Random sorting networks: local statistics via random matrix laws
Vadim Gorin Mustazee Rahman



This paper finds the bulk local limit of the swap process of uniformly random
sorting networks. The limit object is defined through a deterministic procedure, a
local version of the Edelman-Greene algorithm, applied to a two dimensional
determinantal point process with explicit kernel. The latter describes the
asymptotic joint law near 0 of the eigenvalues of the corners in the antisymmetric
Gaussian Unitary Ensemble. In particular, the limiting law of the first time a given
swap appears in a random sorting network is identified with the limiting
distribution of the closest to 0 eigenvalue in the antisymmetric GUE. Moreover,
the asymptotic gap, in the bulk, between appearances of a given swap is the
Gaudin-Mehta law – the limiting universal distribution for gaps between eigenvalues of
real symmetric random matrices.

The proofs rely on the determinantal structure and a double contour integral
representation for the kernel of random Poissonized Young tableaux of arbitrary shape.



 Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients
    
Ling Wan

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Tao WangCorresponding author. Email addresses: ling.wan@whu.edu.cn (L. Wan), tao.wang@whu.edu.cn (T. Wang).

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

DICATAM, Mathematical Division, University of Brescia, Via Valotti 9, 25133 Brescia, Italy

    
=================================================================================================================================================================================================================================================================================================================================================================================





§ INTRODUCTION
 



 §.§ Overview


The main object of this article is the uniformly random sorting network, as
introduced by Angel, Holroyd, Romik, and Virág in <cit.>. Let 𝔖_n denote
the symmetric group and τ_i denote the transposition between i and i+1 for
1 ≤ i ≤ n-1. The τ_i are called adjacent swaps. Let
rev = n, n-1, …, 1 denote the reverse permutation of 𝔖_n.
A sorting network of 𝔖_n is a sequence of permutations σ_0 = id,
σ_1, …, σ_N = rev of shortest length with the property that for
every k, 
    σ_k+1 = σ_k ∘τ_i   for some  i.

In other words, the permutations change by swapping adjacent labels at each step and must go from the
identity to the reverse in the shortest number of swaps. The number of adjacent
swaps required in any sorting network of 𝔖_n is n2. See Figure
<ref> for an example of a sorting network in the wiring diagram
representation. We identify a sorting network of 𝔖_n by its sequence of swaps

    (s_1, …, s_n2),

where s_i denotes the adjacent swap (s_i, s_i+1).

A random sorting network of 𝔖_n is a sorting network of 𝔖_n chosen
uniformly at random. Computer simulations were used to conjecture many beautiful asymptotic
properties of random sorting networks. See <cit.> for an account of these statements and the first
rigorous results, and also <cit.> for other
asymptotic theorems. The proofs of the conjectures from <cit.> were recently announced in <cit.>.



In many examples, random combinatorial structures built out of symmetric groups
are known to exhibit the same asymptotic behavior as random matrices. The most
famous result of this sort due to Baik–Deift–Johansson <cit.> identifies the
fluctuations of longest increasing subsequences of random permutations with the
fluctuations of largest eigenvalue of random Hermitian matrices. Its further
upgrades, <cit.> link fluctuations of
several first rows of the Young diagram distributed according to the Plancherel
measure for symmetric groups to those of several largest eigenvalues. A connection
also exists for “bulk” (i.e. not largest) rows and eigenvalues, see <cit.>.

On the other hand, up to now no such connections were known for random sorting
networks. In the present article we find such a connection. It exists for a sort of local
limit of random sorting networks. Indeed, we find the bulk local limit of random sorting
networks, by proving that it is given by a simple, local, deterministic algorithm
(the local Edelman-Greene algorithm) applied to a specific random point process on
×_≥ 0. In turn, we describe this point process by showing
that its correlation functions have determinantal form and provide explicit
expressions for the corresponding correlation kernel. The very same point process
appeared in the work of Forrester–Nordenstam <cit.> (see also Deffoseux
<cit.>) as the hard edge limit of antisymmetric GUE–corners process; it
describes the asymptotic distribution of the closest to 0 eigenvalues of the corners
of large antisymmetric matrix with i.i.d. (modulo symmetry) Gaussian entries of
mean 0.

A corollary of our theorem is that the rescaled, asymptotic distribution of the first time when the
swap between ⌊n(1+α)/2⌋ and ⌊n(1+α)/2⌋ + 1
appears, for α∈ (-1,1), is the same as the rescaled, asymptotic distribution of the closest to 0
eigenvalue of an antisymmetric-GUE random matrix. Another corollary is that within the bulk,
the asymptotic gap between appearances of the aforementioned swap is described by the
Gaudin–Mehta law — the asymptotic universal distribution of the gap between eigenvalues of
real symmetric random matrices in the bulk. Complete statements are given in the next section.

In an independent and parallel work, Angel, Dauvergne, Holroyd, and Virág <cit.>
also study the bulk local limit of random sorting networks. Their approach is very
different from ours. We deduce explicit formulas for the prelimit local structure of
random sorting networks, and then analyze the asymptotic of these formulas in the
spirit of Integrable Probability, see <cit.> and also <cit.>.
On the other hand, <cit.> argue probabilistically, analyzing a Markov chain (whose
transition probabilities are expressed through the hook formula for dimensions) for
sampling random Young tableaux. The connection to random matrices remains invisible
in the results of <cit.>. It would be interesting to match these two approaches,
but it has not been done so far.




 §.§ Bulk limit of random sorting networks.


We now describe our main result. Informally, we study
the asymptotics of the point process (s_i,i), i=1,…,n2, in a window
of finite height and order n width, so that the number of points in the window
remains finite; see Figure <ref>. Here (s_1,…,s_n2) are
swaps of a random sorting network of 𝔖_n.

In <cit.> it is proven that the point process (s_i,i) is stationary with respect to
the second coordinate. Therefore, it suffices to study windows adjacent to 0
in second coordinate, which we do.

The limiting object S_local is a point process on ×_≥ 0
defined by a two–step procedure. First, we introduce an auxiliary point
process _edge on ×_≥ 0 through its correlation functions.

 
_edge is the (unique) determinantal point process on ×_≥ 0
with correlation kernel
 
    K_edge(x_1,u_1;x_2,u_2) =
    2/π∫_0^1 t^x_2-x_1cos( tu_1 + π/2 x_1) cos( tu_2 + π/2x_2  ) dt,    if   x_2 ≥ x_1; 
       
     - 2/π∫_1^∞ t^x_2-x_1cos(
    tu_1 + π/2 x_1) cos( tu_2 + π/2x_2  ) dt,    if   x_2 < x_1.


We refer to <cit.> and Section <ref> for more detailed discussions of
determinantal point processes. We note that the particles of _ edge on
adjacent lines {x}×_≥ 0 and {x+1}×_≥ 0 almost surely
interlace, see Figure <ref>. The point process _edge has appeared in
the random matrix literature before in <cit.>, <cit.>. In more details, let G be an
infinite random matrix with rows and columns indexed by _> 0, and whose entries are
independent and identically distributed, real-valued, standard Gaussians.
Let A = G - G^T/√(2). The top–left m× m corner of
A almost surely has 2 ⌊ m/2⌋ non-zero eigenvalues of the form

    ±𝐢_1^m,  ±𝐢_2^m ,  …,  ±𝐢_⌊ m/2 ⌋^m,

where 0 < _1^m < _2^m < ⋯ < _⌊ m/2 ⌋^m. Forrester and Nordenstam prove
that _ edge is the weak limit of the point process
{(j,√(2M)_i^2M+j)}⊂ℤ×ℝ_≥ 0, i∈ℤ_> 0, j∈ℤ, as M →∞.

Particle configurations of S_local are obtained from _edge by a
deterministic procedure, which is a local version of the well-known Edelman–Greene
bijection <cit.> between staircase shaped tableaux and sorting networks.
In the following we describe this procedure. A rigorous definition of the procedure
utilizes properties of _edge that are not immediate from Definition <ref>.
We provide the rigorous construction in Section <ref> where the description is given
in the language of Young tableau, which is the more standard setup for defining the Edelman-Greene
bijection.





  
Local Edelman-Greene algorithm
Fix a configuration X of _ edge and suppose that we
want to define the positions of all particles of S_local inside the
rectangle [a,b]× [0,T] with a<0<b. Then almost surely there are two integers
â<2a and b̂>2b such that X has no particles on the segments
â× [0,T] and b̂× [0,T]. The particles of X outside
[â,b̂]×[0,T] are further ignored.

We now define a particle configuration Y – the restriction of  S_local
onto [â,b̂]×[0,T] – through an iterative procedure. Start by declaring
Y=∅, and setting X̂ to be the restriction of X onto
[â,b̂]×[0,T]. Repeat the following until X̂ is empty:


 
  *  Let (x,u) be an element of X̂ with smallest second coordinate.
 The parity of x will be even. Add (x/2,u) to Y, i.e., redefine Y:=Y∪{(x/2,u)}.
 
  * Define the sliding path (x_1,u_1), (x_2,u_2), … as a unique collection of
 points in X̂ (of maximal length) such that
  
   
  * (x_1,u_1)=(x,u),
   
  * u_1<u_2<⋯<u_k and |x_i-x_i+1|=1 for i=1,…,k-1,
   
  * For each i=1,…,k-1, the only points of X̂ in the rectangle
   [x_i-1,x_i+1]× [u_i,u_i+1] are (x_i,u_i) and (x_i+1,u_i+1).
  
  In other words, (x_i+1,u_i+1) is the point in [x_i-1,x_i+1] × (u_i,T], which
  is closest to (x_i,u_i). See Figure <ref> for an illustration.
 
  * Remove the k points (x_1,u_1),…, (x_k,u_k) from X̂ and
 replace them by k-1 points (x_1,u_2), (x_2,u_3), …, (x_k-1,u_k).
 
  * Go back to Step (<ref>), unless X̂ is empty.

The first coordinates of the particles of Y will be integral; this follows from
the interlacing property of the particles of X̂, which is preserved throughout
the steps of the procedure.

One immediate property of the just defined map _edge↦
S_local is that the position of the first particle of S_local
in the ray {a}×_≥ 0 almost surely coincides with the position
of the first particle of _edge in the ray {2a}×_≥ 0.
Therefore, the joint law of the positions of the first particles of S_local
in the rays {a_i}×_≥ 0, for i=1,…,k, can be explicitly evaluated as a
Fredholm determinant. See Corollary <ref> for the case k=1 and
<cit.> for general statements.

We also show that S_local is invariant under translations and reflections
of the first (–valued) coordinate, ergodic with respect to translations of the
first coordinate, and stationary in the second (_≥ 0–valued)
coordinate; see Proposition <ref>.

We are ready to formulate the main result.
 
Fix α∈ (-1,1), and let s_1,s_2,…,s_n 2 be swaps of
a random sorting network of 𝔖_n. Define the point process S_α,n of
rescaled swaps near the point (n(α+1)/2,0) through

    S_α,n={(s_i-⌊n(α+1)/2⌋ ,
    √(1-α^2)·2i/n)}_i=1^n 2.

Then as n→∞, the point process S_α,n converges weakly to S_ local.

It is proven in <cit.> that the global scaling limit of the
space-time swap process of random sorting networks is the product of the semicircle
law and Lebesgue measure. The √(1-α^2) scaling of Theorem
<ref> is consistent with the semicircle result.

We emphasize that Theorem <ref> states both that S_α,n
converges and the limit is obtained by applying the localized Edelman-Greene
algorithm to _edge. Theorem <ref> does not cover the
case |α|=1, where the asymptotic behavior changes. It is plausible that the
methods of the present article can be adapted to this remaining case, but we
not address it here; see <cit.> for another approach to |α|=1 case.

Theorem <ref> implies that the first
swap times in random sorting networks converge to a one–dimensional
marginal of S_ local; the distribution of the latter can be expressed as a
Fredholm determinant. Figure <ref> shows the approximate
sample distribution of the rescaled first swap time and (<ref>) shows the tail asymptotics.
.
Let _FS, α, n be the first time the swap interchanging ⌊n(α+1)/2⌋ with ⌊n(α+1)/2⌋+1 appears in a random sorting network of 𝔖_n. The following
convergence in law holds:

    lim_n→∞ 2 √(1-α^2)/n _FS, α, n = _FS,

where

    _FS > t   = 1 + ∑_k=1^∞(-1)^k/k!∫_[0,t]^k[ K_edge(u_i,u_j)]   du_1 ⋯ du_k,
     and
     K_edge(u_1,u_2)    = sin(u_1-u_2)/π (u_1-u_2) + sin(u_1+u_2)/π (u_1+u_2) .





  
Connection to the Gaudin-Mehta law
A further consequence deals with the limiting law of the gap between swaps on the same horizontal line
in random sorting networks. Fix β∈ (0,1). Given a random sorting network of 𝔖_n,
let _+ be the distance between ⌊βn2⌋ and the closest to its right
swap interchanging ⌊n(α+1)/2⌋ with ⌊n(α+1)/2⌋+1. 
Let _- be the analogous distance to the closest to its left swap.

Due to stationarity of random sorting networks, the joint law _- and _+ is given by

    _- > a, _+ > b = _FS, α, n > a+ b.

Indeed, due to stationarity, both sides of (<ref>) give the probability of the event
that there are no swaps in the interval [-a,b] after the appropriate re-centerings.
Equation <ref> shows that the law of (_+, _-), and hence, of the gap _- +
_+, is determined by the law of the first swap time _FS, α, n. In
particular, their limiting law after rescaling by √(1-α^2)/ n is uniquely
determined from the distribution function (<ref>).

This is connected to the scaling limit of the point process of eigenvalues of GOE random matrices
in the bulk. The scaling limit of the eigenvalues of GOE random matrices near 0 is stationary.
Let -Λ_- and Λ_+ be, respectively, the closest to 0 negative and closest to
0 positive point in the limit process. If the matrices are scaled so that the mean eigenvalue
gap near 0 is 1, then (<ref>) is the distribution function of
(π/2)Λ_+. In other words,  (<ref>) is the asymptotic probability to
see no eigenvalues in an interval of length (2/π)t for large GOE random matrices, normalized so
that the mean eigenvalue gap around the interval is 1; see e.g. <cit.>, <cit.>,
<cit.>. The gap between points, Λ_- + Λ_+, has
its law determined from that of Λ_+ according to (<ref>). This is the
celebrated Gaudin–Mehta law, originally put forward by Wigner as a model for the gap
between energy levels in heavy nuclei and later found in numerous systems.
We arrive at the following corollary.

 
For α∈ (-1,1) and β∈ (0,1), let
Gap_α,β,n be the distance in a random sorting network of
𝔖_n between the two swaps interchanging ⌊n(α+1)/2⌋ with ⌊n(α+1)/2⌋+1: the one closest from the left to time βn2
and the one closest from the right to βn2.
Then, the distributional limit

    lim_n→∞ 4√(1-α^2)/π n Gap_α,β,n

is the Gaudin–Mehta law, i.e. the asymptotic gap in the bulk between eigenvalues of
real symmetric random matrices with mean gap one.




The proof of Theorem <ref> builds upon two ideas. The first one (which
is also used in most of the rigorous results on sorting networks) is to reduce
the study of random sorting networks to uniformly random staircase shaped
standard Young tableaux via the Edelman–Greene bijection <cit.> (see also <cit.>).
Our observation is that if we Poissonize uniformly random standard Young
tableaux (of arbitrary shape!), then the result can be described by a determinantal
point process with an explicit correlation kernel written as a double contour
integral. We further show that the Poissonization does not change the local statistics,
and therefore, the limit theorem is reduced to the asymptotic analysis of the
aforementioned double contour integral, which we perform.

