 1, \qquad \forall x \in {\mathcal{X}}\,. \end{aligned}$$ Using this representation we can write joint distribution after the application of the channel as $$\begin{aligned}
P_{X} \times Q_{Y|X} = (P_{X} \times 1_{Y}) \circ \tilde{Q}_{XY} , \label{eq:schur}\end{aligned}$$ where $\circ$ denotes the Schur product between the vectors and $1_{Y}(y) = 1$ for all $y \in {\mathcal{Y}}$. Note also that the normalization condition enforces that the resulting vector is a probability distribution, and hence every vector with positive elements that satisfies corresponds to a valid channel.
\[lm:choi-covariance\] $Q_{Y^n|X^n}$ is covariant under permutations if and only if $\tilde{Q}_{X^nY^n}$ is permutation invariant. Formally, $$\begin{aligned}
\forall \pi \in S_n:\ Q_{Y^n|X^n} W_{Y^n}[\pi] = W_{X^n}[\pi] Q_{Y^n|X^n}
\quad \iff \quad \forall \pi \in S_n:\ \tilde{Q}_{X^nY^n} W_{X^nY^n}[\pi] = \tilde{Q}_{X^nY^n} \,.
\end{aligned}$$
The following equalities can be verified by close inspection: $$\begin{aligned}
\tilde{Q}_{X^nY^n} W_{X^nY^n}[\pi] &= \big(1_{X^n} \times Q_{Y^n|X^n} \big) \big( W_{X^n}[\pi] \times W_{Y^n}[\pi] \big) \\
&= ( 1_{X^n} W_{X^n}[\pi] ) \times \big( W_{X^n}[\pi^{-1}] Q_{Y^n|X^n} W_{Y^n}[\pi] \big) \\
&= 1_{X^n} \times \big( W_{X^n}[\pi^{-1}] Q_{Y^n|X^n} W_{Y^n}[\pi] \big) \,.
\end{aligned}$$ The equivalence of the two conditions then follows from the fact that $W_{X^n}[\pi^{-1}] W_{X^n}[\pi]$ is the identity channel.
Let $Q_{Y^n|X^n}$ be covariant under permutations. Then from Lemma \[lm:choi-covariance\] we learn that $\tilde{Q}_{X^nY^n}$ is permutation invariant and thus in particular can be written in the form $$\begin{aligned}
\tilde{Q}_{X^nY^n}(x^n,y^n) = \!\! \sum_{\lambda_{XY} \in {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}})} \frac{\tilde{q}_{XY}(\lambda_{XY})}{\# \lambda_{XY}} 1\{ (x^n,y^n) \textrm{ is of type } \lambda_{XY} \} \, , \label{eq:theform}
\end{aligned}$$ where $\tilde{q}_{XY}$ is a probability distribution over joint types $\lambda_{XY}$ and $\#\lambda_{XY}$ denotes the number of sequences of type $\lambda_{XY}$. Moreover, Eq. enforces that for every type $\mu_X \in {\mathcal{T}}_n({\mathcal{X}})$, and any sequence $x^n$ of type $\mu_X$, we have $$\begin{aligned}
1 = \sum_{y^n \in {\mathcal{Y}}^n} \tilde{Q}_{X^nY^n}(x^n,y^n)
&= \sum_{\lambda_{XY} \in {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}}) } \! \frac{\tilde{q}_{XY}(\lambda_{XY})}{\# \lambda_{XY}} \underbrace{ \sum_{y^n \in {\mathcal{Y}}^n} 1\{ (x^n,y^n) \textrm{ is of type } \lambda_{XY} \} }_{ =\ \# \lambda_{Y|X}(x^n) }\,, \label{eq:condition}
\end{aligned}$$ where the number of sequences of type $\lambda_{XY}$ with marginal $x^n$, denoted $\# \lambda_{Y|X}(x^n)$, clearly only depend on the type $\mu_X$ of the marginal. Moreover, if the type of $x^n$ does not correspond to the marginal type $\lambda_X$ of $\lambda_{XY}$ then $\# \lambda_{Y|X}(x^n)$ vanishes. Generally, we have $\# \lambda_{Y|X}(x^n) = 1\{ \mu_x = \lambda_x \} \frac{\# \lambda_{XY} }{ \# \lambda_{X}}$. Hence simplifies to $$\begin{aligned}
\sum_{\substack{\lambda_{XY} \in {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}}) \\ \lambda_X = \mu_X }} \tilde{q}_{XY}(\lambda_{XY}) \frac{1}{\# \lambda_{X}} = 1
\qquad \forall \mu_X \in {\mathcal{T}}_n({\mathcal{X}}) \,. \label{eq:thecondition}
\end{aligned}$$ A direct consequence of this condition is that $\tilde{q}_{XY}(\lambda_{XY}) \leq \# \lambda_X$ for all $\lambda_{XY}$.
Now let us define a universal permutation covariant channel $\tilde{U}_{X^nY^n}$ by the choice $$\begin{aligned}
\tilde{u}_{XY}(\lambda_{XY}) := \frac{\# \lambda_{X}}{ \big| \{ \kappa_{XY} \in {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}}) : \kappa_X = \lambda_X \} \big|}
\end{aligned}$$ which evidently satisfies . Moreover, for all permutation covariant channels with representation $\tilde{Q}_{XY}$ of the form , we have the bound $$\begin{aligned}
\tilde{q}_{XY}(\lambda_{XY}) \leq \# \lambda_X \leq \big| \{ \kappa_{XY} \in {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}}) : \kappa_X = \lambda_X \} \big| \tilde{u}_{XY}(\lambda_{XY}) \leq | {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}}) | \tilde{u}_{XY}(\lambda_{XY}) \,. \end{aligned}$$ Hence also $\tilde{Q}_{XY} \leq | {\mathcal{T}}_n({\mathcal{X}}\times{\mathcal{Y}}) |\, \tilde{U}_{XY}$. The statement of the lemma then follows from the expression in .