Our results on the correlations and limiting behavior of random standard Young tableaux
might be of independent interest, and so we present them in the next section.



 §.§ Random Standard Young Tableaux


A partition λ is a sequence of non-negative integers λ_1 ≥λ_2 ≥⋯≥ 0 such that |λ|:=∑_i=1^∞λ_i<∞.
The length of λ, denoted ℓ(λ), is the number of positive _i
and the size of  is ||.

We identify a partition with a Young diagram (YD), which is the set of lattice points

    { (i,j) ∈^2:  i≥ 1,  1 ≤ j ≤_i}.

The points of the Young diagram  are its cells and we say
the Young diagram has shape .
Given a pair of YDs  and μ, we write ≼μ
if the cells of  are contained within the cells of μ. If the containment
is strict then ≺μ. If ≼μ then μ∖
denotes the cells of μ that are not in . A standard Young tableau (SYT)
of shape  is an insertion of the numbers 1,2, …, || into the cells of 
such that they strictly increase along the rows (from left to right) and also along the
columns (from bottom to top). The numbers within a SYT are its entries.
The set of SYTs of shape  is in bijection with the set of increasing sequences of YDs

    ∅ = ^(0)≺^(1)≺^(2)≺⋯≺^(||) =

such that the entry k is inserted into the singleton cell of ^(k)∖^(k-1).

A staircase shaped SYT of length n-1 (or also n-1 rows) is a SYT of shape
(n-1,n-2,…,2,1), which we denote .
The Edelman–Greene bijection <cit.> gives a one-to-one correspondence
between staircase shaped SYTs and sorting networks; see Section <ref>
for the details. This is the reason for our interest in SYTs.

A Poissonized Young tableau (PYT) of shape λ is an insertion of distinct
real numbers from the interval (0,1) into the cells of  such that they
strictly increase along the rows and  along the columns. Note that if we replace the
entries of a PYT by their relative ranks then we get a SYT. The set of PYTs of
shape  is in bijection with the set of increasing sequences of YDs indicating the times of jumps:

    ∅ = ^(0)t_1≺^(1)t_2≺^(2)t_3≺⋯t_||≺^(||) =

such that the entry t_k is inserted in the singleton cell of ^(k)∖^(k-1). These increasing sequences of Young diagrams
with labels were discussed in <cit.> in the connection to the Young
bouquet; see also <cit.>.

We would like to identify a PYT with a collection of non-intersecting
paths. For that we first map a Young diagram λ to a countable particle
configuration {λ_i-i+1/2}_i=1,2,…⊂ℤ+1/2. This
procedure can be viewed as projecting the boundary of the Young diagram in
Russian notation onto a horizontal line, see Figure <ref>.
The empty Young diagram ∅ corresponds to {-1/2,-3/2,-5/2,…}.



Give a PYT, for each t consider the countable particle configuration corresponding
to the Young diagram filled with the entries ≤ t in the PYT. The trajectories of
particles then form a collection of paths, making jumps to the right at the times
indexed by the entries t_k of the tableau (equivalently, labels in
(<ref>)). Let us draw a cross at a point (x,t), x∈ℤ,
0<t<1, if a particle jumps from (x-1/2) to (x+1/2) at time t; see Figure
<ref>. Although there are infinitely many particles, the only ones that move
are the ℓ() particles that correspond to the rows of  with positive size.



  
Given a finite Young diagram λ, consider the point process
_ of jumps of a uniformly random Poissonized Young tableau of
shape λ. _ is a determinantal point process on × [0,1]
with correlation kernel K_(x_1,t_1; x_2,t_2) as follows.
For x_1,x_2 ∈ and t_1, t_2 ∈ [0,1],

    K_(x_1,t_1; x_2;t_2) = t_2 > t_1,  x_1 > x_2 (t_1-t_2)^x_1-x_2-1/(x_1-x_2-1)! +
       [0,_1-x_2)[0,n+x_1)-w/z+1·G_(z+x_2)/G_(x_1-1-w)·(1-t_2)^z (1-t_1)^w/w+z+x_2-x_1+1,
       where   G_(u) = u+1∏_i=1^∞u+i/u-λ_i+i= u+1+n/∏_i=1^n (u-_i +i),    n≥ℓ(λ).

The contours C_z[0,_1-x_2) and C_w[0,n+x_1) are as shown in Figure
<ref>. Both are counter-clockwise, encloses only the integers in the
respective half open intervals [0,_1-x_2) and [0,n+x_1), and arranged such
that w+z+x_2-x_1+1 remains uniformly bounded away from 0 along the contours.



When t_1 or t_2 equals 1, K_ is to be understood in the limit as
t_1 or t_2 tends to 1. The contours C_z[0,_1-x_2) and C_w[0,n-x_2)
may also be replaced by unbounded contours C_z[0,∞) and C_w[0,∞)
with bounded imaginary parts, respectively.




The proof of Theorem <ref> is through a limit transition in the
correlation kernel of <cit.> (see also <cit.>) for the uniformly random
Gelfand–Tsetlin patterns; the proof is in Section <ref>.
Such a limit transition can be viewed as a degeneration of the combinatorial
structures related to the representation theory of the unitary groups U(N) to
those related to the symmetric groups 𝔖_n, see <cit.> for a discussion.

Let us emphasize that the very same procedure can be used to identify a uniformly random SYT with
a point process of jumps, however, the resulting process will not be
determinantal – this is why we need to pass from SYTs to PYTs.



 §.§ From Poissonized tableaux to local statistics
 

We close the introduction with an outline of the argument that takes us from Poissonized
tableaux to local statistics of sorting networks. By the nature of the Edelman-Greene
bijection, the swaps of a sorting network of 𝔖_n near time 0 are determined by
the location of the largest entries of an SYT of shape . These entries reside within
unit order distance of the edge of , which consist of the cells (i,n-i)
for 1 ≤ i ≤ n-1. As a result, the first step to deriving
local statistics of random sorting networks is to derive the statistics of a uniformly
random PYT of shape  near its edge.

Let _ denote a uniformly random PYT of shape . We are interested in the statistics
of the entries of _ that lie within the following windows. A window is parameterized by a
center α∈ (-1,1) (corresponding to the center (1+α)n/2 of the swaps of a
sorting network), a length L, and an entry height u. The window then consist of entries
_(i,j) that satisfy |i-(1+α)n/2| ≤ L and _(i,j) ≥ 1 - u/n.
In other words, roughly the largest n u entries of _ with row indices in the interval
[(1+α)n/2-L, (1+α)n/2+L].

We study the statistics of _ in a window in terms of its associated process
of jumps _, rescaled accordingly. For each integer n ≥ 1, let c_n be an
integer having the same parity as n and such that |c_n - α n| = O(1) as n →∞.
Consider the rescaled process of jumps

    _α, n = { (x,u) ∈×_≥ 0:  (  x + c_n, 1 - u/n √(1-α^2) ) ∈_}.

Theorem <ref> and Proposition <ref> together imply that _α,n
converges weakly to the point process _edge from Definition <ref>.
It is the building block for the proof of Theorem <ref>.

In Section <ref> we construct the local staircase shaped tableau _edge
by using the local jump process _edge. We prove in Theorem <ref>
that it provides the limiting statistics of _ in local windows.
Using a de-Poissonization argument we conclude in Theorem <ref>
that uniformly random staircase shaped SYTs also converge within local windows
to _edge. Although Poissonization is not important for the local limit,
it is important for the proof.

In Section <ref> we prove Theorem <ref>.
First, we give a proof of Corollary <ref> in Section <ref>.
In Section <ref> we define the local version of the Edelman-Greene algorithm
that maps _edge to S_local. In Section <ref>
we complete the proof of Theorem <ref> and conclude with some
statistical properties of S_local.



 §.§ Acknowledgements
 We would like to thank Balint Virág, who brought
our attention to the problem of identifying the local limit of sorting networks. We are
grateful to Alexei Borodin, Percy Deift, and Igor Krasovsky for valuable discussions
and references. We would also like to thank an anonymous referee for an exceptionally
careful reading of the paper and some helpful feedback.

V. Gorin's research was partially supported by NSF grants DMS-1407562, DMS-1664619, by a Sloan
Research Fellowship, by The Foundation Sciences Mathématiques de Paris, and by NEC
Corporation Fund for Research in Computers and Communications.
M. Rahman's research was partially supported by an NSERC PDF award.



§ PRELIMINARIES
 

This section presents basic facts about Young tableaux, Poissonization and determinantal point processes.
Some material from the Introduction is repeated for convenience.



 §.§ Gelfand-Tsetlin patterns
 

A semi-standard Young tableau of shape = (_1, …, _M), where
_1 ≥_2 ≥⋯≥_M ≥ 0 are integers, is an insertion of
numbers from {1,…, M} into the cells of the YD  such that the entries
weakly increase along each row and strictly increase along every column. It is
important to emphasize that while for the definitions of Young diagrams and standard
Young tableaux the value of M is not important, here the object
substantially depends on M. Semi-standard Young tableaux (SSYTs) are in bijection
with interlacing particle systems, often known as Gelfand-Tsetlin patterns
(or schemes).

A Gelfand-Tsetlin pattern (GTP) with M rows is a triangular array of non-negative
integers [a(i,j)] with row i containing i entries a(i,1), …, a(i,i).
The array satisfies the following order and interlacing constraints.

    Order & Interlace:     a(i,j) ≥ a(i-1,j) ≥ a(i,j+1)   for every  i  and  j.

Let a^(i) = (a(i,1), …, a(i,i)) denote the i-th row of the GTP. Each row
corresponds to a YD due to the order constraints. The interlacing conditions ensure
that a^(i-1)≼ a^(i), and in fact, a^(i)∖ a^(i-1) is a
horizontal strip which means that the cells in any row of a^(i)∖
a^(i-1) are to the left of the cells in the previous row. Figure <ref> provides an example.


The set of GTPs with a fixed top row a^(M) is in bijection with the set of SSYTs
of shape = a^(M). Indeed, given a GTP [a(i,j)] with top row , such a
tableaux is obtained by inserting the value i into the cells of a^(i)∖
a^(i-1) for every 1 ≤ i ≤ M (set a^(0) = ∅). If a^(i)∖ a^(i-1)
is empty then i is not inserted. In the reverse direction, given a SSYT of shape ,
a GTP with top row  is obtained by setting a^(i) to be the YD consisting of the cells
of  with entries ≤ i and removing trailing zero rows to ensure that a^(i) has i entries.

A GTP may also be represented as an interlacing particle system on ( +1/2)×
as follows. Given a GTP [a(i,j)] with M rows, the particle system
[ν(i,j)] associated to it has M rows of particles, with particles on row
i being placed on the horizontal line {y=i} of the plane, and the position of the j-th particle on row i is

    (ν(i,j),  i ) =  (a(i,j) - j + 1/2,   i )  for every   1 ≤ j ≤ i.

The transformation a(i,j) → a(i,j)-j + 1/2 makes the order constraints strict and the interlacing constraints semi-strict:

    ν(i,j) ≥ν(i-1,j) > ν(i,j+1).

The jumps of an interlacing particle system ν with M rows is a set of
points in ×{1, …, M-1}, defined inductively from the top row to the bottom as follows.
Given two consecutive rows [ν(i,·)] and [ν(i-1,·)], the jumps on row i
consist of particles at the positions

    (k,i-1) ∈^2   for every integer  k ∈ [ν(i-1,j), ν(i,j)]   and every   1 ≤ j ≤ i-1.

In other words, the jumps of row i are placed on the horizontal line {y = i-1}
and fill out integers in the intervals [ν(i-1,j),ν(i,j)] for every 1 ≤ j ≤ i-1.
Note that ν may determined from its top row and set of jumps.



 §.§ Poissonized Young tableaux
 

For a YD , let [0,1]^ denote the set of functions from
the cells of  into [0,1]. Let PYT() denote the set of all
functions T  ∈ [0,1]^ that satisfy the following tableau constraints.

    (1)     T(i,j) ≤ T(i,j+1)  for every  (i,j)  and  (i,j+1) ∈, 
     (2)     T(i,j) ≤ T(i+1,j)  for every  (i,j)  and  (i+1,j) ∈.

The Poissonized tableau (PYT) of shape  is an element of
PYT(). The Poissonized staircase shaped tableau of size
N = n2 is an element of PYT().

Let _ denote a uniformly random element of PYT(). Then
_ is related to a uniformly random SYT of shape  in the following way.
First, the entries of _ are distinct with probability 1. Given
that, consider the random SYT _ obtained by inserting k into the cell
that contains the k-th smallest element of _. Then _ is a
uniformly random SYT of shape . In the other direction, _ can be
generated by first sampling _, then independently sampling a uniformly
random Y ∈ [0,1]^, and setting _(i,j) to be the
_(i,j)-th smallest entry of Y.

Throughout the paper, _ denotes a uniformly random element of PYT()
and _ denotes a uniformly random SYT of shape .



 §.§ Jumps of Poissonized tableaux and local limit
 

Any T ∈PYT() can be represented as an interlacing particles system
with a fixed top row in the following manner. Consider 0 ≤ t ≤ 1 and let

    YD^(t) = { (i,j) ∈ : T(i,j) ≤ t}.

The tableau constraints (<ref>) ensure that YD^(t) is a YD for
every t. Recall that a YD can be made to have an infinite number of rows
by appending rows of size 0 after the last positive row. Encode  as
particle configuration on + 1/2 by placing a particle at position

    ν_j = _j - j + 1/2  for   j ≥ 1.

This is an infinite particle configuration on + 1/2 such that ν_1 >
ν_2 > ⋯ and ν_j - ν_j+1 = 1 for j > ℓ() (shown in Figure <ref>).
Let ν^(t) be the particle configuration associated to YD^(t) via (<ref>) and let
ν = (ν^(t); 0 ≤ t ≤ 1) be the particle system on (+ 1/2)
× [0,1] with a particle at position (x,t) if and only if x ∈ν^(t).

The particle system ν^(t) viewed in reverse time, i.e., from t=1 to
t=0, can be interpreted as an ensemble of non-intersecting and non-increasing
paths p(i,u), for 1 ≤ i ≤ℓ(). Let p(i,u) be the (+
1/2)-valued path starting from p(i,0) = ν^(0)_i and decreasing an
integer unit at the times 1-T(i,_i), 1- T(i,_i-1), …, 1-T(i,1). If some
of the entries are equal then p(i,u) decreases by the number of consecutive equal
entries. The paths should be left continuous so that the jumps occur immediately
after the jump times. The path p(i,u) decreases by _i units with final position
p(i,1) = -i + 1/2. Due to the columns of T being non-decreasing –
condition (2) of (<ref>) – the paths are non-intersecting: p(i,u) >
p(i+1,u) for every i and u. Figure <ref> shows the paths associated
to a staircase shaped PYT.

The jumps of p(i,u) consists of points (x,t) ∈× [0,1] such that


  * 1-t is a discontinuity point of p(i,u), i.e., t equals some entry of T on row i.

  * x in an integer in the interval [p(i,(1-t)+),   p(i,1-t) ], where p(i,u+) = lim_s ↓ u p(i,s).

The paths can be reconstructed from their jumps and initial positions.
The jumps of ν, and also of T, is the (possibly) multiset of × [0,1] defined by

    X = { (x,t) : (x,t)   is a jump of some path   p(i,u)}.

X may be a multiset because two adjacent paths may jump at the same time
by amounts that causes some of their jumps to coincide. The coinciding jumps has
be counted with multiplicity. However, if the entries of T are distinct then X is
a simple set. The tableau can be reconstructed from its jumps and the initial position
of the paths.

Let _ denote the jumps of a uniformly random element _
of PYT(). _ is simple almost surely since _
has distinct entries almost surely. Theorem <ref> asserts
that _ is a determinantal point process on × [0,1].



 §.§ Determinantal point processes
 

We describe some basic notions about point processes; for a thorough
introduction see <cit.>. Let S be a locally compact Polish space.
A discrete subset X of S is a countable multiset of S with no
accumulation points. By identifying X with the measure ∑_x ∈ Xδ_x,
the space of discrete subsets can be given the topology of weak convergence
of Borel measures on S. This means that X_n → X_∞ if for every compact subset
C ⊂ S, lim sup_n  # (C ∩ X_n) ≤#(C ∩ X_∞), where cardinality is taken
with multiplicity.








A discrete set is simple if every point in it has multiplicity one. A point process on S
is a Borel-measurable random discrete set of S.
All point processes considered in this paper will be simple almost surely.

Throughout the paper we denote #_ to be counting measure on  and (A)
to be Lebesgue measure on a measurable subset A ⊂. Also, μ_1 ⊗μ_2
denotes the product of measures μ_1 and μ_2, and μ^⊗ k
denotes the k-fold product of μ.

A determinantal point process  on S is a simple point process
for which there is a correlation kernel K : S × S →, and a Radon
measure μ on S, called the reference measure, with the following property.
For every continuous f : S^k → of compact support,

    𝔼  [∑_(x_1,…, x_k) ∈^k 
     x_1, …, x_k  distinct f(x_1, …, x_k)] =
    ∫_S^k [ K(x_i,x_j) ] f(x_1,…,x_k)  μ^⊗ k(dx_1,…,dx_k).

Expectations of the form given by the l.h.s. of (<ref>) determine the law of  under mild
conditions on K <cit.>. This will be the case in this paper as the correlation kernels we
consider will be continuous. If S is discrete then it is customary to take the reference measure to be counting measure.
In this case  is determinantal if for every finite A ⊂ S,

    A ⊂ =  [ K(x,y)  ]_x,y ∈ A.


 
The correlation kernel of a determinantal point process is not unique. If  is a determinantal point process
with correlation kernel K then K may be replaced by g(x)/g(y)K(x,y), for any non-vanishing
function g, without changing determinants on the r.h.s. of (<ref>). Thus the new kernel
determines the same process. This observation will be used multiple times.


The determinantal point processes that we consider will be on spaces of the form
S = ×{1, …, M}, or S = × [0,1], or S = ×_≥ 0,
with reference measures being, respectively, counting measure, #_⊗[0,1]
and #_⊗(_≥ 0). The following lemma records some facts
that will be used in deriving weak limits of determinantal point processes. We do not include the
proof as it is rather standard; see <cit.>.

 
I) Let _M be a determinantal point process on ×{1, …, M}
with correlation kernel K_M. For x_1,x_2 ∈ and 0 ≤ t_1,t_2 ≤ 1, let

    k_M(x_1,t_1; x_2,t_2) = M K_M(x_1, ⌈ Mt_1 ⌉; x_2, ⌈ Mt_2 ⌉).

Suppose that k_M → k uniformly on compact subsets of × (0,1).
Then the point process

    _M^scaled = { (x, t/M): (x,t) ∈_M}

restricted to × (0,1) converges weakly to a determinantal point process  whose
reference measure is #_⊗(0,1) and whose correlation kernel is k.


II) Let _n be a determinantal point process on × (0,1) with reference
measure #_⊗(0,1) and correlation kernel K_n.
For c_n ∈ and β > 0, define a point process on ×_> 0 by

    _n^scaled = {(x- c_n,  β n(1-t) ): (x,t) ∈_n }.

The correlation kernel of _n^scaled with reference measure #_⊗(_> 0) is

    k_n(x_1,u_1; x_2,u_2) = (β n)^-1 K_n  (x_1+c_n, 1- u_1/β n;  x_2+c_n, 1- u_2/β n).

If k_n → k uniformly on compact subsets of ×_> 0 then _n^scaled
converges weakly to a determinantal point process  with reference measure #_⊗(_> 0)
and correlation kernel k.

Extend  to a point process on ×_≥ 0 without additional points.
Then _n^scaled converges weakly to  on ×_≥ 0 if
the points of _n^scaled do not accumulate at the boundary in the sense that for every x ∈,

    lim_→ 0 lim sup_n →∞ _n^scaled∩ ({x}× [0,]) ≠∅ = 0.





§ DETERMINANTAL REPRESENTATION OF POISSONIZED TABLEAUX
 



 §.§ Determinantal representation of discrete interlacing particle systems
 

In order to prove Theorem <ref> we use a determinantal description of discrete
interlacing particle systems due to Petrov. This is the main tool behind the proof.

Let ν = (ν(M,1) + 1/2 > ⋯ > ν(M,M) + 1/2) be a fixed particle
configuration on + 1/2. Here we abuse notation from Section <ref>
to have the ν(M,j)s be integers instead of half-integers. Let ℙ_ν be the
uniform measure on all interlacing particle systems or, equivalently, GTPs as described in
Section <ref>, with fixed top row ν. Let _ν be the point process of jumps
of an interlacing particle system sampled according to ℙ_ν,
where the jumps are as described in Section <ref>.

Petrov <cit.> proves that ℙ_ν is
a determinantal point process on  (+ 1/2) ×{1, …, M} with an explicit correlation kernel.
According to the notation there, particles live on  but we have translated particle
systems by 1/2 so that the jumps are integral. In particular, in the notation of <cit.>,
one has N=M, x ^M,j = ν(M,j) and the variables x_1, x_2 take integer values.
In <cit.> it is explained that the point process of jumps,
_ν, is also determinantal on ×{1, …, M-1} and its correlation kernel
is given in terms of the correlation kernel of ℙ_ν in <cit.>
(up to the translation by 1/2).

In particular, <cit.> proves that the correlation kernel of the jumps is

    K__ν(x_1,m_1;x_2,m_2) = (-1)^x_2-x_1 + m_2-m_1  K_ℙ_ν(x_1-1,m_1+1;x_2,m_2),

where K_ℙ_ν is the kernel presented in <cit.>.
The discussion there is in terms of lozenge tilings of polygonal domains using three types of
lozenges as depicted in Figure <ref>. It is proved that the positions
of any of the three types of lozenges in such a uniformly random tiling is a determinantal point process.
The jumps of an interlacing particle system are given by the positions of the lozenges of the rightmost
type from Figure <ref>, where as the particles themselves are given by the positions of
lozenges of the leftmost type. Jumps occur when a lozenge of the leftmost type is glued along its
bottom diagonal to a lozenge of the rightmost type; see <cit.> for
such a tiling. By Remark <ref>, (-1)^x_2-x_1 + m_2-m_1 K_ℙ_ν(x_1-1,m_1+1;x_2,m_2)
defines the same determinantal point process as K_ℙ_ν(x_1-1,m_1+1;x_2,m_2),
and we will use the latter kernel.



Some notation is needed in order to express the kernel for the point process of jumps.
For integers a and b, let C[a,b) denote a closed, counter-clockwise contour on 
that encloses only the integers a, a+1, …, b-1 if a > b, and empty otherwise.
Throughout the paper, all contours intersect the real line at points which have distance at least
1/10 from the integers. This ensures that the integrands of all contour integrals will be a
uniform distance away from their poles.

For z ∈ and an integer m ≥ 1, let

    (z)_m = z(z+1) ⋯ (z-m+1),    (z)_0 = 1.


 
The process of jumps, _ν, of a uniformly random interlacing particle system
with fixed top row ν =  (ν(M,1) + 1/2 > ν(M,2) + 1/2 > ⋯ > ν(M,M) + 1/2)
is a determinantal point process on ×{1,…,M-1} with correlation kernel K as follows.
For x_1, x_2 ∈ and 1 ≤ m_1, m_2 ≤ M-1,

    K(x_1,m_1; x_2,m_2) = - m_2 ≤ m_1,  x_2 < x_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)! +
        + [x_2,ν(M,1)+1)[x_1-M,ν(M,1)+1) [(z-x_2+1)_M-m_2-1/(w-x_1+1)_M-m_1·(M-m_1-1)!/(M-m_2-1)!×
         ×1/w-z·∏_j=1^M w - ν(M,j)/z - ν(M,j)].

The contour C_w[-M,ν(M,1)+1) contains C_z[x_2,ν(M,1)+1) without intersecting it.




 §.§ Proof of Theorem <ref>
 

Let  be a Young diagram with at most n rows of positive length, that is,
ℓ() ≤ n. For M ≥ n, consider semi-standard Young tableaux of shape

    _M = (_1, …, _n, 0, …, 0_M-n zeroes).

The effect of adding M-n zero rows is to allow the entries in the non-zero rows of
 to be between 1 to M. The law of a uniformly random PYT of shape 
is the weak limit of a uniformly random semi-standard Young tableau of shape _M, as M →∞,
after the entries are rescaled onto the interval [0,1]. Indeed, the law of a uniformly random PYT
of shape  can be approximated by the uniform distribution on points [T(i,j)] ∈ [0,1]^
that satisfy the tableau constraints (<ref>), with each T(i,j) = k/M for some 1 ≤ k ≤ M,
and the column constraints being strict.

The top row of particle systems associated to semi-standard Young tableaux of shape _M
under the bijection described in Section <ref> is

    ν^(M)_ =  ((_1-1) + 1/2, …, (_n-n)+1/2,
    -(n+1) + 1/2,-(n+2) + 1/2,…, -M + 1/2 ).

Due to the approximation scheme above, and the bijection between semi-standard Young
tableaux and interlacing particle systems discussed in Section <ref>,
the process of jumps, _, of a uniformly random PYT of shape  is the weak
limit of the process of jumps, __M, of a uniformly random interlacing particle
system with top row ν^(M) after these jumps are rescaled onto ×{1/(M-1), …, 1}.
Thus, we derive a determinantal description of the rescaled jumps of
__M in the large M limit. Since _ almost surely contains no jumps on the
boundary ×{0, 1}, it suffices to derive the determinantal description with _
restricted to × (0,1).

Let K_M denote the kernel from Theorem <ref> for the process 𝒳__M.
As explained in Remark <ref>, the kernel (M-1)^x_2-x_1K_M(x_1,m_1;x_2,m_2) determines
the same determinantal point process. By Lemma <ref>, in order to prove the theorem it suffices to show that

    (M-1)^x_2-x_1+1 K_M(x_1, ⌈ (M-1)t_1 ⌉; x_2, ⌈ (M-1)t_2 ⌉)  ⟶  K_,

uniformly over compacts subsets of x_1,x_2 ∈ and t_1, t_2 ∈ (0,1).

We begin by deforming the contours in the double contour integral from (<ref>) that defines K_M.
This will simplify the representation of K_M for taking the large M limit.
Deform the w-contour, C_w[x_1-M,ν__M(M,1)+1), by pushing it leftward past the z-contour,
C_z[x_2,ν__M(M,1)+1), so that it encloses the consecutive integers min{x_1,x_2} -1, …, x_1-M.
The deformation results in picking up residues at w = z and also at the consecutive integers w = x_1-1, …, x_2, if x_2 < x_1.

Let J_M(w, x_1,m_1; z, x_2, m_2) denote the integrand of the double contour integral from (<ref>)
but without the factor of 1/(w-z). Note that ν__M(M,1) +1= _1. Calculating the residues at w=z
and leaving the remaining residues as a contour integral provides the following representation of K_M.

    K_M(x_1,m_1;x_2,m_2) = - m_2 ≤ m_1,  x_2 < x_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)!  +     (Ia)
       + 1/2 π𝐢∮_C_z[x_2, _1) dz (z-x_2+1)_M-m_2-1/(z-x_1+1)_M-m_1·(M-m_1-1)!/(M-m_2-1)! +     (Ib)
       + 1/(2 π𝐢)^2∮_C_z[x_2, _1) dz
    ∮_C_w[x_1-M, min{x_1,x_2}) ∪ C_w[x_2, x_1) dw J_M(w,x_1,m_1; z, x_2, m_2)/w-z    (II).


The following lemma simplifies (Ia) + (Ib).

 
For 0 ≤ m_1, m_2 ≤ M-1,

    1/2 π i∮_C_z[x_2, _1) dz (z-x_2+1)_M-m_2-1/(z-x_1+1)_M-m_1·(M-m_1-1)!/(M-m_2-1)!  - m_2 ≤ m_1,  x_2 < x_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)!
        = m_2 > m_1,   x_1 > x_2(m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)!.




The integral (Ib) is evaluated in <cit.> by summing over residues at z = x_2, …, x_1-1
and evaluating the resulting sum in closed form via a hypergeometric identity. We have

    (Ib) = x_1 > x_2(m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)!.

Therefore,

    (Ia) + (Ib) = x_1 > x_2 [ (m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)!
    - m_2 ≤ m_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)!].

If m_2 ≤ m_1 then the above is 0 because both terms in the difference are
equal to (m_1-m_2+x_1-x_2-1)!/(m_1-m_2)!(x_1-x_2-1)!. Hence, (Ia) + (Ib)
in non zero only if x_1 > x_2 and m_2 > m_1 and equals what is given in the statement of the lemma.


Now consider the expression (II) from (<ref>). Observe that the zeroes
of (w-x_1+1)_M-m_1 are at consecutive integers x_1-1, x_1-2, …, x_1-(M-m_1).
On the other hand, the polynomial ∏_j (w-ν__M(M,j)) also has zeroes at ν__M(M,n+i) = -(n+i)
for 1 ≤ i ≤ M-n. Therefore, the only poles of J_M in the w variable are at the integers
x_1-1, x_1-2, …, -n so long as x_1+m_1 ≥ 0. For fixed x_1 and m_1 ≥√(M), say,
the condition x_1+m_1 ≥ 0 is satisfied for all large M. Thus, the contour integral over C_w[x_1-M, min{x_1,x_2})
may be shortened to C_w[-n,min{x_1,x_2}) for all large M if x_1 remains fixed and m_1 ≥√(M).

Having done so, (II) becomes the following integral after changing variables z ↦ z+x_2 and w ↦ -w + x_1-1:

    (II) = [0,_1-x_2)[0,n+x_1)(z+1)_M-m_2-1 (M-m_1-1)!
    ∏_j=1^M -w+x_1-1+ν__M(M,j)/z+x_2-ν__M(M,j)/(-w)_M-m_1 (M-m_2-1)!   (w+z+x_2-x_1+1).

By a slight abuse of notation, let J_M henceforth denote the integrand of (<ref>) without the factor 1/(w+z+x_2-x_1+1).
The following lemma provides the asymptotic form of J_M.

 
Fix integers x_1 and x_2. Suppose 0 < t_1,t_2 < 1 and w and z remain bounded and have distance at
least 1/10 from the integers. Then for m_1 = ⌈ t_1(M-1)⌉ and m_2 = ⌈ t_2(M-1)⌉,

    (M-1)^x_1-x_2-1J_M(w,x_1,m_1;z,x_2,m_2) =
    (1-t_1)^w (1-t_2)^z -w G_(z+x_2)/z+1 G_(x_1-1-w)  (1 + O(M^-1)).