The set ${\mathcal{Q}}$ is clearly compact convex. Note that $$\begin{aligned}
g_s(P_{XYZ} \| P_Y \times P_{X|Y} \times Q_{Z|Y} ) = \sum_{y \in {\mathcal{Y}}} P_Y(y) \, g_s (P_{XZ|Y=y} \| P_{X|Y=y} \times Q_{Z|Y=y}) \,.
\end{aligned}$$ From this we can then deduce, as in , that the optimal channel takes on the form $$\begin{aligned}
\hat{Q}_{Z|Y=y}^s(z) \sim P_{Z|Y=y}(z) g_s \big( P_{X|Y=y,Z=z} \big\| T_{X|Y=y} \big)^{\frac1s}, \label{eq:Qopt2}
\end{aligned}$$ for all $y \in {\mathcal{Y}}$. As such, it is clear that Axioms \[ax:convex\]–\[ax:add\] are satisfied.
It remains to verify Axiom \[ax:universal\]. First note that ${\mathcal{Q}}_n$ is closed under symmetrization. Moreover, any channel $Q_{Z^n|Y^n}$ corresponding to a permutation invariant element of ${\mathcal{Q}}_n^{{\textnormal{sym}}}$ satisfies $$\begin{aligned}
Q_{Z^n|Y^n} W_{Z^n}[\pi] = W_{Y^n}[\pi] Q_{Z^n|Y^n}\,, \qquad \forall \pi \in S_n ,\end{aligned}$$ i.e. $Q_{Z^n|Y^n}$ is permutation covariant. Hence, Lemma \[lm:uni-channel\] applies and guarantees the existence of a sequence of universal channels $\{U^n_{Z^n|Y^n}\}_{n\in\mathbb{N}}$ with $U^n_{Z^n|Y^n} \in {\mathcal{Q}}_n^{{\textnormal{sym}}}$ such that $$\begin{aligned}
P_{XY}^{\times n} \times Q_{Z^n|Y^n}(x^n,y^n,z^n) \leq v(n) P_{XY}^{\times n} \times U^n_{Z^n|Y^n}(x^n,y^n,z^n) \qquad \forall x^n \in {\mathcal{X}}^n, y^n \in {\mathcal{Y}}^n, z^n \in {\mathcal{Z}}^n \,.\end{aligned}$$
Proof of Proposition \[pr:examples-four\] {#sec:proof-examples-4}
-----------------------------------------
The proof proceeds similarly to the proofs of Propositions \[pr:examples-two\] and \[pr:examples-three\]. Axiom \[ax:prod\] holds by definition and Axiom \[ax:universal\] can be verified using the universal distributions $$\begin{aligned}
U_{X^nY^nZ^n}^n = U_{Y^n}^n \times U_{X^n|Y^n}^n \times U_{Z^n|Y^n}^n\end{aligned}$$ with the universal distributions $U_{Y^n}^n$ as in and the universal maps $U_{X^n|Y^n}^n$ and $U_{Z^n|Y^n}^n$ provided by Lemma \[lm:uni-channel\].
Axiom \[ax:para\] is verified with a construction analogous to Proposition \[pr:examples-two\] but this time we need to consider the function $(x,y,z) \mapsto x^{1-s}y^{1-s}z^{1-s}$ which is strictly jointly concave for $s \in (\frac23, 1)$ and strictly jointly convex for $s > 1$.Finally, Axiom \[ax:add\] is again verified using the self-consistency relations.
[^1]: $^\dagger$ Graduate School of Mathematics, Nagoya University, Japan (Email: [[email protected]]([email protected]))
[^2]: $^*$ Centre for Quantum Technologies, National University of Singapore (NUS), Singapore
[^3]: $^{\ddag}$ School of Physics, The University of Sydney, Sydney, Australia (Email: [[email protected]]([email protected]))
[^4]: This paper was presented in part at the 2015 International Symposium on Information Theory in Hong Kong, China.
[^5]: Universal quantum channels were also recently introduced in [@fawzirenner14].
[^6]: However, the converse argument is not true because for any two points $\theta$ and $\theta'$ and $\lambda \in (0,1)$, the distribution $Q_{\lambda \theta+ (1-\lambda) \theta'} \in {\cal Q}$ does not necessarily equal the distribution $\lambda Q_{\theta}+ (1-\lambda) Q_{\theta'} \in {\mathcal{P}}({\mathcal{X}})$. That is, the convex combination based on the parametrization is different from the convex combination in ${\mathcal{P}}({\mathcal{X}})$, in general. Such an example will be given in Section \[sec:ex2\].
[^7]: Verdú [@verdu15] recently surveyed Sibson’s definition and pointed out its favorable mathematical properties in the case of general alphabets.