The big O term is uniform over z,w so long as the stated assumptions hold and t_1,t_2
remain in compact subsets of (0,1). The function G_ is as stated in Theorem <ref>.



We will use the following identity:

    (y)_m = y+m/y,    y ∉{0,-1,-2, …}.

We will also use Stirling's approximation of the Gamma function in the following form:

    y+m/(m-1)! = m^y  (1+O(m^-1)),    m ≥ 1.

The big O term is uniform in m so long as y is bounded and bounded away from negative integers.
Using these two properties, if m_1 =⌈ t_1(M-1)⌉ and m_2 =⌈ t_2(M-1)⌉ then

    (z+1)_M-m_2-1/(-w)_M-m_1·(M-m_1-1)!/(M-m_2-1)! =
    (1-t_1)^w (1-t_2)^z -w/z+1  (M-1)^z+w (1+ O(M^-1))


Now consider the term

    ∏_j=1^M  (w-ν__M(M,j) )    = ∏_j=1^n (w-_j+j) · (w+n+1)_M-n
       = ∏_j=1^n (w - _j+j) ·w+M+1/w+n+1.

Applying (<ref>) to w+M+1 and z+M+1 gives

    ∏_j  w-ν__M(M,j)/z-ν__M(M,j) =
    z+n+1/w+n+1·∏_j=1^n w-_j+j/z- _j+j· M^w-z  (1+O(M^-1)).

Substituting in z+x_2 and -w + x_1-1 then gives

    ∏_j -w+x_1-1-ν__M(M,j)/z+x_2-ν__M(M,j) = G_(z+x_2)/G_(x_1-1-w)  (M-1)^-(w+z) +x_1-x_2-1 (1+O(M^-1)).

Combining (<ref>) with (<ref>) provides the desired conclusion of the lemma.


We now prove that (M-1)^x_2-x_1+1K_M converges to K_.
Suppose x_i are fixed and m_i = ⌈ t_i(M-1) ⌉ for i = 1,2 and t_i ∈ [δ, 1-δ] for some δ > 0.
Recall that K_M is given in (<ref>). For all sufficiently large values of M,
Lemma <ref> and then the identity (<ref>) followed by the estimate
(<ref>) show that (Ia) + (Ib) of (<ref>) equals

    (Ia) + (Ib)     = m_2 > m_1, x_1 > x_2(m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)!
        = t_2 > t_1, x_1 < x_2(t_1-t_2)^x_1-x_2-1/(x_1-x_2-1)!· (M-1)^x_1-x_2-1 (1+ O(M^-1)).


Now we consider (II) in the form given in (<ref>) following the change of variables.
Lemma <ref> implies that as M →∞,

    (M-1)^x_2-x_1+1 J_M(w,x_1,t_1; z, x_2,t_2) → (1-t_1)^w (1-t_2)^z-w G_(z+x_2)/z+1 G_(x_1-1-w).

The convergence is uniform over compact subsets of w and z so long as w and z are
uniformly bounded away from the integers. For all large values of M the contours of integration of
(II) become free of M, namely, z ∈ C_z[0,_1-x_2) and w ∈ C_w[0,n+x_1).
The contours may also be arranged such that they remain bounded away from the
integers and |w+z+x_2-x_1+1| ≥ 1/10 throughout, say. This implies that as M →∞,
(M-1)^x_2-x_1+1· (II) converges to

    [0,_1-x_2)[0,n+x_1) (1-t_1)^w (1-t_2)^z   -w G_(z+x_2)/z+1 G_(x_1-1-w) ×
       × 1/(w+z+x_2-x_1+1).


We have thus concluded that (M-1)^x_2-x_1+1 K_M converges to the kernel K_
given in Theorem <ref>. Moreover, the estimates show that the converge
is uniform over compact subsets of × (0,1). Indeed, so long at t_1,t_2 ∈ [δ,1-δ]
and |x_1|, |x_2| ≤ B, the error term in the convergence is of order O_B,δ(M^-1),
by Lemma <ref>, because the double contours eventually become free of M
and the integrand converges uniformly over the contours. Part (I) of Lemma <ref> now implies
that the rescaled process of jumps, ^scaled__M, converges weakly on × (0,1)
to a determinantal point process with kernel as given in Theorem <ref>.




§ BULK LOCAL LIMIT OF THE JUMPS OF POISSONIZED STAIRCASE SHAPED TABLEAUX
 

In this section we prove that the point process X_α,n from (<ref>)
converges weakly to the point process _edge from Definition <ref>.
This is done in a two-step procedure. First, we prove in Theorem <ref> that the limit of
X_α, n is a determinantal point process whose kernel is given in terms of a double contour integral.
Second, we identifying this kernel with the one from Definition <ref> in Proposition <ref>.

 
The point process _α, n from (<ref>) converges weakly to a
limiting determinantal point process _edge on ×_≥ 0.
The correlation kernel of _edge with respect to reference measure
#_⊗(_≥ 0) is given as follows. For u_1,u_2 ∈_≥ 0 and x_1, x_2 ∈,

    K_edge(x_1,u_1;x_2,u_2) = u_2 < u_1, x_2 < x_1 (u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! +
       [0,∞)[0,∞)π/2·G(w; x_1,u_1)   G(z; x_2,u_2)/w+z +x_2-x_1+1,   where
        G(z; x,u) = u^z/z+1sin(π/2(z+x_2)).

The contours C_z[0,∞) and C_w[0,∞) are unbounded, contain the non-negative integers
but remain uniformly bounded away from them and are arranged such that w+z+x_2-x_1+1 remains
uniformly bounded away from 0. They may also be arranged such that their imaginary parts remain
bounded and C_z contains C_w.

The value of K_edge when u_1 or u_2 equals 0 is to be understood in
the sense of the limit as u_1 or u_2 tends to 0.



The proof proceeds in two steps, each verifying the conditions of part (II) of Lemma <ref>.
First, we will show that the correlation kernel of _α,n converges to K_edge
uniformly on compact subsets of x_1,x_2 ∈ and u_1,u_2 ∈_> 0. Then we
will argue that points of _α,n do not accumulate on the boundary ×{0}
as n →∞.

Let β = √(1-α^2). Part (II) of Lemma <ref> and Theorem <ref>
imply that the correlation kernel of _α,n with reference measure #_⊗(_≥ 0) is

    K_n(x_1,u_1;x_2,u_2) = (β n)^-1 K_(x_1 + c_n, 1- u_1/β n;
    x_1 + c_n, 1- u_1/β n ).

The kernel (β n)^x_1-x_2 K_n determines the same point process by Remark <ref>.
Using part (II) of Lemma <ref> it suffices to show that (β n)^x_1-x_2 K_n
converges uniformly over compact subsets of ×_> 0 to K_edge as n →∞
in order to deduce convergence of _α,n to _edge on ×_>0.

Let G_n denote the function 2^n-1G_, where G_ is as in Theorem <ref>. Then

    G_n(z) = 2^n-1z+n+1/∏_j=1^n-1 (z-n + 2j) = z+n+1/(z-n/2+1)_n-1
    = z+n+1z-n+2/2/z+n/2.

Substitute in x_i +c_n for the variables x_i and 1 - (β n)^-1u_i for the variables t_i in K_.
Then,

    K_n(x_1,u_1;x_2,u_2) = u_2 < u_1, x_2 < x_1 (u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! (β n)^x_2-x_1 +
       [0,n-1-c_n-x_2)[0,n-1+c_n+x_1)-w G_n(z+x_2+c_n)   u_1^w   u_2^z   (β n)^-w-z-1/z+1 G_n(x_1-1+c_n-w) (w+z+x_2-x_1+1).

Using the formula for G_n from (<ref>) and applying the identity

    1-yy   = π/sin(π y),  y ∉{0,-1,-2,…},  to   y = n-c_n-z/2  gives
    
    G_n(z+c_n)    = z+n+c_n+1/n+c_n+z/2n-c_n-z/2·π/sin (π/2(n-c_n-z) ).


The estimate for z from (<ref>) along with the observation that
n ± c_n = (1 ±α)n  + O(1)  → +∞ implies the following asymptotic bevaviour as n →∞.
The symbol ∼ denotes a multiplicative term 1+O(n^-1) where the big O error is uniform over
z in compact subsets of ∖.

    n ±(c_n+z)/2   ∼n ± c_n/2·( n ± c_n/2)^±z/2, 
    n+c_n+z+1   ∼n+c_n· (n+c_n)^z+1.

Therefore,

    n+c_n+z+1/n+c_n+z/2n-c_n-z/2·n+c_n+w/2n-c_n-w/2/n+c_n+w+1   ∼(n+c_n/n-c_n)^w-z/2 (1+α)^z-w  n^z-w
       ∼( 1-α/1+α)^z-w/2 (1+α)^z-w  n^z-w
        = (β n)^z-w.

Substituting in z+x_2 for z and -w+x_1-1 for w in this estimate gives

    G_n(z+x_2+c_n)/G_n(-w+x_1-1+c_n)∼sin (π/2(-w+x_1-1-n-c_n) )/sin (π/2(z+x_2-n-c_n) ) (β n )^w+z + x_2-x_1+1.

The error in the above estimate vanishes as n →∞ so long as w, z and the x_i lie
in compact subsets of their respective domains. We also have that

    sin( π/2(n-c_n-w) )/sin(π/2(n-c_n-w)) =
    sin(π/2 w )/sin(π/2 z )  if   n-c_n   is even.

Since n-c_n is assumed to be even we conclude from (<ref>) that so long
w and z are bounded and remain uniformly bounded away from the integers then

    lim_n →∞  (β n)^x_1-x_2+1 G_n(z+x_2+c_n) (β n)^-w-z-1/G_n(x_1+c_n-1-w) =
    -sin (π/2 (w-x_1+1) )/sin (π/2 (z+x_2)).


The above displays the pointwise limit of the part of the integrand from (<ref>) that depends on n.
In order to interchange the pointwise limit with the contour integral we must show
that the integrand is bounded uniformly over n by a function that is integrable over the
contours z ∈ C_z[0,n-1-c_n-x_2) and w ∈ C_w[0,n-1+c_n+x_1), also uniformly over n.
Then we may apply the dominated convergence theorem.

Towards this end suppose z is such that (1) |(z)| ≤ 2n, (2) |(z)| is uniformly
bounded over n, say by 100, and (3) z remains bounded away from  by distance at least
1/10. In this case Stirling approximation to the Gamma function implies that modulus of
the ratio of the l.h.s. of (<ref>) to its r.h.s. is bounded above and below by exponential factors
of uniform rate in |(z)|. The same holds for the ratio of the l.h.s. of (<ref>) to its r.h.s.
That is, for some constant C,

    e^-C  (|(z)| + 1)≤ | n ±(c_n+z)/2n ± c_n/2·( n ± c_n/2)^±z/2| ≤ e^C  (|(z)| + 1), 
    
    
    e^-C  (|(z)| + 1)≤ | n+c_n+z+1n+c_n· (n+c_n)^z+1 |≤ e^C  (|(z)| + 1).


Throughout the following C denotes a constant that is free of n but its value may change
from line to line. We combine the estimates above with the equation for G_n(z+c_n) from (<ref>)
and observe that there is a C such that 1/C ≤ |sin(n-c_n-z)| ≤ C due to the assumptions
on z. This in turn implies that there is a C such that

    e^-C   (|(z)| +1)≤ | G_n(z+c_n)/(β n)^z| ≤ e^C   (|(z)| + 1).


The contours C_z[0,n-1-c_n-x_2) and C_w[0,n-1+c_n+x_1) can certainly be arranged
such that for fixed x_1 and x_2 the variables z+x_2 and -w+x_1-1 satisfy the
aforementioned assumptions (1)–(3) uniformly over n. Thus, we get the following
uniform estimate over n with z ∈ C_z[0,n-1-c_n-x_2) and w ∈ C_w[0,n-1+c_n+x_1):

    | G_n(z+x_2 + c_n)/G_n(-w+x_1-1+c_n)| ≤ (β n)^(z+w) +x_2-x_1+1  e^C  (|(z)| + |(w)| +1).

The contours may also be arranged such that |w+z+x_2-x_1+1| ≥ 1/10, say. Then the modulus
of the integrand of the double contour integral from (<ref>) satisfies

    (β n)^x_1-x_2 | -w G_n(z+x_2+c_n)   u_1^w   u_2^z   (β n)^-w-z-1/z+1 G_n(x_1-1+c_n-w)  (w+z+x_2-x_1+1)| ≤
                 |-w/z+1 |   |u_1^w|   |u_2^z|   e^C   (|(z)| + |(w)| +1).


Stirling's approximation implies that if (z) ≥ 1/10 and |(z)| remains bounded then

    |u^z|/|z+1| = e^-(z) log(z) + (z)(log u + O(1)).

Applying (<ref>) with y = -w also gives -w = - π [w+1sin(π w)]^-1.
Note that |sin (π w)|^-1≤ C so long as w remains uniformly bounded away from the integers.
Combining this with (<ref>) shows that the r.h.s. of (<ref>) is
integrable over unbounded double contours C_z[0,∞) ∋ z and C_w[0,∞) ∋ w
as long as the contours are arranged such that z,w remain uniformly bounded away from the
integers, have uniformly bounded imaginary parts, and z+w+x_2-x_1+1 remains uniformly bounded
away from 0. Thus, the limit (<ref>), upper bound (<ref>) and the
dominated convergence theorem implies that as n →∞

    (β n)^x_1-x_2  K_n(x_1,u_1; x_2,u_2) →u_2 < u_1, x_2 < x_1 (u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! + 
       [0,∞)[0,∞) u_1^w u_2^z ·--wsin (π/2 (w-x_1+1) )/z+1sin (π/2 (z+x_2) )·1/w+z+x_2-x_1+1.


Furthermore, our estimates show that the convergence is uniform over x_i in compact subsets
of  and u_i in compact subsets of _> 0. (In fact, when some u_i → 0 the integral
contributes only through residues at the origin. Lemma <ref> computes the limit as u_i → 0.)
Comparing the limit integrand with the one presented in (<ref>)
we observe that the proof of the kernel convergence will be complete once it is shown that

    - -wsin (π/2 (w-x_1+1) ) = (π/2)/w+1sin (π/2 (w+x_1) ) .

From (<ref>), --w = π [w+1sin(π w)]^-1. Also, sin ( π/2(w-x+1)) = cos(π/2(w-x)).
Finally, double angle trigonometric formulae imply sin(π w) = 2 sin( π/2(w+x)) cos( π/2(w-x)).
Substituting these equations into the l.h.s. of the above verifies the equality with the r.h.s.



To complete the proof of convergence of _α,n to _edge on ×_≥ 0
it is enough to show, using Lemma <ref>, that for every x ∈,

    lim_→ 0 lim sup_n →∞ # _α,n∩ ({x}× [0,]) = 0.

In other words, points do not accumulate at the boundary in the limit.
From relation (<ref>) for determinantal point processes and (<ref>) we get

    # _α,n∩ ({x}× [0,]) = ∫_0^ K_n(x,t;x,t)  dt 
        = [0,n-1-c_n-x)[0,n-1+c_n+x)-w G_n(z+x+c_n)   (∫_0^ t^w+z  dt )   (β n)^-w-z-1/z+1 G_n(x-1+c_n-w) (w+z+1)
       = [0,n-1-c_n-x)[0,n-1+c_n+x)-w G_n(z+x+c_n)  ^w+z+1  (β n)^-w-z-1/z+1 G_n(x-1+c_n-w) (w+z+1)^2.