[^8]: For $k \geq 2$ we need to consider the function $f(x_1, x_2, \ldots, x_k, y) = \prod_{i=1}^k x_i^{1-s} y^{1-s}$ and the range of $s$ where this function is jointly concave in all its arguments is further restricted to $s > \frac{k}{k+1}$.
|
---
abstract: 'Quantum spin systems with strong geometric restrictions give rise to rich quantum phases such as valence bond solids and spin liquid states. However, the geometric restrictions often hamper the application of sophisticated numerical approaches. Based on the stochastic series expansion method, we develop an efficient and exact quantum Monte Carlo “sweeping cluster” algorithm which automatically satisfies the geometrical restrictions. Here we use the quantum dimer model as a benchmark to demonstrate the reliability and power of this algorithm. Comparing to existing numerical methods, we can obtain higher accuracy results for a wider parameter region and much more substantial system sizes.'
author:
- Zheng Yan
- Yongzheng Wu
- Chenrong Liu
- 'Olav F. Sylju[å]{}sen'
- Jie Lou
- Yan Chen
title: Sweeping cluster algorithm for quantum spin systems with strong geometric restrictions
---
[^1]
[^2]
introduction
============
Frustrated quantum spin systems display rich quantum phases such as valence bond solids[@VBS], resonating valence bond (RVB) states[@Anderson1987], spin ice[@Bramwell2001], and some novel topological states of matter. However, these systems always hamper numerical approaches: exact diagonalization (ED) is limited to finite cluster, quantum Monte Carlo (QMC) has sign problems, and density matrix renormalization group (DMRG)[@White1992] works only for (quasi) one-dimensional lattices. So it is challenging to study numerically three-dimensional spin liquids and other nontrivial phases on larger lattices. Nonetheless, such exciting quantum phases are also found in models without geometrical frustration but with strong geometric restrictions. For example, there is no spin liquid in the J-Q model[@Lou2009], but it can be in the quantum dimer model (QDM)[@RK1988]. These models are similar, but the QDM has a strong geometric restriction, i.e., there must be only one dimer that belongs to one site. Quantum spin models with geometric restrictions are hard problems even by using sophisticated numerical approaches: it is challenging to do sampling in QMC although it has no sign problem, and it is almost impossible to add blocks in DMRG.
![(a). The mapping between the link basis and local spin basis. Every link corresponds to a spin site, then up spin indicates that there is a dimer, and down spin indicates that there is a link without dimer. (b). Flip a plaquette (the bottom one) affects the properties of its surrounding plaquettes.[]{data-label="fig0"}](Fig1.eps){width="8.6cm"}
Usually, the wave function of QDM is written on the link basis(or dimer basis). If the wave function QDM is expressed on the local spin basis which we are familiar with, i.e., every link corresponds to a spin site, then up spin indicates that there is a dimer, and down spin suggests that there is a link without dimer, as depicted in Fig. \[fig0\](a). In terms of local spin basis, geometric restrictions require that six down spins must surround each up spin on a square lattice. This constraint doesn’t exist in conventional spin models, like the Heisenberg model or more complicated spin models with 4 spins ring exchange[@Melko2005]. On the other hand, in terms of dimer basis, because two plaquettes share a common link, flip a plaquette will affect the properties of its surrounding ones, such as from a flippable plaquette, i.e. plaquette with two parallel dimers, to an unflippable one as shown in Fig. \[fig0\](b). So the update of dimer configuration is not a local effect but a global one. In the classical dimer model, one may use a regular loop update to change dimer configurations as illustrated in Fig. \[ctoq\](a). Connect the thick and thin links into a loop, and flip all the links to get a new configuration which obeys the geometric restrictions. It can be seen that the dimer model is a strongly correlated model. Flipping a link at a location will cause links elsewhere to be flipped to ensure geometric constraints.
The world-line quantum Monte Carlo method maps an n-dimension quantum system into an n+1-D classical system. The +1-D here means the imaginary time dimension. If we want to develop a new QMC method for spin models with constraints such as QDM, its schematic diagram of update must be the same as shown in Fig. \[ctoq\](b): The intersection of all imaginary time update lines and each imaginary time surface must be a classic loop update as the blue loops in this figure. The problem now is how to construct an update method as Fig. \[ctoq\](b) shown following the QMC rules.
In this paper, within the stochastic series expansion (SSE) framework [@Sandvik1991; @Sandvik1999], we develop an efficient QMC algorithm which automatically satisfies the geometric restrictions. In principle, this method works as long as the Hamiltonian does not destroy the geometric constraints. This condition allows us to construct novel quantum states through geometric constraints and study them by QMC. In particular, we use the QDM on square and triangular lattices as examples to elaborate the details of this new algorithm and show that it is efficient by calculating the order parameter on large lattices.
QDMs play an important role as low energy effective descriptions of quantum spin systems [@RK1988; @Misguich2003; @Poilblanc2010]. The Rokhsar-Kivelson (RK) QDM was first introduced to study quantum spin liquids, and in particular, the physics of the short-range RVB state is probably related to high-Tc cuprates [@Anderson1987; @Fazekas1974; @Kivelson1987]. Later it was discovered that QDMs also provide particularly simple realizations of topological phases of matter, including a two-dimensional gapped phase with $Z_2$ topological order [@Moessner2001], and a three-dimensional Coulomb phase described by an emergent $U(1)$ symmetry [@Hermele2004; @Huse2003]. Recently, a QDM for the metallic state of the hole-doped cuprates was also proposed to describe the mysterious pseudogap state at low hole density [@Sachdev2015].
Numerical method
================
The QDM Hamiltonian can be written as $$H=-\sum_{\rm plaq}\left(\vphantom{\sum}|{\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}\rangle\langle{\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}|+\rm{H.c.}\right)
+V\sum_{\rm plaq}\left(\vphantom{\sum}|{\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}\rangle\langle{\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}|+
|{\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}\rangle\langle{\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}|\right)
\label{Hamiltonian}$$ where the summations are taken over all elementary plaquettes of the lattice. A dimer represents an $SU(2)$ singlet bond between two spins located at its endpoints, and the kinetic term describes a resonance between the two dimerization of a plaquette. This seemingly simple Hamiltonian contains strong geometric constraint which requires every site on the lattice to be covered by one and only one dimer.
The SSE method is a generalization of Handscomb’s power series expansion method [@Handscomb] for the isotropic $S=1/2$ Heisenberg ferromagnet and antiferromagnet [@Lyklema1982; @Chakravarty1982; @DLee1984] to a much wider range of systems. The starting point of the SSE method is the power series expansion of the partition function in a particular basis $\{|\alpha \rangle\}$. Generally the $S^z$ basis is chosen for spin systems. For QDMs we choose the dimer basis, and write a dimer basis state as $|\alpha\rangle = |D_1,D_2,\ldots,D_N\rangle$, where $D_i$ takes value 1(0) if there is (not) a dimer on link $i$.
![(a). Classical loop update of classical dimer models. After flipping all the links enclosed by the dashed lines, you can get a new configuration that obeys the geometric constraints. (b). Schematic diagram of an update for quantum dimer models. Each imaginary time surface is a classical dimer configuration. Red lines are update-lines of world-line QMC. The blue loops are the intersection of all imaginary time update lines and each imaginary time surface which are the same as the classical loop in (a). []{data-label="ctoq"}](Fig2.eps){width="8.6cm"}
We write the Hamiltonian in terms of plaquette operators $H_p$, $H=-\sum_{p=1}^{N_p}H_p$, where $p$ labels a specific plaquette on the lattice. The plaquette operators are further decomposed into two operators: $H_p = H_{1,p} + H_{2,p}$, where $H_{1,p}$ is diagonal and $H_{2,p}$ is off-diagonal: $$\begin{aligned}
H_{1,p} & = &
-V \left( \vphantom{\sum} | {\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}\rangle \langle {\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}|+| {\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}\rangle \langle {\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}|\right) + V + C,
\label{hb1} \\
H_{2,p} & = & \left( \vphantom{\sum} | {\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}\rangle \langle {\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}| + | {\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}\rangle \langle {\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}| \right).
\label{hb2}\end{aligned}$$ In this Hamiltonian, we have subtracted a constant $N_p(V+C)$ from Eq. (\[Hamiltonian\]), which should be kept in mind when calculating the energy. We do this because the constant $V+C$ makes all matrix elements of $H_{1,p}$ positive provided $C> {\rm min}(-V,0)$. We will choose $C=1$ here for simplicity.
The powers of $H$ in the series expansion of the partition function $Z$ can be expressed as sums of products of the plaquette operators (\[hb1\]) and (\[hb2\]). Such a product is conveniently referred to by an operator-index sequence: $S_n = [a_1,p_1],[a_2,p_2],\ldots,[a_n,p_n]$, where $a_i \in \lbrace 1,2\rbrace$ corresponds to the type of operator ($1$=diagonal, $2$=off-diagonal) and $p_i \in \lbrace
1,\ldots,N_p\rbrace$ is the plaquette index. It is also convenient to work with a fixed-length operator-index list with $M$ entries and to include the identity operator $[0,0]$ as one of the operator types.
The expanded partition function takes then the same form as that for the spin models [@Sandvik1991; @Sandvik1999], $$Z = \sum\limits_\alpha \sum_{S_M} {\beta^n(M-n)! \over M!}
\left \langle \alpha \left | \prod_{i=1}^M H_{a_i,p_i}
\right | \alpha \right \rangle ,
\label{zm}$$ where $n$ is the number of operators $[a_i,p_i] \not= [0,0]$. By inserting complete sets of states between all the plaquette operators, the product can be written as a product of the following non-zero plaquette matrix elements $$\begin{aligned}
\nonumber
&\langle{\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}| H_{1,p} | {\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}\rangle =
\langle{\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}| H_{1,p} | {\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}\rangle = 1,\\
&\langle{\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}| H_{2,p} | {\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}\rangle =
\langle{\hbox{\vbox{{{\hrule height0.05cm width0.2cm depth0pt}}\vskip 0.1cm {{\hrule height0.05cm width0.2cm depth0pt}}}}}| H_{2,p} | {\hbox{{{\vrule height0.2cm width0.05cm depth0pt}}\hskip 0.1cm {{\vrule height0.2cm width0.05cm depth0pt}}}}\rangle = 1,\\
\nonumber
&\langle {\rm others} | H_{1,p} | {\rm others} \rangle = 1+V,
\label{matrelem}\end{aligned}$$ the $| {\rm others}\rangle$ here means that plaquette $p$ has 1 or 0 dimer. Such matrix elements are depicted in Fig. \[vertex\] where the plaquette below(above) is the ket(bra).