The quantity above is of the form I_,n. Arguing exactly as in the derivation of the limit kernel,
I_,n→ I_, where I_ is given by the double contour integral in the definition of K_edge(x,;x,)
from (<ref>) but with an additional factor of w+z+1 in the denominator of the integrand. Indeed,
I_ = ∫_0^ K_edge(x,t;x,t) dt, which is the expected number of points of _edge
on {x}× [0,]. The quantity I_ remains uniformly bounded near = 0 since, as
→ 0, the  contribution to the integral that defines I_ comes from the residues at w,z = 0
and these residues do not depend on . (See Lemma <ref> where lim_→ 0 K_edge(x,;x,)
is derived analogously.) Consequently, I_→ 0 as → 0 and
thus the condition from (<ref>) holds.




 §.§ Integral representation of the edge kernel
 
We begin with an auxiliary lemma.

 
Let C_w [0,∞) and C_z [0,∞) be contours as in the statement of
Theorem <ref> and let G(z;x,u) be as in (<ref>). For t > 0 let

    I(t) = [0,∞)[0,∞)π/2·G(w; x_1,tu_1)   G(z; x_2,tu_2))/w+z +x_2-x_1+1.

Then,

    lim_t → 0 I(t) = 2/π cos (π/2x_1) cos(π/2 x_2) /x_2-x_1+1   if   x_2 ≠ x_1-1
    
    -x_1  even   if   x_2 = x_1-1.

Moreover, I is continuously differentiable on _> 0 and can be differentiated by
interchanging differentiation with the contour integration.



The integrand of I is continuously differentiable in t. Observe that the contours
of integration contain no singularities of the integrand, and in fact, are
arranged to be a positive distance from all zeroes of sin(π/2(z+x_2)) and
sin(π/2(w+x_1)) in the denominator. The estimate for |u^z| / |z+1|
from (<ref>) shows that the derivative of the integrand in the variable t is
absolutely integrable over the contours as long as t lies in a compact subset of _≥ 0.
Consequently, by the dominated convergence theorem, I is continuously differentiable
and the derivative may be interchanged with integration.

Let us now consider the limiting value of I(t) as t → 0. Decomposing
C_w as C_w[0,1) ∪ C_w[1,∞), and similarly for C_z, gives

    ∮_C_z∮_C_w = ∮_C_z[0]∮_C_w[0] +
    ∮_C_z[0]∮_C_w[1,∞]  + ∮_C_z[1,∞]∮_C_w[0] +
    ∮_C_z[1, ∞]∮_C_w[1,∞].

These four contours may also be arranged such that (w+z) > 0 unless both
w ∈ C_w[0] and z ∈ C_z[0]. Recall that

    G(z; x,u) = u^z/z+1sin (π/2(z+x)).

Thus, the integrand of I(t) converges to 0 as t → 0 so long as w ∉ C_w[0] and z ∉ C_z[0].
So each of the double contour integrals above except for the first has a limit
value of 0 as t → 0 (the limit operation may be interchanged with integration as argued above).
To complete the proof it suffices to calculate

    lim_t → 0 [0][0]π/2·G(w; x_1,tu_1)   G(z; x_2,tu_2))/w+z +x_2-x_1+1.


The integral above is evaluated via residues at w=0 and z=0.
If x_2 ≠ x_1-1 then (<ref>) equals

    Res_z=0( Res_w=0 ( (π/2)  G(w;x_1,tu_1) G(z;x_2,tu_2)/w+z+x_2-x_1+1) )
    = 2/πcos (π/2x_2)cos (π/2x_1)/x_2-x_1+1.

This is the limit value of I(t) in the statement of the lemma for x_2 ≠ x_1-1.

Now consider the limit (<ref>) in the case x_2 = x_1-1.
As the contour C_w[0] can be arranged to be contained inside C_z[0], the integral in w equals the
residue of the integrand at the only possible pole at w = 0. This equals

    Res_w=0 ((π/2)   G(w;x_1,tu_1) G(z;x_2,tu_2)/w+z) =
    cos(π/2 x_1) (tu_2)^z/zz+1sin(π/2(z+x_2)).

If x_1 is odd then the above equals 0. Otherwise, cos( (π/2)x_1) = (-1)^x_1/2 and
the integral of the above over C_z[0] is given by its residue at the pole z=0
(note that sin(π/2 x_2) ≠ 0 since x_2 = x_1-1 is odd). The residue equals

    Res_z=0 (cos(π/2 x_1) (tu_2)^z/zz+1sin(π/2(z+x_2))) =
    cos(π/2x_1)/sin(π/2x_2) = -1.

Thus, if x_2 = x_2-1 then (<ref>) equals - x_1  even and this completes the proof.


 
The kernel K_edge has the following form.

    K_edge(x_1,u_1;x_2,u_2) =
    2/π∫_1^0 t^x_2-x_1cos( tu_1 + π/2 x_1) cos( tu_2 + π/2x_2  ) dt,    if   x_2 ≥ x_1; 
     - 2/π∫_1^∞ t^x_2-x_1cos( tu_1 + π/2 x_1) cos( tu_2 + π/2x_2  ) dt,    if   x_2 < x_1.




For t > 0 let

    f(t) = K_edge (x_1, tu_1; x_2, tu_2) = t^x_1-x_2-1u_1 > u_2, x_1 > x_2(u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! + I(t),

where I(t) is as defined in Lemma <ref>. By Lemma <ref>, f
is continuous differentiable on _>0 and the function t^x_2-x_1+1f(t) may
be differentiated by interchanging differentiation with integration.
Differentiating t^x_2-x_1+1f(t) and clearing common powers of t gives

    (x_2-x_1+1)f + tf'    = [0,∞)[0,∞)π/2· G(w; x_1,tu_1)   G(z; x_2,tu_2)) 
       = π/2·1/2 π𝐢∮_C_w[0,∞) dw  G(w; x_1,tu_1) ·1/2 π𝐢∮_C_z[0,∞) dz  G(z; x_2,tu_2)).


The contour integrals can be evaluated by summing over residues of G. Inside the contour
C_z[0,∞], the function G(z,x,u) has simple poles at integers z such that sin (π(z+x)/2) = 0.
In other words, z has to have the same parity as x. Let x̅ =  x  odd.
The residues of the integral come from integers z = 2k + x̅ for k ≥ 0.
Since Res_y=2k(sin(π y/2)^-1) = (2/π)(-1)^k, summing over the residues gives

    1/2 π𝐢∮_C_z[0,∞) dw (tu)^z/z+1sin (π/2(z+x))   = 2(-1)^x+x̅/2/π∑_k ≥ 0  (-1)^k (tu)^2k + x̅/(2k + x̅)!
        = 2(-1)^x+x̅/2/πcos( ut -π/2x̅)
        = 2/πcos( ut +π/2 x).

Consequently,

    (x_2-x_1+1)f(t) + tf'(t) = 2/πcos( u_1t +π/2x_1 ) cos ( u_2t + π/2x_2 ).

Multiplying (<ref>) by t^x_2-x_1 then implies that

    [t^x_2-x_1+1f]' = 2/π t^x_2-x_1cos( u_1t +π/2x_1) cos ( u_2t + π/2x_2 ).


For x_2 ≥ x_1, the r.h.s. of (<ref>) is integrable over t in [0,1].
Moreover, t^x_2-x_1+1f(t) → 0 as t → 0 because lim_t → 0 f(t) = lim_t → 0 I(t),
and the latter limit is finite whereas t^x_2-x_1+1→ 0 as t → 0. Therefore, (<ref>) implies

    f(1) = 2/π∫_0^1
    t^x_2-x_1cos( u_1t +π/2x_1 ) cos ( u_2t + π/2x_2 )  dt.


Next, consider the case x_2 < x_1-1. Now the relation from (<ref>) should be integrated
from 1 to ∞, which is convergent since x_2-x_1 ≤ -2. The formula follows
so long as lim_t →∞ t^x_2-x_1+1f(t) = 0. Rather than derive this limit we take a
slightly indirect approach by considering the limit of f(t) near t=0.
For t > 0 define

    g(t)    = - 2/π∫_1^∞ s^x_2-x_1cos ( u_1st +π/2x_1) cos ( u_2st + π/2x_2)  ds 
       = - 2/π  t^x_1-x_2-1∫_t^∞ s^x_2-x_1cos ( u_1s +π/2x_1) cos ( u_2s + π/2x_2)  ds.

Upon differentiating g it follows readily that g satisfies the same differential equation
as f displayed in (<ref>). Therefore, f(t) = g(t) + C for some constant C.
In order to identify C as zero it suffices to show that lim_t → 0 f(t)-g(t) = 0.
Since t^x_1-x_2-1→ 0 as t → 0, due to x_2 < x_1-1, both f(t) and I(t)
have the same limit as t → 0. Thus, utilizing Lemma <ref>, showing C=0 amounts to proving

    lim_t → 0 g(t) = 2/π cos (π/2x_1) cos(π/2 x_2) /x_2-x_1+1.


The limit of g(t) can be found using L'Hôspital's rule, which shows that

    lim_t → 0 g(t)    = lim_t → 0- 2/π∫_t^∞ s^x_2-x_1cos ( u_1s +π/2x_1)
    cos ( u_2s + π/2x_2) ds/t^x_2-x_1+1
       = lim_t → 0 2/π  t^x_2-x_1cos ( u_1t +π/2x_1)
    cos ( u_2t + π/2x_2)/(x_2-x_1+1) t^x_2-x_1
        = 2/π cos(π/2x_1) cos(π/2x_2) /x_2-x_1+1.

We conclude that g(t) = f(t) for t ∈_> 0, and in particular that f(1) = g(1), as required.

Finally, consider the case x_2 = x_1-1. The r.h.s. of (<ref>) is continuous for
t in [0,1] because one of cos ( u_1t +π/2x_1) or cos ( u_2t + π/2x_2)
has a zero at t=0 depending upon the parity of x_1. From Lemma <ref>,
lim_t → 0 f(t) = u_1 > u_2 -  x_1  even. Therefore, (<ref>) implies that

    f(1) = u_1 > u_2 -  x_1  even +
    2/π∫_0^1 t^-1cos ( u_1t +π/2x_1) cos ( u_2t + π/2x_2) dt.


We now express (<ref>) as an integral over t ∈ [1,∞) as given in the proposition.
First, note K_edge may be modified on the measure zero set
consisting of (x_1,u_1;x_2,u_2) such that u_1=u_2 without changing determinants
in (<ref>), and thus, this does not affect the law of _edge. We will modify
the kernel on this zero set after the following calculations to get the form given in the proposition.

Observe that cos(u_2t + π/2 x_2) = sin(u_2t + π/2x_1) if x_2=x_1-1.
Using trigonometric formulae the integrand of (<ref>) becomes

    2 cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1)/π t = sin((u_1+u_2)t) + sin((u_2-u_1)t)/π t,     x_1  even
    -sin((u_1+u_2)t) + sin((u_2-u_1)t)/π t,     x_1  odd.

Using the fact that ∫_0^∞sin t/t = π/2, we get that for a ∈,

    ∫_0^1 sin (at)/π t dt = sgn(a) ∫_0^|a|sin t/π t dt=
    sgn(a)/2 - ∫_1^∞sin(at)/π t dt.


Using (<ref>) and the representation of the integrand in (<ref>) we infer that if x_1 is even then

    u_1 > u_2 -  x_1  even + 2/π∫_0^1 t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt = 
       u_1 > u_2 - 1 + sgn(u_1+u_2) + sgn(u_2-u_1)/2
    - ∫_1^∞sin((u_1+u_2)t) + sin((u_2-u_1)t)/π t dt = 
        - u_1=u_2 (1 + u_1=0/2) - 2/π∫_1^∞ t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt.

This shows that (<ref>) equals the expression given in the statement of the proposition for
x_2 = x_1-1 and x_1 even except for the additive term - u_1=u_2 (1 + u_1=0/2).
By modifying K_edge on the zero set {u_1=u_2, x_2=x_1-1, x_1  even} we may ignore this term.

For x_1 being odd we argue in the same manner as above to infer that (<ref>) equals

    u_1 > u_2 -  x_1  even + 2/π∫_0^1 t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt = 
       u_1 > u_2  - sgn(u_1+u_2) + sgn(u_1-u_2)/2 +
    ∫_1^∞sin((u_1+u_2)t) + sin((u_1-u_2)t)/π t dt= 
        = -u_1=u_2 > 0/2 - 2/π∫_1^∞ t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt.

Once again, we modify K_edge on the zero set {u_1 = u_2, x_2=x_1-1, x_1  odd}
to ignore the additive term -1/2u_1=u_2 > 0 and get the expression of the kernel
given in the proposition.





 §.§ Statistical properties of _edge
 

This section derives certain properties of _edge, namely, Proposition <ref>,
Proposition <ref>, Lemma <ref> and Lemma <ref>,
that will be used to derive the local limit of staircase shaped tableaux and of sorting networks.

 
The process _edge has the following statistical properties.

I) Translation and reflection invariance: For any integer h the translated process

    _edge + (2h,0) = { (x+2h,u) : (x,u) ∈_edge}

and the reflected process

    (-1,1) * _edge = {(-x,u): (x,u) ∈_edge}

have the same law as _edge.

II) One dimensional marginals: For any x ∈ and u_1, u_2 ∈_≥ 0,

    K_edge(x,u_1; x,u_2) = sin(u_1-u_2)/π  (u_1-u_2) + (-1)^x  sin(u_1+u_2)/π  (u_1+u_2).

Therefore, _edge∩ ({x}×_≥ 0) is a determinantal point process
with reference measure (_≥ 0) and correlation kernel (<ref>).



Part I) From Lemma <ref> the correlation kernel of _edge + (2h,0)
equals K_edge(x_1-2h,u_1; x_2-2h,u_2). The integral representation of
K_edge in Proposition <ref> implies that

    K_edge(x_1-2h,u_1; x_2-2h,u_2) = K_edge(x_1,u_1; x_2,u_2)

upon observing that cos (x + π h) = (-1)^h cos(x), which implies that the integrands do not
change after the kernel is transformed. Consequently, the translated point process has the same law
as the original. Similarly, the correlation kernel for the reflected process is
K_edge(-x_1,u_1;-x_2;u_2) = (-1)^x_1-x_2 K_edge(x_2,u_2;x_1,u_1).
The latter kernel defines the same determinantal point process as _edge in law.
0.1in
Part II) Proposition <ref> gives that

    K_edge(x,u_1;x,u_2)    = 2/π∫_0^1 cos(tu_1 + π/2x) cos(tu_2 + π/2x) dt 
       = 1/π(sin(tu_1-tu_2)/u_1-u_2 + sin(tu_1 + tu_2 + π x)/u_1+u_2 )
     |_t=0^t=1
       = 1/π ( sin(u_1-u_2)/u_1-u_2 + (-1)^x sin(u_1+u_2)/u_1+u_2).

The fact hat _edge∩ ({x}×_≥ 0) is determinantal with kernel as stipulated
follows from the relation (<ref>) for determinantal point processes.