![Some of the vertices and their update prescriptions. The horizontal bar represents the full plaquette operator $H_p$ and the lines of the squares represent the dimer states (thick and thin lines for dimer 1 or 0) on either side of the operator. Update-lines are shown as lines with an arrow. (c) and (d) are different updates of the same configuration.[]{data-label="vertex"}](Fig3.eps){width="8.6cm"}
In the Monte Carlo sampling of the partition function we insert or delete a diagonal operator in the operator-index sequence just like the diagonal update for spins models: We accept the insertion/deletion according to the Metropolis acceptance probabilities, $$\begin{aligned}
P_{\rm ins} & = &
{N_p\beta \langle \alpha| H_{1,p} | \alpha \rangle \over M-n },
\label{diap1} \\
P_{\rm del} & = &
{M-n+1 \over N_p\beta \langle \alpha| H_{1,p} | \alpha\rangle }.
\label{diap2}\end{aligned}$$ The presence of $N_p$ in these probabilities reflects the fact that there are $N_p$ random choices for the plaquette $p$ in converting $[0,0]\to [1,p]$, but only one way to replace $[1,p]\to [0,0]$ when $p$ is given. These diagonal updates are attempted consecutively for all $1,\ldots,M$, and at the same time the state $|\alpha \rangle$ is updated when plaquette flipping operators $[2,p]$ are encountered.
Cluster(loop) updates [@Sandvik1999; @OFS2002] can accomplish substitutions $[1,p] \leftrightarrow [2,p]$ in the standard scheme applied to spin models. There are several kinds of cluster-update schemes: operator loop[@Sandvik1999], directed loop[@OFS2002], cluster-like loop[@Sandvik2003] and others to solve different models. However, due to the geometric restrictions of the QDM, regular cluster updates cannot be applied. The main result described below is a new kind of cluster update obeying imaginary time order to change operators more efficiently. We call it the “sweeping cluster” method. It works as follows.
First, choose a starting operator vertex randomly with flippable plaquettes(FPs) on both sides, either diagonal or off-diagonal. FP means that the plaquette contains two parallel dimers. Next, create a cluster of four update-lines, one for every link of the plaquette, each emanating from the starting vertex in the positive imaginary-time direction. The update-lines serve as guiding lines in the imaginary-time direction on where to change the configuration: The dimer at the end of each update-line is toggled on/off in the proposed new configuration as they sweep simultaneously upwards in imaginary-time. Thus the four initial update-lines rotate the two dimers of the original FP as they go along£¿. The update-lines are extended until they meet another operator vertex from below. Then, after updating the plaquette beneath on the new operator vertex according to the update-lines, we need to decide how to create or destroy update-lines to update the plaquette above and continue sweeping, see Fig. \[vertex\].
{width="17cm"}
For this, there are three different processes to consider: (1) The new plaquette beneath is an FP, and the old plaquette above is not an FP. We can then change the plaquette above into an FP in two ways: either the resulting vertex will become diagonal or off-diagonal. We choose between these two possibilities shown in (c) and (d) in Fig. \[vertex\] with probability $1/2$. (2) The new plaquette beneath is not an FP. Then the change of the upper plaquette is equivalent to the change of one underneath, as shown in (a), (b), (e) and (f) in Fig. \[vertex\] and the operator should be diagonal. (3) Both the new plaquette beneath and the old plaquette above are FPs. Then there are two choices: the cluster-update ends if the number of total lines is four. If not, the four update-lines continue through the vertex and sweep on. The reason that we keep the operator unaltered in the latter case is to keep a detailed balance regarding its reversed process.
As an example, we draw Fig. \[cluster\] where (a) and (b) are the configurations before/after cluster update. Compare the dimer configuration between (a) and (b) at a certain imaginary time, and it returns to loop update in the classical dimer model, i.e., every link passed by loop has to be flipped.
At the end of the sweeping cluster update, when the last four update-lines are deleted, we get a new configuration B with weight $ W_{B} $ to replace the old configuration A with weight $ W_{A} $. To ensure detailed balance, we must invoke a Metropolis accept/reject step[@Metropolis] on the whole cluster update with an acceptance probability $$P_{accept}(A\rightarrow B) = \min( {W(B)P_{\rm select}(B\rightarrow A)\over W(A)P_{\rm select}(A\rightarrow B)}, 1 ),
\label{select-accept}$$ where $P_{\rm select}(A \rightarrow B)$ is the probability for the sweeping cluster update to change configuration A into B. This step involves both the random choice of starting vertex and the random choices in update type (1). If we denote the number of operator vertices in configuration A with FPs on both sides by $N_{\rm FP}$, and the same amount in configuration B by $N_{\rm FP}+\Delta$, then $$P_{accept}(A\rightarrow B) = \min( {\frac}{N_{\rm FP}}{N_{\rm FP}+\Delta} \left( {\frac}{2}{1+V} \right)^\Delta, 1 ).
\label{select-accept2}$$
At low temperature, the first term ${\frac}{N_{\rm FP}}{N_{\rm FP}+\Delta}\approx 1$. At RK point, any new configuration can be accepted. That’s because the wave function of the RK point is an equal weight overlap of all configurations.
Results
=======
To demonstrate the potential of our new method, we first show its efficiency. All the following results were obtained under the condition of $T=0.01$. If we want to solve QDM by the old world-line QMC scheme, we can only use “pair update” which means flipping two FPs face to face [@Sandvik1991]. This update technique is neither ergodic nor efficient, as can be seen from Fig. \[edqmc\], which shows how much the “pair update” and our cluster update deviate from ED for the same number of Monte Carlo steps. Our algorithm matches the ED results much better than the “pair update” does. This is because the “pair update” only changes a few operators which give long autocorrelation times resulting in statistical errors that are smaller than the real error.