 
There is a universal constant C such that for x_1,x_2 ∈ and u_1, u_2 ∈_≥ 0,

    | K_edge(x_1,u_1;x_2,u_2) | ≤C/max{ |x_1-x_2|, |u_1-u_2| } + 1.




Throughout this argument C denotes a universal constant whose value may change from line to line.
We begin with the case x_2 ≠ x_1-1. From the integral representation of K_edge we see
that if x_2 ≥ x_1 then

    |K_edge(x_1,u_1;x_2,u_2)|    = 2/π | ∫_0^1 t^x_2-x_1cos(tu_1+π/2x_1)cos(tu_2+π/2x_2)   dt  | 
       ≤2/π∫_0^1 t^x_2-x_1 dt ≤C/|x_2-x_1|+1.

Similarly, if x_2 < x_1 then x_2 ≤ x_1-2 and

    |K_edge(x_1,u_1;x_2,u_2)|    = 2/π | ∫_1^∞ t^x_2-x_1cos(tu_1+ π/2x_1)cos(tu_2+π/2x_2)   dt  | 
       ≤C/|x_2-x_1|+1.

Combining these bounds we deduce that if x_2 ≠ x_1 -1 then

    | K_edge(x_1,u_1;x_2,u_2)| ≤C/|x_1-x_2|+1.


Now we consider decay in the u-variables, assuming that x_2 ≠ x_1-1.
Define v(t) as

    v(t) = sin (tu_1-tu_2 + π/2 (x_1-x_2))/π(u_1-u_2) +
    sin (tu_1+tu_2 + π/2 (x_1+x_2))/π(u_1+u_2)   if  u_1 ≠ u_2 
    
    t cos(π/2(x_1-x_2))/π +
    sin (tu_1+tu_2 + π/2 (x_1+x_2))/π(u_1+u_2)   if  u_1 = u_2.

Then v'(t) = 2/πcos(tu_1 + π/2x_1)cos(tu_2 + π/2x_2).
Using that |sin(y)/y| ≤ 1, and the formula for v(t), we observe that there is a C such that

    |v(t)| ≤C/|u_1-u_2|+1  if   |u_1-u_2| ≥ 1.


Applying integration by parts to the integral form of K_edge gives, for x_2 ≥ x_1,

    K_edge(x_1,u_1;x_2,u_2) = v(1) -v(0)x_1=x_2 - ∫_0^1 (x_2-x_1)  t^x_2-x_1-1 v(t)   dt.

Now the triangle inequality and (<ref>) imply that if |u_1-u_2| ≥ 1 then

    |K_edge(x_1,u_1;x_2,u_2)|    ≤C/|u_1-u_2|+1 + C   |x_2-x_1|/|u_1-u_2|+1 ∫_0^1 t^x_2-x_1-1  dt 
       ≤2C/|u_1-u_2|+1.

If |u_1-u_2| < 1, then we use the bound (<ref>) to reach the same conclusion as above.
An entirely analogous bound holds when x_2 < x_1 because then x_2 ≤ x_1 -2,
and t^x_2-x_1 is integrable over t ∈ [1,∞). Therefore, for x_2 ≠ x_1-1,

    | K_edge(x_1,u_1;x_2,u_2)| ≤C/|u_1-u_2|+1.

Combining (<ref>) with (<ref>) implies the
required inequality (<ref>) for x_2 ≠ x_1-1.

The case x_2 = x_1-1 requires some care. The representation (<ref>) for
the integrand of K_edge(x_1,u_1;x_1-1,u_2) gives

    K_edge(x_1,u_1;x_1-1,u_2)    =
    - 2/π∫_1^∞ t^-1cos( tu_1 + π/2 x_1) sin( tu_2 + π/2x_1)  dt 
        = - ∫_1^∞(-1)^x_1sin((u_1+u_2)t) + sin((u_2-u_1)t)/π t  dt.


Integration by parts and the triangle inequality imply that for a ≥ 1,

    | ∫_1^∞ dt sin(a t)/t | =  | cos(a)/a - ∫_1^∞ dt cos(at)/at^2| ≤C/a .

For 0 ≤ a ≤ 1, we have

    | ∫_1^∞ dt sin(a t)/t | =  | π/2 - ∫_0^1 dt sin(at)/t | ≤ C + ∫_0^1 dt  a ≤ C.

Together, these bounds imply that for a ∈,

    | ∫_1^∞ dt sin(a t)/t | ≤C/|a|+1.


Separating (<ref>) naturally into two integrals
and applying (<ref>) implies that

    |K_edge(x_1,u_1;x_1-1,u_2)| ≤C/|u_1-u_2|+1.

This establishes (<ref>) for the case x_2 = x_1-1 and completes the proof.




  
Spatial ergodicity of _edge
For h ∈, denote by τ^h the translation that maps (x,u) ↦ (x+2h,u) for (x,u) ∈×_≥ 0.
So _edge is invariant under the action of every τ^h by Proposition <ref>.
An event E associated to _edge is invariant if for every h ∈,
E = τ^h E, where τ^hE = {τ^h(ω): ω∈ E} and τ^h(ω) is the
action of τ^h on a sample outcome ω of _edge. The invariant sigma-algebra
of _edge is the sigma-algebra _inv consisting of all the invariant events.

 
_edge is ergodic w.r.t. spatial translations in that if E ∈_inv then E∈{0,1}.



For A ⊂×_≥ 0, let (A) = σ (_edge∩ A) be the sigma-algebra
generated by the points of _edge restricted to A.  For A, B ⊂×_≥ 0, let

    dist(A,B) = inf {max{|x-y|, |u-v|}: (x,u) ∈ A, (y,v) ∈ B }.

For k ≥ 1, suppose f: (×_≥ 0)^k → is continuous and compactly supported. Let

    N(f) = ∑_(x_1,u_1), …, (x_k,u_k) ∈_edge
     (x_i,u_i) all distinct f(x_1,u_1;⋯; x_k,u_k).

Now suppose f, g : (×_≥ 0)^k → are continuous and compactly supported such that
there are disjoint subsets A, B ⊂×_≥ 0 with
support(f) ⊂ A^k and support(g) ⊂ B^k. This implies that if
(x_1,u_1;⋯; x_k,u_k) ∈support(f) and (x_k+1,u_k+1;⋯; x_2k, u_2k) ∈support(g),
then (x_i,u_i) ≠ (x_k+j,u_k+j) for every 1 ≤ i,j ≤ k. We first show that in this case

    | N(f)N(g) - N(f)N(g) | ≤(2k)! C^2k/dist(A,B)^2 +1  ||f||_1   ||g||_1,

where C is the universal constant from Lemma <ref> and ||f||_1
is the L^1-norm of f with respect to (#_⊗(_≥ 0))^⊗ k.

Indeed, the assumption on the supports of f and g imply from (<ref>) that

    N(f)N(g) = ∫_(×_≥ 0)^2k    [ K_edge(x_i,u_i;x_j,u_j)]_1 ≤ i,j ≤ 2k 
    f(x_1,u_1;⋯; x_k,u_k) ×
        g(x_k+1,u_k+1;⋯; x_2k,u_2k)  d  (#_⊗(_≥ 0) )^⊗ 2k.

Let us expand the determinant of the (2k) × (2k) matrix above as a sum over all permutations.
We break up the permutations into two types: permutations that map the subsets {1,…, k} and {k+1,…, 2k}
into themselves and those that do not. When summed over permutations of the first type the integral above
equals N(f)N(g). For a permutation σ of the second type, observe that there are two indices i and j, with
i ≤ k and j > k, such that σ(i) > k and σ(j) < k. Then for ℓ∈{i,j}, Lemma <ref> gives

    |K_edge(x_ℓ,u_ℓ; x_σ(ℓ), u_σ(ℓ)) | ≤C/max{ |x_ℓ - x_σ(ℓ)|, |u_ℓ-u_σ(ℓ)|} + 1≤C/1 + dist(A,B).

For all other indices ℓ we have |K_edge(x_ℓ,u_ℓ; x_σ(ℓ), u_σ(ℓ))| ≤ C.
Consequently, the term involving σ contributes at most C^2k-2 (1 + dist(A,B))^-2 in absolute value to the determinant
above for every (x_1,u_1; ⋯; x_k,u_k) ∈support(f) and (x_k+1,u_k+1;⋯;x_2k, u_2k) ∈support(g).
Since there are (2k)! -(k!)^2 such permutations σ, we conclude that

    | N(f)N(g) - N(f)N(g) |    ≤(2k)!C^2k/dist(A,B)^2+1∫_(×_≥ 0)^2k |f g|   d  (#_⊗(_≥ 0) )^⊗ 2k
        = (2k)!C^2k/dist(A,B)^2+1  ||f||_1 ||g||_1.


Let ^k(A) be the sigma-algebra generated by the random variables N(f), where f : A^k →
is continuous and compactly supported. The bound (<ref>) implies that if A and B are disjoint,
X is ^k(A)-measurable and Y is ^k(B)-measurable, then,

    | XY - XY| ≤(2k)! C^2k/dist(A,B)^2 + 1 |X||Y| .


The bound in (<ref>) implies ergodicity of _edge as follows.
Let E ∈_inv. Given 0 <  < 1, we may choose an event E' ∈^k([-n,n] ×_≥ 0),
for some k and n, such that E Δ E' <. Since _edge is invariant
under τ^h, we have that τ^h E Δτ^h E' = E Δ E' for every h.
Therefore by the triangle inequality,

    | E' ∩τ^h E' - E ∩τ^h E| ≤E' Δ E + τ^h E Δτ^h E'≤ 2 .

Due to invariance of E this implies that |E' ∩τ^h E' - E| ≤ 2.

Set h = n + m for an integer m ≥ 1. Then τ^h E' ∈^k([n+2m,3n+2m] ×_≥ 0).
We now apply (<ref>) with A = [-n,n] ×_≥ 0 and B = [n+2m,3n+2m] ×_≥ 0,
observing that dist(A,B) = 2m. Since τ^h E' = E' by translation invaraince, we infer that

    | E' ∩τ^h E' - E'^2  | ≤(2k)!C^2k/4m^2.

Since | E' - E| ≤, we conclude that

    |E - E^2| ≤(2k)!C^2k/4m^2 + 5 .

Letting m →∞ followed by → 0 shows that E = E^2, as required.


 
The proof above may be used to deduce that _edge is in fact space-time mixing.


 
Almost surely, _edge has an unbounded collection of points on every line {x}×_≥ 0.
In fact, the following holds. Let N_x(t) = # (_edge∩ ({x}× [0,t]) ).
For every x, the sequence N_x(t)/t → 1/π in probability as t →∞.



Fix x ∈. Using part (II) of Proposition <ref> we see that for any interval [a,b] ⊂_≥ 0,

    N_x([a,b]) = ∫_a^b K_edge(x,u,x,u) du
    = b-a/π + (-1)^x/2 π∫_2a^2b du sin u/u .

Observe from (<ref>) that N_x(t)/t = 1/π + O(1/t) as t →∞.

From (<ref>) we see that K_edge(x,u_1;x,u_2) is symmetric in the variables u_1 and u_2. Thus,

    ρ(u_1,u_2) := K_edge(x,u_1;x,u_2) K_edge(x,u_2;x,u_1) = K_edge(x,u_1;x,u_2)^2 ≥ 0.

From the relation (<ref>) for determinantal point processes we have that

    N_x(t) · (N_x(t)-1)   = ∫_0^t ∫_0^t  [ K_edge(x,u_i;x,u_j)]_i,j=1,2  du_1 du_2
       =  (∫_0^t K_edge(x,u;x,u)  du )^2 - ∫_0^t ∫_0^t ρ(u_1,u_2)   du_1 du_2 
       ≤N_x(t)^2.

This inequality implies that Var(N_x(t)) ≤N_x(t).
Since N_x(t) = (t/π) + O(1), Chebyshev's inequality implies that for any > 0,

    | N_x(t)/t - 1/π | >  =  O(1)/^2   t.

This provides the claimed convergence in probability.

Convergence in probability implies that there is a sequence of times t_k →∞
such that N_x(t_k)/t_k → 1/π almost surely as k →∞.
This in turn implies that there is an unbounded collection of points of _edge on
{x}×_≥ 0 almost surely. An union bound over x provides the claim in the lemma.


 
The following event occurs almost surely. For every
t > 0 there exists a doubly infinite sequence of integers x_i, i ∈,
such that _edge contains no points on each of the segments {2x_i}× [0,t].



By monotocity and an union (or rather intersection) bound over rational values of t,
it suffices to show that the event occurs almost surely for every fixed t > 0. Given a fixed
t, let X_i be the indicator of the event that _edge has no points on
{2i}× [0,t]. It suffices to show that almost surely infinitely many of the X_is equal 1
for i ≥ 0. Then, reflection invariance of _edge and another union bound
imply that almost surely a doubly infinite collection of the X_is are equal to 1, as required.

Due to translation invariance of _edge the sequence X_i, i = 0,1,2,…,
is stationary in that (X_0,X_1, …) has the same law as (X_1,X_2, …). It is
also ergodic by Proposition <ref>. Therefore, by the Ergodic Theorem,

    lim_n →∞ 1/n∑_i=0^n-1 X_i   = X_0 = 1,  almost surely.

The probability that X_0 = 1 is the probability that _edge has no points in {0}× [0,t].
This is strictly positive by (<ref>) below. As a result, an infinite number of the X_is equal
1 whenever the limit in the above holds.





§ THE LOCAL STAIRCASE SHAPED TABLEAU
 

The local staircase shaped tableau, henceforth, local tableau, is a random function on

    = { (x,y) ∈^2: y ≥ 0,   x ≡ y  (mod  2) }.

Figure <ref> provides an illustration. The rows and columns of  are
given by the diagonal lines

    row  2x = {(2x-k,k): k ≥ 0},   column  2x = {(2x+k,k): k ≥ 0}.




In order to define tableaux on  and their convergence, we first explain the topology on [0,∞]
since tableau entries will take values there (we allow the value ∞). The topology on [0,∞] is the
usual topology on _≥ 0 extended in the natural way by stipulating that a sequence
converges to ∞ if its values diverge to ∞, possibly stabilizing to the value ∞. In this case
we will say that the sequence grows to ∞.  For example, 1,2,3,4, … grows to ∞,
as does 1,∞,2,∞, …, as well as 1,∞,∞,∞,….

A tableau is a function T: → [0,∞]  such that it satisfies the tableau constraints

    I)    T(x,y) ≤min{ T(x-1,y+1),   T(x+1,y+1) }  for every   (x,y) ∈. 
    II)    Along every row and column of   T  the entries grow to ∞.


The YD  embeds into  via (i,j) ∈↦ (j-i - n  odd, n-i-j) ∈.
This is a rotation that puts row r of  on row 2(⌊ n/2 ⌋ - r) of ; see Figure <ref>.
In this manner any PYT T of shape  embeds as a tableau
F_T : → [0,∞] by setting

    F_T(x,y) = 
    n   (1-T  ( ⌊n/2⌋ - x+y/2, ⌊n/2⌋ + x-y/2) ),
       if   (⌊n/2⌋ - x+y/2, ⌊n/2⌋ + x-y/2) ∈; 
    ∞,    otherwise.

By an abuse of notation we denote F_T by T.

We say that a sequence of tableaux T_n converges if there is a tableau T such
that, in the aforementioned topology on [0,∞], T_n(x,y) → T(x,y) for every (x,y) ∈Δ_∞.
Note we stipulate that a limit of tableaux remain a tableau.