It is also important to check ergodicity (in a certain winding sector) of the method by tracking the movement of the columnar order parameter as defined in Ref. [@Sachdev1989], $$\begin{split}
\Psi_{col} & = {\frac}{1}{L^2} \sum_{\bf r} \left\{
(-1)^{r_x} [n({\bf r}+{{\bf x}\over 2})-n({\bf r}-{{\bf x}\over 2})]+ \right.\\
&\left. i(-1)^{r_y}[n({\bf r}+{{\bf y}\over 2})-n({\bf r}-{{\bf y}\over 2})] \right\},
\label{ficol}
\end{split}$$ where [**x**]{} and [**y**]{} are unit vectors and $L$ is the linear system size. The dimer number operator n([**r**]{}+[**e**]{}/2) is 1 if the site at [**r**]{} and its nearest neighbor at [**r**]{}+[**e**]{} form a dimer, and zero otherwise. As depicted in the inset of Fig. \[edqmc\], the evolution of $\Psi_{col}$ in a complex plane is circularly distributed even far from the RK point, here we choose $V=0.5$.
![Correctness and ergodicity(in a certain winding sector) check: The energy difference between ED and QMC with two distinct updates, pair update and cluster update, on triangle lattice. Inset: Evolution of $\Psi_{col}$ in a complex plane at $V=0.5$ of 16$\times$16 square lattices by serial computing.[]{data-label="edqmc"}](Fig5.eps){width="8.6cm"}
On the triangular lattice, there is a novel phase called $\sqrt{12} \times \sqrt{12}$ phase between columnar phase and RVB phase of quantum dimer model [@Ralko2005TR]. By employing our algorithm, we calculate the dimer correlation function as Eq.(\[correlationfunction\]) of QDM on triangular lattice and obtain this phase as shown in Fig. \[triangular\]. Red bonds in this figure corresponds to dimers and blue ones mean no dimer. We can clearly observe the periodic $\sqrt{12} \times \sqrt{12}$ structure unit encircled with the black dashed line. According to the principle of Monte Carlo method, the computational complexity of this algorithm is the same order of magnitude on different lattices, because the units are rotated plaquettes.
Hereby we define the dimer correlation function as $$C_{ij}={{\langle n_in_j \rangle - \langle n_i \rangle \langle n_j \rangle}\over{ \langle n_in_i \rangle- \langle n_i \rangle \langle n_i \rangle}},
\label{correlationfunction}$$ $n_i=1(0)$ means link i has a(no) dimer. Furthermore, to verify the accuracy of our algorithm, we also reproduce high precision results for the dimer correlation functions on an $8 \times 8$ square lattice given in Ref. [@Leung1996] which is obtained by ED method. As depicted in Fig. \[correfunction\], we don’t label the error bar since our results are within 0.1 percent difference comparing with the ED results.
![The dimer correlation function of QDM on 12$\times$12 triangle lattice at $V=0.5$. We can see a $\sqrt{12} \times \sqrt{12}$ phase clearly. The dashed line helps us to capture the periodic structure.[]{data-label="triangular"}](Fig6.eps){width="8cm"}
![The correlation function of QDM on $8\times 8$ square lattice at $V=0$. Red bond (positive number) means dimer strength, blue one (negative number) means no-dimer strength.[]{data-label="correfunction"}](Fig7.eps){width="7.5cm"}
Having established its numerical efficiency and accuracy, we use the method to obtain high-precision results for the QDM. The averaged modulus of the columnar order parameter, $\chi_{col} = \sqrt{\langle | \Psi_{col} |^2 \rangle}$, as a function of $V$ is shown in Fig. \[order\] for different lattices sizes. The error bars are smaller than the size of symbol. If long-range columnar order exists, $\chi_{col}$ remains finite as $L \to \infty$. From Fig. \[order\] it is seen that $\chi_{col}$ decreases as $L$ gets larger. However, as shown in the inset of Fig. \[order\], an extrapolation carried out for the special value $V=0$, including the results for larger systems up to $L=160$, indicates that $\chi_{col}$ may converge to a finite value for $L \to \infty$.
![The columnar order parameter as a function of $V$ on $L\times L$ square lattices with $L=8,16,32,64$. Inset shows finite size extrapolation at $V=0$ including also data for $L=128$ and $160$.[]{data-label="order"}](Fig8.eps){width="8.4cm"}
Our new method presented here allows the study of QDM on large lattices at finite temperatures. This method is in contrast to zero temperature projector Monte Carlo methods that have only been applied to QDMs of smaller system sizes than used here to keep the statistical errors under control [@OFS2005; @Trivedi1989; @Baroni1999]. Other quantum cluster algorithm for Ising model with restrictions [@Banerjee2014; @Schlittler2015] can be applicable only on specific lattices and certain parameter regions. Another drawback with these methods is that one must “throw away” configurations which don’t obey the geometric restrictions. This ratio may be as high as $3/4$ [@Banerjee2014].\
Conclusions and Outlook
=======================
Numerical study of the quantum spin model with strong geometric restrictions is important and notoriously difficult. We have introduced the sweeping cluster SSE method to calculate them. The technique keeps the geometric configuration satisfied by sweeping vertices in imaginary-time order. It is the first finite temperature QMC method for QDMs that samples the dimer space directly, which provides a positive all-around solution to this hard problem. The algorithm is valid and efficient for the whole parameter region of QDMs in principle. It works on any lattice geometries and can be generalized to other models such as quantum loop model [@QLM]. Furthermore, all existing numerical algorithms for quantum dimer model can only do sampling in the same winding sector. We have made progress on realizing the sampling of all winding sectors based on our ¡°sweeping cluster¡± algorithm. Besides, our algorithm is a world-line algorithm. This method provides us with access to the (imaginary-time) dynamic behavior of the quantum dimer model and other spin models with strong geometrical restrictions.