A random tableau 𝐓 : → [0,∞] is a Borel probability measure on tableaux
with respect to the topology above. Convergence of a sequence of random tableaux means
weak convergence with respect to this topology.



 §.§ Bulk local limit of staircase shaped tableaux
 

Section <ref> describes how PYTs of a given shape are in
bijection with ensembles of non-increasing and non-intersecting paths whose initial
positions are given in terms of the shape. We describe the bijection explicitly for
tableaux defined on  as it will be useful in the proof of the local limit
theorem.

Consider an ensemble of paths { p(2x,u) }, for x ∈ and u ∈_≥ 0, that
satisfy the following.

    I)     p(x,·) : _≥ 0→ + 1/2  is left continuous, non-increasing with   p(x,0)=2x + 1/2. 
    II)     p(x,·)   are non-intersecting:   p(x,u) > p(x-1,u)   for every   x ∈, u ∈_≥ 0. 
    III)    The jumps of the paths as defined by  (<ref>)  is a discrete subset of  ×_≥ 0.


For a tableau T : → [0,∞], paths satisfying (<ref>)
are obtained by setting p(x,u) = 2x + 1/2 for 0 ≤ u ≤ T(2x,0), and for k ≥ 1,

    p(x,u) = p(x,0) - k   if   T(2x-k+1,k-1) < u ≤ T(2x-k,k).

In other words, p(x,·) is left continuous and decreases by integer units at times indexed by
row 2x of T. The paths are non-intersecting due to the columns of T being non-decreasing.
Indeed, p(x,u) - p(x-1,u) = 2 +N_x-1(u) - N_x(u), where N_x(u) is the number of entries of
T on row 2x with value at most u. Due to the columns being non-decreasing,
N_x-1(u) ≥ N_x(u) -1, and thus, p(x,u) - p(x-1,u) ≥ 1.
The jumps of the paths form a discrete set due to the rows and columns
of T growing to ∞. When a row entry equals ∞ then the corresponding
path jumps only a finite number of times.

Let X denote the jumps for the ensemble of paths associated to a tableau T on .
The jumps can be read off from T in the following manner. For every x ∈, the
jumps on the line {x}×_≥ 0 are the entries of T whose cells have
first coordinate x in . More precisely, if u is the k-th smallest point of X
on {x}×_≥ 0 then

    u =T (x,   2k-1-x  even).

If there are less than k points on {x}×_≥ 0, there is no such
u and the tableau entry above equals ∞.

To see this, observe that u is the time when the path starting at initial position
(x+ 1/2)+2k-1 - x even jumps for the (2k-x even)-th time.
Indeed, this is the k-th path starting at or to the right of position x+ 1/2
and it hits position x - 1/2 after jump number 2k-x even.
The first k jumps on {x}×_≥ 0 are the times when the first k
paths starting at or to the right of position x + 1/2 hits position x - 1/2.
Also, when there is no such u it means that the path starting from
(x+ 1/2)+2k-1 - x even has exhausted its jumps and it does not
get to position x - 1/2.

Let M_T → X denote the map from tableaux defined on  to jumps
of paths satisfying (<ref>). This map is invertible
with the inverse given by the relation (<ref>). Namely, T(x,y)
is the [(y+1+x even)/2 ]–th smallest jump of X on {x}×_≥ 0
with the convention that T(x,y) = ∞ if no such jump exists. Let M_X → T
denote the inverse map.

 
The set of jumps of paths satisfying (<ref>) is closed in the topology
on discrete subsets of ×_≥ 0. The map M_T → X is a homeomorphism
from the set of tableaux on  to the set of jumps of paths satisfying
(<ref>), with inverse given by M_X → T.



The maps M_T → X and M_X → T are inverses by design.
We must show that they are continuous. We begin with continuity of M_T → X.
Let T_n be a sequence of tableaux such that T_n converges to a tableau T_∞.
Let X_n = M_T → X(T_n) and X_∞ = M_T → X(T_∞).
Recall from Section <ref> that convergence of X_n to X_∞
requires that for every x ∈ and k ≥ 1, the k-th smallest point of X_n
on {x}×_≥ 0 must converge to the corresponding point of X_∞,
while accounting for the case that there may be less than k points.

Let y = 2k-1-x  even. Then, by (<ref>), T_n(x,y) is the k-th smallest point
of X_n on {x}×_≥ 0 and similarly for T_∞(x,y).
There are two cases: T_∞(x,y) < ∞ or T_∞(x,y) = ∞.
In the former case, the k-th smallest point of X_n on {x}×_≥ 0
is eventually finite and the same for X_∞. Moreover, we have convergence
of these points since T_n(x,y) → T_∞(x,y). In the latter case, given any
bounded subset of {x}×_≥ 0, the k-th smallest point of X_n
eventually escapes the set or is non-existent due to T_n(x,y) →∞.
This is as required since X_∞ has no k-th smallest point on {x}×_≥ 0.
This proves that M_T → X is continuous.

Now we show that the set of jumps of paths satisfying (<ref>) is
a closed set in the space of discrete subsets of ×_≥ 0, as well as that
that M_X → T is continuous. Suppose X_n is a sequence of such jumps sets and that
it converges to a discrete subset X_∞.

Let T_n = M_X → T(X_n). First, we show that T_n(x,y) converges for every (x,y) ∈.
Indeed, T_n(x,y) is the k-th smallest point of X_n on {x}×_≥ 0 for
k = (y+1+x even)/2. Therefore, convergence of X_n to X_∞
implies that T_n must converge to some function T_∞: → [0,∞].
Note that T_∞(x,y) = ∞ if and only if X_∞ has less than k points
on {x}×_≥ 0.

The function T_∞ is a tableau because the tableau inequalities from (<ref>)
continue to hold in the entry-wise limit, and the rows and columns will grow to ∞ due to
X_∞ being a discrete set. Thus, consider X̂_∞ = M_T → X(T_∞).
By the first part of the proof, X_n →X̂_∞. But then, X̂_∞ = X_∞
because limits of discrete subsets of ×_≥ 0 are unique. This shows both
the closure property of sets of jumps for paths satisfying (<ref>) and
the continuity of M_X → T.


We now state the local limit theorem for Poissonized staircase shaped tableaux in the bulk.
For α∈ (-1,1), let c_n = 2 ( ⌊n/2⌋ - ⌊(1+α)n/2⌋).
For a PYT T having shape , embed it as a tableau on  according to (<ref>)
and consider the rescaled tableau T_α,n : → [0,∞] defined by

    T_α,n(x,y) = √(1-α^2)  T(x+c_n, y).

Let _ be a uniformly random PYT of shape  and denote by _α,n
the random tableau associated to _ by (<ref>).

 
The sequence of random Poissonized tableaux _α,n converges weakly to
a random tableau _edge. Moreover, the law of _edge
is M_X → T( _edge).



Observe that |c_n - α n| ≤ 2 for every n. With this choice of c_n,
the jump process associated to _α,n has law _α,n from (<ref>)
because these jumps are simply the jumps of _ rescaled onto ×_≥ 0
as in (<ref>). Theorem <ref> asserts that _α,n converges
weakly to _edge. Due to being a weak limit of the jumps of ensembles of paths satisfying
(<ref>), _edge is also almost surely the jumps of such an ensemble
of paths by the closure property given in Lemma <ref>. The continuity of M_X → T
then implies that M_X → T(_α,n) converges weakly to M_X → T(_edge).
Therefore, _α,n converges weakly to a random tableau
_edge having the law of M_X → T(_edge).




  
Bulk local limit theorem for random staircase shaped SYT
Recall _ denotes a uniformly random SYT of shape  and N = n2.
Consider the rescaled tableau

    ^rsc_(i,j) = 𝐓_(i,j)/N+1.


 
The random tableau ^rsc_ converges to _edge in the bulk local limit,
that is, under the embedding and rescaling from (<ref>).



Consider the following coupling between _ and _. Given _,
independently sample P_(1) < P_(2) < ⋯ < P_(N) according to the
order statistics of N i.i.d. random variables distributed uniformly on [0,1].
Insert the entry P_(k) into the cell of  that contains entry k of _.
The resulting tableau has the law of _. Using this coupling, and due
to the manner the scaling from (<ref>) is defined, it suffices to show
the following in order to conclude that ^rsc_ converges to _edge
by way of Theorem <ref>.

Fix an L > 0 and consider any (x,y) ∈ such that (x,y) ∈ [-L,L] × [0,L].
Let P_(k) be the entry of _ inside (x,y) under the embedding from (<ref>).
Then as n →∞, we need to show that

    n   | P_(k) - k/N+1 |   ⟶  0   in probability.

The number k is random, its distribution depends on  as well as n and α.

In order to establish (<ref>) we will use the following auxiliary fact, which is
a byproduct of <cit.>. There is a number δ_n of order o(N)
as n →∞, such that with probability tending to 1 as n →∞, every entry of
_ within the cells of ∩ [-L,L] × [0,L] under the embedding (<ref>)
has value at least N - δ_n. As a consequence, k ≥ N - δ_n with probability tending to 1.
We write

    n  | P_(k) - k/N+1| > ≤ n^2/^2 | P_(k)- k/N+1|^2   |  k ≥ N- δ_n + k < N-δ_n.


For a fixed deterministic j,  P_(j) has a Beta distribution with parameters j and N+1-j, which
has mean j / (N+1) and variance

    | P_(j) - j/N+1|^2  = j (N+1-j)/(N+1)^2(N+2).

Since the P_(j)s are independent of , employing the bound above for j ≥ N- δ_n and
summing over the probabilities of k give

    | P_(k)- k/N+1|^2   |  k ≥ N- δ_n≤δ_n/N^2.

The latter quantity is of order o(1)/N as n →∞. Since N= n  2, we conclude that both terms on
the right hand side of (<ref>) tend to 0 as n →∞.




  
Statistical properties of the local staircase shaped tableau
The set  can be made into a directed graph by putting directed
edges from each vertex (x,y) ∈ to the vertices (x-1,y+1) and (x+1,y+1).
The automorphisms of this graph consists of translations ϕ_h,
for h ∈, given by ϕ_h(x,y) = (x+2h,y), as well as a reflection ϕ_-
given by ϕ_-(x,y) = (-x,y). Tableaux are preserved by these automorphisms.

A random tableau 𝐓 is translation invariant if
𝐓∘ϕ_h has the same law as 𝐓 for every translation ϕ_h.
The random tableau is reflection invariant if 𝐓∘ϕ_- has the same law as 𝐓.
The translation invariant sigma-algebra of 𝐓 is the sigma-algebra of events that
remain invariant under every translation:

    _inv = {Events  E  associated to 𝐓 s.t. ϕ_h E = E for every  h ∈. }.

(Recall that ϕ_h E = {ω∘ϕ_h: ω∈ E}.)
We say 𝐓 is ergodic under translations if _inv is the trivial sigma-algebra.

 
The local tableau _edge has the following statistical properties.


  * Almost surely, _edge(x,y) is finite for every (x,y) ∈
and the entries of _edge are all distinct.


  * The law of _edge is both translation and reflection invariant.


  * _edge is ergodic under translations.


  * Almost surely, for every t > 0 there are infinitely many positive and negative
x ∈ such that T_edge(2x,0) > t.




Almost surely, _edge has an infinite and unbounded collection of points
on every line {x}×_≥ 0 by Lemma <ref>. Also,
almost surely, _edge does not contain two points of the form
(x,u) and (y,u) with x ≠ y. To see this, observe from the relation (<ref>)
for determinantal point processes that the expected number of such pairs of points in _edge
is 0 due to the set of such pairs having measure zero with respect to the measure
(#⊗(_≥ 0))^⊗ 2. When both these properties hold,
_edge satisfies (1).

The law of _edge is invariant under translations because for
every translation ϕ_h, the tableau _edge∘ϕ_h is constructed
from the jump process _edge + (2h,0), which has the same law of _edge by
Proposition <ref>. Similarly, reflection invariance of _edge follows from reflection
invariance of _edge. This establishes (2).

The ergodicty of _edge under translations follows from the ergodicity of _edge
under translations (Proposition <ref>). This is because a translation invariant event for
_edge is the image of a translation invariant event for _ edge under the map M_X → T.
Finally, (4) is the statement of Lemma <ref>.




§ RANDOM SORTING NETWORKS
 



 §.§ Sorting networks, Young tableaux and Edelman-Greene bijection
 

Stanley <cit.> enumerated the number of sorting networks of 𝔖_n, which equals

    n2!/∏_j=1^n-1 (2n-1-2j)^j.

Following Stanley, Edelman and Greene <cit.> provided an explicit bijection between sorting
networks and staircase shaped SYT. An account of further combinatorial developments may be found in <cit.>.
We describe the part of the Edelman-Greene bijection that maps staircase shaped tableaux to sorting networks.
The inverse map is a modification of the RSK algorithm; we do not describe it here since it is not used in the paper.
See <cit.> or <cit.> for a full description of the bijection.

Recall that a sorting network of 𝔖_n is identified by its sequence of adjacent swaps
(s_1,…,s_N), where N = n2. For the rest of the paper we will use N to denote n2.
For T ∈SYT(), we adopt the convention that T(i,j) = - ∞ if (i,j) ∉.




  
The Schützenberger operator
Let (i_max(T),j_max(T)) denote the cell containing the maximum entry of a  SYT T.
The Schützenberger operator Φ : SYT() →SYT() is a bijection defined
as follows. Given T ∈SYT(), construct the sliding path of cells
c_0, c_1,…, c_d-1∈ iteratively in the following manner.
Set c_0 = (i_max(T), j_max(T)) and c_d = (1,1). Then set

    c_r+1 = argmax { T(c_r - (1,0)),   T(c_r - (0,1)) }.

Let Φ(T) = [T̂(i,j)] where T̂(c_r) = T(c_r+1) + 1 for 0 ≤ r ≤ d-1,
T̂(c_d) = 1, and T̂(i,j) = T(i,j)+1 for all other cells (i,j) ∈∖{c_0,…, c_d}.
Figure <ref> provides an illustration.



The Edelman-Greene map EG : SYT() ↦{sorting networks of 𝔖_n}
is defined by

    EG(T) =  ( j_max(Φ^k(T)))_0 ≤ k ≤ N-1 ,

where Φ^k is the k-th iterate of Φ. Edelman and Greene <cit.> proved
that EG indeed maps to sorting networks and that is has an inverse.



 §.§ First swap times of random sorting networks: proof of Corollary <ref>
 

Let T_FS(s) be the first time the adjacent swap (s,s+1) appears in a sorting network ω
of 𝔖_n. According to the Edelman-Greene bijection, this time is recorded in
the entry (n-s,s) of EG(ω). Thus,

    T_FS(s) = N+1 - EG(ω)(n-s,s).

In terms of the rescaled tableau _^rsc from (<ref>) we have that

    _FS(s)  law=  (N+1) · (1 - _^rsc(n-s,s)).

This implies the following for _FS, α,n – the first time an adjacent swap between
⌊n(1+α)/2⌋ and  ⌊n(1+α)/2⌋+1 appears in a
random sorting network of 𝔖_n:

    2 √(1-α^2)/n _FS, α,n law= 2N+2/n^2 _α, ^rsc(0,0).

Here, _α, ^rsc is the tableau _^rsc rescaled and embedded into 
according to (<ref>). Theorem <ref> implies that _α, ^rsc(0,0)
converges weakly to _edge(0,0). Since (2N+2)/n^2 → 1, we conclude that the rescaled
_FS, α,n from above converges weakly to _edge(0,0).
Thus, _FS has the law of _edge(0,0).

Now we explain how to get the distribution function of _FS given in
(<ref>). Observe that the event {_edge(0,0) > t} is
the event {_edge∩ ({0}× [0,t]) = ∅}.
The probability of the latter (often known as “gap probability") has the representation given by
(<ref>), which is the Fredholm determinant of K_edge over L^2({0}× [0,t]).
This is a well-known property of determinantal point processes under the condition that the kernel
be of trace class <cit.>. The kernel K_edge is of trace class on L^2({0}× [0,t]) simply because
|K_edge(0,u_1;0,u_2)| ≤ 2/π.

The asymptotic behaviour of the distribution function of _FS is well-known:

    log_FS > t = - 1/4t^2 - 1/2t - 1/8log t
      +7/24log 2+ 3/2ζ'(-1) + o(1)  as  t →∞.

The formula (<ref>) has a history. In theoretical physics literature,
the leading term in (<ref>) was first studied in <cit.>, while the
full expansion was given in <cit.>. The complete mathematical treatment was
developed in <cit.>; the present form of (<ref>) is given
in the last reference.

We will only need the simple corollary of (<ref>) that _FS> t > 0 for every t.



 §.§ Edelman-Greene algorithm on the local tableau
 

The procedure described here is the same as the one given in the Introduction except
that it is in the language of tableaux instead of their jumps. In order to define the
Edelman-Greene algorithm on the local tableau we first introduce some concepts
that allow us to define Edelman-Greene algorithm on tableaux defined on .

A directed path from (x,y) ∈ to (x',y') ∈ is a sequence of cells
c_0 = (x,y), c_1, …, c_k = (x',y') of  such that c_i+1-c_i ∈{ (-1,1), (1,1)} for every i.
The cells of  can be partially ordered as follows: (x,y) ≤ (x',y') if there is a directed path
from (x,y) to (x',y'). Recall that  is a directed graph with edges from (x,y) to (x± 1, y+1).
It can also be thought of as an undirected graph by forgetting the direction of the edges.
A connected subset of  is a connected subgraph of  in the undirected sense.

A Young diagram (YD) of  is a connected subset  that is downward closed in the partial
order, that is, if (x,y) ∈ and (x',y') ≤ (x,y) then (x',y') ∈. For example,  is a YD
of . The boundary of , ∂, consists of cells (x,y) ∉ such that there
is a directed edge from some cell (x',y') ∈ to (x,y). The peaks of  consists
of the maximal cells of  in the partial order.

Let T: → [0,∞] be a tableau as in (<ref>). A sub-tableau
is the restriction of T to a YD ; we say  is the support of the sub-tableau.
Let T^finite = { T(x,y): T(x,y) ≠∞}. We take the support of T to be the support
of T^finite. Observe that T^finite is a countable disjoint union of sub-tableaux of T,
say T_1, T_2, …. Indeed, the support of the T_is are the connected components of the
subgraph spanned by cells (x,y) such that T(x,y) ≠∞. We will call the T_is the
clusters of T. The tableau T is EG-admissible if all the entries of T^finite
are distinct and every cluster T_i is supported on a YD of finite size.




  
Edelman-Greene algorithm on a finite tableau

Let  be a YD of  of finite size and T: →_≥ 0 a tableau
such that all its entries are distinct. The Edelman-Greene map EG takes
as input T and outputs a triple (x,t,T̂), where x ∈, t ∈_≥ 0
and T̂ is a sub-tableau.

The sliding path of T is a directed path c_0, c_1, …, c_k defined by


  * c_0 = argmin {  T(x,y): (x,y) ∈}.

  * c_i+1 = argmin {  T(c_i + (-1,1)), T(c_i + (1,1))  }.

  * c_k = peak of  obtained when both c_k + (± 1,1) belong to ∂.

Let = ∖{c_k} and define T̂ : →_≥ 0 by

    T̂(x,y) = 
    T(x,y),    if   (x,y) ∈∖{c_0, …, c_k-1}; 
    
    T(c_i+1),    if   (x,y) = c_i  for some   0 ≤ i ≤ k-1.

The cell c_0 must be on the bottom level of  and has the form (2x,0) for some x ∈.
Set t = T(c_0). The output is EG(T) = (x,t,T̂), and empty if T is the empty tableau.

The Edelman-Greene algorithm on T outputs a discrete subset S(T) ⊂×_≥ 0,
denoted the swaps of T. Let (x_j, t_j, T̂_j), for 1 ≤ j ≤ ||, be defined
iteratively by (x_1,t_1,T̂_1) = EG(T) and (x_j,t_j,T̂_j) = EG(T̂_j-1)
for 2 ≤ j ≤ ||. Then,

    S(T) = { (x_j,t_j): 1 ≤ j ≤ || }.

If the cell (x,y) ∈ contains the k-th smallest entry of T then its entry is removed
during the k-th iteration of the algorithm. We will say that the entry at (x,y) exits
at time t_k from row x_k. We will also say that (x_k, t_k) originates from cell
(x,y).




  
Edelman-Greene algorithm on an admissible tableau
Let T_1, T_2, … be the clusters of an EG-admissible tableau T.
Observe that for i ≠ j, the swaps of T_i and T_j exit from mutually
disjoint rows. Thus, the swap sets S(T_1), S(T_2), … are row-wise mutually
disjoint. The swaps of T are defined as

    S(T) = ⋃_i  S(T_i).


The local tableau _edge is not EG-admissible. In order to define swaps
for the local tableau we cut off large entries so that it becomes EG-admissible,
and then process the tableau in a graded manner. For this to be successful,
the EG algorithm ought to be consistent in the sense that running it on a tableau, and then
restricting to swaps that originate from a sub-tableau, must produce the same
outcome as the algorithm applied to the sub-tableau. This is not always the case
and the following explains when it may be so.

Given two tableaux T_small and T_big, we say T_small≤ T_big
if the following criteria hold.


  * T_small(x,y) = T_big(x,y) for every (x,y) ∈support(T_small).

  * For every (x,y) ∈support(T_small), and (x',y') ∈support(T_big) ∖support(T_small), if (x,y)
belongs to the same cluster of T_big as (x',y') then T_small(x,y) < T_big(x',y').


 
Let T_small≤ T_big, and suppose that T_big is EG-admissible.
Then, applying the EG algorithm to T_big and restricting to the swaps that originate
from the cells of T_small produces the same outcome as applying the EG algorithm
to T_small. In particular, S(T_small) ⊂ S(T_big).



Observe that the clusters of T_small are contained within the clusters of T_big.
The EG algorithm acts independently on each cluster of T_big is a row-wise
disjoint manner. Fix a particular cluster T of T_big, and suppose
that the clusters of T_small that are contained inside T are T_1, …, T_k.
It suffices to prove that the EG algorithm applied to T, and then restricted to
the swaps that originate from T_1, …, T_k, produces the same outcome as
the algorithm applied to each individual T_i.

Let = support(T) and _i = support(T_i). The assumption is that
each entry of ∖ (∪_i _i) is larger than every entry of ∪_i _i.
Therefore, the EG algorithm applied to T will process every entry of ∪_i _i
before it ever processes an entry from the complement. When some entry
from ∖ (∪_i _i) enters a cell of some _i during
the first ∑_i |_i| steps, the algorithm treats that entry as if it were ∞.
Since T_small agrees with T_big on ∪_i _i, the
EG algorithm will output the swaps of T_1, …, T_k during the
first ∑_i |_i| steps, and then output the remaining swaps of
T ∖ (∪_i T_i). This is what was claimed.


A tableau T is graded EG-admissible if all of its finite-valued entries are distinct
and, if for every t > 0, the sub-tableau

    T^≤ t = { T(x,y): T(x,y) ≤ t}  is EG-admissible.

Observe that T^≤ t_1≤ T^≤ t_2 whenever t_1 ≤ t_2.
Lemma <ref> thus implies that S(T^≤ t_1) ⊂ S(T^≤ t_2).
Therefore, for a graded EG-admissible tableau T, we may define

    S(T) = ⋃_t ≥ 0  S(T^≤ t).


 
Suppose a sequence of tableaux T_n → T_∞, and also that every T_n and T_∞
are graded EG-admissible. Then for every integer x and t ≥ 0, there is a finite
YD  that contains the cluster of (2x,0) in T_n^≤ t for every n.



This follows from a diagonalization argument, more precisely, König's infinity lemma, which
states that every infinite connected graph with finite vertex degrees contains an
infinite path.

Suppose for the sake of a contradiction that the conclusion of the lemma fails.
Let T_n,x,t denote the cluster of (2x,0) in T_n^≤ t. Call a cell (x',y') ∈
bad if there is a undirected path in  from (2x,0) to (x',y') that is contained
in infinitely many of the clusters T_n,x,t. Consider the connected
component of (2x,0) in  that is spanned by the subgraph of bad vertices.
If the component is finite then there is a finite YD  that contains the component.
This implies that for all sufficiently large n, every cell of ∂ lies outside T_n,x,t
because any path from (2x,0) to a cell outside  must pass through ∂.
Therefore, T_n,x,t⊂ for all large n. Since every T_n is
graded EG-admissible, this means that there is a finite YD that contains
every T_n,x,t, which is a contradiction.

Therefore, the connected component of (2x,0) spanned by the bad vertices
is infinite. Since every vertex of  has degree at most 4, König's lemma
provides an infinite path of (distinct) bad vertices (x_0,y_0), (x_1,y_1), …
starting from (x_0,y_0) = (2x,0). By definition of being bad, for every m,
there is a path from (2x,0) to (x_m,y_m) that is contained in some infinite subsequence
of the clusters T_n^m_i,x,t with n^m_i →∞ as i →∞. Let ℓ_m be
the length of this path. Observe that ℓ_m →∞ with m because the distance
from (2x,0) to (x_m,y_m) in  must tend to infinity due to every vertex having degree
at most 4.

The YD _m formed by the cells of  that are at or below the cells on the path from
(2x,0) to (x_m,y_m) must be contained in every cluster T_n^m_i,x,t.
Since T_n converges to T_∞, this implies that _m ⊂ T_∞,x,t
for every m. Since |_m| ≥ℓ_m →∞, we deduce that T_∞,x,t is infinite.
However, this is a contradiction to T_∞ being graded EG-admissible.


 
Suppose a sequence of tableaux T_n → T_∞, and that every T_n as well as T_∞
is graded EG-admissible. Then S(T_n) → S(T_∞) as discrete subsets of ×_≥ 0.



A compact subset of ×_≥ 0 is a finite, disjoint union of sets of the from
{x}× C for x ∈ and compact C ⊂_≥ 0. Therefore, we must
show that for every such x and C,

    lim sup_n  #  [S(T_n) ∩ ({x}× C)  ]  ≤# [ S(T_∞) ∩ ({x}× C) ].

Fix a t > 0 such that C ⊂ [0,t].

Suppose T is a graded EG-admissible tableau. The swaps of T on {x}× [0,t]
are the entries of T^≤ t that exit from row x. Let T_x,t denote the
cluster of (2x,0) in T^≤ t. By Lemma <ref>, the swaps
of T on {x}× [0,t] are completely determined by running the EG algorithm
on T_x,t. We deduce from Lemma <ref> that there is a finite YD  such that

    support(T_n,x,t) ⊂  for every   n  and  support(T_∞,x,t) ⊂.


Since sup_(x',y') ∈ | T_n(x',y') - T_∞(x',y') | → 0,
we conclude that the following must occur for all sufficiently large n.


  * The order of the entries of T_n on  stabilizes to the order of the entries of T_∞ on .

  * For every (x',y') ∈, if T_∞(x',y') ∉ C then T_n(x',y') ∉ C.


Once condition (1) holds then, due to T_n,x,t⊂, a swap from S(T_n)
lies on {x}× C if and only if there is a cell (x',y') ∈ such that
T_n(x',y') ∈ C and, when the EG algorithm is applied to T_∞ restricted to ,
the entry at cell (x',y') exits from row x. The same conclusion holds for swaps of
S(T_∞) on {x}× C. This property along with condition (2) implies that

    S(T_n) ∩ ({x}× C) ⊂ S(T_∞) ∩ ({x}× C)   for all large   n.

This completes the proof.





 §.§ Completing the proof of Theorem <ref>
 


Theorem <ref> will follow from Theorem <ref> once we prove
that the local tableau _edge is graded EG-admissible almost surely. To this end,
first observe that the entries of _edge are finite and distinct by part (1) of
Proposition <ref>. We must show that, almost surely, the clusters
of _edge^≤ t are finite for every t.

By part (4) of Proposition <ref>, the local tableau satisfies the following almost surely:
for every t and x, there are integers a,b ≥ 0 such that _edge(2x-2a,0) > t
and _edge(2x+2b,0) > t. When this property holds the tableau constraints imply that the
cluster of _edge^≤ t containing (2x,0) must be contained within cells whose row
and column indices are both between 2x-2a and 2x+2b. The set of such cells is finite, and so
the cluster of every bottom level cell in _edge^≤ t is finite.
Now if _edge(2x-k,k) ≤ t then cell (2x-k,k) belongs to the same cluster as
(2x,0) in _edge^≤ t since the row entries are non-decreasing.
This implies that, almost surely, _edge^≤ t is EG-admissible for every t, as required.

Finally, we complete the proof.
The law of S_α,n is that of the Edelman-Greene algorithm applied to the
rescaled uniformly random staircase shaped tableau ^rsc_ from (<ref>).
Theorem <ref> asserts that ^rsc_ converges weakly to
_edge as a tableau embedded in . By Skorokhod's representation theorem,
there exists random tableaux _n and _∞ defined on a common probability space
such that _n has the law of ^rsc_, _∞ has the law of
_edge, and _n →_∞ almost surely.

The tableaux _n and _∞ are graded EG-admissible almost surely.
Theorem <ref> then implies that S(_n) converges to S(_∞) almost surely.
This means that S_α,n, which has the law of S(_n), converges weakly to S(_edge),
which is the law of S(_∞).


We conclude with some statistical properties of the local swap process.
 
The process S_local has the following properties.


  * S_local is invariant under translations and reflection of the -coordinate.

  * S_local is stationary in time in that for every t ≥ 0, the process
S_local∩ (×_≥ t) has the same law as (shifted) S_local.

  * S_local is ergodic under translations of the -coordinate in that the sigma-algebra

    _inv = {Events of  S_local that are invariant under every translation}  is trivial .



 We believe that S_local is also ergodic in the time coordinate.
	However, the proof of this is more challenging and, therefore, we leave it as a conjecture.


We have that S_local = S(_edge) in law. Applying a -automorphism to
S_local is the same as first applying its analogue to _edge (the maps ϕ_h and ϕ_-),
and then applying the EG algorithm to the resulting tableau. Thus, the invariance of S_local
under -automorphisms follows from the corresponding invariance of _edge stated
in Proposition <ref>.

Time stationarity of S_local is a consequence of the stationarity of finite
random sorting networks <cit.>, as we explain. If (s_1,…, s_N)
is the sequence of swaps of a random sorting network of 𝔖_n, then
(s_1,…, s_N-1) has the same law as (s_2, …, s_N).

The ergodicity of S_local under -translations is a consequence of the ergodicty of
_edge under translations (part 3 of Proposition <ref>). Indeed,
a translation invariant event for S_local is the image of a translation invariant event of
_edge under the EG algorithm.




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