Acknowledgements
================
We wish to thank T. K. Lee and Wenan Guo for fruitful discussions. ZY acknowledges the support of Nordic Centre. This work was supported by the State Key Programs of China (Grant Nos. 2017YFA0304204 and 2016YFA0300504), the National Natural Science Foundation of China (Grant Nos. 11625416, and 11474064).
[31]{} A. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. **59**, 799 (1987). P. W. Anderson, Science **235**, 1196 (1987). S. T. Bramwell and M. J. P. Gingras, Science **294**, 1495 (2001). S. R. White, Phys. Rev. Lett. **69**, 2863 (1992); Phys. Rev. B **48**, 10345 (1993). J. Lou, A. W. Sandvik, and N. Kawashima, Phys. Rev. B **80**, 180414 (2009); A. W. Sandvik, Phys. Rev. Lett. **98**, 227202 (2007). D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. **61**, 2376 (1988). R. G. Melko and A. W. Sandvik, Phys. Rev. E **72**, 026702 (2005). A. W. Sandvik and J. Kurkij[ä]{}rvi, Phys. Rev. B **43**, 5950 (1991). A. W. Sandvik, Phys. Rev. B **59**, R14157 (1999). G. Misguich, D. Serban, and V. Pasquier, Phys. Rev. B **67**, 214413 (2003). D. Poilblanc, M. Mambrini, and D. Schwandt, Phys. Rev. B **81**, 180402 (2010). P. Fazekas and P. Anderson, Phil. Mag. **30**, 423 (1974). S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Phys. Rev. B **35**, 8865 (1987). R. Moessner and S. L. Sondhi, Phys. Rev. Lett. **86**, 1881 (2001). M. Hermele, M. P. A. Fisher, and L. Balents, Phys. Rev. B **69**, 064404 (2004). D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. **91**, 167004 (2003). M. Punk, A. Allais, and S. Sachdev, Proc. Natl. Acad. Sci. U.S.A. **112**, 9552 (2015).
D. C. Handscomb, Proc. Cambridge Philos. Soc. **58**, 594 (1962). J. W. Lyklema, Phys. Rev. Lett. **49**, 88 (1982). S. Chakravarty and D. B. Stein, Phys. Rev. Lett. **49**, 582 (1982). D. H. Lee, J. D. Joannopoulos, and J. W. Negele, Phys. Rev. B **30**, 1599 (1984). O. F. Sylju[å]{}sen and A. W. Sandvik, Phys. Rev. E **66**, 046701 (2002). A. W. Sandvik, Phys. Rev. E **68**, 056701 (2003). N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. **21**, 1087 (1953). S. Sachdev, Phys. Rev. B **40**, 5204 (1989). A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys. Rev. B 71, 224109 (2005). P. W. Leung, K. C. Chiu, and K. J. Runge, Phys. Rev. B **54**, 12938 (1996).
O. F. Sylju[å]{}sen, Phys. Rev. B **71**, 020401 (2005). N. Trivedi and D. Ceperley, Phys. Rev. B **40**, 2737 (1989). S. Baroni and S. Moroni, Phys. Rev. Lett. **82**, 4745 (1999). D. Banerjee, M. B[ö]{}gli, C. Hofmann, F.-J. Jiang, P. Widmer, and U.-J. Wiese, Phys. Rev. B **90**, 245143 (2014); Phys. Rev. B **94**, 115120 (2016). T. M. Schlittler, T. Barthel, G. Misguich, J. Vidal, and R. Mosseri, Phys. Rev. Lett. **115**, 217202 (2015). F. Pollmann, J. J. Betouras, K. Shtengel, and P. Fulde, Phys. Rev. Lett. **97**, 170407 (2006).
[^1]: Corresponding author
[^2]: Corresponding author
|
---
abstract: 'We sharpen the construction of representation space in the paper “Principal Series Representations of Infinite Dimensional Lie Groups II: Construction of Induced Representations”. We show that the principal series representation spaces constructed there, are completions of spaces of sections of Hilbert bundles rather than completions of quotient spaces of sections.'
address:
- 'Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803'
- 'Department of Mathematics, University of California, Berkeley, CA 94720–3840'
author:
- 'Gestur '' Olafsson'
- 'Joseph A. Wolf'
date: '6 October, 2012'
title: Separating Vector Bundle Sections by Invariant Means
---
[^1]
[^2]
This note is a continuation of [@W], using the same notation. We sharpen the construction of the representation spaces in [@W §§5B, 5C] by proving that the bounded right uniformly continuous sections of a homogeneous Hilbert space bundle $\E_\tau \to G/H$ (defined by a unitary representation $\tau$ of $H$) are separated by means on $G/H$.
**General Setting**
$G$ is a topological group, not necessarily locally compact, and $H$ is a closed amenable subgroup. $\tau$ is a unitary representation of $H$, say on $E_\tau$, and $\E_\tau \to G/H$ is the associated homogeneous Hilbert space bundle. The space $RUC_b(G/H;\E_\tau)$ of bounded right uniformly continuous bounded sections of $\E_\tau \to G/H$ consists of the right uniformly continuous bounded functions $f: G \to E_\tau$ such that $f(xh) = \tau(h)^{-1}f(x)$ for $x \in G$ and $h \in H$. $G$ acts on it by $(\pi_\tau(x)f)(x') = f(x^{-1}x')$. Since $\tau$ is unitary the pointwise norm $||f(xH)||$ is defined. If $\mu$ is a mean on $G/H$ we then have a seminorm on $RUC_b(G/H;\E_\tau)$ defined by $\nu_\mu(f) = \mu(||f||)$. We denote the space of all means on $G/H$ by ${\mathcal{M}}= {\mathcal{M}}(G/H)$.
We use properties of means and amenability from [@Day1], [@Day2] and [@R].
\[5B\] If $0 \ne f \in RUC_b(G/H;\E_\tau)$ then there exists $\mu \in {\mathcal{M}}=
{\mathcal{M}}(G/H)$ such that $\nu_\mu(f) \ne 0$. In other words, in [@W Prop. 5.13 and Cor. 5.14], $\Gamma_{{\mathcal{M}}}(G/H;\E_\tau)$ is the locally convex TVS completion of $RUC_b(G/H;\E_\tau)$.
Let $f \in RUC_b(G/H;\E_\tau)$ be annihilated by all the seminorms $\nu_\mu$, $\mu \in {\mathcal{M}}$. Suppose that $f$ is not identically zero and choose $x \in G/H$ with $f(x) \ne 0$. WE can scale and assume $||f(x)|| = 1$. Evaluation $\delta_x(\varphi) = \varphi(x)$ is a mean on $G$ and $\delta_x(||f||) = 1$. Now the compact convex set $S = \{\sigma \in {\mathcal{M}}(G) \mid \sigma(||f||) = 1\}$ (weak$^*$ topology) is nonempty. Since $H$ is amenable it has a fixed point $\mu_f$ on $S$. Now $\mu_f$ is a mean on $G/H$ and the seminorm $\nu_{\mu_f}(f) = 1$.
**Principal Series**
We specialize Proposition \[5B\] to our setting where $G$ is a real Lie group, e.g. $Sp(\infty;\R)$, and $P$ is a minimal self–normalizing parabolic subgroup. Then the amenably induced representations $\Ind_P^G(\tau)$ of $G$ on the $\Gamma_{{\mathcal{M}}}(G/P;\E_\tau)$, in other words the general principal series representations of $G$, do not require passage to quotient spaces of the $RUC_b(G/P;\E_\tau)$. Further, the argument of [@W Proposition 5.16], that $\Ind_P^G(\tau)|_K = \Ind_M^K$ when the parabolic $P$ is flag-closed, is simplified because we need not compare quotient structures.
**Other Completions**
Here is a Fr' echet space completion of $RUC_b(G/P;\E_\tau)$. Note that $G = \varinjlim G_n$ where the $G_n$ are real reductive groups defined over the rational number field $\Q$ in a consistent way. So we have the rational group $G_\Q := \varinjlim G_{n,\Q}$. The point is that the $G_{n,\Q}$ are countable, so $G_\Q$ is countable, and the evaluations form a countable family $\{\delta_{xP} \mid x \in G_\Q\}$ of means on $G/P$. If $f \in RUC_b(G/P;\E_\tau)$ and $||f||$ is annihilated by each of the “rational” seminorms $\nu_{\delta_{xP}}$, the argument of Proposition \[5B\] shows that $f = 0$. The locally convex TVS structure of $RUC_b(G/P;\E_\tau)$, using only that countable family of seminorms, defines a Fr' echet space completion of $RUC_b(G/P;\E_\tau)$. The action of $G_\Q$ extends by continuity to this completion of $RUC_b(G/P;\E_\tau)$, but it is not clear whether the the action of $G$ extends.
We enumerate $G_\Q$ by the positive integers to define a mean $\mu = \sum_{m \geq 0} 2^{-m} \delta_{xP}^m$ on $G$. The corresponding seminorm $\nu_\mu(f) = \sum_{m \geq 1} 2^{-m} ||f(x_m)||$ is a norm on $RUC_b(G/P;\E_\tau)$. It defines a pre Hilbert space structure on $RUC_b(G/P;\E_\tau)$ by $\langle f, h \rangle = \sum_{m \geq 1} 2^{-m} \langle f(x_m), h(x_m) \rangle$. Again, the action of $G$ on $RUC_b(G/P;\E_\tau)$ does not appear to extend by continuity to the corresponding Hilbert space completion.
[X]{}
D. Beltiţă, Functional analytic background for a theory of infinite–dimensional Lie groups, in “Developments and Trends in Infinite Dimensional Lie Theory”, ed. K.-H. Neeb & A. Pianzola, Birkh" auser Progress in Math [**288**]{} (2011), 367–392.
M. M. Day, Amenable semigroup, Illinois J. Math. [**1**]{} (1957), 509–544.
M. M. Day, Fixed point theorems for compact convex sets, Illinois J. Math. [**5**]{} (1961), 585–590. Correction: Illinois J. Math. [**8**]{} (1964), 713.
N. Rickert, Amenable groups and groups with the fixed point property, Trans. Amer. Math. Soc. [**127**]{} (1967) 221-–232.
J. A. Wolf, Principal series representations of infinite dimensional Lie groups, II: Construction of induced representations. This volume.
[^1]: G' O: Research partially supported by NSF Grant DMS-1101337
[^2]: JAW: Research partially supported by the Simons Foundation
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.